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Je souhaiterais remercier les membres du laboratoire LaMME où j'ai

Je veux exprimer mon entière

Dédicace

À ma mère Marina et ma soeur Evelyn.

Chapter 1 Introduction

Pour l'étude des fluides visqueux incompressibles, les équations de Navier-Stokes sont un modèle fondamental. Dans cette thèse, nous nous proposons d'étudier les équations de Navier-Stokes sur l'espace tout entier, ainsi on oublie les conditions de bord, et on dispose d'un arsenal d'outils provenant de l'analyse harmonique pour son étude. Nous présentons les équations :

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0 ,
où, la vitesse du fluide dénotée par u : [0, +∞) × R 3 → R 3 et la pression du fluide p : [0, +∞) × R 3 → R sont les inconnues, tandis que les données initiales sont la vitesse initiale du fluide dans le temps t = 0 : u 0 : R 3 → R 3 ; et le tenseur de force F = (F i,j ) 1≤i,j≤3 (avec

F i,j : [0, +∞) × R 3 → R) dont la divergence ∇ • F définie par (∑ 3 i=1 ∂ i F i,1 , ∑ 3 i=1 ∂ i F i,2 , ∑ 3 i=1 ∂ i F i,3
), représente la force appliquée au fluide. Dans la littérature, nous trouvons souvent le terme (u • ∇)u réécrit comme ∇ • (u ⊗ u) où l'on définit u ⊗ u = (u i u j ) 1≤i,j≤3 .

Même si les équations (NS) ci-dessus sont un système simplifié après avoir considéré l'espace R 3 tout entier au lieu d'un domaine et d'avoir pris la constante de viscosité du fluide comme étant ν = 1, la problématique de fond continue à être la même, et la compréhension des notions comme la turbulence et l'explosion sont encore le vif du sujet.

Une manière importante de diviser l'étude de ces équations se fait en considérant la notion d'énergie.

Une solution u des équations (NS) sans force extérieure, c'est à dire avec F = 0, issue d'une donné initiale u 0 , est dite une solution d'énergie finie pour une famille de fonctionnelles d'énergie E(t, •), indexée par t, si E(0, u 0 ) < +∞ et t → E(t, u) est décroissante, ces fonctionnelles étant définies positives sur l'espace de fonctions de [0, t] × R 3 dans R 3 , pour tout t ≥ 0.

L'estimation de fonctionnelles d'énergie pour obtenir un contrôle des approximations de solutions et pouvoir ainsi passer à la limite par un argument de compacité de type Aubin-Lions est une idée qui remonte aux travaux fondateurs de Leray en 1934[START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF]. Dans l'étude de solutions du système de Navier-Stokes incompressible, un rôle clé est joué par l'inégalité d'énergie de Leray

u(t) 2 L 2 + 2 t 0 ∇u(s) 2 L 2 ds ≤ u(0) 2 L 2 -2 ∑ 1≤i,j≤3 t 0 F i,j ∂ i u j dx ds,
Leray a construit, pour une donné initiale L 2 , des solutions faibles globales qui satisfont cette inégalité d'énergie, où le terme énergie fait référence à la fonctionnelle E(t, u) = u(t) 2 . L'étude de l'unicité et régularité pour les solutions d'énergie finie de Leray est encore un problème principal.

Récemment, dans [START_REF] Feichtinger | Discretely self-similar solutions for 3D MHD equations and global weak solutions in weighted L 2 spaces[END_REF], H. Feichtinger, K. Gröchenig, Kuijie Li and Baoxiang Wang ont donné un sens à l'unicité du problème pour certains données initiales dans L 2 , en se restreignant aux fonctions dont les fréquences vivent dans le premier octant de l'espace R 3 , et aussi en sortant du cadre de distributions car ils ont utilisé un espace différent de fonctions test.

Pour le cas périodique, c'est à dire, lorsque l'on cherche une solution u : [0, T) × T 3 → R 3 aux équations de Navier-Stokes, il y a eu de nombreuses avancées par rapport a la non-unicité des solutions faibles d'énergie finie.

Quelques travaux qui ont fait partie de ces récentes avancées sont [START_REF] Lellis | On turbulence and geometry: from Nash to Onsager[END_REF], (Buckmaster and Vicol, 2019b) et (Buckmaster and Vicol, 2019a). Ils ont développé une technique, qu'on appelle intégration convexe, pour construire des solutions non uniques.

Cependant, ces résultats ne sont pas exactement liés aux solutions de Leray dans l'espace tout entier. L'appartenance de ces nouvelles solutions non uniques à l'espace L 2 ((0, T), H 1 ) n'est pas connue, et donc le problème de la non-unicité des solutions de Leray dans l'espace tout entier reste ouvert.

Il existe une version locale de l'inégalité d'énergie de Leray liée à l'existence d'une mesure µ, positive et localement finie, pour laquelle la solution u satisfait

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • |u| 2 2 u -∇ • (pu) + u • (∇ • F) -µ.
Cette balance d'énergie locale a été exploitée en 1995 par Farwig et Sohr [START_REF] Farwig | Weighted energy inequalities for the Navier-Stokes equations in exterior domains[END_REF] pour montrer l'existence de solutions faibles dans le cas du domaine extérieur.

Les solutions faibles d'énergie infinie ont été introduites ensuite par Lemarié-Rieusset en 1999 [START_REF] Lemarié-Rieusset | Solutions faibles d'énergie infinie pour les équations de Navier-Stokes dans R 3[END_REF]. Cela a permis de montrer l'existence locale de solutions faibles pour une donnée u 0 uniformément localement de carré intégrable, c'est à dire, telle que

u 0 L 2 uloc = sup x∈R 3 B(x,1)
|u 0 | 2 dy < +∞, et un tenseur de force F ∈ (L 2 t L 2 x ) uloc ((0, 1) × R 3 ), c'est à dire qui satisfait sup

x 0 ∈R 3 1 0 B(x 0 ,1)
|F| 2 dxds < +∞.

Motivation

Nous introduisons une des définitions pionniers.

Définition. (Solution locale de Leray) Un champ de vecteurs u = (u 1 , u 2 , u 3 ) ∈ L 2 loc ([0, T 0 ) × R 3 ) est appelée une solution locale de Leray du problème de Navier-Stokes (NS) associé a une vitesse initiale à divergence nulle u 0 ∈ L 2 uloc et à un tenseur de force F = (F i,j ) ∈ (L 2 t L 2 x ) uloc ((0, T 0 ) × R 3 ) si :

• Pour tout t < T < T 0 , sup 0≤t<T sup

x 0 ∈R 3 B(x 0 ,1)

|u| 2 dx + sup

x 0 ∈R 3 T 0 B(x 0 ,1)
|∇u| 2 dxds < +∞.

• Il existe une distribution p sur (0, T 0 ) × R 3 tel que (u, p) est solution faible du système   

∂ t u = ∆u -∇ • (u ⊗ u) -∇p + ∇ • F ∇ • u = 0,
c'est à dire, pour toute ϕ ∈ D((0, T 0 ) × R 3 ) et pour toute j ∈ {1, 2, 3}, on a

- (0,T)×R 3 u j ∂ t ϕ = (0,T)×R 3 u j ∆ϕ + 3 ∑ i=1 (0,T)×R 3 u i u j ∂ i φ + p, ∂ j ϕ - 3 ∑ i=1 (0,T)×R 3 F i,j ∂ i ϕ et 3 ∑ i=1 (0,T)×R 3 u i ∂ i ϕ = 0 • Pour tout compact K ⊂ R 3 , on a lim t→0 u(t, .) -u 0 L 2 (K) = 0
• u est adaptée au sens de Caffarelli-Kohn-Nirenberg : p ∈ (L

3 2 L 3
2 ) loc et il existe une mesure positive localement finie pour laquelle

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • |u| 2 2 u -∇ • (pu) + u • (∇ • F) -µ.

Motivation

À ce point, nous soulignons le fait qu'un traitement astucieux pour le terme de la pression est nécessaire dans le contexte L 2 uloc , par exemple, pour obtenir les contrôles à priori des solutions du problème approché. En (Lemarié-Rieusset, 2016), on fait varier la définition de la pression, selon la partie de l'espace où l'on veut obtenir un contrôle uniforme pour les solutions approchées.

Dans les dernières années, cette notion de solution locale de Leray a été légèrement modifiée plusieurs fois, soit pour rajouter des conditions sur la pression ou soit pour considérer des données initiales dans des espaces qui ne sont pas contenus dans L 2 uloc et où on ne connaissait pas à priori si la pression était caractérisée. On trouve, dans la littérature l'utilisation des expansions locales de la pression pour analyser l'existence, unicité et régularité des solutions du système de Navier-Stokes, comme l'ont fait (Bradshaw and Tsai, 2020a), (Bradshaw and Tsai, 2020b) et [START_REF] Kang | An -regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data[END_REF].

En réfléchissant à ces aspects, on pourrait se demander sous quelles hypothèses la pression est déterminée, modulo les constantes, par la vitesse et la force. Si on détermine, dans un cadre assez général, une formule globale qui donne la pression en fonction de la vitesse et la force, les différences entre ces récentes définitions des solutions adaptées seront raccourcies. D'autres constructions de ces solutions d'énergie infinie à donnée initiale uniformément localement L 2 ont été données en 2006 par Basson [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF]) et en 2007 par [START_REF] Kikuchi | Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality[END_REF].

Basson a étudié le cas d'une donnée initiale appartenant à L 2 uloc (R 2 ) dans sa thèse [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF], où il a démontré existence globale, unicité, régularité et dépendance continue de la donnée initiale. Basson a aussi étudié un espace de données légèrement plus grand, appelé B 2 (R 2 ), et définie par la norme

u 0 2 B 2 = sup R≥1 1 R 2 B(0,R) |u 0 | 2 dx.
Dans ce cas, pour une solution u ∈ L 2 loc d'un système de Navier-Stokes approché et pour ϕ une fonction test, le terme (∇p • u )ϕdx, qui fait intervenir la pression est divisé en deux morceaux, un où p se contrôle comme |u | 2 et l'autre où ∇p est bien comporté, afin d'obtenir contrôles uniformes pour les solutions approchées u . Avec cette astuce, il arrive à démontrer existence d'une solution locale, mais une formule pour la pression n'est pas explicitée.

Dans les dernières années l'étude du terme de la pression a été un sujet actif, voir par exemple [START_REF] Kukavica | On local uniqueness of solutions of the Navier-Stokes equations with bounded initial data[END_REF], [START_REF] Kukavica | On local uniqueness of weak solutions to the Navier-Stokes system with BMO -1 initial datum[END_REF] et [START_REF] Wolf | On the local pressure of the Navier-Stokes equations and related systems[END_REF]. On dédiera une bonne partie de cette thèse à l'analyse du terme de la pression.

Jia et Šverák ont surpris en 2014 [START_REF] Jia | Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions[END_REF] en donnant une construction de solutions auto-similaires à grandes données régulières. L'ingrédient principal et l'idée originale a été de faire intervenir la théorie du degré de Leray-Schauder. Le résultat a été étendu en 2016 par Lemarié-Rieusset (Lemarié-Rieusset, 2016) pour des solutions à données localement L 2 . L'observation cruciale à ce stade est qu'une donnée auto-similaire (homogène de degré -1) et localement L 2 est automatiquement uniformément localement L 2 .

En 2017 et 2018, [START_REF] Bradshaw | Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations II[END_REF] et Chae et [START_REF] Wolf | Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in L 2 loc (R 3 )[END_REF] ont exploré le cas des solutions auto-similaires pour un sousgroupe discret de dilatations et ils ont donné une réponse positive. Une donnée initiale localement L 2 n'est plus nécessairement uniformément localement L 2 , donc les idées précédentes de Lemarié-Rieusset ne pouvaient pas être appliquées. Dans le travail de Chae et Wolf, où ils considèrent des hypothèses très faibles pour la donnée initiale, juste être discrètement auto-similaire et localement L 2 , une inégalité d'énergie modifiée a été introduite dans les définitions, un détail corrigé en (Bradshaw and Tsai, 2019a). Un beau résumé de la série de résultats en rapport avec la notion d'auto-similarité et d'auto-similarité discrète est donné en (Bradshaw and Tsai, 2019b).

Nous observons à ce point le fait suivant : si on considère un nombre λ > 0 fixé et u un champ de vecteurs tel que λu(λx) = u(x), alors u appartient à L 2 ((1 + |x|) -γ dx) pour tout γ > 1. Cette remarque et les travaux récents de recherche autour des solutions auto-similaires nous ont amené à développer dans (Fernández-Dalgo and Lemarié-Rieusset, 2020b) une extension de la procédure de Leray (pour obtenir des solutions faibles) de L 2 à L 2 ((1 + |x|) -γ dx) avec 0 ≤ γ ≤ 2; on a aussi révisité les résultats en rapport avec les solutions discrètement auto-similaires.

Des améliorations et des raffinements ultérieurs de ces résultats ont été obtenus plus récemment, voir par exemple [START_REF] Bradshaw | Existence of global weak solutions to the Navier-Stokes equations in weighted spaces[END_REF]. Nous présentons dans ce mémoire les résultats ultérieurs.

En rapport avec les solutions axisymétriques, nous savons que Ladyzhenskaya [START_REF] Ladyzhenskaya | On the unique global solvability of the Cauchy problem for the Navier-Stokes equations in the presence of the axial symmetry[END_REF]) à utilisé un contrôle d'énergie sur la vorticité pour démontrer l'existence d'une solution axisymétrique globale sans tourbillon pour une donnée initiale appartenant à H 1 . Bien que ce contrôle soit sous-jacent à la structure des champs de vecteurs axisymétriques sans tourbillon, il est raisonnable de chercher à adapter des résultats analogues pour des espaces à poids radial. C'est ce qu'on a fait en (Fernández-Dalgo and Lemarié-Rieusset, 2021).

Organisation du document

Nous dédions la première partie de cette thèse, le Chapitre 2, à exposer les résultats obtenus en (Fernández-Dalgo and Lemarié-Rieusset, 2020a) qui donnent une caractérisation du terme de la pression dans un contexte assez général pour être appliquée aux solutions d'énergie infinie étudiées jusqu'à présent.

Dans le Chapitre 3, en suivant (Fernández-Dalgo and Lemarié-Rieusset, 2021), nous développons une procédure générale, pour obtenir des solutions faibles dans des espaces L 2 à poids. Cette procédure générale se base en certaines propriétés du poids qui permettent de contrôler le terme de la pression. L'importance de ces résultats se met en évidence dans le chapitre suivant.

Nous présentons des résultats récents d'existence de solutions régulières axisymétriques sans tourbillon dans le Chapitre 4, où nous considérons de données initiales appartenant à des espaces L 2 à poids et dont la vorticité elle aussi appartient à un espace L 2 à poids. Les contrôles obtenus dans le chapitre précédent, le Chapitre 3, sont cruciales parce que les hypothèses sur les poids permettent de considérer des poids avec une plus grande décroissance à l'infini, par rapport à des contrôles obtenus précédemment.

Pour l'étude de l'existence des solutions discrètement auto-similaires nous dédions le Chapitre 5. Nous remarquons que dans le théorème d'existence intervient les espaces à poids L 2 (w γ dx), avec w γ = (1 + |x|) -γ et γ > 1 et la condition γ > 1 est optimale dans le sens qu'une donné initiale discrètement auto-similaire non nulle n'appartient pas à L 2 ((1 + |x|) -1 dx), Ces résultats ont été présentés dans (Fernández-Dalgo and Jarrín, 2021a) dans le contexte des équations de la magnétohydrodynamique.

Dans le chapitre 6, nous revenons sur plusieurs résultats, obtenues pour les équations de Navier-Stokes dans les chapitres précédentes, mais cette fois dans le contexte des équations de la magnétohydrodynamique. Nous aussi ajoutons un nouveau résultat d'unicité fort-faible qui considère des espaces L 2 à poids.

Dans la suite, nous allons décrire de manière plus précise quelques-uns des résultats les plus importants.

Nos premiers efforts visent à donner une formule pour le terme de la pression dans des conditions très générales. Les conditions qu'on a trouvées sont décrites dans le théorème suivant :

Théorème. Considérons la dimension d ∈ {2, 3} et 0 < T < +∞. Soit F(t, x) = F i,j (t, x) 1≤i,j≤d un tenseur qui appartient à L 1 ((0, T), L 1 (R d , w d+1 dx)).

Soit u une solution du problem

   ∂ t u = ∆u -∇ • (u ⊗ u) -S + ∇ • F ∇ • u = 0, ∇ ∧ S = 0, u(0, x) = u 0 (x)
qui satisfait : u appartient à L 2 ((0, T), L 2 w d+1 (R d )), lim t→0 u(t, .) = u 0 ∈ L 2 w d+1 dans D et S appartient à D ((0, T) × R d ).

On considère ϕ ∈ D(R d ) telle que ϕ(x) = 1 dans un voisinage de 0 et on note

A i,j,ϕ = (1 -ϕ)∂ i ∂ j G d .
Alors, il existe g(t) ∈ L 1 ((0, T)) telle que

S = ∇p ϕ + ∂ t g avec p ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * (u i u j -F i,j ) + ∑ i,j
(A i,j,ϕ (xy) -A i,j,ϕ (-y))(u i (t, y)u j (t, y) -F i,j (t, y)) dy.

En plus,

• ∇p ϕ ne dépend pas du choix de ϕ : Si on change ϕ par ψ, on obtient p ϕ (t, x) -p ψ (t, x) = ∑ i,j

(A i,j,ψ (-y) -A i,j,ϕ (-y))(u i (t, y)u j (t, y) -F i,j (t, y)) dy.

• ∇p ϕ est l'unique solution du problème de Poisson

∆w = -∇(∇ • (∇ • (u ⊗ u -F))
qui satisfait lim τ→+∞ e τ∆ w = 0.

• Si on suppose que F appartient à L 1 ((0, T), L 1 w d (R d )) et u ∈ L 2 ((0, T), L 2 w d (R d )), alors on a que g est constante et on peut prendre g = 0, et ∇p ϕ = ∇p 0 où p 0 = ∑ i,j (ϕ∂ i ∂ j G d ) * (u i u j -F i,j ) + ∑ i,j ((1ϕ)∂ i ∂ j G d ) * (u i u j -F i,j ), p 0 ne dépend pas du choix de ϕ et on pouvait définir p 0 = ∑ i,j (∂ i ∂ j G d ) * (u i u j -F i,j ).

Dans le chemin pour arriver à cette caractérisation du terme de la pression, on passe par l'étude de deux problèmes de Poisson, et cela nous amène à un corollaire intéressant par rapport au projecteur de Leray. En effet, on obtient que la définition suivante nous donne bien un projecteur sur les champs de vecteurs à divergence nulle.

Définition. Soit H ∈ L 1 ((0, T), L 1 (R d , w d+1 dx)) un tenseur et soit w = ∇ • H. Le projecteur de Une fois qu'on a éclairé la relation entre le terme de la pression, la vitesse et la force extérieure, nous procédons à établir des contrôles uniformes dans des espaces à poids pour les solutions du système régularisé de Navier-Stokes.

On considère dans un premier temps, un bilan d'énergie pour la vitesse, ce qui nous permet d'obtenir des solutions globales pour des données initiales appartenant à des espaces L 2 à poids. Afin de rendre ces contrôles sur la vitesse réutilisables, pour obtenir des contrôles uniformes pour la vorticité dans des espaces à poids, et ceci sans perte de généralité dans l'ordre de décroissance du poids, nous considérons les propriétés suivantes pour les poids : Définition. Soit Φ une fonction sur R d (2 ≤ d ≤ 4). On dit que Φ est un poids adapté si Φ est une fonction Lipschitz continue et satisfait :

• (H1) 0 < Φ ≤ 1. • (H2) Il existe C 1 > 0 telle que |∇Φ| ≤ C 1 Φ 3 2
• (H3) Il existe r ∈ (1, 2] telle que Φ r ∈ A r (où A r est la classe des poids de Muckenhoupt). Pour d = 4, on a besoin de la condition r < 2.

• (H4) Il existe C 2 > 0 telle que Φ(x) ≤ Φ( x λ ) ≤ C 2 λ 2 Φ(x), pour tout λ ≥ 1. En particulier, Φ est décroissante.

Nos exemples de base, des poids adaptés, sont :

• d = 2, Φ(x) = 1 (1+|x|) γ où 0 ≤ γ < 2 • d = 3 ou d = 4, Φ(x) = 1 (1+|x|) γ où 0 ≤ γ ≤ 2 • d = 3, Φ(x) = 1 (1+r) γ où r = x 2 1 + x 2 2 et 0 ≤ γ < 2.
Notre résultat principal d'existence globale dans le cadre de régularité locale L 2 pour la donné initiale, s'écrit de la manière suivante :

Théorème. Considérons la dimension d ∈ {2, 3, 4} et un poids adapté Φ. Soit u 0 un champ de vecteurs à divergence nulle appartenant à L 2 (Φ dx, R d ). Soit F = (F i,j (t, x)) i,j un tenseur appartenant à L 2 ((0, +∞), L 2 (Φ dx, R d )).
Alors, il existe une solution globale u du problème

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0 qui satisfait : • u appartient à L ∞ ((0, T), L 2 (Φdx)) et ∇u appartient à L 2 ((0, T), L 2 (Φdx)), pour tout T > 0, • p = ∑ 1≤i,j≤d R i R j (u i u j -F i,j ), • la transformation t ∈ [0, +∞) → u(t, .) est faiblement continue de [0, +∞) dans L 2 (Φ dx), et fortement continue en t = 0,
• l'inégalité d'énergie locale pour d ∈ {2, 3} : il existe une mesure positive localement finie µ telle que

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • |u| 2 2 u -pu + u • (∇ • F) -µ,
et dans le cas de la dimension d = 2 on a µ = 0.

En plus, on obtient que

u(t, .) 2 L 2 (Φdx) + 2 t 0 ∇ ⊗ u(s, .) 2 L 2 (Φdx) ds ≤ u 0 2 L 2 (Φdx) - t 0 ∇(|u| 2 ) • ∇Φ dx ds + t 0 (|u| 2 u + 2pu) • ∇Φ dx ds -2 ∑ i ∑ j t 0 F i,j u j ∂ i Φ + F i,j ∂ i u j Φ dx ds.
Dans un deuxième temps, nous étudions un bilan d'énergie pour la vorticité, afin d'obtenir dans le cadre des solutions axisymétriques, des solutions globales pour une donnée initiale appartenant à un espace L 2 à poids et dont la vorticité elle aussi appartient à un espace L 2 à poids.

Ainsi, nous arrivons à notre résultat principal d'existence globale dans le cadre d'une donnée initiale de régularité locale H 1 . Nous observons que les poids Φ

(x) = 1 (1+r) γ et Ψ(x) = 1 (1+r 2 ) δ/2 avec 0 ≤ δ ≤ γ < 2 satisfont les hypothèses du théorème :
Théorème. On considère un poids Φ qui satisfait (H1) -(H4). On assume que Φ ne dépend que de r = x 2 1 + x 2 2 et qu'il existe un poids Ψ qui est une fonction continue qui ne dépend que de r, tel que

Φ ≤ Ψ ≤ 1, Ψ ∈ A 2 et il existe C 1 > 0 tel que |∇Ψ| ≤ C 1 √ ΦΨ and |∆( √ Ψ)| ≤ C 1 √ Φ.
Soit u 0 un champ de vecteurs à divergence nulle sans tourbillon, tel que u 0 appartient à L 2 (Φdx) et ∇ ⊗ u 0 appartient à L 2 (Ψdx). Alors, il existe une solution globale u du problème

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p ∇ • u = 0, u(0, .) = u 0 telle que • u est axisymétrique sans tourbillon, u appartient à L ∞ ((0, T), L 2 (Φ dx)), ∇ ⊗ u ap- partient à L ∞ ((0, T), L 2 (Ψ dx)) et ∆u appartient à L 2 ((0, T), L 2 (Ψ dx)), pour tout T > 0, • les transformations t ∈ [0, +∞) → u(t, .) et t ∈ [0, +∞) → ∇ ⊗ u(t, .) sont faiblement continues de [0, +∞) dans L 2 (Φ dx) et dans L 2 (Ψ dx) respectivement, et sont fortement continues en t = 0.
Après, dans le Chapitre 5 nous analysons les solutions auto-similaires. Il vaut mieux introduire d'abord les notions suivantes :

Définition. On considère λ > 1. On dit que u 0 ∈ L 2 loc (R 3 ) est un champ de vecteurs λ-discrètement auto-similaire (λ- DSS) si λu 0 (λx) = u 0 (x). Un champ de vecteurs dépendant du temps u ∈ L 2 loc ([0, +∞) × R 3 ), est appelé λ-DSS s'il vérifie λu(λ 2 t, λx) = u(t, x).
Un tenseur de force

F ∈ L 2 loc ([0, +∞) × R 3 ), est appelé λ-DSS si λ 2 F(λ 2 t, λx) = F(t, x).
Nous remarquons que dans les contrôles à priori étudiés pour obtenir des solutions discrètement auto-similaires, on considère des espaces convenables pour optimiser l'intégrabilité des solutions obtenues.

On va se rendre compte que l'espace L 10/3 t L 10/3

x apparaît souvent dans cette analyse et c'est cet espace qui nous permet d'obtenir γ proche de 1 dans le théorème cidessous. L'utilité de cette espace L 10/3 t L 10/3 x est un peu naturel comme il est l'espace L p L p plus régulier qui contient l'espace d'énergie naturel pour les équations de Navier-Stokes,

L ∞ t L 2 ∩ L 2 t Ḣ1 .
Un détail important c'est que, comme dans toutes les démonstrations précédentes de l'existence des solutions auto-similaires pour des grandes données initiales, nous faisons intervenir la théorie du dégré de Leray-Schauder. Le résultat est le suivant :

Théorème. Soit γ ∈ (1, 2) et λ ∈ (1, +∞). Si u 0 est un λ-DSS champ de vecteurs à divergence nulle qui appartient à L 2 w γ (R 3 ) et si F est un λ-DSS tenseur de force F(t, x) = F i,j (t, x) 1≤i,j≤3 qui appartient à L 2 loc ((0, +∞), L 2 w γ ), alors les équations de Navier-Stokes avec vitesse initiale u 0 , (NS)    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0
ont une solution globale u qui satisfait les propriétés suivantes :

• u est un λ-DSS champ de vecteurs

• pour tout 0 < T < +∞, u appartient à L ∞ ((0, T), L 2 w γ ) et ∇u ∈ L 2 ((0, T), L 2 w γ )
• la fonction t ∈ [0, +∞) → u(t, .) est faiblement continue de [0, +∞) dans L 2 w γ , et fortement continue en t = 0

• u est adaptée : il existe une mesure positive localement finie µ sur (0, +∞) × R 3 telle que

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • ( |u| 2 2 + p)u + u • (∇ • F) -µ.
Dans le dernier chapitre, le Chapitre 6, nous étudions le système couplé des équations de la magnétohydrodynamique :

(MHD)              ∂ t u = ∆u -(u • ∇)u + (b • ∇)b -∇p + ∇ • F, ∂ t b = ∆b -(u • ∇)b + (b • ∇)u -∇q, ∇ • u = 0, ∇ • b = 0, u(0, •) = u 0 , b(0, •) = b 0 .
Ce système décrit le comportement d'un fluide conducteur du courant électrique en présence de champs électromagnétiques. La vitesse du fluide est notée u et b est un champ magnétique.

La structure des équations (NS) et (MHD) est vraiment très proche. C'est clair que si b = 0 on obtient les équations de Navier Stokes. En plus, ces deux équations ont le même scaling, c'est à dire : Si λ > 0 alors (u, b) est une solution du problème de Cauchy pour le système (MHD) sur (0, T) avec donnée initiale (u 0 , b 0 ) et tenseur de force F, si et seulement si, (u λ , b λ )(t, x) = (λu(λ 2 t, λx), λb(λ 2 t, λx)) est une solution du problème de Cauchy pour le système (MHD) sur (0, T/λ 2 ) avec donnée initiale (u 0,λ , b 0,λ )(x) = (λu 0 (λx), λb 0 (λx)) et tenseur de force

F λ (t, x) = λ 2 F(λ 2 t, λx).
C'est surtout sur le bilan d'énergie que nous sommes intéressés à savoir comme ces deux équations, (NS) et (MHD), se ressemblent. En fait, ce qui nous sera utile c'est que la condition d'être une solution adapté pour le système (MHD) fait intervenir des termes en forme de divergence pour pouvoir contrôler les termes qui proviennent de la partie non-linéaire des équations ou qui proviennent du terme de la pression.

Comme nous allons voir, on va considérer des solutions du système (MHD) pour lesquelles q = 0 et qui satisfont la propriété suivante :

• La solution (u, b, p) est adaptée : il existe une mesure positive localement finie µ sur (0, +∞) × R 3 telle que

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 )u -∇ • (pu) + ∇ • ((u • b)b) + u • (∇ • F) -µ.
Ces remarques essentiellement vont nous permettre d'étudier le système (MHD) en suivant les mêmes idées utilisées pour analyser (NS).

Un résultat du dernier Chapitre qui est nouveau même dans le contexte des équations de Navier-Stokes, est un résultat d'unicité fort-faible dans le cadre des espaces à poids.

Théorème. Soit 0 ≤ γ ≤ 2 et w γ = (1 + |x|) -γ . Soit 0 < T < +∞. Soient u 0 , b 0 ∈ L 2 w γ (R 3 ) = L 2 ((1 + |x|) -γ dx) deux champs de vecteurs à divergence nulle. Aussi, on considère un tenseur de force F(t, x) = F i,j (t, x) 1≤i,j≤3 ∈ L 2 ((0, T), L 2 w γ ).
Soient (u, b, p) et ( ũ, b, p) deux solutions du système (MHD) telles que :

• u, b, ũ, b appartient à l'espace L ∞ ((0, T), L 2 w γ ) et ∇u, ∇ ũ, ∇b, ∇ b ∈ L 2 ((0, T), L 2 w γ ) • les transformations t ∈ [0, T) → (u, b)(t, .) et t ∈ [0, T) → ( ũ, b)(t, .) sont faible- ment continues de [0, T) dans L 2 w γ (R 3 ), et sont fortement continues en t = 0 • (u, b) et ( ũ, b) sont adaptées: il existe deux mesures positives localement finies µ et ν sur (0, T) × R 3 telles que ∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 + p)u + ∇ • ((u • b)b) + u • (∇ • F) -µ, et ∂ t ( | ũ| 2 + | b| 2 2 ) =∆( | ũ| 2 + | b| 2 2 ) -|∇ ũ| 2 -|∇ b| 2 -∇ • ( | ũ| 2 2 + | b| 2 2 + p) ũ + ∇ • (( ũ • b) b) + ũ • (∇ • F) -ν.
Si u, b ∈ L p L q , avec 2 p + 3 q = 1 et p ∈ (2, +∞) alors on a que (u, b, p) = ( ũ, b, p).

