Pedro Alegría

Jesús Gómez

Carlos Gorria

Josu San

Pedro De Même, Je Remercie

Javier Alegría

Duoandikoetxea

Luis

Je Tiens Également À Remercier

Karine Beauchard

Franck Boyer

Jean-Michel Coron

Sylvain Ervedoza

Enrique Fernández-Cara

Marius Tucsnak

Enrique Zuazua

Keywords: contrôlabilité à zéro, contrôlabilité uniforme, équation de la chaleur, inégalité de Carleman, systèmes de Korteweg-de Vries, systèmes de Stokes xvii Carleman inequalities, heat equation, Korteweg-de Vries systems, null controllability, Stokes systems, uniform controllability xix

Pour faire une thèse, il faut des connaissances, la capacité de profiter en apportant de la lumière sur l'inconnu et du financement. Je me considère comme une personne chanceuse, car au cours de ces années, j'ai eu, d'une part, de nombreux enseignants et mentors qui ont contribué à ma formation de mathématicien et qui m'ont fait apprécier les mathématiques, et d'autre part, le soutien de diverses entités publiques et fondations scientifiques. Je tiens à dédier quelques mots de remerciement à tous.

Pour commencer, je remercie

A mis padres, por estar siempre a mi lado, pese a los kilómetros que a veces nos separan.

Science is a cooperative enterprise, spanning the generations.

It's the passing of a torch from teacher, to student, to teacher.

A community of minds reaching back to antiquity and forward to the stars.

Neil deGrasse Tyson

iii Agradecimientos Para hacer una tesis hace falta conocimientos, la capacidad de disfrutar aportando luz a lo desconocido y financiación. Me considero una persona afortunada, pues a lo largo de estos años he tenido por una parte muchos profesores y mentores que han contribuido en mi formación como matemático y que me han hecho disfrutar con las matemáticas, y por otra parte el apoyo de diversos entes públicos y fundaciones científicas. A todos ellos les quiero dedicar unas palabras de agradecimiento.

Para empezar, agradezco a Pedro Alegría, Jesús Gómez, Carlos Gorria y Josu Sangroniz que organicen desde 2004 un taller de matemáticas para alumnos de bachillerato en Bilbao.

Disfruté con las matemáticas en esas sesiones, aprendí mucho y solventó mis dudas sobre si cursar Física o Matemáticas. También les agradezco que actualmente cuenten conmigo para impartir alguna sesión, ya que me encanta enseñar a jóvenes a disfrutar con las matemáticas y a resolver problemas.

A continuación, agradezco al profesorado de la EHU (Universidad Pública Vasca) la profesionalidad con la que han dado las clases y lo mucho que he aprendido durante el grado. En particular, agradezco a Josu Sangróniz todo el tiempo que me ha dedicado, pues siempre tenía alguna actividad que proponerme : recuerdo con especial cariño que me propusiera aprender a usar Latex en primero, resolver desde cero el problema de caracterizar los subgrupos de (Q, +) en segundo o que aceptara co-dirigirme el trabajo de fin de grado en un tema propuesto por mí en tercero. Quiero subrayar que cada vez que redacto un artículo rememoro sus enseñanzas sobre la importancia de la redacción y sentirse orgulloso de la versión final de nuestras obras. Asimismo, también quiero expresar mis agradecimientos a muchos otros profesores de la EHU : a Pedro Alegría, por contribuir a que los alumnos de la EHU puedan participar en la Olimpiada Iberoamericana de v vi Matemáticas Universitarias ; a Gustavo Fernández, por los talleres de Álgebra avanzada que organizó (me pareció bonito el curso sobre los grupos de simetría de árboles infinitos) ; a Marta Macho, tanto por ser consejera de una revista de divulgación de matemáticas llevada por alumnos como por conseguirme que la EHU hiciera un convenio de Erasmus con la Universidad de Lyon Claude Bernard (aunque finalmente la vida me llevó por otro camino) ; a Iraide Mardones, por recomendarme libros buenos de Topología que trabajar durante el verano ; y a Luis Martínez, por co-dirigirme el trabajo de fin de grado y por sus conversaciones sobre problemas de teoría de grafos.

Asimismo, agradezco a Pedro Alegría, Javier Duoandikoetxea, Luis Euscariaza, Gustavo Fernández, Iraide Mardones y Josu Sangróniz por interesarse por cómo me encuentro en París y por mi futuro laboral. También por las conversaciones con ellos sobre nuestra labor docente y la situación actual del mundo investigador. Reconforta saber que 4 años después de abandonar la EHU aún puedo pasarme por su despacho a charlar un rato. En ese sentido, agradezco a mi padre, Juan Carlos Bárcena, profesor titular de Economía en la EHU, todas las conversaciones que tengo con él sobre múltiples aspectos del día a día docente e investigador. Igualmente, agradezco a los compañeros que están cursando la tesis en Bilbao (y especialmente a Daniel Eceizabarrena y a Javier Canto) que me invitasen a ir al BCAM (Basque Center for Applied Mathematics) a contarles en qué temas trabajo y que me inviten a estar con ellos en los congresos en los que coincidimos.

No quisiera desperdiciar la oportunidad de agradecer a Gobierno Vasco por su programa de becas de excelencia para alumnos de grado. Fue un programa con el objetivo de monitorizar y premiar a jóvenes prometedores con independencia de la renta familiar en los años académicos de 2010 a 2013. Sólo pude aprovecharlo un año, y aún así les estoy agradecido porque con ese dinero me financié varios cursos de verano. Quisiera subrayar que cuando recibí esta beca me sentí valorado por los gobernantes de mi tierra.

En cuanto al máster, agradezco a la FSMP (Fundación de Ciencias Matemáticas de París) que me financiase los dos años del máster de Matemáticas en la UPMC (actual Sorbona). También agradezco a Martin Andler que me informase en el curso de verano "Modern Mathematics-International Summer School for Students" sobre la existencia de estas becas y que me animara a solicitarlas, y a Javier Duoandikoetxea y a Josu Sangroniz las cartas de recomendación que me hicieron. Además, agradezco al profesorado de vii la UPMC tanto la buena preparación que me dieron para el doctorado, como lo que disfruté trabajando por entender los contenidos de las materias y por resolver los problemas propuestos. Igualmente, quisiera subrayar el apoyo de algunos profesores en particular.

Para empezar, agradezco a Sylvia Serfaty, mi tutora en la FSMP, el tiempo dedicado, sus consejos a la hora de elegir asignaturas y que me recomendase buenos libros de Análisis que trabajar durante el verano. Igualmente, le agradezco a Sergio Guerrero que el 7 de octubre de 2015, a un mes tras empezar primero de máster, apostase por mí y me ofreciese dirigirme los trabajos del máster y la tesis. También le agradezco el que me guiase a la hora de escoger las asignaturas y que orientase los trabajos del máster para introducirme en sus temas de investigación. Asimismo, quisiera recalcar que el curso de Análisis Funcional de Jean-Yves Chemin y Sergio Guerrero fue un antes y un después en mi pasión hacia esta disciplina, pues llegué a París sin tener claro qué materia me gustaba más y terminé 2015 teniendo claro que quería estudiar problemas teóricos de Análisis.

En cuanto a la financiación del doctorado agradezco al LJLL (Laboratorio Jacques-Louis Lions) y al proyecto DIM Math-Innov el que depositasen su confianza en mí y me financiasen a medias la tesis. Igualmente, agradezco a Benoît Perthame, Didier Smets y Emmanuel Trélat las cartas de recomendación que me hicieron.

Asimismo, agradezco a todos los miembros del LJLL el buen ambiente investigador que ha habido en estos tres años. También agradezco a los evaluadores anónimos cuyos comentarios han contribuido a mejorar los artículos. Igualmente, hay muchas cosas que agradezco a Sergio Guerrero, mi director de tesis : que me proponga unos problemas interesantes, su insistencia en que no me conforme con tener resultados menores, el que se haya preocupado por que me sintiera a gusto en París y por mi futuro laboral, que siempre tenga ganas de conversar sobre sus artículos, que esté trabajando conmigo en la resolución de problemas matemáticos interesantes y que haya compartido sus fondos de investigación conmigo para que pueda ir a congresos y a cursos. Siguiendo en esa línea, agradezco a Enrique Zuazua que se haya interesado por cómo me estaba yendo durante la tesis y que me haya invitado a solicitar uno de los puestos de post-docs que él ofrece en la Universidad Autónoma de Madrid. Igualmente, le agradezco que me esté introduciendo en algunos de sus temas de investigación, que tenga ganas de ponerse a resolver problemas matemáticos interesantes conmigo y la estancia que he realizado (parcialmente financiada por sus fondos de investigación) en el seno de su equipo en la FAU (Friedrich-Alexander viii University Erlangen-Nuremberg). En cuanto a otros investigadores, agradezco a Kévin Le Balc'h y a Ademir Pazoto que estemos trabajando en la resolución de problemas matemáticos interesantes.

Igualmente, agradezco a Karine Beauchard, Franck Boyer, Jean-Michel Coron, Sylvain Ervedoza, Enrique Fernández-Cara, Marius Tucsnak y a Enrique Zuazua por haber accedido a formar parte del tribunal de tesis. En particular, agradezco a Marius Tucsnak y a Enrique Zuazua por haber hecho además los informes de la tesis.

Para terminar, y en un plano más personal, agradezco a mis amigos y familiares el tiempo que hemos pasado juntos durante todos estos años. Agradezco a ex-compañeros del máster, actualmente amigos, que tenga gente con quien quedar a tomar algo o ir al cine en París. Agradezco a amigos y familiares que hayan venido a visitarme a París.

También agradezco a mis tíos y primos de Tres Cantos que me reciban exquisitamente cuando asisto a eventos científicos en Madrid. Y, por último, agradezco de todo corazón a mis familiares y a mis amigos que pueda volver en vacaciones y en algunos fines de semana a mi querida Bilbao y que me hagan sentir tan en casa como hace cinco años. [START_REF] Bárcena-Petisco | Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit[END_REF]) . . . . . . . . . . . 1.2.3 Contrôlabilité à zéro de l'équation de la chaleur dans des pseudocylindres avec un contrôle qui agit à l'intérieur (voir [START_REF] Bárcena-Petisco | Null controllability of the heat equation in pseudocylinders by an internal control[END_REF]) . . . . . .

Introduction

Il y a beaucoup de phénomènes naturels et humains qui peuvent être décrits avec des équations différentielles. De fait, elles sont utilisées pour modéliser, par exemple, des phénomènes issus de la physique (la chaleur, les ondes, la mécanique des fluides, l'élasticité, l'électromagnétisme, etc.), la biologie (les dynamiques des populations, les modèles proieprédateur, la chimiotaxie, etc.), la médecine (la croissance des tumeurs, la tolérance aux antibiotiques, etc.) et l'économie (les jeux à champ moyen, etc.). Pour apprécier l'ampleur des phénomènes qu'on peut décrire avec des équations différentielles on peut consulter [1].

Au cours de cette thèse on travaille sur la résolution de problèmes de contrôlabilité des EDPs qui intéressent aux physiciens. Plus précisément, on travaille dans le Chapitre 2 avec un système qui approche numériquement le système de Stokes, dans le Chapitre 3 avec un système de Stokes qui possède un terme de transport et une viscosité évanescente, dans le Chapitre 4 avec l'équation de la chaleur lorsque le domaine n'est pas C 2 et dans le Chapitre 5 avec un système non-linéaire de deux équations de KdV (Korteweg-de Vries) couplées. On considère dans ce manuscrit surtout des contrôles qui agissent à l'intérieur.

Dans ce chapitre on présente dans la Section 1.1 les notions basiques de contrôlabilité, dans la Section 1.2 un résumé des travaux qui composent ce mémoire, et dans la Section 1.3 quelques perspectives et problèmes ouverts.

Introduction générale à la théorie de contrôle

On présente dans la Section 1.1.1 les notions générales de contrôlabilité et dans la Section 1.1.2 la méthode d'unicité de Hilbert (la HUM).

Les notions générales de contrôlabilité

En termes mathématiques, on considère dans ce manuscrit des EDP (équations aux dérivées partielles) qui peuvent être modélisées par :    y t + Py = f 1 ω pour t > 0, y(0) = y 0 , (1.1) où t → y(t, •) ∈ E(Ω) décrit l'état d'un système pour Ω un domaine (ouvert borné connexe de R d ) et pour E(Ω) un espace de Banach avec une variable spatiale sur Ω (normalement un espace de Lebesgue ou de Sobolev), P est un opérateur différentiel (pas nécessairement linéaire) défini sur (0, T ) × Ω agissant sur la variable spatiale, T > 0 est le temps final de contrôle, ω ⊂ Ω est un domaine (appelé domaine de contrôle) et f est un contrôle qui appartient à un ensemble U ⊂ (L 2 ((0, T ) × ω)) N , où N est le nombre de composantes de y. L'équation (1.1) 1 peut être satisfaite dans plusieurs espaces, comme dans D ((0, T )×Ω) ou dans l'espace de distributions de fonctions à divergence nulle (c'est le cas de Stokes, par example). On suppose aussi que (1.1) est bien posé dans C 0 ([0, T ]; E(Ω)) par rapport à y 0 ∈ E(Ω) et à f ∈ U.

Le problème de la contrôlabilité consiste, pour une condition initiale y 0 donnée et un temps final T > 0, de trouver un contrôle f ∈ U qui mène la solution du système (1.1) vers une cible à l'instant T . Pour une présentation complète et plus générale, on peut consulter [START_REF] Coron | Control and Nonlinearity[END_REF]Section 2.3]. On rappelle les différentes notions de contrôlabilité : Définition 1.3 (Contrôlabilité aux trajectoires). On dit que le système de contrôle (1.1) est contrôlable aux trajectoires au temps T si pour tout y 0 ∈ E(Ω) et pour tout (y 0 , y, f ) solution de (1.1) (tel que y ∈ C 0 ([0, T ]; E(Ω)) et f ∈ U), il existe un contrôle f ∈ U(Ω) tel que la solution de (1.1) satisfait y(T, •) = y(T, •).

Définition 1.4 (Contrôlabilité à zéro). On dit que le système de contrôle (1.1) est contrôlable à zéro au temps T si pour tout y 0 ∈ E(Ω) il existe un contrôle f ∈ U(Ω) tel que la solution de (1.1) satisfait y(T, •) = 0.

Ces définitions sont de type globales, car elles ne restreignent pas la taille de la condition initiale. Hélas, lorsque P n'est pas linéaire l'étude des propriétés globales du système de contrôle (1.1) peut devenir trop complexe, parfois on ne connaît même pas des méthodes pour les étudier. Par conséquent, on peut se contenter de manière provisoire d'obtenir des résultats locaux ; c'est-à-dire, des résultats de contrôlabilité où y 0 est proche de la cible à l'instant initial.

La résolution de chaque problème de contrôlabilité dépend des propriétés particulières de l'équation sur laquelle on travaille. Par exemple, il est connu que la contrôlabilité exacte de l'équation du transport n'est pas assurée si le temps final T est petit puisque la vitesse de propagation est finie. Par contre, si ω est à côté de la partie de la frontière où le flux du transport entre, il est aussi bien connu qu'on a la contrôlabilité exacte pour T suffisamment grand. Également, pour les équations paraboliques, comme l'équation de la chaleur, il n'y a pas d'espoir d'atteindre la contrôlabilité exacte dû à l'effet régularisant, mais c'est habituel d'avoir de la contrôlabilité aux trajectoires et approchée pour tout temps T > 0.

Dans une grande partie de ce mémoire on se concentrera sur la contrôlabilité à zéro des systèmes paraboliques et dispersifs linéaires. On rappelle que pour ces systèmes la contrôlabilité à zéro implique la contrôlabilité aux trajectoires et la contrôlabilité approchée (voir, par exemple, [START_REF] Coron | Control and Nonlinearity[END_REF]). En plus, même pour prouver la contrôlabilité à zéro locale d'un système non-linéaire on peut montrer d'abord la contrôlabilité du système linéarisé, et puis revenir au problème original à l'aide d'un argument d'inversion locale ou de point fixe.

1.1. Introduction générale à la théorie de contrôle Finalement, une question annexe très importante et sur laquelle on va travailler pendant cette thèse est, une fois assurée la contrôlabilité de (1.1), d'estimer le coût du contrôle.

On rappelle que le coût du contrôle à zéro du système (1.1) est donné, en suivant la convention inf ∅ = +∞, par :

K(Ω, ω, P) := sup

y 0 ∈E(Ω)\{0} inf f ∈U :y(T,•)=0 f (L 2 ((0,T )×ω)) N y 0 E(Ω) , (1.2)
où y est la solution de (1.1). Par exemple, c'est pertinent d'obtenir des estimations asymptotiques du coût de la contrôlabilité de (1.1) lorsque T → 0. Également, si l'on a une suite d'opérateurs P ε on peut se demander sur l'évolution du coût de la contrôlabilité lorsque ε → 0, ce qui est pertinent par exemple lorsque les systèmes avec les opérateurs P ε approchent numériquement le système avec l'opérateur P.

Dans la partie suivante on présentera, de manière simple, la méthode utilisée dans le cadre des espaces de Hilbert : la HUM ("Hilbert Uniqueness Method"). C'est une méthode utilisée pour déterminer la contrôlabilité à zéro des systèmes linéaires et, en même temps, si le système est contrôlable, pour avoir des estimations sur le coût du contrôle.

La HUM et l'inégalité d'observabilité

Le but de cette partie est de présenter la HUM ("Hilbert Uniqueness Method"), qui est une méthode classique (voir [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations : recent progress and open questions[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systemes distribués[END_REF]) lorsque P est linéaire pour construire le contrôle v de norme minimale tel que la solution y de (1.1) satisfait y(T, •) = 0. On suppose que E(Ω)

est un sous-espace fermé de (L 2 (Ω)) N , où N ∈ N, et qu'il existe un opérateur différentiel P * tel que le système :

   -ϕ t + P * ϕ = 0 pour t < T, ϕ(T, •) = ϕ T , (1.3)
est bien posé avec une solution dans C 0 ([0, T ]; E(Ω)). On suppose aussi que pour tout f ∈ U, et y 0 , ϕ T ∈ E(Ω) on a :

(0,T )×ω

f • ϕdxdt = Ω y(T, x) • ϕ T (x)dx - Ω y 0 (x) • ϕ(0, x)dx, (1.4)
où y est la solution de (1.1) et ϕ est celle de (1.3). Pour simplifier le problème, on suppose que U = (L 2 ((0, T ) × ω)) N , même si l'on peut adapter cette méthode lorsque U n'est qu'un sous-ensemble de (L 2 ((0, T ) × ω)) N .

On remarque que si le système (1.1) est contrôlable à zéro avec K < +∞ on peut estimer la norme de ϕ(0, •) à l'aide de la norme de ϕ dans (0, T ) × ω. En fait, on considère dans (1.4) la valeur initiale y 0 = ϕ(0, •) et une suite de contrôles f m qui mènent ϕ(0, •) à 0 tels que (voir (1.2)) : lim sup m f m (L 2 ((0,T )×ω)) N ≤ K ϕ(0, •) (L 2 (Ω)) N .

(1.5)

Comme y m (T, •) = 0 on obtient de (1.4) pour tout m ∈ N l'égalité :

Ω |ϕ(0, x)| 2 dx = - (0,T )×ω f m • ϕdxdt.
En conséquence, on obtient par Cauchy-Schwarz et par (1.5) (en prenant la limite lorsque m → ∞) l'inégalité :

Ω |ϕ(0, x)| 2 dx ≤ K ϕ(0, •) (L 2 (Ω)) N ϕ (L 2 ((0,T )×ω)) N ,
ce qui implique :

ϕ(0, •) (L 2 (Ω)) N ≤ K ϕ (L 2 ((0,T )×ω)) N , (1.6)
pour toute solution de (1.3) (avec ϕ T ∈ E(Ω)).

L'inégalité (1.6) est appelée inégalité d'observabilité. On définit : K(Ω, ω, P) = sup

ϕ T ∈(L 2 (Ω)) N \{0}
ϕ(0, •) E(Ω) ϕ (L 2 ((0,T )×ω)) N , (1.7)

en attribuant la valeur +∞ à l'indétermination 0 0 . On a obtenu dans le paragraphe précédant que : K ≤ K.

(1.8)

Grâce à la méthode HUM on peut être plus précis et prouver que :

K = K.
(1.9)

Si K = +∞, (1.8) implique que K = +∞, donc il suffit de considérer K < +∞.

Considérons l'ensemble E(Ω) avec la forme bilinéaire :

φT , ϕ T → (0,T )×ω φ • ϕdxdt.

(1.10)

1.1. Introduction générale à la théorie de contrôle

Comme K < +∞, on a que ϕ T , ϕ T = 0 pour tout ϕ T = 0, et donc (1.10) est un produit scalaire. On déduit du théorème de représentation de Riesz et de la construction de la completitude de E(Ω) l'existence d'une suite ϕ T m ∈ E(Ω) telle que pour tout ϕ T ∈ E(Ω) : lim m→∞ (0,T )×ω

ϕ m • ϕdxdt = - Ω y 0 (x) • ϕ(0, x)dx.
La suite ϕ m est donc une suite de Cauchy dans (L 2 ((0, T ) × ω)) N , et alors elle a une limite f . Grâce à (1.7) on a que ϕ m (0, •) est une suite de Cauchy sur E(Ω), et donc f a une trace sur E(Ω). Alors, en prenant la limite sur :

(0,T )×ω f • ϕ m dxdt = - Ω y 0 (x) • ϕ m (0, x)dx,
on obtient :

f (L 2 ((0,T )×ω)) N ≤ K y 0 E(Ω) .
En plus, la solution de (1.1) satisfait :

0 = Ω y(T, x) • ϕ T (x)dx ∀ϕ T ∈ E(Ω),
ce qui implique que y(T, •) = 0 et que K ≤ K, donc avec (1.8) on a (1.9).

Remarque 1.5. En fait, dans "HUM" il y a le mot "unicitité" car, pour y 0 fixé, le contrôle f qu'on obtient avec cette méthode est l'unique contrôle avec la norme minimale sur (L 2 ((0, T ) × ω)) N . En fait, si f mène y 0 à 0 on obtient par (1.4) que :

(0,T )×ω

f • ϕdxdt = - Ω y 0 (x) • ϕ(0, x)dx ∀ϕ T ∈ E(Ω),
ce qui implique que :

(0,T )×ω

( f -f ) • ϕdxdt = 0 ∀ϕ T ∈ E(Ω).
(1.11)

Alors, en tenant compte du fait que f est la limite de quelques solutions de (1.3) par rapport à la norme L 2 (((0, T ) × ω) N ), on déduit de (1.11) que f -f ⊥ f . Par conséquent, on a : f 2 (L 2 ((0,T )×ω)) N = f -f 2 (L 2 ((0,T )×ω)) N + f 2 (L 2 ((0,T )×ω)) N , ce qui implique que f est l'unique contrôle de norme minimale.

L'inégalité (1.6) est très importante car elle est utilisée pour démontrer la contrôlabilité de nombreuses EDP (consulter [START_REF] Coron | Control and Nonlinearity[END_REF]Section 2] ou [START_REF] Zuazua | Controllability and observability of partial differential equations : some results and open problems[END_REF] pour lire comment elle est utilisée dans les EDP les plus classiques). En fait, cela nous permet d'avoir une méthode générale pour construire un contrôle en réduisant la preuve à la démonstration des inégalités.

Par contre, il n'y a pas de méthode universelle pour démontrer l'existence de K < +∞ tel qu'on a (1.6), mais cela dépend de chaque équation. Dans l'équation de la chaleur, comme c'est expliqué dans la Section 1.2.3, cela peut être démontré de plusieurs manières.

Cependant, la méthode qui permet une généralisation à un plus grand nombre d'équations est celle de Fursikov-Imanuvilov (voir [START_REF] Fursikov | Controllability of evolution equations[END_REF]), qui s'appuie sur les inégalités de Carleman avec des poids du type e -h(x)t -m (T -t) -m , où m ∈ R + et h est une fonction qui dépend de Ω et ω avec certaines propriétés géométriques. De fait, une grande partie de la thèse est basée sur les techniques de [START_REF] Fursikov | Controllability of evolution equations[END_REF] et des poids de ce type.

Résultats principaux et plan de la thèse

Dans cette section on utilise Ω pour noter un domaine (ouvert borné connexe de R d ).

En plus, Q := (0, T ) × Ω (ou Q := (0, T ) × (0, L) quand on travaille dans un intervalle de R), Σ := (0, T ) × ∂Ω et n désigne le vecteur normal unitaire extérieur sur ∂Ω. Également, on note Q ω := (0, T ) × ω pour tout ouvert ω et on utilise le gras pour désigner les espaces à valeurs vectorielles.

On commence par une présentation des résultats principaux obtenus pendant la réalisation de cette thèse :

• Dans le Chapitre 2 on étudie la contrôlabilité du système suivant :

               y ε t -∆y ε + ∇p ε = f ε 1 ω e 1 dans Q, εp ε + ∇ • y ε = 0 dans Q, y ε = 0 sur Σ, y ε (0, •) = y 0 sur Ω.
On prouve que si Ω ⊂ R 2 est un domaine régulier qui satisfait une hypothèse géométrique, le système est contrôlable à zéro avec un coût borné uniformément sur (0, ε 0 ) pour ε 0 suffisamment petit. Pour le prouver, on utilise la stratégie de [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]. La 1.2. Résultats principaux et plan de la thèse difficulté principale est que dans notre système le couplage est fait avec la dérivée croisée et pas avec une dérivée partielle d'ordre 1. Alors, on prouve que la dérivée croisée est coercitive si l'on considère une norme plus forte que L 2 pour la dérivée croisée, des conditions aux limites de Dirichlet, une estimation sur la trace au bord des dérivées d'ordre deux et une hypothèse géométrique sur le domaine. C'est un travail présenté dans [START_REF] Bárcena-Petisco | Null controllability of a penalized Stokes problem in dimension two with one scalar control[END_REF].

• Dans le Chapitre 3 on étudie le coût de la contrôlabilité à zéro du système de Stokes suivant :

               y t -ε∆y + ∂ x d y + ∇q = f 1 ω dans Q, ∇ • y = 0 dans Q, y • n = 0, (Dy • n) tg = 0 sur Σ, y(0, •) = y 0 sur Ω.
Concernant, il est connu (voir [START_REF] Guerrero | Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions[END_REF]) que ce système est contrôlable à zéro, mais la dépendance du coût par rapport au coefficient de viscosité restait à préciser. On montre que dans (0, π) 2 le coût est exponentiellement petit par rapport à ε si T est suffisamment grand, et qu'il est exponentiellement grand par rapport à ε si T est petit et ω est inclus compactement dans (0, π) 2 . En plus, on prouve que dans (0, π) 3 pour tout T > 0 si ω est inclus compactement dans (0, π) 3 , le coût est exponentiellement grand par rapport à ε. Concernant les techniques utilisées, pour démontrer ces résultats on utilise les techniques introduites dans [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF], mais on utilise aussi une technique originale dans des problèmes de transport-diffusion avec une viscosité évanescente : étudier les fonctions propres de l'opérateur elliptique associé au système adjoint. En fait, la décomposition spectrale explique (voir Remarque 1. [START_REF] Bárcena-Petisco | Study of the cost of the controllability of second order parabolic equations with small diffusion and a transport term[END_REF] et Remarque 1. [START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF]) les causes des différences entre la dimension 2 et la dimension 3.

Ce problème est traité dans [START_REF] Bárcena-Petisco | Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit[END_REF].

• Dans le Chapitre 4 on prouve la contrôlabilité à zéro de l'équation de la chaleur dans une famille de domaines avec une partie cylindrique limitée par un graphe Lipschitz.

Pour montrer cela, on montre une inégalité de Carleman qui présente les propriétés d'absorption habituelles en adaptant les preuves de [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Yu | Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications[END_REF]. La difficulté principale est la non-existence des poids habituels pour les domaines C 2 . Pour vaincre cela, on utilise la structure cylindrique et on approxime le système adjoint par le même système posé sur des domaines réguliers. Pour construire ces domaines, on considère les courbes de niveau d'une régularisation de la fonction distance à la frontière. Une autre nouveauté de cet article est que c'est la première fois où les techniques de Fursikov-Imanuvilov sont adaptées à des domaines où les solutions de l'équation de la chaleur avec condition aux limites de Dirichlet et avec données régulières qui sont à support compact n'appartiennent pas à L 2 (0, T ; H 2 (Ω)). C'est un travail présenté dans [START_REF] Bárcena-Petisco | Null controllability of the heat equation in pseudocylinders by an internal control[END_REF].

• Dans le Chapitre 5 on prouve la contrôlabilité locale à 0 du système de KdV couplé suivant : [START_REF] Bárcena-Petisco | Local null controllability of a model system for strong interaction between internal solitary waves[END_REF].

                     u t + uu x + u xxx + av xxx + k 1 vv x + k 2 (uv) x = h 1 dans Q, cv t + rv x + vv x + abu xxx + v xxx + k 2 buu x + k 1 b(uv) x = h 2 + f 1 ω dans Q, u(•, 0) = u(•, L) = u x (•, L) = 0 sur (0, T ), v(•, 0) = v(•, L) = v x (•, L) = 0 sur (0, T ), u(0, •) = u 0 , v(0, •) = v 0 sur (0, L), où k 1 , k 2 , r, a, b, c ∈ R, a, b, c > 0, a 2 b < 1, T > 0, L > 0 et h 1 et h 2 sont
Dans ce qui suit, on présente chaque chapitre de ce mémoire de façon détaillée.

1.2.1 Contrôlabilité à zéro d'un problème de Stokes pénalisé en dimension deux avec un contrôle scalaire (voir [START_REF] Bárcena-Petisco | Null controllability of a penalized Stokes problem in dimension two with one scalar control[END_REF])

Dans cette partie on étudie la contrôlabilité d'un système qui approche numériquement le système de Stokes. Concrètement, on étudie la contrôlabilité à zéro de :

               y ε t -∆y ε + ∇p ε = 0 dans Q, εp ε + ∇ • y ε = 0 dans Q, y ε = 0 sur Σ, y ε (0, •) = y 0 sur Ω.
(1.12)

Résultats principaux et plan de la thèse

Le système (1.12) est utilisé pour approcher numériquement le système de Stokes vu que la condition d'incompressibilité peut causer des problèmes. Cette approximation a été introduite dans [START_REF] Temam | Une méthode d'approximation de la solution des équations des Navier-Stokes[END_REF]. La contrôlabilité de (1.12) a été étudiée dans la littérature (voir [START_REF] Yu | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF][START_REF] Badra | Global Carleman inequalities for Stokes and penalized Stokes equations[END_REF]), mais uniquement quand le contrôle appartient à L 2 (Q ω ). Nous nous intéressons à la contrôlabilité à zéro avec N -1 contrôles scalaires, ce qui n'avait pas encore été étudié.

Plus précisément, on étudie :

               y ε t -∆y ε + ∇p ε = f ε 1 ω e 1 dans Q, εp ε + ∇ • y ε = 0 dans Q, y ε = 0 sur Σ, y ε (0, •) = y 0 sur Ω, (1.13) où Ω ⊂ R 2 est un domaine, e 1 := (1, 0) et y 0 ∈ L 2 (Ω).
Remarque 1.6. L'aspect numérique de (1.13) est intéressant pour les données initiales :

y 0 ∈ H(Ω) := {u ∈ L 2 (Ω) : ∇ • u = 0 dans Ω et u • n = 0 sur ∂Ω}.
Dans ces cas, si (f ε ) ε∈(0,ε 0 ) est une suite de contrôles bornée dans L 2 (Q ω ) telle que L'existence de ces domaines est une difficulté implicite importante car elle implique que si l'on prouve la contrôlabilité de (1.13) pour des domaines réguliers la preuve doit avoir des raisonnements qui ne soient pas applicables dans le cas du losange.

y ε (T, •) = 0 pour tout ε ∈ (0, ε 0 ), il y a une sous-suite f ε k convergente faiblement vers f ∈ L 2 (Q ω ),
Pour traiter la difficulté implicite on formule l'hypothèse suivante :

Hypothèse 1.7. Soit Ω ⊂ R 2 un domaine C 2 dont la frontière ∂Ω est paramétrisée par des fonctions σ 1 , . . . , σ k de longueur d'arc. Alors, pour tout i ∈ {1, . . . , k} et pour tout θ tels que (σ i 1 ) (θ) = 0 ou (σ i 2 ) (θ) = 0, on a que κ i (θ) = 0.

Plus précisément, on prouve le résultat suivant :

Théorème 1.8. Soient Ω ⊂ R 2 un domaine régulier satisfaisant l'Hypothèse 1.7 et ω ⊂ Ω un domaine. Alors, il y a ε 0 > 0 tel que pour tout T > 0 il y a C > 0 tel que si ε ∈ (0, ε 0 )

et y 0 ∈ L 2 (Ω), il y a une fonction scalaire f ε ∈ L 2 (Q ω ) satisfaisant : f ε L 2 (Qω) ≤ C y 0 L 2 (Ω) ,
et telle que la solution de (1.13) satisfait y ε (T, •) = 0.

Remarque 1.9. L'Hypothèse 1.

               -ϕ ε t -∆ϕ ε + ∇π ε = 0 dans Q, επ ε + ∇ • ϕ ε = 0 dans Q, ϕ ε = 0 sur Σ, ϕ ε (T, •) = ϕ T sur Ω.
(1.14)

Concrètement, on prouve l'inégalité de Carleman suivante :

Théorème 1.10. Soient Ω un domaine régulier qui satisfait l'Hypothèse 1.7, ω ⊂ Ω un

ouvert et m ≥ 8. Alors, il y a ε 0 > 0, C > 0 et λ 0 ≥ 1 tels que si T > 0, ε ∈ (0, ε 0 ), ϕ T ∈ L 2 (Ω), λ ≥ λ 0 , et s ≥ e Cλ (T m + T 2m
), on a : Pour prouver le Théorème 1.10 on suit la stratégie de [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]. D'abord, on trouve une inégalité de Carleman pour les solutions de (1.14) avec des conditions aux limites nonhomogènes. Plus précisément, on prouve :

s 15 λ 16 Q e -2sα
Proposition 1.11. Soient Ω un domaine C 4 , ω un sous-domaine de Ω tel que ω 0 ⊂ ω et m ≥ 8. Alors, il y a ε 0 > 0, C > 0 et λ 0 ≥ 1 tels que si T > 0, ε ∈ (0, ε 0 ), ϕ T ∈ L 2 (Ω), h ∈ H 2,5/2 (Σ), λ ≥ λ 0 et s ≥ e Cλ (T m + T 2m
), on a :

s 3 λ 4 Q e -2sα ξ 3 |ϕ ε | 2 + sλ 2 Q e -2sα ξ|∇ϕ ε | 2 ≤ C s 4 λ 5 Q ω e -2sα ξ 4 |ϕ ε | 2 + (1 + T ) (sξ * ) 1/4+1/m e -sα * h 2 H 1,1/2 (Σ) + (sξ * ) -3/4 e -sα * h 2 H 2,5/2 (Σ) , (1.16)
où ϕ ε est la solution du système :

               -ϕ ε t -∆ϕ ε + ∇π ε = 0 dans Q, επ ε + ∇ • ϕ ε = 0 dans Q, ∂ n ϕ ε -π ε n = h sur Σ, ϕ ε (T, •) = ϕ T sur Ω.
(1.17) D'abord, on utilise que ϕ ε satisfait l'équation de la chaleur avec terme source (1+ε) -1 ∇× (∇×ϕ ε ). En plus, on utilise que le rotationnel de ϕ satisfait l'équation de la chaleur, ce qui nous permet d'appliquer l'inégalité de Carleman donnée dans [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF] pour l'équation de la chaleur. Finalement, on utilise les résultats de régularité de (1.17) pour traiter les traces.

Maintenant on va traiter la différence principale avec [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF], qui est que le couplage dans (1.14) est fait par la dérivée croisée et pas par un opérateur différentiel d'ordre 1. On doit donc prouver la coercivité de la dérivée croisée, ce qui est la nouveauté principale de cet article. On sait que la dérivée croisée n'est pas coercitive dans un domaine régulier quelconque, même avec des conditions aux limites de type Dirichlet (on peut considérer, par exemple, le disque unité avec une fonction 1 -x 2 1 -x 2 2 ). Par contre, on peut obtenir sa coercivité si l'on rajoute un terme de bord : Théorème 1.12. Soit Ω un domaine C 4 satisfaisant l'Hypothèse 1.7. Alors, pour tout a 0 > 0 suffisamment petit, il y a C > 0 tel que pour toute fonction u ∈ H 4 (Ω) ∩ H 1 0 (Ω) et pour tout a ∈ [0, a 0 ] on a que :

∂ x 1 u C 0 (Ω) ≤ C( ∂ 2 x 1 x 2 u H 2 (Ω) + L a u H 1 (∂Ω) ), (1.18)
où :

L a u = -a∂ 2 x 2 1 u -∂ 2 x 2 2 u. (1.19)
On ne sait pas si le résultat (1.18) est optimal, mais on sait au moins que le terme au bord et les conditions de Dirchlet sont nécessaires. De même, concernant la norme de la dérivée croisée on peut dire que la norme L 2 (Ω) de la dérivée croisée ne suffit pas :

Proposition 1.13. Soit Ω un domaine C 4 avec une frontière caractérisée par l'équation :

g(x 1 ) + h(x 2 ) = 0, où g, h ∈ C 4 (R). Alors, il n'existe pas de C > 0 tel que pour tout u ∈ H 4 (Ω) ∩ H 1 0 (Ω) on ait : u L 2 (Ω) ≤ C ∂ 2 x 1 x 2 u L 2 (Ω) + L a u H 1 (∂Ω) . (1.20)
On prouve la Proposition 1.13 par contradiction en approximant le Laplacian de la fonction g(x 1 ) + h(x 2 ).

On prouve d'abord le Théorème 1.12 dans le cas où Ω est strictement convexe en utilisant les conditions aux limites de Dirchlet et l'égalité (1. [START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF]. Avec ces équations, on obtient une EDO (équation différentiel ordinaire) sur la frontière, avec un terme qui s'écrit en fonction de ∂ 2 x 1 x 2 u et L a u, après on définit une variable auxiliaire et on fait des estimations, et finalement on estime ∂ x 1 u en termes de la fonction auxiliaire. Un point important de la preuve consiste à démontrer que si l'on a une estimation de ∂ x 1 u en un point p on l'a sur la droite verticale qui contient ce point.

Pour prouver le Théorème 1.12 dans tout domaine Ω qui satisfait l'Hypothèse 1.7 on coupe le domaine dans un nombre fini de morceaux. Après, on donne la preuve dans chaque situation par récurrence (en considérant la borne gauche de chaque morceau) en adaptant la preuve des domaines strictement convexes à chaque situation particulière.

Résultats principaux et plan de la thèse

Soit maintenant ϕ ε la solution de (1.14). En considérant les conditions de Dirichlet, (1.14) 1 et (1.14) 2 on obtient que sur Σ on a :

   -∂ 2 x 2 1 ϕ ε 1 -ε 1+ε ∂ 2 x 2 2 ϕ ε 1 = 1 1+ε ∂ 2 x 1 x 2 ϕ ε 2 , -ε 1+ε ∂ 2 x 2 1 ϕ ε 2 -∂ 2 x 2 2 ϕ ε 2 = 1 1+ε ∂ 2 x 1 x 2 ϕ ε 1 .
Alors, en utilisant le Théorème 1.12, sa version symétrique et l'inégalité de Poincaré on obtient que si Ω satisfait l'Hypothèse 1.7 pour tout t ∈ [0, T ) et ε ∈ (0, ε 0 ] on a :

ϕ ε (t, •) L 2 (Ω) ≤ C ∂ 2 x 1 x 2 ϕ ε (t, •) H 2 (Ω) . (1.21)
Avec (1.21) et (1.16) on suit la méthode de [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF] et on prouve (1.15). En fait, on estime les dérivées croisées en considérant encore plus de dérivées, après on estime les traces avec les résultats de régularité et finalement on estime les termes locaux avec l'inégalité de Cauchy-Schwarz.

Remarque 1.14. C'est dans la preuve du Théorème 1.12 où l'Hypothèse 1.7 est nécessaire.

Pour la plupart des inégalités de Carleman on utilise des estimations paraboliques simples qui n'ont besoin d'aucune hypothèse géométrique autre que la régularité du domaine.

Contrôlabilité uniforme d'un problème de Stokes avec un

terme de transport et une viscosité évanescente (voir [START_REF] Bárcena-Petisco | Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit[END_REF])

Un sujet de recherche important en contrôlabilité est l'étude du coût des contrôles des EDP paraboliques ou dispersives avec une diffusion ou dispersion évanescente. Dans ce sens, il y a des travaux dans la littérature, par exemple, sur l'équation de la chaleur (voir [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF]) et sur l'équation de KdV (voir [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF][START_REF] Glass | Uniform controllability of a transport equation in zero diffusion-dispersion limit[END_REF][START_REF] Carreño | On the non-uniform null controllability of a linear KdV equation[END_REF][START_REF] Carreño | Uniform null controllability of a linear KdV equation using two controls[END_REF]). Dans cette partie on étudie à quel point les résultats des documents cités précédemment sont généralisables au système de Stokes incompressible avec des conditions aux limites de non-glissement, qui fait partie des EDP paraboliques classiques. On rappel que l'espace fonctionnel naturel pour modéliser les fluides incompressibles est :

H(Ω) := {u ∈ L 2 (Ω) : ∇ • u = 0 dans Ω et u • n = 0 sur ∂Ω}.
Dans cette partie on travaille dans Ω d := (0, 1) d . Plus précisément, on étudie dans Ω 2

et Ω 3 la contrôlabilité à zéro du système suivant :

               y t -ε∆y + ∂ x d y + ∇q = f 1 ω dans Q, ∇ • y = 0 dans Q, y • n = 0, (Dy • n) tg = 0 sur Σ, y(0, •) = y 0 sur Ω. (1.22)
On rappel que le gradient symétrisé est donné par Du := 1 2 ∂ x i u j + ∂ x j u i i,j . En plus, on considère les ensembles de contrôles admissibles suivantes : [START_REF] Guerrero | Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions[END_REF] on sait que S 1 et S 3 sont non-vides, tandis qu'on prouve que S 2 est non-vide dans la Section 3.3.1. Quant au coût du contrôle, on le mesure avec les normes habituelles :

         S 1 (y 0 ) := {f ∈ L 2 (Q ω ) : Φ 2 (y 0 , f )(T, •) = 0}, S 2 (y 0 ) := {f 2 ∈ L 2 (Q ω ) : Φ 2 (y 0 , (0, f 2 ))(T, •) = 0}, S 3 (y 0 ) := {f ∈ L 2 (Q ω ) : Φ 3 (y 0 , f )(T, •) = 0}, où y 0 ∈ H(Ω d ), f ∈ L 2 (Q ω ) et Φ d (y 0 , f ) est la solution de (1.22) dans Ω d . Grâce à
                 K 1 (T, ε, ω) := sup y 0 ∈H(Ω 2 )\{0} inf f ∈S 1 (y 0 ) f L 2 (Qω) y 0 L 2 (Ω 2 ) , K 2 (T, ε, ω) := sup y 0 ∈H(Ω 2 )\{0} inf f 2 ∈S 2 (y 0 ) f 2 L 2 (Qω) y 0 L 2 (Ω 2 ) , K 3 (T, ε, ω) := sup y 0 ∈H(Ω 3 )\{0} inf f ∈S 3 (y 0 ) f L 2 (Qω) y 0 L 2 (Ω 3 )
.

(1.23)

Les résultats principaux de ce chapitre sont les suivants :

Theorem 1.15. On a les résultats suivants pour le problème de contrôle (1.22) :

1. Soit ω ⊂ Ω 2 un domaine. Alors, il y a c, C, T 0 > 0 tels que, si T > T 0 et ε ∈ (0, 1) :

K 2 (T, ε, ω) ≤ Ce -cε -1 . 2. Soient h ∈ (0, π) et ω ⊂ (0, π) × (π -h, π) un domaine. Alors, pour tout T ∈ (0, 2(π -h))
, il y a c > 0 tel que, si ε ∈ (0, 1) :

K 1 (T, ε, ω) ≥ ce cε -1 .
1.2. Résultats principaux et plan de la thèse

3. Soient h ∈ (0, π) et ω ⊂ (0, π) × (0, h) un domaine. Alors, pour tout T ∈ (0, π -h),
il y a c > 0 tel que, si ε ∈ (0, 1) :

K 1 (T, ε, ω) ≥ ce cε -1 . 4. Soient h ∈ (0, π) et ω ⊂ (0, π) 2 × (π -h, π
) un domaine. Alors, pour tout T > 0 il y a c > 0 tel que, si ε ∈ (0, 1) : Pour obtenir les résultats de contrôlabilité on travaille avec le système adjoint, qui est donné par :

K 3 (T, ε, ω) ≥ ce cε -1 . (1.
               -ϕ t -ε∆ϕ -∂ x d ϕ + ∇p = 0 dans Q, ∇ • ϕ = 0 dans Q, ϕ • n = 0, (2εDϕ • n + ϕn d ) tg = 0 sur Σ, ϕ(T, •) = ϕ T sur Ω, (1.25)
où d est la dimension de Ω. On obtient avec des techniques classiques (voir la HUM dans la Section 1.1.2) les égalités suivantes :

                   [K 1 (T, ε, ω)] 2 = sup ϕ T ∈H(Ω 2 )\{0} Ω 2 |ϕ(0, x)| 2 dx Qω |ϕ| 2 dxdt , [K 2 (T, ε, ω)] 2 = sup ϕ T ∈H(Ω 2 )\{0} Ω 2 |ϕ(0, x)| 2 dx Qω |ϕ 2 | 2 dxdt , [K 3 (T, ε, ω)] 2 = sup ϕ T ∈H(Ω 3 )\{0} Ω 3 |ϕ(0, x)| 2 dx Qω |ϕ| 2 dxdt
.

(1.26)

Remarque 1.17. Le système (1.25) a des propriétés particulières sur Ω d . Le gradient de la pression de toute solution de (1.25) est nul, ce qui implique que les composantes de toute solution de (1.25) sont des solutions de l'équation de la chaleur.

Pour étudier les solutions de (1.25) on obtient la décomposition spectrale de l'opérateur elliptique associé à (1.25) ; c'est-à-dire, on obtient les solutions de :

         -ε∆u -∂ x d u + ∇p = λu dans Ω, ∇ • u = 0 dans Ω, u • n = 0, (2εDu • n + un d ) tg = 0 sur ∂Ω.
(1.27)

L'approche de faire une décomposition spectrale n'est pas nouvelle en contrôlabilité (voir, par exemple, [START_REF] Coron | Control and Nonlinearity[END_REF]). Concernant les systèmes avec diffusion (ou dispersion) évanescente, cette approche a été utilisée dans [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] mais seulement pour obtenir des bornes du temps T 0 à partir duquel le coût du contrôle décroît exponentiellement avec ε. Par contre, c'est la première fois que la décomposition spectrale a été utilisée pour prouver l'inégalité de dissipation.

Pour obtenir les fonctions propres dans Ω 2 on considère les fonctions à variables séparées compatibles avec les conditions aux limites de (1.27) :

SV (Ω 2 ) := x 1 0 g 1 (s)dsg 2 (x 2 ), -g 1 (x 1 ) x 2 0 g 2 (s)ds : g 1 , g 2 ∈ L 2 0 (0, π) .
On prouve d'abord l'égalité :

span(SV (Ω 2 )) = H(Ω 2 ).
(1.28)

Après, on prouve par une recherche systématique que les vecteurs propres sont :

u m (x) := e -(2ε) -1 x 2 sin(m 1 x 1 ) m 1 (2m 2 ε cos(m 2 x 2 ) -sin(m 2 x 2 )), -2ε cos(m 1 x 1 ) sin(m 2 x 2 ) , où m = (m 1 , m 2 ) ∈ (N * ) 2
, et que ses valeurs propres associées sont :

λ ε m := (m 2 1 + m 2 2 )ε + 1 4ε .
En fait, on a l'égalité :

span(u m ) = H(Ω 2 ), (1.
Ω 2 u m (x) • v m (x)dx = 1 m=m . (1.30)
On rappelle que comme l'opérateur elliptique associé au système (1.27) n'est pas autoadjoint, u m n'est pas orthogonal, donc ce n'est pas une question mineure. Avec une recherche systématique en considérant les bases trigonométriques de L 2 (0, π), on trouve que (1.30) est satisfait pour :

v m (x) := 2 π 2 e (2ε) -1 x 2 m 1 sin(m 1 x 1 ) 2m 2 ε cos(m 2 x 2 ) -sin(m 2 x 2 ) 1 + 4m 2 2 ε 2 , - cos(m 1 x 1 ) sin(m 2 x 2 ) 2ε .
(1.31)

Alors, on est prêts à exprimer les solutions de (1.25) à l'aide de la décomposition spectrale :

Proposition 1.18. Soit ϕ T ∈ H(Ω 2 ). On désigne pour t ≤ T , L T ε (t)ϕ T la valeur dans H(Ω 2 ) de la solution d'énergie du système (1.25) en Ω 2 au temps t. Alors, on a que :

1. Pour tout t < T et ε > 0 : L T ε (t)ϕ T = m∈(N * ) 2 ϕ T , v m L 2 (Ω 2 ) exp (m 2 1 + m 2 2 )ε + 1 4ε (t -T ) u m . (1.32)
En particulier, la série du côté droit de (1.32) est bien définie et absolument convergente dans H(Ω 2 ).

Pour tout

δ > 0 il y a C δ > 0 tel que pour tout T > 0, ϕ T ∈ H(Ω 2 ), ε ∈ (0, 1) et s ≤ T -2π -δ : L T ε (s)ϕ T L 2 (Ω 2 ) ≤ C δ exp s -(T -2π -δ) 4ε ϕ T L 2 (Ω 2 ) . (1.33)
La preuve de la Proposition 1.18 se fait en prouvant les estimations pour ϕ T ∈ span(u m ) et en considérant la continuité par rapport à la valeur initiale. Pour obtenir les estimations on utilise l'inégalité triangulaire et on borne la série par des intégrales faciles à calculer.