Dans le Chapitre 6, nous introduisons aussi les espaces de Morrey locaux :

Définition. Pour γ ≥ 0 et p ∈ (1, ∞). On note B p γ (R d ) l'espace de Banach de toutes les fonctions u ∈ L p loc (R d ) telles que : u B p γ = sup R≥1 1 R γ B(0,R) |u(x)| p dx 1/p < +∞.
Nous notons B p γ L p (0, T) l'espace de Banach de toutes les fonctions u ⊂ (L

p t L p x ) loc ([0, T] × R d ) telles que u B p γ L p (0,T) = sup R≥1 1 R γ T 0 B(0,R) |u(t, x)| p 1 p dx dt < +∞.
lim R→+∞ 1 R γ B(0,R) |u(x)| p dx = 0. On note aussi B 2 2 = B 2 .
L'intérêt des espaces B p γ c'est qu'ils sont très proches des espaces à poids L 2 ((1 + |x|) -γ dx), qui nous ont déjà été utiles. En particulier, pour 2 < δ < +∞, on obtient les inclusions continues

L 2 w 2 ⊂ B 2 2,0 ⊂ B 2 ⊂ L 2 w δ .
Une fois qu'on connaît le résultat d'existence globale d'une solution pour une donné initial qui appartient à L 2 ((1 + |x|) -2 dx) c'est naturel d'essayer à démontrer le résultat pour des données initiales qui appartiennent à B 2 . On fait cet étude et cela conduit au résultat ci-dessous.

Théorème. Considère 0 < T < +∞. Soient u 0 , b 0 ∈ B 2 (R 3 ) deux champ de vecteurs à divergence nulle. Soit F un tenseur qui appartient à B 2 L 2 (0, T). Alors, il existe un temps T 0 ∈ (0, T) tel que le système (MHD) admette une solution (u, b, p, q), avec q = 0, qui satisfait les propriétés suivantes :

• u, b appartient à L ∞ ((0, T 0 ), B 2 ) et ∇u, ∇b appartient à B 2 L 2 (0, T 0 ) • la pression p est liée à u, b et F par la formule : p = ∑ 1≤i,j≤3 R i R j (u i u j -b i b j -F i,j ) • la transformation t ∈ [0, T) → (u(t, •), b(t, •)) est * -faiblement continue de [0, T) dans B 2 (R 3 ), et pour tout compact K ⊂ R 3 , lim t→0 (u(t, •) -u 0 , b(t, •) -b 0 ) L 2 (K) = 0
• la solution (u, b, p) est adaptée : il existe une mesure non-négative localement finie µ sur (0, T) × R 3 telle que

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 + p)u + ∇ • [(u • b)b] + u • (∇ • F) -µ.
On obtient aussi que pour

0 ≤ t ≤ T 0 , max{ (u, b)(t) 2 B 2 , ∇(u, b) 2 B 2 L 2 (0,T 0 ) } ≤ C (u 0 , b 0 ) 2 B 2 + C F 2 B 2 L 2 (0,t) + C t 0 (u, b)(s) 2 B 2 + (u, b)(s) 6 B 2 ds.
En plus, si les données vérifient :

lim R→+∞ R -2 |x|≤R |u 0 (x)| 2 + |b 0 (x)| 2 dx = 0 lim R→+∞ R -2 +∞ 0 |x|≤R |F(t, x)| 2 dx ds = 0,
alors on obtient une solution faible globale (u, b, p).

Chapter 2

The pressure term

The goal of this chapter is to study the auxiliary unknown ∇p in the Cauchy problem for the Navier-Stokes equations on

R d    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0 ,
more specifically, we seek to propose a formula for the pressure. This term is usually interpreted as a Lagrange multiplier for the constraint of incompressibility.

Remark 2.1. We use the following notation : u denotes a vector field (u 1 , u 2 , ...,

u d ), F = (F i,j ) 1≤i,j≤d is a tensor. ∇ • F denotes the vector (∑ i ∂ i F i,1 , ∑ i ∂ i F i,2 , ..., ∑ i ∂ i F i,d ). Thus, for a vector field b such that ∇ • b = 0, we have (b • ∇)u = ∇ • (b ⊗ u).
In order to not assume differentiability of u in our calculations, it is convenient to rewrite the equations as

(NS)    ∂ t u = ∆u -∇ • (u ⊗ u) -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0
If we take the Laplacian in the equations (NS), in view of the identity

-∆w = ∇ ∧ (∇ ∧ w) -∇(∇ • w), we find 0 = -∆∇p -∇(∇ • (∇ • (u ⊗ u -F)) = -∆∇p -∇( ∑ 1≤i,j≤d ∂ i ∂ j (u i u j -F i,j )) and ∂ t ∆u = ∆ 2 u + ∇ ∧ (∇ ∧ (∇ • (u ⊗ u -F))).
Then, we can see that the rotational free unknown ∇p obeys a Poisson equation. We denote G d the following fundamental solution of the operator -∆ (which means -∆G d = δ):

G 2 = 1 2π ln( 1 |x| ), G 3 = 1 4π|x| , we obtain formally ∇p = G d * ∇( ∑ 1≤i,j≤d ∂ i ∂ j (u i u j -F i,j )) + H (2.1)
where H is harmonic in the space variable, in other words ∆H = 0. In the literature, we usually find the hypothesis ∇p vanishes at infinity, which is read as H = 0. Another way to write this assumption would be

∂ t u = G d * ∇ ∧ (∇ ∧ ∂ t u).
The operator

P = G d * ∇ ∧ (∇ ∧ .)
is named as the Leray projection operator and the decomposition (if it is well-defined)

w = Pw -G d * ∇(∇ • w)
the Hodge decomposition of the vector field w.

Thereafter, a relevant issue when dealing with the Navier-Stokes equations is to analyse whether in formula (2.1) the first half of the right-hand term is well-defined, and if so what can we say about the second half.

In order to give meaning to the formal convolution G d * ∇∂ i ∂ j (u i u j ) or to the term ∇∂ i ∂ j G d * (u i u j ), we should demand u i to be locally L 2 t L 2 x (to make u i u j a distribution) and to have small increase at infinity, since ∇∂ i ∂ j G d has small decay at infinity (far from the origin, it belongs to L 1 ∩ L ∞ and is controlled as O(|x| -(d+1) )). We thus will analyse solutions u which belong to L 2 ((0, T), L 2 (R d , w d+1 dx)) with

w γ (x) = (1 + |x|) -γ .
As we will see F i,j plays a role similar to u i u j , thus we will take F belonging to

L 1 ((0, T), L 1 (R d , w d+1 dx)).
The main results in (Fernández-Dalgo and Lemarié-Rieusset, 2020a) are described in this chapter. We first present a Lemma just to clarify the meaning of ∇p in the Navier-Stokes equations, and a definition to be rigorous : Lemma 2.1. Let the dimension d ∈ {2, 3} and consider a real number γ > 0. Let 0 < T < +∞. Let u(t, x) = (u i (t, x)) 1≤i≤d a divergence free vector field which belongs to L 1 ((0, T), L 1 (R d , w γ dx)), and let H = (H i,j (t, x)) be a tensor such that

H(t, x) ∈ L 1 ((0, T), L 1 (R d , w γ dx)).
We define the distribution S as

S = ∆u -∇ • H -∂ t u.
Then, the following statements are equivalents: (A) S is curl-free : ∇ ∧ S = 0. (B) There exists a distribution p ∈ D ((0, T) × R d ) such that S = ∇p.

Remark: to cover the case of the Navier-Stokes equations, we consider

H = u ⊗ u -F, with the hypothesis u ∈ L 2 ((0, T), L 2 (R d , w γ dx)) and F ∈ L 1 ((0, T), L 1 (R d , w γ dx)).
In order to write accurately we introduce a definition, which can be found in [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF].

Definition 2.1. Consider an increasing sequence of compact intervals [a n , b n ], with a n < b n , such that (0, T) = ∪ n∈N [a n , b n ]. Let T n the Frechet space of the functions f ∈ C ∞ ((0, T) × R d ) such that supp f ⊂ [a n , b n ] × R d and the the semi-norms sup a n <t<b n sup x∈R d |x α ∂ β ∂x β ∂ p ∂t p f (t, x)|
are finite, where α, β ∈ N d and p ∈ N.

Let us denote T ((0, T) × R d ) = ∪ n∈N T n the space of test functions on (0, T) × R d which are compactly supported in time and have fast decay in space. We consider the topological structure of T as inductive limit of the Frechet spaces T n . We denote T the dual space of T .

The elements of T are then the distributions ω on (0, T) × R d , such that for all [a, b] ⊂ (0, T) there exists C ≥ 0 and N ∈ N such that for all φ ∈ T with supp

(ϕ) ⊂ [a, b] × R d , we have ω|ϕ ≤ C ∑ |α|,|β|,p≤N, x α ∂ β ∂x β ∂ p ∂t p f (t, x) ∞ .
For f ∈ T , we define the Fourier transform with respect to the spatial variable as

F f (t, ξ) = N d f (t, x)e -ix•ξ dx.
We can define the Fourier transform with respect to the spatial variable for a distribution w ∈ T as usual by duality.

Our principal result with respect to the pressure term in the Navier-Stokes equations is the following one :

Theorem 1. Consider the dimension d ∈ {2, 3}. Let 0 < T < +∞. Consider a tensor F(t, x) = F i,j (t, x) 1≤i,j≤d belonging to L 1 ((0, T), L 1 (R d , w d+1 dx)). Let u be a solution of the problem    ∂ t u = ∆u -∇ • (u ⊗ u) -S + ∇ • F ∇ • u = 0, ∇ ∧ S = 0, u(0, x) = u 0 (x) (2.2)
which satisfies : u belongs to L 2 ((0, T), L 2 w d+1 (R d )), lim t→0 u(t, .) = u 0 ∈ L 2 w d+1 in D and S belongs to D ((0, T) × R d ).

We consider ϕ ∈ D(R d ) such that ϕ(x) = 1 on a neighborhood of 0 and we denote

A i,j,ϕ = (1 -ϕ)∂ i ∂ j G d .
Then, there exist g(t) ∈ L 1 ((0, T)) such that

S = ∇p ϕ + ∂ t g with p ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * (u i u j -F i,j ) + ∑ i,j (A i,j,ϕ (x -y) -A i,j,ϕ (-y))(u i (t, y)u j (t, y) -F i,j (t, y)) dy.
Moreover,

• ∇p ϕ does not depend on the choice of ϕ : If we change ϕ by ψ, we find

p ϕ (t, x) -p ψ (t, x) = ∑ i,j (A i,j,ψ (-y) -A i,j,ϕ (-y))(u i (t, y)u j (t, y) -F i,j (t, y)) dy.
• ∇p ϕ is the unique solution of the Poisson problem

∆w = -∇(∇ • (∇ • (u ⊗ u -F))
which satisfy lim τ→+∞ e τ∆ w = 0 in T .

• If we assume that F belongs to L 1 ((0, T), L 1

w d (R d )) and u ∈ L 2 ((0, T), L 2 w d (R d ))
, then we find that g is constant and we can take g = 0, and ∇p ϕ = ∇p 0 where

p 0 = ∑ i,j (ϕ∂ i ∂ j G d ) * (u i u j -F i,j ) + ∑ i,j ((1 -ϕ)∂ i ∂ j G d ) * (u i u j -F i,j ),
p 0 does not depend on ϕ and we could define

p 0 = ∑ i,j (∂ i ∂ j G d ) * (u i u j -F i,j ).
We remark that the same or similar splits for the pressure term have already been studied by many authors, under different hypothesis, for exemple we refer to [START_REF] Chemin | Fluides parfaits incompressibles. tome 230[END_REF] and [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]. In the literature, we can also find other formulas for the pressure which are useful to deduce more precise controls, for exemple in the two-dimensional case, Gallay gives in [START_REF] Gallay | Infinite energy solutions of the two-dimensional Navier-Stokes equations[END_REF] a formula where vorticity intervenes.

If F = 0, the case g = 0 can be seen as a referential change of the case g = 0, this fact is known as the extended Galilean invariance of the Navier-Stokes equations :

Theorem 2. Consider the dimension d ∈ {2, 3}. Let 0 < T < +∞. Let u be a solution of    ∂ t u = ∆u -∇ • (u ⊗ u) -S ∇ • u = 0, ∇ ∧ S = 0, u(0, x) = u 0 (x) (2.3)
which satisfies : u belongs to L 2 ((0, T), L 2 w d+1 (R d )), lim t→0 u(t, .) = u 0 ∈ L 2 w d in D , and S belongs to D ((0, T) × R d+1 ).

We take ϕ ∈ D(R d ) such that ϕ(x) = 1 on a neighborhood of 0 and we denote

A i,j,ϕ = (1 -ϕ)∂ i ∂ j G d .
We decompose S as

S = ∇p ϕ + ∂ t g with p ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * (u i u j ) + ∑ i,j
(A i,j,ϕ (xy) -A i,j,ϕ (-y))(u i (t, y)u j (t, y)) dy and g(t) ∈ L 1 ((0, T)).

We define

E(t) = t 0 g(λ)dλ
2.1. Curl-free vector fields and w(t, x) = u(t, x -E(t)) + g(t).

Then, w is a solution of the following Navier-Stokes equations

             ∂ t w = ∆w -∇ • (w ⊗ w) -∇q ϕ ∇ • w = 0, w(0, x) = u 0 (x) q ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * (w i w j ) + ∑ i,j
(A i,j,ϕ (xy) -A i,j,ϕ (-y))(w i (t, y)w j (t, y)) dy

(2.4)

with initial data u 0 .

Curl-free vector fields

Below we prove the Lemma 2.1 which clarify the meaning of the curl-free vector field occurring in the Navier Stokes equations, more precisely we prove that the curl-free vector field occurring in the Navier Stokes is necessarily a gradient :

Proof. Let take a partition of unity on (0, T),

∑ j∈Z ω j = 1
where ω j is supported on (2 j-2 T, 2 j T) for j < 0, on (T/4, 3T/4) for j = 0 and on (T -2 -j T, T -2 -(j+2) T) in the case j > 1. We let

V j = -ω j u + t 0 ω j ∆u -ω j ∇ • H + (∂ t ω j )u ds.
Therefore V j is a sum of the form A + t 0 ∆B + ∇ • C + D ds where A, B, C and D belong to L 1 ((0, T), L 1 (R d , w γ dx)) (the fact of belonging to L 1 in time is stable by integration); so that, by the Fubini theorem, we can see this distribution as a timedependent tempered distribution.

We have ∂ t V j = ω j S, V j is equal to 0 for t on a neighborhood of the origin, and ∇ ∧ V j = 0. Furthermore, S = ∑ j∈Z ∂ t V j .

We take Φ ∈ S(R d ) such that the Fourier transform of Φ has compact support and is equal to 1 on a neighborhood of 0. Then, Φ * V j is well-defined and ∇ ∧ (Φ *

V j ) = 0. Let us write X j = Φ * V j and Y j = V j -X j .
Thus,

Y j = ∇ 1 ∆ ∇ • Y j
and by Poincaré's Lemma,

X j = ∇( 1 0 x • X j (t, λx)dλ).
Therefore, we conclude that S = ∇p with

p = ∂ t ∑ j∈Z ( 1 0 x • X j (t, λx)dλ + 1 ∆ ∇ • Y j ).

The Poisson problem

-∆U = ∂ k ∂ i ∂ j h
In this section we consider a simple but crucial Poisson problem :

Proposition 2.1. If h ∈ L 1 (R d , (1 + |x|) -(d+1) dx), then U = U 1 + U 2 = (∂ k (ϕ∂ i ∂ j G d )) * h + ∂ k ((1 -ϕ)∂ i ∂ j G d ) * h. is a distribution, U 2 belongs to L 1 (R d , (1 + |x|) -(d+1) dx) and U is a solution of the problem -∆U = ∂ k ∂ i ∂ j h.
In fact, U is the unique solution in S such that lim τ→0 e τ∆ U = 0 in S .

Proof. We can verify that ∂ j G d satisfies

∂ j G d = +∞ 0 ∂ j W t dt where W t (x) is the heat kernel W t (x) = (4πt) -d 2 e -|x| 2 4t .
In particular, on R d \ {0}, we have

∂ j G d = c d x j |x| d with c d = -1 2(4π) d/2 +∞ 0 e -1 4u du u d+2 2 . Remark that U 1 = (∂ k (ϕ∂ i ∂ j G d )) * h, is well-defined since ∂ k (ϕ∂ i ∂ j G d
) is a compactly supported distribution. In order to control U 2 , let us write

1 (1 + |x|) d+1 |∂ k ((1 -ϕ)∂ i ∂ j G d (x -y))||h(y)|dydx ≤ 1 (1 + |x|) d+1 C (1 + |x -y|) d+1 |h(y)|dydx ≤ C 1 (1 + |y|) d+1 |h(y)|dy
where the last inequality is justified by the following fact

1 (1 + |x|) d+1 1 (1 + |x -y|) d+1 dx ≤ |x|> |y| 2 1 (1 + |x|) d+1 1 (1 + |x -y|) d+1 dx + |x-y|> |y| 2 1 (1 + |x|) d+1 1 (1 + |x -y|) d+1 dx ≤ 2 d+1 (1 + |y|) d+1 1 (1 + |x -y|) d+1 dx + 2 d+1 (1 + |y|) d+1 1 (1 + |x|) d+1 dx ≤ C 1 (1 + |y|) d+1 .
As U is well-defined, we compute -∆U. We find -∆U 1 is equal to

(-∆∂ k (ϕ∂ i ∂ j G d )) * h = ∂ k (ϕ∂ i ∂ j h) -∂ k ((∆ϕ)∂ i ∂ j G d ) * h -2 ∑ 1≤l≤d ∂ k ((∂ l ϕ)∂ l ∂ i ∂ j G d ) * h. 2.2. The Poisson problem -∆U = ∂ k ∂ i ∂ j h
To calculate -∆U 2 , we verify that it is possible to differentiate under the integration sign to obtain

-∆U 2 = ∂ k ((1 -ϕ)∂ i ∂ j h) + ∂ k ((∆ϕ)∂ i ∂ j G d ) * h + 2 ∑ 1≤l≤d ∂ k ((∂ l ϕ)∂ l ∂ i ∂ j G d ) * h.
Therefore, U is a solution of the Poisson problem.

A computation for e τ∆ U shows that

e τ∆ U = (e τ∆ ∂ k ∂ i ∂ j G d ) * h, hence |e τ∆ U(x)| ≤ C 1 ( √ τ + |x -y|) d+1 |h(y)| dy.
By dominated convergence we find lim τ→+∞ e τ∆ U = 0 in

L 1 (R d , (1 + |x|) -(d+1) dx).
We suppose that V is another solution of the Poisson problem with V ∈ S and lim τ→+∞ e τ∆ V = 0 in S , then ∆(U -V) = 0 and U -V ∈ S , so that U -V is a polynomial; since the assumption lim τ→0 e τ∆ (U -V) = 0, we deduce that this polynomial is 0.

If we consider better integrability of h, we expect to enhance integrability of U 2 . For instance, we find :

Proposition 2.2. If h ∈ L 1 (R d , (1 + |x|) -d dx)) then U 2 = ∂ k ((1 -ϕ)∂ i ∂ j G d ) * h belongs to L 1 (R d , (1 + |x|) -d ).
Proof. We know that

1 (1 + |x|) d |∂ k ((1 -ϕ)∂ i ∂ j G d (x -y))||h(y)| dy dx ≤ C 1 (1 + |x|) d 1 (1 + |x -y|) d+1 |h(y)| dy dx.
For |y| ≤ 1, we just observe that

1 (1 + |x|) d 1 (1 + |x -y|) d+1 dx ≤ C 1 (1 + |x|) d+1 dx ≤ C while for |y| > 1, as the real number |x|< 1 2 1 |x| d-1 1 |x-y |y| | 2 dx does not depend on y, we can write 1 (1 + |x|) d 1 (1 + |x -y|) d+1 dx ≤ |x|> |y| 2 1 (1 + |x|) d 1 (1 + |x -y|) d+1 dx + |x|< |y| 2 1 (1 + |x|) d 1 (1 + |x -y|) d+1 dx ≤ 2 d (1 + |y|) d 1 (1 + |x -y|) d+1 dx + 2 d-1 (1 + |y|) d-1 |x|< |y| 2 1 |x| d-1 1 |x -y| 2 dx ≤ C 1 (1 + |y|) d + C 1 (1 + |y|) d-1 1 |y| |x|< 1 2 1 |x| d-1 1 |x -y |y| | 2 dx ≤ C 1 (1 + |y|) d .
This fact proves the proposition.

The Poisson problem

-∆V = ∂ i ∂ j h Proposition 2.3. Let h ∈ L 1 ((1 + |x|) -d-1 dx) and let A ϕ = (1 -ϕ)∂ i ∂ j G d . Then, V = V 1 + V 2 = (ϕ∂ i ∂ j G d ) * h + (A ϕ (x -y) -A ϕ (-y))h(y)dy is a distribution such that V 2 belongs to L 1 ((1 + |x|) -γ ), for γ > d + 1, and V is a solution of the problem -∆V = ∂ i ∂ j h.
Proof. We know that V 1 is well defined because ϕ∂ i ∂ j G d is a compactly supported distribution. Below we verify that V 2 is well defined.

For |y| ≤ 1, we have

1 (1 + |x|) γ |A ϕ (x -y) -A ϕ (-y)|dx ≤ C A ϕ L ∞ 1 (1 + |x|) γ dx.
For |y| > 1, by the mean value inequality we obtain

|x|< |y| 2 1 (1 + |x|) γ |A ϕ (x -y) -A ϕ (-y)|dx ≤ C 1 |y| d+1 |x|< |y| 2 |x| (1 + |x|) γ dx
and the other part can be controlled in the following way

|x|> |y| 2 1 (1 + |x|) γ |A ϕ (-y)|dx ≤ C 1 |y| d |x|> |y| 2 1 |x| γ ≤ C 1 |y| γ and for ε > 0 such that γ -ε ≥ d + 1, we have that |x|> |y| 2 1 (1 + |x|) γ |A ϕ (x -y)|dx ≤ C |x|> |y| 2 1 |x| γ 1 (1 + |x -y|) d dx ≤ C |x|> |y| 2 1 |x| γ 1 |x -y| d-ε dx ≤ C 1 |y| γ-|x|> 1 2 1 |x| γ 1 |x -y |y| | d-ε dx ≤ C 1 |y| d+1 .

Characterisation of the pressure term.

We slightly generalize Theorem 1 in order to make it applicable to other classical incompressible equations as the magnetohydrodynamics system or the Euler equations. Remark that for p ∈ (1, +∞),

L p (R d , w d+1 dx) ⊂ L 1 (R d , w d+1 dx) but L p (R d , w d dx) is not contained in L 1 (R d , w d dx).
The theorem of characterisation is stated as follows:

Theorem 3. Consider the dimension d ∈ {2, 3}. Let 0 < T < +∞. Consider a tensor H(t, x) = H i,j (t, x) 1≤i,j≤d belonging to L 1 ((0, T), L 1 (R d , w d+1 dx)). Let u be a solution of    ∂ t u = ∆u -∇ • H -S ∇ • u = 0, ∇ ∧ S = 0, u(0, x) = u 0 (x) (2.5)
which satisfies : u belongs to L 1 ((0, T), L 1 w d+1 (R d )), S belongs to D ((0, T) × R d ), and u(t) converges to u 0 ∈ L 1 w d+1 in D . We consider ϕ ∈ D(R d ) such that ϕ(x) = 1 on a neighborhood of 0 and we define

A i,j,ϕ = (1 -ϕ)∂ i ∂ j G d .
Then, there exist g(t) ∈ L 1 ((0, T)) such that

S = ∇p ϕ + ∂ t g with p ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * (H i,j ) + ∑ i,j (A i,j,ϕ (x -y) -A i,j,ϕ (-y))(H i,j (t, y)) dy.

Moreover,

• ∇p ϕ does not depend on the choice of ϕ : If we change ϕ by ψ, we find p ϕ (t, x) -p ψ (t, x) = ∑ i,j (A i,j,ψ (-y) -A i,j,ϕ (-y))(H i,j (t, y)) dy.

• ∇p ϕ is the unique solution of the Poisson problem

∆w = -∇(∇ • H)
with lim τ→+∞ e τ∆ w = 0 in T .

• If we assume that H belongs to L 1 ((0, T), L 1 w d (R d )) and u ∈ L 1 ((0, T), L p w d (R d )), for some p ∈ [1, +∞), we have g is constant and we can take g = 0, and ∇p ϕ = ∇p 0 where p 0 = ∑ i,j

(ϕ∂ i ∂ j G d ) * (H i,j ) + ∑ i,j ((1 -ϕ)∂ i ∂ j G d ) * (H i,j ),
p 0 does not depend on ϕ and we could define p 0 = ∑ i,j (∂ i ∂ j G d ) * (H i,j ).

Proof. Taking the divergence in the differential equation

∂ t u = ∆u -∇ • H -S, we find -∑ i,j ∂ i ∂ j H i,j -∇ • S = 0 and -∆S = ∇( ∑ i,j ∂ i ∂ j (H i,j )). We denote A i,j,ϕ = (1 -ϕ)∂ i ∂ j G d . By Proposition 2.3, we can define p ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * h i,j + ∑ i,j (A i,j,ϕ (x -y) -A i,jϕ (-y))h i,j (y)dy and U = U 1 + U 2 = ∇ ∑ i,j (ϕ∂ i ∂ j G d ) * h i,j + ∇ ∑ i,j ((1 -ϕ)∂ i ∂ j G d ) * h i,j = ∇p ϕ .
We can define Ũ = S -U. First, remark that ∆U = ∆S so that ∆ Ũ = 0, thus Ũ is harmonic in space variable. Now, we consider a test function α ∈ D(R) such that α(t) = 0 for all |t| ≥ ε, and a test function

β ∈ D(R 3 ). For t ∈ (ε, T -ε) we have Ũ(t) * t,x (α ⊗ β) =(u * (-∂ t α ⊗ β + α ⊗ ∆β) + (-H) • * (α ⊗ ∇β))(t, •) -∑ i,j ((h ij ) * (∇(ϕ∂ i ∂ j G d ) * (α ⊗ β)))(t, •) -(U 2 * (α ⊗ β))(t, •). By Proposition 2.1, we conclude that Ũ * (α ⊗ β)(t, .) belongs to L 1 (R d , (1 + |x|) -d-1
), and then it is a tempered distribution. Since it is harmonic it must to be a polynomial. Belonging to the set L 1 (R d , (1 + |x|) -d-1 ) this polynomial is constant.

If H belongs to L 1 ((0, T), L Thus, we have k = 0. Now, using an approximation of the identity Φ ε = 1 ε 4 α( t ε )β( x ε ) and taking the limit when goes to 0, we obtain a similar result for Ũ. We thus have

S = ∇p ϕ + f (t), with f (t) = 0 if H belongs to L 1 ((0, T), L 1 w d (R d )) and u ∈ L 1 ((0, T), L p w d (R d )).
As f does not depend on x, for a function β ∈ D(R d ) such that β dx = 1 we have f = f * x β; then we can write

f (t) = ∂ t (u 0 * β -u * β + t 0 u * ∆β -(u ⊗ u -F) • * ∇β -p ϕ * ∇β ds) = ∂ t g.
Since ∂ t ∂ j g = ∂ j f = 0 y ∂ j g(0, .) = 0, we find that g depends only on t; moreover, the formula above shows that g ∈ L 1 ((0, T)).

The proof of Theorem 2 is classic, and this property is called the extended Galilean invariance of the Navier-Stokes equations :

Proof. We have assumed that

∂ t u = ∆u -(u • ∇)u -∇p ϕ - d dt g(t),
with g ∈ L 1 ((0, T)). We define

E(t) = t 0 g(λ)dλ and w = u(t, x -E(t)) + g(t).
Then, we compute ∂ t w to find

∂ t w =∂ t u(t, x -E(t)) -g(t) • ∇u(t, x -E(t)) + d dt g(t) =∆u(t, x -E(t)) -[(u • ∇)u](t, x -E(t)) -∇p ϕ (t, x -E(t)) - d dt g(t) -g(t) • ∇u(t, x -E(t)) + d dt g(t) =∆w -(w • ∇)w -∇p ϕ (t, x -E(t)).
If we define q ϕ (t, x) = p ϕ (t, x -E(t)), then we obtain

q ϕ = ∑ i,j (ϕ∂ i ∂ j G d ) * (w i w j ) + ∑ i,j
(A i,j,ϕ (xy) -A i,j,ϕ (-y))(w i (t, y)w j (t, y)) dy, which proves the theorem.

Some Applications

Proposition 2.1 and 2.3 have a nice implication. We may define the Leray projection operator on the divergence of tensors that belong to L 1 ((0, T), L 1 (R d , w d+1 dx)) :

Definition 2.2. Let H ∈ L 1 ((0, T), L 1 (R d , w d+1 dx)) and w = ∇ • H. The Leray projection P(w) of w on solenoidal vector fields is defined by

Pw = w -∇p ϕ
where ∇p ϕ is the unique solution of

-∆∇p = -∇(∇ • w)
such that lim τ→+∞ e τ∆ ∇p = 0.

Remark Note that Proposition 2.3 implies ∇ • P(w) = 0.

After applying the Leray projector, the Navier-Stokes equations take the following special form :

(MNS) ∂ t u = ∆u -P∇ • (u ⊗ u -F), u(0, .) = u 0 .
Then the following integro-differential equation arise

u = e t∆ u 0 - t 0 e (t-s)∆ P∇ • (u ⊗ u -F) ds,
where a solution u can be seen as a fixed point of an application on a suitable space.

The study of the convolution operator e (t-s)∆ P∇• is the core of the method of mild solutions of Kato and Fujita [START_REF] Fujita | The initial value problem for the Navier-Stokes equations with data in L p[END_REF]. The kernel of the operator e ∆ P is called the Oseen kernel. Thus, we will call equations (MNS) a mild formulation of the Navier-Stokes equations.

The mild formulation together with the local Leray energy inequality has been as well a key tool for extending Leray's theory of weak solutions in L 2 to the setting of weak solutions with infinite energy. We may propose a general definition of suitable Leray-type weak solutions :

Definition 2.3 (Suitable Leray-type solution). Consider F ∈ L 2 ((0, T), L 2 (R d , 1 (1+|x|) d+1 )) and u 0 ∈ L 2 (R d , 1 (1+|x|) d+1 ) with ∇ • u 0 = 0. We consider the Navier-Stokes problem on (0, T) × R d :    ∂ t u = ∆u -P(∇ • (u ⊗ u -F)) ∇ • u = 0, u(0, x) = u 0 (x) (2.6)
A suitable Leray-type solution u of the Navier-Stokes equations is a vector field u defined on (0, T) × R d satisfying the equations above and such that :

• u is locally L 2 t H 1 x on (0, T) × R d • sup 0<t<T |u(t, x)| 2 1 (1+|x|) d+1 dx < +∞ • (0,T)×R d |∇ ⊗ u(t, x)| 2 1 (1+|x|) d+1 dx dt < +∞ • the application t ∈ [0, T) → u(t, x) • w(x)
dx is continuous for every smooth compactly supported vector field w

• for every compact subset K of R d , lim t→0 K |u(t, x)u 0 (x)| 2 dx = 0.

• defining p ϕ as (the) solution of -∆p ϕ = ∑ i,j ∂ i ∂ j (u i u j -F i,j ) given by Proposition 2.3, u is suitable in the sense of Caffarelli, Kohn and Nirenberg : there exists a nonnegative locally bounded Borel measure µ on (0, T) × R d such that

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇ ⊗ u| 2 -∇ • (( |u| 2 2 + p ϕ )u) + u • (∇ • F) -µ Remarks :
a) Under those hypotheses, p ϕ belongs locally to L 3/2 t,x and u belongs locally to L 3 t,x so that ( |u| 2 2 + p ϕ )u is well-defined as a distribution.

b) [START_REF] Caffarelli | Partial regularity of suitable weak solutions of the Navier-Stokes equations[END_REF][START_REF] Caffarelli | Partial regularity of suitable weak solutions of the Navier-Stokes equations[END_REF][START_REF] Caffarelli | Partial regularity of suitable weak solutions of the Navier-Stokes equations[END_REF] introduced suitability to get estimates on partial regularity for weak Leray solutions. Suitability is a local assumption. When we consider a solution of the Navier-Stokes equations on a small domain with no specifications on the behaviour of u at the boundary, the estimates on the pressure (and the Leray projection operator) are no longer available. However, Wolf described in [START_REF] Wolf | On the local pressure of the Navier-Stokes equations and related systems[END_REF] a local decomposition of the pressure; he could generalize the notion of suitability to this new description of the pressure. We refer the paper [START_REF] Chamorro | The role of the pressure in the partial regularity theory for weak solutions of the Navier-Stokes equations[END_REF] for more information on the equivalence of various notions of suitability. c) In the case of uniformly locally square integrable solutions, the relationship between the mild formulation (MNS) and the system (NS) described in Theorem 1 has been studied in [START_REF] Furioli | Unicité dans L 3 (R 3 ) et d'autres espaces limites pour Navier-Stokes[END_REF][START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF], other paper we refer is [START_REF] Dubois | What is a solution to the Navier-Stokes equations?[END_REF]. Their results show the equivalence between (NS) and (MNS) in the case when u and F decay at infinity (more precisely, when u belongs to the closure of test functions in (L 2 t L 2 x ) uloc and when F belongs to the closure of test functions in (L 1 t L 1 x ) uloc ). Earlier results for mild solutions in L p t L q x with 2 p + d q ≤ 1 and d < q < +∞, were proved in 1972 by Fabes, Jones and Riviere [START_REF] Fabes | The initial value problem for the Navier-Stokes equations with data in L p[END_REF], it is a simpler case where the theory of singular integrals may be directly applied. d) Non-decaying solutions has been discussed in [START_REF] Kukavica | On local uniqueness of solutions of the Navier-Stokes equations with bounded initial data[END_REF] for the study of the Cauchy problem with initial value in L ∞ and in [START_REF] Kukavica | On local uniqueness of weak solutions to the Navier-Stokes system with BMO -1 initial datum[END_REF] for the study of the Cauchy problem with initial value in BMO -1 . The systems (NS) and (MNS) are no longer equivalent and if F = 0, general solutions are better described through the extended Galilean invariance of the equations. In this paper, we find again such a description in the case of more general weak solutions, for which the integral formulation does not provide any existence or uniqueness results, in contrast to the case of solutions in L ∞ or BMO -1 data. More precisely, a suitable Leray-type solution (u, p) arise to the (MNS) formulation if and only if ∇p = ∇p ϕ while for general weak solutions we may have solutions such that ∇p = ∇p ϕ + f (t).

We list here a few examples to be found in the literature :

1. Solutions in L 2 : in 1934, Leray studied the Navier-Stokes problem (NS) with an initial data u 0 ∈ L 2 and a forcing tensor F ∈ L 2 t L 2 x [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF]. He found solutions belonging to u ∈ L ∞ L 2 ∩ L 2 Ḣ1 . Leray's construction by mollification provides suitable solutions which are automatically a solution of the mild formulation of the Navier-Stokes equations (MNS).

Solutions in L 2

uloc : in 1999, Lemarié-Rieusset studied the Navier-Stokes problem (MNS) with an initial data u 0 ∈ L 2 uloc [START_REF] Lemarié-Rieusset | Solutions faibles d'énergie infinie pour les équations de Navier-Stokes dans R 3[END_REF][START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]. He obtained (local in time) existence of a suitable solution u on (0, T 0 ) × R d satisfying sup

x 0 ∈R d sup 0<t<T 0 B(x 0 ,1) |u(t, x)| 2 dx < +∞ and sup x 0 ∈R d T 0 0 B(x 0 ,1) |∇ ⊗ u(t, x)| 2 dx < +∞. Remark that we have u ∈ L 2 ((0, T 0 ), L 2 (R d , 1 (1+|x|) d+1 dx)) but u does not be- long to L 2 ((0, T 0 ), L 2 (R d , 1
(1+|x|) d dx)); thus, in this setting, problems (NS) and (MNS) are not equivalent.

Various reformulations of local Leray solutions in L 2

uloc have been provided, two examples are those of [START_REF] Kikuchi | Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality[END_REF] and (Bradshaw and Tsai, 2019a). However, as we have seen the formulas proposed for the pressure are actually equivalent, as they all imply that u is solution to the (MNS) problem.

Basson proved in 2006 that, for d = 2 the solution u is indeed global (i.e. T 0 = T) and is unique [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF] .

3. Solutions in weighted L 2 spaces will be studied in the following chapters. For such solutions, (NS) and (MNS) are equivalents, and the fact that ∇p = ∇p ϕ can be deduced from the weighted integrability of the solution. In particular we will expose the results in (Fernández-Dalgo and Lemarié-Rieusset, 2020b) where the data satisfy

u 0 ∈ L 2 (R 3 , w γ dx) and F ∈ L 2 ((0, +∞), L 2 (R 3 , w γ dx))
with 0 < γ ≤ 2. They proved (global in time) existence of a suitable solution u such that, for all T 0 < +∞,

sup 0<t<T 0 |u(t, x)| 2 w γ (x) dx < +∞ and T 0 0 |∇ ⊗ u(t, x)| 2 w γ (x), dx < +∞.
By Theorem 1, we can see that the formula proposed by Bradshaw and Tsai for the pressure can be derived from the weighted integrability of the solution and u is a solution of the (MNS).

4. Homogeneous Statistical Solutions : in 1977, Vishik and Fursikov considered the (MNS) problem with a random initial value u 0 (ω) [START_REF] Vishik | Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes[END_REF]. The statistics of the initial distributions were supposed to be invariant through translation of the arguments of u 0 , more precisely : for every Borel subset B of L 2 loc (R 3 ) and every x 0 ∈ R 3 ,

Pr(u 0 (• -x 0 ) ∈ B) = Pr(u 0 ∈ B).
Another crucial assumption was that u 0 has a bounded mean energy density :

e 0 = E |x|≤1 |u 0 | 2 dx |x|≤1 dx < +∞.
Then, we have Pr(u 0 ∈ L 2 and u = 0) = 0 while, for any > 0,

Pr( |u 0 | 2 1 (1 + |x|) 3+ dx < +∞) = 1.
In (Vishik and Fursikov, 1988), they solved the Navier-Stokes equation for almost every initial value u 0 (ω), and the solution belonged almost surely to the space

L ∞ t L 2 x ( 1 (1+|x|) 3+ dx) with ∇ ⊗ u ∈ L 2 t L 2 x ( 1 (1+|x|) 3+ dx).
Basson gave a description of the pressure in those equations (which is equivalent to our description through the Leray projection operator). He proved the suitability of these solutions as well (Basson, 2006a).

Chapter 3

Solutions in weighted L 2 spaces

We place ourselves in the context of the weighted L 2 setting, in dimension d with d ∈ {2, 3, 4}, when the weight Φ satisfies some basic hypothesis that allow us to use the Leray's projection operator and energy controls.

First, we recall the definition of Muckenhoupt weights : for 1 < q < +∞, a positive weight w belongs to A q (R d ) if and only if sup

x∈R d ,ρ>0 1 |B(x, ρ)| B(x,ρ) Φ dy 1 q 1 |B(x, ρ)| B(x,ρ) Φ -1 q-1 dy 1-1 q < +∞. (3.1) Definition 3.1. Let Φ be a function on R d (2 ≤ d ≤ 4).
We say that Φ is an adapted weight function if it is a continuous Lipschitz function such that :

• (H1) 0 < Φ ≤ 1. • (H2) There exists C 1 > 0 such that |∇Φ| ≤ C 1 Φ 3 2
• (H3) There exists r ∈ (1, 2] such that Φ r ∈ A r (where A r is the Muckenhoupt class of weights). For d = 4, we require r < 2 as well.

• (H4) There exists C 2 > 0 such that Φ(x) ≤ Φ( x λ ) ≤ C 2 λ 2 Φ(x), for all λ ≥ 1. We can verify that w γ = 1 (1+|x|) γ belongs to A q (R d ) if and only if -d(q -1) < γ < d. Then, Φ = w γ is an adapted weight if 0 ≤ γ ≤ 2 and γ < d.
To analyze weights that depend only on some variable, it is convenient replace in inequality (3.1) the balls B(x, ρ) by the cubes

Q(x, ρ) =]x 1 -ρ, x 1 + ρ[× • • • ×]x d - ρ, x d + ρ[.
Thus, it is easy to verify that if Φ(x) = Ψ(x 1 , x 2 ) and 1 < q < +∞, we have

Φ ∈ A q (R 3 ) if and only if Ψ ∈ A q (R 2 ). In fact, for I 1 =]x 1 -ρ, x 1 + ρ[, I 2 =]x 2 - ρ, x 2 + ρ[, I 3 =]x 3 -ρ, x 3 + ρ[ we have ( Q(x,ρ) Φ dy) 1 q ( Q(x,ρ) Φ -1 q-1 dy) 1-1 q = 2ρ( I 1 ×I 2 Ψ dy) 1 q ( I 1 ×I 2 Ψ -1 q-1 dy) 1-1 q .
As corollary, Φ(x) = 1 (1+r) γ , where r = x 2 1 + x 2 2 , is an adapted weight on R 3 if and only if 0 ≤ γ < 2.

In summary, the following slowly decaying functions are adapted weights :

• d = 2, Φ(x) = 1 (1+|x|) γ where 0 ≤ γ < 2 • d = 3 or d = 4, Φ(x) = 1 (1+|x|) γ where 0 ≤ γ ≤ 2 • d = 3, Φ(x) = 1 (1+r) γ where r = x 2 1 + x 2 2 and 0 ≤ γ < 2.
Our main result in this chapter concerns the existence of weak suitable solutions belonging to a weighted L 2 space. Observe that the weight allows us to consider initial data with a weak decay at infinity. Theorem 4. Consider the dimension d ∈ {2, 3, 4} and an adapted weight Φ. Let u 0 be a divergence free vector field which belongs to L

2 (Φ dx, R d ). Let F = (F i,j (t, x)) i,j be a tensor such that F ∈ L 2 ((0, +∞), L 2 (Φ dx, R d )).
Then, there exists a global solution u of the problem

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0 which satisfies • u belongs to L ∞ ((0, T), L 2 (Φdx)) and ∇u belongs to L 2 ((0, T), L 2 (Φdx)), for all T > 0, • p = ∑ 1≤i,j≤d R i R j (u i u j -F i,j ), • the map t ∈ [0, +∞) → u(t, .
) is weakly continuous from [0, +∞) to L 2 (Φ dx), and strongly continuous at t = 0,

• the local energy inequality for d ∈ {2, 3} : there exists a locally finite non-negative measure µ such that

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • |u| 2 2 u -pu + u • (∇ • F) -µ,
and in the case of dimension d = 2, we have µ = 0. Moreover, we find

u(t, .) 2 L 2 (Φdx) + 2 t 0 ∇ ⊗ u(s, .) 2 L 2 (Φdx) ds ≤ u 0 2 L 2 (Φdx) - t 0 ∇(|u| 2 ) • ∇Φ dx ds + t 0 (|u| 2 u + 2pu) • ∇Φ dx ds -2 ∑ i ∑ j t 0 F i,j u j ∂ i Φ + F i,j ∂ i u j Φ dx ds.

Some lemmas on weights.

As a consequence of the Hölder inequality, we have

A q (R d ) ⊂ A r (R d ) if q ≤ r.
Lemma 3.1. If Φ ∈ A s then for all θ ∈ (0, 1), we have Φ θ ∈ A p where θ = p-1 s-1 . In particular, if we suppose Φ to satisfy (H3), we obtain Φ ∈ A p with p = 1 + r-1 r = 2 -1 r < 2, and then Φ ∈ A 2 .

Proof. We use the Hölder inequality, as 1 p = 1 s + s-p ps we have

( Q Φ p-1 s-1 dx) 1 p ( Q Φ -( p-1 s-1 )( 1 p-1 ) dx) 1-1 p = ( Q (Φ 1 s (Φ -1 s-1 ) s-p ps ) p dx) 1 p ( Q Φ -( p-1 s-1 )( 1 p-1 ) dx) 1-1 p ≤ ( Q Φ dx) 1 s ( Q Φ -1 s-1 dx) 1 p -1 s +1-1 p
The behavior of adapted weights in the Sobolev espaces W 1,r is favorable :

Lemma 3.2. Let Φ be a weight satisfying (H1) and (H2) and let 1 ≤ r < +∞.

Then : a) √ Φ f ∈ H 1 if and only if f ∈ L 2 (Φ dx) and ∇ f ∈ L 2 (Φ dx); and we have √ Φ f H 1 ≈ Φ(| f | 2 + |∇ f | 2 ) dx 1/2 b) Φ f ∈ W 1,r if and only if f ∈ L r (Φ r dx) and ∇ f ∈ L r (Φ r dx); and Φ f W 1,r ≈ Φ r (| f | r + |∇ f | r ) dx 1/r Proof. We just need to observe that |∇Φ| ≤ C 1 Φ 3/2 ≤ C 1 Φ and |∇( √ Φ)| = 1 2 |∇Φ| Φ √ Φ ≤ 1 2 C 1 √ Φ.
It is well known that for a weight w ∈ A q (1 < q < +∞), the Riesz transforms and the Hardy-Littlewood maximal function are bounded on L q (w dx), we refer to [START_REF] Stein | Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals[END_REF]. In our setting the following inequalities will be useful : Lemma 3.3. Consider the dimension d ∈ {2, 3, 4} and let Φ be a weight satisfying (H1), (H2) and (H3). Then : a) for j = 1, . . . , d, the Riesz transforms R j satisfy that

√ ΦR j f 2 ≤ C √ Φ f 2 and √ ΦR j f H 1 ≤ C √ Φ f H 1 ; b) for j = 1, . . . , d the Riesz transforms R j fulfills ΦR j f r ≤ C Φ f r and ΦR j f W 1,r ≤ C Φ f W 1,r ; c) if P is the Leray projection operator then for a vector field u we have √ ΦPu 2 ≤ C √ Φu 2 and √ ΦPu H 1 ≤ C √ Φu H 1 ; d) for a vector field u we have √ Φ u H 1 ≈ √ Φ u 2 + √ Φ∇ • u 2 + √ Φ∇ ∧ u 2 . e) Let θ (x) = 1 d θ( x )
, where θ ∈ D(R d ), θ is non-negative and radially decreasing and θ dx = 1. Then, there exists a constant C, which does not depend on nor f , such that

√ Φ (θ * f ) 2 ≤ C √ Φ f 2 and √ Φ (θ * f ) H 1 ≤ C( √ Φ f L 2 + √ Φ ∇ f L 2 ). Proof. a) Since ∂ k (R j f ) = R j (∂ k f
) and using Φ ∈ A 2 and Lemma 3.2 we obtain the inequality.

b) Using Φ r ∈ A r and Lemma 3.2 we can conclude the inequality. c) Can be obtained from a) since for v = Pu, we have

v j = ∑ d k=1 R j R k (u k ). d) If R = (R 1 , . . . , R d ), we have the identity -∆u = ∇ ∧ (∇ ∧ u) -∇(∇ • u) so that ∂ k u = R k R ∧ (∇ ∧ u) -R k R(∇ • u),
then applying a) we obtain d) e) We recall the classical inequality (see Theorem 2.1.10 in Chapter 2 of [START_REF] Grafakos | Classical harmonic analysis[END_REF]) which states that we have |θ * f | ≤ M f , where M f is the Hardy-Littlewood maximal function of f , and, similarly, |∂ k (θ * f )| ≤ M ∂ k f . Then, as Φ ∈ A 2 and using Lemma 3.2 we conclude e).

Our assumptions on adapted weights, also imply slowly decaying at infinity : Lemma 3.4. Let Φ be a weight which satisfy (H1) and (H2). Then there exists a constant

C 3 such that 1 (1 + |x|) 2 ≤ C 3 Φ.
Moreover, if d = 3 and Φ depends only on r = x 2 1 + x 2 2 , then we obtain

1 (1 + |r|) 2 ≤ C 3 Φ. Proof. Let us denote x 0 = 1 |x| x and g(λ) = Φ(λx 0 ). We remark that g (λ) = x 0 • ∇Φ(λx 0 ) ≥ -C 1 (Φ(λx 0 )) 3/2 = -C 1 g(λ) 3/2 .
Therefore,

C 1 λ ≥ - λ 0 g (µ)g(µ) -3/2 dµ = 2(g(λ) -1/2 -g(0) -1/2 ) and hence Φ(x) -1/2 ≤ Φ(0) + C 1 2 |x| ≤ C 3 (1 + |x|).
If Φ depends only on r, we just need to observe that

1 (1 + |r|) 2 ≤ C 3 Φ(x 1 , x 2 , 0) = C 3 Φ(x).

Proof of Theorem 4 3.2 A priori estimates

Consider φ ∈ D(R d ) a real-valued test function which is equal to 1 on a neighborhood of 0 and let φ (x) = φ( x). Let

u 0, = P(φ u 0 ) and F = φ F.
Then, u 0, is divergence free and since Φ ∈ A 2 , u 0, converges to u 0 in L 2 (Φ dx). We also check, by dominated convergence, that F converges to the tensor

F in L 2 ((0, T), L 2 (Φ dx)). Let θ (x) = 1 d θ( x )
, where θ ∈ D(R d ), θ is non-negative and radially decreasing and θ dx = 1.

Observe that u 0, belongs to L 2 . Then, a classical result for the mollified Navier-Stokes equations gives the existence of a unique global solution u of the problem

(NS )    ∂ t u = ∆u -((u * θ ) • ∇)u -∇p + ∇ • F ∇ • u = 0, u (0, .) = u 0, which belongs to C([0, +∞), L 2 (R d )) ∩ L 2 ((0, +∞), Ḣ1 (R d )).
For the dimension d ∈ {2, 3}, we look to demonstrate that

√ Φu (t) 2 L 2 + t 0 √ Φ∇ ⊗ u 2 L 2 ds ≤ √ Φu 0, 2 L 2 + c √ ΦF 2 L 2 ((0,+∞),L 2 ) + C Φ t 0 √ Φu 2 L 2 + √ Φu 2d L 2 ds, (3.2)
where C Φ does not depend on nor on u 0 . When d = 4, we will obtain the next useful variant of this inequality, which holds under the assumption

√ Φ u 2 < 0 with C 0 < 1 8 where C > 0 is a fixed constant; the variant is √ Φu (t) 2 L 2 + t 0 √ Φ∇ ⊗ u 2 L 2 ds ≤ √ Φu 0, 2 L 2 + c √ ΦF 2 L 2 ((0,+∞),L 2 ) + C Φ t 0 √ Φu 2 L 2 ds. (3.3) Let us denote b = u * θ . As we have seen √ Φ, ∇ √ Φ ∈ L ∞ , then pointwise multiplication by √ Φ maps boundedly H 1 to H 1 and H -1 to H -1 . Thus, we know that √ Φu ∈ L 2 H 1 and √ Φ∂ t u ∈ L 2 H -1
, so that we can calculate ∂ t u • u Φ dx and more precisely we find :

|u (t, x)| 2 2 Φ dx + t 0 |∇ ⊗ u | 2 Φdx ds = |u 0, (x)| 2 2 Φ dx - t 0 (∇ ⊗ u ) • (∇Φ ⊗ u ) dx ds + t 0 ( |u | 2 2 b + p u ) • ∇Φ dx ds -∑ i ∑ j t 0 F ,i,j u ,j ∂ i Φ + F ,i,j ∂ i u ,j Φ dx ds. We use the fact that |∇Φ| ≤ C 0 Φ 3 2 ≤ C 0 Φ, in order to control the following term |- t 0 (∇ ⊗ u ) • (∇Φ ⊗ u )dx ds| ≤ 1 8 t 0 √ Φ ∇ ⊗ u 2 L 2 + C t 0 √ Φ u 2 L 2 ,
and also we can easily control the term

|-∑ i ∑ j t 0 F ,i,j u ,j ∂ i Φ + F ,i,j ∂ i u ,j Φ dx ds| ≤ 1 8 t 0 √ Φ ∇ ⊗ u 2 L 2 + c t 0 √ Φ F 2 L 2 + C Φ t 0 √ Φ u 2 L 2 .
Now, we analyze the integrals containing the pressure term. Strategically, we distinguish two cases :

• Case 1: d = 2 and r ∈ (1, 2], or d = 3 and r ∈ [ 6 5 , 2], or d = 4 and r ∈ [ 4 3 , 2). We observe that for those values of d and r we have

0 ≤ d 2 - d 2r ≤ 1 and Ḣ d 2 -d 2r ⊂ L 2r and 0 ≤ d r - d 2 ≤ 1 and Ḣ d r -d 2 ⊂ L r r-1 .
Using the continuity of the Riesz transforms on L r (Φ r dx), we observe that

t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ t 0 Φ(|u | |b | + |p |) r √ Φu r r-1 ≤ C t 0 √ Φu 2r √ Φb 2r √ Φu r r-1 ds Using the Sobolev embedding Ḣ d 2 -d 2r ⊂ L 2r , the fact that |∇ √ Φ| ≤ C √ Φ,
and the continuity of the maximal function operator on L 2 (Φdx), we obtain

√ Φb 2r ≤ C √ Φb 1-( d 2 -d 2r ) 2 ∇ ⊗ ( √ Φb ) d 2 -d 2r 2 ≤ C √ Φb 1-( d 2 -d 2r ) 2 ( √ Φb 2 + √ Φ∇ ⊗ b 2 ) d 2 -d 2r ≤ C √ Φu 1-( d 2 -d 2r ) 2 ( √ Φu 2 + √ Φ∇ ⊗ u 2 ) d 2 -d 2r ,
and

√ Φu 2r ≤ C √ Φu 1-( d 2 -d 2r ) 2 ( √ Φu 2 + √ Φ∇ ⊗ u 2 ) d 2 -d 2r .
Using the embedding

Ḣ d r -d 2 ⊂ L r r-1 , we also have √ Φu r r-1 ≤ C √ Φu 1-( d r -d 2 ) 2 ∇ ⊗ ( √ Φu ) d r -d 2 L 2 ≤ C √ Φu 1-( d r -d 2 ) 2 ( √ Φu 2 + √ Φ∇ ⊗ u L 2 ) d r -d 2 .
Therefore,

t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ C t 0 √ Φu 3-d 2 2 ( √ Φu 2 + √ Φ∇ ⊗ u L 2 ) d 2 ds.
Using the Young inequality, we then find for

d = 2 or d = 3 t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ 1 8 t 0 √ Φ∇ ⊗ u 2 L 2 ds + C Φ t 0 √ Φu 2 L 2 + √ Φu 12-2d 4-d L 2 ds,
where, we have 12-2d 4

-d = 2d since d ∈ {2, 3}. When we consider d = 4, provided that √ Φ u 2 < 0 with C 0 < 1 8 we find t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ 1 8 t 0 √ Φ∇ ⊗ u 2 L 2 ds + 1 8 t 0 √ Φu 2 L 2 ds,
• Case 2: d = 3 andr ∈ (1, 6 5 ), or d = 4 andr ∈ (1, 4 3 ). Let q = dr d-r ; for those values of d, r and q, we have

W 1,r ⊂ L q 0 ≤ d - d r ≤ 1 and Ḣd(1-1 r ) ⊂ L 2r 2-r . and 0 ≤ d r - d 2 -1 ≤ 1 and Ḣ d r -d 2 -1 ⊂ L q q-1 .
Using the continuity of the Riesz transforms on L r (Φ r dx), we can observe that

t 0 ( |u | 2 |b | 2 + |p||u |) |∇Φ| dx ds ≤ t 0 Φ|u | 2 q √ Φb q q-1 ds + t 0 Φp q √ Φu q q-1 ds ≤C t 0 Φ|u | 2 W 1,r √ Φb q q-1 ds + ∑ ij t 0 Φb ,i u ,j W 1,r √ Φu q q-1
ds.

We remark that

Φb ,i u ,j W 1,r ≤ Φb ,i u ,j r + ∑ k ( b ,i u ,j ∂ k Φ L r + Φ b ,i ∂ k u ,j L r + Φ u ,i ∂ k b ,j L r ) ≤C( √ Φu 2r 2-r √ Φb 2 + √ Φb 2r 2-r √ Φ∇ ⊗ u 2 + √ Φu 2r 2-r √ Φ∇ ⊗ b 2 ), ≤C (
√ Φu L 2 + √ Φ∇ ⊗ u L 2 )( √ Φu Ḣd(1-1 r ) + √ Φb Ḣd(1-1 r ) ). and √ Φb Ḣd(1-1 r ) ≤ C √ Φb 1-(d-d r ) 2 ∇ ⊗ ( √ Φb ) d-d r 2 ≤ C √ Φb 1-(d-d r ) 2 ( √ Φb 2 + √ Φ∇ ⊗ b 2 ) d-d r ≤ C √ Φu 1-(d-d r ) 2 ( √ Φu L 2 + √ Φ∇ ⊗ u L 2 ) d-d r .
Therefore, we get

∑ i,j Φb ,i u ,j W 1,r + Φ|u | 2 W 1,r ≤ C √ Φu 1-(d-d r ) 2 ( √ Φu L 2 + √ Φ∇ ⊗ u L 2 ) 1+d-d r .
On the other hand, we have

√ Φb q q-1 ≤ C √ Φb 2-( d r -d 2 ) 2 ∇ ⊗ ( √ Φb ) d r -d 2 -1 2 ≤ C √ Φu 2-( d r -d 2 ) 2 ( √ Φu L 2 + √ Φ∇ ⊗ u L 2 ) d r -d 2 -1 .
Hence, we find again

t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ C t 0 √ Φu 3-d 2 2 ( √ Φu 2 + √ Φ∇ ⊗ u L 2 ) d 2 ds.
and we conclude in the same way as for the first case.

In the Case 1 and Case 2, we have found for d ∈ {2, 3},

t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ 1 8 t 0 √ Φ∇ ⊗ u 2 L 2 ds + 1 8 t 0 √ Φu 2 L 2 + √ Φu 2d L 2 ds and for d = 4, under the assumption √ Φ u 2 < 0 with C 0 < 1 8 where C > 0 is a fixed constant, we find t 0 ( |u | 2 |b | 2 + |p ||u |) |∇Φ| dx ds ≤ 1 8 t 0 √ Φ∇ ⊗ u 2 L 2 ds + C Φ t 0 √ Φu 2 L 2 ds.
From these controls, we get inequalities (3.2) and (3.3). Next, we will see that these inequalities give us a control on the size of √ Φ u 2 on an interval of time that does not depend on : Lemma 3.5. Suppose that α is a continuous non-negative function on [0, T) which satisfies, for three constants A, B ∈ (0, +∞)

and b ∈ [1, ∞), α(t) ≤ A + B t 0 α(s) + α(s) b ds. • For b > 1, let 0 < T 1 < T and T 0 = min(T 1 , 1 3 b B(A b-1 +(BT 1 ) b-1 ) ). Then, we have, for every t ∈ [0, T 0 ], α(t) ≤ 3A.
• We suppose b = 1. Then, we have, for every t ∈ [0, 1 4B ], the estimate α(t) ≤ 2A. Proof. First, consider b > 1. We seek to estimate the first time T * < T 1 (if it exists) for which we have α(T * ) = 3A.