Avec la Proposition 1.18 et (1.26) on est prêts à prouver les trois premiers points du Théorème 1.15 :

• Pour le premier point on suit la stratégie de [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF] : avec l'inégalité de dissipation (1.33) et une inégalité de Carleman qu'on prouve, on obtient que le coût du contrôle décroît exponentiellement avec ε. Pour prouver l'inégalité de Carleman on utilise que la deuxième composante des solutions de (1.25) satisfait l'équation de la chaleur (voir la Remarque 1.17) et après on obtient l'inégalité pour la première composante en utilisant la condition d'incompressibilité.

• Pour le deuxième point on considère ϕ(t, x) = e -λ ε 1,1 (T -t) u 1,1 (x), qui est une solution de (1.25), et on fait des estimations assez simples.

• Pour le troisième point on adapte la technique de [71, Théorème 1]. Dans [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF] les auteurs utilisent le fait qu'une solution à valeur initiale positive reste positive, propriété qu'on n'a pas pour les solutions de (1.25). Par contre, on vérifie que si l'on prend une valeur initiale ϕ T telle que supp(ϕ T ) ⊂ (0, π) × (π -δ, π), il existe c > 0 tel que pour tout ε ∈ (0, 1) on a :

ϕ ε (0, •) L 2 (Ω) ≥ c,
inégalité qu'on montre par contradiction. Ensuite, on obtient des estimations de ϕ 2 sur Q ω à l'aide d'une inégalité d'Agmon comme dans [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF] en utilisant que ϕ 2 est une solution de la chaleur (voir Remarque 1.17). Pour conclure, on utilise des estimations de régularité parabolique pour avoir l'estimation de ∂ x 2 ϕ 2 , et donc avec la condition d'incompressibilité et Poincaré on obtient des estimations de ϕ 1 .

Finalement, pour prouver le quatrième point du Théorème 1.15 on cherche des solutions de (1.27) dans Ω 3 . De nouveau, on considère les fonctions à variables séparées compatibles avec les conditions aux limites de (1.27) :

SV (Ω 3 ) := SV 1 (Ω 3 ) ∪ SV 2 (Ω 3 ) ∪ SV 3 (Ω 3 ) := 0, g 1 (x 1 ) x2 0 g 2 (s)dsg 3 (x 3 ), -g 1 (x 1 )g 2 (x 2 ) x3 0 g 3 (s)ds : g 1 ∈ L 2 (0, π); g 2 , g 3 ∈ L 2 0 (0, π) ∪ - x1 0 g 1 (s)dsg 2 (x 2 )g 3 (x 3 ), 0, g 1 (x 1 )g 2 (x 2 ) x3 0 g 3 (s)ds : g 2 ∈ L 2 (0, π); g 1 , g 3 ∈ L 2 0 (0, π) ∪ x1 0 g 1 (s)dsg 2 (x 2 )g 3 (x 3 ), -g 1 (x 1 ) x2 0 g 2 (s)dsg 3 (x 3 ), 0 : g 3 ∈ L 2 (0, π); g 1 , g 2 ∈ L 2 0 (0, π) .
(1. On peut prouver assez facilement que : 1.2.3 Contrôlabilité à zéro de l'équation de la chaleur dans des pseudo-cylindres avec un contrôle qui agit à l'intérieur (voir [START_REF] Bárcena-Petisco | Null controllability of the heat equation in pseudocylinders by an internal control[END_REF])

u(x) = 2e -ε -1 x 3 π ε (1 -e -2πε
La contrôlabilité à zéro de l'équation de la chaleur avec des contrôles internes est un des problèmes historiques en contrôlabilité des équations paraboliques. Pour prouver la contrôlabilité à zéro de cette équation il y a trois méthodes :

• Utiliser les estimations d'observabilité de l'équation des ondes (voir, par exemple, [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF]).

• Prouver des inégalités spectrales et après considérer les propriétés des fonctions analytiques (voir, par exemple, [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]).

• Prouver des inégalités de Carleman avec des poids spécifiques (voir, par exemple, [START_REF] Fursikov | Controllability of evolution equations[END_REF]). Cette dernière méthode est connue comme la méthode de Fursikov-Imanuvilov.

Ω = U ({(x, z) : x ∈ B, z ∈ (0, H(x))}) . (1.35)
En plus, on désigne sa partie cylindrique par :

C := U B × 0, inf B H .
(1.36)

Si U = I (l'endomorphisme de l'identité), on dit que le pseudo-cylindre est orienté canoniquement.

Plus précisément, on prouve :

Théorème 1.21. Soient Ω un pseudo-cylindre et ω ⊂ Ω un sous-domaine. Alors, il y a

C > 0 tel que si T > 0, A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q) et y 0 ∈ L 2 (Ω), il y a un contrôle f ∈ L 2 (Q ω ) tel que la solution du système :          y t -∆y + A • ∇y + ay = f 1 ω dans Q, y = 0 sur Σ, y(0, •) = y 0 sur Ω,
satisfait y(T, •) = 0, et tel que le contrôle satisfait l'estimation :

f L 2 (Qω) ≤ Ce Cκ(T,a,A) y 0 L 2 (Ω) , avec : κ(T, a, A) := 1 + T -1 + T a L ∞ (Q) + a 2/3 L ∞ (Q) + (1 + T ) A 2 (L ∞ (Q)) d+1 .
Remarque 1.22. On peut appliquer les techniques de [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] pour voir que la contrôlabilité à zéro avec le coût Ce Cκ(T,a,A) implique la contrôlabilité aux trajectoires (et donc approchée)

du système non-linéaire de l'équation de la chaleur si la non-linéarité est d'ordre 0 et suffisamment faible.

Remarque 1.23. On prouve que les pseudo-cylindres ne sont pas contenus dans les domaines considérés dans [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF], donc c'est une contribution dans ce sens. En plus, notre résultat permet d'étudier, entre autres, la contrôlabilité de l'équation de la chaleur non-linéaire.

Finalement, dans [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF] les auteurs utilisent des inégalités spectrales et les propriétés des fonctions analytiques, tandis que nous, on le fait en adaptant la méthode de Fursikov-Imanuvilov.

1.

{u ∈ H 1 0 ( Ω) : ∆u ∈ L 2 ( Ω)} ⊂ H 2 ( Ω) continûment, (1.40)
∂ ñη ≤ 0 sur ∂ Ω. (1.41)
Alors, on considère les poids :

α := e λ 2m+1 2m k -e λ(k+η) t m (T -t) m , ξ := e λ(k+η) t m (T -t) m , ( 1
lim ε→0 + sup Ω\Ωε d(•, ∂Ω) = 0. (1.43)
Alors, il y a C > 0 tel que pour tout

T > 0, g ∈ L 2 (Q), G ∈ (L 2 (Q)) d , u T ∈ L 2 (Ω), k > 2(m + 1) η L ∞ (Ω)
, et pour les poids définis dans (1.42), on a que :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 , (1.44) pour tout λ ≥ C, s ≥ C(T m + T 2m
) et où u est la solution de :

         -u t -∆u = g + ∇ • G dans Q, u = 0 sur Σ, u(T, •) = u T sur Ω.
(1.45)

La preuve de la Proposition 1.25 utilise les techniques habituelles (voir [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF][START_REF] Yu | Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications[END_REF]) et le lemme suivant :

Lemme 1.26. Soient Ω un domaine Lipschitz, Ω ε ⊂ Ω des domaines Lipschitz satisfaisant

(1.43), u 0 ∈ L 2 (Ω) et g ∈ L 2 (Q). Alors, S Ω ε , u 0 1 Ωε , g1 Ωε 1 Ωε → L 2 (Q) S(Ω, u 0 , g), (1.46) où :
S(Ω, u 0 , g)(t, x), est l'unique solution dans L 2 (0, T ;

H 1 0 (Ω)) ∩ H 1 (0, T ; H -1 (Ω)) de :          u t -∆u = g dans Q, u = 0 sur Σ, u(0, •) = u 0 sur Ω.
(1.47)

On prouve le Lemme 1.26 en considérant la formulation variationnelle de (1.47).

Résultats principaux et plan de la thèse

On peut construire assez facilement la fonction auxiliaire η lorsque Ω est un pseudocylindre canoniquement orienté et ω ⊂ C. On peut supposer que :

ω = B(x, r) × (z -r, z + r) ⊂⊂ C, (1.48)
pour r > 0 et (x, z) ∈ Ω tels que :

z < inf B H. (1.49)
Alors, on considère :

η(x, z) := η B (x) -c(z -z)
c ≥ c 0 , il y a C > 0 tel que pour tout T > 0, A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), g ∈ L 2 (Q), G ∈ (L 2 (Q)) d+1 , u T ∈ L 2 (Ω), k > 2(m + 1) η L ∞ (Ω)
, et pour les poids définis par (1.42), on a que :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 , (1.52) pour tout λ ≥ C, s ≥ C T m + T 2m (1 + a 2/3 L ∞ (Q) + A 2 (L ∞ (Q)) d+1 ) , (1.53)
et pour u la solution de :

         -u t -∆u -∇ • (Au) + au = g + ∇ • G dans Q, u = 0 sur Σ, u(T, •) = u T sur Ω.
(1.54)

On rappelle que la Proposition 1.28 implique le Théorème 1.21 pour tout ω ⊂ C (voir, par exemple, [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]).

Finalement, on prouve le cas ω ⊂ Ω \ C. Dans ce cas, on peut supposer que :

ω = B(x, r) × (z -r, z + r) ⊂⊂ Ω \ C, (
ω * = B(x, r) × (2r, 4r) ⊂⊂ Ω ∩ C.
Alors, par la Proposition 1.28 on a pour toute solution de (1.53) :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Q ω * e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 . (1.59)
Ensuite, on considère une fonction auxiliaire η satisfaisant (1.37) sur ( Ω, ω) et telle que :

   η ≤ η dans ω * , η ≤ η dans Ω \ B(x, 2r) × (r, z + 2r) .
(1.60)

En plus, on considère une fonction positive régulière χ telle que : supp(χ) ⊂⊂ Ω, χ = 1 dans B(x, 2r) × (r, z + 2r).

(1.61)

Pour transmettre l'information on utilise l'inégalité de Carleman suivante : [START_REF] Fernández-Cara | Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability[END_REF], on a :

s 3 λ 4 Q e -2s α ξ 3 |χu| 2 + sλ 2 Q e -2s α ξ|∇(χu)| 2 ≤ C s 3 λ 4 Qω e -2s α ξ 3 |χu| 2 + Q e -2s α| -2∇χ • ∇u -∆χu -(∇χ • A)u -∇χ • G + χg -aχu| 2 + s 2 λ 2 Q e -
C > 0 telle que si T > 0, A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), g ∈ L 2 (Q), G ∈ (L 2 (Q)) d+1 , u T ∈ L 2 (Ω), k > 2(m + 1) max{ η L ∞ (Ω) , η L ∞ (Ω) }, λ ≥ C, et s satisfait (1.
s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 + s 3 λ 4 Q e -2s α ξ 3 |χu| 2 + sλ 2 Q e -2s α ξ|∇(χu)| 2 ≤ C s 3 λ 4 Qω e -2s α ξ 3 |u| 2 + Q e -2sα |g| 2 + Q e -2s α|χg| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 + s 2 λ 2 Q e -2s
                     u t + uu x + u xxx + av xxx + k 1 vv x + k 2 (uv) x = h 1 dans Q, cv t + rv x + vv x + abu xxx + v xxx + k 2 buu x + k 1 b(uv) x = h 2 + f 1 ω dans Q, u(•, 0) = u(•, L) = u x (•, L) = 0 sur (0, T ), v(•, 0) = v(•, L) = v x (•, L) = 0 sur (0, T ), u(0, •) = u 0 , v(0, •) = v 0 sur (0, L), (1.64)
où u 0 , v 0 ∈ L 2 (0, L), h 1 , h 2 sont deux fonctions exponentiellement petites près de T (voir

(1.66)) et a, b, c, r, k 1 , k 2 ∈ R tels que : a, b, c > 0 et a 2 b < 1. (1.65)
On peut consulter [START_REF] Gear | Weak and strong interactions between internal solitary waves[END_REF] et [START_REF] Bona | A model system for strong interaction between internal solitary waves[END_REF] pour comprendre sa déduction et son interprétation d'un point de vue physique.

La plupart des résultats sur la contrôlabilité de (1.63) (et de ses versions linéarisées) traitent la contrôlabilité frontière. Les premiers résultats datent de 1997 (voir [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]). Pour une résumé des résultats de contrôlabilité sur l'équation de KdV avec des contrôles qui agissent sur la frontière on peut consulter [START_REF] Rosier | Control and stabilization of the Korteweg-de Vries equation : recent progresses[END_REF]. Quant à la contrôlabilité de (1.63) avec des contrôles qui agissent à l'intérieur, c'est un sujet de recherche relativement récent (voir [START_REF] Capistrano-Filho | Internal controllability of the Korteweg-de Vries equation on a bounded domain[END_REF]).

En ce qui concerne les systèmes du type (1.64), il n'a été étudié dans la littérature qu'avec des contrôles qui agissent sur le bord (voir [START_REF] Capistrano-Filho | Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictions on the spatial domain[END_REF][START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF][START_REF] Cerpa | A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF][START_REF] Micu | On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF]). Dans cette partie, on s'intéresse à la contrôlabilité de (1.64), où le contrôle agit à l'intérieur et uniquement sur la deuxième composante. Plus précisément, on prouve le résultat suivant :

Théorème 1.30. Soient L > 0, T > 0 et supposons que (1.65) est satisfait. Alors, il y a

C > 0 tel que pour tout h 1 , h 2 satisfaisant e C (T -t) 12 (h 1 , h 2 ) ∈ L 2 (0, T ; H -1 (0, L)), (1.66) il y a δ > 0 tel que pour tout U 0 < δ il y a f ∈ L 2 (Q ω ) et U ∈ C([0, T ]; L 2 (0, L)) satisfaisant que (U, f ) est une solution de (1.64) et U (T, •) = 0.
Dans cette partie on utilise la notation U := (u, v) et U 0 := (u 0 , v 0 ).

La preuve du Théorème 1.30 repose sur un théorème d'inversion locale (version surjective) et sur un résultat de contrôlabilité à zéro pour le système linéarisé autour de 0 avec un terme source :

         LU = (h 1 , h 2 + f 1 ω ) dans Q, U (•, 0) = U (•, L) = U x (•, L) = 0 sur (0, T ), U (0, •) = U 0 sur (0, L), (1.67)
où :

LU := (u t + u xxx + av xxx , cv t + rv x + abu xxx + v xxx ).
(1.68)

Pour prouver la contrôlabilité de (1.67) on utilise la HUM et on prouve une inégalité de Carleman pour le système adjoint de (1.67), qui est donné par :

         L * Φ = G dans Q, Φ(•, 0) = Φ(•, L) = Φ x (•, 0) = 0 sur (0, T ), Φ(T, •) = Φ T sur (0, L), (1.69) où Φ := (ϕ, φ), Φ T = (ϕ T , φ T ) ∈ L 2 (0, L), G ∈ L 2 (Q) et : L * Φ := (-ϕ t -ϕ xxx -abφ xxx , -cφ t -rφ x -aϕ xxx -φ xxx ).
(1.70)

Plus précisément, on montre :

Proposition 1.31. Soient L > 0, ω ⊂ (0, L) un ouvert, m ≥ 12 et supposons que (1.65) Pour prouver la Proposition 1.31 on utilise la stratégie de [START_REF] Carreño | Local null controllability of the N -dimensional Navier-Stokes system with N -1 scalar controls in an arbitrary control domain[END_REF]. Cette stratégie consiste de façon globale à décomposer les solutions du système linéarisé autour de 0 multipliées par des poids qui décroissent exponentiellement en 0 et en T . On les décompose en une première solution avec la source à laquelle on applique des estimations de régularité et en une deuxième solution régulière à laquelle on applique des estimations de Carleman comme dans [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF].

est satisfait. Alors, il y a C > 0 tel que pour tout λ ≥ C, s ≥ C(T m + T 3m ), Φ T ∈ L 2 (0, L) et G ∈ L 2 (Q) on a l'estimation : s 6 λ 6 T 0 e -2sα * -10sβ * (ξ * ) 6 Φ 2 H 2 (0,L) ≤ C Qω e -sα * -10sβ * φ 2 + Q e -8sβ
Pour pouvoir utiliser cette méthode, il faut avoir des résultats de régularité de (1.69).

Pour cela, on combine les stratégies présentées dans [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF] et dans [START_REF] Micu | On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF] et on prouve :

Proposition 1.32. Soit L > 0 et supposons que (1.65) est satisfait. Alors :

1. Pour tout T > 0, Φ T ∈ L 2 (0, L) et G ∈ L 2 (0, T ; H -1 (0, L)) le système (1.69) admet
une unique solution :

Φ ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)).
En plus, il y a C > 0 indépendant de T , Φ T et G tels que :

Φ C([0,T ];L 2 (0,L)))∩L 2 (0,T ;H 1 (0,L)) ≤ C(1 + T ) Φ T L 2 (0,L) + G L 2 (0,T ;H -1 (0,L)) .
2. Il y a C > 0 tel que pour tout T > 0 et G ∈ L 2 (Q) on a :

Φ ∈ L 2 (0, T ; H 2 (0, L)) ∩ H 1 (0, T ; H -1 (0, L))
satisfaisant l'estimation :

Φ L 2 (0,T ;H 2 (0,L))∩H 1 (0,T ;H -1 (0,L)) ≤ C(1 + T ) G L 2 (Q) ,
pour Φ la solution d'énergie de (1.69) avec Φ T = 0.

3. Il y a C > 0 tel que pour tout

T > 0 et G ∈ L 2 (0, T ; H 1 0 (0, L)) on a Φ ∈ H 1,3 (Q) satisfaisant l'estimation : Φ H 1,3 (Q) ≤ C(1 + T 3 ) G L 2 (0,T ;H 1 0 (0,L)) ,
pour Φ la solution d'énergie de (1.69) avec Φ T = 0.

En plus, on généralise ces résultats à des normes d'ordre supérieur par récurrence.

Il y a deux différences importantes entre notre preuve et celle de [START_REF] Carreño | Local null controllability of the N -dimensional Navier-Stokes system with N -1 scalar controls in an arbitrary control domain[END_REF].

T 0 e -14sβ * Φ 2 H 1 0 (0,L) + e -7sβ * Φ 2 C([0,T ];L 2 (0,L)) ≤ C Qω e -10sβ * φ 2 + Q e -8sβ * |G| 2 .
(1.72)

On cherche des solutions de (1.67) dans l'espace E de couples (U, f ) tels que :

• U ∈ L 2 (e 4sβ * (0, T ); L 2 (0, L)), • U ∈ L 2 (e 7 2 sβ * (0, T ); H 1 0 (0, L)) ∩ C(e 7 2 sβ * [0, T ]; L 2 (0, L)), • f 1 ω ∈ L 2 (e 5sβ * (0, T ); L 2 (0, L)),
• LU -(0, f 1 ω ) ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)).

Concrètement, on a :

Proposition 1.33. Soient U 0 ∈ L 2 (0, L), h 1 , h 2 ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)) et supposons que (1.65) est satisfait. Alors, il y a une solution de (1.67) qui satisfait (U, f ) ∈ E.

Remarque 1.34. Si (U, f ) ∈ E, on a que U (T, •) = 0.

La preuve de la Proposition 1.33 suit les arguments de [START_REF] Fursikov | Controllability of evolution equations[END_REF]. On considère P 0 le sous-

espace de C 3 ([0, T ] × [0, L]) des fonctions Φ telles que : Φ(•, 0) = Φ(•, L) = Φ x (•, 0) = 0 sur [0, T ],
et on définit sur P 0 :

   a( Φ, Φ) := Qω e -10sβ * φφ + Q e -8sβ * (L * Φ) • (L * Φ), (Φ) := L 0 U 0 • Φ(0, x)dx + T 0 (h 1 , h 2 ), Φ H -1 (0,L)×H 1 0 (0,L) .
1.

U = e -8sβ * L * Φ, f = -e -10sβ * φ1 ω ,
est une solution de (1.67) qui appartient à E.

Finalement, on considère h 1 , h 2 comme énoncés dans (1.66), avec C suffisamment grand tels que h 1 , h 2 ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)). Soit :

G : E → L 2 (e 7sβ * (0, T ); H -1 (0, L)) × L 2 (0, L),
l'opérateur défini par :

G(U, f ) := (u t + uu x + u xxx + av xxx + k 1 vv x + k 2 (uv) x -h 1 , cv t + rv x + vv x + abu xxx + v xxx + k 2 buu x + k 1 b(uv) x -h 2 -f 1 ω , u(0, •), v(0, •)). (1.73)
Pour finir la preuve du théorème 1.30 il suffit d'utiliser le théorème d'inversion locale qui dit que si G est C 1 et si DG[0] est surjective, alors il y a δ > 0 tel que si U 0 < δ il y a (U, f ) ∈ E tel que :

G(U, f ) = (0 R 2 , U 0 ).

Un travail complémentaire et des problèmes ouverts

On finit ce chapitre introducteur en faisant une brève présentation d'un travail complémentaire concernant le coût de la contrôlabilité de l'équation de la chaleur avec des conditions aux limites mixtes et une diffusion évanescente, ainsi qu'en énonçant quelques perspectives et problèmes qui demeurent ouverts.

1.3.1 Travail complémentaire : étude du coût de la contrôlabilité d'une équation parabolique d'ordre deux avec diffusion évanescente et un terme de transport (voir [START_REF] Bárcena-Petisco | Study of the cost of the controllability of second order parabolic equations with small diffusion and a transport term[END_REF])

Un travail complémentaire réalisé pendant la thèse est l'étude du coût de la contrôllabilité à zéro de l'équation de la chaleur avec diffusion évanescente, un terme de transport et des conditions aux limites mixtes, de type Fourier et de type Dirichlet. Plus précisément, on étudie la contrôlabilité du système suivant :

               y t -ε∆y + ∂ x 1 y = 1 ω f, dans (0, T ) × Ω, ∂ n y + ay = 0, sur (0, T ) × Γ, y = 0, sur (0, T ) × Γ * , y(0, •) = y 0 , sur Ω, (1.74)
où Γ ⊂ ∂Ω est relativement ouvert, Γ * := ∂Ω \ Γ et a(x, ε) est une fonction telle que a(•, ε) ∈ L ∞ (Γ) pour tout ε ∈ (0, ε 0 ). Quant au coût du contrôle, on le définit par :

K(Ω, ω, T, ε) := sup y 0 ∈L 2 (Ω)\{0} inf f ∈S(y 0 ) f L 2 (Qω) y 0 L 2 (Ω)
, C'est un travail présenté dans [START_REF] Bárcena-Petisco | Study of the cost of the controllability of second order parabolic equations with small diffusion and a transport term[END_REF].

La motivation de cette étude est de mieux comprendre les résultats du Chapitre 3 (voir [START_REF] Bárcena-Petisco | Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit[END_REF]). De fait, les résultats obtenus au Chapitre 3 suggèrent que le comportement du coût du contrôle par rapport à ε dépend des conditions aux limites, et avec ce travail on vérifie que cette conjecture est vraie :

• On prouve que, sous certaines conditions, le coût décroît exponentiellement :

Theorem 1.35. Soient Ω un domaine C 2 , ω ⊂ Ω un sous-domaine et supposons que (Γ, a) satisfait (a + (2ε) -1 n 1 )1 Γ ≥ 0. Alors, il y a T 0 , c, C > 0 tels que pour tout ε ∈ (0, ε 0 ) et T ≥ T 0 on obtient que : K(Ω, ω, T, ε) ≤ Ce -cε -1 .
Les hypothèses du Théorème 1.35 incluent des systèmes avec des conditions aux limites qui sont presque de Dirichlet et avec des conditions aux limites mixtes qui uniquement ont des conditions de Robin (ou de Neumann) sur la partie du bord où le flux du transport sort.

• On prouve qu'avec des conditions aux limites de Neumann au moins dans la partie du bord où le flux du transport entre, le coût du contrôle peut croître exponentiellement :

Theorem 1.36. Soient Ω un domaine, Γ = ∂Ω, a = 0 et ω ⊂ Ω un sous-domaine compactement inclus. Alors, pour tout T > 0 il y a c > 0 tel que pour tout ε > 0 on a que :

K(Ω, ω, T, ε) ≥ ce cε -1 . Theorem 1.37. Soient L, h > 0, Ω = (-L, 0), Γ = {-L}, a = 0 et ω ⊂ (-L + h, 0)
un sous-domaine. Alors, pour tout T > 0 il y a c, ε 0 > 0 tels que pour tout ε ∈ (0, ε 0 ) on a l'estimation :

K(Ω, ω, T, ε) ≥ ce cε -1 .
• On étudie le coût de la contrôlabilité avec des conditions aux limites de Neumann lorsque le domaine de contrôle est le domaine entier (lorsque ω = Ω) :

Theorem 1.38. Soient L > 0, Ω = (-L, 0), Γ = {-L, 0} et a = 0. Alors, il y a c, C > 0 tels que pour tout T ≥ 4L on a que :

cT -1/2 ≤ lim inf ε→0 K(Ω, Ω, T, ε) ≤ lim sup ε→0 K(Ω, Ω, T, ε) ≤ CT -1/2 .
Pour prouver les Théorèmes 1.35, 1.36, 1.37 et 1.38 on utilise la méthode de HUM.

Plus précisément, on adapte la méthode de [START_REF] Bárcena-Petisco | Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit[END_REF] et on se concentre dans la décomposition spectrale de l'opérateur elliptique associé au système adjoint de (1.74). En plus, dans le Carleman on adapte les preuves de [START_REF] Fursikov | Controllability of evolution equations[END_REF] et [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF] au système adjoint de (1.74) en vainquant les difficultés des conditions aux limites et de la manque de régularité de ces solutions (compte tenu du fait qu'on doit faire des estimations précises par rapport à ε).

Perspectives et problèmes ouverts

Dans cette partie, on fait quelques commentaires et on présente quelques problèmes ouverts.

Par rapport au Chapitre 2, on remarque que les problèmes suivants restent ouverts :

• Déterminer si pour tout domaine Ω qui satisfait l'Hypothèse 1.7, le système de Navier-Stokes avec la pénalisation (1.13) 2 est localement contrôlable à zéro avec un contrôle scalaire.

• Déterminer si l'hypothèse géométrique du Théorème 1.8 (respectivement du Théorème 1.12) est nécessaire pour ε > 0 (respectivement pour a ≥ 0) suffisamment petit.

• Déterminer si le système pénalisé de Stokes en dimension 3 avec un contrôle qui agit sur deux composantes est contrôlable à zéro.

• Déterminer la contrôlabilité du système de Stokes (et de Navier-Stokes) avec d'autres conditions de compressibilité, comme ∇ • y = -εp t ou ∇ • y = ε∆p (voir [START_REF] Shen | Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations[END_REF]).

Par rapport au Chapitre 3, on peut mentionner : • Déterminer le T 0 optimal à partir duquel le coût de la contrôlabilité en dimension 2 décroît de façon exponentielle.

Par rapport au Chapitre 4, on remarque que les problèmes suivants restent ouverts :

• Déterminer la contrôlabilité à zéro de l'équation de la chaleur pour tout domaine Ω Lipschitz.

• Déterminer la contrôlabilité à zéro de l'équation de la chaleur dans des pseudocylindres avec d'autres conditions aux limites, par exemple des conditions aux limites de type Neumann.

• Déterminer la contrôlabilité à zéro de l'équation de la chaleur semi-linéaire dans des pseudo-cylindres lorsque la non-linéarité dépend du gradient de la solution.

• Établir la contrôlabilité à zéro d'autres systèmes paraboliques lorsque le domaine n'est pas C 2 ni un produit tensoriel de domaines C 2 ; par exemple, le système de Stokes.

Par rapport au Chapitre 5, on peut mentionner :

• Déterminer la contrôlabilité aux trajectoires du système (1.64).

• Déterminer la contrôlabilité locale à zéro de (1.64) avec une force agissant sur la frontière et sur une seule composante.

Chapitre 2

Null controllability of a penalized

Stokes problem in dimension two with one scalar control

In this chapter we consider a penalized Stokes system defined in a regular domain Ω ⊂ R 2 and with Dirichlet boundary conditions. We prove that our system is null controllable using a scalar control localized in an open subset inside Ω and whose cost is bounded uniformly with respect to the parameter that converges to 0. This chapter is included in [START_REF] Bárcena-Petisco | Null controllability of a penalized Stokes problem in dimension two with one scalar control[END_REF].

Introduction

Main results

Let T > 0 and Ω ⊂ R 2 a regular domain. Throughout this paper we use the term "domain" to refer to a bounded connected non-empty open set. As usual, we denote Q := (0, T )×Ω and Σ := (0, T )×∂Ω. In this paper we work on the Stokes penalized system with Dirichlet boundary conditions. This system is given by the following equations :

               v ε t -∆v ε + ∇q ε = f in Q, εq ε + ∇ • v ε = 0 in Q, v ε = 0 on Σ, v ε (0, •) = v 0 in Ω.
(2.1)

Here f : Q → R 2 is a source term, v 0 : Ω → R 2 is an initial condition and ε > 0.

This system approximates the classical Stokes problem, which is given by the following equations :

               v t -∆v + ∇q = f in Q, ∇ • v = 0 in Q, v = 0 on Σ, v(0, •) = v 0 in Ω.
(2.

2)

The main objective of this paper is to prove that system (2.1) is null controllable with a one-dimensional control whose cost is uniformly bounded with respect to ε. We prove it for almost every direction, being these directions different for each Ω. In addition, if Ω is strictly convex we prove it for all the directions.

First of all, we state what hypothesis Ω must satisfy to be controllable by a force parallel to e 1 := (1, 0). In order to do so, if Ω ⊂ R 2 is a C 2 domain, we use the standard convention to parametrize ∂Ω : we denote the arc-length parametrization of each connected component of ∂Ω by σ i = (σ i 1 , σ i 2 ) and the signed curvature of ∂Ω on the point σ i (θ) by κ i (θ). In both terms the superscript i is omitted if ∂Ω is connected. We suppose that each component is parametrized in the standard way ; that is, for U (x, y) := (-y, x) (U is the rotation of 90 degrees to the left) and for all p = σ i (θ) ∈ ∂Ω, there is δ 0 (p) > 0 such that if δ ∈ (0, δ 0 (p)), then p + δU ((σ i ) (θ)) ∈ Ω.

Remark 2.1. Since the σ i are arc-length, we have the well-known equalities :

κ i = (σ i 2 ) (σ i 1 ) -(σ i 1 ) (σ i 2 ) = (σ i 2 ) (σ i 1 ) = - (σ i 1 ) (σ i 2 )
.

(2.3) Hypothesis 2.1. Let Ω ⊂ R 2 be a C 2 domain of boundary ∂Ω parametrized by functions σ 1 , . . . , σ k as explained in the previous paragraph. Then, for any i ∈ {1, . . . , k} and for any θ such that (σ i 1 ) (θ) = 0 or (σ i 2 ) (θ) = 0, we have that κ i (θ) = 0.

Remark 2.2. Hypothesis 2.1 means that if Ω is a C 2 domain, then on all the points of the boundary of horizontal or vertical tangent line the curvature is not null. We use it to avoid pathologies near those points, since in that case we do not know how to proceed.

Hypothesis 2.1 is not restrictive at all, thanks to the following lemma, which we prove at the beginning of Subsection 2.4.1 :

Lemma 2.3.
Let Ω be a C 2 domain. Then, there is an orthogonal R 2 -endomorphism U such that the domain Ω := U (Ω) satisfies Hypothesis 2.1. In fact, if we denote U ψ the endomorphism characterized by e 1 := (1, 0) → (cos(ψ), sin(ψ)) and e 2 := (0, 1) →

(-sin(ψ), cos(ψ)), then, for almost every ψ in [-π, π], U ψ (Ω) satisfies Hypothesis 2.1.
With Lemma 2.3 in mind, we state one of the main results of this paper :

Theorem 2.4. Let Ω ⊂ R 2 be a regular domain that satisfies Hypothesis 2.1 and let ω ⊂ Ω be a non-empty open set. Then, there is ε 0 > 0 such that for all T > 0 there is

C > 0 such that if ε ∈ (0, ε 0 ) and y 0 ∈ L 2 (Ω), there is a scalar-valued function f ε ∈ L 2 ((0, T ) × ω)
satisfying :

f ε L 2 ((0,T )×ω) ≤ C y 0 L 2 (Ω) ,
and such that the solution of the following system :

               y ε t -∆y ε + ∇p ε = f ε 1 ω e 1 in Q, εp ε + ∇ • y ε = 0 in Q, y ε = 0 on Σ, y ε (0, •) = y 0 in Ω, (2.4
)

satisfies y ε (T, •) = 0.
As usual, in Theorem 2.4 and throughout this paper L p and H s denote respectively the Lebesgue and Sobolev spaces of vector-valued functions.

Remark 2.5. From Lemma 2.3 and since system (2.1) is invariant with respect to rotations, we actually have for almost all directions e θ := (sin(θ), cos(θ)) that there is ε 0 > 0 such that for all T > 0 there is C > 0 such that for all ε ∈ (0, ε 0 ) system (2.1) is null controllable with a force f ε 1 ω e θ satisfying :

f ε L 2 ((0,T )×ω) ≤ C y 0 L 2 (Ω) .
Remark 2.6. A natural question that may arise is the relation between the control problem (2.4) and the control problem :

               y t -∆y + ∇p = f 1 ω e 1 in Q, ∇ • y = 0 in Q, y = 0 on Σ, y(0, •) = y 0 in Ω, (2.5)
for f ∈ L 2 ((0, T ) × ω) and for y 0 ∈ H(Ω) (the subspace of L 2 (Ω) of functions of null divergence and null normal trace). Using weak compactness, we have that there is a sequence f ε k which converges weakly in L 2 ((0, T ) × ω) to some function f 0 . Moreover, using the techniques presented in the proof of [102, Theorem I.2], we have that

y ε k (t, •)
converges in the H -1 (Ω)-norm for all t ∈ [0, T ] to y(t, •) (the solution of (2.5)). In particular, since y ε k (T, •) = 0, we have that y(T, •) = 0. Consequently, this provides an alternative way of proving the well-known result that the system (2.5) is null controllable with a one-dimensional control supported in any regular domain (see [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]). In that sense,

an interesting problem that remains open is if the control of minimal L 2 -norm of (2.4)
converges to the control of minimal L 2 -norm of (2.5).

In order to prove the null controllability of (2.4), we consider as usual its adjoint system :

               -ϕ ε t -∆ϕ ε + ∇π ε = 0 in Q, επ ε + ∇ • ϕ ε = 0 in Q, ϕ ε = 0 on Σ, ϕ ε (T, •) = ϕ T in Ω. (2.6)
Indeed, it is a well-known result (see [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations : recent progress and open questions[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systemes distribués[END_REF]) that the existence in (2.4) of a control f ε bounded uniformly in ε is equivalent to proving that there is ε 0 > 0 such that for all

T > 0 there is C > 0 such that if ε ∈ (0, ε 0 ) and ϕ T ∈ L 2 (Ω) we have : Ω |ϕ ε (0, •)| 2 ≤ C (0,T )×ω |ϕ ε 1 | 2 , ( 2.7)
for ϕ ε the solution of (2.6). The proof of Theorem 2.4 is thus reduced to the proof of the observability inequality (2.7), and we focus on it from now on.

In order to prove estimate (2.7) we prove a Carleman inequality. Before presenting it, let us define the weights we use throughout the paper as follows :

α(t, x) = e 2λ η 0 ∞ -e λη 0 (t(T -t)) m , ξ(t, x) = e λη 0 (t(T -t)) m , α * (t) = max x∈Ω α(t, x), ξ * (t) = min x∈Ω ξ(t, x).
(2.8)

These weights are classical and widely used in the literature. Here, λ and m are positive real numbers to be fixed later on. Moreover, η 0 ∈ C 4 Ω is a fixed function that satisfies η 0 > 0 in Ω, η 0 = 0 on ∂Ω, and |∇η 0 | > 0 in Ω \ ω 0 for some non-empty domain ω 0 compactly included in ω. We know that such function η 0 exists as long as Ω ∈ C 4 . Indeed, a classical proof of the existence of such function is given in [START_REF] Fursikov | Controllability of evolution equations[END_REF], whereas an alternative

recent proof is given in [73, Lemma 2.1]. Theorem 2.7.
Let Ω be a regular domain that satisfies Hypothesis 2.1, let ω ⊂ Ω be an open set, and let m ≥ 8. Then, there is ε 0 > 0, C > 0 and λ 0 ≥ 1 such that if T > 0, ε ∈ (0, ε 0 ), λ ≥ λ 0 , and s ≥ e Cλ (T m + T 2m ), we have :

s 15 λ 16 Q e -2sα * (ξ * ) 15 |ϕ ε | 2 ≤ Cs 34 λ 35 (0,T )×ω e -2sα ξ 34 |ϕ ε 1 | 2 , (2.9)
for any ϕ ε regular solution of (2.6) and for the weights defined in (2.8).

Proving (2.7) from (2.9) is mainly done by an energy estimate on χϕ ε , for χ ≥ 0 a regular cut-off function such that χ = 1 in [0, T /2] and χ = 0 in [3T /4, T ]. This is a classic and easy procedure, so it is omitted in this paper.

We remark that in (2.6) we can simplify the pressure and get the following equations

for ϕ ε :    -ε 1+ε ∂ t ϕ ε 1 -∂ xx ϕ ε 1 -ε 1+ε ∂ yy ϕ ε 1 = 1 1+ε ∂ xy ϕ ε 2 , -ε 1+ε ∂ t ϕ ε 2 -ε 1+ε ∂ xx ϕ ε 2 -∂ yy ϕ ε 2 = 1 1+ε ∂ xy ϕ ε 1 .
(2.10)

Using standard methods for coupled parabolic equations we know that the important term is ∂ xy ϕ ε 2 , which appears in the equation satisfied by ϕ ε 1 . The main difficulty on proving (2.9) is that ∂ xy • L 2 (Ω) is not necessarily a norm (see Remark 2.31 and Proposition 2.32 below). However, if we add some information on the first and second order derivatives of ∂ xy ϕ ε and on the boundary, we do have a norm. In that sense, we consider the following operator :

L a u = -a∂ xx u -∂ yy u.
(2.11)

Theorem 2.8. Let Ω be a C 4 domain that satisfies Hypothesis 2.1. Then, for a 0 > 0 small enough, there is C > 0 such that for any function u ∈ H 4 (Ω) ∩ H 1 0 (Ω) and for any a ∈ (0, a 0 ] we have that :

∂ x u C 0 (Ω) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ).
(2.12)

Remark 2.9. By continuity, (2.12) remains true for a = 0.

Remark 2.10. Thanks to Poincaré inequality, there is C > 0 and a 0 > 0 such that for all a ∈ [0, a 0 ] and for all u ∈ H 4 (Ω) ∩ H 1 0 (Ω), we have : Indeed, in that case, there is C > 0 and a 0 > 0 such that for all a ∈ [0, a 0 ] and for any function u ∈ H 4 (Ω) ∩ H 1 0 (Ω), we have :

u C 0 (Ω) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ). ( 2
∂ y u C 0 (Ω) ≤ C( ∂ xy u H 2 (Ω) + La u H 1 (∂Ω) ).
In particular, in this case we also have an estimate of u C 0 (Ω) similar to (2.13).

The reason why (2.12) is useful to prove (2.9) is the following one :

Remark 2.12. Let us consider ϕ ε a solution of (2.6). Using (2.10) and the Dirichlet boundary conditions, we have that, on ∂Ω, for all t ∈ [0, T ) :

   -∂ xx ϕ ε 1 -ε 1+ε ∂ yy ϕ ε 1 = 1 1+ε ∂ xy ϕ ε 2 , -ε 1+ε ∂ xx ϕ ε 2 -∂ yy ϕ ε 2 = 1 1+ε ∂ xy ϕ ε 1 .
Thus, by Remark 2.10 and Remark 2.11, we get that there is C > 0 and ε 0 > 0 such that for all t ∈ [0, T ) and ε ∈ (0, ε 0 ] :

ϕ ε (t, •) L 2 (Ω) ≤ C ∂ xy ϕ ε (t, •) H 2 (Ω) . (2.15)
Finally, let us make some remarks about possible extensions of the work :

Remark 2.13. The case of Theorem 2.4 and Theorem 2.7 for Ω ⊂ R 3 is left for future research. The main complication that arises is to prove an analogous result to Theorem 2.8 because there is a larger variety of domains in R 3 than in R 2 . Indeed, in R 3 there is one curvature for each direction.

Remark 2.14. The construction provided in Section 2.2.1 for Lipschitz domains in which (2.4) is not null-controllable just gives one problematic ε for each Ω. Thus, it is an open problem to know if for all Lipschitz domain Ω ⊂ R 2 there is ε 0 small enough such that if ε ∈ (0, ε 0 ) system (2.4) is null controllable.

Historical background

Getting an approximation of the Stokes and the Navier-Stokes systems by approximating the incompressibility condition by a term involving the pressure was made for the first time in [START_REF] Temam | Une méthode d'approximation de la solution des équations des Navier-Stokes[END_REF], where the author considered the almost incompressible Navier-Stokes system. Many other ways of approximating the Navier-Stokes equations have been presented throughout the years. In the survey [START_REF] Shen | Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations[END_REF] the author presents different ways of approximating the Navier-Stokes system through the incompressibility condition and compares them. Moreover, there are physical systems which satisfy in some ways the property of being almost incompressible, as shown in [START_REF] Sussman | A finite element formulation for nonlinear incompressible elastic and inelastic analysis[END_REF] and the references therein.

The interior null controllability of system (2.1) was first proved in [73, Section 4] with a control bounded uniformly with respect to ε, for ε small enough. Then, in [START_REF] Badra | Global Carleman inequalities for Stokes and penalized Stokes equations[END_REF], this same property is proved with an additional first order term. Moreover, in [START_REF] Badra | Global Carleman inequalities for Stokes and penalized Stokes equations[END_REF] the author also proves the local controllability to trajectories of the penalized Navier-Stokes system uniformly on ε for ε small enough.

There is an extensive literature on controllability of partial differential equations uniformly with respect to a vanishing parameter. For a transport equation with a small diffusion term, see [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF] (see also [START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF] and [START_REF] Lissy | A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation[END_REF]). The case of the KdV equation is treated in [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF], [START_REF] Carreño | On the non-uniform null controllability of a linear KdV equation[END_REF] and [START_REF] Carreño | Uniform null controllability of a linear KdV equation using two controls[END_REF], while a chemotaxis system is presented in [START_REF] Chaves-Silva | A uniform controllability result for the Keller-Segel system[END_REF].

As for the restriction of having controls with a reduced number of components, it is not new in the Navier-Stokes mathematical context. This same property has already been proved for the Stokes problem (2.2) in [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]. Consequently, in this paper we prove that a system which approximates the Stokes system conserves that property after choosing a valid reference system. Moreover, controllability results with controls having one null component have been proved for other systems : for instance, the local null controllability of the Navier-Stokes system (see [START_REF] Carreño | Local null controllability of the N -dimensional Navier-Stokes system with N -1 scalar controls in an arbitrary control domain[END_REF]), the local controllability to the trajectories of the

Some previous and intermediary results

Navier-Stokes and the Boussineq system when the domain "touches" the boundary (see [START_REF] Fernández-Cara | Some controllability results for the N -Dimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls[END_REF]), or the existence of insensitizing controls (see [START_REF] Guerrero | Controllability of systems of Stokes equations with one control force : existence of insensitizing controls[END_REF][START_REF] Carreño | Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system[END_REF]). Similarly, the approximate controllability of the Stokes system in a cylindrical domain with a control having two null components is proved in [START_REF] Lions | A generique uniqueness result for the Stokes system and its control theoretical consequences. Partial differential equations and applications[END_REF]. Finally, the local null controllability of the Navier-Stokes system in dimension three with one scalar control is proved in [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF].

Outside the Navier-Stokes context, there is a huge literature on controllability results

with controls having a reduced number of components. For instance, the null controllability in the context of linear thermoelasticity (see [START_REF] Lebeau | Null-Controllability of a System of Linear Thermoelasticity[END_REF]), the existence of insensitizing controls for the heat equation (see [START_REF] De Teresa | Identification of the class of initial data for the insensitizing control of the heat equation[END_REF]), the controllability to trajectories in phase-field models (see [START_REF] Ammar-Khodja | Controllability to the trajectories of phase-field models by one control force[END_REF]), the controllability in cascade-like systems (see [START_REF] Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF]) and the controllability in reaction-diffusion systems (see [START_REF] Ammar-Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF]). For more results on the controllability of parabolic systems with a reduced number of control, see the survey [START_REF] Ammar-Khodja | Recent results on the controllability of linear coupled parabolic problems : a survey[END_REF] and the references therein.

Remark 2.15. The main difference of the problem we consider in this paper with respect to the above cited papers is the coupling. Indeed, in all the papers cited above (and in the literature as far as we know) the coupling is constituted by a zero, first or second order term which induces a norm in the subset of H 2 (Ω) which satisfies their respective boundary conditions. However, in our situation this is clearly not the case since the differential operator ∂ xy with Dirichlet boundary conditions does not induce a norm for some Ω (see Remark 2.31 below).

The rest of the paper is organized as follows : in Section 2.2 we present some analytical results ; in Section 2.3 we prove Theorem 2.8 when Ω is strictly convex ; in Section 2.4 we prove Lemma 2.3 and Theorem 2.8 ; and in Section 2.5 we end the proof of Theorem 2.7.

Finally, in the Appendix we prove some technical results stated in Section 2.2.

Some previous and intermediary results

In this section we present some results that are either interesting for understanding the problem or needed later. The section is split in three parts : first, in Subsection 2.2.1 we prove that there is a domain Ω which is not C 2 and where we do not even have approximate null controllability ; then, in Subsection 2.2.2 we present several results on Cauchy problems and a classical result on linear ordinary differential equations ; finally, in Subsection 2.2.3 we present some Carleman estimates.

A negative controllability result

In this subsection we provide a counterexample on null controllability with one component of (2.4) when Ω is not C 2 , even if ω = Ω. With that purpose, we show that we do not even have the approximate null controllability.

Definition 2.16. We recall that system (2.4) is approximately null controllable if for all y 0 ∈ L 2 (Ω) and for all η > 0 there is a function

f ε ∈ L 2 (0, T ) × ω such that the solution y ε of (2.4) satisfies y ε (T, •) L 2 (Ω) ≤ η.
Proposition 2.17. Let ε > 0. Then, there is Ω ⊂ R 2 a domain that is just Lipschitz such that, even for ω = Ω, (2.4) is not approximately null controllable.

In order to prove Proposition 2.17 we use the technique presented in [85, Section 3].

We give the proof for completeness, as it is an easy consequence of the results presented in [START_REF] Zuazua | A uniqueness result for the linear system of elasticity and its control theoretical consequences[END_REF] and [START_REF] Sweers | On the nonexistence of some special eigenfunctions for the Dirichlet Laplacian and the Lamé system[END_REF]. Indeed, the Stokes penalized system and the Lamé system has the same elliptic operator after an easy reparametrization, being the first one its parabolic version (that is, the associated abstract heat equation) and the other one the hyperbolic version (that is, the associated abstract wave equation).

Proof of Proposition 2.17. It is a classical result that system (2.4) is not approximately null controllable if there is ϕ ε = 0 solution of (2.6) such that ϕ ε 1 = 0 in (0, T ) × ω. Suppose that the following scalar-valued system has a nonzero solution u ε :

         -∂ xx u ε -1+ε ε ∂ yy u ε = λu ε in Ω, u ε = 0 on ∂Ω, ∂ xy u ε = 0 in Ω.
(2.16)

Then, ϕ ε t, (x, y) := (0, e λt u ε (x, y)) is a non-trivial solution of (2.6) which satisfies ϕ ε 1 = 0. Consequently, it suffices to find a domain Ω with a nonzero solution of (2.16).

The third equation of (2.16)

is satisfied if u ε (x, y) = f (x)+g(y) for any f, g ∈ C 2 (R; R).
Moreover, u ε satisfies the first equation of (2.16) if :

-f (x) -λf (x) = 1 + ε ε g (y) + λg(y).
(2.17)

Since both sides of equation (2.17) depend on independent variables, they must be constant.

So we have to solve an ordinary differential equation with constant coefficients. We can suppose that they are equal to 0. Otherwise, if they are equal to some other value α,

we have that f := f + α λ and g := g -α λ are solutions for the case α = 0 such that u(x, y) = f (x) + g(y).

The solutions (f, g) of the system :

   f (x) + λf (x) = 0, 1+ε ε g (y) + λg(y) = 0,
are exponential, affine or trigonometric functions, depending on the value of λ. Since we need u ε to be null on a bounded boundary, then necessarily they must be trigonometric ; that is, λ > 0. In fact, the first equation of (2.16) is satisfied by :

u ε (x, y) = sin √ λx -sin ελ 1 + ε y . (2.18)
Finally, we have to consider that the function u ε given in (2.18) is null on the lines :

               x = ε 1+ε y, x = ε 1+ε y + 2π √ λ , x = - ε 1+ε y + π √ λ , x = - ε 1+ε y -π √ λ . (2.19)
Consequently, the function u ε given in (2.18) is a solution of (2.16) in R ε,λ , for R ε,λ the domain limited by (2.19), which is a rhombus.

Remark 2.18. With this method we can find for R ε,λ (for ε and λ fixed) a sequence of eigenfunctions which satisfy (2.16). The sequence is given by :

u ε n (x, y) = sin (2n + 1) √ λx -sin (2n + 1) ελ 1 + ε y , n ∈ N,
and their respective eigenvalues are λ n = (2n + 1) 2 λ.

Remark 2.19. The "reason" why unique continuation fails is that ∂R ε,λ contains lines of a specific slope. Indeed, if for ε fixed we try to replicate the proof of Theorem 2.8 for the rhombus R ε,λ we find out that the information that (2.6) provides on the boundary is equivalent to the information provided by the Dirichlet condition. More precisely, for the case of Ω = R ε,λ we are blocked in (2.38) below, since we have κ = 0 and (σ 1 ) 2 -ε 1+ε (σ 2 ) 2 = 0. This problem somehow persists when we try to generalize the proof to regular domains and the only solution we have found is to exclude a few directions (a subset of [0, 2π) of null measure).