We have

α ≤ A BT 1 + ( BT 1 A ) b-1 α b .
We thus find

α(T * ) ≤ 2A + BT * (3A) b (1 + ( BT 1 A ) b-1 ) which implies BT * 3 b (A b-1 + (BT 1 ) b-1 ) ≥ 1. Now, consider b = 1, so that α(t) ≤ A + 2B t 0 α(s).
We look to estimate the first time T * < T (if it exists) for which we have

α(T * ) = 2A.
We have

2A = α(T * ) ≤ A + 2BT * 2A
which implies

T * ≥ 1 4B
Applying Lemma 3.5 to the inequalities (3.2) and (3.3), we thus find that there exists a constant C Φ ≥ 1 such that if T 0 satisfies

• if d = 2, C Φ 1 + u 0 2 L 2 (Φdx) + F 2 L 2 ((0,+∞),L 2 (Φdx)) T 0 = 1 • if d = 3, C Φ 1 + u 0 2 L 2 (Φdx) + F 2 L 2 ((0,+∞),L 2 (Φdx)) 2 T 0 = 1 • if d = 4 and c Φ ( u 0 2 L 2 (Φ dx) + F 2 L 2 ((0,+∞),L 2 (Φdx)) ) ≤ 1 C Φ ,
with C Φ T 0 = 1, and where c Φ > 0 is a constant such that for all > 0,

u 0, 2 L 2 (Φ dx) + c F 2 L 2 ((0,+∞),L 2 (Φdx)) ≤ c Φ ( u 0 2 L 2 (Φ dx) + F 2 L 2 ((0,+∞),L 2 (Φdx)) ), then sup 0≤t≤T 0 u (t, .) 2 L 2 (Φdx) + T 0 0 ∇ ⊗ u 2 L 2 (Φ dx) ds ≤ C Φ ( u 0 2 L 2 (Φ dx) + F 2 L 2 ((0,+∞),L 2 (Φdx)) ).
(3.4)

Passage to the limit and local existence

We know that u is bounded in

L ∞ ((0, T 0 ), L 2 (Φ dx)) and ∇ ⊗ u is bounded in L 2 ((0, T 0 ), L 2 (Φ dx)
). This will alow us to use the following version of the Aubin-Lions theorem :

Lemma 3.6 (Aubin-Lions compactness theorem). Let s > 0, q > 1 and σ < 0. Let ( f n ) n be a sequence of functions on (0, T) × R d such that, for all T 0 ∈ (0, T) and all ϕ ∈ D(R d ),

• ϕ f n is bounded in L 2 ((0, T 0 ), H s ) • ϕ∂ t f n is bounded in L q ((0, T 0 ), H σ ) .
Then, there exists a subsequence ( f n k ) such that f n k is strongly convergent in L 2 loc ([0, T) × R d ), more precisely : if we denote f ∞ the limit, then for all T 0 ∈ (0, T) and for all R 0 > 0, lim

n k →+∞ T 0 0 |x|≤R 0 | f n k -f ∞ | 2 dx dt = 0.
For a proof of the Lemma, we refer the books [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models[END_REF]Lemarié-Rieusset, 2016).

We want to verify that ϕ∂ t u is bounded in L α ((0, T 0 ), H -s ) for some s ∈ (-∞, 0) and some α > 1.

In Case 1, we have seen that Φb ⊗ u and

Φp = ∑ 3 i=1 ∑ 3 j=1 R i R j (b ,i u ,j ) are bounded in L α 1 ((0, T 0 ), L r ), where α 1 = 2r dr-d , so that α 1 ∈ [2, ∞) if d = 2, α 1 ∈ [ 4 3 , 4] if d = 3 and α 1 ∈ (1, 2] if d = 4.
In Case 2, we have seen that Φb ⊗ u and Φp are bounded in L α 2 ((0, T 0 ), W 1,r ), where α 2 = 2r r+dr-d and thus it is bounded in L α 2 L q , with q = dr d-r . We observe that if

d = 3 then α 2 ∈ ( 4 3 , 2) and if d = 4 then α 2 ∈ (1, 2). Let ϕ ∈ D(R d ).
We have that ϕu is bounded in L 2 ((0, T 0 ), H 1 ), it is the first assumption to apply the Aubin-Lions theorem. Now, writing

∂ t u = ∆u -( 3 ∑ j=1 ∂ j (b ,j u ) + ∇p ) + ∇ • F and using the embeddings L r ⊂ Ḣ d 2 -d r ⊂ H -1 (in Case 1) or L dr d-r ⊂ H -( d r -d 2 -1) ⊂ H -1 (in Case 2) we see that ϕ∂ t u is bounded in L α i ((0, T 0 ), H -2 ).
Thus, by the Aubin-Lions theorem, there exist u and a sequence ( k ) k∈N converging to 0 such that u k converges strongly to u in L 2 loc ([0, T 0 ) × R 3 ): for every T ∈ (0, T 0 ) and every R > 0, we have

lim k→+∞ T 0 |y|<R |u k -u| 2 dx ds = 0.
Then, we have that

u k converges *-weakly to u in L ∞ ((0, T 0 ), L 2 (Φdx)), ∇ ⊗ u k converges weakly to ∇ ⊗ u in L 2 ((0, T 0 ), L 2 (Φdx)), and u k converges weakly to u in L 3 ((0, T 0 ), L 3 (Φ 3 2 dx)). We deduce that b k ⊗ u k is weakly convergent in (L 6/5 L 6/5 ) loc to b ⊗ u and thus in D ((0, T 0 ) × R d ); as in Case 1, it is bounded in L α 1 ((0, T 0 ), L r ),
and in Case 2 it is bounded in L α 2 ((0, T 0 ), W 1,r ), it is weakly convergent in these spaces respectively (as D is dense in their dual spaces).

By the continuity of the Riesz transforms on L r (Φ r dx) and on W 1,r (Φ r dx), we find that in Case 1 and Case 2, p k is convergent to the distribution

p = 3 ∑ i=1 3 ∑ j=1 R i R j (u i u j ).
We thus have obtained

∂ t u = ∆u + (u • ∇)u -∇p + ∇ • F.
Moreover, we have seen that ∂ t u is locally in L 1 H -2 , and thus u has representative such that t → u(t, .) is continuous from [0, T 0 ) to D (R d ) and coincides with u(0, .) + t 0 ∂ t u ds. In the sense of distributions, we have u(0, .)

+ t 0 ∂ t u ds = u = lim k→+∞ u k = lim k→+∞ u 0, k + t 0 ∂ t u n k ds = u 0 + t 0 ∂ t u ds,
hence, u(0, .) = u 0 , and thus u is a solution of (NS).

Now, we want to prove the energy balance. In the case of dimension 2, we remark that, since

√ Φu ∈ L ∞ L 2 ∩ L 2 H 1 , we have by interpolation that √ Φu ∈ L 4 L 4
, and then we can define ((u • ∇)u) • u. Then the equality

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • |u| 2 2 u -pu + u • (∇ • F) can be proved.
Let us consider the case d = 3. We define

A = -∂ t ( |u | 2 2 ) + ∆( |u | 2 2 ) -∇ • |u | 2 2 u -p u + u • (∇ • F ) = |∇ ⊗ u | 2 .
Remark that, as u k is locally strongly convergent in L 2 L 2 ; and locally bounded in

L ∞ L 2 , it is then locally strongly convergent in L p L 2 , for all p ∈ [2, +∞). Then, as √ Φ∇ ⊗ u is bounded in L 2 ((0, T), L 2 )
, by the Gagliardo-Nirenberg interpolation inequalities we obtain u k is locally strongly convergent in L p L q with 2 p + 3 q > d 2 . In Case 1, we know that p k is locally weakly convergent in L α 1 L r and we know by the remark above that u k is locally strongly convergent in L α 1 α 1 -1 L r r-1 , and hence p k u k converges in the sense of distributions.

In Case 2, we know that p k is locally weakly convergent in L α 2 L q and by the observation above, u k is locally strongly convergent in L α 2 α 2 -1 L q q-1 , and hence p k u k converges in the sense of distributions.

Thus,

A k is convergent in D ((0, T) × R 3 ) to A = -∂ t ( |u| 2 2 ) + ∆( |u| 2 2 ) -∇ • |u| 2 2 u -pu + u • (∇ • F),
and

A = lim k→+∞ |∇ ⊗ u k | 2 . If θ ∈ D((0, T) × R d ) is non-negative, we have that √ θ∇ ⊗ u k is weakly convergent in L 2 L 2 to √ θ∇ ⊗ u, so that Aθ dx ds = lim k →+∞ A k θ dx ds = lim k→+∞ |∇ ⊗ u k | 2 θ dx ds ≥ |∇ ⊗ u| 2 θ dx ds.
Hence, in the case d = 3, there exists a non-negative locally finite measure µ on (0, T) × R 3 such that A = |∇u| 2 + µ, i.e. such that

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • |u| 2 2 u -pu + u • (∇ • F) -µ.

Convergence to the initial data

Let us consider inequalities (3.2) and (3.3), together with the inequality (3.4). On (0, T 0 ) we have a uniform control, in and t, of u (t, .) 2 L 2 (Φdx) . Thus, from (3.2) and (3.3) we get

u (t, .) 2 L 2 (Φ dx) ≤ u 0, 2 L 2 (Φ dx) + C Φ t(1 + u 0 2d L 2 (Φ dx) + F 2d L 2 ((0,+∞),L 2 (Φdx)) ).
We remember that u 0, k is strongly convergent to u 0 in L 2 (Φ dx). Moreover, as

u k = u 0, k + t 0 ∂ t u k ds, we observe that u k (t, .) is convergent to u(t, .) in D (R d ),
and we can deduce that it is weakly convergent in

L 2 (Φ dx) (as it is bounded in L 2 (Φdx)), so that u(t, .) 2 L 2 (Φ dx) ≤ u 0 2 L 2 (Φ dx) + C Φ t(1 + u 0 2d L 2 (Φ dx) + F 2d L 2 ((0,+∞),L 2 (Φdx)) ).
This remark implies lim sup

t→0 u(t, .) 2 L 2 (Φdx) ≤ u 0 2 L 2 (Φdx) .
For reciprocal inequality, we recall that u is weakly continuous in L 2 (Φdx), therefore

u 0 2 L 2 (Φdx) ≤ lim inf t→0 u(t, .) 2 L 2 (Φdx) .
We conclude that u 0 2

L 2 (Φdx) = lim t→0 u(t, .) 2 L 2 (Φdx)
, It allows us to turn the weak convergence into a strong convergence in the Hilbert space.

Global existence using a scaling argument

We fix λ > 0. Then u is a solution of the Cauchy initial value problem for the approximated Navier-Stokes system (NS ) on (0, T) with initial value u 0, and forcing tensor F if and only if u ,λ (t, x) = λu (λ 2 t, λx) is a solution for the approximated Navier-Stokes equations (NS λ ) on (0, T/λ 2 ) with initial value u 0, ,λ (x) = λu 0, (λx) and forcing tensor F ,λ (x) = λ 2 F (λ 2 t, λx). We shall denote u 0,λ = λu 0 (λx) and F λ (x) = λ 2 F(λ 2 t, λx).

We have proved that for d ∈ {2, 3},

√ Φu ,λ (t) 2 L 2 + t 0 √ Φ∇ ⊗ u ,λ 2 L 2 ≤ √ Φu 0, ,λ 2 L 2 + C √ ΦF ,λ 2 L 2 ((0,+∞),L 2 ) + C Φ t 0 √ Φu ,λ 2 L 2 + √ Φu ,λ 2d L 2 ds
where C Φ does not depend on , λ nor on u 0 . When d = 4, under the assumption

√ Φ u 2 < 0 with C 0 < 1 8 where C > 0 is a fixed constant, we have found √ Φu ,λ (t) 2 L 2 + t 0 √ Φ∇ ⊗ u ,λ 2 L 2 ds ≤ √ Φu 0, ,λ 2 L 2 + c √ ΦF ,λ 2 L 2 ((0,+∞),L 2 ) + C Φ t 0 √ Φu ,λ 2 L 2 ds.
By Lemma 3.5, we thus found that there exists a constant

C Φ ≥ 1 such that if T λ satisfies • if d = 2, C Φ 1 + u 0,λ 2 L 2 (Φdx) + F λ 2 L 2 ((0,+∞),L 2 (Φdx)) T λ = 1 • if d = 3, C Φ 1 + u 0,λ 2 L 2 (Φdx) + F λ 2 L 2 ((0,+∞),L 2 (Φdx)) 2 T λ = 1 • if d = 4 and c Φ ( u 0,λ 2 L 2 (Φ dx) + F λ 2 L 2 ((0,+∞),L 2 (Φdx)) ) ≤ 1 C Φ with C Φ T λ = 1 then sup 0≤t≤T λ u ,λ (t, .) 2 L 2 (Φdx) + T λ 0 ∇ ⊗ u ,λ 2 L 2 (Φ dx) ds ≤ C Φ ( u 0,λ 2 L 2 (Φ dx) + F λ 2 L 2 ((0,+∞),L 2 (Φdx)) ).
It permits to deduce that the solutions u are controlled, uniformly in , on (0, λ 2 T λ ). In fact, we get for t ∈ (0, T λ ),

|u ,λ (t, x)| 2 Φ(x) dx = |u (λ 2 t, y)| 2 Φ( y λ )λ 2-d dy ≥ λ 2-d |u (λ 2 t, x)| 2 Φ(x) dx and T λ 0 |∇ ⊗ u ,λ (t, x)| 2 Φ(x) dx dt = λ 2 T λ 0 |∇ ⊗ u ,λ (s, y)| 2 Φ( y λ )λ 2-d dy ds ≥λ 2-d λ 2 T λ 0 |∇ ⊗ u (t, x)| 2 Φ(x) dx dt.
Moreover, we get lim

λ→+∞ λ 2 T λ = +∞ when 2 ≤ d ≤ 4, and if d = 4, lim λ→+∞ u 0,λ 2 L 2 (Φ dx) + F λ 2 L 2 ((0,+∞),L 2 (Φdx)) = 0.
Indeed, we observe that

λ 2 |u 0 (λx)| 2 Φ(x) dx = λ 2-d |u 0 (x)| 2 Φ( x λ ) dx = λ 4-d |u 0 (x)| 2 Φ( x λ ) λ 2 Φ(x) Φ(x) dx and +∞ 0 λ 4 |F(λ 2 x, λx)| 2 Φ(x) dx = λ 2-d +∞ 0 |F(t, x)| 2 Φ( x λ ) dx = λ 4-d +∞ 0 |F(t, x)| 2 Φ( x λ ) λ 2 Φ(x) Φ(x) dx. Since Φ( x λ ) λ 2 Φ(x) ≤ min{C 2 , 1 λ 2 Φ(x)
} by hypothesis (H4), we find by dominated convergence that u

0,λ L 2 (Φ dx) = o(λ 4-d 2 ) and also F λ L 2 ((0,+∞),L 2 (Φ dx)) = o(λ 4-d 2 ), so that lim λ→+∞ λ 2 T λ = +∞.
Therefore, if we consider a finite time T and a sequence k converging to 0, we may choose λ such that

λ 2 T λ > T (if d = 4 we demand λ to satisfy c Φ ( u 0,λ L 2 (Φ dx) + F λ L 2 ((0,+∞),L 2 (Φ dx))
) < 0 as well). As we have seen, we obtain a uniform control of u and of ∇ ⊗ u on (0, λ 2 T λ ), hence a uniform control of u and of ∇ ⊗ u on (0, T). We thus may exhibit a solution on (0, T) using again the Rellich-Lions theorem, by extracting a subsequence k n . A diagonal argument conducts then to exhibit a global solution.

Thus we have proved Theorem 4.

Global Regular Solutions in dimension 2

In dimension d = 2, we can treat the existence problem in higher regularity, while, in the case d = 3, one will restrict the analysis to the case of axisymmetric flows with no swirl which will be treated in the following chapter.

Theorem 5. Consider d = 2 and a weight Φ satisfying (H1) -(H4). Let u 0 be a divergence free vector field, such that u 0 , ∇ ⊗ u 0 belong to L 2 (Φdx). Let F = (F i,j (t, x)) be a tensor such that F, ∇ • F belong to L 2 ((0, +∞), L 2 (Φdx)). Then, there exists a global solution u of the problem

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0 such that • u and ∇u belong to L ∞ ((0, T), L 2 (Φ dx)) and ∆u belongs to L 2 ((0, T), L 2 (Φ dx)), for all T > 0 • the maps t ∈ [0, +∞) → u(t, .
) and t ∈ [0, +∞) → ∇u(t, .) are weakly continuous from [0, +∞) to L 2 (Φdx), and are strongly continuous at t = 0.

In the case of dimension d = 2, the Navier-Stokes equations are well-posed in H 1 and we don't need to mollify the equations. More precisely, we may approximate the Navier-Stokes equations with the system

(NS )    ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u (0, .) = u 0,
where u 0, = P(φ u 0 ) and F = φ F.

Observe that the vorticity ω = ∇ ∧ u is solution of

   ∂ t ω = ∆ω -(u • ∇)ω + ∇ ∧ (∇ • F ) ∇ • ω = 0, ω (0, .) = ω 0,
where ω 0, = ∇ ∧ (φ u 0 ).

Since u 0, belongs to H 1 , we know that we have a global solution u . We then just have to prove that, for every finite time T 0 , the norms ω L ∞ ((0,T 0 ),L 2 (Φ dx)) and ∇ω L 2 ((0,T 0 ),L 2 (Φ dx)) are uniformly controlled.

We can compute ∂

t ω • ω Φ dx, which gives |ω (t, x)| 2 2 Φ dx + t 0 |∇ω | 2 Φdx ds = |ω 0, (x)| 2 2 Φ dx - t 0 ∇( |ω | 2 2 ) • ∇Φdx ds + t 0 |ω | 2 2 u • ∇Φ dx ds + t 0 (∇ • F ) 2 (-∂ 1 ω Φ -∂ 1 Φ ω )) + (∇ • F ) 1 (∂ 2 ω Φ + ∂ 2 Φ ω )) dx ds, where (∇ • F ) j is the j-coordinate of ∇ • F , more precisely (∇ • F ) j = ∑ i ∂ i F ,j . As t 0 |ω | 2 2 u • ∇Φ dx ds ≤ t 0 √ Φω 2 L 8 3 √ Φu L 4 ≤ t 0 ( √ Φω 3/4 L 2 ∇( √ Φω ) 1/4 L 2 ) 2 √ Φu L 4 we obtain √ Φω (t) 2 L 2 + t 0 √ Φ∇ω 2 L 2 ≤ √ Φω 0, 2 L 2 + C √ Φ(∇ • F ) 2 L 2 ((0,+∞),L 2 ) + C Φ t 0 √ Φω 2 L 2 (1 + √ Φu 4 3 L 4 ) ds
We can conclude that, for all T > 0 and for all t ∈ (0, T),

√ Φω (t) 2 L 2 + t 0 √ Φ∇ω 2 L 2 ≤ ( √ Φω 0, 2 L 2 + C √ Φ(∇ • F ) 2 L 2 ((0,+∞),L 2 ) )e C Φ sup >0 t 0 (1+ √ Φu L 4 ) 4 3 ds
Thus, we have uniform controls on (0, T).

Chapter 4

Regular Axisymmetric Solutions without swirl 4.1 Axisymmetry.

We consider the usual coordinates (x 1 , x 2 , x 3 ) in the Euclidean space R 3 , and the cylindrical coordinates (r, θ, z) provided by the formulas x 1 = r cos θ, x 2 = r sin θ and x 3 = z.

Let us denote (e 1 , e 2 , e 3 ) the usual canonical basis e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1).

We attach to each point x (with r = 0) another orthonormal basis

e r = ∂x ∂r = cos θe 1 + sin θe 2 , e θ = 1 r ∂x ∂θ = -sin θe 1 + cos θe 2 , e z = ∂x ∂z = e 3 .
Let us consider a vector field u = (u 1 , u 2 , u 3 ) = u 1 e 1 + u 2 e 2 + u 3 e 3 . Then, we find their coordinates in the new basis u = (u 1 cos θ + u 2 sin θ) e r + (-u 1 sin θ + u 2 cos θ) e θ + u 3 e z .

We denote (u r , u θ , u z ) p the coordinates of u in the basis (e r , e θ , e z ). In what follows in this chapter we will consider axially symmetric (axisymmetric) vector fields u without swirl and axisymmetric scalar functions a, which means that u = u r (r, z) e r + u z (r, z) e z and a = a(r, z).

Our goal is to adapt the energy estimates in weighted spaces to the framework of axisymmetric solutions of the Navier-Stokes equations. We look to generalize the following well known result is due to Ladyzhenskaya, we refer to [START_REF] Ladyzhenskaya | On the unique global solvability of the Cauchy problem for the Navier-Stokes equations in the presence of the axial symmetry[END_REF]Lemarié-Rieusset, 2016). Proposition 4.1. Consider a divergence free axisymmetric vector field u 0 without swirl which belongs to H 1 . Then, the problem

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p ∇ • u = 0, u(0, .) = u 0 has a unique solution u ∈ C([0, +∞), H 1
). This solution is axisymmetric without swirl. Moreover, u, ∇ ⊗ u belong to L ∞ ((0, +∞), L 2 ), and ∇ ⊗ u, ∆u belong to L 2 ((0, +∞), L 2 ).

If u 0 ∈ H 2 then the following inequality is fulfilled

|ω(t)| 2 r 2 dx ≤ |ω 0 | 2 r 2 ≤ ∇ ⊗ ω 0 2 2 .
This result is our starting point.

The uniqueness of the solutions in the statement above can be seen as a consequence of the Serrin's Weak-Strong uniqueness criterion. The problem of the uniqueness of solutions, when we treat with weighted spaces, is non-trivial and we leave it as an open problem.

Below, we present our main results with respect to axisymetric without swirl solutions : Theorem 6. Consider a weight Φ satisfying (H1) -(H4). Assume that Φ depends only on r = x 2 1 + x 2 2 and there exists a continuous weight Ψ (that depends only on r) such that

Φ ≤ Ψ ≤ 1, Ψ ∈ A 2 and there exists C 1 > 0 such that |∇Ψ| ≤ C 1 √ ΦΨ
Let u 0 be a divergence free axisymmetric vector field without swirl, such that u 0 belongs to L 2 (Φ dx) and ∇ ⊗ u 0 belongs to L 2 (Ψ dx). Then there exists a time T > 0, and a local solution u on (0, T) of the problem

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p ∇ • u = 0, u(0, .) = u 0 such that
• u is axisymmetric without swirl, u belongs to L ∞ ((0, T), L 2 (Φ dx)), ∇ ⊗ u belongs to L ∞ ((0, T), L 2 (Ψ dx)) and ∆u belongs to L 2 ((0, T), L 2 (Ψ dx)),

• the maps t → u(t, .) and t → ∇u(t, .) are weakly continuous from [0, T) to L 2 (Φ dx), and are strongly continuous at t = 0, Remark: If the vorticity is more integrable at time t = 0, it will remain in positive times as well.

Example : The weights Φ(x) = 1 (1+r) γ and Ψ(x) = 1 (1+r) δ with 0 ≤ δ ≤ γ < 2 fulfill these hypothesis.

If we suppose the weight Ψ to be a little more regular, we obtain a global existence result. The next theorem precise these conditions on the weight.

Theorem 7. Consider a weight Φ satisfying (H1) -(H4). Assume that Φ depends only on r = x 2 1 + x 2 2 and there exists a continuous weight Ψ ∈ W 2,∞ (that depends only on r) such that Φ ≤ Ψ ≤ 1, Ψ ∈ A 2 and there exists C 1 > 0 such that

|∇Ψ| ≤ C 1 √ ΦΨ and |∆( √ Ψ)| ≤ C 1 √ Φ.
Let u 0 be a divergence free axisymmetric vector field without swirl, such that u 0 belongs to L 2 (Φdx) and ∇ ⊗ u 0 belongs to L 2 (Ψdx). Then there exists a global solution u of the problem

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p ∇ • u = 0, u(0, .) = u 0 such that • u is axisymmetric without swirl, u belongs to L ∞ ((0, T), L 2 (Φ dx)), ∇ ⊗ u belongs to L ∞ ((0, T), L 2 (Ψ dx))
and ∆u belongs to L 2 ((0, T), L 2 (Ψ dx)), for all T > 0,

• the maps t ∈ [0, +∞) → u(t, .) and t ∈ [0, +∞) → ∇ ⊗ u(t, .) are weakly continuous from [0, +∞) to L 2 (Φ dx) and to L 2 (Ψ dx) respectively, and are strongly continuous at t = 0.

Example : The weights Φ(x) = 1 (1+r) γ and Ψ(x) = 1 (1+r 2 ) δ/2 with 0 ≤ δ ≤ γ < 2 fulfill the hypothesis of the last theorem. Remark also that 1

(1+r 2 ) δ/2 is equivalent to 1 (1+r) δ .

A priori controls and local existence

Let us take a real-valued radial function φ ∈ D(R 2 ) which is equal to 1 on a neighborhood of 0 and let φ (x) = φ( (x 1 , x 2 )). For ∈ (0, 1], we denote u 0, = P(φ u 0 ).

Thus, u 0, is a divergence free axisymmetric without swirl vector field and u 0, belongs to H 1 . We have

ω 0, = ∇ ∧ u 0, = ∇ ∧ (φ u 0 ) = φ ω 0 + (∇φ)( x) ∧ u 0 .
Then, using the fact that Φ ∈ A 2 and | ∇φ(

x)| ≤ C 1 r 1 r≥ 1 C ≤ C 1 r≥ 1 C √ Φ, we can conclude that lim ε→0 u 0 -u 0, L 2 (Φ dx) + ω 0 -ω 0, L 2 (Ψ dx) = 0.
Let u be the global solution of the following approximated problem

(NS )    ∂ t u = ∆u -(u • ∇)u -∇p ∇ • u = 0, u (0, .) = u 0,
given by the Proposition 4.1. We denote ω = ∇ ∧ u , therefore

∂ t u = ∆u + (u • ∇)u -∇p and ∂ t ω = ∆ω + (ω • ∇)u -(u • ∇)ω . (4.1) Since √ Ψω ∈ L 2 H 1 (because √ Ψ, ∇ √ Ψ ∈ L ∞ ) and √ Ψ∂ t ω ∈ L 2 H -1 , we can compute ∂ t ω • ω Ψ dx. More precisely, using (4.1) we find |ω (t, x)| 2 2 Ψ dx + t 0 |∇ ⊗ ω | 2 Ψdx ds = |ω 0, (x)| 2 2 Ψ dx - t 0 ∇( |ω | 2 2 ) • ∇Ψdx ds + t 0 |ω | 2 2 u • ∇Ψ -(ω • u )ω • ∇Ψ dx - t 0 ((ω • ∇)ω ) • u Ψ dx ds ≤ |ω 0, (x)| 2 2 Ψ dx + 1 8 t 0 |∇ ⊗ ω | 2 Ψdx ds + C t 0 √ Ψ ω 2 2 ds + C t 0 √ Ψ ω 2 √ Ψ ω 6 √ Φu 3 ds - t 0 ((ω • ∇)ω ) • u Ψ dx ds ≤ |ω 0, (x)| 2 2 Ψ dx + 1 4 t 0 |∇ ⊗ ω | 2 Ψdx ds + C t 0 √ Ψ ω 2 2 ds + C t 0 √ Ψ ω 2 2 ( √ Φu 3 + ( √ Φu 4/3 3 ) ds - t 0 ((ω • ∇)ω ) • u Ψ dx ds
As ω = ω ,θ e θ , we obtain

ω • ∇ω = - ω 2 ,θ r e r .
Then, in order to control u • (ω • ∇ω ), we split the domain of integration in two parts, one where r is small and the other one where r is large. The support of φ 1 is contained in {x / r < R} for some R > 0, and the support of 1φ 1 is contained in the set {x / r > R 0 } for some R 0 > 0. We observe that

inf r<R Φ(x) = inf √ x 2 1 +x 2 2 <R Φ(x 1 , x 2 , 0) > 0 and inf r<R Ψ(x) = inf √ x 2 1 +x 2 2 <R Ψ(x 1 , x 2 , 0) > 0.
On the other hand, we see that

inf r>R 0 r 2 Φ(x) = inf √ x 2 1 +x 2 2 >R 0 (x 2 1 + x 2 2 )Φ(x 1 , x 2 , 0) ≥ inf |x|>R 0 |x| 2 Φ(x) > 0.
Then, we write :

- t 0 ((ω • ∇)ω ) • u Ψ dx ds = t 0 φ 1 ( (ω • ∇)u ) • ω ) Ψ dx ds + t 0 (ω • u )(ω • ∇φ 1 )Ψ dx ds + t 0 φ 1 (ω • u )ω • ∇Ψdx ds - t 0 (1 -φ 1 )(u • (ω • ∇ω ))Ψdx ds ≤C t 0 |ω | 2 |∇ ⊗ u | Ψ 3/2 dx ds + C t 0 |ω | 2 |u | √ Φ Ψ dx ds. Since Ψ ∈ A 2 we have √ Ψ∇ ⊗ u 2 ≈ √ Ψω 2 ; moreover, we know that ∇ ⊗ ( √ Φu ) 2 ≤ C( √ Φu 2 + √ Ψω 2 ) and ∇ ⊗ ( √ Ψω ) 2 ≤ C( √ Ψω 2 + √ Ψ∇ ⊗ ω 2 ).
Therefore, we get

- t 0 ((ω • ∇)ω ) • u Ψ dx ds ≤C t 0 √ Ψ∇ ⊗ u L 2 √ Ψω L 3 √ Ψω L 6 ds + C t 0 √ Φu L 6 √ Ψω L 3 √ Ψω L 2 ds ≤C t 0 √ Ψω 3 2 L 2 ( √ Ψω L 2 + √ Ψ∇ ⊗ ω L 2 ) 3 2 ds + C t 0 √ Φu L 2 √ Ψω 3 2 L 2 ( √ Ψω L 2 + √ Ψ∇ ⊗ ω L 2 ) 1 2 ds ≤C t 0 ( √ Φu 2 + √ Φu 4/3 2 ) √ Ψω 2 2 + √ Ψω 3 2 + √ Ψω 6 2 ds + 1 8 t 0 √ Ψ∇ ⊗ ω 2 2 ds
and finally,

√ Ψω (t) 2 L 2 + t 0 √ Ψ∇ ⊗ ω 2 L 2 ds ≤ √ Ψω 0, 2 L 2 + C (1 + √ Φu 3 + ( √ Φu 4/3 3 ) √ Ψω 2 2 ds + C t 0 ( √ Φu 2 + √ Φu 4/3 2 ) √ Ψω 2 2 + √ Ψω 3 2 + √ Ψω 6 2 ds ≤ √ Ψω 0, 2 L 2 + C t 0 (1 + √ Φu 2 + √ Φu 4/3 2 ) √ Ψω 2 2 + √ Ψω 6 2 ds
We already know that the quantity √ Φu (t) L 2 remains bounded (independently of ) on every bounded interval. Therefore, we may again use Lemma 3.5

to control sup 0≤t≤T 0 ω (t, .) 2 L 2 (Ψdx) + T 0 0 ∇ω 2 L 2 (Ψdx) ds
for some T 0 , independently of . After, this control is transferred to the limit ω since ω = lim ω k = lim ∇ ∧ u k . We thus have proved local existence of a regular solution and Theorem 6 is proved.

Very regular initial value.

Next, we present an apparently more restrictive result than our main Theorem (Theorem 7), however we will see that it implies almost directly our main Theorem.

Proposition 4.2. Consider a weight Φ satisfying (H1) -(H4). Assume that Φ depends only on r = x 2 1 + x 2 2 and there exists a continuous weight Ψ (that depends only on r) such that Φ ≤ Ψ ≤ 1, Ψ ∈ A 2 and there exists C 1 > 0 such that

|∇Ψ| ≤ C 1 √ ΦΨ and |∆( √ Ψ)| ≤ C 1 √ Φ.
Consider a divergence free axisymmetric vector field u 0 without swirl, such that u 0 belongs to L 2 (Φdx), and ∇ ⊗ u 0 , ∆u 0 belong to L 2 (Ψdx). Then there exists a global solution u of the problem

(NS)    ∂ t u = ∆u -(u • ∇)u -∇p ∇ • u = 0, u(0, .) = u 0 such that
• u is axisymmetric without swirl, u belongs to L ∞ ((0, T), L 2 (Φ dx)), ∇ ⊗ u belongs to L ∞ ((0, T), L 2 (Ψ dx)) and ∆u belongs to L 2 ((0, T), L 2 (Ψ dx)), for all T > 0,

• the maps t ∈ [0, +∞) → u(t, .) and t ∈ [0, +∞) → ∇ ⊗ u(t, .) are weakly continuous from [0, +∞) to L 2 (Φ dx) and to L 2 (Ψ dx) respectively, and are strongly continuous at t = 0, Proof. We retake the notation used in the proof of the last theorem. Ladyzhenskaya's inequality for axisymmetric fields with no swirl gives

|ω (t)| 2 r 2 dx ≤ |ω 0, | 2 r 2 dx.
Since

∂ i ω 0, = φ ∂ i ω 0 + ∂ i φ( x)∂ i ω 0 + (∇φ)( x) ∧ ∂ i u 0 + 2 (∇∂ i φ)( x) ∧ u 0 ,
we obtain that lim

→0 ∇ ⊗ ω 0, -∇ ⊗ ω 0 L 2 (Ψ dx) = 0. As |ω 0, -ω 0 | 2 r 2 dx ≤ C( 0<r<1 |∇ ⊗ ω 0, -∇ ⊗ ω 0 | 2 Ψ dx + 1<r<+∞ |ω 0, -ω 0 | 2 Ψ dx),
we also find

lim →0 |ω 0, -ω 0 | 2 r 2 dx = 0.
We have observed that

|ω (t, x)| 2 2 Ψ dx + t 0 |∇ ⊗ ω | 2 Ψdx ds = |ω 0, (x)| 2 2 Ψ dx - t 0 ∇( |ω | 2 2 ) • ∇Ψdx ds + t 0 |ω | 2 2 u • ∇Ψ dx ds - t 0 (ω • u )ω • ∇Ψ dx ds - t 0 u (ω • ∇ω ) Ψ dx ds.
Therefore,

√ Ψω (t) 2 L 2 + 2 t 0 √ Ψ∇ω 2 L 2 ≤ √ Ψω 0, 2 L 2 + 2 t 0 √ Ψω L 2 √ Ψ∇ω L 2 + t 0 √ Φu L 3 √ Ψω 2 L 3 + t 0 1 r |u r, ||ω | 2 Ψ dx ds Now, we have t 0 1 -φ 1 (x) r |u r, ||ω | 2 Ψ dx ds ≤ t 0 √ Φu L 3 √ Ψω 2 L 3 and t 0 φ 1 (x) r |u ,r ||ω | 2 Ψ dx ds ≤ C t 0 ω r L 2 √ Ψu L ∞ √ Ψω L 2 , Moreover, ω r L 2 ≤ C ω 0, r L 2 ≤ C( √ Ψω 0, L 2 + √ Ψ∇ ⊗ ω 0, L 2 ) ≤ C ( √ Φu 0 L 2 + √ Ψω 0 L 2 + √ Ψ∇ ⊗ ω 0 L 2 ) and √ Ψu 2 L ∞ ≤ C ∇ ⊗ ( √ Ψu ) 2 ∆( √ Ψu ) 2 ≤ C ( √ Φu L 2 + √ Ψω L 2 + √ Ψ∇ ⊗ ω L 2 ) 2 . Let us denote A 0 = √ Φu 0 L 2 + √ Ψω 0 L 2 + √ Ψ∇ ⊗ ω 0 L 2 . Then, we find √ Ψω (t) 2 L 2 + t 0 √ Ψ∇ ⊗ ω 2 L 2 ≤ √ Ψω 0, 2 L 2 + C t 0 √ Φu 2 L 2 + C Φ,Ψ t 0 √ Ψω 2 L 2 (1 + A 0 + A 2 0 + √ Φu L 3 + √ Φu 2 L 3 ) ds
We can thus conclude that, for all T > 0 and for all t ∈ (0, T),

√ Ψω (t) 2 L 2 + t 0 √ Ψ∇ ⊗ ω 2 L 2 ≤( √ Ψω 0, 2 L 2 + C sup >0 T 0 √ Φu 2 L 2 )e C Φ,Ψ sup >0 t 0 (1+A 2 0 + √ Φu L 3 + √ Φu 2 L 3 ) ds
Therefore, we can obtain a solution on (0, T) using the Aubin-Lions Theorem. A diagonal argument allows to get a global solution.

Existence of global regular axisymmetric solutions

Proof of Theorem 7 Theorem 6 permits to find a local solution v on (0, T 0 ) with initial value u 0 , which is continuous from (0, T 0 ) to D . Then, we take T 1 ∈ (0, T 0 ) such that ∇ ⊗ (∇ ∧ v)(T 1 , .) ∈ L 2 (Φdx). Using Proposition 4.2, we get a solution w on (T 1 , +∞) and initial value v(T 1 ). Our global solution is then defined as u = v on (0, T 1 ) and u = w on (T 1 , +∞).

In fact, if φ is a test function of R 3 , then ∂ t (φv) belongs to L 1 H -1 and φv is continuous from [0, T 1 ] to H -1 . Similarly, for the solution w on t > T 1 , with v(T 1 ) = w(T 1 ),

we have that φw is continuous from [T 1 , +∞[ to H -1 .
Then, for a test function ψ(t, x) we find

t<T 1 v∂ t ψ dtdx = - t<T 1 ψ∂ t v dtdx + v(T 1 , x)ψ(T 1 , x) dx and t>T 1 w∂ t ψ dtdx = - t>T 1 ψ∂ t w dtdx -v(T 1 , x)ψ(T 1 , x) dx,
so that for u = 1 t<T 1 v + 1 t>T 1 w, we have

∂ t u = 1 t<T 1 ∂ t v + 1 t>T 1 ∂ t w.
Thus, Theorem 7 is proved.

In summary, we have generalized the Ladyzhenskaya's result (Proposition 4.1), on axisymmetric solutions with initial data in H 1 , and we preserve the local regularity properties. However, note that we do not address the uniqueness problem, despite having good local regularity properties for these new axisymmetric solutions. We leave it as a nice open problem.

The difficulty of this problem is that the weight is no longer derived in the estimation of the bilinear term and thus we cannot conclude directly without additional hypothesis.

Chapter 5

Discretely Selfsimilar Solutions

In this section, we propose a new simple proof of the results of Chae and Wolf (Chae and [START_REF] Wolf | Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in L 2 loc (R 3 )[END_REF] and Bradshaw and Tsai (Bradshaw and Tsai, 2019a) on the existence of discretely selfsimilar solutions of the Navier-Stokes problem (and of Jia and Šverák [START_REF] Jia | Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions[END_REF] for self-similar solutions)

We specify the concepts related to the notion of self-similarity.

Definition 5.1. We consider λ > 1.

We say that

u 0 ∈ L 2 loc (R 3 ) is a λ-discretely selfsimilar (λ-DSS) vector field if λu 0 (λx) = u 0 (x). A time dependent vector field u ∈ L 2 loc ([0, +∞) × R 3 ) is called λ-DSS if λu(λ 2 t, λx) = u(t, x). A forcing tensor F ∈ L 2 loc ([0, +∞) × R 3 ) is called λ-DSS if λ 2 F(λ 2 t, λx) = F(t, x).
We speak of selfsimilarity if u 0 , u or F are λ-DSS for all real λ > 1.

The next examples permit us to link the discretely selfsimilarity notion with the weighted spaces L 2 w γ = L 2 ((1 + |x|) -γ dx). We observe that a very important feature of the weights w γ (x) = (1 + |x|) -γ is the control of their gradients : if γ > 0 then

|∇w γ (x)| = γ w γ (x) 1 + |x| . In particular, |∇w γ (x)| ≤ C γ w γ (x), and if γ ≤ 2 then we have |∇w γ (x)| ≤ C γ w 3γ 2 (x). Examples : • Let γ > 1 and λ > 1. Then, if u 0 ∈ L 2 loc (R 3 ) is λ-DSS, then u 0 ∈ L 2
w γ and there exist two positive constants, A γ,λ and B γ,λ , such that

A γ,λ 1<|x|≤λ |u 0 (x)| 2 dx ≤ |u 0 (x)| 2 w γ (x) dx ≤ B γ,λ 1<|x|≤λ |u 0 (x)| 2 dx. • u 0 ∈ L 2 loc is self-similar if and only if it is of the form u 0 = w 0 ( x |x| ) |x| with w 0 ∈ L 2 (S 2 ). • Consider γ > 1. F belongs to L 2 ((0, +∞), L 2 w γ ) is self-similar if and only if it is of the form F(t, x) = 1 t F 0 ( x √ t ) with |F 0 (x)| 2 1 |x| dx < +∞.
Proof.

• If u 0 is λ-DSS, we have for k ∈ Z

λ k <|x|<λ k+1 |u 0 (x)| 2 w γ (x) dx ≤ λ k (1 + λ k ) γ 1<|x|<λ |u 0 (x)| 2 dx
and we know that ∑ k∈Z

λ k (1+λ k ) γ < +∞ for γ > 1.
• If u 0 is self-similar, we find u 0 (x) = 1 |x| u 0 ( x |x| ). Thus, for all λ > 1 1<|x|<λ

|u 0 (x)| 2 dx = (λ -1) S 2 |u 0 (σ)| 2 dσ. • If F is self-similar, we find F(t, x) = 1 t F 0 ( x √ t ). Then, +∞ 0 |F(t, x)| 2 w γ (x) dx ds = +∞ 0 |F 0 (x)| 2 w γ ( √ t x) dx dt √ t = C γ |F 0 (x)| 2 dx |x| where C γ = +∞ 0 1 (1+ √ θ) γ dθ √ θ < +∞.
Our main result in the context of discretely selfsimilar initial data is the following one.

Theorem 8. Let γ ∈ (1, 2) and λ ∈ (1, +∞). If u 0 is a divergence free λ-DSS vector field which belongs to L 2 w γ (R 3 ) and if F is a λ-DSS forcing tensor F(t, x) = F i,j (t, x) 1≤i,j≤3 belonging to L 2 loc ((0, +∞), L 2 w γ ), then the Navier-Stokes equations with initial data u 0 , (NS)

   ∂ t u = ∆u -(u • ∇)u -∇p + ∇ • F ∇ • u = 0, u(0, .) = u 0
have a weak global solution u which satisfy the following properties :

• u is a λ-DSS vector field

• For all 0 < T < +∞, u belongs to L ∞ ((0, T), L 2 w γ ) and ∇u belongs to L 2 ((0, T), L 2 w γ )

• The function t ∈ [0, +∞) → u(t, .
) is weakly continuous from [0, +∞) to L 2 w γ , and strongly continuous in t = 0

• u is suitable : there exist a locally finite non-negative measure µ on (0, +∞) × R 3 such that

∂ t ( |u| 2 2 ) = ∆( |u| 2 2 ) -|∇u| 2 -∇ • ( |u| 2 2 + p)u + u • (∇ • F) -µ.

A priori estimates for the linearized problem

In this section, we investigate a priori controls for the linearized Navier-Stokes problem, also known as advection-diffusion problem.

Lemma 5.1. Let 0 ≤ γ ≤ 2 and 0 < T < +∞. Let u 0 ∈ L 2 w γ (R 3 ) be a divergencefree vector field and let F be a tensor

F(t, x) = F i,j (t, x) 1≤i,j≤3 ∈ L 2 ((0, T), L 2 w γ ). Let α ∈ [3, 10
3 ] and let v ∈ L α ((0, T), L α w αγ/2 ) be a time-dependent divergence free vector-field. Let (u, p) be a solution of the following advection-diffusion problem

(LNS)    ∂ t u = ∆u -(v • ∇)u -∇p + ∇ • F, ∇ • u = 0, u(0, •) = u 0 which satisfies : • u belongs to L ∞ ((0, T), L 2 w γ ) and ∇u belongs to L 2 ((0, T), L 2 w γ ) and p ∈ D ((0, T) × R 3 ) • the map t ∈ [0, +∞) → u(t) is weakly continuous from [0, +∞) to L 2
w γ , and is strongly continuous at t = 0

• the solution (u, p) is suitable : there exist a non-negative locally finite measure µ on (0, T) × R 3 such that

∂ t ( |u| 2 2 ) =∆( |u| 2 2 ) -|∇u| 2 -∇ • ( |u| 2 2 v) -∇ • (pu) + u • (∇ • F) -µ. (5.1)
Then we have the following controls:

• If 0 < γ ≤ 2,
for almost every τ ≥ 0, and for τ = 0, we have for all t ≥ τ,

u(t) 2 L 2 wγ + 2 t τ ∇u(s) 2 L 2 wγ ds ≤ u(τ) 2 L 2 wγ - t τ ∇(|u| 2 ) • ∇w γ dx ds + t τ (|u| 2 v + 2pu) • ∇w γ dx ds -2 ∑ 1≤i,j≤3 ( t τ F i,j (∂ i u j )w γ dx ds + t τ F i,j u j ∂ i (w γ ) dx ds),
(5.2)

and the map t → u(t) from [0, +∞) to L 2 w γ is right strongly continuous almost everywhere and

u(t) 2 L 2 wγ + t τ ∇u(s) 2 L 2 wγ ds ≤ u(τ) 2 L 2 wγ + C γ t τ F(s) 2 L 2 wγ ds + C α,γ t τ v(s) 2α 2α-3 L α w αγ/2 u(s) 2 L 2 wγ ds.
(5.3)

• If γ = 0, for almost all τ ≥ 0 and for τ = 0, we have for all t ≥ τ,

u(t) 2 L 2 + 2 t τ ∇u(s) 2 L 2 ds ≤ u(τ) 2 L 2 -2 ∑ 1≤i,j≤3 t τ F i,j ∂ i u j dx ds,
and the map t → u(t) from [0, +∞) to L 2 is right strongly continuous almost everywhere.

Proof. We consider the case 0 < γ ≤ 2 (the changes required for the case γ = 0 are obvious).

First, remark that we know a formula for the pressure. And by the continuity of the Riesz transforms we even can write p = ∑ i ∑ j R i R j (u i u j -F i,j ). In fact, we prove below that : for β ∈ 5 4 , 6α 10+α ⊂ ( 5 4 , 3 2 ) and a verifying 2 a + 3

β = 5 α + 3 2 (hence a ∈ (2, 20α 50-9α )), ∑ 1≤i,j≤3 R i R j (u i v j ) ∈ L a ((0, T), L β w βγ (R d )) (5.4) and ∑ 1≤i,j≤3 R i R j F i,j ∈ L 2 ((0, T), L 2 w γ ),
(5.5) so that the pressure given by Theorem 1 is necessarily p

= ∑ i ∑ j R i R j (u i u j -F i,j ).
As discussed in Section 3.1, we know that the Riesz transforms are continuous on L q (w δ dx) for 0 ≤ δ < 3 and q ∈ (1, +∞). Then, as γβ < 3 we have

R i R j (u i v j ) L a ((0,T),L β w βγ (R 3 )) ≤ C γ,β u i v j L a ((0,T),L β w βγ (R 3 )) ,
so, taking ã and b such that 1 a = 1 ã + 1 α and 1 β = 1 b + 1 α , by the Hölder inequalities, and moreover, as we also have 2 ã + 3 b = 3 2 (and thus b ∈ ( 5α 4α-5 , 6α 4+α ) ⊂ (2, 6)), we find

u i v j L a ((0,T),L β w βγ (R 3 )) ≤ C γ,β √ w γ u i L ã ((0,T),L b (R 3 )) • √ w γ v j L α ((0,T),L α (R 3 )) < +∞,
To justify that the right side is finite, remark that we have

√ w γ u ∈ L ∞ ((0, T), L 2 )
and √ w γ ∇u ∈ L 2 ((0, T), L 2 ), then by interpolation in the space variable, for ã and b, such that b ∈ [2, 6] and 2 ã + 3 b = 3 2 we find

w γ 2 u ∈ L ã L b, (5.6) in particular w γ 2 u ∈ L 4 L 3 ∩ L 10 3 L 10 3 ⊂ L α L α .
Indeed, we have

√ w γ u j L ã ((0,T),L b (R 3 )) ≤ Cγ,ã, b u ã-2 ã L ∞ ((0,T),L 2 wγ (R 3 )) T 0 ( u(s) L 2 wγ (R 3 ) + ∇u(s) L 2 wγ (R 3 ) ) 2 ds 1 ã < +∞.
For the remaining term, we directly observe that

R i R j F ij L 2 ((0,T),L 2 wγ (R 3 )) ≤ C F ij L 2 ((0,T),L 2 wγ (R 3 )) .
Remark that under the hypothesis in Theorem 5.1, all the terms in the local energy balance are well defined. In (5.10) below we will justify that pu ∈ L 1 loc .

Let 0 < t 0 < t 1 < T, we take a non-decreasing function α ∈ C ∞ (R) equal to 0 on (-∞, 1

2 ) and equal to 1 on (1, +∞). For 0

< η < min( t 0 2 , T -t 1 ), let α η,t 0 ,t 1 (t) = α( t -t 0 η ) -α( t -t 1 η
).

(5.7)

Observe that α η,t 0 ,t 1 converges almost everywhere to 1 (t 0 ,t 1 ) when η → 0 and ∂ t α η,t 0 ,t 1 is the difference between two identity approximations, the first one in t 0 and the second one in t 1 .

Consider a non-negative function φ ∈ D(R 3 ) which is equal to 1 for |x| ≤ 1 and to 0 for |x| ≥ 2. We define

φ R (x) = φ( x R
).

(5.8)

For > 0, we let w γ, = 1 (1+ √ 2 +|x| 2 ) γ (if γ = 0, we let w γ, = 1 ). We have α η,t 0 ,t 1 (t)φ R (x)w γ, (x) ∈ D((0, T) × R 3 ) and α η,t 0 ,t 1 (t)φ R (x)w γ, (x) ≥ 0. Thus, using the local energy balance (5.1), we find

- |u| 2 2 ∂ t α η,t 0 ,t 1 φ R w γ, dx ds + |∇u| 2 α η,t 0 ,t 1 φ R w γ, dx ds ≤ - 3 ∑ i=1 ∂ i u • u α η,t 0 ,t 1 (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + 3 ∑ i=1 [( |u| 2 2 )v i + pu i ]α η,t 0 ,t 1 (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ 1≤i,j≤3 ( F i,j u j α η,t 0 ,t 1 (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + F i,j ∂ i u j α η,t 0 ,t 1 φ R dx ds). We denote A R, (t) = |u(t, x)| 2 φ R (x)w γ, (x) dx. Since - |u| 2 2 ∂ t α η,t 0 ,t 1 φ R w γ, dx ds = - 1 2 ∂ t α η,t 0 ,t 1 A R, (s) ds,
we find for all t 0 and t 1 Lebesgue points of the measurable functions A R, ,

lim η→0 - |u| 2 2 ∂ t α η,t 0 ,t 1 φ R w γ, dx ds = 1 2 (A R, (t 1 ) -A R, (t 0 )).
Thus, letting η goes to 0 to find

-lim η→0 |u| 2 2 ∂ t α η,t 0 ,t 1 φ R w γ, dx ds + t 1 t 0 |∇u| 2 φ R w γ, dx ds ≤ - 3 ∑ i=1 t 1 t 0 (∂ i u • u) (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + 3 ∑ i=1 t 1 t 0 [( |u| 2 2 )v i + pu i ](w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ 1≤i,j≤3 ( t 1 t 0 F i,j u j (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + t 1 t 0 F i,j ∂ i u j φ R dx ds).
By continuity, we can let t 0 goes to 0 and thus replace t 0 by 0 in the inequality. Moreover, if we let t 1 goes to t, then by weak continuity, we find that A R, (t) ≤ lim inf t 1 →t A R, (t 1 ), so that we may as well replace t 1 by t ∈ (t 0 , T). Hence, for almost every τ ∈ (0, T), also for τ = 0, and for all t ∈ (τ, T) :

1 2 (A R, (t) -A R, (τ)) + t τ |∇u| 2 φ R w γ, dx ds ≤ - 3 ∑ i=1 t τ (∂ i u • u) (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + 3 ∑ i=1 t τ [( |u| 2 2 )v i + pu i ](w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ 1≤i,j≤3 ( t τ F i,j u j (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds - t τ F i,j ∂ i u j φ R dx ds),
(5.9)

Note that all terms in the right side are dominated. In fact, we will treat the most difficult term. Note that as γ ≤ 2, there exists C γ > 0, which does not depend on R > 1 nor on > 0, such that

|w γ, ∂ i φ R | + |φ R ∂ i w γ, | ≤ C γ w γ (x) 1 + |x| ≤ C γ w 3γ/2 (x), thus we have |[( |u| 2 2 )v i + pu i ](w γ, ∂ i φ R + φ R ∂ i w γ, )| ≤ C(|u| 2 |v| + |p| |u|)w 3γ 2 ,
where we must study the expression in the right side. For the term (|p| |u|)w 3γ 2 , by

(5.4) and by (5.5) (for β ∈ 5 4 , 6α 10+α and a such that 2 a + 3 β = 5 α + 3 2 ) we have

w γ p ∈ L a L β + L 2 L 2 .
Thereafter, by (5.6) taking b such that 1

β + 1 b = 1 and ã such that 2 ã + 3 b = 3 2 we get w γ 2 |u| ∈ L ã L b ∩ L 2 L 2 ⊂ L a a-1 L β β-1 ∩ L 2 L 2 . Thus, |p||u|w 3γ 2 ∈ L 1 .
(5.10)

Similarly, for the term |u| 2 |v|w 3γ 2 , as |v| |u|w γ ∈ L a L β , and

w γ 2 |u| ∈ L a a-1 L β β-1 , we conclude |u| 2 |v|w 3γ 2 ∈ L 1 .
Thus, by dominated convergence, letting R goes to +∞ and then goes to 0, we find the energy control (5.2). We let t goes to τ in (5.2), so that lim sup

t→τ u(t) 2 L 2 wγ ≤ u(τ) 2 L 2 wγ . Also, as u is weakly continuous in L 2 w γ , u(τ) 2 L 2 wγ ≤ lim inf t→τ u(t) 2 L 2 wγ . Thus u(τ) 2 L 2 wγ = lim t→τ u(t) 2 L 2 wγ .
It is known that in a Hilbert space X, if x n converges weakly to x, and x n n converges to x , then x n converges strongly to x, in effect we can take the limit when n goes to ∞ in the equality

x n -x 2 X = x n 2 X -2 x n , x X + x 2 X .
By the weak continuity of the map t → u(t) ∈ L 2 w γ and as lim t→τ + u(t) 2

L 2 wγ = u(τ) 2 L 2
wγ , for almost every τ ∈ [0, T), and for τ = 0, we conclude the right strongly continuity almost everywhere, and continuity at the initial time 0. Now, in order to obtain (5.3). Consider the energy control (5.2) and the following estimates:

t τ ∇|u| 2 • ∇w γ ds ds ≤2γ t τ |u||∇u| w γ dx ds ≤ 1 4 t τ ∇u 2 L 2 wγ ds + 4γ 2 t τ u 2 L 2 wγ ds,
For the pressure, we write p = p 1 + p 2 with

p 1 = 3 ∑ i=1 3 ∑ j=1 R i R j (v i u j -c i b j ), p 2 = - 3 ∑ i=1 3 ∑ j=1 R i R j (F i,j ). Since w 2γα 2+α ∈ A 2α 2+α
the continuity of the Riesz transforms gives the following control

t τ (|u| 2 v + 2p 1 u) • ∇w γ dx ds C γ t τ (|u| 2 |v| + 2|p 1 ||u|) w 3/2 γ dx ds ≤C α,γ t τ w 1/2 γ u 2α α-2 w γ |v||u| 2α 2+α ds ≤C α,γ t τ w γ/2 u 2α α-2 w γ/2 u 2 w γ/2 v α ds.
We remark that 2α α-2 ∈ [5, 6]. Thus, using the Gagliardo-Nirenberg inequality we obtain

w γ/2 u 2α α-2 ≤ C α,γ ∇(w γ/2 u) 3 α 2 w γ/2 u α-3 α 2 ≤ C α,γ ( w γ/2 u 2 + w γ/2 ∇u 2 ) 3 α w γ/2 u α-3 α
and by the Young inequality with 3 2α + 2α-3 2α = 1,

C α,γ t τ w γ/2 u 2α α-2 w γ/2 u 2 w γ/2 v α ds ≤ C α,γ t τ ( w γ/2 u 2 + w γ/2 ∇u 2 ) 3 α w γ/2 u α-3 α 2 w γ/2 u 2 w γ/2 v α ds ≤ 1 16 t τ ( w γ/2 ∇u 2 2 + w γ/2 u 2 2 ) ds + C α,γ t τ ( w γ/2 u α-3 α 2 w γ/2 u 2 w γ/2 v α ) 2α 2α-3 ds ≤ 1 16 t τ ( w γ/2 ∇u 2 2 + w γ/2 u 2 2 ) ds + C α,γ t τ v 2α 2α-3 L α w αγ/2 u 2 L 2 wγ ds.
We have found

| t τ (|u| 2 v + 2p 1 u) • ∇(w γ ) dx ds| ≤ 1 4 t τ ( ∇u(s) 2 L 2 wγ + u(s) 2 L 2 wγ ) ds + C α,γ t τ v(s) 2α 2α-3 L α w αγ/2 u(s) 2 L 2 wγ ds.
On the other hand, since w γ ∈ A 2 , we can write

t τ p 2 u • ∇w γ dx ds ≤C γ t τ |p 2 ||u|w γ dx ds ≤C γ t τ u 2 L 2 wγ + p 2 2 L 2 wγ ds ≤C γ t τ u 2 L 2 wγ + F 2 L 2 wγ ds.
Finally, for the other terms, we have

2 ∑ 1≤i,j≤3 t τ (F i,j (∂ i u j )w γ + F i,j u j ∂ i (w γ )) dx ds ≤ C γ t τ |F|(|∇u| + |u|)w γ dx ds ≤ 1 4 t τ ∇u 2 L 2 wγ ds + C γ t τ u 2 L 2 wγ ds + C γ t τ F 2 L 2 wγ ds.
Hence we have found the estimate (5.3) and Lemma 5.1 is proven. Using the Grönwall inequality, the following corollary is a direct consequence of Theorem 5.1: Corollary 5.1. Under the assumptions of Theorem 5.1, we have

sup 0<t<T u 2 L 2 wγ + ∇u 2 L 2 ((0,T),L 2 wγ ) ≤ u 0 2 L 2 wγ + C γ F L 2 ((0,T),L 2 wγ ) e C α,γ (T+T 2α-5 2α-3 v 2 2α-3 L α ((0,T),L α w αγ/2 ) )
Another important consequence is the following uniqueness result for the advectiondiffusion problem.

Corollary 5.2. . Let 0 ≤ γ ≤ 2. Let 0 < T < +∞. Let u 0 ∈ L 2 w γ (R 3 ) be a divergence-free vector field and F(t, x) = F i,j (t, x) 1≤i,j≤3 be a tensor such that F(t, x) ∈ L 2 ((0, T), L 2 w γ ). Let α ∈ [3, 10 3 ] and let v ∈ L α ((0, T), L α w αγ/2 ) be a time-dependent divergence free vectorfields. Assume moreover that v ∈ L 2 t L ∞ x (K) for every compact subset K of (0, T) × R 3 .

Let (u 1 , p 1 ) and (u 2 , p 2 ) be two solutions of the advection-diffusion problem

   ∂ t u = ∆u -(v • ∇)u -∇p + ∇ • F, ∇ • u = 0, u(0, •) = u 0 ,
which satisfies for k ∈ {1, 2} :

• u k belongs to L ∞ ((0, T), L 2 w γ ) and ∇u k belongs to L 2 ((0, T), L 2 w γ ) and p k is a distribution on (0, T) × R 3

• the map t ∈ [0, +∞) → u k (t) is weakly continuous from [0, +∞) to L 2
w γ , and is strongly continuous at t = 0 : Then (u 1 , p 1 ) = (u 2 , p 2 ).

Proof.