Results on Cauchy problems

In this subsection we present some results about the Stokes penalized problem : first with Dirichlet boundary conditions and then with Neumann boundary conditions. We also present a classical estimate about a linear differential equation. But before, we recall the definition of the interpolation spaces, for p, q ≥ 0 :

H p,q (Q) := H p 0, T ; L 2 (Ω) ∩ L 2 0, T ; H q (Ω) , H p,q (Σ) := H p 0, T ; L 2 (∂Ω) ∩ L 2 0, T ; H q (∂Ω) . Lemma 2.20. Let i ∈ N, Ω ∈ C 2i . Then, there is ε 0 > 0 and C > 0 such that if T > 0, ε ∈ (0, ε 0 ), v 0 = 0 and f ∈ H i-1,2i-2 (Q) satisfying ∂ m t f (t, •) = 0 for all m ∈ N ∩ [0, i -2], we have that the solution v ε of (2.1) satisfies v ε ∈ H i,2i (Q) with the estimate : v ε H i,2i (Q) + ε -1 ∇ • v ε H i-1,2i-1 (Q) ≤ C f H i-1,2i-2 (Q) . (2.20)
The proof of Lemma 2.20 is mainly by induction. The base case (i = 0) can be proved by Galerkin method (we just have to replicate the method in [START_REF] Evans | Partial Differential Equation[END_REF]Chapter 7.1] and see that the constants are independent of ε). As for the inductive case, we get the regularity in time by considering that v ε t is a solution of (2.1) with (f, 0) replaced by (f t , 0) and using again the Galerkin method. Moreover, we get the regularity in space by using the estimate for the steady Stokes problem given in [103, Proposition I.2.2].

Let us now state the Stokes penalized system with non-homogeneous Neumann boun-

dary conditions :                v ε t -∆v ε + ∇q ε = f in Q, εq ε + ∇ • v ε = 0 in Q, ∂ n v ε -q ε n = h on Σ, v ε (0, •) = v 0 in Ω.
(2.21)

We have the following regularity and existence results, which are proved in Annex 2.A :

Lemma 2.21. Let Ω ∈ C 2 . Then, there is ε 0 > 0 and C > 0 such that if T > 0, ε ∈ (0, ε 0 ), v 0 ∈ H 1 (Ω), f ∈ L 2 (Q) and h ∈ H 1,1/2 (Σ), system (2.21
) has a unique solution :

(v ε , q ε ) ∈ H 1,2 (Q) × H 0,1 (Q).
In addition, that solution satisfies the estimate :

v ε H 1,2 (Q) + q ε H 0,1 (Q) ≤ C √ 1 + T f L 2 (Q) + h H 1,1/2 (Σ) + C v 0 H 1 (Ω) + ∇ • v 0 ε L 2 (Ω) + h(0, •) L 2 (Σ) + h(T, •) L 2 (Σ) . (2.22)
Remark 2.22. It is not necessary to assume that ε is small enough if we just want to prove existence and uniqueness of the energy solution of (2.21). Indeed, we prove collaterally that for all ε ∈ R + , (2.21) has a solution in H 1,1 (Q) and that the norm

v ε H 1,1 (Q) can be estimated by the right-hand side of (2.22) for a constant C independent of ε (see (2.77) below). Lemma 2.23. Let Ω ∈ C 4 . Then, there is ε 0 > 0 and C > 0 such that if T > 0, ε ∈ (0, ε 0 ), v 0 = 0, f ∈ H 1,2 (Q) satisfies f (0, •) = 0 and h ∈ H 2,5/2 (Σ) satisfies : h(0, •) = 0, h(T, •) = 0, ∂ t h(0, •) = 0 and ∂ t h(T, •) = 0,
we have that the solution v ε of (2.21) belongs to H 2,4 (Q) with the estimate :

v ε H 2,4 (Q) ≤ C √ 1 + T f H 1,2 (Q) + h H 2,5/2 (Σ) .
(2.23)

Remark 2.24. These results are not optimal in terms of the regularity imposed on h, but they are enough for our purpose.

Finally, we recall the following classical estimate for a linear ordinary differential equation :

Lemma 2.25. Let T > 0 and let x be the solution in C 0 ([0, T ]) of the following ordinary differential equation :

   a(t)x(t) + x (t) = g(t) t ∈ (0, T ), x(0) = x 0 , for x 0 ∈ R, a ∈ L 1 (0, T ) and g ∈ L 1 (0, T ).
Then, we have the estimate :

x C 0 ([0,T ]) ≤ (|x 0 | + g L 1 (0,T ) )e a L 1 (0,T ) (2.24)

Results about Carleman estimates

In this subsection we present some Carleman estimates that are needed later. We first state a Carleman estimate which concerns a parabolic equation with non-homogeneous Neumann boundary conditions. More precisely, we consider the following system :

         -δϕ t -∆ϕ = f in Q, ∂ n ϕ = h on Σ, ϕ(T, •) = ϕ T in Ω, (2.25) for ϕ T ∈ L 2 (Ω), f ∈ L 2 (Q) and δ ∈ (0, 1]. Lemma 2.26. Let Ω be a C 4 domain, let ω be an open set included in Ω such that ω 0 ⊂ ω, let m ≥ 1 and let r ∈ R. Then, there is C > 0 and λ 0 ≥ 1 such that if T > 0, δ ∈ (0, 1], ϕ T ∈ L 2 (Ω), f ∈ L 2 (Q), h ∈ L 2 (Σ), λ ≥ λ 0 and s ≥ e Cλ (T m + T 2m
) we have :

s 3+r λ 4+r Q e -2sα ξ 3+r |ϕ| 2 + s 1+r λ 2+r Q e -2sα ξ 1+r |∇ϕ| 2 ≤ C    s r λ r Q e -2sα ξ r |f | 2 + s 3+r λ 4+r (0,T )×ω e -2sα ξ 3+r |ϕ| 2 + s 1+r λ 1+r Σ e -2sα ξ 1+r |h| 2    , (2.26)
for ϕ is the solution of (2.25).

The case m = 1, δ = 1 and r = 0 of Lemma 2.26 is proved in [52, Theorem 1]. We get the case for m ≥ 1 and r ∈ R repeating all the steps in [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF] and we get uniformity on δ following the steps of, for instance, [START_REF] Yu | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF].

Next, we also need the following elliptic inequality, whose proof can be found in [38, Lemma 3] :

Lemma 2.27. Let Ω be a C 4 domain, let m ≥ 1 and let r ∈ R. Then, there is C > 0 and

λ 0 ≥ 1 such that if T > 0, λ ≥ λ 0 , s ≥ CT 2m and u ∈ L 2 0, T ; H 1 (Ω)
, we have :

s 2+r λ 3+r Q e -2sα ξ 2+r |u| 2 ≤ C    s r λ 1+r Q e -2sα ξ r |∇u| 2 + s 2+r λ 3+r (0,T )×ω 0 e -2sα ξ 2+r |u| 2    .
(2.27)

Finally, we need a Carleman inequality for the backwards solution of (2.21) (see (2.31) below). For a simpler statement of the Carleman inequality, we define the weights :

η(t) := (sξ * (t)) 1/4+1/m e -
ε 0 > 0, C > 0 and λ 0 ≥ 1 such that if T > 0, ε ∈ (0, ε 0 ), ϕ T ∈ L 2 (Ω), h ∈ H 2,5/2 (Σ), λ ≥ λ 0 and s ≥ e Cλ (T m + T 2m
), we have :

s 3 λ 4 Q e -2sα ξ 3 |ϕ ε | 2 + sλ 2 Q e -2sα ξ|∇ϕ ε | 2 ≤ C s 4 λ 5 (0,T )×ω e -2sα ξ 4 |ϕ ε | 2 + (1 + T ) ηh 2 H 1,1/2 (Σ) + ηh 2 H 2,5/2 (Σ) , (2.30)
for ϕ ε the solution of the following system :

               -ϕ ε t -∆ϕ ε + ∇π ε = 0 in Q, επ ε + ∇ • ϕ ε = 0 in Q, ∂ n ϕ ε -π ε n = h on Σ, ϕ ε (T, •) = ϕ T in Ω.
(

The proof of this Carleman estimate is presented in Annex 2.B.

Remark 2.29. Proposition 2.28 with h = 0 implies that there is ε 0 > 0 and C > 0 such that for all T > 0, ε ∈ (0, ε 0 ] and y 0 ∈ L 2 (Ω), there is

f ε ∈ L 2 (0, T ) × ω such that f ε L 2 ((0,T )×ω) ≤ C y 0 L 2 (Ω) ;
and such that the solution of :

               y ε t -∆y ε + ∇p ε = f ε 1 ω in Q, εp ε + ∇ • y ε = 0 in Q, ∂ n y ε -p ε n = 0 on Σ, y ε (0, •) = y 0 in Ω, satisfies y ε (T, •) = 0.
Up to our knowledge the result presented in Remark 2.29 is new.

Proof and optimality of Theorem 2.8 when Ω is strictly convex

In Section 2.3 we first give some remarks about how much Theorem 2.8 can be improved and we then prove Theorem 2.8 when Ω is strictly convex. The proof is simpler, clearer and more explicit than when we are in a general domain. We recall that Ω strictly convex means that its boundary consists of one connected component and that :

min θ∈[0,|∂Ω|] κ(θ) > 0. (2.32)
Moreover, we remark that a strictly convex domain always satisfies Hypothesis 2.1.

When Ω is a strictly convex convex domain, we do not need a 0 to be small. We state this in the following proposition :

Proposition 2.30. Let Ω be a strictly convex C 4 domain. Then, there is C > 0 such that for all a ∈ (0, 1] and for any real valued function u ∈ H 4 (Ω) ∩ H 1 0 (Ω), we have :

∂ x u C 0 (Ω) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ). (2.33)
Throughout this section we prove Proposition 2.30, which automatically implies Theorem 2.8 if Ω is strictly convex.

Remark 2.31. Estimate (2.33) is false if we remove the term L a u H 1 (∂Ω) . Indeed, we just have to consider the domain Ω = {(x, y) :

x 2 + y 2 ≤ 1} and the function u(x, y) = 1 -x 2 -y 2 .
Although it might be possible that the spaces we give on the right of (2.33) are not optimal, the statement is false if we replace

∂ xy u H 2 (Ω) by ∂ xy u L 2 (Ω) in the right-hand side of (2.33), even if we replace ∂ x u C 0 (Ω) by u L 2 (Ω)
in the left-hand side of (2.33).

Proposition 2.32. Let Ω a C 4 domain with a boundary characterized by an equation of the following kind :

g(x) + h(y) = 0, for g, h ∈ C 4 (R).
Then, there is no C > 0 such that for any u ∈ H 4 (Ω) ∩ H 1 0 (Ω), we have :

u L 2 (Ω) ≤ C ∂ xy u L 2 (Ω) + L a u H 1 (∂Ω) .
(2.34)

Remark 2.33. The hypothesis of Proposition 2.32 includes circles, ellipses and p-norm spheres (for p ≥ 2) among others.

Proof. [Proof of Proposition 2.32] We prove this assertion by contradiction. Let us suppose that there is C > 0 such that, for any u ∈ H 4 (Ω) ∩ H 1 0 (Ω), we have (2.34). Let us consider the function w(x, y) := g(x) + h(y), for g and h the functions stated in the hypothesis of Proposition 2.32. Then, w ∈ H 2 (Ω) ∩ H 1 0 (Ω) and ∂ xy w = 0.

We consider f n ∈ D(Ω) a sequence such that f n → L a w in L 2 (Ω) and we define u n as the solution of the equations :

   L a u n = f n in Ω, u n = 0 on ∂Ω.
By usual theorems on elliptic regularity we have u n ∈ H 4 (Ω) ∩ H 1 0 (Ω). Thus, applying (2.34) and since f n vanishes on the boundary, we get that :

u n L 2 (Ω) ≤ C ∂ xy u n L 2 (Ω) . (2.35)
Moreover, by continuity with respect to the force in the elliptic problem, we have that

u n → u in H 2 (Ω)
, where u is the solution of

   L a u = L a w in Ω, u = 0 on ∂Ω. Consequently, u n → w in H 2 (Ω).
Thus, if we take limits in (2.35) we get :

w L 2 (Ω) ≤ C ∂ xy w L 2 (Ω) = 0,
which is absurd.

Proof. [Proof of Proposition 2.30] In order to make the proof more understandable we split it in three steps : first, we obtain a differential equation on the boundary in terms of ∂ xy u and L a u ; then, we define an auxiliary function and perform estimates on it ; finally, we estimate ∂ x u in terms of the auxiliary function.

Step 1 : Getting an equation on the boundary.

In order to get a differential equation on the boundary, we consider that because of the Dirichlet boundary condition u satisfies the equation :

u σ 1 (θ), σ 2 (θ) = 0, ∀θ ∈ [0, |∂Ω|].
If we differentiate this, we have :

σ 1 ∂ x u + σ 2 ∂ y u = 0, ∀θ ∈ [0, |∂Ω|]. (2.36)
Moreover, if we differentiate (2.36), we get :

σ 1 ∂ x u + σ 2 ∂ y u + (σ 1 ) 2 ∂ xx u + (σ 2 ) 2 ∂ yy u + 2σ 1 σ 2 ∂ xy u = 0, ∀θ ∈ [0, |∂Ω|]. (2.37)
The idea is to get an equality from (2.37) in which we only have ∂ x u, ∂ xx u, ∂ xy u and L a u. In order to get it, we multiply (2.37) by σ 2 and use (2.36) and (2.11). We get that :

(σ 1 σ 2 -σ 2 σ 1 )∂ x u + σ 2 (σ 1 ) 2 -a(σ 2 ) 2 ∂ xx u = -2σ 1 (σ 2 ) 2 ∂ xy u + (σ 2 ) 3 L a u, ∀θ ∈ [0, |∂Ω|].

Recalling (2.3) we can rewrite the previous equation as follows :

-

κ∂ x u + σ 2 (σ 1 ) 2 -a(σ 2 ) 2 ∂ xx u = -2σ 1 (σ 2 ) 2 ∂ xy u + (σ 2 ) 3 L a u, ∀θ ∈ [0, |∂Ω|]. (2.38)
Thanks to (2.32), we can divide (2.38) by κ :

-∂ x u + σ 2 κ (σ 1 ) 2 -a(σ 2 ) 2 ∂ xx u = - 2σ 1 (σ 2 ) 2 κ ∂ xy u + (σ 2 ) 3 κ L a u, ∀θ ∈ [0, |∂Ω|]. (2.39)
In order to shorten this expression, we introduce the following notation :

A(θ) := σ 2 (θ) κ(θ) (σ 1 (θ)) 2 -a(σ 2 (θ)) 2 = σ 2 (θ) κ(θ) 1 -(a + 1)(σ 2 (θ)) 2 . (2.40)
Thus, (2.39) turns into :

-∂ x u + A∂ xx u = - 2σ 1 (σ 2 ) 2 κ ∂ xy u + (σ 2 ) 3 κ L a u ∀θ ∈ [0, |∂Ω|]. (2.41)
Step 2 : Defining an auxiliary function.

We now consider the lower part of the boundary :

Γ := {(σ 1 (θ), σ 2 (θ)) : σ 1 (θ) ≥ 0}.
We can extend the functions κ, σ i and σ ii (i = 1, 2) to Ω. In order to do so, we define Θ h (x) as the only value θ ∈ [0, |∂Ω|] such that σ 1 (θ) = x and σ(θ) ∈ Γ. We consider the following auxiliary function in Ω :

g(x, y) := -∂ x u(x, y) + A(Θ h (x))∂ xx u(x, y). (2.42)
Besides, for any set S ⊂ R 2 , we define :

O(S) := S + Re 2 ∩ Ω, P h (S) := σ(Θ h (S)).
We estimate g and ∂ x g on horizontal segments ; that is, on segments of the type :

l := [x l , x r ] × {y} ⊂ Ω. (2.43)
First, we estimate the L 1 -norm of ∂ x g on any horizontal segment l ⊂ Ω. We consider the following equality :

∂ x g(x, y) = ∂ x g(σ(Θ h (x))) + y σ 2 (Θ h (x))
∂ xy g(x, z)dz, ∀(x, y) ∈ Ω.

(2.44)

The second term in this equality is clearly estimated in L 1 (l, dx)-norm by

∂ xy u H 2 (Ω)
(see (2.42)). In order to estimate the first term in the previous equality, we differentiate (2.41) in the direction θ :

σ 1 ∂ x g = -σ 2 ∂ y g + ∂ θ - 2σ 1 (σ 2 ) 2 κ ∂ xy u + (σ 2 ) 3 κ L a u , θ ∈ [ θ, θ].
Since σ 1 (Θ h (x)) = x, we have that :

Θ h (x) = 1 σ 1 (Θ h (x))
.

So, if we combine this with the fact that |σ | = 1, with the fact that Θ h (σ 1 (θ)) = θ, and recalling the notation presented in (2.43), we have that :

l 1 |σ 1 | -σ 2 ((∂ y g) • σ) + ∂ θ - 2σ 1 (σ 2 ) 2 κ ((∂ xy u) • σ) + (σ 2 ) 3 κ ((L a u) • σ) (Θ h (x))dx = l d σ(Θ h (x)) dx -σ 2 ∂ y g -∂ θ 2σ 1 (σ 2 ) 2 κ ∂ xy u - 2σ 1 (σ 2 ) 2 κ (σ 1 ∂ x + σ 2 ∂ y )∂ xy u + ∂ θ (σ 2 ) 3 κ L a u + (σ 2 ) 3 κ (σ 1 ∂ x + σ 2 ∂ y )L a u (σ(Θ h (x)))dx = P h (l) -σ 2 ∂ y g -∂ θ 2σ 1 (σ 2 ) 2 κ ∂ xy u - 2σ 1 (σ 2 ) 2 κ (σ 1 ∂ x + σ 2 ∂ y )∂ xy u + ∂ θ (σ 2 ) 3 κ L a u + (σ 2 ) 3 κ (σ 1 ∂ x + σ 2 ∂ y )L a u ,
for the geometric functions in the second and third integral above evaluated in Θ h (x), when we are in a point p = (x, y) ∈ Γ. Thus, recalling that ∂ y g = -∂ xy u + A∂ xxy u and recalling (2.44) we get that :

∂ x g L 1 (l,dx) ≤ C ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) . (2.45)
Next, we estimate the C 0 -norm of g on any horizontal segment l ⊂ Ω. Indeed, if we use (2.39), (2.42) and the formula :

g(x, y) = g(σ(Θ h (x))) + y σ 2 (Θ h (x)) ∂ y g(x, z)dz, ∀(x, y) ∈ Ω,
we get, for some p ∈ l, an estimate of g(p) in terms of ∂ xy u H 2 (Ω) and L a u H 1 (∂Ω) with a constant depending on l. So, if we also consider (2.45), we get that :

g C 0 (l) ≤ C(l)( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ).
(2.46)

Step 3 : Getting the information from the ordinary differential equation (2.42).

We split Ω in different subsets depending on the sign of A (Θ h (x)). With that purpose, we denote θ 0 , θ ± a and θ ± 1 the values such that :

         σ (θ 0 ) = e 1 , σ (θ ± a ) = a a+1 , ± 1 a+1 , σ (θ ± 1 ) = ±e 2 .
(2.47)

Since the domain is strictly convex, all θ i are uniquely determined for a fixed. We set :

x 0 := σ 1 (θ 0 ) ; x ± a := σ 1 (θ ± a ) and x ± 1 := σ 1 (θ ± 1 ).
Because of (2.40) we have that A(θ 0 ) = 0 and A(θ ± a ) = 0. By (2.42) and (2.46), this implies that we can estimate ∂ x u in the vertical segments given by {x = x 0 } and {x = x ± a }. More generally, if we have an estimate of |∂ x u(x * , y * )|, we have an estimate of |∂ x u(x * , y)| for all y such that (x * , y) ∈ Ω because ∂ xy u C 0 (Ω) can be estimated by ∂ xy u H 2 (Ω) .

Consequently, we just have to transmit horizontally the punctual estimates of |∂ x u|. In order to do so, in the rest of the proof we fix some appropriate horizontal segments and regard equality (2.42) as an ordinary differential equation.

We first prove estimate (2.33) in Ω ∩ {x ≥ x 0 }. We consider a sequence of values

s 0 < • • • < s n such that s 0 := x 0 , s n := x +
1 and such that for any i ∈ {1, . . . , n} there exists an horizontal segment l i ⊂ Ω such that the abscissa of its left endpoint (respectively its right endpoint) is s i-1 (respectively s i ). Because we are in a regular bounded convex domain, all this can be done (see Figure 2.1 for one such example). We denote by y i the second coordinate of any point of l i . Since the segments l 1 , . . . , l n do not depend on a, we have, due to (2.46), the following estimate in those segments with a constant that only depends on Ω :

g C 0 (∪ n i=1 l i ) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ). (2.48)
Let us set j a ∈ {1, . . . , n} such that θ + a ∈ Θ h (l ja ). If there are two such segments, we choose the one on the right. We first get an estimate in O({x ≥ x + a }), where we have :

A (Θ h (x)) ≤ 0 (2.49)
(see (2.40)). For that purpose, we define l r a :=

l ja ∩ O({x ≥ x + a }).
First, considering that ∂ x u(x + a , y) = -g(x + a , y) and (2.48), we obtain the following estimate :

|∂ x u(x + a , y ja )| ≤ C ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) .
(2.50)

Figure 2.1 -An illustration of the strictly convex case Next, we have that, for any z ∈ [x + a , s ja ] :

- z x + a (x -z -2)(∂ x u∂ xx u)(x, y ja )dx = 1 2 z x + a |∂ x u(x, y ja )| 2 dx + (x + a -z -2) 2 |∂ x u(x + a , y ja )| 2 -(-1)|∂ x u(z, y ja )| 2 . (2.51)
Thus, considering (2.42), (2.50) and (2.51), we have that for a constant C independent of z :

1 2 z x + a |∂ x u(x, y ja )| 2 dx + z x + a (x -z -2)A (Θ h (x)) |∂ xx u(x, y ja )| 2 dx + |∂ x u(z, y ja )| 2 ≤ z x + a (x -z -2)(g∂ xx u)(x, y ja )dx + C ∂ xy u 2 H 2 (Ω) + L a u 2 H 1 (∂Ω) . (2.52)
So, the last term to be estimated in (2.52) is the one with g. In order to do that, we integrate by parts :

z x + a (x -z -2)(g∂ xx u)(x, y ja )dx = - z x + a (x -z -2)(∂ x g∂ x u)(x, y ja )dx - z x + a (g∂ x u)(x, y ja )dx -(x + a -z + 2)g(x + a , y ja )∂ x u(x + a , y ja ) -2g(z, y ja )∂ x u(z, y ja ) ≤ C ∂ x g L 1 (l r a ,dx) + g C 0 (l r a ) ∂ x u C 0 (l r a ) . (2.53)
Using that ab ≤ 1 4η a 2 + ηb 2 and estimating the norms of g by (2.45) and (2.48), we obtain the following from (2.52) :

z x + a |∂ x u(x, y ja )| 2 dx + |∂ x u(z, y ja )| 2 ≤ C η ∂ xy u 2 H 2 (Ω) + L a u 2 H 1 (∂Ω) + η ∂ x u 2 C 0 (l r a ) .
Since z is arbitrary, we deduce that :

∂ x u C 0 (l r a ) ≤ C η ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) + η ∂ x u C 0 (l r a ) .
Thus, taking η small enough, we can absorb the last term on the right-hand side. Moreover,

as ∂ xy u C 0 (Ω) is estimated in terms of ∂ xy u H 2 (Ω)
, we get :

∂ x u C 0 (O(l r a )) ≤ C ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) . (2.54)
This method also works for l ja+1 , . . . , l n because the trace of ∂ x u on the left of these segments is obtained first by (2.54) and then inductively. Indeed, in l i it suffices to multiply at both sides of the identity (2.42) by (x -s i -2)∂ xx u and integrate by parts as above.

Therefore, we can get inductively the estimate :

∂ x u C 0 (O(l r a )) + n i=ja+1 ∂ x u C 0 (O(l i )) ≤ C ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ,
which implies that :

∂ x u C 0 (O({x≥x + a )}) ≤ C ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) .
As for getting the estimate in O({x ∈ [x 0 , x + a ]}), we can obtain it analogously as before : first in l l a := l ja ∩ O({x ≤ x + a }), then inductively in l ja-1 , . . . , l 1 . The only difference is that this time we multiply by x + x + a + 2 and we spread the estimates to the left. The reason for these changes is that for

x ∈ [x 0 , x + a ], A (Θ h (x)) is positive instead of negative (see (2.40)).
Finally, it is quite clear that we can get the estimate in a similar way for Ω ∩ {x < x 0 }. Thus, (2.33) is true.

Proof of Theorem 2.8 and some geometrical results

In this section we present the proof of Theorem 2.8 as well as some strongly related results. First, in Subsection 2.4.1 we prove Lemma 2.3. Second, in Subsections 2.4.2 and 2.4.3 we state and prove some geometrical consequences of Hypothesis 2.1. Finally, in Subsection 2.4.4 we prove Theorem 2.8, using, among others, Section 2.3.

We recall that some of the notation used in this section has been introduced above Hypothesis 2.1.

Proof of Lemma 2.3

Lemma 2.3 is a consequence of Sard's Theorem :

Theorem 2.34 (Sard's Theorem). Let f : R → R a C 1 function. Let X := {x ∈ R : f (x) = 0}.
Then, f (X) has zero measure.

Proof. [Proof of Lemma 2.3] In order to apply Sard's Theorem, we consider the functions :

(σ 1 1 ) , . . . , (σ k 1 ) , (σ 1 2 ) , . . . , (σ k 2 ) . Let us denote Z ⊂ [-1, 1] the set of values such that if (σ i 1 ) (θ) = z or if (σ i 1 ) (θ) = -z or if (σ i 2 ) (θ) = z or if (σ i 2 ) (θ) = -z ; then, κ i (θ) = 0. Since κ i (θ) = 0 implies that (σ i 1 ) (θ) = 0 and (σ i 2 ) (θ) = 0 (see (2.
3)), thanks to Sard's Theorem (and to the fact that a finite union of null-measure sets and translations of those sets are still of null measure) we get that

the measure of [-1, 1] \ Z is 0. Let us consider ψ ∈ sin -1 (Z). Then, we have that ∂ (U ψ (Ω)) = U ψ (∂Ω) and it can be parametrized by σi := U ψ (σ i ). Moreover, we have (U ψ (σ i )) = U ψ ((σ i ) ). Consequently, (σ i ) = ±e 2 if and only if (σ i ) (θ) = U (-ψ)(0, ±1) ; that is, if and only if (σ i ) (θ) = ±(sin(ψ), cos(ψ)). Similarly, (σ i ) = ±e 1 if and only if (σ i ) (θ) = ±(cos(ψ), -sin(ψ)).
Furthermore, if κi (θ) is by definition the curvature in σi (θ), we have, by the non-variation of the curvature by rotations, κi (θ) = κ i (θ). Thus, by the definition of Z, taking z = sin ψ, we have that all the points of ∂ (U ψ (Ω)) with horizontal or vertical tangent vector have a non zero curvature.

Finally, the measure of R \ sin -1 (Z) is null because, for all k ∈ Z, the sinus is a diffeomorphism from (π(k -1/2), π(k + 1/2)) to (-1, 1) (and because a countable union of sets of null measure has null measure).

Geometrical consequences of Hypothesis 2.1

In order to prove Theorem 2.8, we need to define equivalent notions to the ones presented in the convex case (see Section 2.3). Definition 2.35. We define Γ as the subset of ∂Ω such that p = σ i (θ) ∈ Γ if and only if at least one of the following properties is satisfied :

• ∃δ 0 (p) > 0 : ∀δ ∈ (0, δ 0 (p)), p + δe 2 ∈ Ω, • (σ i ) (θ) = ±e 2 .
When Ω is convex we have that Γ is the bottom of Ω. Moreover, for an illustration on what Γ may look like in a non-convex domain, we can regard Figure 2.2 below.

Remark 2.36. The relative boundary of Γ is given by points of tangent vectors ±e 2 . Indeed, the components of ∂Ω are closed curves ; thus, by regularity, having a vertical tangent vector is the only possibility.

Definition 2.37. Let (x, y) ∈ Ω. We define :

P h (x, y) := (x, y) -λe 2 such that λ := min{λ ∈ R + : (x, y) -λe 2 ∈ Γ}.
We remark that when Ω is convex P h represents the vertical projection on Γ, is continuous and does not depend on y. In the general situation, though, there is a dependence on y and P h is not continuous when Γ is not connected (see B 1 in Figure 2.2 below). Yet, we can define an application P h that coincides with the one given in Section 2.3 when Ω is convex :

Definition 2.38. Let l i = [x i l , x i r ] × {y i } ⊂ Ω a segment. Then, P h (l i ) := P h ((x i l , x i r ) × {y i }).
Moreover, we see that Hypothesis 2.1 implies the existence of segments like in the case of a convex domain :

Lemma 2.39. Let Ω be a domain that satisfies Hypothesis 2.1. Then, there is a subset S ⊂ Ω such that :

• S is a finite union of horizontal segments

l i := [x i l , x i r ] × {y i }. • P h (S) = Γ.
• P h is continuous in the relative interior of each segment l i .

Example 2.40. In Figure 2.2 S is given by the segments :

[A 1 , A 2 ], [A 2 , A 5 ], [C 1 , C 5 ], [D 1 , D 6 ] and [E 1 , E 2 ].
The proof of Lemma 2.39 is postponed to Section 2.4.3. We first prove some geometrical results : Let Ω be a domain that satisfies Hypothesis 2.1. We have :

1. If (σ i 1 ) (θ) = 0 or if (σ i 2 ) (θ) = 0, then, for some δ(θ) > 0, κ i does not change of sign in (θ -δ(θ), θ + δ(θ)).
2. The number of points in ∂Ω with tangent vectors ±e 1 or ±e 2 is finite.

3. Given any c ∈ R, the number of points in ∂Ω ∩ {x = c} or in ∂Ω ∩ {y = c} is finite. 4. Given any c ∈ R, there is δ(c) > 0 such that :

• We have ([c -δ(c), c + δ(c)] × R) ∩ ∂Ω = p=σ ip (θp)∈∂Ω∩{x=c} σ ip (I p ), for I p = (θ 1 p , θ 2 p ), for some θ 1 p < θ p < θ 2 p . • In the set [c -δ(c), c + δ(c)] \ {c} × R ∩ ∂Ω, we do not have p = σ i (θ) with (σ i ) (θ) = ±e 2 .
5. There is some η > 0 such that for all points p = σ i (θ p ) ∈ ∂Ω with

(σ i ) (θ p ) = ±e 1 , there exists a neighbourhood V p = σ i (I p ) ⊂ ∂Ω (I p = (θ 1 p , θ 2 p ), for some θ 1 p < θ p < θ 2 p ) such that σ i 2 (θ 1 p ) = σ i 2 (θ 2
p ) and such that |κ i | > η. 6. There exists a 0 > 0 small enough such that, for all a ∈ (0, a 0 ), for each point

p = σ i (θ) ∈ ∂Ω with (σ i (θ)) = ±e 2 there is a neighbourhood U p ⊂ ∂Ω which has exactly a point of tangent vector ± a 1+a , 1 1+a
and exactly another one of tangent vector ± a 1+a , -

1 1+a . Reciprocally, if p a = σ i (θ a ) ∈ ∂Ω satisfies (σ i ) (θ a ) = ± a 1+a , ± 1 1+a
, then p a ∈ U p , for U p one of the above defined neighbourhoods. Finally, we can suppose that for some η > 0, |κ i | > η on those neighbourhoods.

Proof. [Proof of Lemma 2.41] Firstly, implication 1 is an easy consequence of Ω being at least C 2 .

Secondly, we prove implication 2 for points of tangent vector ±e 2 by contradiction. If they are not finite, by (pre-)compactness and regularity of Ω, there is a point p = σ i (θ) and θ n → θ such that (σ i 1 ) (θ n ) = 0. Obviously (σ i 1 ) (θ) = 0. But, because of the regularity of Ω, we also have (σ i 1 ) (θ) = 0. Indeed,

(σ i 1 ) (θ) = lim s→θ (σ i 1 ) (s) -(σ i 1 ) (θ) s -θ = lim n→∞ (σ i 1 ) (θ n ) -(σ i 1 ) (θ) θ n -θ = 0. (2.55)
Thus, we get by (2.3) that κ i (θ) = 0, which contradicts Hypothesis 2.1. The proof for points with tangent vector ±e 1 is analogous.

Thirdly, given any line x = c, we prove by contradiction that there is a finite number of points in ∂Ω ∩ {x = c}. Indeed, if we have an infinite number of points, by regularity and compactness we can write a sequence of distinct elements as σ i (θ n ) with θ n → θ. Since

σ i 1 (θ n ) = c
, by an equality similar to (2.55), (σ i 1 ) (θ) = 0. Since κ i (θ) = 0 by Hypothesis 2.1, σ i (θ n ) cannot be in {x = c} for n large enough, contradicting the choice of θ n . The proof for y = c is analogous.

Fourthly, statement 4 is a consequence of assertion 2. Indeed, the only possibility is that there is an infinite number of curves of ∂Ω that approach the line x = c and then move away (like a parabola). But this implies that there is an infinite number of points of tangent vector ±e 2 , which contradicts statement 2.

Fifthly, assertion 5 is an easy consequence of statements 1 and 2 and of picking the neighbourhoods small enough.

Finally, statement 6 is a consequence of assertion 2. Indeed, we consider U p j some neighbourhoods of p j , for p j the points of tangent vector ±e 2 . We have the bound :

inf i inf θ:σ i (θ) ∈Up j ∀j |(σ i 1 ) (θ)| > 0.
Moreover, by making them smaller if necessary, we have due to Hypothesis 2.1 and the continuity of the κ i :

inf i inf θ:∃j:σ i (θ)∈Up j |κ i (θ)| > 0.
Consequently, we can fix some smaller neighbourhoods U p j and the parameters a 0 and η.

Proof of Lemma 2.39 and some remarks

In this subsection we first present the proof of Lemma 2.39 ; and then, we state some direct consequences.

First of all, we define some useful notation :

Definition 2.42. Let Ω ⊂ R 2 be a domain and x ∈ R. We define :

Ω x := Ω ∩ (-∞, x) × R .
Now we are ready to present the proof :

Proof. [Proof of Lemma 2.39] Without loss of generality we can suppose that : 0 = min{x : ∃y with (x, y) ∈ Ω}.

We consider :

I = {x ∈ R + : ∀s ∈ [0, x] ∃S s ⊂ Ω s :
S s satisfies the conclusion of Lemma 2.39 with Γ replaced by Γ ∩ Ω s }. Summing up, since all this happens for a finite number of situations, for δ(c) sufficiently small we have (c, c + δ(c)) ⊂ I.

Remark 2.43. Because of the conclusion of Lemma 2.39, the left endpoint of each segment

l i is either a point p = σ i (θ) ∈ Γ with (σ i ) (θ) = ±e 2 and κ i (θ) > 0 (the case of A 1 in Figure 2.
2) or it can be joined by a vertical segment (including degenerated segments)

inside Ω with some other segment l j such that x j l < x i l ≤ x j r (the case of A 2 , C 1 , D 1 and E 1 in Figure 2 Remark 2.45. Given any segment l i as defined in Lemma 2.39, because of its third property, it makes sense to define Θ i h (x), A i (x) and g i (x, y) as in the convex case (see Section 2.3) by looking at P h (l i ). Indeed, they make sense in the domain limited superiorly by l i and inferiorly by P h (l i ). Moreover, we recall that :

g i (x, y) := -∂ x u(x, y) + A i (Θ i h (x))∂ xx u(x, y). (2.56)
We also remark that if min l i |κ(Θ i h (x))| > 0, we can prove the following estimate as in the convex case (see (2.45) and (2.46)) for a constant C depending only on S :

g i C 0 (l i ) + ∂ x g i L 1 (l i ,dx) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ).
(2.57)

Proof of Theorem 2.8

In order to prove Theorem 2.8 we get an estimate in each segment l i given by Lemma 2.39. Indeed, we prove inductively that :

∂ x u C 0 (l i ) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ). (2.58)
By inductively, we mean that (2.58) is proved for any other segment l j such that x j l < x i l (see Lemma 2.39 for the notation).

First, for getting a pointwise estimate on x i l , we consider the two situations given in Remark 2.43. In the first case, because of statement 6 on Lemma 2.41, we can get an estimate on x i l by ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) as in the convex case (see Section 2.3). In the second case, by the induction hypothesis and the fact that ∂ xy u C 0 (Ω) can be estimated by ∂ xy u H 2 (Ω) , we get the estimate on x i l by ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) . So, in both cases we have : • Let us first deal with the case 1. We recall that by Remark 2.45, functions like g i or Θ i h or A i make sense in l i . We start in the subcase in which (σ i 1 ) (Θ i h (x i r )) = 0. In Figure 2.3, Ω 1 illustrates an example in which the surface is locally convex and Ω 2 an example in which it is locally concave. We remark that, in the concave case, the ratio

|∂ x u(x i l , y i )| ≤ C ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) . ( 2
(σ i 2 )
κ i has the same sign as in the convex case (by the criteria fixed above Hypothesis 2.1, (σ i 2 ) < 0, (σ i 1 ) ≥ 0 and κ i < 0 in that part of Γ). Analogously to the convex case (see Section 2.3), we define p a ∈ V q as the point such that

A i (Θ h (p a )) = 0.
Moreover, q a is the point in l i such that P h (q a ) = p a . Thus, in both cases, we can multiply (2.56) by x + C for C large enough in the segment [q 1 , q a ] and by x -C for C large enough in [q a , q 2 ] and then follow the procedure of the convex case. The case in which (σ i 1 ) (Θ i h (x i l )) = 0 is analogous. Consequently, we have (2.58) in the case 1.

• If P h (l i ) belongs neither to U p nor to V p (situation 2), we have, by statement 6 in Lemma 2.41, that there is η > 0 such that :

|A i (Θ i h (l i ))| > η (see
g i (x, y) := - 1 A i (Θ i h (x)) ∂ x u(x, y) + ∂ xx u(x, y). (2.60)
In that sense, we can get the same estimate for g i as the one for g i in (2.57) by following similar steps. Thus, seeing (2.60) as an ordinary linear differential equation in l i whose initial value is in (x i l , y i ), we get (2.58) by Lemma 2.25. • As for the case 3 (see Figure 2.4 for the notation), we mainly replicate the method of the previous section. In this paragraph and in the following one, we do all the estimates by ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) even if we do not explicitely write it. We already have by (2.59) a pointwise estimate in q -1 , so we can prove (2.58) in [q -1 , q] following the method of the convex case. As for [q, q 1 ], to replicate the method of the convex case, we need an estimate of |∂ x u(q 1 )|. In order to get it, first we use

ting ∂ xy u C 0 ([p 1 ,q 1 ]
) , we have an estimate of |∂ x u(q 1 )|, so we can replicate the method of the previous section to prove (2.58) in [q, q 1 ].

• The last case is that P h (l i ) is locally concave (see Figure 2.5 for the notation).

The technique is the same one as in the case 3 (recall that the ratio In order to do it, we consider that in each component of ∂Ω we have the following equality :

aκ i ∂ y u -(σ i 1 ) (1 -(a + 1)((σ i 2 ) ) 2 )∂ yy u = ((σ i 1 ) ) 3 L a u -2a((σ i 1 ) ) 2 (σ i 2 ) ∂ xy u.
This equality can be obtained similarly to (2.38). So, similarly as in Remark 2.45,

we define locally at the left of σ i ((θ 1 p , θ p )) (see statement 5 in Lemma 2.41 for the notation) a function Θ i v (y) and a function :

ĝ(x, y) := aκ i (Θ i v (y))∂ y u(x, y) -(σ i 1 ) (1 -(a + 1)((σ i 2 ) ) 2 ) (Θ i v (y))∂ yy u(x, y). (2.61)
Moreover, we can obtain as (2.57) that (see Figure 2.5 for the notation) :

ĝ L 1 ([q -1 ,k -1 ],dy) ≤ C( ∂ xy u H 2 (Ω) + L a u H 1 (∂Ω) ).
So, by seeing (2.61) as an ordinary differential equation whose initial data is p -1 , because there is η > 0 such that (σ Finally, by Remark 2.39, since we have (2.58) for all segments in S, we have (2.12).

i 1 ) (1 -(a + 1)((σ i 2 ) ) 2 ) ≥ η in V p ,

Proof of Theorem 2.7

For the proof of this theorem we define a subdomain ω compactly contained in ω such that ω 0 ⊂ ω. Moreover, we consider a cut-off function χ ≥ 0 satisfying supp(χ) ⊂ ω and χ = 1 in ω. In addition, we define D i as the tensor which contains all the derivatives of order i with respect to the x and y variable. In order to clarify the proof we divide it in several steps.

Step 1 : Estimates of the crossed derivative.

First of all, we consider estimate (2.15) squared, multiplied by (sξ * ) 15 λ 16 e -2sα * and integrated in time. If we also bound the weights (see (2.8)), we get that :

s 15 λ 16 Q e -2sα * (ξ * ) 15 |ϕ ε | 2 ≤ Cs 15 λ 16 2 i=0 Q e -2sα ξ 15 |D i ∂ xy ϕ ε | 2 . (2.62)
Next, we apply the elliptic estimate (2.27) to D i ∂ xy ϕ ε for i = 0, . . . , 7 (we take D 0 ∂ xy ϕ ε := ∂ xy ϕ ε ). We get that :

7 i=0 s 19-2i λ 20-2i Q e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2 ≤ C s 3 λ 4 Q e -2sα ξ 3 |D 8 ∂ xy ϕ ε | 2 + 7 i=0 s 19-2i λ 20-2i (0,T )×ω 0 e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2 . (2.63)
Moreover, since under our hypothesis 1 ≤ Csξ, we combine that fact with (2.63) and

(2.62), and we get that :

s 15 λ 16 Q e -2sα * (ξ * ) 15 |ϕ ε | 2 + 7 i=0 s 19-2i λ 20-2i Q e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2 ≤ C    s 3 λ 4 Q e -2sα ξ 3 |D 8 ∂ xy ϕ ε | 2 + 7 i=0 s 19-2i λ 20-2i (0,T )×ω 0 e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2    .
(2.64)

To continue with, we deal with each term of D 8 ∂ xy ϕ ε . In order to do so, we use Proposition 2.28 (on each term of D 8 ∂ xy ϕ ε ) and get that :

sλ 2 Q e -2sα ξ|D 9 ∂ xy ϕ ε | 2 + s 3 λ 4 Q e -2sα ξ 3 |D 8 ∂ xy ϕ ε | 2 ≤ C    s 4 λ 5 (0,T )×ω e -2sα ξ 4 |D 8 ∂ xy ϕ ε | 2 + (1 + T ) ηh 2 H 1,1/2 (Σ) + ηh 2 H 2,5/2 (Σ)    , (2.65)
for h := ∂ n D 8 ϕ ε + ε -1 ∇ • D 8 ϕ ε .
Remark 2.46. It is well-known since [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF] that by taking enough derivatives we can absorb the trace. Indeed, each time we use (2.27), the weight is, up to a constant, divided by s 2 λ 2 ξ 2 . Moreover, with the weights α * and ξ * (the weights on the boundary), we can formally "remove a derivative" by multiplying the weight by C(sξ * ) 1+1/m (this is rigorously done in the next step). Consequently, it is clear that we can absorb the trace using Proposition 2.28 by taking enough derivatives. In our case, if we take less derivatives, what happens when we remove the derivatives of the trace is that we get something with a weight larger than s 15 λ 16 e -2sα * (ξ * ) 15 , which cannot be absorbed by the left-hand side of (2.62).

Step 2 : Absorbing the trace terms.

Let us start absorbing (1 + T ) ηh 2 H 1,1/2 (Σ) . We consider the continuous injections :

H 6,12 (Q) ⊂ H 1 (0, T ; H 10 (Ω)) and H 5,11 (Q) ⊂ H 1 (0, T ; H 8+4/5 (Ω)).
Consequently, we have that :

ηh H 1,1/2 (Σ) ≤ C ηϕ ε H 1 (0,T ;H 10 (Ω)) + ε -1 ∇ • (ηϕ ε ) H 0,9 (Q) + ε -1 ∇ • (ηϕ ε ) H 1 (0,T ;H 8+4/5 (Ω)) ≤ C ηϕ ε H 6,12 (Q) + ε -1 ∇ • (ηϕ ε ) H 5,11 (Q) . (2.66)
We recall that :

(t, x) → η(T -t)ϕ ε (T -t, x)
is a solution of (2.1) with null initial value and force -η (T -t)ϕ ε (T -t, x). Thus, applying estimate (2.20) with i = 6 to (2.66), we have that :

ηϕ ε H 6,12 (Q) + ε -1 ∇ • (ηϕ ε ) H 5,11 (Q) ≤ C η ϕ ε H 5,10 (Q) .
If we repeat this reasoning five times, we get that :

ηh H 1,1/2 (Σ) ≤ C η vi) ϕ ε L 2 (Q) .
We have that, if s ≥ e Cλ (T m + T 2m ) and m ≥ 8 (see (2.28)) :

(1 + T 1/2 )|η vi) | ≤ C(sξ * ) 6+1/4+7/m e -sα * ≤ C(sξ * ) 15/2 e -sα * .
Consequently, if we also have λ ≥ λ 0 , we can absorb the term (1 + T ) ηh H 1,1/2 (Σ) by the left-hand side of (2.64).

Finally, we have to absorb (1 + T 1/2 ) ηh H 2,5/2 (Σ) . In order to do so, we recall that :

H 7,14 (Q) ⊂ H 2 (0, T ; H 10 (Ω)
) and H 6,13 (Q) ⊂ H 2 (0, T ; H 8+2/3 (Ω)).

Thus, we have that :

ηh H 2,5/2 (Σ) ≤ C ηϕ ε H 0,12 (Q) + ηϕ ε H 2 (0,T ;H 10 (Ω)) + ε -1 ∇ • (ηϕ ε ) H 0,11 (Q) + ε -1 ∇ • (ηϕ ε ) H 2 (0,T ;H 8+2/3 (Ω)) ≤ C ηϕ ε H 7,14 (Q) + ε -1 ∇ • (ηϕ ε ) H 6,13 (Q) .
Consequently, using estimate (2.20) seven times, we get that :

ηh H 2,5/2 (Σ) ≤ ηvii) ϕ ε L 2 (Q) ;
which is a term that can be absorbed by the left-hand side of (2.64) if λ ≥ λ 0 , s ≥ e Cλ (T m + T 2m ) and m ≥ 8, because under those hypothesis (see (2.29)) :

(1 + T 1/2 )|η vii) | ≤ C(sξ * ) 6+1/4+7/m e -sα * ≤ C(sξ * )
s 15 λ 16 Q e -2sα * (ξ * ) 15 |ϕ ε | 2 + 9 i=0 s 19-2i λ 20-2i Q e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2 ≤ C    7 i=0 s 19-2i λ 20-2i (0,T )×ω 0 e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2 + s 4 λ 5 (0,T )×ω e -2sα ξ 4 |D 8 ∂ xy ϕ ε | 2    .
(2.68)

Step 3 : Bounding the local terms.

In order to bound the local terms, we start estimating everything by a local term of ∂ xy ϕ ε . We do it with the usual technique : we bound each 1 ω by χ (which is defined at the beginning of this section) to a sufficiently high power, we integrate by parts and we use properly weighted Cauchy-Schwarz inequalities. After all this process, we get from (2.68) that :

s 15 λ 16 Q e -2sα * (ξ * ) 15 |ϕ ε | 2 + 9 i=0 s 19-2i λ 20-2i Q e -2sα ξ 19-2i |D i ∂ xy ϕ ε | 2 + 8 i=1 s 28-3i λ 29-3i (0,T )×ω χ 4+2i e -2sα ξ 28-3i |D i ∂ xy ϕ ε | 2 ≤ Cs 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 |∂ xy ϕ ε | 2 .
(2.69) Indeed, when i = 1, . . . , 8, we have that :

s 28-3i λ 29-3i (0,T )×ω χ 4+2i e -2sα ξ 28-3i |D i ∂ xy ϕ ε | 2 = s 28-3i λ 29-3i (0,T )×ω D χ 4+2i e -2sα ξ 28-3i D i ∂ xy ϕ ε • D i-1 ∂ xy ϕ ε ≤ C δ s 28-3(i-1) λ 29-3(i-1) (0,T )×ω χ 4+2(i-1) e -2sα ξ 28-3(i-1) |D i-1 ∂ xy ϕ ε | 2 + δ s 28-3i λ 29-3i (0,T )×ω χ 4+2i e -2sα ξ 28-3i |D i ∂ xy ϕ ε | 2 + s 28-3(i+1) λ 29-3(i+1) (0,T )×ω χ 4+2(i+1) e -2sα ξ 28-3(i+1) |D i+1 ∂ xy ϕ ε | 2 . (2.70)
The exponents of s, ξ and λ in (2.69) might look strange. The reason is that we have s 4 λ 5

in the last local term of (2.68) instead of s 3 λ 4 , which is the usual term.