We know that p k satisfy

p k = ∑ 1≤i,j≤3 R i R j (u k,i v j -F i,j ). Let w = u 1 -u 2 , p = p 1 -p 2 . Then we have    ∂ t w = ∆w -(v • ∇)w -∇p, ∇ • w = 0, w(0, •) = 0, Consider β ∈ 5 4 , 6α
10+α and a verifying 2 a + 3 β = 5 α + 3 2 . For all compact subset K of (0, T) × R 3 , we have w ⊗ v to L 2 t L 2 x (K), and to L a ((0, T), L β w βγ ) by (5.4). We will verify that ∂ t w is locally L 2 H -1 . Let ϕ, ψ ∈ D((0, T) × R 3 ) such that ψ = 1 on the neighborhood of the support of ϕ. We write

ϕp = ϕR ⊗ R(ψ(v ⊗ w)) + ϕR ⊗ R((1 -ψ)(v ⊗ w)). We get ϕR ⊗ R(ψ(v ⊗ w)) L 2 L 2 ≤ C ϕ,ψ ψ(v ⊗ w) L 2 L 2 (Supp(ψ)) < +∞ and ϕR ⊗ R((1 -ψ)(v ⊗ w)) L a L ∞ ≤ C ϕ,ψ (v ⊗ w) L a L β w βγ < +∞ with C ϕ,ψ ≤ C ϕ ∞ 1 -ψ ∞ sup x∈Supp ϕ   y∈Supp (1-ψ) (1 + |y|) γ |x -y| 3 β β-1   1-1 β < +∞,
where we have used the fact that (3γ) β β-1 > 3. Thus, we may take the scalar product of ∂ t w with w in order to find that

∂ t ( |w| 2 2 ) =∆( |w| 2 2 ) -|∇w| 2 -∇ • ( |w| 2 2 )v -∇ • (pw).
The assumptions of Theorem 5.1 are satisfied then we use Corollary 5.1 to find that w = 0 and consequently p = 0.

We will use the following control.

Corollary 5.3. Assume the hypothesis of Theorem 5.1. If v is controlled by u in the following sense: for every t ∈ (0, T),

v(t) 2 L α w αγ/2 ≤ C 0,α,γ u(t) 2( 3 2 -3 α ) L 2 wγ ( u(t) 2 L 2 wγ + ∇u(t) 2 L 2 wγ ) ( 3 α -1 2 ) , then there exists a constant C α,γ ≥ 1 such that if T 0 < T satisfies C α,γ 1 + u 0 2 L 2 wγ + T 0 0 F 2 L 2 wγ ds 2 T 0 ≤ 1 we have sup 0≤t≤T 0 u(t) 2 L 2 wγ + T 0 0 ∇u(s) 2 L 2 wγ ds ≤ C α,γ (1 + u 0 2 L 2 wγ + T 0 0 F 2 L 2 wγ ds).
Proof. Since (5.3),

u(t) 2 L 2 wγ + t 0 ∇u(s) 2 L 2 wγ ds ≤ u(0) 2 L 2 wγ + C γ t 0 F(s) 2 L 2 wγ ds + C α,γ t 0 v(s) 2α 2α-3 L α w αγ/2 u(s) 2 L 2 wγ ds.

As we have v(s)

2α 2α-3 L α w αγ/2 ≤ C 0,α,γ u 3(α-2) 2α-3 L 2 wγ ( u 2 L 2 wγ + ∇u 2 L 2 wγ ) 6-α 2(2α-3) ,
and using again the Young inequalities with 6-α 2(2α-3) + 5α-12 2(2α-3) = 1 we obtain v(s)

2α 2α-3 L α w αγ/2 u(s) 2 L 2 wγ ≤C α,γ u 7α-12 2α-3 L 2 wγ ( u 2 L 2 wγ + ∇u 2 L 2 wγ ) 6-α 2(2α-3) ≤ 1 16 ( u 2 L 2 wγ + ∇u 2 L 2 wγ ) + C α,γ u 2(7α-12) 5α-12 L 2 wγ .
Thus, we find

u(t) 2 L 2 wγ + 1 2 t 0 ∇u 2 L 2 wγ ds ≤ u 0 2 L 2 wγ + C γ T 0 0 F 2 L 2 wγ ds + C α,γ t 0 u 2 L 2 wγ + u 2(7α-12) 5α-12
L 2 wγ ds.

As 2(7α-12) 5α-12 ≤ 6, and moreover, using Lemma 3.5 with the function

α(t) = u(t) 2 L 2 wγ + 1 2 t 0 ∇u 2 L 2
wγ ds, we get for a constant C α,γ > 0, if

T 0 ≤ 1 C α,γ 1 + u 0 2 L 2 wγ + T 0 0 F 2 L 2 wγ ds then sup 0≤t≤T 0 u(t) 2 L 2 wγ + T 0 0 ∇u(s) 2 L 2 wγ ds ≤ C α,γ (1 + u 0 2 L 2 wγ + T 0 0 F 2 L 2 wγ ds).

Solutions to the advection-diffusion problem

In this section, we establish the existence of solutions for the linearized problem. First, we consider the following stability result:

Theorem 9. Let 0 ≤ γ ≤ 2. Let 0 < T < +∞. Let u 0,n ∈ L 2 w γ (R 3 ) be divergence-free vec- tor fields. Let F n ∈ L 2 ((0, T), L 2
w γ ) be tensors. Let α ∈ [3, 10 3 ] and let v n be time-dependent divergence free vector-fields wich belong to L α ((0, T), L α w αγ/2 ).

Let (u n , p n ) be solutions of the following advection-diffusion problems

(LNS n )    ∂ t u n = ∆u n -(v n • ∇)u n -∇p n + ∇ • F n , ∇ • u n = 0, u n (0, •) = u 0,n
verifying the same hypothesis of Theorem 5.1.

If u 0,n is strongly convergent to u 0,∞ in L 2 w γ , if the sequence F n is strongly convergent to F ∞ in L 2 ((0, T), L 2
w γ ), and moreover, if the sequence v n is bounded in L α ((0, T), L α w αγ/2 ), then there exists u ∞ , v ∞ , p ∞ and an increasing sequence (n k ) k∈N with values in N such that • u n k converges *-weakly to u ∞ in L ∞ ((0, T), L 2 w γ ), ∇u n k converges weakly to ∇u ∞ in L 2 ((0, T), L 2 w γ ).

• v n k converges weakly to v ∞ in L α ((0, T), L α w αγ/2 ), p n k converges weakly to p ∞ in L a ((0, T 0 ), L β w γβ ) + L 2 ((0, T 0 ), L 2
w γ ), for a parameter β ∈ 5 4 , 6α 10+α and for a parameter a verifying 2 a + 3 β = 5 α + 3 2 .

• u n k converges strongly to u ∞ in L 2 loc ([0, T) × R 3 ) : for every T 0 ∈ (0, T) and every R > 0, we have

lim k→+∞ T 0 0 |y|<R |u n k (s, y) -u ∞ (s, y)| 2 ds dy = 0. Moreover, (u ∞ , p ∞ ) is a solution of the problem (LNS ∞ )    ∂ t u ∞ = ∆u ∞ -(v ∞ • ∇)u ∞ -∇p ∞ + ∇ • F ∞ , ∇ • u ∞ = 0, u ∞ (0, •) = u 0,∞ ,
and verifies the hypothesis of Theorem 5.1.

Proof.

By Theorem 5.1 and Corollary 5.1, we know that u n is bounded in L ∞ ((0, T), L 2 w γ ) and ∇u n is bounded in L 2 ((0, T), L 2 w γ ). In particular, writing p n = p n,1 + p n,2 with

p n,1 = 3 ∑ i=1 3 ∑ j=1 R i R j (v n,i u n,j ), p 2 = - 3 ∑ i=1 3 ∑ j=1 R i R j (F n,i,j ),
we obtain, by (5.4) and (5.5), that p n,1 is bounded in L a ((0, T), L β w βγ ) and p n,2 is bounded in L 2 ((0, T), L 2 w γ ).

Let ϕ ∈ D(R 3 ). We have that the sequence ϕu n is bounded in L 2 ((0, T), H 1 ). Moreover, by the controls for p n , we get that ϕ∂ t u n is bounded in

L 2 L 2 + L 2 W -1,β + L 2 H -1 ⊂ L 2 ((0, T), H 1 2 -3 β ) ⊂ L 2 ((0, T), H -2
). Thus, by a Rellich-Lions lemma there exists u ∞ ∈ L 2 loc ([0, T) × R 3 ) and an increasing sequence (n k ) k∈N with values in N such that u n k converges strongly to u ∞ in L 2 loc ([0, T) × R 3 ) : for every T 0 ∈ (0, T) and every R > 0, we have

lim k→+∞ T 0 0 |y|<R |u n k (s, y) -u ∞ (s, y)| 2 dy ds = 0. As u n is bounded in L ∞ ((0, T), L 2 w γ ) and ∇u n is bounded in L 2 ((0, T), L 2 w γ ) we have that u n k converges *-weakly to u ∞ in L ∞ ((0, T), L 2
w γ ) and we have that ∇u n k converges weakly to ∇u ∞ in L 2 ((0, T), L 2 w γ ).

Using the Banach-Alaoglu's theorem, there exists v ∞ such that v n k converges weakly to v ∞ in L α ((0, T), L α w αγ/2 ). In particular, we have that the term v n k ,i u n k ,j is weakly convergent in (L 6/5 L 6/5 ) loc and thus in D ((0, T) × R 3 ). As by (5.4), this term is bounded in L a ((0, T), L β w βγ ), it is weakly convergent in L a ((0, T), L β w βγ ).

Let us write

p ∞ = p ∞,1 + p ∞,2 with p ∞,1 = 3 ∑ i=1 3 ∑ j=1 R i R j (v ∞,i u ∞,j ), p 2 = - 3 ∑ i=1 3 ∑ j=1 R i R j (F ∞,i,j ).
As the Riesz transforms are bounded on the spaces L β w βγ and L 2 w γ , we find that p n k ,1 is weakly convergent in L a ((0, T), L β w βγ ) to p ∞,1 , and moreover, we find that

p n k ,2 is strongly convergent in L 2 ((0, T), L 2 w γ ) to p ∞,2 .
With those facts, we obtain that (u ∞ , p ∞ ) verifies in D ((0, T) × R 3 ):

∂ t u ∞ = ∆u ∞ -(v ∞ • ∇)u ∞ -∇p ∞ + ∇ • F ∞ , ∇ • u ∞ = 0,
In particular, ∂ t u ∞ belongs locally to the space L 2 t H -2

x , and then this function has a representative such that t → u ∞ (t, •) is continuous from [0, T) to D (R 3 ) and coincides with u ∞ (0, •) + t 0 ∂ t u ∞ ds. We have necessarily u ∞ (0, •) = u 0,∞ and thus u ∞ is a solution of (LNS ∞ ).

Following, we define

A n k = -∂ t ( |u n k | 2 2 ) + ∆( |u n k | 2 2 ) -∇ • ( |u n k | 2 2 )v n k -∇ • (p n k u n k ) + u n k • (∇ • F n k ),
and in order to take the limit when n k goes to ∞, we remark that, as u k is locally strongly convergent in L 2 L 2 ; and locally bounded in L ∞ L 2 , it is locally strongly convergent in L p L 2 , for p < ∞. Then, since w γ 2 ∇ ⊗ u n k is bounded in L 2 ((0, T), L 2 ), by the Gagliardo-Nirenberg interpolation inequalities we obtain u n k is locally strongly convergent in L p L q with 2 p + 3 q > 3 2 . We know that p n k is locally weakly convergent in L a L β and u n k is locally strongly

convergent in L a a-1 L β β-1 since 2(1 -1 a ) + 3(1 -1 β ) = 13 2 -5 α > 3 2 , then p n k u n k con- verges in the sense of distributions.
With these remarks we conclude that A n k converges to A ∞ in D ((0, T) × R 3 ) where

A ∞ = -∂ t ( |u ∞ | 2 2 ) + ∆( |u ∞ | 2 2 ) -∇ • ( |u ∞ | 2 2 )v ∞ -∇ • (p ∞ u ∞ ) + u ∞ • (∇ • F ∞ ).
Moreover, by hypothesis there exists µ n k a non-negative locally finite measure on (0, T) × R 3 such that

∂ t ( |u n k | 2 2 ) =∆( |u n k | 2 2 ) -|∇u n k | 2 -∇ • ( |u n k | 2 2 )v n k -∇ • (p n k u n k ) + u n k • (∇ • F n k ) -µ n k .
Since the definition of A n k we can write A n k = |∇u n k | 2 + µ n k , and thus

A ∞ = lim n k →+∞ |∇u n k | 2 + µ n k .
By weak convergence, we have for a non-negative function Φ ∈ D((0, T) × R 3 )

A ∞ Φ dx ds = lim n k →+∞ A n k Φ dx ds ≥ lim sup n k →+∞ |∇u n k | 2 Φ dx ds ≥ |∇u ∞ | 2 Φ dx ds.
Thus, there exists a non-negative locally finite measure µ ∞ on (0,

T) × R 3 such that A ∞ = |∇u ∞ | 2 + µ ∞
, and then we obtain

∂ t ( |u ∞ | 2 2 ) =∆( |u ∞ | 2 2 ) -|∇u ∞ | 2 -∇ • ( |u ∞ | 2 2 )v ∞ -∇ • (p ∞ u ∞ ) + u ∞ • (∇ • F ∞ ) -µ ∞ .
As in (5.9) with the functions (u n k , p n k ) and with a = 0, and moreover, taking the limsup when n k → +∞ we have lim sup

n k →+∞ |u n k (t, x)| 2 2 φ R w γ,ε dx + t 0 |∇u n k | 2 φ R w γ, dx ds ≤ |u 0,∞ (x)| 2 2 φ R w γ,ε dx - 3 ∑ i=1 t 0 ∂ i u ∞ • u ∞ (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + 3 ∑ i=1 t 0 [( |u ∞ | 2 2 )v ∞,i + p ∞ u ∞,i ](w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ 1≤i,j≤3 ( t 0 F ∞,i,j u ∞,j (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds - t 0 F ∞,i,j ∂ i u ∞,j φ R dx ds). Now, recall that u n k = u 0,n k + t 0
∂ t u n k ds and then we see that for all t ∈ (0, T),

u n k (t, •) converges to u ∞ (t, •) in D (R 3 ). Moreover, as u n k (t, •) is bounded in L 2 w γ (R 3 ) we get that u n k (t, •) converges to u ∞ (t, •) in L 2 loc (R 3 ). Then, we have obtained that |u ∞ (t, x)| 2 2 φ R w γ, dx ≤ lim sup n k →+∞ |u n k (t, x)| 2 2 φ R w γ,ε dx.
On the other hand, as ∇u n k is weakly convergent to ∇u ∞ in L 2 t L 2 w γ , we have

t 0 |∇u ∞ (s, x)| 2 2 φ R w γ,ε dx ds ≤ lim sup n k →+∞ t 0 |∇u n k | 2 φ R w γ, dx ds.
Thus, taking the limit when R → 0 and ε → 0, for every t ∈ (0, T) we get :

u ∞ (t) 2 L 2 wγ + 2 t 0 ∇u ∞ (s) 2 L 2 wγ ds ≤ u 0,∞ 2 L 2 wγ - t 0 (∇|u ∞ | 2 ) • ∇w γ dx ds + t 0 [( |u ∞ | 2 2 )v] • ∇w γ dx ds + 2 t 0 p ∞ u ∞ • ∇w γ dx ds -∑ 1≤i,j≤3 ( t 0 F ∞,i,j (∂ i u ∞,j )w γ dx ds + t 0 F ∞,i,j u ∞,i ∂ j (w γ ) • ∇w γ dx ds).
Now, in this estimate we take the limsup when t goes to 0 in order to find

lim t→0 u ∞ (t) 2 L 2 wγ = u 0,∞ 2 L 2 wγ .
which implies strongly convergence of the solution to the initial data (since we have weak convergence and convergence of the norms in a Hilbert space). The proof is thus finished.

Theorem 10. Let α ∈ [3, 10 3 ] and γ ∈ [0, 6 α ). Let u 0 be divergence-free vector fields such that u 0 ∈ L 2 w γ (R 3 ). Let F be a tensor such that F ∈ L 2 ((0, +∞), L 2 w γ ). Let v be a time dependent divergence free vector-field which fulfills for every T > 0, v ∈ L α ((0, T), L α w αγ/2 ).

Then, the problem

(LNS)    ∂ t u = ∆u -(v • ∇)u -∇p + ∇ • F, ∇ • u = 0, u(0, •) = u 0 ,
has a solution (u, p) which satisfies the assumptions of Lemma 5.1.

Proof. For the initial data u 0,R = P(φ R u 0 ) ∈ L 2 (R 3 ), for the forcing tensors F R = φ R F ∈ L 2 ((0, T), L 2 ), and for v R = P(φ R v), we consider the solution (u R, , p R, ) of the approximated system

   ∂ t u R, = ∆u R, -((v R * θ ) • ∇)u R, -∇p R, + ∇ • F R , ∇ • u R, = 0, u R, (0, •) = u 0,R , where u R, ∈ C([0, T), L 2 (R 3 )) ∩ L 2 ([0, T), Ḣ1 (R 3
)) for every 0 < T < +∞, and (u R, , p R, ) verify all the assumptions of Theorem 5.1 (with energy equality). Since

αγ 2 < 3, we have v R * θ L α ((0,T),L α w αγ/2 ) < C v R L α w αγ/2 < C v L α w αγ/2 .
(5.11)

Let R n be a sequence converging to +∞ and n a sequence converging to 0 and let us denote

u 0,n = u 0,R n , F n = F R n , v n = v R n * n and u n = u R n , n .
As u 0,n is strongly convergent to u 0 in L 2 w γ , F n is strongly convergent to F in L 2 ((0, T), L 2 w γ ), and moreover, as v n is bounded in L α ((0, T), L α w αγ/2 ) (since we have (5.11)), we can apply Theorem 9, so there exists (u, V, p) and there exists an increasing sequence (n k ) k∈N with values in N such that:

• u n k converges *-weakly to u in L ∞ ((0, T 0 ), L 2 w γ ), ∇u n k converges weakly to ∇u in L 2 ((0, T 0 ), L 2 w γ ).

• v n k converges weakly to V in L α ((0, T 0 ), L α w αγ/2 ), the sequence p n k converges weakly to p in L a ((0, T 0 ),

L β w γβ ) + L 2 ((0, T 0 ), L 2 w γ ), for a parameter β ∈ 5 4 , 6α 10+α
and a parameter a verifying 2 a + 3

β = 5 α + 3 2 . • u n k converges strongly to u in L 2 loc ([0, T 0 ) × R 3 ),
and moreover, (u, p) is a solution of the advection-diffusion problem

   ∂ t u = ∆u -(V • ∇)u -∇p + ∇ • F, ∇ • u = 0, u(0, •) = u 0 .
which verifies:

• the map t ∈ [0, T 0 ) → u(t) is weakly continuous from [0, T 0 ) to L 2 w γ , and is strongly continuous at t = 0.

• there exists a non-negative locally finite measure µ on (0, T) × R 3 such that we have the local energy balance

∂ t ( |u| 2 2 ) =∆( |u| 2 2 ) -|∇u| 2 -∇ • ( |u| 2 2 V) -∇ • (pu) + u • (∇ • F) -µ. If we verify that V = v the proof is finished. As v n = θ n * (v n -v) + θ n * v, then we verify that v n k is convergent to v in D ((0, T 0 ) × R 3 ). Thus we have V = v.

The mollified linearized problem.

We fix 1 < λ < +∞.

Let θ be an radially decreasing function in D(R 3 ) with θ dx = 1, in particular θ is non-negative. We define

θ ,t (x) = 1 ( √ t) 3 θ( x √ t
).

(5.12)

We look for a discretely selfsimilar solution of the mollified problem

(NS )    ∂ t u = ∆u -((u * θ ,t ) • ∇)u -∇p + ∇ • F, ∇ • u = 0, u(0, •) = u 0 ,
we refocus the problem in the search of a fixed point for the application v → L (v) where L (v) is a solution of the mollified linearized problem

(LNS )    ∂ t u = ∆u -((v * θ ,t ) • ∇)u -∇p + ∇ • F, ∇ • u = 0, u(0, •) = u 0 , Lemma 5.2. Let α ∈ [3, 10 3 ] and γ ∈ [1, 6 α )
. Let u 0 be a λ-DSS divergence-free vector field which belong to L 2 w γ (R 3 ). Let F be a λ-DSS tensor wich satisfies F ∈ L 2 loc ((0, +∞), L 2 w γ ). Moreover, let v be a λ-DSS time-dependent divergence free vector-field such that for every T > 0, v ∈ L α ((0, T), L α w αγ/2 ).

Then, the linearized mollified problem (LNS ) has a unique solution (u, p) which satisfies all the conclusions of Theorem 10. Moreover, the function u is a λ-DSS vector field.

Proof. As we have |v(t, .) * θ ,t | ≤ M v(t,.) , we find

v(t) * θ ,t L α ((0,T),L α w αγ/2 ) ≤ C α,γ v L α ((0,T),L α w αγ/2 ) .
Theorem 10 provides a solution (u, p) on the interval of time (0, T). Moreover, as v * θ ,t belongs the space to L 2 t L ∞ x (K) for every compact subset K of (0, T) × R 3 , we can use Corollary 5.2 to conclude that this solution (u, p) is unique.

We will prove that this solution is λ

-DSS. Let ũ(t, x) = 1 λ u( t λ 2 , x λ ) and p(t, x) = 1 λ 2 p( t λ 2 , x λ ). Remark that v * θ ,t is λ-DSS. In fact, λ(v * θ ,t )(λ 2 t, λx) = λ R 3 v(λ 2 t, λx -y) 1 ( √ λ 2 t) 3 θ( y √ λ 2 t )dy = R 3 λv(λ 2 t, λx -λy) 1 ( √ λ 2 t) 3 θ( λy √ λ 2 t )λ 3 dy = R 3 v(t, x -y) 1 ( √ t) 3 θ( y √ t ) = (v * θ ,t )(t, x).
Then, we get ( ũ, p), is a solution of (LNS ) on (0, T). Thus, we have the identities ( ũ, p) = (u, p) from which we conclude that (u, p) is λ-DSS.

The mollified Navier-Stokes equations.

For α ∈ [3, 10 3 ] and for v ∈ L α ((0, T), L α w αγ/2 ) the terms u of the solution provided by Lemma 5.2 belongs to L α ((0, T), L α w αγ/2 ) by interpolation. Then the map L ,α : v → u where L ,α v = u is well defined from

X T,γ,α = {v ∈ L α ((0, T), L α w αγ/2 ) / v is λ -DSS} to X T,γ,α .
At this point, we introduce the following technical lemmas:

Lemma 5.3. For γ > 10 α -2, X T,γ,α is a Banach space, and the norms v L α ((0,T),L α

w αγ/2
) and v L α ((0,T/λ 2 )×B(0, 1 λ )) are equivalents. Proof. Using a change of variables, we get

T 0 B(0,1) |v(t, x)| α dx dt = λ 5-α T λ 2 0 B(0, 1 λ ) |v(t, x)| α dx dt and, for k ∈ N, T 0 λ k-1 <|x|<λ k |v(t, x)| α dx dt = λ (5-α)(k+1) T λ 2(k+1) 0 1 λ 2 <|x|< 1 λ |v(t, x)| α dx dt.
then splitting the integral in spatial variable we find

v α L α ((0,T),L α w αγ/2 ) = T 0 B(0,1) |v(t, x)| α w αγ/2 dx dt + +∞ ∑ k=1 T 0 λ k-1 <|x|<λ k |v(t, x)| α w αγ/2 dx dt ≤ λ 5-α-αγ 2 v α L α ((0,T/λ 2 ),×B(0, 1 λ )) + C +∞ ∑ k=1 λ (5-α-αγ 2 )(k+1) v α L α ((0,T/λ 2 ),×B(0, 1 λ ))
Since γ > 10 α -2 we have 5α -αγ 2 < 0. Thus, ∑ +∞ k=1 λ (5-α-αγ 2 )(k+1) < +∞, and

v α L α ((0,T),L α w αγ/2 ) ≤ C v α L α ((0,T/λ 2 ),×B(0, 1 λ )) .
The other estimate is clear

v α L α ((0,T/λ 2 ),×B(0, 1 λ )) ≤ v α L α ((0,T),L α w αγ/2
) .

Remark 5.1. In the condition γ > 10 α -2 in Lemma 5.3, we observe that values of α close to 10 3 permit to consider values of γ close to 1.

Lemma 5.4. Let α ∈ [3, 10 3 ) and γ ∈ ( 10 α -2, 6 α ), the mapping L ,α is continuous and compact on X T,γ,α .

Proof. Let v n be a bounded sequence in X T,γ,α and let u n = L ,α v n . Remark that, as αγ 2 < 3 the sequence v n (t) * θ ,t is bounded in X T,γ,α . Using Theorem 5.1 and Corollary 5.1 we have that the sequence u n is bounded in L ∞ ((0, T), L 2 w γ ) and moreover ∇u n is bounded in L 2 ((0, T), L 2 w γ ).

Thus, by Theorem 9 there exists u ∞ , p ∞ , V ∞ , and an increasing sequence (n k ) k∈N with values in N such that we have:

• u n k converges *-weakly to u ∞ in L ∞ ((0, T), L 2 w γ ), ∇u n k converges weakly to ∇u ∞ in L 2 ((0, T), L 2 w γ ). • v n k * θ ,t converges weakly to V ∞ in L α ((0, T), L α w αγ/2 ).
• p n k converges weakly to p in L a ((0, T 0 ),

L β w γβ ) + L 2 ((0, T 0 ), L 2 w γ ), for a parameter β ∈ 5 4 , 6α
10+α and a parameter a verifying 2 a + 3 β = 5 α + 3 2 .

• u n k converges strongly to u ∞ in L 2 loc ([0, T) × R 3 ) : for every T 0 ∈ (0, T) and every R > 0, we have

lim k→+∞ T 0 0 |y|<R |u n k (s, y) -u ∞ (s, y)| 2 ds dy = 0, • and    ∂ t u ∞ = ∆u ∞ -(V ∞ • ∇)u ∞ -∇p ∞ + ∇ • F, ∇ • u ∞ = 0, u 0,∞ = u 0 .
We will prove the compactness of L ,α . As before √ w γ u n is bounded in L 10/3 ((0, T) × R 3 ) by interpolation hence strong convergence of u n k in L 2 loc ([0, T) × R 3 ) implies the strong convergence of u n k in L α loc ((0, T) × R 3 ). Moreover, we have that u ∞ is still λ-DSS (a stable property under weak limits). With these information we obtain that u ∞ ∈ X T,γ,α and lim

n k →+∞ T λ 2 0 B(0, 1 λ )
|u n k (s, y)u ∞ (s, y)| α ds dy = 0, which proves that L ,α is compact. Now, we prove the continuity of L ,α . Let v n be such that v n is convergent to v ∞ in X T,γ,α . Then if we take a convergent subsequence of u n = L ,α v n with limit u ∞ , we necessarily have u ∞ = L ,α (v ∞ ), thus the relatively compact sequence u n can have only one limit point which is

L ,α (v ∞ ), hence u n is converges to L ,α (v ∞ ). Then L ,α is continuous. Lemma 5.5. Let α ∈ [3, 10 3 ). Let γ ∈ ( 10 α -2, 6 α ). If µ ∈ [0, 1] and u solves u = µL ,α (u) then u X T,γ,α ≤ C u 0 ,F,γ,α,T,λ
where the constant C u 0 ,F,γ,α,T,λ depends only on u 0 , F, γ, α, T and λ (but not on µ nor on ).

Proof. We let ũ = 1 µ u, so that

   ∂ t ũ = ∆ ũ -((u * θ ,t ) • ∇) ũ -1 µ ∇p + ∇ • F, ∇ • ũ = 0, ũ(0, •) = u 0 , where ∇p = ∇(∑ i ∑ j R i R j [(u i * θ ,t )u j ] -µF i,j ).
Multiplying by µ, we obtain

   ∂ t u = ∆u -((u * θ ,t ) • ∇)u -∇p + ∇ • µF, ∇ • u = 0, u(0, •) = µu 0 .
We consider the constant C α,γ given in the Corollary 5.3, and we take T 0 ∈ (0, T) such that

C α,γ 1 + u 0 2 L 2 wγ + T 0 0 F 2 L 2 wγ ds 2 T 0 ≤ 1, which implies C α,γ 1 + µu 0 2 L 2 wγ + T 0 0 µF 2 L 2 wγ ds 2 T 0 ≤ 1, then we have the controls sup 0≤t≤T 0 u(t) 2 L 2 wγ + T 0 0 ∇u 2 L 2 wγ ds ≤ C α,γ (1 + µ 2 u 0 2 L 2 wγ + µ 2 T 0 0 F 2 L 2 wγ ds) ≤ C α,γ (1 + u 0 2 L 2 wγ + T 0 0 F 2 L 2 wγ ds)
In particular, by interpolation

T 0 0 u α L α w αγ/2
ds is bounded by a constant C u 0 ,F,γ,α and we can go back from T 0 to T, using the selfsimilarity property.

Lemma 5.6. Let α ∈ [3, 10 3 ). Let γ ∈ ( 10 α -2, 6 α ). There is at least one solution u of the problem u = L ,α (u ).

Proof. The uniform estimates for the fixed points of the application µL for 0 ≤ µ ≤ 1 given by Lemma 5.5, and Lemma 5.4 permit to apply Leray-Schauder principle and Schaefer theorem to find a solution of the problem u = L ,α (u ).

Existence of discretely self-similar solutions.

Proof of Theorem 8

As γ ∈ (1, 2) we know that u 0 ∈ L 2 w γ . We take α ∈ [3, 10 3 ) such that γ ∈ ( 10 α -2, 6 α ).