In order to get in the right-hand side of (2.69) only a weighted local L 2 -norm of ϕ ε 1 we must treat ∂ xy ϕ ε 1 and ∂ xy ϕ ε 2 differently. As for ∂ xy ϕ ε 1 , we can deal with it quite easily. Indeed, when we integrate by parts twice, we have that :

s 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 |∂ xy ϕ ε 1 | 2 = s 28 λ 29 (0,T )×ω ∂ xy χ 4 e -2sα ξ 28 ∂ xy ϕ ε 1 ϕ ε 1 + s 28 λ 29 (0,T )×ω ∂ x χ 4 e -2sα ξ 28 ∂ xyy ϕ ε 1 ϕ ε 1 + s 28 λ 29 (0,T )×ω ∂ y χ 4 e -2sα ξ 28 ∂ xxy ϕ ε 1 ϕ ε 1 + s 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 ∂ xxyy ϕ ε 1 ϕ ε 1 . (2.71)
We can deal with all the term of (2.71) as usual. In the end, for δ > 0 as small as needed, after an absorption, we get that :

s 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 |∂ xy ϕ ε 1 | 2 ≤ C δ s 34 λ 35 (0,T )×ω e -2sα ξ 34 |ϕ ε 1 | 2 + δ    s 22 λ 23 (0,T )×ω χ 8 e -2sα ξ 22 |∂ xxyy ϕ ε 1 | 2 + s 24 λ 25 (0,T )×ω χ 6 e -2sα ξ 24 |D 1 ∂ xy ϕ ε 1 | 2    . (2.72)
Finally, we have to estimate the term of ∂ xy ϕ ε 2 . Indeed, by (2.6), we find out that :

s 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 |∂ xy ϕ ε 2 | 2 = s 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 ∂ xy ϕ ε 2 (-ε∂ t ϕ ε 1 -(1 + ε)∂ xx ϕ ε 1 -ε∂ yy ϕ ε 1 ).
We can deal with the term in the right-hand side integrating by parts in space and time and using weighted Cauchy-Schwarz inequalities. In order to deal with the term of ε∂ txy ϕ ε 2 that appears after the integration by parts, we have to consider that :

ε∂ txy ϕ ε 2 = -(ε∂ xxxy ϕ ε 2 + (1 + ε)∂ xyyy ϕ ε 2 + ∂ xxyy ϕ ε 1 ) .
Consequently, we get that, after an absorption :

s 28 λ 29 (0,T )×ω χ 4 e -2sα ξ 28 |∂ xy ϕ ε 2 | 2 ≤ C δ s 34 λ 35 (0,T )×ω e -2sα ξ 34 |ϕ ε 1 | 2 + δ    s 22 λ 23 (0,T )×ω χ 8 e -2sα ξ 22 |D 2 ∂ xy ϕ ε | 2 + s 24 λ 25 (0,T )×ω χ 6 e -2sα ξ 24 |D 1 ∂ xy ϕ ε | 2    . (2.73)
Summing up, if m ≥ 8, λ ≥ λ 0 , and s ≥ e Cλ (T m + T 2m ), combining (2.69), (2.72) and

(2.73) we get (2.9).

2.A Existence, uniqueness and regularity of (2.21)

In this section we first prove Lemma 2.21 and then prove Lemma 2.23. Our proofs are classical, since they use Galerkin method and elliptic estimates (see Lemma 2.47 below).

We follow the steps of [START_REF] Evans | Partial Differential Equation[END_REF]Chapter 7.1], but we do the necessary adaptations due to the different boundary conditions.

Proof of Lemma 2.21 : uniqueness. In order to prove the uniqueness, we just have to show that for f = 0, h = 0 and v 0 = 0, the unique solution is v ε = 0. Indeed, by multiplying by v ε the first equation of (2.21) and by integrating in (0, t) × Ω (by parts),

we have, for all t ∈ [0, T ] :

Ω |v ε (t, •)| 2 2 + [0,t]×Ω |∇v ε | 2 + ε [0,t]×Ω |q ε | 2 = 0, which implies that v ε = 0.
Proof of Lemma 2.21 : existence. As for the existence, we consider the Galerkin method. It is well-know that there is a set of eigenvalues {λ i } i∈N → +∞ and a set of

L 2 (Ω)-orthonormal and H 1 (Ω)-orthogonal eigenvectors w i such that {w i } i∈N ⊂ H 2 (Ω)
and that -∆w i = λ i w i . In that sense, for u ∈ L 2 (Ω), we denote P n u the orthonormal projection of u into w 1 , . . . , w n . We consider the Galerkin sub-problems, for n ∈ N, t ∈ [0, T ] :

Ω ∂ t v ε n (t, •)•w i + Ω ∇v ε n (t, •) : ∇w i + Ω (∇ • v ε n (t, •))(∇ • w i ) ε = Ω f (t, •)•w i + ∂Ω h(t, •)•w i , (2.74)
2.A. Existence, uniqueness and regularity of (2.21)

for all i = 1, . . . , n. We look for a solution which belongs to C 1 [0, T ]; w 1 , . . . , w n ;

that is, we look for

a ε i,n ∈ C 1 ([0, T ]) such that a ε i,n (0) = v 0 , w i L 2 (Ω) and that v ε n (t, x) := n i=1 a ε i,n (t)w i (x) is a solution of (2.74).
Energy estimates. It is easy to see that each set of components (a ε i,n ) n i=1 is the solution of a linear ordinary differential equation of n equations and n unknowns. Therefore, system (2.74) together with the initial condition has a well-defined solution. Moreover, adding up (2.74) multiplied by the coefficients a ε i,n and integrating in time, we get that, for any t ∈ [0, T ], provided that v ε n is defined :

Ω |v ε n | 2 (t, •) 2 + [0,t]×Ω |∇v ε n | 2 + [0,t]×Ω (∇ • v ε n ) 2 ε = [0,t]×Ω f • v ε n + [0,t]×∂Ω h • v ε n + Ω |P n v 0 | 2 2 .
We have to consider that :

[0,t]×Ω f • v ε n ≤ C(1 + T ) [0,t]×Ω |f | 2 + 1 4(1 + T ) [0,t]×Ω |v ε n | 2 .
Moreover, we have that :

[0,t]×∂Ω h • v ε n ≤ C(1 + T ) [0,t]×∂Ω |h| 2 + 1 4(1 + T ) [0,t]×Ω |v ε n | 2 + |∇v ε n | 2 .
Consequently, due to Gronwall's inequality and usual absorptions, we get that :

Ω |v ε n | 2 (t, •) + [0,t]×Ω |∇v ε n | 2 + [0,t]×Ω (∇ • v ε n ) 2 ε ≤ C    (1 + T ) [0,t]×Ω |f | 2 + (1 + T ) [0,t]×∂Ω |h| 2 + Ω |P n v 0 | 2    . (2.75)
Thanks to estimate (2.75), all the solutions (v ε n ) n∈N are bounded uniformly in C 0 ([0, t]; L 2 (Ω))∩ L 2 (0, t; H 1 (Ω)). This implies, due to extension theorems related with the Cauchy-Lipschitz systems, that all the v ε n are defined in [0, T ] and that we can take a weak limit in L 2 (0, T ; H 1 (Ω)).

Estimates on H 1,1 (Q). Next, in order to take limits in (2.74), we need to prove that the v ε n are also uniformly bounded in H 1 (0, T ; L 2 (Ω)). By multiplying (2.74) by (a ε i,n ) , adding all up and integrating in time, we have that :

Q |∂ t v ε n | 2 + T 0 ∂ t 2 Ω |∇v ε n | 2 + T 0 ∂ t 2 Ω (∇ • v ε n ) 2 ε = Q f • ∂ t v ε n + Σ h • ∂ t v ε n .
In order to deal with the term Σ h • ∂ t v ε n we have to integrate by parts in time. Then, using also (2.75), we get for a constant C that does not depend on n :

v ε n H 1,1 (Q) ≤ C √ 1 + T f L 2 (Q) + h H 1,1/2 (Σ) + C v 0 H 1 (Ω) + ∇ • v 0 ε L 2 (Ω) + h(0, •) L 2 (Σ) + h(T, •) L 2 (Σ) . (2.76)
So, up to extracting a subsequence, we have that (v ε n ) n∈N converges weakly in H 1,1 (Q) to some function v ε which satisfies :

v ε H 1,1 (Q) ≤ C √ 1 + T f L 2 (Q) + h H 1,1/2 (Σ) + C v 0 H 1 (Ω) + ∇ • v 0 ε L 2 (Ω) + h(0, •) L 2 (Σ) + h(T, •) L 2 (Σ) . (2.77)
Thus, we can take limits in (2.74). Indeed, we have for every i ∈ N, as functions of

L 2 (0, T ) : Ω v ε t • w i + Ω ∇v ε : ∇w i + Ω (∇ • v ε )(∇ • w i ) ε = Ω f • w i + ∂Ω h • w i . ( 2.78)
We recall that H 1 (0, T ;

L 2 (Ω)) is compactly embedded in C 0 ([0, T ]; H -1 (Ω)). Thus, weak convergence in H 1 (0, T ; L 2 (Ω)) implies strong convergence in C 0 ([0, T ]; H -1 (Ω)), so v ε (0, •) = v 0 .
Estimates on H 1,2 (Q). In order to prove that the solution is in L 2 (0, T ; H 2 (Ω)), we use that it satisfies for any i ∈ N, as functions of L 2 (0, T ), that :

Ω ∇v ε : ∇w i + Ω (∇ • v ε )(∇ • w i ) ε = Ω (f -v ε t ) • w i + ∂Ω h • w i . (2.79)
So, we use the following lemma, whose proof can be found in [START_REF] Boyer | Mathematical tool for the study of the incompressible Navier-Stokes equations and related models[END_REF]Theorem IV.7.1] :

Lemma 2.47. Let us consider Ω ∈ C 2 and the system :

               -∆u + ∇g = f 1 in Ω, ∇ • u = f 2 in Ω, ∂ n u -gn = f 3 on ∂Ω, Ω u = 0, (2.80
)

for f 1 ∈ L 2 (Ω), f 2 ∈ L 2 (Ω) and f 3 ∈ H 1/2 (∂Ω).
Then, if we have as a vector equation :

Ω f 1 + ∂Ω f 3 = 0, (2.81)
the solution (u, g) of (2.80) is unique and

D 2 u L 2 (Ω) + g H 1 (Ω) ≤ C f 1 L 2 (Ω) + f 2 H 1 (Ω) + f 3 H 1/2 (∂Ω) .
In order to apply Lemma 2.47 it suffices to take f 1 (t,

•) := f (t, •) -v ε t (t, •), f 2 (t, •) := ∇ • v ε (t, •) and f 3 (t, •) := h(t, •).
In addition to that, (2.81) is satisfied because :

Ω v ε t = Ω (∆v ε -∇q) + Ω f = ∂Ω (∂ n v ε -qn) + Ω f = ∂Ω h + Ω f .
Thus, since :

u(t, •) = v ε (t, •) -1 |Ω| Ω v ε (t, •) and g(t, •) = ∇•v ε ε (t, •) = q ε (t, •),
if we combine (2.77) and Lemma 2.47, remarking that D 2 u = D 2 v ε , we get the estimate :

v ε H 1,2 (Q) + q ε H 0,1 (Q) ≤ C √ 1 + T f L 2 (Q) + h H 1,1/2 (Σ) + C v 0 H 1 (Ω) + ∇ • v ε H 0,1 (Q) + ∇ • v 0 ε L 2 (Ω) + h(0, •) L 2 (Σ) + h(T, •) L 2 (Σ) .
(2.82)

This expression can be simplified since for ε small enough we can absorb the term

∇ • v ε H 0,1 (Q) by q ε H 0,1 (Q) . So estimate (2.22) is established.
Proof of Lemma 2.23. As for the proof of Lemma 2.23, it consists of repeating the Galerkin method for v ε t , since v ε t is a solution of (2.21) with (f, h, 0) replaced by (f t , h t , 0). Indeed, we first get an estimate for each ∂ t v ε n and then pass to the limit. Finally, we use a more complete version of Lemma 2.47, which can be found in [12, Theorem IV.7.1].

2.B Proof of Proposition 2.28

Throughout this proof we consider ω some open subdomain of Ω compactly contained in ω such that ω 0 ⊂ ω (see Proposition 2.28 for the definition of ω). In order to make the reading of the proof more comfortable we split it in several steps : first, we bound left of (2.30) by a trace and a local term with the help of the rotational ; then, we deal with the trace and local terms as usual.

Step 1 : Bounding by a trace and a local term.

To begin with, we have that ∇×ϕ ε is a solution of the heat equation, since ∇×(∇π ε ) = 0. So, using Lemma 2.26 for r = -1 and δ = 1, we get that if λ ≥ λ 0 and s ≥ e Cλ (T m +T 2m ) :

s 2 λ 3 Q e -2sα ξ 2 |∇ × ϕ ε | 2 + λ Q e -2sα |∇(∇ × ϕ ε )| 2 ≤ C    Σ e -2sα |∂ n (∇ × ϕ ε )| 2 + s 2 λ 3 (0,T )×ω e -2sα ξ 2 |∇ × ϕ ε | 2    . (2.83)
Next, we consider that the divergence satisfies :

∇(∇ • ϕ ε ) = ∆ϕ ε + ∇ × (∇ × ϕ ε ).
This implies that ϕ ε satisfies :

- ε 1 + ε ∂ t ϕ ε -∆ϕ ε = 1 1 + ε (∇ × (∇ × ϕ ε )).
Thus, using again Lemma 2.26 for ω defined as before, r = 0 and now δ = ε 1+ε , we get that if λ ≥ λ 0 and s ≥ e Cλ (T m + T 2m ) :

s 3 λ 4 Q e -2sα ξ 3 |ϕ ε | 2 + sλ 2 Q e -2sα ξ|∇ϕ ε | 2 ≤ C    s 3 λ 4 (0,T )×ω e -2sα ξ 3 |ϕ ε | 2 + Q e -2sα |∇(∇ × ϕ ε )| 2 + sλ Σ e -2sα ξ|∂ n ϕ ε | 2    . (2.84)
Next, we remark that the term of ∇(∇ × ϕ ε ) on the right-hand side of (2.84) can be absorbed by the left-hand side of (2.83) for λ ≥ λ 0 . Thus, we have that :

s 3 λ 4 Q e -2sα ξ 3 |ϕ ε | 2 + sλ 2 Q e -2sα ξ|∇ϕ ε | 2 + s 2 λ 3 Q e -2sα ξ 2 |∇ × ϕ ε | 2 + λ Q e -2sα |∇(∇ × ϕ ε )| 2 ≤ C s 3 λ 4 (0,T )×ω e -2sα ξ 3 |ϕ ε | 2 + s 2 λ 3 (0,T )×ω e -2sα ξ 2 |∇ × ϕ ε | 2 + sλ Σ e -2sα ξ|∂ n ϕ ε | 2 + Σ e -2sα |∂ n (∇ × ϕ ε )| 2 .
(2.85)

Step 2 : Absorption of the trace.

In this step we absorb the traces with the estimates established in Lemma 2.21. We recall that on ∂Ω : α = α * and ξ = ξ * .

Let us first bound the third integral on the right-hand side of (2.85). First, we consider that, integrating by parts :

sλ Σ e -2sα * ξ * |∂ n ϕ ε | 2 ≤ C (sξ * ) 5/4-1/m λ 2 e -sα * ϕ ε 1/2 L 2 (Q) (sξ * ) 1/4+1/m e -sα * ϕ ε 3/2 H 0,2 (Q) .
Using Young's inequality we get that :

sλ Σ e -2sα * ξ * |∂ n ϕ ε | 2 ≤ C (sξ * ) 5/4-1/m λ 2 e -sα * ϕ ε 2 L 2 (Q) + (sξ * ) 1/4+1/m e -sα * ϕ ε 2 H 0,2 (Q) . ( 2

.86)

We can absorb the first term on the right-hand side of (2.86) by the left-hand side of (2.85) by taking s ≥ CT 2m and λ ≥ 1.

We can bound the fourth integral at the right-hand side of (2.85) similarly. Indeed, integrating by parts, we get that, if s ≥ CT 2m :

Σ e -2sα * |∂ n (∇ × ϕ ε )| 2 ≤ C (sξ * ) 1/4 e -sα * ϕ ε 3/2 H 0,2 (Q) (sξ * ) -3/4 e -sα * ϕ ε 1/2 H 0,4 (Q) ≤ C (sξ * ) 1/4 e -sα * ϕ ε 2 H 0,2 (Q) + (sξ * ) -3/4 e -sα * ϕ ε 2 H 0,4 (Q) . (2.87)
So, we first deal with the term ηϕ ε 2 H 1,2 (Q) (see (2.28) for the definition of η). We remark that

(t, x) → η(T -t)ϕ ε (T -t, x)
is a solution of (2.21) with null initial value, force -η (T -t)ϕ ε (T -t, x) and boundary Neumann term η(T -t)h(T -t, x). Consequently, because of (2.22), we get that :

ηϕ ε 2 H 1,2 (Q) ≤ C(1 + T ) η ϕ ε 2 L 2 (Q) + ηh 2 H 1,1/2 (Σ) . (2.88)
Moreover, the term of η ϕ ε can be absorbed by the left of (2.85) if m ≥ 8, λ ≥ λ 0 , and s ≥ e Cλ (T m + T 2m ), since in that case :

(1 + T 1/2 )|η | ≤ C(sξ * ) 1+1/4+2/m e -sα * ≤ C(sξ * ) 3/2 e -sα * . (2.89)
Let us now estimate the term ηϕ ε 2 H 0,4 (Q) (see (2.29) for the definition of η). We have that

(t, x) → η(T -t)ϕ ε (T -t,
ηϕ ε 2 H 0,4 (Q) ≤ C(1 + T ) η ϕ ε 2 H 1,2 (Q) + ηh 2 H 2,5/2 (Σ) . (2.90)
Let us now estimate the first norm at the right-hand side of (2.90). To begin with, since m ≥ 8 and s ≥ e Cλ (T m + T 2m ) imply that (1 + T 1/2 )|η | ≤ Cη, we have that :

(1 + T ) η ϕ ε 2 H 0,2 (Q) ≤ C ηϕ ε 2 H 0,2 (Q) , (2.91)
which is estimated by the left-hand side of (2.88). To continue with, we have that, if m ≥ 8 and s ≥ e Cλ (T m + T 2m ) :

(1 + T ) η ϕ ε 2 L 2 (Q) ≤ C (sξ * ) 1+1/4+2/m e -sα * ϕ ε 2 L 2 (Q) ≤ C (sξ * ) 3/2 e -sα * ϕ ε 2 L 2 (Q) , (2.92)
a term which can be absorbed by the left-hand side of (2.85) for λ large enough. Finally, we have that, if m ≥ 8 and s ≥ e Cλ (T m + T 2m ) :

(1 + T 1/2 )|η ϕ t | ≤ |ηϕ ε t | ≤ |(ηϕ ε ) t | + |η ϕ ε |,
which implies that :

(1 + T ) η ϕ ε t 2 L 2 (Q) ≤ C ηϕ ε 2 H 1,0 (Q) + η ϕ ε 2 L 2 (Q) , ( 2
s 3 λ 4 Q e -2sα ξ 3 |ϕ ε | 2 + sλ 2 Q e -2sα ξ|∇ϕ ε | 2 + s 2 λ 3 Q e -2sα ξ 2 |∇ × ϕ ε | 2 + λ Q e -2sα |∇(∇ × ϕ ε )| 2 ≤ C s 3 λ 4 (0,T )×ω e -2sα ξ 3 |ϕ ε | 2 + s 2 λ 3 (0,T )×ω e -2sα ξ 2 |∇ × ϕ ε | 2 + (1 + T ) ηh 2 H 1,1/2 (Σ) + ηh 2 H 2,5/2 (Σ)
.

Finally, we remove the derivative from the local terms. We do it with the usual localizing techniques : we multiply by a cut-off function χ, integrate by parts and use Cauchy-Schwarz weighted inequalities. So, if λ ≥ λ 0 , s ≥ e Cλ (T m + T 2m ) and m ≥ 8, we get estimate (2.30).

Chapitre 3

Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit

In this chapter we consider a Stokes system with no-slip boundary conditions. The main results concern the behaviour of the cost of null controllability with respect to the diffusion coefficient when the control acts in the interior. In particular, we prove in (0, π) 2 that for a sufficiently large time the cost decays exponentially as the diffusion coefficient vanishes, whereas in (0, π) 3 we prove that for most of the control domains and for any time T > 0 the cost explodes exponentially as the diffusion coefficient vanishes. This chapter is included in [START_REF] Bárcena-Petisco | Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit[END_REF].

Introduction

Main results

Throughout the paper we are interested in understanding the following control problem :

               y t -ε∆y + ∂ x d y + ∇q = f 1 ω in Q, ∇ • y = 0 in Q, y • n = 0, (Dy • n) tg = 0 on Σ, y(0, •) = y 0 on Ω. (3.1)
As usual, here and throughout the paper, T > 0, Ω ⊂ R d and ω ⊂ Ω are domains,

Q := (0, T ) × Ω, Q ω := (0, T ) × ω, Σ := (0, T ) × ∂Ω, Du := 1 2 ∂ x i u j + ∂ x j u i i,j
for all u ∈ L 2 (Ω) (in the distributional sense), n denotes the normal vector pointing outwards ∂Ω, "∂ n " denotes the normal outward partial derivative on ∂Ω, and

v tg := v -(v • n)n.
The bold notation is used to denote vectorial spaces. As for initial data, y 0 , we take it in H(Ω), which is defined by :

H(Ω) := {u ∈ L 2 (Ω) : ∇ • u = 0 in Ω and u • n = 0 on ∂Ω}.
We recall that we can define a normal trace which belongs to H -1/2 (∂Ω) in the (closed) subspace of L 2 (Ω) of functions that have a divergence in L 2 (Ω). As for the control force, f , we want it at least in L 2 (Q ω ) and, if possible, of null first component. Finally, throughout the document Ω d := (0, π) d .

Let us denote the sets of admissible controls as :

         S 1 (y 0 ) := {f ∈ L 2 (Q ω ) : Φ 2 (y 0 , f )(T, •) = 0}, S 2 (y 0 ) := {f 2 ∈ L 2 (Q ω ) : Φ 2 (y 0 , (0, f 2 ))(T, •) = 0}, S 3 (y 0 ) := {f ∈ L 2 (Q ω ) : Φ 3 (y 0 , f )(T, •) = 0}, for y 0 ∈ H(Ω d ), for f ∈ L 2 (Q ω )
and for Φ d (y 0 , f ) the solution of (3.1) in Ω d . Thanks to [START_REF] Guerrero | Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions[END_REF], we know that S 1 and S 3 are non-empty, whereas we obtain in Section 3.3.1 that S 2 is non-empty by proving the corresponding observability estimate. As for the cost of the control, we measure it with the usual norms. In fact, we denote :

                 K 1 (T, ε, ω) := sup y 0 ∈H(Ω 2 )\{0} inf f ∈S 1 (y 0 ) f L 2 (Qω) y 0 L 2 (Ω 2 ) , K 2 (T, ε, ω) := sup y 0 ∈H(Ω 2 )\{0} inf f 2 ∈S 2 (y 0 ) f 2 L 2 (Qω) y 0 L 2 (Ω 2 ) , K 3 (T, ε, ω) := sup y 0 ∈H(Ω 3 )\{0} inf f ∈S 3 (y 0 ) f L 2 (Qω) y 0 L 2 (Ω 3 ) . (3.2)
The main results that we prove in the paper are the following ones :

Theorem 3.1. We have the following results for the control system (3.1) :

1. Let ω ⊂ Ω 2 a domain. Then, there are some c, C, T 0 > 0 such that, if T > T 0 and ε ∈ (0, 1) :

K 2 (T, ε, ω) ≤ Ce -cε -1 .
2. Let h ∈ (0, π) and ω ⊂ (0, π) × (π -h, π) a domain. Then, for any T ∈ (0, 2(π -h)),

there is c > 0 such that, if ε ∈ (0, 1) :

K 1 (T, ε, ω) ≥ ce cε -1 .
3. Let h ∈ (0, π) and ω ⊂ (0, π) × (0, h) a domain. Then, for any T ∈ (0, π -h), there is c > 0 such that, if ε ∈ (0, 1) :

K 1 (T, ε, ω) ≥ ce cε -1 .
4. Let h ∈ (0, π) and ω ⊂ (0, π) 2 × (π -h, π) a domain. Then, for any T > 0 there is c > 0 such that, if ε ∈ (0, 1) :

K 3 (T, ε, ω) ≥ ce cε -1 .
Remark 3.2. The results in dimension 2 are not surprising as they are similar to those in [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF][START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF][START_REF] Lissy | A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF]. The result in dimension 3 (see Item 4), however, is surprising because it holds for all T > 0 and for a wide range of control domains (in particular, for all the control domains compactly supported in Ω 3 ). This result is explained because we have much more freedom to construct eigenfunctions of the adjoint operator (see (3.5) and Remark 3.21 in Section 3.4 below). Indeed, we can construct in dimension 3 a continuous family of eigenfunctions whose respective eigenvalues do not explode when ε → 0 + . The existence of a system with this property is a novelty with respect to the existing literature.

Remark 3.3. The analogous results to the ones stated in Theorem 3.1 and in the lemmas and propositions throughout the paper are true for any rectangle or cuboids and with any velocity parallel to their edges. Indeed, all the operations can be replicated under those assumptions. The reason of restricting to (0, π) 2 or (0, π) 3 is to reduce the number of operations so that the reader can focus on the main ideas of the proof.

In order to estimate the cost of the control, we study the adjoint system :

               -ϕ t -ε∆ϕ -∂ x d ϕ + ∇p = 0 in Q, ∇ • ϕ = 0 in Q, ϕ • n = 0, (2εDϕ • n + ϕn d ) tg = 0 on Σ, ϕ(T, •) = ϕ T on Ω, (3.3)
for d the dimension of Ω. We recall the following classical result (see, for instance, [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations : recent progress and open questions[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systemes distribués[END_REF]) :

                   [K 1 (T, ε, ω)] 2 = sup ϕ T ∈H(Ω 2 )\{0} Ω 2 |ϕ(0, x)| 2 dx Qω |ϕ| 2 dxdt , [K 2 (T, ε, ω)] 2 = sup ϕ T ∈H(Ω 2 )\{0} Ω 2 |ϕ(0, x)| 2 dx Qω |ϕ 2 | 2 dxdt , [K 3 (T, ε, ω)] 2 = sup ϕ T ∈H(Ω 3 )\{0} Ω 3 |ϕ(0, x)| 2 dx Qω |ϕ| 2 dxdt . (3.4)
While working in Ω 2 we use the following notation :

Σ lr := (0, T ) × {0, π} × (0, π), and Σ bt := (0, T ) × (0, π) × {0, π}.

In order to understand the (energy) solutions of (3.3), we do a spectral decomposition.

In particular, we look for the solutions of :

         -ε∆u -∂ x d u + ∇p = λu in Ω, ∇ • u = 0 in Ω, u • n = 0, (2εDu • n + un d ) tg = 0 on ∂Ω, (3.5)
for d the dimension of Ω.

Remark 3.4. One interesting property of system (3.3) considered in Ω 2 is that, as long as ϕ T ∈ H(Ω 2 ), the gradient of the pressure of the solution of (3.3) is null for any given time. Indeed, by taking the divergence in (3.3) 1 we obtain that -∆p = 0, so the result is proved if we show that ∂ n p = 0 on Σ. For that purpose, we multiply (3.3) 1 by n. First, ϕ t • n = 0 as ϕ • n = 0. Second, we have that :

-(ε∆ϕ + ∂ x 2 ϕ) • n1 Σ lr = -(ε∂ 2 x 1 ϕ 1 + ε∂ 2 x 2 ϕ 1 + ∂ x 2 ϕ 1 )n 1 1 Σ lr = ε∂ x 2 ∂ x 1 ϕ 2 n 1 1 Σ lr = 0.
We have used on the previous equality that n 2 = 0 on Σ lr , (3.3) 2 and that ϕ 1 = ∂ x 1 ϕ 2 = 0 on Σ lr because of (3.3) 3 . And finally, we have that :

-(ε∆ϕ+∂ x 2 ϕ)•n1 Σ bt = -(ε∂ 2 x 1 ϕ 2 +ε∂ 2 x 2 ϕ 2 +∂ x 2 ϕ 2 )n 2 1 Σ bt = (ε∂ x 1 ∂ x 2 ϕ 1 +∂ x 1 ϕ 1 )n 2 1 Σ bt = 0.
We have used that n 1 = 0 on Σ bt , (3.3) 2 and that ϕ 2 = ε∂ x 2 ϕ 1 + ϕ 1 = 0 on Σ bt because of (3.3) 3 .

Remark 3.5. An immediate consequence of Remark 3.4 is that if ϕ T ∈ H(Ω 2 ), ϕ 1 (the first component of the solution ϕ of (3.3) in Ω 2 ) satisfies :

               -z t -ε∆z -∂ x 2 z = 0 in Q, z = 0 on Σ lr , ε∂ x 2 z + z = 0 on Σ bt , z(T, •) = z T on Ω 2 , (3.6)
for z T = ϕ T 1 . In addition, ϕ 2 satisfies :

               -z t -ε∆z -∂ x 2 z = 0 in Q, ∂ x 1 z = ∂ n z = 0 on Σ lr , z = 0 on Σ bt , z(T, •) = z T on Ω 2 , (3.7)
for z T = ϕ T 2 .

Throughout the paper we use c and C to denote strictly positive constants, which might be different each time. We denote the small constants by c and the large ones by C.

Historical background

The first control system with a small diffusion and a transport term that was analysed was the heat equation in dimension 1 with Dirichlet boundary conditions in [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF].

Afterwards, the same problem but in any dimension and with any speed belonging to W 1,+∞ (R + × Ω) was studied in [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF]. More recently, better approximations of the optimal time in which the cost of the control decays have been given : the upper bound was improved in [START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF][START_REF] Lissy | A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation[END_REF] (in the first one through complex analysis and in the second one by transforming the original equation into the pure heat equation), and the lower bound was improved in [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF][START_REF] Laurent | Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller[END_REF] through complex analysis and properties of the entire functions in the first one and spectral analysis on the second. As for similar results, work has been done in the the Burgers equation (see [START_REF] Glass | On the uniform controllability of the Burgers equation[END_REF]), in the KdV equation (see [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF][START_REF] Glass | Uniform controllability of a transport equation in zero diffusion-dispersion limit[END_REF][START_REF] Carreño | On the non-uniform null controllability of a linear KdV equation[END_REF][START_REF] Carreño | Uniform null controllability of a linear KdV equation using two controls[END_REF]) and in an artificial advection-diffusion problem (see [START_REF] Cornilleau | Controllability and observability of an artificial advection-diffusion problem[END_REF][START_REF] Cornilleau | On the cost of null-control of an artificial advectiondiffusion problem[END_REF]). As for the Stokes system with small diffusion and a transport term, this is the first time that such a system has been studied and is, indeed, one of the contributions of the paper.

The study of control problems associated to Stokes systems with Navier-slip boundary condition (see (3.1)) is not new in the literature : the existence of d dimensional controls leading to null controllability was proved in [START_REF] Guerrero | Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions[END_REF], while the existence of d -1 dimensional controls has recently been proved in [START_REF] Guerrero | Local null controllability of the N-dimensional Navier-Stokes system with nonlinear Navier-slip boundary conditions and N -1 scalar controls[END_REF]. The need of working in a specific domain to simplify the problem is not new either. For example, recently, in [START_REF] Coron | Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force[END_REF], they prove that the Navier-Stokes is (globally) null controllable in any rectangle with boundary controls at two opposing edges and with a phantom force (see also [START_REF] Guerrero | A result concerning the global approximate controllability of the Navier-Stokes system in dimension 3[END_REF] for a weaker result in cuboids).

Finally, the idea of using spectral decomposition is not new in Control Theory (see, for instance, [START_REF] Coron | Control and Nonlinearity[END_REF]). Indeed, for the heat and the Stokes context alone, there are many documents which deal with eigenfunctions of the elliptic operator, for proving the existence of some control (see, for example, [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Lions | A generique uniqueness result for the Stokes system and its control theoretical consequences. Partial differential equations and applications[END_REF][START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]), for estimating the cost of the control (see, for instance, [START_REF] Ervedoza | Sharp observability estimates for heat equations[END_REF] and, very recently, [START_REF] Laurent | On uniform observability of gradient flows in the vanishing viscosity limit[END_REF]) , and for giving negative answer to the existence of a control (see, for example, [START_REF] Lions | A generique uniqueness result for the Stokes system and its control theoretical consequences. Partial differential equations and applications[END_REF]). As for a system with small diffusion and a transport term, a spectral decomposition indirectly appears in [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF], when getting lower bounds for the optimal time T 0 in which the cost of the control decays exponentially with ε.

However, as far as the author knows, this is the first time that a spectral decomposition has been used to get the dissipation estimate in a transport-diffusion system.

The rest of the paper is organized as follows : in Section 3. 3.2 Spectral decomposition of H((0, π) 2 ) and some immediate consequences

In this section we look for solutions of (3.5) in Ω 2 and use them to extract information about the solutions of (3.3) in Ω 2 . In particular, in Section 3.2.1 we provide the proof of a technical result, in Section 3.2.2 we get some eigenfunctions (solutions of (3.5)) which form a total set in H(Ω 2 ), and in Section 3.2.3 we use those eigenfunctions to extract information about the solutions of (3.3). We avoid using Remark 3.4 and Remark 3.5 to have a proof that can be generalized to other Stokes systems and because it does not spare us many calculations.

As for the notation, given any real Banach space (V, • ) and S ⊂ V , we recall that span(S) denotes the set of all linear combinations of the elements of S. Moreover, we recall that S is a total set if span(S) = V . Also,

l 2 := (a k ) k∈N * : a k ∈ R and k∈N * a 2 k < +∞ .
Besides, for any bounded domain Θ ⊂ R d , we denote :

L 2 0 (Θ) := g ∈ L 2 (Θ) : Θ g(x)dx = 0 .
Moreover, we use m = (m 1 , m 2 ) ∈ (N * ) 2 to index the eigenfunctions. Finally, we denote the set of functions with separated variables having null normal trace and null divergence as :

SV (Ω 2 ) := x 1 0 g 1 (s)dsg 2 (x 2 ), -g 1 (x 1 ) x 2 0 g 2 (s)ds : g 1 , g 2 ∈ L 2 0 (0, π) . (3.8)

Some functional analysis results

In order to look for eigenfunctions, we look among those which have a simple structure ; in particular, among those which belong to SV (Ω 2 ). Indeed, by construction, SV (Ω 2 ) ⊂ H(Ω 2 ). Moreover, the interest of SV (Ω 2 ) is that it is a total set : Proposition 3.6. We have :

span(SV (Ω 2 )) = H(Ω 2 ).
Remark 3.7. This result is not surprising since it is well-known that functions with separated variables form a total set in L 2 (Ω 2 ).

In order to prove Proposition 3.6, we first need some technical results whose proofs, up to the author's knowledge, do not appear explicitly in any book or paper :

Lemma 3.8. Let A 1 , A 2 , B 1 , B 2 ∈ R. Let Θ := (A 1 , A 2 ) × (B 1 , B 2 ), p ∈ (1, +∞) and h ∈ L p (Θ) a function such that : Θ h(x 1 , x 2 )g 1 (x 1 )g 2 (x 2 )dx = 0, (3.9)
for all g 1 ∈ L ∞ 0 (A 1 , A 2 ) and for all g 2 ∈ L ∞ 0 (B 1 , B 2 ). Then, there are

h 1 ∈ L p (A 1 , A 2 ) and h 2 ∈ L p (B 1 , B 2 ) such that : h(x 1 , x 2 ) = h 1 (x 1 ) + h 2 (x 2 ) almost everywhere in Θ,
and such that :

A 2 A 1 h 1 (s)ds = 0.
(3.10)

For the proof, we follow the classical scheme of first proving a regular version of the lemma, and then generalizing it to weaker spaces thanks to mollifiers.

Lemma 3.9. Let A 1 , A 2 , B 1 , B 2 ∈ R. Let Θ := (A 1 , A 2 )×(B 1 , B 2 ), N ≥ 0 and h ∈ C N Θ a function such that : Θ h(x 1 , x 2 )g 1 (x 1 )g 2 (x 2 )dx = 0, (3.11
)

for all g 1 ∈ L ∞ 0 (A 1 , A 2 ) and for all g 2 ∈ L ∞ 0 (B 1 , B 2 ). Then, there are h 1 ∈ C N [A 1 , A 2 ] and h 2 ∈ C N [B 1 , B 2 ] such that : h(x 1 , x 2 ) = h 1 (x 1 ) + h 2 (x 2 ) in Θ,
and such that :

A 2 A 1 h 1 (s)ds = 0.
(3.12)

We remark that (3.10) and (3.12) are required to have some continuity in the decomposition (h 1 , h 2 ). Moreover, they also provide uniqueness.

Proof. [Proof of Lemma 3.9] Since the Lebesgue measure works well with translations and dilations, it suffices to prove the result for Θ = (0, 1) 2 . In particular, it suffices to prove that for all x 1 , x 2 , x1 , x2 ∈ (0, 1) such that x 1 = x1 and x 2 = x2 :

h(x 1 , x 2 ) + h(x 1 , x2 ) = h(x 1 , x2 ) + h(x 1 , x 2 ). ( 3
h(x 1 , x 2 ) = h(x 1 , 0) + h(0, x 2 ) -h(0, 0).
Let us suppose that there exist a 1 , a 2 , b 1 , b 2 ∈ (0, 1) such that a 1 = a 2 and b 1 = b 2 and such that (3.13) is not true. By symmetry, we can suppose that :

h(a 1 , b 1 ) + h(a 2 , b 2 ) > h(a 1 , b 2 ) + h(a 2 , b 1 ).
Then, for some δ > 0 small enough (in particular smaller than

a 1 , a 2 , b 1 , b 2 , 1-a 1 , 1-a 2 , 1- b 1 , 1 -b 2 )
, by the continuity of h, we have for all s = (s 1 , s 2 ) ∈ [-δ, δ] 2 the inequality :

h(a 1 + s 1 , b 1 + s 2 ) + h(a 2 + s 1 , b 2 + s 2 ) > h(a 1 + s 1 , b 2 + s 2 ) + h(a 2 + s 1 , b 1 + s 2 ).
Let us define :

g 1 := 1 (a 1 -δ,a 1 +δ) -1 (a 2 -δ,a 2 +δ) ,
and :

g 2 := 1 (b 1 -δ,b 1 +δ) -1 (b 2 -δ,b 2 +δ) .
Then, we have that g 1 , g 2 ∈ L ∞ 0 (0, 1), but :

Θ h(x 1 , x 2 )g 1 (x 1 )g 2 (x 2 )dx = [-δ,δ] 2 h(a 1 + s 1 , b 1 + s 2 ) + h(a 2 + s 1 , b 2 + s 2 ) -h(a 1 + s 1 , b 2 + s 2 ) -h(a 2 + s 1 , b 1 + s 2 ) ds 1 ds 2 > 0,
which contradicts (3.11).

For the following proof we use the notation

I k := 1 k , 1 -1 k and I 2 k := I k × I k , for k sufficiently large.
Proof. [Proof of Lemma 3.8] Again, since the Lebesgue measure works well with translations and dilations, it suffices to prove the result for Θ = (0, 1) 2 . In particular, in order to prove the existence of such h 1 , h 2 , it suffices to prove that, for all k ∈ N * , there is

h 1,k ∈ L p 0 (I k ) and h 2,k ∈ L p (I k ) with h 1,k L p (I k ) and h 2,k L p (I k )
bounded uniformly, such that :

h(x 1 , x 2 ) = h 1,k (x 1 ) + h 2,k (x 2 ) almost everywhere in I 2 k . (3.14)
Indeed, we just have to take weak limits (up to extracting a subsequence) at both sides of :

h(x 1 , x 2 )1 I 2 k (x 1 , x 2 ) = h 1,k (x 1 )1 I k (x 1 ) + h 2,k (x 2 )1 I k (x 2
), and use that the unit ball of L p (0, 1) is weakly compact.

In order to prove the existence of h 1,k and h 2,k , let us consider ξ an even mollifier

(a positive C ∞ (R) function such that ξ L 1 (R) = 1) supported in [-1, 1]. We denote ξ ε (s) := ε -1 ξ (ε -1 s).
Let us now consider g 1 , g 2 ∈ L ∞ 0 (I k ) (and null outside I k ). We have, for all ε < k -1 , by Fubini : g 1 ξ ε , g 2 ξ ε ∈ L ∞ 0 (0, 1). Thus, by (3.9), Fubini, the even symmetry of ξ, and the support of g 1 and g 2 , we have that :

0 = Θ h(x 1 , x 2 ) 1 0 g 1 (s 1 )ξ ε (x 1 -s 1 )ds 1 1 0 g 2 (s 2 )ξ ε (x 2 -s 2 )ds 2 dx = Θ g 1 (s 1 )g 2 (s 2 ) Θ h(x 1 , x 2 )ξ ε (s 1 -x 1 )ξ ε (s 2 -x 2 )dx ds 1 ds 2 = I 2 k g 1 (s 1 )g 2 (s 2 ) h (ξ ε ⊗ ξ ε ) (s 1 , s 2 )ds 1 ds 2 . (3.15) Let us denote h ε := h (ξ ε ⊗ ξ ε ). Since ξ is a mollifier, h ε ∈ C ∞ I 2 k and : h ε → h in L p (I 2 k ). (3.16)
Moreover, since identity (3.15) is satisfied for any g 1 , g 2 in L ∞ 0 (I k ), we have that, because of Lemma 3.9, the existence of h 1,ε and h 2,ε in C ∞ I k such that :

h ε (x 1 , x 2 ) = h 1,ε (x 1 ) + h 2,ε (x 2 ) in I 2 k ,
and :

I k h 1,ε (s)ds = 0. (3.17)
Because of (3.16) and the justifications given in the previous paragraph it suffices to prove that h 1,ε L p (I k ) and h 2,ε L p (I k ) are uniformly bounded. In the case of h 2,ε , using (3.17), we get that :

k -2 k h 2,ε (x 2 ) = I k h ε (x 1 , x 2 )dx 1 in I k .
Thus, by Fubini, Hölder and Young, we get for k ≥ 4 the estimate :

h 2,ε L p (I k ) ≤ 2 h ε L p (I 2 k ) ≤ 2 h L p (Θ) . (3.18)
As for h 1,ε , since :

k -2 k h 1,ε (x 1 ) = I k h ε (x 1 , x 2 ) -h 2,ε (x 2 ) dx 2 ,
by the triangular inequality, Fubini, Hölder, Young and (3.18), we get for k ≥ 4 the estimate : So, let us consider u ∈ H(Ω 2 ) such that :

h 1,ε L p (I k ) ≤ 6 h L p (Θ) .
Ω 2 u • vdx = 0,
for any v ∈ SV (Ω 2 ). Recalling (3.8) and with an integration by parts, we have that, for any g 1 , g 2 ∈ L 2 0 (0, π) :

0 = Ω 2 u 1 (x 1 , x 2 ) x 1 0 g 1 (s)ds g 2 (x 2 )dx - Ω 2 u 2 (x 1 , x 2 )g 1 (x 1 ) x 2 0 g 2 (s)ds dx = Ω 2 - x 1 0 u 1 (s, x 2 )ds + x 2 0 u 2 (x 1 , s)ds g 1 (x 1 )g 2 (x 2 )dx. (3.20)
Thus, using Lemma 3.8 for p = 2 and Θ = Ω 2 , we have that there are h 1 , h 2 ∈ L 2 (0, π) such that :

-

x 1 0 u 1 (s, x 2 )ds + x 2 0 u 2 (x 1 , s)ds = h 1 (x 1 ) + h 2 (x 2 ).
If we differentiate with respect to ∂ x 1 x 2 , we have that :

0 = -∂ x 2 u 1 (x 1 , x 2 ) + ∂ x 1 u 2 (x 1 , x 2 ) = ∇ × u.
In addition, since u ∈ H(Ω 2 ), we also have that ∇ • u = 0 and u • n = 0. Consequently, since Ω 2 is Lipschitz and simply connected, we have that u = 0 (see, for instance, [12, Lemma IV.4.6]).

A total set of H(Ω 2 ) formed by solutions of (3.5)

Throughout this section we focus on getting all the solutions of (3.5) (in Ω 2 ). In particular, we first get the candidates and then prove that they form a total set : Proposition 3.10. For all m = (m 1 , m 2 ) ∈ (N * ) 2 let us set :

u m (x) := x 1 0 g 1,m 1 (s)dsg 2,m 2 (x 2 ), -g 1,m 1 (x 1 ) x 2 0 g 2,m 2 (s)ds , (3.21)
for

g 1,m 1 (s) := cos(m 1 s), g 2,m 2 (s) := (2m 2 ε cos(m 2 s) -sin(m 2 s)) e -(2ε) -1 s . (3.22)
Then, u m is a solution of the system (3.5) (in Ω 2 ) such that :

u m L 2 (Ω 2 ) = (1 -e -πε -1 )m 2 2 πε 3 (1 + 4ε 2 (m 2 1 + m 2 2 )) m 2 1 + 4m 2 1 m 2 2 ε 2 , (3.23)
whose associated pressure is constant and whose associated eigenvalue is :

λ m = (m 2 1 + m 2 2 )ε + 1 4ε . (3.24)
Remark 3.11. We have the equality :

u m (x) = e -(2ε) -1 x 2 sin(m 1 x 1 ) m 1 (2m 2 ε cos(m 2 x 2 ) -sin(m 2 x 2 )), -2ε cos(m 1 x 1 ) sin(m 2 x 2 ) . (3.25)
Moreover, using that m 1 , m 2 ≥ 1 and ε ∈ (0, 1), we obtain from (3.25) the estimate :

u m L ∞ (Ω 2 ) ≤ C(m 2 ε + 1). (3.26)
Proof. We can check directly the conclusions of Proposition 3.10 for the functions given in (3.25). In addition, we can get all the eigenfunctions of the elliptic system associated to (3.7) and (3.6) and find the pairs of compatibility. However, we show an "intuitive" way to get the u m to provide a better insight to the reader which can help him/her to understand what happens in other Stokes systems. With that purpose, we look for eigenfunctions in

SV (Ω 2 ) (see (3.8)) such that g 1 , g 2 ∈ C ∞ ([0, π]).
We can prove easily that in that case the condition (3.5) 3 is equivalent to (if g 1 = 0 and g 2 = 0) :

g 1 (0) = g 1 (π) = εg 2 (0) + g 2 (0) = εg 2 (π) + g 2 (π) = 0. (3.27)
If we apply the divergence operator to (3.5) 1 , we obtain that ∆p = 0. In particular, we can apply the Laplacian operator to (3.5) 1 to get possible solutions. Indeed, since :

∆u 1 = g 1 (x 1 )g 2 (x 2 ) + x 1 0 g 1 (s)dsg 2 (x 2 ), ∂ x 2 ∆u 1 = g 1 (x 1 )g 2 (x 2 ) + x 1 0 g 1 (s)dsg 2 (x 2 ), ∆ 2 u 1 = g 1 (x 1 )g 2 (x 2 ) + 2g 1 (x 1 )g 2 (x 2 ) + x 1 0 g 1 (s)dsg iv) 2 (x 2 ) ;
we have that :

-εg 1 (x 1 )g 2 (x 2 ) -2εg 1 (x 1 )g 2 (x 2 ) -ε

x 1 0 g 1 (s)dsg iv) 2 (x 2 ) -g 1 (x 1 )g 2 (x 2 ) - x 1 0 g 1 (s)dsg 2 (x 2 ) -λg 1 (x 1 )g 2 (x 2 ) -λ x 1 0 g 1 (s)dsg 2 (x 2 ) = 0.
We have that g 1,m 1 ∈ L 2 0 (0, π) (see (3.22)), that g 1,m 1 satisfies the first two conditions of (3.27), and that g 1,m 1 is an eigenfunction of the Laplacian ; so it is a reasonable function to do an attempt. Under that choice, the functions g 2 ∈ L 2 0 (0, π) have to satisfy the two last conditions of (3.27) and the ODE :

-εm 4 1 g 2 + 2εm 2 1 g 2 -εg iv) 2 + m 2 1 g 2 -g 2 + λm 2 1 g 2 -λg 2 = 0. (3.28)
In order to solve (3.28) we calculate the roots of its characteristic polynomial :

-ε(r 2 -m 2 1 ) 2 -r(r 2 -m 2 1 ) -λ(r 2 -m 2 1 ) = 0. (3.29)
Let us focus on the roots of :

r 2 + r ε + λ ε -m 2 1 = 0, (3.30)
which are given by :

r = -1 ε ± 1 ε 2 -4 λ ε + 4m 2 1 2 .
These roots are complex if λ is sufficiently large with respect to ε -1 . We then have that the associated solutions for λ large enough are :

g 2,m 2 (s) := (A cos(m 2 s) + B sin(m 2 s))e -(2ε) -1 s , (3.31)
where m 2 > 0 is defined by :

m 2 := m 2 (ε, λ, m 1 ) := λ ε - 1 4ε 2 -m 2 1 . (3.32)
Consequently, the g 2,m 2 that we get are solutions of :

-εg 2 -g 2 = (λ -εm 2 1 )g 2 . (3.33)
We are going to see that there are some g 2,m 2 ∈ L 2 0 (0, π) that also satisfy the last two conditions of (3.27). Indeed, we have that :

g 2,m 2 (s) = - A 2ε + m 2 B cos(m 2 s) + - B 2ε -Am 2 sin(m 2 s) e -(2ε) -1 s .
From g 2,m 2 (0) = -εg 2,m 2 (0), we obtain that :

A = -2m 2 εB. (3.34)
Moreover, from g 2,m 2 (π) = -εg 2,m 2 (π) and (3.34) we have that :

-2m 2 εB cos(m 2 π) + B sin(m 2 π) = -2m 2 εB cos(m 2 π) + 1 2 -2m 2 2 ε 2 B sin(m 2 π).
Consequently,

B sin(m 2 π) = -4m 2 2 ε 2 B sin(m 2 π
). Hence, we have either m 2 ∈ N * or B = 0, which by (3.34) implies that g 2,m 2 = 0, which just gives the null solution. Consequently, the possible values for g 2,m 2 are (we have taken

B = -1) : g 2,m 2 (s) = 2m 2 ε cos(m 2 s) -sin(m 2 s) e -(2ε) -1 s = d 2ε sin(m 2 s)e -(2ε) -1 s ds , ∀m 2 ∈ N * , (3.35)
which clearly belong to L 2 0 (0, π). Thus, our candidates are those defined in (3.22) and λ m satisfies (3.24) because of (3.32).

In order to see that u m satisfies the first equation of (3.5) 1 with a constant pressure for any given time, we have to use (3.21), that g 2,m 2 is a solution of (3.33) and that g 1,m 1 is a solution of -εg 1,m 1 = εm 2 1 g 1,m 1 . Moreover, in order to see that they satisfy the second equation of (3.5) 1 , we consider (3.33) and that εg 2,m 2 (0) + g 2,m 2 (0) = 0, which imply that : (3.36)

(λ -εm 2 1 ) s 0 g 2,m 2 (s )ds = -εg 2,m 2 -g 2,m 2 = -ε d 2 s 0 g 2,m 2 (s )ds ds 2 - d s 0 g 2,m 2 (s )ds ds .
In addition, recalling that m 1 ∈ N * , we have the identity :

π 0 sin 2 (m 1 x 1 )dx 1 = π 2 . (3.37)
Similarly, we have that :

(2m 2 ε cos(m 2 x 2 ) -sin(m 2 x 2 )) 2 = 2m 2 2 ε 2 (1 + cos(2m 2 x 2 )) -2m 2 ε sin(2m 2 x 2 ) + (1 -cos(2m 2 x 2 )) 2 = 4m 2 2 ε 2 + 1 2 + 4m 2 2 ε 2 -1 2 cos(2m 2 x 2 ) -2m 2 ε sin(2m 2 x 2 ). (3.38)
Consequently, considering that m 2 ∈ N * , (3.36) and (3.38) we obtain the equality :

π 0 (2m 2 ε cos(m 2 x 2 ) -sin(m 2 x 2 )) 2 e -ε -1 x 2 dx 2 = π 0 4m 2 2 ε 2 + 1 2 + 4m 2 2 ε 2 -1 2 cos(2m 2 x 2 ) -2m 2 ε sin(2m 2 x 2 ) e -ε -1 x 2 dx 2 = ε 4m 2 2 ε 2 + 1 2 (1 -e -πε -1 ) + 4m 2 2 ε 2 -1 2 ε -1 (1 -e -πε -1 ) 4m 2 2 + ε -2 -2m 2 ε 2m 2 (1 -e -πε -1 ) 4m 2 2 + ε -2 = (4m 2 2 ε 3 + ε)(4m 2 2 ε 2 + 1) + 4m 2 2 ε 3 -ε -8m 2 2 ε 3 8m 2 ε 2 + 2 (1 -e -πε -1 ) = 2m 2 2 ε 3 (1 -e -πε -1 ). (3.39)
Moreover, since m 1 ∈ N we have the identity :

π 0 cos 2 (m 1 x 1 )dx 1 = π 2 . (3.40)
In addition, from m 2 ∈ N * we obtain the equality :

π 0 sin 2 (m 2 x 2 )e -ε -1 x 2 dx 2 = π 0 1 -cos(2m 2 x 2 ) 2 e -ε -1 x 2 = 1 2 ε - ε 4m 2 2 ε 2 + 1 (1-e -πε -1 ) = 2m 2 2 ε 3 4m 2 2 ε 2 + 1 (1 -e -πε
u m 2 L 2 (Ω) = π m 2 1 m 2 2 ε 3 (1 -e -πε -1 ) + 4πm 2 2 ε 5 4m 2 2 ε 2 + 1 (1 -e -πε -1 ) = (1 -e -πε -1 )π 4m 4 2 ε 5 + m 2 2 ε 3 + 4m 2 2 m 2 1 ε 5 4m 2 2 m 2 1 ε 2 + m 2 1 = (1 -e -πε -1 )m 2 2 πε 3 (1 + 4ε 2 (m 2 1 + m 2 2 )) m 2 1 + 4m 2 1 m 2 2 ε 2
, which implies (3.23).

Proposition 3.12. The set {u m } m∈(N * ) 2 is a total set.

Proof. Thanks to Proposition 3.6, it suffices to prove that :

SV (Ω 2 ) ⊂ span{u m }. (3.42)
We prove this inclusion constructively ; that is, writing an element of SV (Ω 2 ) as a limit of linear combinations of the eigenfunctions u m . We consider a function u ∈ SV (Ω 2 ) (see

(3.8)).
Let us recall that cos(ks) √ 2 -1 π k∈N * ∪ {π -1 } is an orthonormal basis of L 2 (0, π) (the basis obtained by the diagonalization of the Laplacian with Neumann boundary conditions in (0, π)). So, since g 1 ∈ L 2 0 (0, π), we have that, for some a k ∈ l 2 :

g 1 (s) = k≥1 a k cos(ks) √ 2 -1 π . (3.43)
That series must be understood as a limit in L 2 (0, π).

Next, we remark that :

γ ε k √ 2 -1 π √ 1 + 4k 2 ε 2 k≥1 e -(2ε) -1 s ε(1 -e -π/ε ) , (3.44)
for :

γ ε k (s) := 2kε cos(ks) -sin(ks), (3.45) is the orthonormal basis obtained by diagonalization of the Laplacian in (0, π) with Robin boundary conditions :

   -2εg (0) = g(0), -2εg (π) = g(π). (3.46)
Indeed, we have that the Laplacian is self-adjoint in the subspace of H 2 (Ω 2 ) that satisfies (3.46). Thus, we have, for some b k ∈ l 2 :

g 2 (s) = g 2 (s)e (2ε) -1 s e -(2ε) -1 s = k≥1 b k γ ε k (s)e -(2ε) -1 s √ 2 -1 π √ 1 + 4k 2 ε 2 . (3.47)
We have used that g 2 ∈ L 2 0 (0, π) implies that g 2 e (2ε) -1 s is orthogonal to e -(2ε) -1 s . Moreover, we have to understand the series as a limit in L 2 (0, π), not as a pointwise one. In that sense, we can place the exponential into the series because the multiplication of the functions of L 2 (0, π) by an element of L ∞ (0, π) is a continuous linear form.

Next, we recall that the operator s 0 is a continuous endomorphism in L 2 (0, π). Thus, we have that :

s 0 g 1 (s )ds = k≥1 a k sin(ks) k √ 2 -1 π . (3.48)
Similarly, we have that :

s 0 g 2 (s )ds = k≥1 b k 2ε sin(ks)e -(2ε) -1 s √ 2 -1 π √ 1 + 4k 2 ε 2 . (3.49)
The limits in (3.48) and (3.49) are both pointwise and in L 2 (0, π).

We finally recall that the tensor product of two convergent sequences of L 2 (0, π) converges in L 2 (Ω 2 ) to the tensor product of their respective limits. Thus, we have that (3.43), (3.47), (3.48) and (3.49) imply :

x 1 0 g 1 (s)dsg 2 (x 2 ) = lim k→∞ k m 1 ,m 2 ≥1 2a m 1 b m 2 π 1 + 4m 2 2 ε 2 sin(m 1 x 1 ) m 1 γ ε m 2 (x 2 )e -(2ε) -1 x 2 ,
and that :

-g 1 (x 1 )

x 2 0 g 2 (s)ds = lim k→∞ k m 1 ,m 2 ≥1 2a m 1 b m 2 π 1 + 4m 2 2 ε 2 -2ε cos(m 1 x 1 ) sin(m 2 x 2 )e -(2ε) -1 x 2 .
Consequently, we have that :

u = lim k→∞ k m 1 ,m 2 ≥1 2a m 1 b m 2 π 1 + 4m 2 2 ε 2 u m ,
which proves (3.42).

Properties of the solution of

(3.3) in Ω 2
Since our operator is not self-adjoint, the total set {u m } given in (3.25) does not have to be orthogonal (and it is not). In addition, it is not immediate to write an element of H(Ω 2 ) as a series of elements proportional to those in {u m }. Indeed, there are total sets in l 2 (for instance {e 1 } ∪ {e 1 + e k } k≥2 ) such that it is not possible to write any element of l 2 (for instance (k -1 ) k≥1 ) as a series of elements proportional to those in the total sets.

In our particular situation we do not even know if this is possible (probably yes), but we will prove that at least we can express the solutions of (3.3) in Ω 2 with the help of {u m }.

The next step is given any u ∈ span{u m }, to get its coordinates in our generating system. In order to do so, we get a set of functions v m ∈ L 2 (Ω 2 ) such that {(u m , v m )} is a bi-orthogonal system ; that is, such that :

Ω 2 u m (x) • v m (x)dx = 1 m=m . (3.50)
Usually, we can get those functions v m by diagonalization of the adjoint operator, but it is easier to do a systematic search in L 2 (Ω 2 ). In that sense we define (recall (3.45)) :

v m (x) := 2 π 2 e (2ε) -1 x 2 m 1 sin(m 1 x 1 ) γ ε m 2 (x 2 ) 1 + 4m 2 2 ε 2 , - cos(m 1 x 1 ) sin(m 2 x 2 ) 2ε . ( 3
v m L 2 (Ω 2 ) ≤ C m 1 + 1 ε e (2ε) -1 π . (3.52)
Now we are ready to get the solutions of (3.3) through a spectral decomposition.

Proposition 3.13. Let ϕ T ∈ H(Ω 2 ). Let us denote, for t ≤ T , L T ε (t)ϕ T the value in H(Ω 2 ) of the energy solution of system (3.3) in Ω 2 on time t. Then, we have that :

1. For all t < T and ε > 0 :

L T ε (t)ϕ T = m∈(N * ) 2 ϕ T , v m L 2 (Ω 2 ) exp (m 2 1 + m 2 2 )ε + 1 4ε (t -T ) u m . (3.53)
In particular, the series of the right hand-side of (3.53) is well-defined and absolutely convergent in H(Ω 2 ).

2. For all δ > 0 there is C δ > 0 such that for all T > 0, ϕ T ∈ H(Ω 2 ), ε ∈ (0, 1) and

s ≤ T -2π -δ : L T ε (s)ϕ T L 2 (Ω 2 ) ≤ C δ exp s -(T -2π -δ) 4ε ϕ T L 2 (Ω 2 ) . (3.54)
Throughout the proof we denote C ε,t a generic constant which might be different each time and which just depends on ε and t.

Proof. First of all, we have that (3.53) is true if ϕ T ∈ span{u m } as a consequence of Proposition 3.10 (see (3.24)) and (3.50). Moreover, for t > 0 and ε > 0 fixed, L T ε (t) is a continuous endomorphism on H(Ω 2 ). Consequently in order to prove Item 1, it suffices to prove that the series of the right-hand side of (3.53) is well-defined, absolutely convergent and continuous. We shall prove the three properties at once. Recalling estimate (3.26), estimate (3.52) and that xe -x 2 is bounded in R + , we have that, if ε ∈ (0, 1) :

m∈(N * ) 2 ϕ T , v m L 2 (Ω 2 ) exp (m 2 1 + m 2 2 )ε + 1 4ε (t -T ) u m L 2 (Ω 2 ) ≤ C ε,t ϕ T L 2 (Ω 2 ) m∈(N * ) 2 m 1 m 2 exp (m 2 1 + m 2 2 )ε(t -T ) ≤ C ε,t ϕ T L 2 (Ω 2 ) m∈(N * ) 2 exp (m 2 1 + m 2 2 ) ε(t -T ) 2 . (3.55)
The series in the right-hand side of (3.55

) is convergent because, in (R + ) 2 : exp (m 2 1 + m 2 2 ) ε(t -T ) 2 1 [m 1 -1,m 1 ]×[m 2 -1,m 2 ] (x 1 , x 2 ) ≤ exp (x 2 1 + x 2 2 ) ε(t -T ) 2 1 [m 1 -1,m 1 ]×[m 2 -1,m 2 ] (x 1 , x 2 ),
which implies that :

m∈(N * ) 2 exp (m 2 1 + m 2 2 ) ε(t -T ) 2 ≤ (R + ) 2 exp (x 2 1 + x 2 2 ) ε(t -T ) 2 dx = 2 ε(T -t) π 4 .
(3.56)

Combining (3.55) and (3.56), we have that : Finally, let us prove (3.54). Indeed, using (3.26), the triangular inequality, and Cauchy-Schwarz, we have that, for all ε ∈ (0, 1) :

m∈(N * ) 2 ϕ T , v m L 2 (Ω 2 ) exp (m 2 1 + m 2 2 )ε + 1 4ε (t -T ) u m L 2 (Ω 2 ) ≤ C ε,t ϕ T L 2 (Ω 2 ) , (3.57
L T ε (s)ϕ T L 2 (Ω 2 ) ≤ C ϕ T L 2 (Ω 2 ) m∈(N * ) 2 m 2 v m L 2 (Ω 2 ) exp (m 2 1 + m 2 2 )ε + 1 4ε (s -T ) . (3.58)
Next, using (3.52) and that xe -x 2 is bounded in R + , if s ≤ T -2 and if ε ∈ (0, 1), we have that (3.58) turns into :

L T ε (s)ϕ T L 2 (Ω 2 ) ≤ C ϕ T L 2 (Ω 2 ) ε 2   m∈(N * ) 2 exp -(m 2 1 + m 2 2 )ε   exp s -(T -2π) 4ε .
(3.59)

Using (3.56) for t = T -2, we have that there is C δ > 0 such that for all ε ∈ (0, 1) :

1 ε 2 m∈(N * ) 2 exp -(m 2 1 + m 2 2 )ε ≤ π 4ε 3 ≤ C δ e δ(4ε) -1 .
Consequently, (3.59) implies (3.54).

Remark 3.14. Let T > 0 and ϕ T ∈ H(Ω 2 ). Then, the application :

R + * ε → ϕ ε (0, •) ∈ H(Ω 2 )
(for ϕ ε the solution of (3.3) in Ω 2 and initial value ϕ T ) is continuous. To prove the continuity, we fix ε 0 > 0 and > 0 and verify that if δ > 0 is small enough, we have for all ε ∈ (ε 0 -δ, ε 0 + δ) the following inequality :

ϕ ε (0, •) -ϕ ε 0 (0, •) L 2 (Ω 2 ) ≤ . (3.60)
First, using (3.26), (3.52) and that se -s 2 ∈ L ∞ (R + , ds) we obtain that for all ε ∈ (ε 0 /2, 3ε 0 /2) and M ∈ N :

m 1 +m 2 ≥M ϕ T , v m L 2 (Ω 2 ) exp -(m 2 1 + m 2 2 )ε + 1 4ε T u m L 2 (Ω 2 ) ≤ C(T, ϕ T ) 1 + 1 ε 2 exp -T + 2π 4ε m 1 +m 2 ≥M exp -(m 2 1 + m 2 2 ) εT 2 ≤ C(T, ϕ T , ε 0 ) m 1 +m 2 ≥M exp -(m 2 1 + m 2 2 ) ε 0 T 4 . (3.61)
Consequently, since the series in the right-hand of (3.61) is convergent, there is M (T, ϕ T , ε 0 ) large enough such that for all ε ∈ (ε 0 /2, 3ε 0 /2) :

m 1 +m 2 ≥M ϕ T , v m L 2 (Ω 2 ) exp -(m 2 1 + m 2 2 )ε + 1 4ε T u m L 2 (Ω 2 ) ≤ 3 . (3.62)
So, using (3.53) and the triangular inequality, we obtain for all ε ∈ (ε 0 /2, 3ε 0 /2) the estimate :

ϕ ε (0, •)-ϕ ε 0 (0, •) L 2 (Ω 2 ) ≤ 2 3 + m 1 +m 2 <M ϕ T , v m L 2 (Ω 2 ) exp -(m 2 1 + m 2 2 )ε + 1 4ε T -exp -(m 2 1 + m 2 2 )ε 0 + 1 4ε 0 T u m L 2 (Ω 2 )
. (3.63)

Finally, considering that the sum in the right-hand side of (3.63) is finite and its terms are continuous with respect to ε, we have (3.60) for all ε ∈ (ε 0 -δ, ε 0 + δ) if δ is small enough.

3.3

The cost of the control in (0, π) 2

In this section we prove the first three items of Theorem 3.1. In the three proofs we use the equivalent definitions of the costs given in (3.4). In addition, for the proof of Item 2 we use the eigenfunctions given in (3.25), whereas for the other two proofs we use Remark 3.5 and then inspire in [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF]. The three proofs are independent.

Proof of Item 1 of Theorem 3.1

In order to prove that the cost of the control decays, we first state a Carleman inequality for system (3.7), then use that the divergence is null, and finally apply usual parabolic estimates. We recall that ω ⊂ Ω 2 is any arbitrary subdomain.

We consider ω 0 an open ball centered at some point x = (x 1 , x 2 ) and whose closure is contained in ω. We consider the auxiliary functions :

η ± (x) := 2π 2 -1 ±G(x 1 ) -(x 2 -x 2 ) 2 ,
for G a regular (positive) function such that G(0) = G(π) = 0, G < 0, G (x 1 ) = 0 and

G L ∞ (0,π) ≤ π 2 .
Finally, let us consider the following classical weights, for T > 0 :

α ± (t, x) := e 8λ -e λ(η ± (x)+6) t( T -t) , ξ(t, x) := e λ(η ± (x)+6) t( T -t) , α * ± (t) = max x∈Ω α ± (t, x), ξ * ± (t) = min x∈Ω ξ ± (t, x).
(3.64)

Remark 3.15. By the choice of G we have that η ± L ∞ (Ω 2 ) ≤ 1, the weights α ± are positive. This way of defining the weights is classical (see, for instance, [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Coron | Control and Nonlinearity[END_REF]). Proposition 3.16. There is a constant C > 0 such that for any domain ω ⊂ Ω 2 , T > 0 and z T ∈ L 2 (Ω 2 ) we have that :

τ λ 2 (0, T )×Ω 2 e -2τ α + ξ + |∇z| 2 dxdt + τ 3 λ 4 (0, T )×Ω 2 e -2τ α + ξ 3 + |z| 2 dxdt ≤ Cτ 3 λ 4 (0, T )×ω e -2τ α + ξ 3 + |z| 2 dxdt, (3.65)
for z the solution of (3.7) (in (0, T ) instead of in (0, T )), for the weights defined in (3.64)

and for any ε ∈ (0, 1), λ ≥ C and τ ≥ C( T + T 2 )ε -1 .

The proof of Proposition 3.16 is essentially a combination of the proofs given in [71,

Proposition 1] and [52, Lemma 1]. Since the ideas of the proof are not original and since the proof is rather long, we just sketch it in Appendix 3.A.

A straight consequence of Proposition 3.16 and Remark 3.5 is that there is C > 0 such that for any ϕ T ∈ L 2 (Ω 2 ), if ϕ denotes the solution of (3.3) in Ω 2 , for any λ ≥ C, for any τ ≥ Cε -1 , and for the weights (3.64) with T = 1, we have that :

τ λ 2 (0,1)×Ω 2 e -2τ α + ξ + |∇ϕ 2 (t + T -1, x)| 2 dxdt + τ 3 λ 4 (0,1)×Ω 2 e -2τ α + ξ 3 + |ϕ 2 (t + T -1, x)| 2 dxdt ≤ Cτ 3 λ 4 (0,1)×ω e -2τ α + ξ 3 + |ϕ 2 (t + T -1, x)| 2 dxdt. (3.66)
Moreover, because of (3.6) 2 , we have by Poincaré inequality (and Fubini) that for all t < T :

τ λ 2 Ω 2 e -2τ α * + (t) ξ * + (t)|ϕ 1 | 2 (t, x)dx = τ λ 2 e -2τ α * + (t) ξ * + (t) π 0 ϕ 1 (t, •, x 2 ) 2 L 2 (0,π;dx 1 ) dx 2 ≤ Cτ λ 2 e -2τ α * + (t) ξ * + (t) π 0 ∂ x 1 ϕ 1 (t, •, x 2 ) 2 L 2 (0,π;dx 1 ) dx 2 = Cτ λ 2 Ω 2 e -2τ α * + (t) ξ * + (t)|∂ x 1 ϕ 1 (t, x)| 2 dx. (3.67)
Thus, since ∇ • ϕ = 0, combining (3.66) with (3.67), we have for λ ≥ C, τ ≥ Cε -1 the estimate :

τ λ 2 (0,1)×Ω 2 e -2τ α * + ξ * + |ϕ(t + T -1, x)| 2 dxdt ≤ Cτ 3 λ 4 (0,1)×ω e -2τ α + ξ 3 + |ϕ 2 (t + T -1, x)| 2 dxdt.
(3.68)

So, fixing λ large enough, and τ = τ 0 ε -1 for τ 0 large enough, we have that (3.68) implies that :

ϕ L 2 ((T -2/3,T -1/3)×Ω 2 ) ≤ Ce Cε -1 ϕ 2 L 2 ((T -1,T )×ω) . (3.69)
Let T > 7, using Item 2 of Proposition 3.13 there is C > 0 such that for δ = 7-2π-2/3, for all t ∈ (T -2/3, T -1/3) and for s = 0, we have that :

ϕ(0, •) L 2 (Ω 2 ) ≤ C exp 7 -2/3 -t 4ε ϕ(t , •) L 2 (Ω 2 ) ≤ C exp 7 -T 4ε ϕ(t , •) L 2 (Ω 2 ) .
(3.70) So, combining (3.69) and (3.70), we get that, for any T > 7 :

ϕ(0, •) L 2 (Ω 2 ) ≤ C exp 7 -T 4ε ϕ L 2 ((T -2/3,T -1/3)×Ω 2 ) ≤ C exp C -T 4ε ϕ 2 L 2 ((T -1,T )×ω) .
In particular, if T 0 is sufficiently large, Item 1 of Theorem 3.1 is true.

Proof of Item 2 of Theorem 3.1

Let us fix h ∈ (0, π) and a control domain ω ⊂ (0, π) × (π -h, π). We prove that, for T ∈ (0, 2(π -h)) fixed, the cost of the control is at least exponentially large with respect to ε -1 . We recall that :

u(x) := ce -(2ε) -1 x 2 (sin(x 1 )(2ε cos(x 2 ) -sin(x 2 )), -2ε cos(x 1 ) sin(x 2 )) ,
for :

c := 1 + 4ε 2 (1 -e -πε -1 )πε 3 (1 + 8ε 2 )
, is a solution of (3.5) for λ ε := 2ε + 1 4ε and such that u L 2 (Ω 2 ) = 1 (see Proposition 3.10). We remark that since ω ⊂ (0, π) × (π -h, π) we have for all δ > 0 and ε ∈ (0, 1) the estimate :

u L ∞ (ω) ≤ Cε -3/2 e -(2ε) -1 (π-h) ≤ C δ e -(2ε) -1 (π-h-δ) .
(3.71)

Indeed, we have that ε -3 ≤ C δ e δε -1 for all δ > 0 and ε ∈ R + .

For proving (3.71) we have calculated the maximum of e -2ε -1 x 2 in (0, π) × (π -h, π).

Moreover, we have that :

ϕ(t, x) := u(x)e -λε(T -t) ,
is a solution of (3.3) in Ω 2 for ϕ T = u. On the one hand, we have that :

Ω 2 |ϕ(0, x)| 2 dx = e -2λεT . (3.72)
On the other hand, using (3.71) and bounding e -2λε(T -t) by 1 we obtain the estimate :

Qω |ϕ| 2 dxdt = Qω |u(x)| 2 e -2λε(T -t) dxdt ≤ u 2 L ∞ (ω) Qω dxdt ≤ C δ exp h -π + δ ε . (3.73)
So, combining (3.4), (3.72) and (3.73) we get that :

[K 1 (T, ε, ω)] 2 ≥ c δ exp π -h -δ -4ε 2 T -T /2 ε ,
which implies Item 2 of Theorem 3.1 by choosing δ ∈ (0, π -h -T /2).

Proof of Item 3 of Theorem 3.1

Let h ∈ (0, π), a control domain ω ⊂ (0, π) × (0, h) and T ∈ (0, π -h) fixed. In order to prove that the cost of the control blows out exponentially with respect to ε, we shall adapt the proof given in [71, Theorem 1]. Indeed, unlike in [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF], ϕ is a vectorial function, the boundary conditions are not purely Dirichlet, and we cannot pick in (3.3) an initial value ϕ T such that Ω ϕ T 2 > 0 because for all u ∈ H(Ω) we have that Ω u 1 = Ω u 2 = 0.

To make the proof more clear, we split it with the help of two lemmas :

Lemma 3.17. Let h ∈ (0, π), T ∈ (0, π -h) and δ ∈ (0, π -T -h). Let ϕ T ∈ H(Ω 2 ) ∩ C ∞ Ω 2 \ {0} such that supp(ϕ T ) ⊂ (0, π) × (π -δ, π
) and let ϕ ε be the solution of (3.3) in Ω 2 and with initial value ϕ T . Then, there is c > 0 such that, for all ε ∈ (0, 1) :

ϕ ε (0, •) L 2 (Ω) ≥ c. (3.74)
Remark 3.18. For any δ > 0 there are functions ϕ T which satisfy the hypothesis of Lemma 3.17. For example, if

χ δ ∈ C ∞ ([0, π]) is a function supported in (π -δ, π) of null mean,
we can consider :

ϕ T (x) := x 1 0 χ δ (s)dsχ δ (x 2 ), -χ δ (x 1 ) x 2 0 χ δ (s)ds . (3.75)
Consequently, neither Lemma 3.17 nor Lemma 3.19 below are empty results.

Proof. We prove (3.74) by contradiction. First of all, we remark that, thanks to forward uniqueness of (3.3) (which can be proved by contradiction with backward uniqueness and (3.53)), ϕ ε (0, •) L 2 (Ω) > 0 for all ε ∈ (0, 1]. In addition, because of Remark 3.14, the only problematic situation arises if there is

ε k → 0 such that ϕ ε k (0, •) L 2 (Ω) → 0. Let us
suppose the existence of such sequence ε k and derive a contradiction.

We have that {ϕ ε 2 : ε ∈ (0, 1)} is bounded in L 2 (Q) (see Remark 3.5) ; thus, we can suppose that ϕ ε k 2 converges weakly in L 2 (Q) to some function γ. Let us consider ψ ∈ D([0, T ] × Ω 2 ). Using Remark 3.5 we have that :

0 = Q ϕ ε k 2 (∂ t ψ-ε k ∆ψ+∂ x 2 ψ)dxdt+ ϕ ε k 2 (0, •), ψ(0, •) L 2 (Ω 2 ) -ϕ T 2 , ψ(T, •) L 2 (Ω 2 ) . (3.76)
Thirdly, since ϕ ε 2 satisfies (3.7) 3 we have the equality :

-2 Ω 2 e 2( δ(π-x 2 )-( δ+( δ) 2 )(T -t))ε -1 ϕ ε 2 ∂ x 2 ϕ ε 2 dx = -2 δε -1 Ω 2 e 2( δ(π-x 2 )-( δ+( δ) 2 )(T -t))ε -1 |ϕ ε 2 | 2 dx. (3.82)
Consequently, considering (3. So, we obtain from (3.79) the following Agmon identity :

T 0 Ω 2 e ( δ(π-x 2 )-( δ+( δ) 2 )(T -t))ε -1 ϕ ε 2 2 dxdt +2ε T 0 T t Ω 2 ∇ e ( δ(π-x 2 )-( δ+( δ) 2 )(T -s))ε -1 ϕ ε 2 (s, x) 2 dxdsdt = T Ω 2 e δ(π-x 2 )ε -1 ϕ T 2 2
dx.

(3.83)

On the one hand, using that the support of ϕ T is included in (0, π) × (π -δ, π), we have that :

T Ω 2 e δ(π-x 2 )ε -1 ϕ T 2 2 dx ≤ πe 2 δδε -1 Ω 2 ϕ T 2 2 dx. (3.84)
On the other hand, recalling that ω ⊂ (0, π) × (0, h), we find that :

e 2 δ(π-h-T -δT )ε -1 Qω |ϕ ε 2 | 2 dxdt ≤ T 0 Ω 2 e ( δ(π-x 2 )-( δ+( δ) 2 )(T -t))ε -1 ϕ ε 2 2 dxdt. (3.85)
Combining (3.83), (3.84) and (3.85), we have that :

Qω |ϕ ε 2 | 2 dxdt ≤ π exp 2 δ(T + δT + h + δ -π) ε ϕ T 2 2 L 2 (Ω) ; (3.86)
that is, we get the decay of the L 2 norm of ϕ 2 in the control domain.

Remark 3.20. So far, combining (3.4), Lemma 3.17, Lemma 3.19 and Remark 3.18, we have that for a given h ∈ (0, π), for all ω ⊂ (0, T ) × (0, h), and for all T ∈ (0, π -h), there is c > 0 such that for all ε ∈ (0, 1) :

K 2 (T, ε, ω) ≥ ce cε -1 .
Now we have the tools to end the proof of Item 3 of Theorem 3.1. For that proof, we denote, for h ∈ (0, π) and T ∈ (0, π -h) both fixed parameters :

δ := (π -h -T )/3, ω 1 := (0, π) × (0, h), ω 2 := (0, π) × (0, h + δ). (3.87)
Proof. [End of the proof of Item 3 of Theorem 3.1] Let ϕ ε be the solution of (3.3) in Ω 2 of initial value (3.75) and with δ as in (3.87). Thanks to (3.4), Lemma 3.17 and Lemma 3.19, the only thing left to prove is that Qω |ϕ ε 1 | 2 dxdt decays exponentially. For that purpose, we get an estimate of ∂

x 2 ϕ ε 2 in L 2 (Q ω 1 ).
Let us fix θ(s) a positive regular scalar cut-off function supported in (-∞, h + δ) and such that θ = 1 in (-∞, h]. We define :

ψ ε (t, x) := θ(x 2 )ϕ ε 2 (t, x). (3.88)
Recalling the support of ϕ T (see (3.75)), we have that ψ ε satisfies :

               -∂ t ψ ε -ε∆ψ ε -∂ x 2 ψ ε = εθ (x 2 )ϕ ε 2 -2ε∂ x 2 θ (x 2 )ϕ ε 2 -θ (x 2 )ϕ ε 2 in Q, ψ ε = 0 on Σ b,t , ∂ x 1 ψ ε = 0 on Σ l,r , ψ ε (T, •) = 0 on Ω 2 .
(3.89)

In particular, if we multiply (3.89) by ψ ε and we integrate by parts, it is not difficult to deduce that :

Q |∇ψ ε | 2 dxdt ≤ C ε Qω 2 |ϕ ε 2 | 2 dxdt. (3.90)
Using Lemma 3.19 for h = h + δ, T = T and δ = δ and using that for all δ > 0 there is

C δ > 0 such that ε -1 ≤ C δ e δε -1
in R + , we get from (3.90) that :

Qω 1 |∂ x 2 ϕ ε 2 | 2 dxdt ≤ Ce -cε -1 .
In order to conclude, using that the divergence of ϕ ε is null, we get that :

Qω 1 |∂ x 1 ϕ ε 1 | 2 dxdt ≤ Ce -cε -1 .
In addition, since ϕ ε 1 is null on x 1 = 0 (see (3.6)), we have, thanks to Poincaré inequality :

Qω |ϕ ε 1 | 2 dxdt ≤ Qω 1 |ϕ ε 1 | 2 dxdt ≤ Ce -cε -1 . ( 3

3.4

The control problem (3.1) in (0, π) 3 In this section we first give some solutions of (3.5) in Ω 3 and then, we prove Item 4 of Theorem 3.1.

3.4.1

Brief study of the spectral decomposition of H((0, π) 3 )

In order to do the spectral decomposition, we consider the set of separated variables with null divergence, null normal trace, and a null component :

SV (Ω 3 ) := SV 1 (Ω 3 ) ∪ SV 2 (Ω 3 ) ∪ SV 3 (Ω 3 ) := 0, g 1 (x 1 ) x 2 0 g 2 (s)dsg 3 (x 3 ), -g 1 (x 1 )g 2 (x 2 ) x 3 0 g 3 (s)ds : g 1 ∈ L 2 (0, π); g 2 , g 3 ∈ L 2 0 (0, π) ∪ - x 1 0 g 1 (s)dsg 2 (x 2 )g 3 (x 3 ), 0, g 1 (x 1 )g 2 (x 2 ) x 3 0 g 3 (s)ds : g 2 ∈ L 2 (0, π); g 1 , g 3 ∈ L 2 0 (0, π) ∪ x 1 0 g 1 (s)dsg 2 (x 2 )g 3 (x 3 ), -g 1 (x 1 ) x 2 0 g 2 (s)dsg 3 (x 3 ), 0 : g 3 ∈ L 2 (0, π); g 1 , g 2 ∈ L 2 0 (0, π) .
Remark 3.21. Unlike in SV (Ω 2 ), now the mean of one of the three auxiliary functions is not necessarily 0. This difference is crucial in explaining why the solutions of (3.3) behaves differently in Ω 3 . Indeed, this difference allows us to construct eigenfunctions whose associated eigenvalue does not explode with ε (see (3.93)), which is something that we cannot do in Ω 2 .

In order to look for eigenfunctions in SV (Ω 3 ), we remark that if g 1 , g 2 , g 3 are regular functions, the boundary conditions of (3.3) translates into :

g 1 (0) = g 1 (π) = g 2 (0) = g 2 (π) = εg 3 (0) + g 3 (0) = εg 3 (π) + g 3 (π) = 0. (3.92)
Let us focus on SV 3 (Ω 3 ). We can again consider for m 1 , m 2 ∈ N * , g 1,m 1 (s) := cos(m 1 s) and g 2,m 2 (s) := cos(m 2 s). As for g 3 , we can look at the solutions of :

g + g ε + λ ε -m 2 1 -m 2 2 g = 0,
an equation that can be obtained as in the proof of Proposition 3.10. Since the mean of g 3 is not necessarily 0 and since s 0 g 3 (s )ds does not have to be an eigenfunction of 3.4. The control problem (3.1) in (0, π) 3

-ε∆+∂ x 3 (because the third component of an element of SV 3 (Ω 3 ) is null), we can consider small values of λ. In particular we get for any m 1 , m 2 ∈ N * the following eigenfunction :

1 m 1 sin(m 1 x 1 ) cos(m 2 x 2 )e -ε -1 x 3 , - 1 m 2 cos(m 1 x 1 ) sin(m 2 x 2 )e -ε -1 x 3 , 0 , (3.93)
whose associated eigenvalues is :

λ ε m 1 ,m 2 := ε(m 2 1 + m 2 2 ), (3.94)
and whose associated pressure term is constant.

Remark 3.22. We can get for dimension 3 similar results to those of Remark 3.4, Proposition 3.6, Proposition 3.10 and Proposition 3.12, with the only difference that some eigenvalues do not explode with ε. In that sense, we can get similar results as Item 1 of Proposition 3.13, but we cannot get a general dissipation result as the one in Item 2 of Proposition 3.13.

Proof of Item 4 of Theorem 3.1

For this proof we consider the function :

u(x) := 2e -ε -1 x 3 π ε (1 -e -2πε
-1 ) (sin(x 1 ) cos(x 2 ), -cos(x 1 ) sin(x 2 ), 0) , which by (3.93) and (3.94) (we have taken m 1 = m 2 = 1) is a solution of (3.5) in Ω 3 for λ = 2ε and a pressure which is constant. Moreover, we can easily check that u L 2 (Ω) = 1 and that ϕ(t, x) := u(x)e -2ε(T -t) is a solution of (3.3) in Ω 3 . In addition, since ω ⊂ (0, π) 2 × (π -h, π) we can bound e -ε -1 x 3 by e -(π-h)ε -1 and obtain for all δ > 0 and ε ∈ (0, 1) that :

u L ∞ (ω) ≤ Cε -1/2 e -(π-h)ε -1 ≤ C δ e -ε -1 (π-h-δ) . (3.95)
Indeed, we have that ε -1/2 ≤ C δ e δε -1 for all δ > 0 and ε ∈ R + .

On the one hand, we have that :

Ω 3 |ϕ(0, x)| 2 dx = e -4εT . (3.96)
On the other hand, using (3.95) and bounding e -2ε(T -t) by 1 we obtain that :

Qω |ϕ| 2 dxdt = Qω |u(x)| 2 e -4ε(T -t) dxdt ≤ u 2 L ∞ (ω) Qω dxdt ≤ T C δ exp 2(-π + h + δ) ε . (3.97)
So, if we combine (3.96) and (3.97), we have that for all δ ∈ (0, π -h) (remember (3.4) and that ε ∈ (0, 1)) :

[K 3 (T, ε, ω)] 2 ≥ Ω 3 |ϕ(0, x)| 2 dx Qω |ϕ| 2 dxdt ≥ c δ e -4T T exp 2(π -h -δ) ε ;
that is, for T fixed the cost of the control explodes exponentially when ε → 0 + . Remark 3.23. A difference between dimension 3 and dimension 2 can be seen by comparing (3.72) and (3.96). Indeed, for T fixed (3.72) decays with ε, whereas (3.96) does not.

Further comments and open problems

• About the boundary control. The case where we have a control on the boundary of (3.1) (as proposed in [START_REF] Coron | On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions[END_REF]) instead of in the interior remains an open problem.

• The spectral method in the transport-diffusion elemental equation. Let us consider the control problem studied in [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF] :

         y t + L ε y = 1 ω f in Q, y = 0 on Σ, y(0, •) = y 0 on Ω, (3.98)
for

L ε := -ε∆ + ∂ x d ,
for Ω any open regular domain. In order to prove the dissipation result for the adjoint system, we can use a spectral decomposition. In particular, if {w m } is the spectral basis when we diagonalize the Dirichlet Laplacian in Ω, of eigenvalues λ m , for wm := w m e (2ε) -1 x d we have that { wm } is a linearly independent total set such that L ε wm = (ελ m + (4ε) -1 ) wm , and such that each wm satisfies Dirichlet boundary conditions. Consequently, we can replicate the procedure of Section 3.2.1 and get the dissipation result, with an accurate result (see (3.54)), which looking at the eigenvalues

Further comments and open problems

clearly is the optimal one. As for the optimal dissipation rate, there is another proof in [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF]Lemma 4] which uses a representation theorem and a comparison theorem.

• The control problem (3.1) when we change the boundary conditions. We can attempt for Ω = (0, π) 2 and d = 2 to replicate this method to get similar results with a boundary condition of the type :

(-2εDu • n + γn 2 u) tg = 0, (3.99)
for γ ∈ R (it is specially interesting the case γ = 1/2, when the operator is self-adjoint).

Even if it actually produces eigenfunctions, it does it by taking into account the other two roots of (3.29) ; that is, r = ±m 1 . In particular, we get eigenfunctions with g 2,m 2 of the type (see (3.31)) :

A cos(m 2 πs)e (2ε) -1 s + B sin(m 2 πs)e (2ε) -1 s + Ce m 1 s + De -m 1 s , and with m 2 not necessarily in N * (and not necessarily easy to calculate explicitly). The root r = m 1 does not depend on ε or, more astonishingly, on λ, because :

u(x 1 , x 2 ) := (sin(m 1 x 1 )e m 1 x 2 , -cos(m 1 x 1 )e m 1 x 2 ) = ∇ -cos(m 1 x 1 )e m 1 x 2 m 1 .
Therefore, that term is an eigenfunction independent of m 2 or ε because of the help of the pressure term. A similar thing happens with the term of De -m 1 s . All this, of course, adds some difficulties in proving that they form a total set (if they actually do), difficulties which we have been unable to overcome. Therefore, getting the eigenfunctions of the Stokes system with a boundary condition of the type (3.99) (together with u • n = 0) can be considered for future work.

As for the control problem (3.1) with Dirichlet boundary conditions, replicating the method of this paper has several problems. For instance, in dimension 2, using the notation of (3.8), we now have the boundary conditions :

g 1 (0) = g 1 (π) = g 2 (0) = g 2 (π) = 0.
So, the analogous choice for g 1,m 1 would be sin(m 1 s) (see (3.22) 1 ) ; but, for m 1 odd it does not belong to L 2 0 (0, π). Moreover, for any m 1 ∈ N * , s 0 sin(m 1 s )ds = m -1 1 (1 -cos(m 1 s)), which is not an eigenfunction of the Laplacian. Consequently, it is very likely that another method must be used to get the cost of the control.

• The control problem (3.1) in other domains. It is clear that working in (0, π) 2 or (0, π) 3 is really helpful, not only for the dissipation, but for the cost of the control as well. Indeed, we have used several times in Section 3.3 that the solutions of (3.3) are solutions of the heat equation. Consequently, it would be interesting to know what results can be obtained in other domains.

• The cost of the control in dimension

3 if ω is near x 3 = 0. It is an open question if K 3 (T, ε, ω) (see (3.
2)) also explodes with ε for large times T if there is x ∈ (0, π) 2 × {0} and δ > 0 such that B(x, δ) ∩ {x 3 > 0} ⊂ ω.

• The optimal T 0 in which the cost of the control decays with ε. The optimal T 0 in which the cost of the control decays with ε is an open problem (see Item 1 of Theorem 3.1). Thanks to Item 2 and Item 3 of Theorem 3.1 we can get a lower bound. Moreover, we could have got an upper bound by doing all the operations of the proof of the Carleman (Proposition 3.16) explicitly as in [START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF], but it would be far from optimal, so is not worth the effort. A more subtle technique would be to transform the problem in a fast-control problem as in [START_REF] Lissy | A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation[END_REF] ; but, on the one hand, when we multiply by a weight which depends on a spatial variable, the fact that the divergence is null is lost ; and, on the other hand, the fact that fast controls have a cost of e CT -1 (without specifying the C) in the Stokes system is just a recent result (see [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the Stokes system[END_REF] and [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]). In addition, this problem is probably more difficult than getting the optimal T 0 for (3.98), which is still an open problem, even in dimension 1.

• Relation of the cost of control with a force of d -1 components and with a force of d components. An interesting question is to see if the fact that K 1 (T, ε, ω) decays or explodes with ε is equivalent to the fact that K 2 (T, ε, ω) decays or explodes with ε (see (3.2) for the definitions).

3.A Sketch of the proof of Proposition 3.16

We recall that Ω 2 := (0, π) 2 , ω is any open domain contained in Ω 2 , z is the solution of (3.7) (in (0, T ) instead of in (0, T )) and the weights are given in (3.64). In addition, in this section Q := (0, T ) × Ω 2 , Q ω := (0, T ) × ω, Σ l,r := (0, T ) × {0, π} × (0, π) and Σ b,t := (0, T ) × (0, π) × {0, π}.

We first remark that the auxiliary functions η ± satisfy :

η + = η -, ∂ n η + + ∂ n η -= 0, |∇η + | = |∇η -| on {0, π} × (0, π), ∂ n η ± < 0 on (0, π) × {0, π}. (3.100)
In addition, there is δ > 0 a constant that just depends on the control domain and fixed for this section such that :

|∇η ± | ≥ δ > 0 in Ω 2 \ ω 0 . (3.101)
As for the weights, we have the usual bounds (see Remark 3.15) :

|∂ x i α ± | = |∂ x i ξ ± | ≤ Cλξ ± , |∂ t α ± | ≤ C T ξ 2 ± , |∂ 2 t 2 α ± | ≤ Cξ 2 ± (1 + T 2 ξ ± ). (3.102)
In order to prove the Carleman inequality, as explained above, there is nothing original in the proof, since we do a mixture of the scheme of [71, Proposition 1] and [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF]Lemma1].

In particular, we also consider the change of variables :

ψ ± := e -sα ± z. (3.103)
We remark that :

α + = α -, ξ + = ξ -, ψ + = ψ -on Σ lr . (3.104)
Moreover, ψ has Dirichlet boundary conditions on Σ bt . As for Σ lr , since z has Neumann boundary conditions we have the equality :

∂ n ψ ± = sλξ ± ∂ n η ± ψ ± . (3.105)
In addition, combining (3.100) 1 , (3.104) and (3.105), we have the equality :

|∇ψ + | = |∇ψ -| on Σ lr . (3.106)
We finally remark that :

L ± 1 ψ ± + L ± 2 ψ ± = L ± 3 ψ ± , (3.107)
for :

         L ± 1 ψ ± := -2ετ λ 2 |∇η ± | 2 ξ ± ψ ± -2ετ λξ ± ∇η ± • ∇ψ ± + ∂ x 2 ψ ± + ψ ± t , L ± 2 ψ ± := ετ 2 λ 2 |∇η ± | 2 ξ 2 ± ψ ± + ε∆ψ ± + τ ∂ t (α ± )ψ ± -τ λ∂ x 2 η ± ξ ± ψ ± , L ± 3 ψ ± := ετ ∆η ± ξ ± ψ ± -ετ λ 2 |∇η ± | 2 ξ ± ψ ± . (3.108)
As usual, we denote (L ± i ψ) j the j-th term of L ± i ψ and calculate the product :

L ± 1 ψ ± , L ± 2 ψ ± L 2 (Q) .
Most of the operations here are repetitions of [71, Proposition 1] with the exception of what we do with the boundary term when integrating by parts (and with the difference that z is a solution of the direct equation in [START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF]). Thus, we skip some operations.

To begin with, we have for λ ≥ C and τ ≥ C T 2 (see (3.101) for the definition of δ)

the estimate :

(L ± 1 ψ ± ) 1 + (L ± 1 ψ ± ) 2 , (L ± 2 ψ ± ) 1 L 2 (Q) = ε 2 τ 3 λ 4 Q |∇η ± | 4 ξ 3 ± |ψ ± | 2 dxdt + O ε 2 s 2 λ 3 Q ξ 3 ± |ψ ± | 2 dxdt -ε 2 τ 3 λ 3 Σ lr |∇η ± | 2 ξ 3 ± ∂ n η ± |ψ ± | 2 dxdt ≥ 3δ 4 4 ε 2 τ 3 λ 4 Q ξ 3 ± |ψ ± | 2 dxdt -δ 4 ε 2 τ 3 λ 4 Qω 0 ξ 3 ± |ψ ± | 2 dxdt -ε 2 τ 3 λ 3 Σ lr |∇η ± | 2 ξ 3 ± ∂ n η ± |ψ ± | 2 dxdt. (3.109)
Moreover, after an integration by parts, considering that ψ ± = 0 on Σ bt and (3.102), we obtain for λ ≥ 1 and τ ≥ C( T + T 2 ) the bound :

(L ± 1 ψ ± ) 3 +(L ± 1 ψ ± ) 4 , (L ± 2 ψ ± ) 1 L 2 (Q) = O ετ 2 λ 3 Q ξ 2 ± |ψ ± | 2 dxdt + ε T τ 2 λ 2 Q ξ 3 ± |ψ ± | 2 dxdt . ( 3
≥ C( T + T 2 )ε -1 : i∈{+,-} L i 1 ψ i , (L i 2 ψ i ) 1 L 2 (Q) ≥ i∈{+,-} δ 4 2 ε 2 τ 3 λ 4 Q ξ 3 i |ψ i | 2 dxdt-δ 4 ε 2 τ 3 λ 4 Qω 0 ξ 3 i |ψ i | 2 dxdt. (3.111)
To continue with, we have that, integrating by parts, with Cauchy-Schwarz inequality and by (3.100) :

(L ± 1 ψ ± ) 1 , (L ± 2 ψ ± ) 2 L 2 (Q) = 2ε 2 τ λ 2 Q |∇η ± | 2 ξ ± |∇ψ ± | 2 dxdt -2ε 2 τ λ 2 Σ lr |∇η ± | 2 ξ ± ∂ n ψ ± ψ ± dx 2 dt + O ε 2 τ 2 λ 4 Q ξ 2 ± |ψ ± | 2 dxdt + ε 2 Q (τ ξ + λ 2 )|∇ψ ± | 2 dxdt . (3.112)
Next, we have again by integration by parts that :

(L ± 1 ψ ± ) 2 , (L ± 2 ψ ± ) 2 L 2 (Q) = -2ε 2 τ λ Σ bt ∂ n η ± ξ ± |∂ n ψ ± | 2 dx 1 dt -2ε 2 τ λ Σ lr (∂ tg η ± ∂ tg ψ ± + ∂ n η ± ∂ n ψ ± )ξ ± ∂ n ψ ± dx 2 dt + O ε 2 τ λ Q ξ ± |∇ψ ± | 2 dxdt + 2ε 2 τ λ 2 Q ξ ± |∇η ± • ∇ψ ± | 2 dxdt + ε 2 τ λ Q ξ ± ∇η ± • ∇|∇ψ ± | 2 dxdt. (3.113)
As for the term of the gradient, we have the equality :

ε 2 τ λ Q ξ ± ∇η ± • ∇|∇ψ ± | 2 dxdt = -ε 2 τ λ 2 Q |∇η ± | 2 ξ ± |∇ψ ± | 2 dxdy + O ε 2 τ λ Q ξ ± |∇ψ ± | 2 dxdt + ε 2 τ λ Σ bt ∂ n η ± ξ ± |∂ n ψ ± | 2 dx 1 dt + ε 2 τ λ Σ lr ∂ n η ± ξ ± |∇ψ ± | 2 dx 2 dt. (3.114)
Next, we have that :

(L ± 1 ψ ± ) 3 , (L ± 2 ψ ± ) 2 L 2 (Q) = O ε Σ bt |∂ n ψ ± | 2 dx 1 dt + ε Σ lr ∂ x 2 ψ ± ∂ n ψ ± dx 2 dt. (3.115)
To continue with, we have the equality :

(L ± 1 ψ ± ) 4 , (L ± 2 ψ ± ) 2 L 2 (Q) = ε Σ lr ψ ± t ∂ n ψ ± dx 2 dt. (3.116)
So, before adding up, because of (3.100) 2 , we have for λ ≥ C and τ ≥ C( T + T 2 )ε -1 the bound :

-2ε 2 τ λ Σ bt ∂ n η ± ξ ± |∂ n ψ ± | 2 dx 1 dt + ε 2 τ λ Σ bt ∂ n η ± ξ ± |∂ n ψ ± | 2 dx 1 dt + O ε Σ bt |∂ n ψ ± | 2 dx 1 dt ≥ 0. ( 3
i∈{+,-} L i 1 ψ i , (L i 2 ψ i ) 2 L 2 (Q) ≥ i∈{+,-} O ε 2 τ 2 λ 4 Q ξ 2 i |ψ i | 2 dxdt + i∈{+,-} δ 2 2 ε 2 τ λ 2 Q ξ i |∇ψ i | 2 dxdt -δ 2 ε 2 τ λ 2 Qω 0 ξ i |∇ψ i | 2 dxdt. (3.118)
As for the rest of the terms, it is easy to verify for λ ≥ 1 and τ ≥ C( T + T 2 )ε -1 the bound :

i∈{+,-} L i 1 ψ i , (L i 2 ψ i ) 3 + (L i 2 ψ i ) 4 L 2 (Q) = i∈{+,-} O ε 2 τ 3 λ 3 Q ξ 3 i |ψ i | 2 dxdt . (3.119)
So, if we add (3.112), (3.118) and (3.119), we get after absorptions for λ ≥ C and τ ≥

C( T + T 2 )ε -1 that : 2 i∈{+,-} L i 1 ψ i , L i 2 ψ i L 2 (Q) + 2δ 4 ε 2 τ 3 λ 4 Qω 0 ξ 3 i |ψ i | 2 dxdt + 2δ 2 ε 2 τ λ 2 Qω 0 ξ i |∇ψ i | 2 dxdt ≥ i∈{+,-} δ 4 2 ε 2 τ 3 λ 4 Q ξ 3 i |ψ i | 2 dxdt + δ 2 2 ε 2 τ λ 2 Q ξ i |∇ψ i | 2 dxdt.
So, considering (3.107) and (3.108) 3 we find after some easy absorptions for λ ≥ C and

τ ≥ C( T + T 2 )ε -1 the estimate : i∈{+,-} L i 1 ψ i 2 L 2 (Q) + L i 1 ψ i 2 L 2 (Q) + δ 4 4 ε 2 τ 3 λ 4 Q ξ 3 i |ψ i | 2 dxdt+ δ 2 2 ε 2 τ λ 2 Q ξ i |∇ψ i | 2 dxdt ≤ i∈{+,-} 2δ 4 ε 2 τ 3 λ 4 Qω 0 ξ 3 i |ψ i | 2 dxdt + 2δ 2 ε 2 τ λ 2 Qω 0 ξ i |∇ψ i | 2 dxdt.
Moreover, if we consider in (3.108) the equations of ∆ψ ± and ∂ t ψ ± , we have, after usual absorptions, for λ ≥ C and τ ≥ C( T + T 2 )ε -1 , the inequality :

i∈{+,-} τ -1 Q ξ -1 (ε -2 |ψ i t | 2 + |∆ψ i | 2 )dxdt + δ 4 16 τ 3 λ 4 Q ξ 3 i |ψ i | 2 dxdt + δ 2 8 τ λ 2 Q ξ i |∇ψ i | 2 dxdt ≤ i∈{+,-} 2δ 4 τ 3 λ 4 Qω 0 ξ 3 i |ψ i | 2 dxdt + 2δ 2 τ λ 2 Qω 0 ξ i |∇ψ i | 2 dxdt. (3.120)
From (3.120) it is well-known how to obtain (3.65) (see, for instance, [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF]).