We consider u solutions of the problem u = L ,α (u ) given by Lemma 5.6. Lemma 5.5, shows that u * θ ,t is bounded in L α ((0, T), L α w αγ/2 ). Theorem 5.1 and Corollary 5.1 allow us to conclude that u is bounded in L ∞ ((0, T), L 2 w γ ) and ∇u is bounded in L 2 ((0, T), L 2 w γ ). Theorem 9 provides u, v, p and a decreasing sequence ( k ) k∈N converging to 0, such that

• u k converges *-weakly to u in L ∞ ((0, T), L 2 w γ ), ∇u k converges weakly to ∇u in L 2 ((0, T), L 2 w γ ) • u k * θ k ,t converges weakly to v in L α ((0, T), L α w αγ/2 ) • p k converges weakly to p in L a ((0, T 0 ), L β w γβ ) + L 2 ((0, T 0 ), L 2 w γ ), for a parameter β ∈ 5 4 , 6α
10+α and a parameter a verifying 2 a + 3 β = 5 α + 3 2 .

• u k converges strongly to u in

L 2 loc ([0, T) × R 3 ) • and    ∂ t u = ∆u -(v • ∇)u -∇p + ∇ • F, ∇ • u = 0, u 0 = u 0 . The proof is finished if v = u. As we have u k * θ k ,t = (u k -u) * θ k ,t + u * θ k ,t .
We remark that u * θ ,t converges strongly in L 2 loc ((0, T) × R 3 ) as goes to 0 and then u k * θ k ,t converges to u in D .

Chapter 6

The incompressible magneto-hydrodynamics equations

It is natural to look for extend the results obtained for the (NS) equations to the more general setting of the coupled magneto-hydrodynamics system:

(MHD)              ∂ t u = ∆u -(u • ∇)u + (b • ∇)b -∇p + ∇ • F, ∂ t b = ∆b -(u • ∇)b + (b • ∇)u -∇q, ∇ • u = 0, ∇ • b = 0, u(0, •) = u 0 , b(0, •) = b 0 .
where the fluid velocity u : [0, +∞) × R 3 → R 3 , the fluid magnetic field b : [0, +∞) × R 3 → R 3 , the fluid pressure p : [0, +∞) × R 3 → R, and q : [0, +∞) × R 3 → R are the unknowns. On the other hand, the data of the problem are given by the fluid velocity at t = 0: u 0 : R 3 → R 3 ; the magnetic field at t = 0, b 0 : R 3 → R 3 ; and the forcing tensor F.

Observe that Theorem 3 gives sufficient conditions to determine the pressure terms through the formulas

p = ∑ i,j R i R j (u i u j -b i b j -F i,j ) q = ∑ i,j R i R j (u i b j -b i u j ) = 0.
Note that the proofs in this chapter also work if one introduces different constants in front of ∆u and ∆b.

DSS solutions

It is not complicated to verify that the theory developed for discretely self-similar solutions in the Chapter 5 can be generalized to the case of (MHD) equations. We will only give a light about how it works.

First, we look for solutions of the linearized system

(MHDL)              ∂ t u = ∆u -(v • ∇)u + (c • ∇)b -∇p + ∇ • F, ∂ t b = ∆b -(v • ∇)b + (c • ∇)u -∇q, ∇ • u = 0, ∇ • b = 0, u(0, •) = u 0 , b(0, •) = b 0 . Proposition 6.1. Let α ∈ [3, 10 3 ] and γ ∈ [0, 6 α ). Let u 0 , b 0 be divergence-free vector fields such that u 0 , b 0 ∈ L 2 w γ (R 3 ). Let F = (F i,j
) 1≤i,j≤3 be a tensor belonging to L 2 ((0, +∞), L 2 w γ ). Let v, c be time dependent divergence free vector-fields such that, v, c ∈ L α ((0, T), L α w αγ/2 ), for every T > 0. Then, the advection-diffusion problem (MHDL) has a global solution (u, b, p, q) which satisfies

• u, b belong to L ∞ ((0, T), L 2
w γ ) and ∇u, ∇b belong to L 2 ((0, T), L 2 w γ )

• The distributions p and q are related with u, b, v, c and F by

p = ∑ i,j R i R j (v i u j -c i b j -F i,j ) and q = ∑ i,j R i R j (v i b j -c i u j ) • the map t ∈ [0, +∞) → (u(t), b(t)) is weakly continuous from [0, +∞) to L 2
w γ , and is strongly continuous at t = 0 :

• the solution (u, b, p, q) is suitable : there exists a non-negative locally finite measure µ on (0, +∞) × R 3 such that

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 )v -∇ • (pu) -∇ • (qb) + ∇ • ((u • b)c) + u • (∇ • F) -µ. (6.1)

Sketch of the proof

We consider the mollified problem

             ∂ t u R, = ∆u R, -((v R * θ ) • ∇)u R, + ((c R * θ ) • ∇)b R, -∇p R, + ∇ • F R , ∂ t b R, = ∆b R, -((v R * θ ) • ∇)b R, + ((c R * θ ) • ∇)u R, -∇q R, , ∇ • u R, = 0, ∇ • b R, = 0, u R, (0, •) = u 0,R , b R, (0, •) = b 0,R , where (u 0,R , b 0,R ) = (P(φ R u 0 ), P(φ R b 0 )) ∈ L 2 (R 3 ), F R = φ R F ∈ L 2 ((0, T), L 2 ), and (v R , c R ) = P(φ R v, φ R c). We find solutions (u R, , b R, ) ∈ C([0, T), L 2 (R 3 )) ∩ L 2 ([0, T), Ḣ1 (R 3
)) which satisfy the following energy equality, where we omit the subscripts,

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 )(v * θ ) -∇ • (pu) -∇ • (qb) + ∇ • ((u • b)(c * θ )) + u • (∇ • F)
Then, we pass to the limit when R goes to +∞ and goes to 0, using the a priori controls, described in the following Proposition, and compactness.

To simplify the notation, for a Banach space X ⊂ D of vector fields endowed with a norm • X , we denote

(u, v) 2 X = u 2 X + v 2 X , and ∇(u, v) 2 X = ∇u 2 X + ∇v 2 X . Proposition 6.2. Let 0 ≤ γ ≤ 2 and 0 < T < +∞. Let u 0 , b 0 ∈ L 2 w γ (R 3
) be a divergencefree vector fields and let F ∈ L 2 ((0, T), L 2 w γ ) be a tensor. Consider α ∈ [3, 10 3 ] and v, c ∈ L α ((0, T), L α w αγ/2 ) two time-dependent divergence free vector-fields. Let (u, b, p, q) be a solution of the following advection-diffusion problem (MHDL). which satisfies :

• u, b belong to L ∞ ((0, T), L 2 w γ ) and ∇u, ∇b belong to L 2 ((0, T), L 2 w γ )

• p, q are distributions on (0, T) × R 3

• the map t ∈ [0, +∞) → (u(t), b(t)) is weakly continuous from [0, +∞) to L 2
w γ , and is strongly continuous at t = 0

• the solution (u, b, p, q) is suitable : there exists a non-negative locally finite measure µ on (0, +∞) × R 3 such that

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 )v -∇ • (pu) -∇ • (qb) + ∇ • ((u • b)c) + u • (∇ • F) -µ.
Then we have the following controls:

• If 0 < γ ≤ 2, then for almost every τ ≥ 0, and for τ = 0, we have for all t ≥ τ,

(u, b)(t) 2 L 2 wγ + 2 t τ ( ∇(u, b)(s) 2 L 2 wγ )ds ≤ (u, b)(a) 2 L 2 wγ - t τ ∇(|u| 2 + |b| 2 ) • ∇w γ dx ds + t τ [( |u| 2 2 + |b| 2 2 )v] • ∇w γ dx ds + 2 t τ pu • ∇w γ dx ds + 2 t τ qb • ∇w γ dx ds + 2 t τ [(u • b)c] • ∇w γ dx ds -2 ∑ 1≤i,j≤3 ( t τ F i,j (∂ i u j )w γ dx ds + t τ F i,j u j ∂ i w γ dx ds), (6.2)
the map t → (u(t), b(t)), from [0, +∞) to L 2 w γ , is right strongly continuous almost everywhere, and moreover

(u, b)(t) 2 L 2 wγ + t τ ∇(u, b)(s) 2 L 2 wγ ds ≤ (u, b)(τ) 2 L 2 wγ + C γ t τ F(s) 2 L 2 wγ ds + C α,γ t τ (v, c)(s) 2α 2α-3 L α w αγ/2 (u, b)(s) 2 L 2 wγ ds.
(6.3)

• If γ = 0, then for almost all τ ≥ 0 and for τ = 0, we have for all t ≥ τ,

(u, b)(t) 2 L 2 + 2 t τ ( ∇(u, b)(s) 2 L 2 )ds ≤ (u, b)(τ) 2 L 2 + 2 ∑ 1≤i,j≤3 t τ F i,j ∂ i u j dx ds, the map t → (u(t), b(t)), from [0, +∞) to L 2 w γ
, is right strongly continuous almost everywhere.

Sketch of the proof

As all the terms of order three in the right side of the energy balance are written in a divergence form, we can introduce the weights w γ as for (NS) to obtain a priori controls.

We will just to illustrate how to pass from (6.2) to (6.3). The rest is very similar to the case of the Navier-Stokes equations.

First, remark that the gradient terms (∇p, ∇q) are necessarily related to (u, b, v, c) and F through the Riesz transforms

R i = ∂ i √ -∆ by the formulas ∇p = ∇ ∑ 1≤i,j≤3 R i R j (u i v j -b i c j -F i,j ) , and ∇q = ∇ ∑ 1≤i,j≤3 R i R j (v i b j -c i u j ) ,
where :

For

β ∈ 5 4 , 6α 10+α ⊂ ( 5 4 , 3 2 ) and a verifying 2 a + 3 β = 5 α + 3 2 (so that a ∈ (2, 20α 50-9α )) we have ∑ 1≤i,j≤3 R i R j (u i v j -b i c j ) ∈ L a ((0, T), L β w βγ (R d )), ∑ 1≤i,j≤3 R i R j (v i b j -c i u j ) ∈ L a ((0, T), L β w βγ (R d )). and ∑ 1≤i,j≤3 R i R j F i,j ∈ L 2 ((0, T), L 2 w γ ).
We write p = p 1 + p 2 where

p 1 = 3 ∑ i=1 3 ∑ j=1 R i R j (v i u j -c i b j ), p 2 = - 3 ∑ i=1 3 ∑ j=1 R i R j (F i,j ), Since w 2γα 2+α ∈ A 2α 2+α
we have the following control

t τ (|u| 2 v + |b| 2 v + 2((u • b)c) + 2p 1 u + 2qb) • ∇w γ dx ds ≤C γ t τ (|u| 2 |v| + |b| 2 |v| + 2|u||b||c| + 2|p 1 ||u| + 2|q 1 ||b|) w 3/2 γ dx ds ≤C α,γ t τ w 1/2 γ u 2α α-2 ( w γ |v||u| 2α 2+α + w γ |c| |b| 2α 2+α ) ds + C α,γ t τ w 1/2 γ b 2α α-2 ( w γ |b||v| 2α 2+α + w γ |c| |u| 2α 2+α ) ds ≤C α,γ t τ w γ/2 u 2α α-2 ( w γ/2 u 2 w γ/2 v α + w γ/2 b 2 w γ c α ) ds + C α,γ t τ w γ/2 b 2α α-2 ( w γ/2 b 2 w γ/2 v α + w γ/2 u 2 w γ/2 c α ) ds.
At this point we need to estimate each term above but, for the sake of simplicity, we will only treat one term in the right side since the other terms follow the same estimates. We remark that 2α α-2 ∈ [5, 6], thus, using the Gagliardo-Nirenberg inequality we find

w γ/2 u 2α α-2 ≤ C α,γ ∇(w γ/2 u) 3 α 2 w γ/2 u α-3 α 2 ≤ C α,γ ( w γ/2 u 2 + w γ/2 ∇u 2 ) 3 α w γ/2 u α-3 α 2 ,
and then, using the Young inequalities with 3 2α + 2α-3 2α = 1 we obtain

C α,γ t τ w γ/2 u 2α α-2 w γ/2 b 2 w γ/2 c α ds ≤ C α,γ t τ ( w γ/2 u 2 + w γ/2 ∇u 2 ) 3 α w γ/2 u α-3 α 2 w γ/2 b 2 w γ/2 c α ds ≤ 1 16 t τ ( w γ/2 ∇u 2 2 + w γ/2 u 2 2 ) ds + C α,γ t τ ( w γ/2 u α-3 α 2 w γ/2 b 2 w γ/2 c α ) 2α 2α-3 ds ≤ 1 16 t τ ( w γ/2 ∇u 2 2 + w γ/2 u 2 2 ) ds + C α,γ t τ c 2α 2α-3 L α w αγ/2 ( u 2 L 2 wγ + b 2 L 2 wγ ) ds.
Treating the other terms in the same way we are able to write

| t τ (|u| 2 v + |b| 2 v + 2((u • b)c) + 2p 1 u + 2q 1 b) • ∇(w γ ) dx ds| ≤ 1 4 t τ ( ∇(u, b)(s) 2 L 2 wγ + (u, b)(s) 2 L 2 wγ ) ds + C α,γ t τ (v, c)(s) 2α 2α-3 L α w αγ/2 (u, b)(s) 2 L 2 wγ ds.
On the other hand, since w γ ∈ A 2 , we can write

t τ p 2 u • ∇w γ dx ds ≤C γ t τ |p 2 ||u|w γ dx ds ≤C γ t τ u 2 L 2 wγ + p 2 2 L 2 wγ ds ≤C γ t τ (u, b) 2 L 2 wγ + F 2 L 2 wγ ds.
The other terms are easier to treat.

The rest of procedure to obtain discretely selfsimilar solutions works without significant changes. We apply the Leray-Schauder principle and Schaefer theorem to find a point fixe of the application L ,α which sends a pair of discretely selfsimilar vector fields (v, c) into the discretely selfsimilar solution (u, b) of the mollified linearized problem

(LMHD )              ∂ t u = ∆u -((v * θ ,t ) • ∇)u + ((c * θ ,t ) • ∇)b -∇p + ∇ • F, ∂ t b = ∆b -((v * θ ,t ) • ∇)b + ((c * θ ,t ) • ∇)u -∇q ∇ • u = 0, ∇ • b = 0, u(0, •) = u 0 , b(0, •) = b 0 ,
and we passe to the limit. As in the case of the Navier-Stokes equations, we find

p = ∑ i,j R i R j (u i u j -b i b j -F i,j ) and q = ∑ i,j R i R j (u i b j -b i u j ) = 0
Remark that the structure of the function θ ,t defined in (5.12) permits to transfer the selfsimilarity property to the solution. Then, we arrive as in the case of the Navier-Stokes equations to the following conclusion.

Theorem 11. We consider a real number λ > 1 and we let u 0 , b 0 be two λ-DSS vector fields, and be locally L 2 .

Let γ ∈ (1, 2). We consider a λ-DSS tensor F which belongs to L 2 loc ((0, +∞), L 2 w γ ). Then, the (MHD) equations has a global weak solution (u, b, p, q), with q = 0, such that :

• u, b are λ-DSS vector fields • the distributions p and q is related with u, b and F by

p = ∑ i,j R i R j (u i u j -b i b j -F i,j ) q = ∑ i,j R i R j (u i b j -b i u j ) = 0 • for every 0 < T < +∞, u, b belong to L ∞ ((0, T), L 2 w γ ) and ∇u, ∇b belong to L 2 ((0, T), L 2 w γ ) • the map t ∈ [0, +∞) → (u(t), b(t)) is weakly continuous from [0, +∞) to L 2
w γ , and is strongly continuous at t = 0 • (u, b, p) is suitable : it verifies the local energy inequality (6.1).

In the next section we present a new result which we have not still addressed in the context of the Navier-Stokes equations.

Weak-strong uniqueness in weighted spaces

The study of uniqueness of Leray weak solutions for the Navier-stokes equations remains an outstanding open problem, so research community has attired the attention to look for supplementary assumptions in order to ensure the uniqueness of solutions. This kind of statements are known as weak-strong uniqueness results.

We complement the study of the (MHD) equations in the framework of weighted L 2 spaces with a weak-strong uniqueness theorem. This result, is obtained in the setting of the multiplier space X T which we introduce as follows:

For a time 0 < T < +∞ fix, let us denote E T the energy space of the timedependent vector fields v belonging to L ∞ ((0, T), L 2 ) and such that ∇v belongs to L 2 ((0, T), L 2 ). E T is doted by the norm

v 2 E T = sup 0≤t≤T v(t, .) 2 L 2 + T 0 ∇v(s, .) 2 L 2 ds.
Then, we define X T the space of pointwise multipliers on (0, T) × R 3 from E T to L 2 ((0, T), L 2 ), which is a Banach space with the norm:

u X T = sup v E T ≤1 uv L 2 ((0,T),L 2 ) .
Moreover, we define X (0)

T the space of multipliers u ∈ X T such that for every t 0 ∈ [0, T) we have lim

t 1 →t + 0 1 (t 0 ,t 1 ) (t)u(t, •) X T = 0.
The multiplier space X (0)

T gives us a natural and general framework to prove a weak-strong uniqueness criterion. More precisely, based on the classical Prodi-Serrin's type condition [START_REF] Prodi | Un teorema di unicitá per le equazioni de Navier-Stokes[END_REF][START_REF] Serrin | The initial value problem for the Navier-Stokes equations[END_REF] for the (NS) equations, we obtain the following result.

Theorem 12 (Weak-strong uniqueness). Let 0 ≤ γ ≤ 2. Let 0 < T < +∞. Let u 0 , b 0 ∈ L 2
w γ (R 3 ) be divergence-free vector fields, and moreover, consider a forcing tensor F ∈ L 2 ((0, T), L 2 w γ ).

Let (u, b, p) and ( ũ, b, p) two solutions of the (MHD) system, with initial data u 0 , b 0 , forcing tensor F, and such that :

• u, b, ũ, b belong to the space L ∞ ((0, T), L 2 w γ ) and ∇u, ∇ ũ, ∇b, ∇ b ∈ L 2 ((0, T), L 2 w γ ) • the maps t ∈ [0, T) → (u, b)(t, .) and t ∈ [0, T) → ( ũ, b)(t, .
) are weakly continuous from [0, T) to L 2 w γ (R 3 ), and are strongly continuous at t = 0 :

lim t→0 (u(t, .) -u 0 , b(t, .) -b 0 ) L 2 wγ = 0, and lim t→0 ( ũ(t, .) -u 0 , b(t, .) -b 0 ) L 2 wγ = 0.
• there exist non-negative locally finite measures µ and ν on (0, T) × R 3 such that

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • [ |u| 2 2 + |b| 2 2 + p]u + ∇ • [(u • b)b] + u • (∇ • F) -µ, (6.4) and ∂ t ( | ũ| 2 + | b| 2 2 ) =∆( | ũ| 2 + | b| 2 2 ) -|∇ ũ| 2 -|∇ b| 2 -∇ • [ | ũ| 2 2 + | b| 2 2 + p] ũ + ∇ • [( ũ • b) b] + ũ • (∇ • F) -ν. (6.5) If u, b ∈ X (0)
T and if the product u • ∇ p is well defined as a distribution, in the sense that ∇ p belongs to L 1 loc and the pointwise product u • ∇ p ∈ L 1 loc , then we have (u, b, p) = ( ũ, b, p). P.G. Lemarié-Rieusset used the space X

T to prove a weak-strong uniqueness criterion of weak Leray solutions (see the Theorem 12.4, page 359 of the book (Lemarié-Rieusset, 2016)). He proves some examples of embeddings E ⊂ X (0) T .

We may observe that we also need the assumption that u • ∇ p is well-defined in the distributional sense, which essentially is a technical requirement. However, in some particular cases this assumption is no longer required. For example, this is the case of the space L p ((0, T), L q ) ⊂ X (0) T (with 2/p + 3/q = 1 and 2 < p < +∞) where the products above are well defined. Indeed, we remark that ∇ p ∈ L p((0, T), L q loc ) where 1 p = 1 -1 p and 1 q = 1 -1 q . To verify that, let 1 p = 1 p -1 2 and 1 q = 1 q -1 2 , by interpolation we find that √ w γ ũ, √ w γ b ∈ L p ((0, T), L q ). Then, for a test function ϕ ∈ D (R 3 ), using the continuity of the Riesz transforms, and moreover, assuming F = 0 (only for the sake of simplicity) we have

ϕ∇ p L p L q ≤ C ϕ w γ ∇ p L p L q ≤ C ϕ ∑ i,j,k w γ ∂ k ( ũi ũj ) + ∂ k ( bi bj ) L p L q ≤ C( √ w γ ũ L p L q √ w γ ∇ ũ L 2 L 2 + √ w γ b L p L q √ w γ ∇ b L 2 L 2 ). Thus, if u ∈ L p ((0, T), L q ) then we have u • ∇ p ∈ L 1 loc ([0, T] × R 3 ).

Proof of Theorem 12

By the characterization of the pressure term given in Chapter 2 (with the dimension d = 3), we know that p can be taken as follows, let us choose ϕ ∈ D(R 3 ) such that ϕ(x) = 1 on a neighborhood of the origin and let us denote h i,j = u i u jb i b j -F i,j , and

Φ i,j,ϕ = (1 -ϕ)∂ i ∂ j G 3 , with G 3 (x) = 1
|x| , then we can take

p = ∑ i,j (ϕ∂ i ∂ j G 3 ) * h i,j + ∑ i,j (Φ i,j,ϕ (x -y) -Φ i,jϕ (-y))h i,j (y)dy, As √ w γ u, √ w γ b ∈ L ∞ ((0, T), L 2 ) and √ w γ ∇u, √ w γ ∇b ∈ L 2 ((0, T), L 2 ), we
obtain by interpolation that w γ u and w γ b belong to

L â L b with 2 â + 3 b = 3 2 and â ∈ [2, +∞]. Taking r ∈ 1, min{ 3 2 , 3 γ } and â satisfying 2 â + 3 r = 3, we get that R i R j (u i u j ), R i R j (b i b j ) ∈ L â((0, T), L r w rγ (R d )),
and by the continuity of the Riesz transforms on L 2 w γ (R 3 ) we have

R i R j F i,j ∈ L 2 ((0, T), L 2 w γ (R d )).
Indeed, the following estimate holds: taking b given by 2 â + 3 b = 3 2 , we can write

R i R j (u i u j ) L â ((0,T),L r wrγ (R 3 )) ≤ C γ,r u i u j L â ((0,T),L r wrγ (R 3 )) ≤ C γ,r √ w γ u i L ∞ ((0,T),L 2 (R 3 )) • √ w γ u j L â ((0,T),L b (R 3 )) ≤ γ 1 â Cγ,r u 1+ â-2 â L ∞ ((0,T),L 2 wγ (R 3 )) × T 0 ( u(s) L 2 wγ (R 3 ) + ∇u(s) L 2 wγ (R 3 ) ) 2 ds 1 â .
Thus, as the Riesz transforms are well-defined for all the terms composing the pressure terms, we have necessarily the identity

p = ∑ 1≤i,j≤3 R i R j (u i u j -b i b j -F i,j ).
The corresponding identity hold true for the pressure term p. Now, let v = uũ, w = b -b, and a = pp. So, we will prove the identities v = 0, w = 0 and a = 0.

Using the identity

|v| 2 + |w| 2 2 = |u| 2 + |b| 2 2 + | ũ| 2 + | b| 2 2 -u • ũ -b • b, (6.6)
we write

∂ t ( |v| 2 + |w| 2 2 ) =∂ t ( |u| 2 + |b| 2 2 ) + ∂ t ( | ũ| 2 + | b| 2 2 ) -u • ∂ t ũ -ũ • ∂ t u -b • ∂ t b -b • ∂ t b.
(6.7)

Recall that by assumption, the terms u • ∂ t ũ and b • ∂ t b are well-defined as distribu- tions so it remains to verify that the terms ũ • ∂ t u and b • ∂ t b are also well-defined in the distributional sense. For this we have the following simple lemma.

Lemma 6.1. Within the framework of Theorem 12, as we have u, b ∈ X T then we get

ũ • ∂ t u ∈ L 1 loc ([0, T] × R 3 ) and b • ∂ t b ∈ L 1 loc ([0, T] × R 3 ).
Proof. We shall verify that we have ũ

• ∂ t u ∈ L 1 loc ([0, T] × R 3 ).
The treatment for the other term b • ∂ t b follows the same lines.

As we have ∂

t u = ∆u -(u • ∇)u + (b • ∇)b -∇p + ∇ • F, then we formally write ũ • ∂ t u = ũ • ∆u -ũ • ((u • ∇)u) + ũ • ((b • ∇)b) -ũ • ∇p + ũ • (∇ • F),
and we must prove that each term in the right side belong to L 1 loc ([0, T] × R 3 ). We detail the computations for the terms ũ • ((u • ∇)u) and ũ • ∇p. Let 0 < t ≤ T and let ϕ ∈ D(R 3 ) be an arbitrary test function. We set ψ ∈ D(R 3 ) such that 0 ≤ ψ ≤ 1 and ψ = 1 on supp(ϕ).

For the term ũ • ((u • ∇)u), as div(u) = 0 we have (u • ∇)u = ∇ • (u ⊗ u) and then ϕ ũ • ((u • ∇)u) dx ds = ∑ i,j ϕ ũi ∂ j (u j u i ) dx ds = ∑ i,j ϕ ũi ∂ j (ψ u j u i ) dx ds, therefore we can write t 0 ϕ ũ • ((u • ∇)u) dx ds ≤ C ∑ i,j t 0 ϕ ũi Ḣ1 ∂ j (ψu j u i ) Ḣ-1 ds ≤C t 0 ∇(ϕ ũ) L 2 ψ(u ⊗ u) L 2 ds ≤ C ∇(ϕ ũ) L 2 t L 2 x u ⊗ ( √ w γ u) L 2 t L 2 x ≤C γ,T ( √ w γ ũ L ∞ L 2 x + √ w γ ∇ ũ L 2 L 2 x ) u X T √ w γ u E T ds ≤C γ,T ( √ w γ ũ L ∞ t L 2 x + √ w γ ∇ ũ L 2 t L 2 x ) × u X T ( √ w γ u L ∞ t L 2 x + √ w γ ∇u L 2 t L 2
x ) < +∞.

(6.8)

Now, we study the term ũ • ∇p. Let us write

∇p =∇ ∑ 1≤i,j≤3 R i R j (u i u j -b i b j -F i,j ) =∇ ∑ 1≤i,j≤3 R i R j (u i u j -b i b j ) -∇ ∑ 1≤i,j≤3 R i R j (F i,j ) =∇p 1 + ∇p 2 .
The term ũ • ∇p 2 is easily estimated by the hypothesis on the tensor F and the computations above. Thereafter, for the term ∇p 1 , following the same estimates performed in (6.8), and using the fact that R i R j is a bounded operator in the space L 2 , we find

t 0 ϕ ũ • ∇p 1 dx ds ≤C γ,T ( √ w γ ũ L ∞ t L 2 x + √ w γ ∇ ũ L 2 t L 2 x ) × u X T ( √ w γ u L ∞ t L 2 x + √ w γ ∇u L 2 t L 2 x ) + b X T ( √ w γ b L ∞ t L 2 x + √ w γ ∇b L 2 t L 2 x ) < +∞.
The lemma is proved.

Once all the terms in (6.7) are well-defined as distributions, we use the locally energy balances (6.4) and (6.5) to obtain

∂ t ( |v| 2 + |w| 2 2 ) + µ + ν = ∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • [ |u| 2 2 + |b| 2 2 + p]u + ∇ • [(u • b)b] + u • (∇ • F) + ∆( | ũ| 2 + | b| 2 2 ) -|∇ ũ| 2 -|∇ b| 2 -∇ • [ | ũ| 2 2 + | b| 2 2 + p] ũ + ∇ • [( ũ • b) b] + ũ • (∇ • F) -u • ∂ t ũ -ũ • ∂ t u -b • ∂ t b -b • ∂ t b,
which we order as

∂ t ( |v| 2 + |w| 2 2 ) + µ + ν = ∆( |u| 2 + |b| 2 2 ) + ∆( | ũ| 2 + | b| 2 2 ) (1) -|∇u| 2 -|∇b| 2 -|∇ ũ| 2 -|∇ b| 2 -u • ∂ t ũ -ũ • ∂ t u -b • ∂ t b -b • ∂ t b (2) -∇ • [ |u| 2 2 + |b| 2 2 + p]u -∇ • [ | ũ| 2 2 + | b| 2 2 + p] ũ + ∇ • [(u • b)b] + ∇ • [( ũ • b) b] + u • (∇ • F) + ũ • (∇ • F).
Now, in order to rewrite that in a most convenient form, we use (6.6) to treat the terms (1) and (2), so that

∂ t ( |v| 2 + |w| 2 2 ) + µ + ν = ∆( |v| 2 + |w| 2 2 ) -|∇v| 2 -|∇w| 2 -(∂ t u -∆u) • ũ -(∂ t ũ -∆ ũ) • u -(∂ t b -∆b) • b -(∂ t b -∆ b) • b (3) -∇ • [ |u| 2 2 + |b| 2 2 + p]u -∇ • [ | ũ| 2 2 + | b| 2 2 + p] ũ + ∇ • [(u • b)b] + ∇ • [( ũ • b) b] + u • (∇ • F) + ũ • (∇ • F).
Thereafter, to study (3) we use the fact that (u, b, p) and ( ũ, b, p) are two solutions of the equations (MHD). Thus, we find

∂ t ( |v| 2 + |w| 2 2 ) + µ + ν = ∆( |v| 2 + |w| 2 2 ) -|∇v| 2 -|∇w| 2 -∇ • |u| 2 2 u + | ũ| 2 2 ũ + ((u • ∇)u) • ũ + (( ũ • ∇) ũ) • u (4) -∇ • |b| 2 2 u + | b| 2 2 ũ + (( ũ • ∇) b) • b + ((u • ∇)b) • b (5) + ∇ • ((u • v)b) -(( b • ∇) b) • u -((b • ∇)b) • ũ + ∇ • (( ũ • ṽ) b) -(( b • ∇) ũ) • b -((b • ∇)u) • b -∇ • (av).
We look for rewrite the right side as a sum of terms of the form

∇ • ((x • y)z)
where at least two elements of {x, y, z} belong to {v, w}, or terms of the form

((x • ∇)y) • z
where y ∈ {v, w} and at least one element of {x, z} belongs to {u, b}. As we will see, this fact will us permit to use the hypothesis u, b ∈ X T to get a good control and apply the Grönwall inequality.