1973 (see [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF]) and recently, it has been used to get accurate bounds on the cost of the null controllability of the heat equation (see, for instance, [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF], [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF] and [START_REF] Ervedoza | Sharp observability estimates for heat equations[END_REF]). As for other parabolic equations, it has also been used to get estimate of the Stokes equation (see, for instance, [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the Stokes system[END_REF]).

• The second one is based on spectral inequalities and on properties of analytic functions. The first time this approach is used for the heat equation was in 1995 in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF],

where the authors proved the controllability in C ∞ domains. More recently, the null controllability of the heat equation has been studied with these techniques when the control is localized in an arbitrary measurable set of positive measure (see, for instance, [START_REF] Wang | L ∞ -null controllability for the heat equation and its consequences for the time optimal control problem[END_REF], [START_REF] Zhang | An observability estimate for the heat equation from a product of two measurable sets[END_REF] and [START_REF] Apraiz | Null-control and measurable sets *[END_REF]), which is one of the advantages of this method. In addition, these results have recently been extended to Lipschitz domains which are locally star-shaped (see [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF]). As for other parabolic equations, this method has been used recently to study the controllability of the Stokes equation (see [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]) and higher order parabolic equations (see, for instance, [START_REF] Escauriaza | Observation from measurable sets for parabolic analytic evolutions and applications[END_REF] and [START_REF] Escauriaza | Analyticity of solutions to parabolic evolutions and applications[END_REF]).

• The third one is based on multiplying the adjoint function by an exponential weight and using the PDE satisfied by the new function. Indeed, Fursikov and Imanuvilov proved with this method in 1996 the controllability of the heat equation in C 2 domains (see [START_REF] Fursikov | Controllability of evolution equations[END_REF]) when the control domain is an open set. This method is really flexible and it is the base to study the controllability of countless parabolic differential equations, both linear and non-linear (see, for instance, [START_REF] Fursikov | Controllability of evolution equations[END_REF]). As a continuation of the work made in [START_REF] Fursikov | Controllability of evolution equations[END_REF], Carleman estimates of the heat equation have been established when the system is not regular, which proves the null controllability of some related linear and non-linear systems. For instance, in [START_REF] Yu | Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications[END_REF] and [START_REF] Fernández-Cara | Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability[END_REF] the authors consider non-regular source terms. Moreover, in [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF] and [START_REF] Yu | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF] the authors study some boundary conditions which imply that the solution is not in

L 2 (0, T ; H 2 (Ω)).
Additional examples are given in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF], [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF] and [START_REF] Rousseau | Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF], where the authors analyse parabolic equations in which the diffusion coefficient is not continuous. Finally, it is well-known that this method can be easily generalized to cubes or, more generally, to any cartesian product of C 2 domains (see, for instance, [START_REF] González-Burgos | Some results on controllability for linear and nonlinear heat equations in unbounded domains[END_REF]).

The main objective of this paper is to extend the method of Fursikov and Imanuvilov to additional domains which are not C 2 when there is a lack of regularity caused by the domain and the control acts on an internal subdomain. In particular, we aim to get a Carleman inequality with a source term. It is a relevant problem because, unlike in [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF],

Null controllability in pseudo-cylinders 123 this method allows us to consider lower order terms with coefficients that just belong to L ∞ , and hence treat non-linearities. We recall that the authors in [START_REF] Fursikov | Controllability of evolution equations[END_REF] prove a Carleman inequality with the help of an auxiliary function η which satisfies :

η ∈ C 2 Ω , η = 0 on ∂Ω, η > 0 in Ω, inf Ω\ω |∇η| > 0, (4.1)
for ω the control domain. Since in a Lipschitz domain the heat equation has a unique energy solution, the existence of some Carleman estimate seems reasonable. However, two difficulties arise :

• The first and main difficulty is that the construction of the function η given in [START_REF] Fursikov | Controllability of evolution equations[END_REF] does not work when Ω is not C 2 . Indeed, if Ω has corners and if ω is compactly included in Ω, the three conditions in (4.1) are incompatible. The only solution known so far for other domains (like cylinders) is to construct manually an auxiliary function which satisfies the following assumptions (which are verified by any η satisfying (4.1)) :

η ∈ C 2 Ω , ∂ n η ≤ 0 on ∂Ω, inf Ω\ω |∇η| > 0, (4.2)
for n the outward unit normal vector on ∂Ω. In the case of cylinders, the construction of η is done for instance as a sum of functions of different variables.

• The second difficulty is that [START_REF] Fursikov | Controllability of evolution equations[END_REF] uses that the Laplacian behaves well in Ω in the sense that D Ω (∆) ⊂ H 2 (Ω) continuously, for

D Ω (∆) := {w ∈ H 1 0 (Ω) : ∆w ∈ L 2 (Ω)},
omitting the Ω if it is clear. However, when Ω is Lipschitz this is not always true, since in a general Lipschitz domain the most we can ensure is that D(∆) ⊂ H 3/2 (Ω) continuously. Indeed, the fact that D(∆) ⊂ H 3/2 (Ω) continuously is proved in [START_REF] Jerison | The Neumann problem on Lipschitz domains[END_REF],

and for all ε > 0 a counter-example of a domain Ω ε for which there is w ∈ H [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. In particular, if u is a solution of the backwards heat equation with Dirichlet boundary conditions, initial value in H 1 0 (Ω) and source term in L 2 ((0, T ) × Ω), the most that we can expect is that u ∈ H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; D(∆)). However, while doing the Carleman in [START_REF] Fursikov | Controllability of evolution equations[END_REF] the authors do some calculations with the second order derivatives, so they need that the regular solutions of the heat equation belong to L 2 (0, T ; H 2 (Ω)). One of the contributions of this paper on the Fursikov-Imanuvilov method is reducing the difficulty that the L 2 -norm of the Laplacian is not equivalent

1 0 (Ω ε ) such that -∆w ∈ C ∞ Ω ε but w ∈ H 3/2+ε (Ω) is given for instance in
Ω = U ({(x, z) : x ∈ B, z ∈ (0, H(x))}) . (4.3)
Moreover, we denote the cylindrical part by :

C := U B × 0, inf B H . (4.4)
If U = I (the identity endomorphism), we say that the pseudo-cylinder is canonically oriented. In that case, we have :

C = B × 0, inf B H . (4.5)
Remark 4.2. If Ω is a pseudo-cylinder, we can split ∂Ω into three parts :

U (B × {0}) , L := U ({(x, z) : x ∈ ∂B, z ∈ (0, H(x))}) , T := U ({(x, H(x)) : x ∈ B}) .
If Ω is canonically oriented, those parts are given by :

B × {0}, L = {(x, z) : x ∈ ∂B, z ∈ (0, H(x))}, T = {(x, H(x)) : x ∈ B}.
Example 4.3. See Figure 4.1 for a canonically-oriented pseudo-cylinder in R 2 .

Remark 4.4. The pseudo-cylinders are relevant domains, for example, in fluid mechanics (see, for instance, [START_REF] Pedlosky | Geophysical fluid dynamics[END_REF] and [START_REF] Guerrero | On the controllability of the hydrostatic stokes equations[END_REF]).

In this paper we solve the previously presented difficulties for pseudo-cylinders and get some Carleman estimates (see Proposition 4. Before presenting the main result of this paper we introduce the following notation :

Definition 4.5.
Let Ω be any domain and let ω ⊂ Ω be a subdomain. We define as usual

Q := (0, T ) × Ω, Q ω := (0, T ) × ω, Σ := (0, T ) × ∂Ω
and n the outward unit normal vector on ∂Ω. Similarly, if the domain is denoted by Ω, we define Q := (0, T )× Ω, Σ := (0, T )×∂ Ω and ñ as the outward unit normal vector on ∂ Ω. Moreover, we denote by ω ⊂⊂ Ω if ω is compactly included in Ω. Finally, in order to shorten the notation we omit the "dt", "dx"

and "dz" when we are integrating and the integrated variables can be easily deduced by looking at the integration domain.

Now, we may state the main result of this paper :

Theorem 4.6. Let Ω be a pseudo-cylinder and ω ⊂ Ω be a subdomain. Then, there is

C > 0 such that if T > 0, A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), and
y 0 ∈ L 2 (Ω), there is a control v ∈ L 2 (Q ω )
such that the solution of the system :

         y t -∆y + A • ∇y + ay = v1 ω in Q, y = 0 on Σ, y(0, •) = y 0 on Ω,
satisfies y(T, •) = 0, and such that the control satisfies the estimate :

v L 2 (Qω) ≤ Ce CK(T,a,A) y 0 L 2 (Ω) ,
for :

K(T, a, A) := 1 + T -1 + T a L ∞ (Q) + a 2/3 L ∞ (Q) + (1 + T ) A 2 (L ∞ (Q)) d+1 .
Remark 4.7. Since the Laplacian is invariant under rotations (and translations), it suffices to prove Theorem 4.6 when Ω is canonically oriented. Moreover, we remark that if we prove Theorem 4.6 for some control domain ω, the conclusions of Theorem 4.6 are true for all ω ⊃ ω, as it suffices to consider v = ṽ1 ω, for ṽ the control supported in ω.

Using Theorem 4.6 we can get positive results about the controllability of the semi-linear heat equation when the non-linearity depends only on y (see Comment 6 on Section 4.3.3

for a remark about non-linearities depending on ∇y) :

         y t -∆y + f (y) = v1 ω in Q, y = 0 on Σ, y(0, •) = y 0 on Ω. (4.6)
Indeed, under weak non-linearities we have controllability to trajectories and approximate controllability :

Corollary 4.8.
Let Ω be a pseudo-cylinder, ω ⊂ Ω be a subdomain, T > 0 and y * ∈ C([0, T ]; L 2 (Ω)) be a solution of (4.6) corresponding to some data (y 0 ) * ∈ L 2 (Ω) and

v * ∈ L ∞ (Q ω ).
Assume that f : R → R is locally Lipschitz and satisfies almost everywhere in R :

|f (s)| ≤ C(1 + |s| 1+4/d ), (4.7)
and :

lim |s|→∞ f (s) |s| log 3/2 (1 + |s|) = 0. (4.8)
Then, for all y 0 ∈ L 2 (Ω) there is v ∈ L ∞ (Q ω ) such that the solution of (4.6) belongs to

C([0, T ]; L 2 (Ω)) and satisfies y(T, •) = y * (T, •).
Corollary 4.9.

Let Ω be a pseudo-cylinder, ω ⊂ Ω be a domain, T > 0 and let us assume that (4.6) has a solution y * ∈ C([0, T ]; L 2 (Ω)) corresponding to some initial state in (y 0 ) * ∈ L 2 (Ω) and to some control function v * ∈ L ∞ (Q ω ). Assume that f is locally Lipschitz and satisfies (4.7) almost everywhere in R and (4.8). Then, for all y 1 ∈ L 2 (Ω)

and all ε > 0 there is v ∈ L ∞ (Q ω ) such that (4.6) has a solution in C([0, T ]; L 2 (Ω))
satisfying :

y(T, •) -y 1 ≤ ε.
In order to see that Theorem 4.6 implies Corollaries 4.8 and 4.9 it suffices to follow step by step the proof given in [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] for the case in which Ω is a C 2 domain. Indeed, in order to see that results on Sobolev spaces used in [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] remain true when Ω is just Lipschitz, the reader can consult for instance [START_REF] Adams | Sobolev spaces[END_REF], [START_REF] Boyer | Mathematical tool for the study of the incompressible Navier-Stokes equations and related models[END_REF] and [START_REF] Simon | Compact sets in the space L p (O, T ; B)[END_REF]. In addition, we can find in [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] some facts about controllability of the semi-linear heat equation which remain true in pseudocylinders, like that (4.6) is not approximately controllable for f (s) = s 0 log p (1 + σ)dσ, p > 2 and ω ⊂⊂ Ω.

Remark 4.10. The results presented in this paper differ from [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF] because of the following reasons :

• There are some pseudo-cylinders which are not locally star-shaped (see Appendix 4.B) ; thus, this is the first time that the null controllability of the heat equation is proved in these domains. In addition, there are locally star-shaped domains which are not pseudo-cylinders (for example, a disk or a pentagon). Of course, there also are pseudo-cylinders which are locally star-shaped (for instance, the rectangle).

• The coefficient of the lower order terms in Theorem 4.6 can depend on t and x, whereas in [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF] the coefficients only depend on t.

• Thanks to the fact that we can establish controllability results for the heat equation with coefficients that depend on t and x, we can prove controllability results for the semi-linear heat equation, which is an open problem of [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF].

• In [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF] the control is localized in an arbitrary measurable subset of strictly positive measure, whereas in this paper the control is localized in open subsets.

In this document we consider the following weights defined in Q :

α η := e λ 2m+1 2m k -e λ(k+η) t m (T -t) m , ξ η := e λ(k+η) t m (T -t) m , (4.9)
for m ≥ 1 a real constant, η some auxiliary C 2 Ω function (which, unlike it is usually done in the literature, may take negative values) and k > 2(m+1) η L ∞ (Ω) a real constant, all of them to be fixed later. In (4.9) the subindex η might be omitted if it is clear.

Moreover, if we denote some function by η, we use the notation α and ξ to refer to α η and ξ η, respectively. We recall that we have the following estimates for the weights given in (4.9), for some C > 0 that is universal (independent of any variable like Ω, η, m, k, s, λ, ...) :

|ξ t | + |α t | ≤ CmT ξ (m+1)/m , |α tt | ≤ C(mξ (m+1)/m + m(m + 1)T 2 ξ (m+2)/m ). (4.10)
Weights similar to (4.9) were first introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF]. However, our version is more similar to the one presented in [START_REF] Fernández-Cara | Global carleman inequalities for parabolic systems and applications to controllability[END_REF], though we have done slight modifications to also consider the situations in which η is not a positive function.

Remark 4.11. We need k > 2(m + 1) η L ∞ (Ω) to assure that the numerator of α is strictly positive, so that we can integrate by parts in the time variable with a null boundary term when proving the Carleman inequality. Moreover, we also need it to prove (4.10).

Remark 4.12. These weights, stated as in (4.9), have the advantage that can be compared easily when using two different auxiliary functions. In particular, if we have η 1 and η 2 defined in a canonically oriented pseudo-cylinder, for any

m ≥ 1, k > 2(m + 1) max{ η 1 L ∞ (Ω) , η 2 L ∞ (Ω) }, t ∈ (0, T ) and (x, z)
∈ Ω, we find :

ξ η 1 (t, x, z) ≤ ξ η 2 (t, x, z) if and only if η 1 (x, z) ≤ η 2 (x, z),
and :

-α η 1 (t, x, z) ≤ -α η 2 (t, x, z) if and only if η 1 (x, z) ≤ η 2 (x, z).
Let us explain briefly how to solve the two problems presented at the beginning of this section on pseudo-cylinders :

• Concerning the construction of the auxiliary function, which is the main difficulty, we consider different approaches depending on the control domain. If ω ⊂ Ω we use an approach consisting of expressing η as a sum of two different variables, whereas if ω ⊂ Ω \ C, we "transmit" the estimate from one control domain to another with a second auxiliary function. This is done in Section 4.3.

• As for dealing with the fact that D(Ω) ⊂ H 2 (Ω), we just have to approximate Ω by C 2 domains which are compatible with η (once η is constructed). This is explained in Section 4.2 for Lipschitz domains.

The rest of the paper is organised as follows : in Section 4.2 we explain how to deal with Ω being just Lipschitz, in Section 4.3 we get the Carleman inequalities, in Appendix 4.A we prove the compatibility of the proposed auxiliary function with the domain and in Appendix 4.B we show that there are pseudo-cylinders which are not locally star-shaped.

A Carleman inequality for some Lipschitz domains

Let Ω be a Lipschitz domain. We define :

W Ω := L 2 (0, T ; H 1 0 (Ω)) ∩ H 1 (0, T ; H -1 (Ω)),
omitting the subindex Ω if it is clear. In addition, we define :

Definition 4.13.
Let Ω be a Lipschitz domain, let u 0 ∈ L 2 (Ω) and let g ∈ L 2 (Q). Then, we denote by : S(Ω, u 0 , g)(t, x), the only solution in W of :

         u t -∆u = g in Q, u = 0 on Σ, u(0, •) = u 0 on Ω. (4.11)
Finally, we define the convergence of domains in the following way :

Definition 4.14. Let ε 0 > 0, Ω be a domain and (Ω ε ) ε∈(0,ε 0 ) be domains such that Ω ε ⊂ Ω for all ε ∈ (0, ε 0 ). We say that Ω ε → d Ω if :

lim ε→0 + sup Ω\Ωε d(•, ∂Ω) = 0. (4.12) Remark 4.15. If Ω ε → d Ω, then Ω ε converges to Ω in measure ; that is, 1 Ω\Ωε L 1 (R d ) → 0,
because of the Dominated Convergence Theorem.

We have the following technical result :

Lemma 4.16.
Let Ω be a Lipschitz domain, Ω ε ⊂ Ω be some Lipschitz domains such that :

Ω ε → d Ω, u 0 ∈ L 2 (Ω) and g ∈ L 2 (Q). Then, S Ω ε , u 0 1 Ωε , g1 Ωε 1 Ωε → L 2 (Q) S(Ω, u 0 , g). (4.13)
As far as we know, the proof of Lemma 4.16 is not available in the literature, so we give a proof, whose originality we do not claim. For the proof of Lemma 4.16 we need the following interpolation result, proved for instance in [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] and [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systemes distribués[END_REF] :

Lemma 4.17. Let Ω be a Lipschitz domain. Then,

1. The injection W ⊂ L 2 (Q) is compact. 2. The injection W ⊂ C([0, T ]; L 2 (Ω)) is continuous.
3. For all u, w ∈ W, we have u(t, •), w(t, •) L 2 (Ω) ∈ W 1,1 (0, T ) and :

∂ t u(t, •), w(t, •) L 2 (Ω) = ∂ t u(t, •), w(t, •) H -1 (Ω)×H 1 0 (Ω) + ∂ t w(t, •), u(t, •) H -1 (Ω)×H 1 0 (Ω) .
Proof. [Proof of Lemma 4.16] In order to prove Lemma 4.16 we recall that for any Lipschitz domain Ω, any ũ0 ∈ L 2 ( Ω) and any g ∈ L 2 (Q), S( Ω, ũ0 , g) is characterized as the only element in W Ω such that :

   T 0 ∂ t S( Ω, ũ0 , g), φ H -1 ( Ω),H 1 0 ( Ω) + Q ∇S( Ω, ũ0 , g) • ∇φ = Q gφ ∀φ ∈ D( Q), S( Ω, ũ0 , g)(0, •) = ũ0 . (4.14)
We define :

S 0 := S(Ω, u 0 , g), S ε := S Ω ε , u 0 1 Ωε , g1 Ωε 1 Ωε and Q ε := (0, T ) × Ω ε ,
for Ω, u 0 , g and Ω ε as stated in Lemma 4.16. We recall that, since w → w1 Ωε is an isometric embedding from W Ωε to W Ω , we have S ε ∈ W Ω and the equality :

S ε W Ω = S ε W Ωε . (4.15)
Using the density of D(Q ε ) in W Ωε we find the equality :

S ε (T, •) 2 L 2 (Ωε) + 2 Qε |∇S ε | 2 = 2 Q gS ε + u 0 2 L 2 (Ωε) .
In particular, if we use the Poincaré inequality on Ω and Cauchy-Schwarz, we obtain for a constant C(diam(Ω)) > 0 :

S ε L 2 (0,T ;H 1 0 (Ωε)) ≤ C g L 2 (Qε) + u 0 L 2 (Ωε) ≤ C g L 2 (Q) + u 0 L 2 (Ω) .
In addition, looking at (4.11) 1 we obtain that ∂ t S ε ∈ L 2 (0, T ; H -1 (Ω ε )) and the existence of a constant C(diam(Ω)) > 0 such that :

S ε W Ωε ≤ C g L 2 (Q) + u 0 L 2 (Q) .
Consequently, S ε is uniformly bounded in W Ω due to (4.15) ; thus, S ε has at least a weakly convergent sequence in W Ω . Moreover, thanks to Item 1 of Lemma 4.17, we have that the sequence is convergent in L 2 (Q).

In order to end the proof, it suffices to see that for all u ∈ W Ω such that there is a sequence S ε i satisfying that S ε i u in W Ω (with ε i → 0), then u = S 0 . To get the equality, we prove that u satisfies (4.14) for ( Ω, ũ0 , g) = (Ω, u 0 , g) :

• In order to prove the variational condition of (4.14), we pick φ ∈ D(Q). We remark that d(supp x φ, ∂Ω) > 0, for :

supp x φ := {x : ∃t ∈ [0, T ] : φ(t, x) = 0}.
In particular, if i is sufficiently large, we have that supp x φ ⊂⊂ Ω ε i . Consequently, by

(4.14) 1 (for ( Ω, ũ0 , g) = (Ω ε i , u 0 1 Ωε i , g1 Ωε i ))
and by taking into account the support of φ, we obtain for i large enough the equality :

T 0 ∂ t S ε i , φ H -1 (Ω),H 1 0 (Ω) + Q ∇S ε i • ∇φ = Q gφ.
Thus, if we take the weak limit in W Ω , we get that :

T 0 ∂ t u, φ H -1 (Ω),H 1 0 (Ω) + Q ∇u • ∇φ = Q gφ.
Since φ is arbitrary, u satisfies (4.14) 1 .

• As for the initial condition of (4.14), we recall that because of Item 2 of Lemma 4.17,

w ∈ W → w(0, •) ∈ L 2 (Ω),
is a continuous operator. Therefore, since the weak limit is preserved by linear continuous operators between Hilbert spaces, we have that

S ε i (0, •) u(0, •) in L 2 (Ω). Moreover, S ε i (0, •) = u 0 1 Ωε i → u 0 in L 2 (Ω) by Remark 4.15. Consequently,
from the uniqueness of the weak limit we obtain that u(0, •) = u 0 .

In order to prove a Carleman inequality for Ω, the approximation must be made not only by C 2 domains, but also these domains must be compatible with the function η in the following way :

Definition 4.18.
Let Ω be a domain, let ω ⊂ Ω be a subdomain and let η be a real valued function satisfying :

η ∈ C 2 Ω , inf Ω\ω |∇η| > 0. (4.16)
We say that Ω is a compatible open set with respect to Ω, ω and η if we have that :

Ω is a Lipschitz open set such that ω ⊂ Ω ⊂ Ω, (4.17)
D Ω(∆) ⊂ H 2 ( Ω) continuously, (4.18)
∂ ñη ≤ 0 on ∂ Ω. ( 4
( Q), u T ∈ L 2 ( Ω), k > 2(m + 1) η L ∞ (Ω)
, and for the weights defined in (4.9), we have that :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 + s -1 Q e -2sα ξ -1 (|∆u| 2 + |u t | 2 ) ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 , (4.20)
for any λ ≥ C, s ≥ C(T m + T 2m ) and for u the solution of :

         -u t -∆u = g in Q, u = 0 on Σ, u(T, •) = u T on Ω.
Proof. Lemma 4.22 is a direct consequence of the proof of the usual Carleman estimate, which can be found for instance in [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Global carleman inequalities for parabolic systems and applications to controllability[END_REF]. We remark that since inf Ω\ω |∇η| > 0, there exists a domain ω 0 ⊂⊂ ω which satisfies inf Ω\ω 0 |∇η| > 0. The only additional difficulty is to prove that the constant C is independent of Ω. Indeed, the proofs presented in [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Global carleman inequalities for parabolic systems and applications to controllability[END_REF] show that C depends continuously only on :

• η C 2 Ω (which can be bounded by η C 2 (Ω) ),

• m (see (4.10)),

• inf Ω\ω 0 |∇η| (which can be inferiorly bounded by inf Ω\ω 0 |∇η| > 0),

• ω and ω 0 . Now we are ready to prove a Carleman inequality for domains that can be approximated properly by regular domains :

Lemma 4.23.
Let Ω be a Lipschitz domain, ω ⊂ Ω be a subdomain, η be a function satisfying (4.16) and m ≥ 1. Let us suppose that there are ε 0 > 0 and (Ω ε ) ε∈(0,ε 0 ) satisfying (4.17)-(4. [START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF]) and such that Ω ε → d Ω. Then, there is C > 0 such that for all T > 0,

g ∈ L 2 (Q), u T ∈ L 2 (Ω), k > 2(m + 1) η L ∞ (Ω)
, and for the weights defined in (4.9), we have that :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 + s -1 Q e -2sα ξ -1 (|∆u| 2 + |u t | 2 ) ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 , (4.21)
for any λ ≥ C, s ≥ C(T m + T 2m ) and for u the solution of :

         -u t -∆u = g in Q, u = 0 on Σ, u(T, •) = u T on Ω. (4.22)
In order to prove Lemma 4.23 we first get the estimate on the zero-order term, and then get the higher order terms as in [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions : the linear case[END_REF] and [START_REF] Fernández-Cara | Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability[END_REF].

Proof. By Lemma 4.22 we have that for all m ≥ 1 there is C > 0 such that for all T > 0,

u T ∈ L 2 (Ω), g ∈ L 2 (Q)
, and ε ∈ (0, ε 0 ) :

s 3 λ 4 Q e -2sα ξ 3 |u ε 1 Ωε | 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u ε | 2 + Q e -2sα |g1 Ωε | 2 , (4.23)
for any λ ≥ C, s ≥ C(T m + T 2m ) and for :

u ε (t, x) := S Ω ε , u T 1 Ωε , g(T -•, •)1 Ωε (T -t, x).
Considering Lemma 4.16, we can take the limit in (4.23) and get the estimate :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 . (4.24)
Next, multiplying (4.22) 1 by sλ 2 e -2sα ξu we get the term of ∇u. Indeed, integrating by parts we get that : we get for any λ ≥ C and s ≥ C(T m + T 2m ) that :

sλ 2 Q e -2sα ξ|∇u| 2 = sλ 2 Q e -2sα gu -sλ 3 Q e -2sα ξ∇η • ∇uu + 2s 2 λ 3 Q e -2sα ξ 2 ∇η • ∇uu - sλ 2 2 Q e -2sα ξ t |u| 2 + s 2 λ 2 Q e -
sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 . (4.26)
To continue with, we assume that u T ∈ H 1 0 (Ω). Multiplying (4.22) 1 by -s -1 e -2sα ξ -1 u t we obtain that :

s -1 Q e -2sα ξ -1 |u t | 2 = -s -1 Q e -2sα ξ -1 gu t - 1 2 s -1 Q (e -2sα ξ -1 ) t |∇u| 2 + s -1 Q ∇(e -2sα ξ -1 ) • ∇uu t . (4.27)
We have used that the solutions of the heat equation with initial value in H 1 0 (Ω) and source term in L 2 (Q) belongs to the closure of H 1 (0, T ; H 1 0 (Ω)) ∩ L 2 (0, T ; D(∆)) with norm H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; D(∆)) when Ω is Lipschitz. Using weighted Cauchy-Schwarz inequalities, (4.10) and (4.26) we find for any λ ≥ C and s ≥ C(T m + T 2m ) that :

s -1 Q e -2sα ξ -1 |u t | 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 .
(4.28)

Finally, considering that -∆u = u t + g we obtain from (4.28) that : Finally, we remark that we can extend the result presented in Lemma 4.23 to situations in which there is a source term in L 2 (0, T ; H -1 (Ω)) reproducing step by step the proof given in [START_REF] Yu | Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications[END_REF] (the reader can consult for instance [START_REF] Adams | Sobolev spaces[END_REF] and [START_REF] Boyer | Mathematical tool for the study of the incompressible Navier-Stokes equations and related models[END_REF] to see that the results of the Sobolev spaces used in [START_REF] Yu | Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications[END_REF] remain true when Ω is Lipschitz) : Proposition 4.24. Let Ω be a Lipschitz domain, ω ⊂ Ω be a subdomain, η be a function satisfying (4.16) and m ≥ 1. Let us suppose that there are ε 0 > 0 and (Ω ε ) ε∈(0,ε 0 ) satisfying (4.17)-(4. [START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF]) and such that Ω ε → d Ω. Then, there is C > 0 such that for all T > 0,

s -1 Q e -2sα ξ -1 |∆u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -
g ∈ L 2 (Q), G ∈ (L 2 (Q)) d , u T ∈ L 2 (Ω), k > 2(m + 1) η L ∞ (Ω)
, and for the weights defined in (4.9), we have that :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 , (4.30)
for any λ ≥ C, s ≥ C(T m + T 2m ) and for u the solution of :

         -u t -∆u = g + ∇ • G in Q, u = 0 on Σ, u(T, •) = u T on Ω. ( 4

The case ω ⊂ C

Let Ω be a canonically oriented pseudo-cylinder (see Definition 4.1) and let ω ⊂ C. By taking a smaller control domain if necessary, we can suppose that :

ω = B(x, r) × (z -r, z + r) ⊂⊂ C, (4.32)
for some r > 0 and some (x, z) ∈ Ω such that :

z < inf B H. (4.33)
We construct an auxiliary function η satisfying (4.16). Considering that B is C 2 and (4.1), there is η B ∈ C 2 B satisfying : So, we define in Ω : Let Ω be a canonically oriented pseudo-cylinder, ω be given by (4.32)

η B = 0 on ∂B, η B > 0 in B, inf B\B(x,r) |∇η B | > 0. ( 4
η(x, z) := η B (x) -c(z -z) 2 , ( 4
and m ≥ 1. Then, there exists c 0 (Ω, ω, η B ) > 0 such that if η is defined as in (4.36) with c ≥ c 0 , there is C > 0 such that for all

T > 0, A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), g ∈ L 2 (Q), G ∈ (L 2 (Q)) d+1 , u T ∈ L 2 (Ω), k > 2(m + 1) η L ∞ (Ω)
, and for the weights defined in (4.9), we have that :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 , (4.37) for any λ ≥ C, s ≥ C T m + T 2m (1 + a 2/3 L ∞ (Q) + A 2 (L ∞ (Q)) d+1 ) , (4.38)
and for u the solution of : Remark 4.30. It is a classical result (see, for instance, [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations : recent progress and open questions[END_REF], [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systemes distribués[END_REF], [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]) that Proposition 4.28 (taking m = 1) implies that Theorem 4.6 is true for all canonically oriented pseudo-cylinders and all ω given by (4.32). By Remark 4.7 this implies Theorem 4.6 for all pseudo-cylinders and all w ⊂ C.

         -u t -∆u -∇ • (Au) + au = g + ∇ • G in Q, u = 0 on Σ, u(T, •) = u T on Ω. ( 4
s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + |au| 2 + s 2 λ 2 Q e -2sα ξ 2 (|G| 2 + |Au| 2 ) ≤ C s 3 λ 4 Qω e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 + a 2 L ∞ (Q) T 6m Q e -2sα ξ 3 |u| 2 + s 2 λ 2 A 2 L ∞ (Q) T 2m Q e -2sα

The case ω ⊂ Ω \ C

Let Ω be a canonically oriented pseudo-cylinder and let ω ⊂ Ω \ C. For this case we suppose, by making again the control domain smaller if necessary, that :

ω = B(x, r) × (z -r, z + r) ⊂⊂ Ω \ C, (4.42)
for some (x, z) ∈ Ω \ C such that : and some :

z > inf B H,
r ∈ 0, inf B H 4 , (4.43)
such that : Ω := B(x, 3r) × (0, z + 3r) ⊂ Ω. (4.44) In this case we cannot consider an auxiliary function similar to (4.36) because now we have that T ∩ {(x, z) : z = z} = ∅. However, thanks to Proposition 4.28 our problem can be seen as a problem of transmitting the estimate from one control domain to another.

Indeed, we define :

ω * := B(x, r) × (z -r, z + r), (4.45)
for x := x and z := 3r. Thus, considering also (4.43) we have that : We fix m ≥ 1 and define η as in (4.36) with c ≥ c 0 (Ω, ω * , η B ). We obtain by Proposition 4.28 a constant C > 0 such that for all

ω * = B(x, r) × (2r, 4r) ⊂⊂ Ω ∩ C.
T > 0, A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), g ∈ L 2 (Q), G ∈ (L 2 (Q)) d+1 , u T ∈ L 2 (Ω)
, and k > 2(m + 1) η L ∞ (Ω) we have that :

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 ≤ C s 3 λ 4 Q ω * e -2sα ξ 3 |u| 2 + Q e -2sα |g| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 , (4.46)
for λ ≥ C, s satisfying (4.38) and u the solution of (4.39).

Next, we estimate the weighted L 2 (Q ω * )-norm by a weighted L 2 (Q ω )-norm. To do this, we consider a function whose maximum is in ω but which is large in ω * . Specifically, we consider :

η(x, z) := -η 1 (|x -x|) -η2 (z) + 2 sup Ω |η| , (4.47)
for η1 a positive C ∞ function supported in (r/2, +∞) such that :

         inf (r,+∞) η 1 > 0, η1 < sup Ω |η| /2 in (r/2, r), η1 > 3 sup Ω |η| in (2r, 3r), (4.48)
and for η2 a positive C ∞ function such that :

               sup (0,z-r) η 2 < 0, inf (z+r,z+3r) η 2 > 0, η2 < sup Ω |η| /2 in (2r, 4r), η2 > 3 sup Ω |η| in (0, r) ∪ (z + 2r, z + 3r). (4.49)
We remark that η ∈ C 2 Ω and satisfies : With the function η we can define the weights α and ξ as in (4.9). Let us fix :

∂ n η < 0 on ∂ Ω, inf Ω\ω |∇η| > 0. ( 4
k > 2(m + 1) max{ η L ∞ (Ω) , η L ∞ (Ω) } = 2(m + 1) η L ∞ (Ω) . (4.52)
Thanks to (4.51) and Remark 4.12 we have that :

   ξ ≤ ξ; -α ≤ -α in Q ω * , ξ ≤ ξ; -α ≤ -α in (0, T ) × Ω \ B(x, 2r) × (r, z + 2r) . (4.53)
Next, we consider that u is the solution of (4.39) and that χ(x, z) is a positive regular cut-off function such that :

supp(χ) ⊂⊂ Ω, χ = 1 in B(x, 2r) × (r, z + 2r). (4.54)
Then, v := χu is the solution of :

         -v t -∆v -∇ • (Av) + av = F (∇χ, u) + χg + ∇ • (χG) in Q, v = 0 on Σ, v(T, •) = χu T on Ω, (4.55)
for :

F (∇χ, u) := -2∇χ • ∇u -∆χu -(∇χ • A)u -∇χ • G. ( 4
(η, m) > 0 such that if A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), g ∈ L 2 (Q), G ∈ (L 2 (Q)) d+1 , u T ∈ L 2 (Ω), λ ≥ C and s ≥ C(T m + T 2m ) : s 3 λ 4 Q e -2s α ξ 3 |v| 2 + sλ 2 Q e -2s α ξ|∇v| 2 ≤ C s 3 λ 4 Qω e -2s α ξ 3 |v| 2 + Q e -2s α|F (∇χ, u) + χg -av| 2 + s 2 λ 2 Q e -2s α ξ 2 (|χG -Av| 2 ) . (4.57)
Considering that v := χu and (4.57), after some easy absorptions and bounds we obtain for λ ≥ C and s satisfying (4.38) that :

s 3 λ 4 Q e -2s α ξ 3 |χu| 2 + sλ 2 Q e -2s α ξ|∇(χu)| 2 ≤ C s 3 λ 4 Qω e -2s α ξ 3 |χu| 2 + Q e -2s α(|χg| 2 + |F (∇χ, u)| 2 ) + s 2 λ 2 Q e -2s α ξ 2 |χG| 2 . (4.58)
To continue with, as χ = 1 in ω and ω * , χ ∈ D Ω , and we have (4.53) 1 , we may combine (4.46) and (4.58) and get that : Let Ω be a canonically oriented pseudo-cylinder, ω be given by (4.42), Ω be given by (4.44), η be given by (4.36) (with x := x and z := 3r), η be given by (4.47), χ be given by (4.54) and m ≥ 1. Then, there exists a constant C > 0 such that if T > 0,

s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 + s 3 λ 4 Q e -2s α ξ 3 |χu| 2 + sλ 2 Q e -2s α ξ|∇(χu)| 2 ≤ C s 3 λ 4 Qω e -2s α ξ 3 |u| 2 + Q e -2sα |g| 2 + Q e -2s α(|χg| 2 + |F (∇χ, u)| 2 ) + s 2 λ 2 Q e -2sα ξ 2 |G| 2 + s 2 λ 2 Q e -2s α( ξ)
s 3 λ 4 Q e -2sα ξ 3 |u| 2 + sλ 2 Q e -2sα ξ|∇u| 2 + s 3 λ 4 Q e -2s α ξ 3 |χu| 2 + sλ 2 Q e -2s α ξ|∇(χu)| 2 ≤ C s 3 λ 4 Qω e -2s α ξ 3 |u| 2 + Q e -2sα |g| 2 + Q e -2s α|χg| 2 + s 2 λ 2 Q e -2sα ξ 2 |G| 2 + s 2 λ 2 Q e -2s α( ξ)
A ∈ (L ∞ (Q)) d+1 , a ∈ L ∞ (Q), g ∈ L 2 (Q), G ∈ (L 2 (Q)) d+1 , u T ∈ L 2 (Ω), k satisfies (4.52),
λ ≥ C and s satisfies (4.38), we have (4.60) for the weights defined in (4.9) and u the solution of (4.39).

Remark 4.33. It is a classical result (see, for instance, [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations : recent progress and open questions[END_REF], [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systemes distribués[END_REF], [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]) that Pro- • On B × {0} we have that ∂ n η = -∂ z η = -2cz < 0.

• On L we have by (4.35) that

∂ n η = ∂ n B η B < 0.
• On T we find that :

n (x, H(x)) = 1 1 + |∇H(x)| 2 (-∇H(x), 1).
Thus, on T the normal derivative is given by :

∂ n η (x, H(x)) = -∇H(x) • ∇η B (x) -2c(H(x) -z) 1 + |∇H(x)| 2 .
This implies that the condition :

c ≥ ∇H L ∞ (B) ∇η B C(B) 2(inf B H(x) -z) , ( 4

4.A Proof of Proposition 4.29

We recall that Ω is a canonically oriented pseudo-cylinder, ω is given by (4.32) and η is given by (4.36). Similarly, in this proof we use c and C to denote respectively small and large strictly positive constants that may be different each time and only depend on Ω, ω and η B , and ε is taken as a small strictly positive parameter.

The structure of the proof is the following : first we present an auxiliary result from differential geometry, second we introduce the regularized measure, third we approximate H (see Definition 4.1) by smaller regular functions, fourth we introduce some domains Ω ε and prove that they satisfy (4.40), fifth we prove that the domains Ω ε satisfy (4.17) and (4.18) for all ε <<< ε 0 (Ω, ω), and finally we assume that c >>> c 0 (Ω, ω, η B ) and ε <<< ε 0 (Ω, ω, η B , c) and prove (4.19). We recall that once the open sets Ω ε satisfy (4.17)-(4.19), Remark 4.20 implies that they are connected.

In the proof we state that several things happen if c >>> c 0 (Ω, ω, η B ) and ε <<< ε 0 (Ω, ω, η B , c). We do not calculate explicitly the value of c 0 (Ω, ω, η B ) and of ε 0 (Ω, ω, η B , c)

the same way we do not calculate explicitly the values of C or c. Indeed, note that the number of times that we require such a bound is finite (as c 0 and ε 0 do not depend on the specific (x, z) but at most on a finite division of subcases), so it suffices to consider the maximum of all the c 0 and the minimum of all the ε 0 .

Step 1 : an auxiliary result. In order to construct these domains, we need the following basic result from differential geometry, whose proof can be found for instance in [START_REF] Csikós | Differential geometry[END_REF] :

Lemma 4.34. Let Ω be a domain and let Φ : Ω → R be a C 2 function such that Φ ≤ 0 on ∂ Ω. Then, for all θ > 0 satisfying Φ -1 (θ, +∞) = ∅ and :

Φ -1 ({θ}) ∩ (∇Φ) -1 ({0}) = ∅, (4.62)
we have that Φ -1 (θ, +∞) is a C 2 open set. Moreover, the boundary of Φ -1 (θ, +∞) is given by Φ -1 ({θ}). Finally, the outward normal unit vector, which we denote by n θ , satisfies on Φ -1 ({θ}) :

n θ (x) = - ∇Φ(x) |∇Φ(x)| . ( 4.63)
A first approach can be to use Lemma 4.34 on the distance function from (x, z) to ∂Ω.

Nonetheless, this is not possible because that function is not differentiable. On the one hand, even in regular domains d(•, ∂Ω) may not be C 1 in the interior (for instance, in the disk the distance to the boundary is not differentiable at its center). On the other hand, near T the function d(•, ∂Ω) is not C 1 when T is just Lipschitz. However, it is possible to construct a function which behaves similarly to the distance. In particular, to face the first problem we regularize the distance with the tools introduced in [START_REF] Lieberman | Regularized distance and its applications[END_REF] and to face the second one we approximate H by regular functions.

Step 2 : the regularized measure. We define the function :

d : x ∈ B → d(x, ∂B),
and the sets :

B ε := d -1 (ε, +∞). (4.64)
Since B is C 2 , we have for some ε(Ω) > 0 the following properties :

• d is a W 2,∞ (d -1 ([0, ε])) function such that |∇d| = 1 in d -1 ([0, ε]).
• For all x ∈ d -1 ([0, ε]) there is a unique point P (x) such that :

d (x, P (x)) = d(x). (4.65)
Moreover, the function

x → P (x) is continuous in d -1 ([0, ε]).
The proof of these two assertions can be found for instance in [START_REF] Boyer | Mathematical tool for the study of the incompressible Navier-Stokes equations and related models[END_REF]Section III.3]. Also, it can be proved easily that for all ε <<< ε 0 (Ω) the set B ε is a domain satisfying :

∂B ε = d -1 ({ε}).
In order to regularize d in the interior we use the function introduced in [START_REF] Lieberman | Regularized distance and its applications[END_REF], whose properties can be consulted for instance in [START_REF] Boyer | Mathematical tool for the study of the incompressible Navier-Stokes equations and related models[END_REF]Section III.3.3]. Indeed, we consider :

ρ : x ∈ B → R,
which is defined as the only fixed point of τ → G(x, τ ), for :

G(x, τ ) := B(0,1) d x + τ 2 x ς(x)dx,
for ς a mollifier in R d ; that is, a positive function belonging to D(B(0, 1)) such that :

B(0,1) ς(x)dx = 1.
We have that ρ satisfies the following properties, which are proved for instance in [12, Section III.3.3] :

ρ ∈ W 2,∞ (B) ∩ C ∞ (B) , (4.66)
ρ ≥ 0 in B and ρ = 0 if and only if x ∈ ∂B, (4.67)

1 2 ≤ d(x) ρ(x) ≤ 3 2 , ∀x ∈ B, (4.68)
∇ρ(x) = -n B (x), ∀x ∈ ∂B. (4.69)
Step 3 : approximating H by smaller regular functions. We consider again ς a mollifier in R d , and we define ς ε := ε -d ς • ε , which is supported in B(0, ε) and satisfies :

B(0,ε) ς ε (x)dx = 1. (4.70)
In addition, we define in B ε the function

H ε := (1 - √ ε) H * ς ε , which is well-defined because x ∈ B ε implies that B(x, ε) ⊂ B (see (4.64)). Furthermore, if x ∈ B ε : H ε (x) = 1 - √ ε B(0,ε) H(x -x)ς ε (x)dx ≤ 1 - √ ε H(x) + ε H W 1,∞ (B) B(0,ε) ς ε (x)dx = H(x) - √ εH(x) + ε H W 1,∞ (B) -ε 3/2 H W 1,∞ (B) . (4.71)
This follows from (4.70) and from the fact that for all x ∈ B(0, ε) we have that :

H(x -x) = H(x) - 1 0 ∇H(x -τ x) • xdτ ≤ H(x) + |x| H W 1,∞ (B) ≤ H(x) + ε H W 1,∞ (B) .
So, since inf B H > 0, from (4.71) we obtain for all ε <<< ε 0 (Ω) that :

H ε ≤ H in B ε . (4.72)
Remark 4.35. We have included a term 1 -√ ε in the definition of H ε to have (4.72).

Similarly, we have for all x ∈ B ε the lower bound :

H ε (x) ≥ 1 - √ ε H(x) -ε H W 1,∞ (B) . (4.73)
Indeed, this follows from (4.70) and from the fact that for all x ∈ B(0, ε) we have that :

H(x -x) = H(x) - 1 0 ∇H(x -τ x) • xdτ ≥ H(x) -|x| H W 1,∞ (B) ≥ H(x) -ε H W 1,∞ (B) .
One important consequence of (4.71) and (4.73) is the convergence :

inf Bε H ε → inf B H. (4.74)
A last important property of the functions H ε is that they are uniformly Lipschitz for ε <<< ε 0 (Ω). Indeed, from Young's convolution inequality and (4.70) we obtain that :

∇H ε L ∞ (Bε) = 1 - √ ε ς ε * ∇H L ∞ (Bε) ≤ ∇H L ∞ (B) . (4.75)
Step 4 : approximating Ω by some regular domains Ω ε . We define in Ω for all ε satisfying that B 2ε = ∅ the following functions :

D ε (x, z) := χ ε (x)ρ(x)(H ε (x) -z)z, ∀(x, z) ∈ Ω, (4.76)
for χ ε a positive C ∞ (R d ) cut-off function which takes values in [0, 1], supported in B ε and such that χ ε = 1 in B 2ε (for instance 1 B (3/2)ε * ς ε/4 ).

Remark 4.36. Since χ ε = 1 in B 2ε we obtain from (4.76) that :

∇D ε (x, z) = zρ(x)∇H ε (x) + z(H ε (x) -z)∇ρ(x), ρ(x) (H ε (x) -2z) , ∀(x, z) ∈ B 2ε . (4.77)
We approximate Ω by :

Ω ε := D -1 ε ε H 2 L ∞ (B) , +∞ ,
which is well-defined if ε <<< ε 0 (Ω) (as in that case B 2ε = ∅). We first prove (4.40). For that, we fix ε <<< ε 0 (Ω), we also fix (x, z) ∈ Ω \ Ω ε and we get a bound of d((x, z), ∂Ω).

Since (x, z) ∈ Ω \ Ω ε , we have that D ε (x, z) ≤ ε H 2 L ∞ (B)
, which implies that :

min{χ ε (x), ρ(x), (H ε (x) -z), z} ≤ ε 1/4 H 2 L ∞ (B) ,
so we are in at least one of the following cases :

• If χ ε (x) ≤ ε 1/4 H 2 L ∞ (B)
, then from ε <<< ε 0 (Ω) we have that x ∈ B 2ε , which implies that :

d((x, z), ∂Ω) ≤ d(x) ≤ 2ε. (4.78) • If ρ(x) ≤ ε 1/4 H 2 L ∞ (B)
, we have by (4.68) that :

d((x, z), ∂Ω) ≤ d(x) ≤ 3 2 ε 1/4 H 2 L ∞ (B) . (4.79) • If H ε (x) -z ≤ ε 1/4 H 2 L ∞ (B)
, using (4.71) we have that :

d((x, z), ∂Ω) ≤ d((x, z), T) = H(x) -z = (H(x) -H ε (x)) + H ε (x) -z ≤ ε 1/4 H 2 L ∞ (B) + C √ ε. (4.80) • If z ≤ ε 1/4 H 2 L ∞ (B)
we have that :

d((x, z), ∂Ω) ≤ d((x, z), B × {0}) ≤ ε 1/4 H 2 L ∞ (B) . ( 4
• If (x, z) ∈ B × {0}, then D ε (x, z) = 0. • If (x, z) ∈ L, then x ∈ ∂B, so χ ε (x) = 0 (by the support of χ ε ), so D ε (x, z) = 0. • If (x, z) ∈ T, then z = H(x). If D ε (x, z) = 0 we have that x ∈ supp(χ ε ) ⊂ B ε , so H ε (x) ≤ H(x) by (4.72), so D ε (x, z) < 0 ; otherwise, D ε (x, z) = 0. Consequently, if (x, z) ∈ T we have D ε (x, z) ≤ 0.
Thus, to apply Lemma 4.34 it remains to check that :

D -1 ε ε H 2 L ∞ (Ω) ∩ (∇D ε ) -1 ({0}) = ∅. (4.82)
For that purpose we fix (x, z) ∈ Ω satisfying :

D ε (x, z) = ε H 2 L ∞ (B) , ∂ z D ε (x, z) = 0, (4.83)
and prove that ∇ x D ε (x, z) = 0.

We first remark that from (4.83) 1 we have that :

x ∈ B 2ε . (4.84) Indeed, from (4.72) we obtain that :

χ ε (x)(H ε (x) -z)z ≤ H 2 L ∞ (B)
ρ(x) = 4ε H 2 L ∞ (B) H 2 ε (x) , (4.86)
which implies together with (4.77) the equality :

∇D ε (x, z) = 2ε H 2 L ∞ (B) H ε (x) ∇H ε (x) + H 2 ε (x) 4 ∇ρ(x), 0 .
So, because of (4.74) and (4.75) the previous equality turns into :

∇D ε (x, z) = O(ε) + H 2 ε (x) 4 ∇ρ(x), 0 . ( 4
∂Ω ε = D -1 ε (ε H 2 L ∞ (B) ). ( 4
ρ(x)(H ε (x) -z)z = ε H 2 L ∞ (B) , ∀(x, z) ∈ ∂Ω ε . (4.89)
Similarly, if ε <<< ε 0 (Ω) and (x, z) ∈ ∂Ω ε , using (4.63), (4.77) and (4.84) we obtain that :

n ε (x, z) = - ∇D ε (x, z) |∇D ε (x, z)| = -zρ(x)∇H ε (x) -z(H ε (x) -z)∇ρ(x), ρ(x) (2z -H ε (x)) | -zρ(x)∇H ε (x) -z(H ε (x) -z)∇ρ(x), ρ(x) (2z -H ε (x)) | , (4.90)
where n ε denotes the outward unit normal vector on ∂Ω ε .

Step 6 : the domains Ω ε satisfy (4.19). Considering (4.90) and (4.36), we have to prove that if c >>> c 0 (Ω, ω, η B ), ε <<< ε 0 (Ω, ω, η B , c) and (x, z) ∈ ∂Ω ε , we have that : Rigorously, we fix first c ≥ c o (Ω, ω, η B ), second ε <<< ε 0 (Ω, ω, η B , c) and third (x, z) ∈ ∂Ω ε , for c 0 and ε 0 to be obtained during the proof. We prove (4.91) for those values. As stated before, we follow different (but a finite number of) approaches depending on (x, z).

-zρ(x)∇H ε (x) • ∇η B (x) -z(H ε (x) -z)∇ρ(x) • ∇η B (x) -2ρ(x)(2z -H ε (x))c(z -z) ≤ 0. ( 4
A first partition of the boundary is the following :

   ∂Ω ε,b := ∂Ω ε ∩ {(x, z) : z ∈ (0, H ε (x)/2]}, ∂Ω ε,t := ∂Ω ε ∩ {(x, z) : z ∈ (H ε (x)/2, H ε (x))}. ( 4
ε H L ∞ (B) ≤ zρ(x) ≤ 2ε H 2 L ∞ (B) inf Bε H ε . (4.93)
Thanks to (4.74), from ε <<< ε 0 (Ω) we have the inequality :

1 inf Bε H ε ≤ 2 inf B H
; so if we define :

κ 1 := H L ∞ (B) , κ 2 := 4 H 2 L ∞ (B) inf B H , (4.94)
we have that (4.93) and ε <<< ε 0 (Ω) imply that :

κ 1 ε ≤ zρ(x) ≤ κ 2 ε. ( 4
z ≥ κ 1 κ -1 2 ε. (4.96)
We first focus on the sign of :

-zρ(x)∇H ε (x) • ∇η B (x) -z(H ε (x) -z)∇ρ(x) • ∇η B (x).
From (4.95) we obtain the estimate : -

|zρ(x)∇H ε (x) • ∇η B (x)| ≤ Cε. ( 4
z(H ε (x) -z)∇ρ(x) = z(H ε (x) -z)n B (P (x)) + z(H ε (x) -z) (-n B (P (x)) -∇ρ(x)) . (4.98)
Recalling (4.65)-(4.69), from ε <<< ε 0 (Ω) we obtain the estimate :

| -n B (P (x)) -∇ρ(x)| = |∇ρ(P (x)) -∇ρ(x)| ≤ ρ W 2,∞ (B) |P (x) -x| = ρ W 2,∞ (B) d(x) ≤ 3 2 ρ W 2,∞ (B) ρ(x). (4.99)
So, combining (4.99) and (4.95) we get the bound :

|z(H ε (x) -z) (-n B (P (x)) -∇ρ(x)) • ∇η B (x)| ≤ C|zρ(x)| ≤ Cε. (4.100)
Finally, we have the equality :

z(H ε (x) -z)n B (P (x)) • ∇η B (x) = z(H ε (x) -z)n B (P (x)) • ∇η B (P (x)) + z(H ε (x) -z)n B (P (x)) • (-∇η B (P (x)) + ∇η B (x)) . (4.101)
Since η B ∈ C 2 B , arguing similarly to (4.99) we obtain the estimate : Consequently, if we combine (4.97)-(4.103), we find that :

|z(H ε (x) -z)n B (P (x)) • (-∇η B (P (x)) + ∇η B (x)) | ≤ Cε. ( 4
-zρ(x)∇H ε (x) • ∇η B (x) -z(H ε (x) -z)∇ρ(x) • ∇η B (x) ≤ Cε -c √ ε. (4.104)
Let us now analyze the term :

-2ρ(x)(2z -H ε (x))c(z -z) = 2ρ(x)(H ε (x) -2z)c(z -z).
• On the one hand, if z ≤ z (see (4.32) for the definition of z), we have that (see

(4.92) 1 ) : 2ρ(x)(H ε (x) -2z)c(z -z) ≤ 0,
which, together with (4.104) implies (4.91) since ε <<< ε 0 (Ω, ω, η B ).

• On the other hand, if z ≥ z, we get from (4.95) the inequality :

ρ(x) ≤ Cε. (4.105)
Thus, we obtain the estimate :

|2ρ(x)(H ε (x) -2z)c(z -z)| ≤ Ccε. (4.106)
Therefore, if we combine (4.104) and (4.106), we get the bound : and (4.74), we obtain from ε <<< ε 0 (Ω, ω) the inequality :

-zρ(x)∇H ε (x) • ∇η B (x) -z(H ε (x) -z)∇ρ(x) • ∇η B (x) -2ρ(x)(2z -H ε (x))c(z -z) ≤ C(1 + c)ε -c √ ε, which
-zρ(x)∇H ε (x) • ∇η B (x) -z(H ε (x) -z)∇ρ(x) • ∇η B (x) -2ρ(x)(2z -H ε (x))c(z -z) ≤ Cε + Cε 1/2 -ccε 1/2 .
Consequently, as c >>> c 0 (Ω, ω, η B ) and ε <<< ε 0 (Ω, ω, η B , c), we have (4.91).

Case 2 : if (x, z) ∈ ∂Ω ε,t . As for ∂Ω ε,t the situation is very similar. Indeed, from (4.89)

we find the bounds :

κ 1 ε ≤ (H ε (x) -z)ρ(x) ≤ κ 2 ε, (4.109)
for κ 1 and κ 2 defined in (4.94). As before, we distinguish the cases ρ(x) ≤ √ κ 2 ε and

H ε (x) -z ≤ √ κ 2 ε.
H ε (x) -z ≥ κ 1 κ -1 2 ε. (4.110)
Arguing as in the case 1.1, we get the estimate :

-z(H ε (x) -z)∇ρ(x) • ∇η B (x) ≤ Cε -cε 1/2 . ( 4

.111)

In order to continue, as before, we make a distinction depending on how close z is to

H ε (x) :
• We start with the subcase :

z ≥ max z inf B H , 2 3 H ε (x). (4.112)
We obtain from (4.33) and ε <<< ε 0 (Ω, ω) the inequality :

H ε (x) ≥ 4 z inf B H H(x). ( 4
-2ρ(x)(2z -H ε (x))c(z -z) ≤ -ccρ(x). (4.115)
Moreover, we clearly have the estimate : - Thus, we find the estimate :

-zρ(x)∇H ε (x) • ∇η B (x) ≤ Cρ(x). ( 4
zρ(x)∇H ε (x) • ∇η B (x) -2ρ(x)(2z -H ε (x))c(z -z) ≤ 0. ( 4
-zρ(x)∇H ε (x) • ∇η B (x) -2ρ(x)(2z -H ε (x))c(z -z) < Ccε. ( 4
ρ(x) ≥ κ 1 κ -1 2 ε. (4.121)
Moreover, from (4.75), H ε (x) -z ≤ √ κ 2 ε and (4.121) we find the upper bound :

-zρ(x)∇H ε (x) • ∇η B (x) -z(H ε (x) -z)∇ρ(x) • ∇η B (x) ≤ C ρ(x) + ε 1/2 ≤ Cρ(x).
(4.122)

In addition, since H ε (x) -z ≤ √ κ 2 ε and ε <<< ε 0 (Ω, ω) we have that :

z > max inf Bε H ε z z, 2 3 H ε (x) . (4.123)
Indeed, because of (4.74), H ε (x) -z ≤ √ κ 2 ε and ε <<< ε 0 (Ω) we have that :

z ≥ H ε (x) - √ κ 2 ε ≥ H ε (x) - 1 6 inf B H ≥ H ε (x) - 1 3 inf Bε H ε (x) ≥ 2 3 H ε (x).
Similarly, as a consequence of (4.74), (4.33) and ε <<< ε 0 (Ω, ω) we have that :

z ≥ H ε (x) - √ κ 2 ε ≥ inf Bε H ε - √ κ 2 ε ≥ inf Bε H ε - 1 2 1 - z inf B H inf B H ≥ inf Bε H ε -1 - z inf Bε H ε inf Bε H ε = z inf Bε H ε inf Bε H ε = inf Bε H ε z z.
4.B. The existence of a pseudo-cylinder which is not locally star-shaped Thus, its local minimums are taken at x = 0 and x = 3 -i for all i ∈ N (with H(3 -i ) = 0).

In addition, its local maximums are taken at

x = 2 • 3 -i , for all i ∈ N * (with H(2 • 3 -i ) = 3 • (2 • 3 -i )).
In fact, H(x) ≤ 3x for all x ∈ [0, 1], being equal if and only if x = 0 or x = 2 • 3 -i , for all i ∈ N * .

In order to prove that Ω is not locally star-shaped we show that for all q ∈ B(0, 1/3)∩Ω there is a segment of length |q| with one end in q and such that ∩Ω is not connected. This

shows that the alternative definition given in Remark 4.41 is not satisfied for p = (0, 0) and r = 1/3. Hence, we fix q = (x q , z q ) ∈ B(0, 1/3) ∩ Ω and follow different approaches depending on q :

Case 1 : if z q < 3x q (see q 1 in Figure 4.5). If z q < 3x q , we consider (x q > 0 because

q ∈ Ω) : = x, z q x q x : x ∈ [0, x q ] .
Clearly, is of length |q| and one of its ends is q. In addition, we have that leaves and enters Ω an infinite number of times, as we have for all i ∈ N * that :

z q x q • (3 -i ) > 0 = H(3 -i ), z q x q • (2 • 3 -i ) < 3 • (2 • 3 -i ) = H(2 • 3 -i ), so ∩ Ω is not connected.
Case 2 : if z q ≥ 3x q and if there is i ∈ N * such that z q ∈ [2/3 i+1 , 2/3 i ) and z q < 2 3 i -6x q (see q 2 in Figure 4.5). Under these hypotheses the point q is in the following triangle :

T 1,i := (x, z) ∈ R 2 : z ≥ 2 3 i+1 , x > 0, z < 2 3 i -6x , (4.126)
whose vertices are :

0, 2 3 i+1 , 2 3 i+2 , 2 3 i+1 , 0, 2 3 i . ( 4

.127)

A consequence of (4.126) is that : have received wide attention from the applied science community due to their importance in applications and have also been studied extensively from the theoretical point of view, providing a framework for development of analytical ideas and tools.

T 1,i ⊂ (x, z) : x < 2 3 i+2 . ( 4
In this context, the interaction of internal gravity waves propagating in one horizontal direction has been considered by Gear and Grimshaw [START_REF] Gear | Weak and strong interactions between internal solitary waves[END_REF]. More precisely, the authors derived a model to describe strong interactions of two long internal gravity waves in a stratified fluid, where the two waves are assumed to correspond to different modes of the linearized equations of motion. It can be written as We also refer to [START_REF] Bona | A model system for strong interaction between internal solitary waves[END_REF] for an extensive discussion on the physical relevance of the system in its full structure.

u t + uu x + u xxx + av xxx + k 1 vv x + k 2 (uv) x = 0, cv t + rv x + vv x + abu xxx + v xxx + k 2 buu x + k 1 b(uv) x = 0, ( 5

Setting of the problem

In this paper, we are concerned with the study of the controllability properties of the system (5.1), posed on a bounded interval (0, L), with homogeneous boundary conditions.

The main purpose is to steer the solutions of the system to the rest by means of a control supported on an interior open subset ω of (0, L), acting on one equation only. As usual, we denote

Q := (0, T ) × (0, L), Q ω := (0, T ) × ω, for any open subset ω of (0, L),
1 ω , the characteristic function on ω.

(5.2)

With the notation above, our attention is given to the following distributed control

Introduction

This phenomenon, the so-called critical length phenomenon, was observed for the first time by Rosier in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]. Roughly speaking, it was proved the existence of a finite dimensional subspace M of L 2 (0, L), which is not reachable by the KdV equation, when starting from the origin, if L belongs to a countable set of lengths.

More recently, by using the approach used in the present paper, the internal null controllability for the Hirota-Satsuma system was addressed in [START_REF] Carreño | Internal null controllability of the generalized Hirota-Satsuma system[END_REF]. However, due to the complexity of the system (5.1), which has coupling constituted by third order terms, the control problem presents new difficulties we have to deal with. Then, we develop techniques applicable to this more complicated situation, which allow to slightly improve the result present in [START_REF] Carreño | Internal null controllability of the generalized Hirota-Satsuma system[END_REF] and, in addition, can also be applied to other dispersive and parabolic systems.

To conclude, we can mention the work [START_REF] Capistrano-Filho | Pointwise control of the linearized gear-grimshaw system[END_REF] where the authors consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported at a given point, the linear system associated with (5.1) on the unit circle. The results are obtained by means of spectral analysis and Fourier expansion of the solutions.

Main results

Before stating our main result, we introduce the space H p,p (Q) := H p (0, T ; L 2 (0, L)) ∩ L 2 (0, T ; H p (0, L)), for p, p ≥ 0, endowed with its natural norm, where Q was defined in (5.2). In addition, to shorten the notation we denote

• U := (u, v), U 0 := (u 0 , v 0 ),
• V = V 2 , for V any Banach space (like L 2 , H 1 0 or H p,p ), • Φ := (ϕ, φ) the adjoint variable,

• C (resp. q) a generic positive constant (resp. polynomial of positive coefficients) that may be different each time.

The main result that we prove in this paper concerns the local null controllability of (5.3) : Theorem 5.1. Let L > 0, T > 0 and assume that (5.4) is satisfied. Then, there is C > 0

such that for all h 1 , h 2 satisfying e C (T -t) 12 (h 1 , h 2 ) ∈ L 2 (0, T ; H -1 (0, L)),
One of the main contributions of this paper is to make it clear the difficulty of getting Carleman estimates for any well-posed dispersive or parabolic system with a force G ∈ L 2 (Q) or G = 0. Indeed, to overcome the fact that solution Φ is not regular enough (because G ∈ L 2 (Q)) the proof of the Carleman estimate presents the novelty that we decompose Φ as the sum of the solution of three Cauchy problems. In the literature, the adjoint variable has been decomposed as the sum of the solution of two Cauchy problems (see, for instance, [START_REF] Carreño | Local null controllability of the N -dimensional Navier-Stokes system with N -1 scalar controls in an arbitrary control domain[END_REF]), but for (5.8) we clearly need more. This implies some changes in the proof with respect to the existing results, which are explained in Section 5.3.2 and can be applied for many other dispersive and parabolic systems.

Historical background

There is a huge literature on controllability results with controls having a reduced number of components concerning parabolic equations (see the survey [START_REF] Ammar-Khodja | Recent results on the controllability of linear coupled parabolic problems : a survey[END_REF] and the references therein). For instance, there are a lot of results about Navier-Stokes-like systems, such as [START_REF] Bárcena-Petisco | Null controllability of a penalized Stokes problem in dimension two with one scalar control[END_REF][START_REF] Carreño | Local null controllability of the N -dimensional Navier-Stokes system with N -1 scalar controls in an arbitrary control domain[END_REF][START_REF] Carreño | Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system[END_REF][START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF][START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF][START_REF] Fernández-Cara | Some controllability results for the N -Dimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls[END_REF][START_REF] Guerrero | Controllability of systems of Stokes equations with one control force : existence of insensitizing controls[END_REF][START_REF] Lions | A generique uniqueness result for the Stokes system and its control theoretical consequences. Partial differential equations and applications[END_REF]. It has also been studied, for example, the controllability in cascade-like systems [START_REF] Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF][START_REF] Carreño | Boundary controllability of a cascade system coupling fourth-and second-order parabolic equations[END_REF], the null controllability in the context of linear thermoelasticity [START_REF] Lebeau | Null-Controllability of a System of Linear Thermoelasticity[END_REF], the controllability to trajectories in phase-field models [START_REF] Ammar-Khodja | Controllability to the trajectories of phase-field models by one control force[END_REF], the existence of insensitizing controls for the heat equation [START_REF] De Teresa | Identification of the class of initial data for the insensitizing control of the heat equation[END_REF], and the controllability in reactiondiffusion systems [START_REF] Ammar-Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF]. However, as for KdV-like systems, it is a very recent research topic which has been studied only in [START_REF] Capistrano-Filho | Pointwise control of the linearized gear-grimshaw system[END_REF] and [START_REF] Carreño | Internal null controllability of the generalized Hirota-Satsuma system[END_REF].

The main difference of the problem we consider in this paper with respect to the above cited papers, except [START_REF] Capistrano-Filho | Pointwise control of the linearized gear-grimshaw system[END_REF], is the coupling. Indeed, in the papers cited above (and in the literature, as far as we know) the coupling is constituted by a zero, first or second order term. However, in this paper the coupling is constituted by a third order term. In this sense, the main contribution of this paper is in the technicalities of the Carleman, which provides a technique to move from the controllability of the dipersive linearized system with null force to the controllability of the linearized system with a force decaying in T (usually a necessary step to prove the local null controllability of some nonlinear systems).

In [START_REF] Capistrano-Filho | Pointwise control of the linearized gear-grimshaw system[END_REF], the authors consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, the linearized system on the unit circle. More precisely, by means of spectral analysis and Fourier expansion they prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability by using one control function. It is also possible to find the sharp time of the controllability.

By contrast, the study of the boundary controllability properties is considerably more developed. The first result was obtained in [START_REF] Micu | On the controllability of a linear coupled system of Korteweg-de Vries equations[END_REF], when the model is posed on a periodic domain and r = 0. In this case, a diagonalization of the main terms allows to decouple the corresponding linear system into two scalar KdV equations and to use the previous results available in the literature. Later on, in [START_REF] Micu | On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF], the authors proved the local exact boundary controllability property for the nonlinear system (5.1) posed on a bounded interval. Their result was improved in [START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF], [START_REF] Cerpa | A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF] and in [START_REF] Capistrano-Filho | Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictions on the spatial domain[END_REF], where the same boundary control problem was addressed with a different set of boundary conditions. In all those works, except in [START_REF] Cerpa | A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF], the results were proved by applying the duality approach and some ideas introduced by Rosier in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]. Then, it was discovered that whether the system is exactly controllable depends on the length of the spatial domain, i.e., if L does not lie in the so-called countable set of critical lengths. More precisely, the observability inequality is proved with the aid of a compactness argument that leads the issue to a nonstandard unique continuation principle for the eigenvalues of the state operator associated with the model. Then, the problem is solved by using the Paley-Wiener method introduced by Rosier, which allows the authors determine explicitly all the critical lengths for which the exact controllability fails for the linearized system. The same problem was addressed in [START_REF] Cerpa | A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF] when only two controls act on the Neumann boundary conditions. In this case, the analysis of the linearized system is much more complicated. Therefore, the authors used a direct approach based on the multiplier technique that gives the observability inequality for small values of the length L and large time of control T .

Before closing this section, we mention the work [START_REF] Capistrano-Filho | Control of a boussinesq system of kdv-kdv type on a bounded interval[END_REF], in which the boundary controllability problem was addressed for a Boussinesq system of KdV type.

The rest of the paper is organized as follows : in Section 5.2 we study the regularity estimates of some elliptic and evolution problems, in Section 5.3 we prove a Carleman inequality for (5.8), and finally in Section 5.4 we end the proof of Theorem 5.1.

Some elliptic and evolution problems

In this section we first prove a Poincaré inequality and secondly we prove some existence, uniqueness and regularity results for (5.7) and (5.8).

A Poincaré inequality

In this subsection we prove the following Poincaré inequality : Proposition 5.2. Let L > 0 and i ∈ N. Then, there is C > 0 such that for all w ∈ H 2+i (0, L) satisfying w(0) = w(L) = w (0) = 0, (5.10)

we have the estimate Proposition 5.2, for i ≥ 1, is well-known and has been used in the literature (see for instance [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF]). However, as far as we know, there is no reference for i = 0.

w H 2+i (0,L) ≤ C w xxx H i-1 (0,L) . ( 5
In order to prove Proposition 5.2, we use the following interpolation result :

Lemma 5.4. Let L > 0. Then, there is C > 0 such that for all z ∈ H 1 (0, L) we have that

z x L 2 (0,L) ≤ C( z L 2 (0,L) + z xx H -1 (0,L) ). (5.12)
Proof. [Proof of Lemma 5.4] We fix χ ∈ D(0, L) such that L 0 χ = 1. Then, for any ψ ∈ D(0, L) we have the estimate below

L 0 z x ψ = L 0 z x x 0 ψ(s) -χ(s) L 0 ψ(z)dz ds x dx + L 0 z x χ L 0 ψ(s)dsdx = -z xx , x 0 ψ(s) -χ(s) L 0 ψ(z)dz ds H -1 (0,L)×H 1 0 (0,L) - L 0 zχ x L 0 ψ(s)ds ≤ C( z xx H -1 (0,L) + z L 2 (0,L) ) ψ L 2 (0,L) . (5.13)
Consequently, (5.12) is obtained from Riesz theorem and the density of D(0, L) in L 2 (0, L).

Proof. [Proof of Proposition 5.2] It suffices to prove Proposition 5.2 for i = 0. Indeed, once that case is proved, we have the following estimate

w H 2+i (0,L) ≤ C( w H 2 (0,L) + w xxx H i-1 (0,L) ) ≤ C( w xxx H -1 (0,L) + w xxx H i-1 (0,L) ) ≤ C w xxx H i-1 (0,L) .
We prove the case i = 0 by contradiction.

If Proposition 5.2 is false for i = 0, there is a sequence w j ∈ H 2 (0, L) satisfying (5.10) such that

w j H 2 (0,L) = 1, (5.14)
and

w j xxx H -1 (0,L) → 0. (5.15)
By compactness, we can suppose that w j is convergent in H 7/4 (0, L) to some function w. The convergence implies that w satisfies (5.10) and that w xxx = 0 as a distribution.

Hence, we can prove easily that w = 0. Consequently,

w j H 1 (0,L) → 0, (5.16)
which together with Lemma 5.4 for z = w x and (5.15) implies that

w j xx L 2 (0,L) → 0. (5.17)
Thus, we find from (5.16) and (5.17) that w j H 2 (0,L) → 0, contradicting (5.14).

Existence, uniqueness and regularity properties of the solutions of some dissipative evolution problems

We have the following result for (5.7), which is a consequence of the results presented in [90, Section 2.1] and classical semigroup theory :

Lemma 5.5. Let L > 0, T > 0 and assume that (5.4) is satisfied. Then, for any U 0 ∈ L 2 (0, L) and G ∈ L 1 (0, T ; L 2 (0, L)) system (5.7) admits a unique solution

U ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)).
Moreover, there is C > 0 independent of U 0 and G such that

U C([0,T ];L 2 (0,L))∩L 2 (0,T ;H 1 (0,L)) ≤ C U 0 L 2 (0,L) + G L 1 (0,T ;L 2 (0,L)) . In addition, if U 0 ∈ D(0, L) and G ∈ D(Q) we have that U ∈ C ∞ (Q).
In a similar way, we have an analogous result for the adjoint system : Lemma 5.6. Let L > 0, T > 0 and assume that (5.4) is satisfied. Then, for any Φ T ∈ L 2 (0, L) and G ∈ L 1 (0, T ; L 2 (0, L)) system (5.8) admits a unique solution Φ ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)).

Moreover, there is C > 0 independent of Φ T and G such that Φ C([0,T ];L 2 (0,L)))∩L 2 (0,T ;H 1 (0,L)) ≤ C Φ T L 2 (0,L) + G L 1 (0,T ;L 2 (0,L)) .

In addition, if Φ T ∈ D(0, L) and G ∈ D(Q) we have that Φ ∈ C ∞ (Q).

To continue with, we prove more accurate results about the regularity of the solutions of (5.8) :

Proposition 5.7. Let L > 0 and assume that (5.4) is satisfied. Then :

1. For all T > 0, Φ T ∈ L 2 (0, L) and G ∈ L 2 (0, T ; H -1 (0, L)) system (5.8) admits a unique solution Φ ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)).

Moreover, there is C > 0 independent of T , Φ T and G such that Φ C([0,T ];L 2 (0,L)))∩L 2 (0,T ;H 1 (0,L)) ≤ C(1+T ) Φ T L 2 (0,L) + G L 2 (0,T ;H -1 (0,L)) . (5.18)

2. There is C > 0 such that for all T > 0 and G ∈ L 2 (Q) we have Φ ∈ L 2 (0, T ; H 2 (0, L)) ∩ H 1 (0, T ; H -1 (0, L))

satisfying the estimate Φ L 2 (0,T ;H 2 (0,L))∩H 1 (0,T ;H -1 (0,L)

) ≤ C(1 + T ) G L 2 (Q) , (5.19)
for Φ the energy solution of (5.8) with initial value Φ T = 0.

3. There is C > 0 such that for all T > 0 and G ∈ L 2 (0, T ; H 1 0 (0, L)) we have Φ ∈ H 1,3 (Q) satisfying the estimate Φ H 1,3 (Q) ≤ C(1 + T 3 ) G L 2 (0,T ;H 1 0 (0,L)) , (5.20) for Φ the energy solution of (5.8) with initial value Φ T = 0.

For the proof of Proposition 5.7 we need the following result, which is proved in [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF] : Proof. [Proof of Proposition 5.7.] Step 1 : some first facts and presentation of the proof.

The uniqueness result of statement 1 is a consequence of the linearity of the problem and of Lemma 5.6 applied to Φ T = 0 and G = 0. Thus, we just have to focus on the existence results and the regularity estimates (5.18)- (5.20). For that purpose, thanks to linearity and density results, it actually suffices to prove (5.18)-(5.20) when Φ T ∈ D(0, L)

(or Φ T = 0) and G ∈ D(Q). We remind the reader that under those hypothesis thanks to Lemma 5.6 the energy solution of (5.8) belongs to C ∞ (Q). Hence, the rest of the proof is structured as follows : we first diagonalize (5.8) and secondly we prove the estimates (5.18)-(5.20).

Step 2 : diagonalization of (5.8). We first remark that (5.8) is given by We have that the eigenvalues of M are

               -Φ * t -M Φ * xxx -   0
λ ± := 1 + 1 c ± 1 + 1 c 2 -4 1-a 2 b c 2 = 1 + 1 c ± 1 -1 c 2 + 4 a 2 b c 2 , (5.23)
Two consequences of Proposition 5.7 are the following :

Corollary 5.9. Let L > 0, p ∈ N * and assume that (5.4) is satisfied. Then, there is a polynomial q such that for all T > 0 and G ∈ H p-1 (0, T ; H 1 0 (0, L)) ∩ L 2 (0, T ; H 3p-3 (0, L)) satisfying ∂ i t G(T, •) = 0, ∀i ∈ {0, 1, . . . , p -2}, we have that Φ ∈ H p,3p (Q) and the following estimate holds Φ H p,3p (Q) ≤ q(T )( G H p-1 (0,T ;H 1 0 (0,L))∩L 2 (0,T ;H 3p-3 (0,L)) ), (5.34) for Φ the energy solution of (5.8) with initial value Φ T = 0.

Corollary 5.10. Let L > 0, p ∈ N * and assume that (5.4) is satisfied. Then, there a polynomial q such that for all T > 0 and G ∈ H p (0, T ; H -1 (0, L)) ∩ L 2 (0, T ; H 3p-2 (0, L)) satisfying ∂ i t G(T, •) = 0, ∀i ∈ {0, 1, . . . , p -1}, we have that Φ ∈ H p (0, T ; H 1 (0, L)) ∩ L 2 (0, T ; H 3p+1 (0, L)) and the following estimate holds Φ H p (0,T ;H 1 (0,L))∩L 2 (0,T ;H 3p+1 (0,L)) ≤ q(T ) G H p (0,T ;H -1 (0,L))∩L 2 (0,T ;H 3p-2 (0,L)) , (5.35) for Φ the energy solution of (5.8) with initial value Φ T = 0.

Corollary 5.9 is proved by induction, using the inductive hypothesis on Φ t and diagonalising L * as in (5.25). The time estimate of Corollary 5.10 is proved by using (5.18) on ∂ p t Φ. The space estimate of Corollary 5.10 is proved by diagonalising L * as in (5.25) and induction : first by using (5.18) on Φ t , and then by applying the inductive hypothesis on Φ t . Indeed, we have the following continuous inclusions : H p (0, T ; H -1 (0, L)) ∩ L 2 (0, T ; H 3p-2 (0, L)) ⊂ H 1 0, T ; H 3(p-1)-2 p-1 p -1 p (0, L) ⊂ H 1 0, T ; H 3(p-1)-2 (0, L) .

Remark 5.11. We have for (5.7) regularity results analogous to those in Proposition 5.7, Corollary 5.9 and Corollary 5.10.

A Carleman inequality for the solutions of (5.8)

In order to shorten the notation, we define for a function g ∈ C([0, L]) and any subset X ⊂ [0, L] the integral : The structure of this section is the following : first we present the weights, and secondly we state and prove the Carleman inequality.

Definition of the weights and an auxiliary result

for m ∈ R + and λ > 0 to be fixed later. Similar weights were first defined in [START_REF] Fursikov | Controllability of evolution equations[END_REF], but this version is taken from [START_REF] Fernández-Cara | Global carleman inequalities for parabolic systems and applications to controllability[END_REF]. We remark that (5.36) 3 and (5.36) 4 imply the equalities :

α * (t) = α(t, 0) = α(t, L), ξ * (t) = ξ(t, 0) = ξ(t, L).

(5.38)

We recall that for all i ∈ N * there is C > 0 such that if λ ≥ C we have the estimate : In general, for all i ∈ N * there is C > 0 such that if λ ≥ C we have the estimates :

|∂ i t ξ| + |∂ i t α| ≤ CT i (ξ * ) 1+i/m . ( 5
|∂ i t β * | ≤ CT i (β * ) 1+i/m , |∂ i t β * | ≤ CT i (ξ * ) 1+i/m .
(5.44)

Finally, for all δ > 0 and k ∈ R + there is C > 0 such that for all s ∈ R + :

(sβ * ) k ≤ Ce δsβ * .

(5.45) Indeed, x k e -δx ∈ L ∞ (R + , dx).

In order to prove a Carleman inequality for (5.8) we need the following technical result :

Proposition 5.12. Let L > 0, i ≥ 0, j, k 1 , k 2 ∈ R, k 3 ∈ R \ {0}, m ≥ 1/2 and ω ⊂ (0, L).

There is C > 0 such that for all λ ≥ C, s ≥ C(T m + T 2m ), and w ∈ H (5.47) Proposition 5.12 can be proved for i = j = k 1 = k 2 = 0 by following, step by step, with some small changes the proof presented in [22, Proposition 3.1] and in [8, Theorem 3.1] (which are based on the proof presented in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]). In addition, we have for any i ≥ 0 that s i/2 e -sα ξ i/2 w ∈ H 2,3 (Q), so we just have to replace w by s i/2 ξ i/2 w in (5.46) and do the usual absorptions. As for j, it suffices to multiply (5.46) by λ j at both sides of the inequality. Finally, we can prove it for any k 1 , k 2 ∈ R by considering that w t + k 3 w xxx = g -k 1 w x -k 2 w xx , and then absorbing the first and second order term with the left-hand side of (5.46).

The Carleman estimate

In this section we prove the following Carleman estimate : for Φ the energy solution of (5.8).

Remark 5.14. We have that s ≥ C(T m + T 3m ) implies that s ≥ CT 2m , which implies that sξ * ≥ 1.

The structure of the proof is the following : first we reduce it to the case in which the solution is regular, second we decompose Φ as the sum of the solutions of three Cauchy problems, third we get regularity estimates for the first two, fourth we deduce a PDE that just contains the second component of the third auxiliary variable, fifth we get a Carleman estimate for the third auxiliary variable, sixth we estimate the trace terms, seventh we estimate the local terms, and finally we deduce a Carleman estimate just in terms of a local term of φ.

Remark 5.15. As explained in the introduction, the proof of Proposition 5.13 presents the novelty that we decompose Φ as the sum of the solutions of three Cauchy problems to overcome the fact that the adjoint variable is not regular enough. The main difference in considering three Cauchy problems instead of two (as it has been done in the literature) is that in many parts of the proof we need to use the weights (5.40) instead of (5.37) as a way to ensure that the solutions of those Cauchy problems belong to L 2 (Q). In addition, there is a second novelty which is to use a factorization to simplify the sixth order differential equation that we get when we uncouple.

Proof. [Proof of Proposition 5.13] Step 1 : reducing to regular solutions. By linearity arguments, density arguments and Lemma 5.6 it suffices to prove (5.48) for G ∈ D(Q) and Φ T ∈ D(0, L). In those cases we have by Lemma 5.6 that Φ ∈ C ∞ (Q). Thus, from now we suppose that G ∈ D(Q) and Φ T ∈ D(0, L).

Moreover, using (5.18) for system (5.50) and (5.41) 3 , we have that for all δ > 0 there is a polynomial q of positive coefficients such that, for all t ∈ [0, T ], Φ(t, •) L 2 (0,L) ≤ q(T -t) e -5sβ * G L 2 ((t,T )×(0,L)) ≤ q(T )e -δsβ * (t) e -(5-δ)sβ * G L 2 (Q) . (5.55) This allows us to estimate sβ * tt Φ L 2 (Q) and (sβ * t ) 2 Φ L 2 (Q) in terms of e -(5-δ)sβ * G L 2 (Q) . Consequently, looking at the system satisfied by β * t Φ, from (5.19), (5.44) 1 , (5.45) and (5.55) we get sβ * t Φ L 2 (0,T ;H 2 (0,L))∩H 1 (0,T ;H -1 (0,L)) ≤ q(T ) e -(5-δ)sβ * G L 2 (Q) .

(5.56) Thus, using (5.35) for p = 1, from (5.56) if follows that Φ H 1 (0,T ;H 1 (0,L))∩L 2 (0,T ;H 4 (0,L)) ≤ q(T ) e -(5-δ)sβ * G L 2 (Q) . Consequently, taking δ > 0 small enough and keeping the interesting terms, for λ ≥ C and s ≥ C(T m + T 3m ) we have Φ L 2 (0,T ;H 2 (0,L)) + Φ H 1 (0,T ;H 1 (0,L))∩L 2 (0,T ;H 4 (0,L)) ≤ C e -4sβ * G L 2 (Q) .

(5.59)

Step 4 : uncoupling (5.53). We now get an equation containing just φ. By applying the operator -∂ t -∂ 3 x to the second equation of (5.53) 1 and using the first equation of (5.53) 1 , we obtain the PDE : c φtt + c φtxxx + r φtx + r∂ 4

x φ -a 2 b∂ 6 x φ + φtxxx + ∂ 6 With the objective of simplifying the left-hand side of (5.60), we show the existence of κ 1 , κ 2 , κ1 , κ2 ∈ R such that the following equality is satisfied :

c(∂ t + κ1 ∂ x + κ 1 ∂ 3 x )(∂ t + κ2 ∂ x + κ 2 ∂ 3 x ) φ = κ1 κ2 ∂ 2 x φ -5sβ * t ∂ 3 x (ϕ + φ) -5s(∂ t + ∂ 3 x )(β * t (φ + φ)).
(5.61)

The higher order terms form the system :

   κ 1 + κ 2 = 1 + 1 c , κ 1 κ 2 = 1-a 2 b
c .

(5.62) Thus, we search for κ 1 such that :

κ 2 1 - c + 1 c κ 1 + 1 -a 2 b c = 0.
(5.63)

The discriminant of (5.63) is given by :

(c + 1) 2 c 2 -4 1 -a 2 b c = c 2 + 2c + 1 -4c + 4a 2 bc c 2 = (c -1) 2 + 4a 2 bc c 2 > 0;
hence, equation (5.63) and system (5.62) have two real solutions. As for the coefficients of the terms of order 4, we find the system    κ1 + κ2 = r c , κ 2 κ1 + κ 1 κ2 = r c , which clearly has at least a real solution (κ 1 , κ2 ). Consequently, the decomposition (5.61) is possible. In particular, we define the auxiliary variable θ := (∂ t + κ2 ∂ x + κ 2 ∂ 3

x ) φ.

(5.64)

Step 5 : a Carleman estimate for Φ. We have the following Carleman estimates :

• If we use Proposition 5.12 on φ for (k 1 , k 2 , k 3 ) = (κ 2 , 0, κ 2 ) and consider the homogeneous Dirichlet boundary conditions, we obtain that : (5.65)

• If we use Proposition 5.12 on φx and recall the boundary conditions of Φ, we get : • If we use Proposition 5.12 on θ for (k 1 , k 2 , k 3 ) = (κ 1 , 0, κ 1 ), we get from c > 0, (5.64) and (5.61) the following estimate : • From (5.64) we have :

s 4 λ 5
s 2 λ 3 Q e -
s 4 λ 5 Q e -2sα ξ 4 | φt | 2 ≤ Cs 4 λ 5 Q e -2sα ξ 4 (| φxxx | 2 + | φx | 2 + θ 2 ).
(5.68)

• If we consider the second equation of (5.53) 1 , it follows that :

s 4 λ 5 Q e -2sα ξ 4 | φxxx | 2 ≤ Cs 4 λ 5 Q e -2sα ξ 4 (| φxxx | 2 + | φx | 2 + | φt | 2 + s 2 (β * t ) 2 φ 2 + s 2 (β * t ) 2 φ2
). (5.69)

• Thanks to the second equation of (5. Step 6 : absorption of the trace terms. Let us first show how to absorb the trace term of φ. We recall (5.38). From trace interpolation estimates and Young inequality we get : |(s 2/3 λ 2/3 e -sα * (ξ * ) 2/3 ) t | 2 φ 2 H 2 (0,L) + s 2/3 λ 2/3 e -sα * (ξ * ) 2/3 φ 2 H 1 (0,T ;H 2 (0,L)) .

s 2 λ 2
(5.74)

It can be proved easily that the first term in the right-hand side of (5.74) can be absorbed by the left-hand side of (5.72) using that m ≥ 4. As for the second one, using (5.19) on Φ t we have : s 2/3 λ 2/3 e -sα * (ξ * ) 2/3 φ 2 H 1 (0,T ;H 2 (0,L)) ≤ (1 + T 2 ) (s 2/3 λ 2/3 e -sα * (ξ * ) 2/3 ) t Φ 2 H 1,0 (Q) + s 2/3 λ 2/3 e -sα * (ξ * ) 2/3 sβ * t Φ 2 H 1,0 (Q) + s 2/3 λ 2/3 e -sα * (ξ * ) 2/3 sβ * t Φ 2 H 1,0 (Q) . (5.75)

The third term in the right-hand side of (5.75) can be estimated by (5.59). Moreover, the first two terms in the right-hand side of (5.75) can be absorbed by the left-hand side of (5.72) if m ≥ 12. Indeed, the most problematic term is

s 10/3 λ 4/3 Q e -2sα * (|α * t | 2 + |β * t | 2 )(ξ * ) 4/3 | Φt | 2 ,
which can be absorbed by the left-hand side of (5.72) by considering (5.39), (5.44) 2 and that CT 2+2 = CT 4 ≤ s 2/m . The other terms can be estimated similarly.

Let us now estimate the trace term of θ. Because of (5.64) and the Dirichlet boundary conditions, we have that :

θ t = κ2 φtx + κ 2 φtxxx , on {0, L} × (0, T ).

In addition, considering that the source term in (5.52) satisfies Dirichlet boundary conditions and that L * can be diagonalize as in (5.25), we obtain from (5.53) that there is (5.77)

κ 3 ∈ R such that
The second term on the right hand-side of (5.77) can be bounded with (5.35) for p = 1 : e -13 12 sα * φ 2 L 2 (0,T ;H 4 (0,L)) ≤ C (e -13 12 sα * ) t Φ 2

H 1 (0,T ;H -1 (0,L))∩L 2 (0,T ;H 1 (0,L))

+ sβ * t e -13 12 sα * Φ 2 H 1 (0,T ;H -1 (0,L))∩L 2 (0,T ;H 1 (0,L))

+ sβ * t e -13 12 sα * Φ 2 H 1 (0,T ;H -1 (0,L))∩L 2 (0,T ;H 1 (0,L)) . (5.78)

The third term in the right-hand side of (5.78) can be estimated with (5.59). As for the other two terms, they can be absorbed by the left-hand side of (5.76).

In order to study the local term of θ, we substitute θ by the right-hand side of (5.64).

The spatial terms can be treated similarly as before. As for φt we obtain for λ ≥ C and s ≥ C(T m + T 3m ) the estimate We can treat the last term at the left hand-side of (5.79) as before thanks to interpolation results, Corollary 5.9 (for p = 2), Corollary 5.10 (for p = 1), (5.58) and (5.42). As for the first term in the right-hand side of (5.79), we can simply integrate by parts : The last term in the right-hand side of (5.80) can be estimated as before with the help of Corollary 5.9 (for p = 2), Corollary 5.10 (for p = 1), (5.58) and (5.42).

So, summing up and keeping the interesting terms, from (5.77) we get is very similar. Indeed, the careful reader can remark that in the proof of Proposition 5.13 the terms of Φ has been absorbed easily because of the presence of an exponential weight, whereas it is in the terms of Φ where we have done more work.

The nonlinear control problem

In the following, we denote L r (ρ(t)(0, T ); V ) := {w : ρ(t)w ∈ L r (0, T ; V )}, for V a Banach space, ρ a measurable function and r ∈ [1, +∞]. Moreover, we define w L r (ρ(t)(0,T );V ) := ρ(t)w L r (0,T ;V ) .

Similarly, we define C(ρ(t)[0, T ]; V ) := {w : ρ(t)w ∈ C([0, T ]; V )}.

In addition, in this section we fix (m, λ, s) so that Proposition 5.13 is satisfied and In that sense, in the following the generic constant C may also depend on s, λ and T .

|sβ * t | ≤ Ce
The structure of this section is the following : first we study the controllability of the linearized problem, second we end the prove of Theorem 5.1, and thirdly we make some remarks about similar control problems which can be studied by using the technique presented in this paper.

5.4.1 Null controllability of (5.5)

Let E be the couples (U, f ) such that :

• U ∈ L 2 (e 4sβ * (0, T ); L 2 (0, L)),

• U ∈ L 2 (e • f 1 ω ∈ L 2 (e 5sβ * (0, T ); L 2 (0, L)),

• LU -(0, f 1 ω ) ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)).

Then, we have the following controllability result :

Proposition 5.17. Let U 0 ∈ L 2 (0, L), h 1 , h 2 ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)) and assume that (5.4) is satisfied. Then, there is a solution of (5.5) satisfying (U, f ) ∈ E.

Remark 5.18. If (U, f ) ∈ E, we have that U (T, •) = 0.

For the proof of Proposition 5.17 we follow the approach introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF] which is based on duality. In particular, we first prove the following auxiliary result : (h 1 , h 2 ), Φ H -1 (0,L)×H 1 0 (0,L) .

(5.87)

Notice that the observability inequality (5.84) holds for every Φ ∈ P 0 taking G = L * Φ.

Consequently, a is a scalar product in P 0 and there is C > 0 such that for all Φ ∈ P 0 we have that | (Φ)| ≤ C( e 7sβ * (h 1 , h 2 ) L 2 (0,T ;H -1 (0,L)) + (u 0 , v 0 ) L 2 (0,L) ) a(Φ, Φ).

In the following we denote P the completion of P 0 with respect to a (and keep denoting a and to their only continuous extensions). By Riesz theorem there is Φ ∈ P such that a( Φ, Φ) = (Φ), ∀Φ ∈ P.

We define (U, f ) by U := e -8sβ * L * Φ, f := -e -10sβ * φ1 ω .

(5.88)

  générale à la théorie de contrôle . . . . . . . . . . . . . . . . . 1.1.1 Les notions générales de contrôlabilité . . . . . . . . . . . . . . . . 1.1.2 La HUM et l'inégalité d'observabilité . . . . . . . . . . . . . . . . . 1.2 Résultats principaux et plan de la thèse . . . . . . . . . . . . . . . . . . . . 1.2.1 Contrôlabilité à zéro d'un problème de Stokes pénalisé en dimension deux avec un contrôle scalaire (voir [15]) . . . . . . . . . . . . . . . 1.2.2 Contrôlabilité uniforme d'un problème de Stokes avec un terme de transport et une viscosité évanescente (voir

1. 2 . 4

 24 Contrôlabilité à zéro locale d'un modèle d'interaction forte entre deux ondes internes solitaires (voir [17]) . . . . . . . . . . . . . . . 1.3 Un travail complémentaire et des problèmes ouverts . . . . . . . . . . . . . 1.3.1 Travail complémentaire : étude du coût de la contrôlabilité d'une équation parabolique d'ordre deux avec diffusion évanescente et un terme de transport (voir [16]) . . . . . . . . . . . . . . . . . . . . . 1.3.2 Perspectives et problèmes ouverts . . . . . . . . . . . . . . . . . . . xiii Chapitre 1

  et donc, y ε k tend vers y, la solution du système de Stokes avec valeur initiale y 0 et contrôle f 1 ω e 1 , qui satisfait y(T, •) = 0. Cependant, comme les solutions de (1.13) pour y 0 ∈ H(Ω) n'appartiennent pas à C 0 ([0, T ]; H(Ω)) mais à C 0 ([0, T ]; L 2 (Ω)), pour pouvoir appliquer la HUM (voir Section 1.1.2) on travaille avec toutes les valeurs initiales dans L 2 (Ω), même si les seules conditions initiales intéressantes sont celles dans H(Ω). D'abord, on prouve que pour tout ε > 0 il y a un losange Ω ε ⊂ R 2 tel que (1.13) n'est pas contrôlable à zéro avec un seul contrôle scalaire. On prouve ce résultat en montrant que la propriété de continuation unique dans ces domaines Ω ε est fausse, et pour montrer cela on fait une recherche systématique dans l'ensemble des fonctions à variables séparées.

Figure 2 . 2 -

 22 Figure 2.2 -An illustration of S in a non-convex domain

First, we remark

  Figure 2.2).• As for the points in {x = c} ∩ Γ of tangent vector ±e 2 , we might not be allowed to extend the segment because P h becomes discontinuous and we need to start two new segments (for instance, in A 2 in Figure2.2 we must stop and start two new segments : one with left endpoint A 2 , the other one with left endpoint C 1 ). It might also be the case that we need to start a new segment above the point (see near D 1 in Figure2.2).

Figure 2 . 4 -

 24 Figure 2.4 -Case 3 of the proof of Theorem 2.8

Figure 2 . 5 -

 25 Figure 2.5 -Case 4 of the proof of Theorem 2.8

  have the same sign as in the convex case), but this time we face the extra difficulty of showing that having the estimate of |∂ y u(p -1 )| implies having the estimate of |∂ y u(p 1 )|.

  we can use Lemma 2.25 to get an estimate of |∂ y u(k -1 )|. Then, because ∂ xy u C 0 ([k -1 ,k 1 ]) can be estimated, we have an estimate of |∂ y u(k 1 )|. Finally, we can propagate the pointwise estimate of |∂ y u| in the segment [k 1 , p 1 ] in an analogous way as in the segment [p -1 , k -1 ]. Consequently, we have the estimate of |∂ y u(p 1 )| and we can replicate the method of the case 3 to obtain (2.58).

  2 we study systems (3.3) and (3.5) for Ω 2 , in Section 3.3 we study the cost of the control for Ω 2 , in Section 3.4 we study the control problem (3.1) and its adjoint system for Ω 3 , in Section 3.5 we do some further comments about the techniques and point out some open problems, and in Appendix 3.A we prove a technical result.

Now we are ready to prove Proposition 3. 6 :

 6 Proof. [Proof of proposition 3.6] Since span(SV (Ω 2 )) is a closed vectorial subspace of H(Ω 2 ), it suffices to prove that : span(SV (Ω 2 ))

Finally, let us prove ( 3 .

 3 [START_REF] Carreño | Local null controllability of the N -dimensional Navier-Stokes system with N -1 scalar controls in an arbitrary control domain[END_REF]) using(3.25). We recall that, if a, b ∈ R such that a = 0 or b as )e bs ds = e bs (b sin(as)-a cos(as))+a a 2 +b 2 , s 0 cos(as )e bs ds = e bs (b cos(as)+a sin(as))-b a 2 +b 2 .

  -1 ). (3.41) So, combining (3.25), (3.37), (3.39), (3.40) and (3.41) we obtain that :

  ) which implies that the series in the right-hand side of (3.53) is absolutely convergent and thus well-defined. Moreover, since the right-hand side of (3.53) is linear, (3.57) implies that the right-hand side of (3.53) is continuous in H(Ω 2 ).

7 )

 7 1 and (3.80)-(3.82) we obtain (3.79).

. 91 )

 91 So, combining (3.4), (3.74), (3.78) and (3.91) we end the proof of Item 3 of Theorem 3.1.

  .110) Summing up, thanks to (3.109), (3.110), (3.100), (3.101) and (3.104), we have for λ ≥ C and τ

  .117) Summing up, if we consider (3.111)-(3.117), (3.100), (3.101) and (3.104), we get that :

  to the H 2 -norm to the existence of a proper auxiliary function. We do it in Section 4.2 by approximating the domain with regular subdomains.As a first step to prove the null controllability of the heat equation by an internal control in any Lipschitz domain, in this paper we prove the null controllability in pseudo-cylinders.By pseudo-cylinder we mean the following : Definition 4.1. We say that Ω ⊂ R d+1 (d ≥ 1) is a pseudo-cylinder if there are B ⊂ R d a C 2 domain, U the composition of a rotation and a translation and H : B → R + a Lipschitz function satisfying inf B H > 0 such that :

  28 and Proposition 4.32 below) which present the usual absorption properties. However, as we explain in Comment 5 of Section 4.3.3, our method does not work in every Lipschitz domain, so the general case remains an interesting open problem.

Figure 4 .

 4 Figure 4.1 -A canonically oriented pseudo-cylinder.

. 36 )

 36 for c a large positive constant to be fixed later on. It is evident that η ∈ C 2 Ω . As ∇η = (∇η B , -2c(z -z)), it follows from (4.32) and (4.34) 3 the inequality : inf Ω\ω |∇η| > 0. Example 4.27. See Figure 4.2 for an illustration of a control domain and of ∇η in the pseudo-cylinder introduced in Example 4.3.

Figure 4 . 2 -

 42 Figure 4.2 -A canonically oriented pseudo-cylinder with its auxiliary function.

ξ 3 |u| 2 .

 2 (4.41) Thus, taking λ ≥ C and s satisfying (4.38) we can absorb the the last two terms in the right-hand side of (4.41).

Figure 4 . 3 -

 43 Figure 4.3 -A canonically oriented pseudo-cylinder with a control domain in Ω \ C.

Example 4 . 31 .

 431 See Figure 4.3 for an illustration of the pseudo-cylinder given in Example 4.3 when ω ⊂⊂ Ω \ C.

. 50 )

 50 Moreover, from (4.45), (4.48) and (4.49) we get that :   η ≤ η in ω * , η ≤ η in Ω \ B(x, 2r) × (r, z + 2r) .

position 4 .

 4 [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] (taking m = 1) implies that Theorem 4.6 is true for all canonically oriented pseudo-cylinders and all ω given by (4.42). Together with Remarks 4.30 and 4.7, this result ends the proof of Theorem 4.6 (up to proving Proposition 4.29).

Figure 4 . 4 - 2 .

 442 Figure 4.4 -A canonically oriented pseudo-cylinder with a control domain Γ ⊂ T.

. 97 )

 97 Moreover, we have the equality (see (4.65)) :

  .116) Thus, we get from c >>> c 0 (Ω, ω, η B ), ρ ≥ 0, (4.115) and (4.116) the upper bound :

Figure 4 . 5 -

 45 Figure 4.5 -A pseudo-cylinder which is not locally star-shaped (approximate figure).

  .1) where a, b, c, r, k 1 , k 2 are real parameters, the unknowns u and v are real valued functions of the variables x and t and subscripts indicate partial differentiation. Observe that (5.1) has the structure of a pair of Korteweg-de Vries (KdV) equations with both linear and nonlinear coupling terms.

. 11 ) 5 . 3 .

 1153 Remark By symmetry, for all i ∈ N we also have(5.11) for all w ∈ H 2+i (0, L) satisfying w(0) = w(L) = w (L) = 0.

Lemma 5 . 8 .

 58 Let L > 0 and k > 0. Then, there is C > 0 such that for all T > 0 andg ∈ L 2 (Q) we have w ∈ L 2 (0, T ; H 2 (0, L)) satisfying the estimate w L 2 (0,T ;H 2 (0,L)) ≤ C g L 2 (Q) , -kw xxx = g, in Q, w(•, 0) = w(•, L) = w x (•, 0) = 0, on (0, T ),w(T, •) = 0 on (0, L).



  , in Q, Φ * (•, 0) = Φ * (•, L) = Φ * x (•, 0) = 0, on (0, T ), Φ * (T, •) = (Φ T ) * , on (0, L),

  Let L > 0 and ω ⊂ (0, L) be an open subdomain. We consider η ∈ C 3 ([0, L]) a function satisfying :η ∞ = 1, inf (0,L)\ω |η | > 0, sup (0,L) η < 0, η(0) = η(L) = 0.(5.36)With that auxiliary function we define the following classical weights :α(t, x) := e λ(2m+2) -e λ(2m+η(x)) t m (T -t) m , ξ(t, x) := e λ(2m+η(x)) t m (T -t)m , α * (t) := sup x∈(0,L) α(t, x), ξ * (t) := inf x∈(0,L) ξ(t, x),

. 39 )

 39 Next, we define the weights :β(t, x) := e λ(2m+2) -e λ(2m+η(x)) (T 2 l(t/T )) m , β * (t) := sup x∈(0,L) β(t, x),(5.40)for l(t) ∈ C ∞ ([0, 1]) a positive function satisfying :l(t) = t(1 -t), in [3/4, 1], l(t) > t(1 -t), in [0, 3/4), l (t) < 0, in [0, 1]. (5.41)An easy consequence of (5.41) is thatβ * ≤ α * in [0, T ].(5.42)As for these weights, we have the following bound for λ ≥ C :|β * t | = mT l (t/T )e λ(2m+2) -e 2mλ (T 2 l(t/T )) m+1 ≤ CT (β * ) 1+1/m . (5.43)

2 , 3 e -2sα ξ 5+i w 2 ≤-2sα ξ 5+i w 2 +

 2322 (Q) we have the estimates 1+i λ 2+j Q e -2sα ξ 1+i |w xx | 2 + s 3+i λ 4+j Q e -2sα ξ 3+i |w x | 2 + s 5+i λ 6+j Q C s i λ j Q e -2sα ξ i g2 + s 1+i λ 2+j Qω e -2sα ξ 1+i |w xx | 2 + s 5+i λ 6+j Qω e s -1+i λ -1+j T 0 {0,L} e -2sαξ -1+i |w t | 2 , (5.46) for g := w t + k 1 w x + k 2 w xx + k 3 w xxx .

Proposition 5 . 13 . 6 λ 6 T 0 e

 513660 Let L > 0, ω ⊂ (0, L) be an open subset, m ≥ 12 and assume that(5.4) is satisfied. There is C > 0 such that for all λ ≥ C, s ≥ C(T m +T 3m ), Φ T ∈ L 2 (0, L)and G ∈ L 2 (Q) we have the estimates -2sα * -10sβ * (ξ * ) 6 Φ 2 H 2 (0,L) ≤ C Qω e -sα * -10sβ * φ 2 + Q e -8sβ * |G| 2 ,(5.48)

(5. 57 )

 57 Finally, since λ ≥ C and s ≥ C(T m + T 3m ), from (5.45) and (5.41) 3 we obtain the estimateT ≤ C s 1/m T 2 ≤ C(sβ * (0)) 1/m ≤ Ce sδβ * (0) ≤ Ce sδβ * (t) , ∀t ∈ [0, T ].(5.58)

3 x

 3 (ϕ + φ) -5s(∂ t + ∂ 3 x )(β * t (φ + φ)). (5.60)

s 6 λ 7 Qe -2sα ξ 6 | φxx | 2 + s 8 λ 9 Qe -2sα ξ 8 | 11 Qe -2sα ξ 10 φ2≤ C s 5 λ 5 Qe -2sα ξ 5 θ 2 + s 6 λ 7

 762981110527 φx | 2 + s 10 λ Qω e -2sα ξ 6 | φxx | 2 + s 10 λ 11 Qω e -2sα ξ 10 φ2 .

Q e -2sα ξ 4 | φxxx | 2 + s 6 λ 7 Qe -2sα ξ 6 | φxx | 2 + s 8 λ 9 Qe -2sα ξ 8 | φx | 2 ≤ C s 3 λ 3 Qe -2sα ξ 3 |θ x | 2 + s 4 λ 5 Qωe -2sα ξ 2 | φxxx | 2 + s 8 λ 9 Qωe -2sα ξ 8 | φx | 2 + s 2 λ 2 T 0

 427629823252298220 {L} e -2sα ξ 2 | φtx | 2 . (5.66)

s 6 λ 6 T 0 e 2 H 2 ≤ Cs 6 λ 6 Qe 7 /2 λ 4 Qe 2 ≤ 7 /2 λ 4 Qes 4 λ 5 Qe 3 i=0s 10 - 2 + s 7 /2 λ 4 Qe 2 + s 4 λ 5 Qe -2sα ξ 4 | φxxx | 2 + s 6 λ 6 T 0 e 2 H 2 ( 2 i=0s 6 - 2 H 1 C 3 i=0s 10 - 2 s 6 -

 6022674274531027425426022262131026 [START_REF] Fernández-Cara | Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability[END_REF] 1 ,(5.64) and Poincaré inequality (Proposition 5.2 for i = 0) we obtain : -2sα * (ξ * ) 6 φ (0,L) -2sα (ξ* ) 6 (| φxx | 2 + | φx | 2 + φ2 + θ 2 + s 2 (β * t ) 2 φ 2 + s 2 (β * t ) 2 φ2). (5.70)• Considering the first equation of (5.53) 1 the following holds :s -2sα * (ξ * ) 7/2 | φt | Cs -2sα * ((ξ * ) 7/2 (| φxxx | 2 + | φxxx | 2 + (5sβ * t ) 2 ( φ2 + ϕ 2 )). (5.71)We recall that because of (5.44) 2 we have for λ ≥ C and s ≥ C(T m + T 3m ) the estimate (sβ * t ) 2 ≤ Cs 2+2/m (ξ * ) 2+2/m . Thus, the zero-order term of φ in the right-hand side of (5.71) can be absorbed by the left-hand side of (5.70) if m ≥ 4, λ ≥ C and s ≥ C(T m + T 3m ). So, considering (5.44) 2 , (5.59) and (5.65)-(5.71), after some easy absorptions and majorations we obtain for m ≥ 4, λ ≥ C and s ≥ C(T m + T 3m ) that :-2sα ξ 4 | φt | 2 + 2i λ 11-2i Q e -2sα ξ 10-2i |∂ i x φ| -2sα * (ξ * ) 7/2 | φt | -2sα * (ξ * ) 6 φ 2i λ 7-2i Q e -2sα ξ 6-2i |∂ i x θ| 2 + Φ 2 L 2 (0,T ;H 2 (0,L)) + Φ (0,T ;H 1 (0,L))∩L 2 (0,T ;H 4 (0,L)) ≤ 2i λ 11-2i Qω e -2sα ξ 10-2i |∂ i x φ| 2 + i=0,2i λ 7-2i Qω e -2sα ξ6-2i |∂ i x θ| 2 + T 0 {0,L} e -2sα |θ t | 2 + s 2 λ 2 T 0 {L} e -2sα ξ 2 | φtx | 2 + Q e -8sβ * |G| 2 . (5.72)

e -2sα ξ 2 | φtx | 2 ≤ Cs 2 λ 2 T 0 e 2 H 2 (≤ C s 4 λ 4 Qe 3 T 0 e 2 H 2 3 T 0 e

 2220224302230 -2sα * (ξ * ) 2 φt 3/-2sα * (ξ * ) 4 | φt | 2 + s 4/3 λ 4/-2sα * (ξ * ) 4/3 φt (0,L) . (5.73)It is clear that the first term in the right-hand side of (5.73) can be absorbed by the left-hand side of (5.72) for λ ≥ C. As for the second term, we deduce that :s 4/3 λ 4/-2sα * (ξ * )

θ 2 T 0 s 4 λ 5 Qe -2sα ξ 4 | 3 i=0s 10 - 2 + s 7 /2 λ 4 Qe 2 + s 4 λ 5 Qe -2sα ξ 4 | φxxx | 2 + s 6 λ 6 T 0 e 2 H 2 ( 2 L 2 2 H 1 C 3 i=0s 10 - 2 s 6 - 3 i=0s 10 - 0 e -15 8 sα * φ 2 H 3 2 H 4

 20543102742542602222213102631002324 t (•, 0) = 0, on (0, T ), θ t (•, L) = κ 3 φtx (•, L), on (0, T ). Thus, by Remark 5.14 we obtainT 0 {0,L} e -2sα |θ t | 2 ≤ Cs 2 λ {L} e -2sα ξ 2 | φtx | 2 ,a term which has already been treated.Summing up, after the operations on this step and keeping the interesting terms in (5.72) we get if m ≥ 12, λ ≥ C and s ≥ C(T m + T 3m ) the following estimate : φt| 2 + 2i λ 11-2i Q e -2sα ξ 10-2i |∂ i x φ| -2sα * (ξ * ) 7/2 | φt | -2sα * (ξ * ) 6 φ (0,T ;H 2 (0,L)) + Φ (0,T ;H 1 (0,L))∩L 2 (0,T ;H 4 (0,L)) ≤ 2i λ 11-2i Qω e -2sα ξ 10-2i |∂ i x φ| 2 + i=0,2i λ 7-2i Qω e -2sα ξ 6-2i |∂ i x θ| 2 + Q e -8sβ* |G| 2 . (5.76) Step 7 : absorption of the local terms. Using space interpolation and Young inequality, we obtain that : 2i λ 11-2i Qω e -2sα ξ 10-2i |∂ i x φ| 2 ≤ C T (ω) ≤ C Qω e -sα * φ2 + e -13 12 sα * φ 2 L 2 (0,T ;H 4 (0,L)) .

i=0, 2 s 6 -* φt 2 H 3

 2623 2i λ 7-2i Qω e -2sα ξ 6-2i (∂ i x ∂ t φ) 2 ≤ C Q (0,L) . (5.79)

Qω e -15 8 sα * | φt | 2 4

 24 sα * | φtt | 2 . (5.80)

s 6 λ 6 T 0 e 2 L 2 that s 6 λ 6 T 0 e 2 L 2

 60226022 -2sα * (ξ * ) 6 Φ 2 H 2 (0,L) + Φ 2 L 2 (0,T ;H 2 (0,L)) + Φ (0,T ;H 2 (0,L))≤ C Qω e -sα * φ2 + Q e -8sβ * |G| 2 . (5.81)Step 8 : returning to the variable Φ. Considering (5.49) and (5.59), from (5.81) wededuce -2sα * (ξ * ) 6 Φ 2 H 2 (0,L) + Φ 2 L 2 (0,T ;H 2 (0,L)) + Φ (0,T ;H 2 (0,L)) ≤ C Qω e -sα * -10sβ * φ 2 + Q e -8sβ* |G| 2 . (5.82) Therefore, from (5.82) and (5.49) we obtain (5.48). Remark 5.16. It is clear that the technique for decomposing e -5sβ * Φ as the solution of three Cauchy problem can be generalized to decomposing e -5sβ * Φ as the solution of N Cauchy problems. This could be useful in situations in which we need to have higher regularity for the solutions of the last two Cauchy problems. It is thanks to this technique why in parabolic and dissipative linear systems with the usual regularity estimates (that is, which satisfy the analogue regularity estimates of either the heat equation or the KdV equation) the effort of getting a Carleman estimate with source term G = 0 or G ∈ L 2 (Q)

7 2 7 2

 77 sβ * (0, T ); H 1 0 (0, L)) ∩ C(e sβ * [0, T ]; L 2 (0, L)),

Lemma 5 . 19 . 0 e -14sβ * Φ 2 H 1 0T 3T / 4 e -14sβ * Φ 2 H 1 0 0 U 0

 51902142100 Let L > 0, ω ⊂ (0, L) be an open subset and assume that (5.4) is satisfied.There is C > 0 such thatT (0,L) + e -7sβ * Φ 2 C([0,T ];L 2 (0,L)) ≤ C Qω e -10sβ * φ 2 + Q e -8sβ * |G| 2 ,(5.84)for Φ the energy solution of (5.8) and (m, λ, s) fixed in the beginning of this section.Proof. [Proof of Lemma 5.19.] It is a straight consequence of (5.40), (5.41) 1 , Proposition 5.7 applied to e -7sβ * Φ in [3T /4, T ] and (5.48) that the following estimate holds :(0,L) + e -7sβ * Φ 2 C([3T /4,T ];L 2 (0,L)) ≤ C Qω e -10sβ * φ 2 + Q e -8sβ * |G| 2 .(5.85)Let χ be a positive function supported in [0, 7/8] such that χ = 1 in [0,3/4]. By considering(5.40), the problem satisfied by χ(t/T )Φ, (5.18) and (5.85) we easily obtain (5.84).Proof. [Proof of Proposition 5.17] Let P 0 be the subspace ofC 3 ([0, T ] × [0, L]) of the functions Φ such that Φ(•, 0) = Φ(•, L) = Φ x (•, 0) = 0, on [0, T ].Let a : P 0 × P 0 → R be the bilinear form a( Φ, Φ) = Qω e -10sβ * φφ + Q e -8sβ * (L * Φ) • (L * Φ) (5.86) and : P 0 → R the linear form (Φ) = L • Φ(0, x)dx + T 0

  ) 15 |ϕ ε | 2 ≤ Cs 34 λ 35

	e -2sα ξ 34 |ϕ ε 1 | 2 ,	(1.15)
	Qω
	où ϕ ε est la solution de (1.14).

* (ξ * Les poids α et ξ sont des fonctions du type (C -h(x))t -m (T -t) -m et h(x)t -m (T -t) -m

respectivement, où h est une fonction qui dépend de Ω et ω avec certaines propriétés géométriques. En plus, ξ * (t) est le minimum de ξ(t, •) sur Ω et α * (t) est le maximum de α(t, •) sur Ω. On peut consulter (2.8) pour avoir la définition exacte des poids.

24 )

 24 Remarque 1.[START_REF] Bárcena-Petisco | Study of the cost of the controllability of second order parabolic equations with small diffusion and a transport term[END_REF]. Les trois premières propriétés du Théorème 1.15 sont des propriétés analogues à celles présentes dans[START_REF] Coron | Singular optimal control : a linear 1-D parabolichyperbolic example[END_REF][START_REF] Guerrero | Singular optimal control for a transport-diffusion equation[END_REF][START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF][START_REF] Glass | Uniform controllability of a transport equation in zero diffusion-dispersion limit[END_REF][START_REF] Carreño | On the non-uniform null controllability of a linear KdV equation[END_REF][START_REF] Carreño | Uniform null controllability of a linear KdV equation using two controls[END_REF]. Par contre, la quatrième propriété est différente et montre (en lisant sa preuve) que les conditions aux limites sont importantes pour avoir de la contrôlabilité avec un coût exponentiellement petit lorsque la viscosité tend vers 0 et le temps est suffisamment grand. La cause de la différence entre Ω 2 et Ω 3

est que la condition d'incompressibilité est plus restrictive en dimension 2 qu'en dimension 3 pour construire des solutions (car en dimension 2 c'est une équation pour deux composantes, tandis qu'en dimension 3 c'est une équation pour trois composantes).

  Une fois les fonctions propres calculées, on cherche une suite v m dans L 2 (Ω 2 ) telle que {u m , v m } soit bi-orthogonal ; c'est-à-dire, tel que :

	1.2. Résultats principaux et plan de la thèse
	29)
	ce qui n'est pas un résultat classique car l'opérateur elliptique associé au système (1.27)
	n'est pas auto-adjoint. Pour prouver (1.29) on prouve d'abord que SV (Ω 2 ) ⊂ span(u m )
	en utilisant des bases trigonométriques connues de L 2 (0, π), ce qui, avec (1.28), implique
	(1.29).

  2. Résultats principaux et plan de la thèseIl suffit de prouver le Théorème 1.21 pour les pseudo-cylindres orientés canoniquement car le laplacien est invariant par rotations. Alors, on prouve le Théorème 1.21 dans ces cas en adaptant la méthode de Fursikov-Imanuvilov. En fait, cette méthode a deux difficultés à vaincre pour être généralisable aux ensembles Lipschitz. La première, et la plus

	importante, est le fait que la construction des fonctions auxiliaires proposée dans [57] ne
	marche que pour les domaines C 2 . En fait, on a besoin d'une fonction η telle que :
	η ∈ C 2 Ω , ∂ n η ≤ 0 sur ∂Ω, inf	|∇η| > 0.	(1.37)
		Ω\ω
	Pour vaincre cette difficulté on construit une fonction auxiliaire en considérant la structure
	des domaines lorsque ω ⊂ C et on transmet l'information d'un domaine de contrôle	à
	l'autre lorsque ω ⊂ Ω \ C.
	La deuxième difficulté est le fait que les solutions à données régulières (et à support
	compact) de l'équation de la chaleur n'appartiennent pas toujours à L 2 (0, T ; H 2 (Ω)). Pour
	la résoudre on approche les domaines avec des domaines réguliers d'une façon compatible
	avec la fonction auxiliaire. Plus précisément :
	Définition 1.24. Soient Ω un domaine, ω ⊂ Ω un sous-domaine et η une fonction satisfai-
	sant :
	η ∈ C 2 Ω , inf	|∇η| > 0.	(1.38)
	Ω\ω
	On dit que Ω est un ouvert compatible avec Ω, ω et η si :
	Ω est un ouvert Lipschitz tel que ω ⊂ Ω ⊂ Ω,	(1.39)

  α ξ

	1.2. Résultats principaux et plan de la thèse
	la direction horizontale :
	2	|χG| 2 ,
	où les poids sont définis dans (1.42) et u est la solution de (1.54).
	On rappelle que la Proposition 1.29 implique le Théorème 1.21 lorsque ω ⊂ Ω \ C (voir,
	par exemple, [57, 56]), ce qui conclut la preuve du Théorème 1.21.
	1.2.4 Contrôlabilité à zéro locale d'un modèle d'interaction forte
	entre deux ondes internes solitaires (voir [17])
	L'équation de Korteweg-de Vries (KdV en abrégé) est un modèle mathématique pour
	les vagues dans un canal peu profond. Elle est donnée par :
	y t + y xxx + yy x = 0, pour t > 0, x ∈ I,	(1.63)
	où I ⊂ R est un intervalle (fini ou infini) et y(t, x) est l'amplitude de l'eau au temps t au
	point x. Évidemment, il faut ajouter la valeur initiale et les conditions aux limites. On
	peut consulter [106] pour comprendre sa déduction et l'interprétation d'un point de vue
	physique.
	Dans cette partie on s'intéresse aux propriétés de contrôle de la version de (1.63) sui-
	vante, qui modélise les interactions fortes de deux ondes de gravité qui se propagent sur

  3. Un travail complémentaire et des problèmes ouvertsOn montre à l'aide de (1.72) que a est un produit scalaire et que est continue sur (P 0 , a). Donc, si P est la fermeture de P 0 par rapport à a, a et admettent un unique prolongement continu sur P . En utilisant le théorème de représentation de Riesz on obtient

	que le problème variationnel suivant :
	a( Φ, Φ) = (Φ), ∀Φ ∈ P,
	admet une unique solution Φ ∈ P . Alors, on prouve que :

•

  Les techniques qu'on utilise pour prouver le Théorème 1.15 sont très spécifiques.

	De fait, on ne peut pas généraliser la preuve à d'autres domaines ou avec d'autres
	conditions aux limites. Comme le cas qu'on a étudié est un cas particulier, les
	résultats obtenus dans le Théorème 1.15 risquent de ne pas pouvoir être généralisés.
	Cela serait intéressant d'étudier le coût de la contrôlabilité dans d'autres domaines
	et avec d'autres conditions aux limites. Par example, le problème (1.22) mais avec
	des conditions aux limites de type Dirichlet reste ouvert.

  .2). 2.44. Another easy consequence of Lemma 2.39 is that, if Ω satisfies Hypothesis 2.1, since P h (S) = Γ, for all p ∈ Ω there is λ ∈ R such that [p, p + λe 2 ] ⊂ Ω and such that p + λe 2 ∈ S. Consequently, since ∂ xy u C 0 (Ω) is estimated by ∂ xy u H 2 (Ω) , if suffices to get an estimate of ∂ x u C 0 (S) to prove Theorem 2.8.

	Remark

  .[START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF] So, once we have(2.59), we have to propagate the estimate in l i . Indeed, we can split, extend and move the segments l i (see the points A 3 , A 4 , C 3 , C 4 , D 2 , D 3 , D 4 in Figure2.2, which allow us to split their respective segment into smaller ones) so that we only have one of the four following possibilities for P

h (l i ) : Figure 2.3 -Case 1 of the proof of Theorem 2.8 1. P h (l i ) is the intersection of Γ with one of the neighbourhoods U p (see statement 6 in Lemma 2.41).

2. P h (l i ) has null intersection with all the neighbourhoods U p and V p . 3. P h (l i ) is one of the neighbourhoods V p (see statement 5 in Lemma 2.41) which has a positive curvature. 4. P h (l i ) is one of the neighbourhoods V p (see statement 5 in Lemma 2.41) which has a negative curvature.

  15/2 e -sα * . (2.67) Summing up, if we combine (2.64) and (2.65), and then do the corresponding absorptions, we have the estimate :

  x) is a solution of (2.21) with null initial value, force -η (T -t)ϕ ε (T -t, x) and boundary Neumann term η(T -t)h(T -t, x). Consequently, if we use (2.23), we get that :

  Summing up, if we combine (2.85)-(2.93) we get that, if m ≥ 8, λ ≥ λ 0 and s ≥ e Cλ (T m + T 2m ) :

	2.B. Proof of Proposition 2.28
	.93)
	terms which can be estimated by the left-hand side of (2.88) and (2.85) respectively.

  .[START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF] Remark 4.19. It is possible for η to take negative values. Indeed, in[START_REF] Fursikov | Controllability of evolution equations[END_REF] they do not need that η ≥ 0 for proving the controllability of the heat equation with Dirichlet boundary conditions. They need it, though, to prove the controllability of other systems, like the Stokes equation or the heat equation with Neumann boundary conditions. Remark 4.20. One consequence of Definition 4.18 is that Ω is connected. Indeed, if Ωi is a connected component, then ω ∩ Ωi = ∅. Otherwise, because of (4.19) and (4.16) 2the maximum of η in Ωi is not reached on the boundary (the tangential derivative of η is non-null), but again because of (4.16) 2 the maximum is not reached in the interior, getting an absurd. Consequently, all the connected components of Ω intersect ω. Since ω is connected, Ω has only one connected component, so Ω is connected.Remark 4.21. Every C 2 domain satisfies(4.18). Moreover, there are some Lipschitz domains which also satisfy it, like (0, 1) d or, in general, any cartesian product of C 2 domains.

	Let us now state the Carleman inequality for domains which satisfy (4.17)-(4.19) :
	Lemma 4.22. Let Ω be a Lipschitz domain, ω ⊂ Ω be a subdomain, η be a function
	satisfying (4.16) and m ≥ 1. Then, there is C > 0 such that for all T > 0, Ω satisfying
	(4.17)-(4.19), g ∈ L 2

  2sα α t ξ|u| 2 . (4.25) By doing weighted Cauchy-Schwarz estimates in (4.25) and considering (4.10) and (4.24)

  2sα |g| 2 .

	(4.29)
	So, combining (4.24), (4.26), (4.28) and (4.29) we obtain (4.21) for all u T ∈ H 1 0 (Ω). Finally,
	we obtain (4.21) for all u T ∈ L 2 (Ω) by density.

  .31) Remark 4.25. As the source term in (4.31) is just in L 2 (0, T ; H -1 (Ω)), we do not (and cannot) have neither the Laplacian nor the time derivative on the left-hand side of (4.30). This section is split in three parts : in Subsection 4.3.1 we prove Theorem 4.6 in the case ω ⊂ C, in Subsection 4.3.2 we prove Theorem 4.6 in the case ω ⊂ Ω \ C, and in Subsection 4.3.3 we make some comments about some technical motivations and possible extensions.

	Remark 4.26. The constant C in Lemma 4.23 and Proposition 4.24 is independent of the
	sequence Ω ε .
	4.3 Proof of Theorem 4.6

  we have that the tangential derivative of η B is null on ∂B. Thus, using (4.34) 2 and that B(x, r) ⊂⊂ B, we obtain that ∂ n B η B (x) = 0 for all x ∈ ∂B.Moreover, from (4.34) 1 and (4.34) 2 we find that ∂ n B η B (x) < 0 for all x ∈ ∂B, which implies (4.35) by compactness.

			.34)
	One important consequence of (4.34) is the inequality :
	sup	∂ n B η B < 0.	(4.35)
	∂B
	Indeed, using (4.34) 1

  .[START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF] In order to prove Proposition 4.28 we state the following technical result whose proof is postponed to Appendix 4.A. Proposition 4.29. Let Ω be a canonically oriented pseudo-cylinder, ω be given by(4.32) and η B ∈ C 2 (B) satisfying (4.34). Then, there is c 0 (Ω, ω, η B ) > 0 such that if η is given by (4.36) with c ≥ c 0 , there exist ε 0 (Ω, ω, η B , c) > 0 and some domains (Ω ε ) ε∈(0,ε 0 ) satisfying Proof. [Proof of Proposition 4.28] Using Propositions 4.29 and 4.24, we obtain that for all λ ≥ C and s ≥ C(T m + T 2m ) :

	(4.17)-(4.19) and :
	Ω ε → d Ω.	(4.40)

  2 |χG| 2 . (4.59) Next, we recall that χ ∈ C 2 Ω , that : supp(∆χ), supp(∇χ) ⊂ Ω \ B(x, 2r) × (r, z + 2r) , and (4.53) 2 . These assertions imply that by taking λ ≥ C and s satisfying (4.38) the term of F (∇χ, u) in the right-hand side of (4.59) can be absorbed. Thus, we get :

  2 |χG| 2 . (4.60)

	Finally, let us state what we have proved :
	Proposition 4.32.

3 .

 3 Validity of the partial results in any pseudo-cylinder. Even if we have stated and proved the partial results (seePropositions 4.28, 4.29 and 4.32) in canonically oriented pseudo-cylinders, these results hold in any pseudo-cylinder since the Laplacian operator is invariant under rotations (and translations).4. The dependence of c with respect to η B . It seems impossible to approximate a domain Ω by some subdomains Ω ε satisfying (4.17)-(4.19) without having ∂ n η ≤ 0 on ∂Ω (though this is an interesting open question). This implies that if Ω is a pseudo-cylinder, if ω is given by (4.32) and if η is the auxiliary function defined in(4.36), we need to take c large enough with respect to η B . Indeed, let us compute ∂ n η and verify that it is negative on all the subsets of the boundary given by Remark 4.2.

  if we had a function η satisfying (4.16) which is much bigger in the interior of the cylinders than close to L. However, it seems incompatible with picking in (4.36) c large with respect to η B . 6. Semi-linear heat equation. A problem that remains open is the controllability of the semi-linear heat equation stated in pseudo-cylinders with a non-linearity acting also in the first order term. Indeed, we cannot just follow the steps in[START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] because they require for Ω to be C 2 when they prove that the Carleman estimate implies the null-controllability of the semi-linear heat equation (more specifically, they use that D(∆) ⊂ H 2 (Ω) continuously and the analogous results for L p -spaces).

	idea could work
	.61)
	is necessary to ensure that ∂ n η ≤ 0 on T (the denominator in (4.61) is not null
	by (4.33)).
	5. Problems when extending this method to Lipschitz domains. A problem
	that remains open is the null controllability of the heat equation in any Lipschitz
	domain. A natural attempt following the Fursikov-Imanuvilov approach is to split
	the domain into a finite amount of pseudo-cylinders and a single compactly included
	C 2 subdomain, and then to apply a Carleman estimate in each subdomain (with
	the help of some cut-off functions). To that end, we need to absorb what is in a
	neighbourhood L (see Remark 4.2) with estimates in some other pseudo-cylinders
	and then use the same technique as Section 4.3.2 to pass from a finite number of
	control domains that are compactly contained in Ω to a single control domain. This

  Step 5 : proof of the regularity of the domains Ω ε . Next, let us prove that if ε <<< ε 0 (Ω, ω) the open sets Ω ε satisfy (4.17) and (4.18). Indeed, if ε <<< ε 0 (Ω, ω) we have by which implies that ω ⊂ Ω ε . Consequently, we only have to prove that if ε <<< ε 0 (Ω) the domains Ω ε are C 2 (see Remark 4.21), which we will prove by using Lemma 4.34. So we fix ε <<< ε 0 (Ω) and check that the domain Ω ε satisfies the requirements of Lemma 4.34 : First, it is clear that D ε is a function which belongs to C 2 Ω (ρ ∈ C 2 (supp(χ ε )) by(4.66)). Moreover, it satisfies D ε ≤ 0 on ∂Ω. Indeed, we consider (4.76) and the division of the boundary as stated in Remark 4.2 :

	(4.40) that :
	sup	d(•, ∂Ω) < d(ω, ∂Ω),
	Ω\Ωε

.81) So, from (4.78)-(4.81) we obtain for all ε <<< ε 0 (Ω) and (x, z) ∈ Ω \ Ω ε that : d((x, z), ∂Ω) ≤ Cε 1/4 , which implies (4.40).

  Consequently, thanks to Lemma 4.34, if ε <<< ε 0 (Ω), we have that Ω ε is C 2 and :

	.87)
	So, from (4.87), (4.86), ε <<< ε 0 (Ω), (4.74), (4.66), (4.68) and (4.69) we obtain that
	(4.83) implies that ∇ x D ε (x, z) = 0, which proves (4.82).

  .88) Remark 4.37. In order to prove (4.84) we have only used (4.83) 1 and that ε <<< ε 0 (Ω).

	Consequently, if ε <<< ε 0 (Ω), considering (4.88), (4.84) and that B 2ε ⊂ χ -1 ε ({1}) we
	find that :

  .[START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] Intuitively, the proof consists on using that ∂Ω ε is near ∂Ω and that n ε somehow approximates n. Indeed, we follow different (but a finite number of) approaches depending if we are close to B × {0}, L or T (see Remark 4.2 for the notation). This is necessary because η and n ε have different behaviours in different parts of the boundary (they do

	not present any symmetry), so considering only (4.85) does not suffice and we have to be
	more precise.

  The first one concerns the points near L, whereas the second one concerns the points near T.Case 2.1 : if (x, z) ∈ ∂Ω ε,t and if ρ(x) ≤ √ κ 2 ε. Due to (4.109) we have the bound :

  .120) So, from (4.111), (4.120) and ε <<< ε 0 (Ω, ω, η B , c) we obtain (4.91).

	Remark 4.39. Again, note that for some control domains there are (x, z) ∈ ∂Ω ε,t satisfying ρ(x) ≤ √ κ 2 ε and (4.119). In fact, (4.119) provides us some additional information because √ ε >>> ε for ε small. Moreover, this does not suppose a contradiction with (4.85) because
	it just means that C > 4 in (4.119).

Case 2.2 : if (x, z) ∈ ∂Ω ε,t and if H ε (x) -z ≤ √ κ 2 ε.

We have from (4.109) the inequality :

  2sα ξ 2 |θ xx | 2 + s 4 λ 5 Q e -2sα ξ 4 |θ x | 2 + s 6 λ 7 Q e -2sα ξ 6 θ 2 Qω e -2sα ξ 2 |θ xx | 2 + s 6 λ 7 Qω e -2sα ξ 6 θ 2 + -2sα |θ t | 2 .(5.67)

	≤ C sλ	e -2sα ξ κ1 κ2 φxx -5sβ * t (ϕ xxx + φxxx ) -5s(∂ t + ∂ 3 x ) β * t (φ + φ)	2
	Q
		T
	+ s 2 λ 3
		0	{0,L}

e

Remerciements

By taking the weak limit of ϕ ε k 2 , the strong limit of ε k ∆ψ (to 0) and the strong limit of ϕ ε k 2 (0, •) (to 0), we get from (3.76) that :

We have that equation (3.77) is true in particular for ψ(t, x) = ϕ T 2 (x 1 , x 2 + T -t), which belongs to D([0, T ] × Ω). Consequently, we get that ϕ T 2 L 2 (Ω 2 ) = 0. Since ϕ T ∈ H(Ω 2 ) ∩ C 1 Ω 2 this implies that ϕ T 1 = 0, contradicting the fact that ϕ T = 0.

Lemma 3.19. Let h ∈ (0, π), T ∈ (0, π -h) and δ ∈ (0, π -T -h). Then, if ϕ T ∈ H(Ω 2 )

such that supp(ϕ T ) ⊂ (0, π) × (π -δ, π) there is c > 0 such that for all ε ∈ (0, 1) :

for ϕ ε the solution of (3.3) in Ω 2 and for initial value ϕ T .

For the following proof we fix a parameter δ > 0 so that T ∈ 0, π-h-δ 1+ δ

.

Proof. First of all, we recall that ϕ ε 2 satisfies (3.7) for z T := ϕ T 2 . Multiplying (3.7) 1 by

and integrating in Ω 2 we deduce that for all t ∈ [0, T ] we have the following equality :

Chapitre 4

Null controllability of the heat equation in pseudo-cylinders by an internal control

In this chapter we prove the null controllability of the heat equation in domains with a cylindrical part and limited by a Lipschitz graph. The proof consists mainly on getting a Carleman estimate which presents the usual absorption properties. The main difficulty we face is the loss of existence of the usual weighted function in C 2 smooth domains. In order to deal with this, we use its cylindrical structure and approximate the system by the same system stated in regular domains. Finally, we show some applications like the controllability of the semi-linear heat equation in those domains. This chapter is presented in [START_REF] Bárcena-Petisco | Null controllability of the heat equation in pseudocylinders by an internal control[END_REF].

Introduction

The null controllability of the heat equation has been an interesting research topic in the last 40 years. Regarding this topic, in this paper we focus on domains ; that is, on bounded connected non-empty open sets of R d . There are three main approaches in the literature :

• The first one uses the observability estimates of the abstract wave equation to prove the null controllability of the abstract heat equation. This method dates back to 121 So, from (4.123), (4.74) and (4.33) we find that :

Consequently, from (4.122), (4.124) and c >>> c 0 (Ω, ω, η B ) we obtain (4.91).

4.B

The existence of a pseudo-cylinder which is not locally star-shaped

We show in this section the existence of a pseudo-cylinder in R 2 which is not locally star-shaped. The structure of this section is the following : first we recall the definition of locally star-shaped as stated in [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF], but adapted to our notation, second we construct the pseudo-cylinder, and thirdly we prove that it is not locally star-shaped.

Definition 4.40. A Lipschitz domain Ω is locally star-shaped if for all p ∈ ∂Ω there is r p > 0 and q p ∈ Ω such that |p -q p | < r p and B(q p , r p ) ∩ Ω is star-shaped with center in

Remark 4.41. As a consequence of [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF]Remark 3.5] we can choose r p as small as needed.

This implies that a locally star-shaped domain can also be defined as a Lipschitz domain Ω such that for all p ∈ ∂Ω and all r > 0 there is some r ∈ (0, r) and q ∈ Ω satisfying |p -q| < r and that B(q, r) ∩ Ω is star-shaped with center in q.

We define :

We have that Ω clearly is a pseudo-cylinder (take U (x, z) = (x, 3 -z), B = (0, 1) and

for an approximate illustration of Ω near each point

Let us first prove that :

We remark that for all (x, z) ∈ T 1,i we have that :

Therefore, proving (4.129) reduces to finding the maximum of G 1 on T 1,i , which is a simplex. Since G 1 is an affine function, it is well-known that the maximum is reached on the vertices, which in this case are given by (4.127). Since : < |(x, z)| 2 for all (x, z) ∈ T 1,i , which in particular implies (4.129).

We define :

We have that g is well-defined because of (4.128). In fact, is the segment with one end in q and containing 2 3 i+2 , 2 3 i+1 because of (4.129) and because g 2 3 i+2 = 2 3 i+1 . Moreover, we recall that :

In addition, using (4.126) 3 and (4.128) we obtain that :

Considering that in a neighbourhood of 2 3 i+2 , 2 3 i+1 the set ∂Ω is a segment of slope -6 (see (4.125) for i + 2 instead of i) and (4.133), we get by continuity for some ε q > 0 small enough that :

Consequently, is a segment of length |q|, with one end in q and such that ∩ Ω is not connected.

Case 3 : if z q ≥ 3x q and if there is i ∈ N * such that z q ∈ [2/3 i+1 , 2/3 i ) and z q ≥ 2 3 i -6x q (see q 3 in Figure 4.5). Under these hypotheses q is in the triangle :

whose vertices are given by :

It can be deduced from (4.134) that :

We remark that the horizontal distance of the points in the line z = 2 3 i -6x with respect to the line z = 2 3 i-1 -6x is given by 2 3 i+1 (both lines are parallel). This implies that the vertical distance of q to the line z = 2 3 i-1 -6x is bounded by 2 3 i+1 . Moreover, because of (4.135) we have that |(x, z)| > 2 3 i+1 for all (x, z) ∈ T 2,i . Thus, the segment = {(x, z q ) : x ∈ [x q , x q + |q|]} is of length |q|, one of its ends is q, and satisfies that ∩ Ω is not connected, as 2 3 i+1 , z q ∈ ∩ (R 2 \ Ω) and 1 3 i -zq 6 + ε q , z q ∈ ∩ Ω for some ε q > 0 small enough (see (4.125) for i + 1 instead of i).

Remark 4.43. The construction given in this section can clearly be generalized to higher dimensions.

Chapitre 5

Local null controllability of a model system for strong interactions between internal solitary waves

In this chapter we prove the local null controllability property for a nonlinear coupled system of two Korteweg-de Vries equations posed on a bounded interval and with a source term decaying exponentially on t = T . The system was introduced by Gear and Grimshaw to model the interactions of two dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in L 2 have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system. This chapter is included in [START_REF] Bárcena-Petisco | Local null controllability of a model system for strong interaction between internal solitary waves[END_REF].

Introduction

Nonlinear dispersive partial differential equations naturally appear as models describing wave phenomena in various branches of physics and engineering, such as quantum mechanics, nonlinear optics, plasma physics, water waves, and atmospheric sciences. They system with initial data (u 0 , v 0 ), source term (h 1 , h 2 ) and control f :

(5.3) Under these conditions, considerations will be given to the following null controllability problem :

Given T > 0, an initial state (u 0 , v 0 ) in a certain space, and a source term (h 1 , h 2 ) in a certain space, can one find an appropriate control f , so that the system (5.3) admits a solution (u, v) which satisfies (u(T, •), v(T, •)) = (0, 0) ?

Our main result gives a positive answer to this question. The proof combines the analysis of the linearized system and an inverse function theorem for the full system. In order to analyze the linearized system, we follow the classical duality approach [START_REF] Dolecki | A general theory of observation and control[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF],

which reduces the null controllability property to the proof of an observability inequality for the solutions of the corresponding adjoint system. Here, this is done by using Carleman estimates.

In order to overcome some technical difficulties that will become clear later, we assume that the coefficients a, b and c satisfy a, b, c > 0 and a 2 b < 1.

(5.4)

To our knowledge, this problem has not been addressed in the literature and the existing developments do not allow to give an immediate answer to it. Indeed, the controllability problem for system (5.1) was first solved when the control acts through the boundary conditions [START_REF] Capistrano-Filho | Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictions on the spatial domain[END_REF][START_REF] Capistrano-Filho | Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain[END_REF][START_REF] Cerpa | A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF][START_REF] Micu | On the controllability of a coupled system of two Korteweg-de Vries equations[END_REF]. The results are obtained combining the analysis of the linearized system and the Banach's fixed-point theorem. In order to study the linear problem, the authors follow closely the ideas introduced by Rosier [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] while studying the boundary controllability of the KdV equation (based on compactness arguments, a spectral analysis of the state operator and multipliers). But, being different from other systems, the length L of the spatial domain may play a crucial role in determining the controllability of the system, specially when some configurations of the controls input are allowed to be used.

there is δ > 0 such that for all U 0 < δ there is f ∈ L 2 (Q ω ) and U ∈ C([0, T ]; L 2 (0, L))

satisfying that (U, f ) is a solution of (5.3) and U (T, •) = 0.

As usual, we obtain the null controllability of (5.3) by proving the null controllability of the linearized system around 0. That system is given by

for h 1 , h 2 ∈ L 2 (0, T ; H -1 (0, L)) decaying exponentially at t = T as specified in Section 5.4.1, U 0 ∈ L 2 (0, L) and

First, in Section 5.2 we study the main properties of the associated Cauchy problem, which is given by

for U 0 ∈ L 2 (0, L) and G ∈ L 2 (0, T ; H -1 (0, L)). Afterwards, in order to show the null controllability of (5.5) we follow the classical duality approach [START_REF] Dolecki | A general theory of observation and control[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF], which reduces the problem to prove an observability inequality for the solutions of the corresponding adjoint system, which is given by

for Φ T ∈ L 2 (0, L), G ∈ L 2 (Q) and

(5.9)

In fact, we study the main properties of (5.8) in Section 5.2, and then we prove a Carleman estimate in Section 5.3.

which are strictly positive and different by (5.4). Thereby, there is a matrix P such that

Then, if Φ := P Φ * , (5. [START_REF] Carreño | On the non-uniform null controllability of a linear KdV equation[END_REF] we have that Φ satisfies the system

for

.

(5.26)

Clearly, it suffices to get the estimates for system (5.25), which is easier because the coupling appears only on the first-order spatial term.

Step 3 : proof of (5.18). If we multiply (5.25) by Φ, after integrating by parts and applying Cauchy-Schwarz inequality we obtain

L 2 (0,T ;H 1 0 (0,L)) .

(5.27)

Thus, if we multiply (5.25) by (L + 1 -x) Φ, (5.18) is obtained by using classical estimates, integration by parts, (5.27) to deal with the first order term and (5.24).

Step 4 : proof of (5.19). We recall that from now on ΦT = P (Φ T ) * = 0.

(5.28)

Thanks to (5.25) we know that system (5.8) behaves in the higher order terms as two independent KdV equations. In particular, by considering (5.28), (5.21) (g is taken as the components of G + M Φx ) and (5.18) we find that

(5.29)

Hence, we obtain from (5.24), (5.26) and (5.29) that

(5.30)

We can easily get (5.19) from (5.30) and (5.8).

Step 5 : proof of (5.20). Considering (5.28), we remark that Ψ := Φxxx is the solution

Using that G ∈ L 2 (0, T ; H 3 0 (0, L)) and considering that the solution can be defined by transposition as in [62, Proposition 1] we find the following estimate : (5.31) Recalling that Ψ = Φxxx and (5.29) we obtain from (5.31) that

(5.32)

Consequently, using the following trace estimate

we find from (5.24), (5.26) and (5.32) that

In addition, the L 2 -norm of Φ t is estimated from (5.8) and (5.33), and thus we obtain (5.20).

Step 2 : Decomposition of Φ. We consider that e -5sβ * Φ = Φ + Φ + Φ, (

for Φ := ( φ, φ) the solution of the system

Φ := (ϕ, φ) the solution of the system

and Φ := ( φ, φ) the solution of the system

We remark that because of (5.49), Φ satisfies :

(5.53) Moreover, from Proposition 5.7, Corollary 5.9 and Corollary 5.10 we deduce that

Step 3 : estimates of the solutions of (5.50) and (5.51). First of all, we obtain from (5.19) that : Φ L 2 (0,T ;H 2 (0,L)) ≤ q(T ) e -5sβ * G L 2 (Q) .

(5.54)

We recall that q stands for a generic polynomial of positive coefficient that might be different each time it appears.

It can be proved easily that U is the unique solution by transposition of

(5.89)

Indeed, let Φ be the solution of (5.8) for G ∈ D(Q) and for Φ T = 0. Then, we have by Lemma 5.6 that Φ ∈ C ∞ (Q) ∩ P 0 , which implies that

So we just have to use the density of D(Q) in L 2 (0, T ; H 1 0 (0, L)). Consequently, since f, h 1 , h 2 ∈ L 2 (0, T ; H -1 (0, L)), we obtain from Remark 5.11 that

As for the estimates, from (5.88) we get

Moreover, since U is the solution by transposition of (5.89) we have that LU -(0, f 1 ω ) = (h 1 , h 2 ) ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)).

(5.91)

Finally, we consider that Ũ = e 7 2 sβ * U satisfies :

Moreover, from U ∈ L 2 (e 4sβ * (0, T ); H -1 (0, L)) and (5.83) we obtain

Consequently, from Remark 5.11 we find U ∈ L 2 (e (5.92)

Thereby, we obtain from (5.90)-(5.92) that (U, f ) ∈ E, which along with being the solution of (5.89) completes the proof.

Proof of Theorem 5.1

We now prove the local null controllability of system (5.3) by a local inversion argument.

We recall that the components of U are denoted by u and v. We consider h 1 , h 2 as in the statement of Theorem 5.1, with C large enough so that h 1 , h 2 ∈ L 2 (e 7sβ * (0, T ); H -1 (0, L)).

Let G : E → L 2 (e 7sβ * (0, T ); H -1 (0, L)) × L 2 (0, L) be an operator defined by :

Hence, by Remark 5.18 we just have to check those two hypotheses to conclude the proof of Theorem 5.1 :

• In order to see that G is C 1 , since G is the sum of an affine form and some bilinear forms, it suffices to see that it is a bounded operator. The fact that the affine form is continuous is a straight consequence of the definition of E. As for the bilinear terms of (5.93), we can prove that they are bounded by considering that for all w 1 ∈ L ∞ (e • As for the surjectivity of DG[0], we remark that

thus, it is a consequence of Proposition 5.17.

Some additional remarks

The technique presented in this paper can be used to solve other controllability problems :

5.4. The nonlinear control problem

• We can prove the local null controllability of the following control system assuming (5.4). Indeed, we can do that by replicating the proof of Theorem 5.1 step by step.

• We can slightly improve the result presented in [START_REF] Carreño | Internal null controllability of the generalized Hirota-Satsuma system[END_REF] and show that the general Hirota-Satsuma system can be controlled with two forces supported in any two control domains (in [START_REF] Carreño | Internal null controllability of the generalized Hirota-Satsuma system[END_REF] the authors need that one of the control domains touches the boundary). The proof is very similar to the one of Theorem 5.1, as we only have to do some minor adjustments.