For the term (4), we observe that

((u • ∇)u) • ũ + (( ũ • ∇) ũ) • u =((u • ∇)v) • ũ + ((u • ∇) ũ) • ũ -(( ũ • ∇)v) • u + (( ũ • ∇)u) • u =∇ • | ũ| 2 2 u + |u| 2 2 ũ -(( ũ • ∇)v) • u -((u • ∇)v) • v + ((u • ∇)v) • u =∇ • | ũ| 2 2 u + |u| 2 2 ũ - |v| 2 2 u + ((v • ∇)v) • u, hence (4) = -∇ • v • (u + ũ) 2 v + |v| 2 2 u + ((v • ∇)v) • u
We treat (5) in a similar way. As

(( ũ • ∇) b) • b + ((u • ∇)b) • b = -(( ũ • ∇)w) • b + (( ũ • ∇)b) • b + ((u • ∇)w) • b + ((u • ∇) b) • b =∇ • |b| 2 2 ũ + | b| 2 2 u -(( ũ • ∇)w) • b + ((u • ∇)w) • b we find (5) = -∇ • w • (b + b) 2 v + |w| 2 2 u + ((v • ∇)w) • b
With these identities on (4) and (5), we get

∂ t ( |v| 2 + |w| 2 2 ) + µ + ν = ∆( |v| 2 + |w| 2 2 ) -|∇v| 2 -|∇w| 2 -∇ • v • (u + ũ) 2 v + |v| 2 2 u + ((v • ∇)v) • u -∇ • w • (b + b) 2 v + |w| 2 2 u + ((v • ∇)w) • b +∇ • ((u • b)b) -(( b • ∇) b) • u -((b • ∇)b) • ũ (6) +∇ • (( ũ • b) b) -(( b • ∇) ũ) • b -((b • ∇)u) • b (7)
, -∇ • (av).

In order to obtain (6) and (7), remark that

∇•((u • b)b) + ∇ • (( ũ • b) b) =∇ • ((u • w)w) + ∇ • ((u • w) b) + ∇ • ((u • b)b) + ∇ • ((v • b)w) -∇ • ((v • b)b) + ∇ • ((u • b) b) =∇ • ((u • w)w) + ∇ • ((v • b)w) + ∇ • ((u • b) b) + ∇ • (( ũ • b)b) which gives (6) + (7) =∇ • ((u • w)w) + ∇ • ((v • b)w) + ∇ • ((u • b) b) + ∇ • (( ũ • b)b) -(( b • ∇) b) • u -((b • ∇)b) • ũ -(( b • ∇) ũ) • b -((b • ∇)u) • b =∇ • ((u • w)w) + ∇ • ((v • b)w) + (( b • ∇u) • b) + (( b • ∇b) • u) + ((b • ∇ ũ) • b) + ((b • ∇ b) • ũ) -(( b • ∇) b) • u -((b • ∇)b) • ũ -(( b • ∇) ũ) • b -((b • ∇)u) • b =∇ • ((u • w)w) + ∇ • ((v • b)w) + ∇ • ((v • w)b) -((w • ∇)w) • u -((w • ∇)v) • b
Thus, we get

∂ t ( |v| 2 + |w| 2 2 ) + µ + ν = ∆( |v| 2 + |w| 2 2 ) -|∇v| 2 -|∇w| 2 -∇ • v • (u + ũ) 2 v + |v| 2 2 u + ((v • ∇)v) • u -∇ • w • (b + b) 2 v + |w| 2 2 u + ((v • ∇)w) • b + ∇ • ((v • b)w) + ∇ • ((u • w)w) + ∇ • ((v • w)b) -((w • ∇)w) • u -((w • ∇)v) • b -∇ • (av),
which we rewrite in the following way

∂ t ( |v| 2 + |w| 2 2 ) + |∇v| 2 + |∇w| 2 + µ + ν =∆( |v| 2 + |w| 2 2 ) -∇ • v • (u + ũ) 2 v + w • (b + b) 2 v + |v| 2 + |w| 2 2 u A 1 + ∇ • (v • b)w + (u • w)w + (v • w)b A 2 -((w • ∇)w)) • u -((w • ∇)v)) • b + ((v • ∇)v) • u + ((v • ∇)w) • b A 3 -∇ • (av).
(6.9)

We will apply (6.9) to a suitable test function. First, we consider the function α η,t 0 ,t 1 defined in (5.7), which converges almost everywhere to 1 [t 0 ,t 1 ] when η → 0 and such that ∂ t α η,t 0 ,t 1 is the difference between two identity approximations, the first one in t 0 and the second one in t 1 .

Thereafter, we consider a non-negative function φ ∈ D(R 3 ) which is equal to 1 for |x| ≤ 1 and to 0 for |x| ≥ 2 and we set

φ R (x) = φ( x R
).

For > 0, we denote w γ, = 1 (1+ √ 2 +|x| 2 ) δ . We may observe that α η,a,s (t)φ R (x)w γ, (x) belongs to D((0, T) × R 3 ) and α η,a,s (t)φ R (x)w γ, (x) ≥ 0. Thus, applying (6.9) to this particular test function, we obtain

- |v| 2 + |w| 2 2 ∂ t α η,t 0 ,t 1 φ R w γ, dx ds + (|∇v| 2 + |∇w| 2 ) α η,t 0 ,t 1 φ R w γ, dx ds ≤ -∑ i ∂ i (v • v + w • w) α η,t 0 ,t 1 (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ i (A 1 + A 2 ) i α η,t 0 ,t 1 (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + A 3 α η,t 0 ,t 1 φ R w γ, dx ds + ∑ i (av) i α η,t 0 ,t 1 (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds.
In this inequality, we take the limit when η → 0. By the dominated convergence theorem we obtain (when the limit in the left side is well-defined)

-lim η→0 |v| 2 + |w| 2 2 ∂ t α η,t 0 ,t 1 φ R w γ, dx ds + t 1 t 0 (|∇v| 2 + |∇w| 2 ) φ R w γ, dx ds ≤ -∑ i t 1 t 0 ∂ i (v • v + w • w) (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ i t 1 t 0 (A 1 + A 2 ) i (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + t 1 t 0 A 3 φ R w γ, dx ds + ∑ i t 1 t 0 (av) i (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds.
We take t 0 and t 1 two Lebesgue points of the measurable function

A R, (t) = (|v(t, x)| 2 + |w(t, x)| 2 )φ R (x)w γ, (x) dx, so that we have - |v| 2 + |w| 2 2 ∂ t α η,a,s φ R w γ, dx ds = - 1 2 ∂ t α η,a,s A R, (s) ds,
and

lim η→0 - |v| 2 + |w| 2 2 ∂ t α η,a,s φ R w γ, dx ds = 1 2 (A R, (t 1 ) -A R, (t 0 )).
Thereafter, the continuity at 0 of v and w allows us to let t 0 goes to 0 and thus we replace t 0 by 0 in this inequality. Additionally, if we let t 1 goes to t, where t ∈ (0, T), then by weak continuity we know that A R, (t) ≤ lim inf

t 1 →t
A R, (t 1 ), so that we may replace t 1 by t as well. In conclusion, for every t ∈ (0, T) we have

|v(t, •)| 2 + |w(t, •)| 2 2 φ R w γ, dx + t 0 (|∇v| 2 + |∇w| 2 ) φ R w γ, dx ds ≤ -∑ i t 0 ∂ i (v • v + w • w) (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds -∑ i t 0 (A 1 + A 2 ) i (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds + t 0 A 3 w γ dx ds + ∑ i t 0 (av) i (w γ, ∂ i φ R + φ R ∂ i w γ, ) dx ds.
In this inequality, we take now the limit when R → +∞, and after that, the limit when → 0 to get:

|v(t, •)| 2 + |w(t, •)| 2 2 w γ dx + t 0 (|∇v| 2 + |∇w| 2 ) w γ dx ds ≤ - t 0 ∇(v • v + w • w) • ∇w γ dx ds - t 0 (A 1 + A 2 ) • ∇w γ dx ds I 1 + t 0 A 3 w γ dx ds I 2 + t 0 av • ∇w γ dx ds I 3
. Now, we shall estimate the terms I 1 , I 2 and I 3 . Recall we denote

(|v| 2 + |w| 2 )w γ dx = √ w γ (v, w) 2 L 2 , (|∇v| 2 + |∇w| 2 )w γ dx = √ w γ ∇(v, w) 2 L 2 . Lemma 6.2. |I 1 | ≤ C γ t 0 √ w γ (v, w)(s, •) 2 L 2 ds + 1 4 t 0 √ w γ ∇(v, w)(s, •) 2 L 2 ds. Proof. As |∇w γ | ≤ C γ w 3 2 γ we have |I 1 | ≤ C γ t 0 |A 1 + A 2 |w 3/2γ dx ds.
Then, as each term in the expression A 1 + A 2 writes down as the product of three vectors: (x • y)z where at least two vectors belong to {v, w} and the third one belongs to {u, b, ũ, b}. So, we will estimate the generic expression (x • y)z, where, we assume that x, z ∈ {v, w} and y ∈ {u, b, ũ, b}. Remark that for δ > 0 (which we will fix later), using the Hölder inequalities and the Young inequalities we obtain

|(x • y)z|w 3/2γ dx ≤ ( √ w γ |x|)( √ w γ |z|)( √ w γ |y|)dx ≤ √ w γ x L 3 √ w γ z L 6 √ w γ y L 2 ≤ δ -1 √ w γ x 2 L 3 + δ √ w γ z 2 L 6 √ w γ y 2 L 2 (a)
.

Interpolation inequalities gives

√ w γ x 2 L 3 ≤ δ -2 √ w γ x 2 L 2 + δ 2 √ w γ x 2 L 6 , therefore we can write (a) ≤ δ -3 √ w γ x 2 L 2 + δ √ w γ x 2 L 6 + δ √ w γ z 2 L 6 √ w γ y 2 L 2 = (b).
At this point, we apply the Sobolev embedding to estimate √ w γ x 2 L 6 and √ w γ z 2 L 6 , and we find

(b) ≤ δ -3 √ w γ x 2 L 2 + δ( √ w γ x 2 L 2 + √ w γ ∇x 2 L 2 ) + δ( √ w γ z 2 L 2 + √ w γ ∇z 2 L 2 ) √ w γ y 2 L 2 .
Using the previous estimate we get

(b) ≤δ -3 √ w γ (v, w) 2 L 2 + Cδ( √ w γ (v, w) 2 L 2 + √ w γ ∇(v, w) 2 L 2 ) + Cδ( √ w γ (v, w) 2 L 2 + √ w γ ∇(v, w) 2 L 2 ) sup 0<s<T √ w γ y 2 L 2 .
We set the parameter δ small enough to find max Cδ, Cδ sup 0<s<T

√ w γ y 2 L 2
≤ 1/64, and thus we finally get

|(x • y)z|w 3/2γ dx ≤ C γ √ w γ (v, w) 2 L 2 + 1 32 √ w γ ∇(v, w) 2 L 2 .
Integration in temporal variable gives us the desired estimate.

To study I 2 , we use the assumption u, b ∈ X T .

Lemma 6.3. For all 0 < t < T, we have

|I 2 | ≤C u X T sup 0≤s≤t √ w γ (v, w)(s) 2 L 2 + t 0 √ w γ ∇(v, w)(s, •) 2 L 2 ds + C b X T sup 0≤s≤t √ w γ (v, w)(s, •) 2 L 2 + t 0 √ w γ ∇(v, w)(s, •) 2 L 2 ds .
Proof. Each term in the expression A 3 is written in the form 3 ∑ i,j=1

x j (∂ j z i )y i , where, {x, z} belong to {v, w} and y ∈ {u, b}. By the Cauchy-Schwarz inequalities (in the spatial and temporal variables) we find

t 0 w γ |x||∇z||y|(s, •) L 1 ds ≤ t 0 √ w γ |x||y|(s, •) 2 L 2 ds 1/2 × t 0 √ w γ |∇z|(s, •) 2 L 2 ds 1/2 .
Thereafter, by definition of the multiplier space X T , and by definition of the energy space E T we have

t 0 w γ |x||∇z||y|(s, •) L 1 ds ≤ y X T √ w γ x E T t 0 √ w γ |∇z|(s, •) 2 L 2 ds 1/2 ≤C y X T √ w γ x 2 E T + t 0 √ w γ |∇z|(s, •) 2 L 2 ds ,
that is the desired estimate.

Finally, we study I 3 .

Lemma 6.4. |I 3 | ≤ C t 0 √ w γ (v, w)(s, •) 2 L 2 ds + 1 4 t 0 √ w γ ∇(v, w)(s, •) 2 L 2 ds. Proof. Remark we have p -p = ∑ i,j R i R j (u i u j -ũi ũj -b i b j + bi bj ). As v = u - ũ
and w = b -b, we have

u i u j -ũi ũj -b i b j + bi bj = v i u j + ũi v j -w j b j -bi w j .
Then, since |∇w γ | ≤ C γ w 3 2 γ the Hölder inequalities gives

|I 3 | ≤C γ w γ ∑ i,j R i R j (u i u j -ũi ũj -b i b j + bi bj ) L 6 5 √ w γ v L 6 ≤C γ w γ ∑ i,j R i R j (v i u j + ũi v j -w j b j -bi w j ) L 6 5 √ w γ v L 6 ≤C γ ( w γ (|u| + | ũ|)|v| L 6 5 + w γ (|b| + | b|)|w| L 6 5 ) √ w γ v L 6 ≤C γ √ w γ (|u| + | ũ|) L 2 √ w γ v L 3 √ w γ v L 6 + C γ √ w γ (|b| + | b|) L 2 √ w γ w L 3 √ w γ v L 6 ≤C γ √ w γ v L 3 √ w γ v L 6 √ w γ (|u| + | ũ|) L 2 + C γ √ w γ w L 3 √ w γ v L 6 √ w γ (|b| + | b|) L 2 = (a).
For δ > 0, using the Young inequalities, and thereafter the Sobolev embedding we find

(a) ≤C γ δ -1 √ w γ v 2 L 3 + C γ δ √ w γ v 2 L 6 √ w γ (|u| + | ũ|) 2 L 2 + C γ δ -1 √ w γ w 2 L 3 + Cδ √ w γ v 2 L 6 √ w γ (|b| + | b|) 2 L 2 ≤Cδ -3 √ w γ v 2 L 2 + C γ δ( √ w γ v 2 L 2 + √ w γ |∇v| 2 L 2 ) + C γ δ( √ w γ v 2 L 2 + √ w γ |∇v| 2 L 2 ) sup 0≤s≤T √ w γ (|u| + | ũ|) L 2 + C γ δ -2 √ w γ w 2 L 2 + C γ δ( √ w γ w 2 L 2 + √ w γ |∇w| 2 L 2 ) + C γ δ( √ w γ v 2 L 2 + √ w γ |∇v| 2 L 2 ) sup 0≤s≤T √ w γ (|b| + | b| L 2 .
We set the parameter δ small enough and the lemma is proved. Now, we resume the proof of the theorem. We consider 0 ≤ t 0 < t 1 < T if we suppose that v = w = 0 on [0, t 0 ], for all t ∈ [t 0 , t 1 ]. we have obtained

(v, w)(t, •) 2 L 2 wγ + 1 2 t 0 ∇(v, w) 2 L 2 wγ ds ≤C t 0 (v, w)(s) 2 L 2 wγ ds + C 1 (t 0 ,t 1 ) u X T sup 0≤s≤t 1 (v, w)(s) 2 L 2 wγ ds + t 0 ∇(v, w)(s) 2 L 2 wγ ds + C 1 (t 0 ,t 1 ) b X T sup 0≤s≤t 1 (v, w)(s) 2 L 2 wγ + t 0 ∇(v, w)(s) 2 L 2 wγ ds .
We then take the supremum on [0, t 1 ], to get

sup 0≤t≤t 1 (v, w)(t, •) 2 L 2 wγ + 1 2 t 1 0 ∇(v, w) 2 L 2 wγ ds ≤C t 1 0 (v, w)(s) 2 L 2 wγ ds + C 1 (t 0 ,t 1 ) u X T sup 0≤s≤t 1 (v, w)(s) 2 L 2 wγ + t 1 0 ∇(v, w)(s) 2 L 2 wγ ds + C 1 (t 0 ,t 1 ) b X T sup 0≤s≤t 1 (v, w)(s) 2 L 2 wγ + t 1 0 ∇(v, w)(s) 2 L 2 wγ ds . Let us denote f (t 1 ) = sup 0≤t≤t 1 (v, w)(t) 2 L 2 wγ + 1 2 t 1 0 ∇(v, w)(s) 2 L 2 wγ ds.
We take T 0 ∈ (t 0 , T) such that for t 1 ∈ (t 0 , T 0 ) the quantities 1 (t 0 ,t 1 ) u X T and 1 (t 0 ,t 1 ) b X T are small enough so that for all t 0 < t 1 < T 0 :

f (t 1 ) ≤ C t 1 0 f (s)ds.
The Grönwall's lemma permits us to conclude (u, b, p) = ( ũ, b, p) on [t 0 , T 0 ]. As t 0 ∈ [0, T) is arbitrary we have (u, b, p) = ( ũ, b, p) on [0, T).

Local Morrey spaces

There exists a slight generalisation of Theorem 4 in the case of dimension 2 and 3. It is obtained by consider a bigger space than L 2 (w 2 ), and close to that. Instead of dealing with weighted Lebesgue spaces, one may deal with a kind of local Morrey space, the space B 2 γ . In this section we state some previous results on weighted spaces and local Morrey spaces. We state these results in R d . Definition 6.1. For γ ≥ 0 and p ∈ (1, ∞). We denote B p γ (R d ) the Banach space of all functions u ∈ L p loc (R d ) such that :

u B p γ = sup R≥1 1 R γ B(0,R) |u(x)| p dx 1/p < +∞.
Moreover, for 0 < T ≤ +∞, we define B p γ L p (0, T) as the Banach space of all functions u ⊂ (L

p t L p x ) loc ([0, T] × R d ) such that u B p γ L p (0,T) = sup R≥1 1 R γ T 0 B(0,R) |u(t, x)| p 1 p dx dt < +∞.
|u| p (1 + |x|) δ dx = |x|≤1 |u| p (1 + |x|) δ dx + ∑ n∈N 2 n-1 ≤|x|≤2 n |u| p (1 + |x|) δ dx ≤ |x|≤1 |u| p dx + ∑ n∈N 1 (1 + 2 n-1 ) δ 2 n-1 ≤|x|≤2 n |u| p dx ≤ |x|≤1 |u| p dx + c ∑ n∈N 1 2 δn 2 n-1 ≤|x|≤2 n |u| p dx ≤ (1 + c ∑ n∈N 1 2 (δ-γ)n ) sup R≥1 1 R γ |x|≤R |u| p dx, therefore, B p γ ⊂ L p w δ .
Similarly, for all δ > γ, B p γ L p (0, T) ⊂ L p ((0, T), L p w δ ).

To prove that L p ((0, T), L

p w γ ) ⊂ B p γ,0 L p (0, T). Consider λ > 1, n ∈ N and denote u n (t, x) = u(t, λ n x). As we have sup R≥1 1 (λ n R) γ T 0 |x|≤λ n R |u(t, x)| p dx dt = sup R≥1 λ (d-γ)n R γ
< γ < δ < ∞ we have B p γ = [L p , L p w δ ] γ δ ,∞ and • B p γ and • [L p ,L p w δ ] γ δ ,∞
are equivalent norms.

Proof. Consider f ∈ B p γ . For A < 1, we let f 0 = 0 and f 1 = f . Then, f = f 0 + f 1 and f 1 L p w δ ≤ CA γ δ -1 f B p γ . For A > 1, we denote R = A p δ > 1. We write f 0 = f 1 |x|≤R and f 1 = f 1 |x|>R . Then, we find f 0 p ≤ C f B p γ R γ p = CA γ δ f B p γ and f 1 p L p w δ = ∑ n∈N 2 n-1 R≤|x|≤2 n R |u| p (1 + |x|) δ dx ≤ CR γ-δ ∑ n∈N 1 2 (δ-γ)n f p B p γ = CA ( γ δ -1)p f p B p γ . Therefore, B p γ → [L p , L p w δ ] γ δ ,∞ . Now, consider f ∈ [L p , L p w δ ] γ
δ ,∞ , then there exists c > 0 such that for all A > 0, there exist

f 0 ∈ L p and f 1 ∈ L p w δ such that f = f 0 + f 1 , f 0 p ≤ cA γ δ and f 1 L p w δ ≤ cA γ δ -1 .
For each j ∈ N, we take A = 2 jδ p so that

1 2 jγ |x|<2 j | f | p dx ≤ C 1 2 jγ |x|<2 j | f 0 | p dx + 1 2 jγ |x|<2 j | f 1 | p dx ≤ C 1 2 jγ f 0 p p + C 2 jγ |x|≤1 | f 1 | p (1 + |x|) δ dx + C j ∑ k=1 2 kδ 2 jγ 2 k-1 <|x|<2 k | f 1 | p (1 + |x|) δ dx ≤ C 1 2 jγ f 0 p p + C 2 j(δ-γ) f 1 p L p w δ ≤ C .
We have found sup j∈N

1 2 jγ |x|<2 j | f | p dx < +∞ and hence sup R≥1 1 R γ |x|<R | f | p dx < +∞.
An important corollary of Theorem 13 reads as follows: ,∞ , for some δ < δ 0 < d. Then, we conclude the continuity of these operators on L p w δ 0 .

Solutions in local Morrey spaces

Local Morrey spaces B 2 d occur naturally in the setting of homogeneous statistical solutions, we refer to [START_REF] Vishik | Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes[END_REF] and [START_REF] Dostoglou | Homogeneous measures and spatial ergodicity of the Navier-Stokes equations[END_REF] for exemple.

For the (NS) equations, in the paper [START_REF] Bradshaw | Existence of global weak solutions to the Navier-Stokes equations in weighted spaces[END_REF], the main theorem on global existence given in (Fernández-Dalgo and Lemarié-Rieusset, 2020b) is improved with respect to the initial data. We relax the method developed in the second paper to enlarge the initial data space and thus we generalize the previous results to the framework of the (MHD) equations.

Our main result in this direction reads as follows:

Theorem 14 (Local and global existence). Let 0 < T < +∞. Let u 0 , b 0 ∈ B 2 (R 3 ) be divergence-free vector fields. Let F be a tensor belonging to B 2 L 2 (0, T). Then, there exists a time T 0 ∈ (0, T) such that the system (MHD) admits a solution (u, b, p) with the following properties :

• u, b belong to L ∞ ((0, T 0 ), B 2 ) and ∇u, ∇b belong to B 2 L 2 (0, T 0 )

• the pressure p is related to u, b and F by the formula:

p = ∑ 1≤i,j≤3 R i R j (u i u j -b i b j -F i,j ) • the map t ∈ [0, T) → (u(t, •), b(t, •)) is * -weakly continuous from [0, T) to B 2 (R 3 ), and for all compact set K ⊂ R 3 , lim t→0 (u(t, •) -u 0 , b(t, •) -b 0 ) L 2 (K) = 0,
• the solution (u, b, p) is suitable : there exists a non-negative locally finite measure µ on (0, T) × R 3 such that

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 + p)u + ∇ • [(u • b)b] + u • (∇ • F) -µ.
We also obtain for all 0 ≤ t ≤ T 0 ,

max{ (u, b)(t) 2 B 2 , ∇(u, b) 2 B 2 L 2 (0,T 0 ) } ≤ C (u 0 , b 0 ) 2 B 2 + C F 2 B 2 L 2 (0,t) + C t 0 (u, b)(s) 2 B 2 + (u, b)(s) 6 B 2 ds.
(6.10)

Moreover, if the data verify:

lim R→+∞ R -2 |x|≤R |u 0 (x)| 2 + |b 0 (x)| 2 dx = 0 lim R→+∞ R -2 +∞ 0 |x|≤R |F(t, x)| 2 dx ds = 0,
then we get a global weak solution(u, b, p).

We prove a global control on the solutions (6.10) which is not exhibited in (Bradshaw, [START_REF] Bradshaw | Existence of global weak solutions to the Navier-Stokes equations in weighted spaces[END_REF]. In the setting of the space B 2 (R 3 ), the control on the pressure p is a little more technical.

Getting back to the (NS) equations, the global existence and uniqueness of solutions for the 2D case with initial data u 0 ∈ B 2 (R 2 ) was an open problem proposed by A. Basson in [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF]. In Section 6.5 we make a discussion on problematic arising on the local and global existence for the 2D case, and moreover, we give a sketch of the proof of a result analogous to Theorem 14 in dimension 2.

Our main theorem bases on results for the equations:

(MHD )              ∂ t u = ∆u -(v • ∇)u + (c • ∇)b -∇p + ∇ • F, ∂ t b = ∆b -(v • ∇)b + (c • ∇)u -∇q, ∇ • u = 0, ∇ • b = 0, u(0, •) = u 0 , b(0, •) = b 0 .
where we will consider (v, c) = (u * θ , b * θ ), where, for 0 < ε < 1 and for a fixed, non-negative and radially non increasing test function θ ∈ D(R 3 ) which is equals to 0 for |x| ≥ 1 and θ dx = 1; we define θ ε (x) = 1 ε 3 θ(x/ε).

A priori estimates

Theorem 15. Let 0 < T < +∞. Let u 0 , b 0 ∈ B 2 be a divergence-free vector fields and let F be a tensor such that F ∈ B 2 L 2 (0, T). Moreover, let (u, b, p, q) be a solution of the problem (MHD ) or a solution of the problem (MHD).

We suppose that:

• u, b belongs to L ∞ ((0, T), B 2 ) and ∇u, ∇b belongs to B 2 L 2 (0, T).

• The map t ∈ [0, T) → u(t, •) is * -weakly continuous from [0, T) to B 2 , and for all compact set K ⊂ R 3 we have:

lim t→0 (u(t, •) -u 0 , b(t, •) -b 0 ) L 2 (K) = 0.
In particular, we can take p and q related to u, b and F by

p = ∑ 1≤i,j≤3 R i R j (v i u j -c i b j -F i,j ) and q = ∑ 1≤i,j≤3 R i R j (v i b j -c j u i ).
• The solution (u, b, p, q) is suitable : there exists a non-negative locally finite measure µ on (0, T) × R 3 such that

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • ( |u| 2 2 + |b| 2 2 )v + pu + ∇ • ((u • b)c + qb) + u • (∇ • F) -µ. (6.11)
Then, there exists a constant C ≥ 1, which does not depend on T, and not on u 0 , b 0 u, b, F nor , such that:

• We have on [0, T):

max{ (u, b)(t) 2 B 2 , ∇(u, b) 2 B 2 L 2 (0,t) } ≤ C (u 0 , b 0 ) 2 B 2 + C F 2 B 2 L 2 (0,t) + C t 0 (u, b)(s) 2 B 2 + (u, b)(s) 6 B 2 ds.
(6.12)

• Moreover, if T 0 < T is small enough:

C 1 + (u 0 , b 0 ) 2 B 2 + F 2 B 2 L 2 (0,T 0 ) 2 T 0 ≤ 1,
then the following estimate with respect to the data holds:

sup 0≤t≤T 0 max{ (u, b)(t, .) 2 B 2 , ∇(u, b) 2 B 2 L 2 (0,t) } ≤ C 1 + (u 0 , b 0 ) 2 B 2 + F 2 B 2 L 2 (0,T 0 ) . (6.13)
Proof. We focus only in the case (v, c) = (u * θ ε , b * θ ε ), as the case (v, c) = (u, b) can be treated in a similar way.

We start by proving (6.12). As usual, we look to apply the energy balance (6.11) to a suitable test function.

Consider the function α η,t 0 ,t 1 defined in (5.7) and take a non-negative function φ ∈ D(R 3 ) which is equals to 1 for |x| ≤ 1/2 and is equals to 0 for |x| ≥ 1; and for R ≥ 1 we set

φ R (x) = φ( x R
).

(6.14)

Applying the energy balance (6.11) to the test function α η,t 0 ,t 1 φ R , we obtain

- |u| 2 2 + |b| 2 2 ∂ t α η,t 0 ,t 1 φ R dx ds + |∇u| 2 + |∇b| 2 α η,t 0 ,t 1 φ R dx ds ≤ |u| 2 + |b| 2 2 α η,t 0 ,t 1 ∆φ R dx ds + 3 ∑ i=1 [( |u| 2 2 + |b| 2 2 )v i + pu i ]α η,t 0 ,t 1 ∂ i φ R dx ds + 3 ∑ i=1 [(u • b)c i + qb i ]α η,t 0 ,t 1 ∂ i φ R dx ds -∑ 1≤i,j≤3 ( F i,j u j α η,t 0 ,t 1 ∂ i φ R dx ds + F i,j ∂ i u j α η,t 0 ,t 1 φ R dx ds).
We let η goes to 0, when the limit in the left side above is well-defined, we obtain by the dominated convergence :

-

lim η→0 |u| 2 2 + |b| 2 2 ∂ t α η,t 0 ,t 1 φ R dx ds + t 1 t 0 |∇u| 2 + |∇b| 2 φ R dx ds ≤ t 1 t 0 |u| 2 + |b| 2 2 ∆φ R dx ds + 3 ∑ i=1 t 1 t 0 [( |u| 2 2 + |b| 2 2 )v i + pu i ]∂ i φ R dx ds + 3 ∑ i=1 t 1 t 0 [(u • b)c i + qb i ]∂ i φ R dx ds -∑ 1≤i,j≤3 (
t 1 t 0 F i,j u j ∂ i φ R dx ds + t 1 t 0 F i,j ∂ i u j φ R dx ds). We denote A R (t) = (|u(t, x)| 2 + |b(t, x)| 2 )φ R (x) dx, so that - ( |u| 2 2 + |b| 2 2 )∂ t α η,t 0 ,t 1 φ R dx ds = - 1 2 ∂ t α η,t 0 ,t 1 A R (s) ds, Thus, if t 0 and t 1 are Lebesgue points of A R (t) we find lim η→0 - ( |u| 2 2 + |b| 2 2 )∂ t α η,t 0 ,t 1 φ R dx ds = 1 2 (A R (t 1 ) -A R (t 0 )).
As φ R is a support compact function we can let t 0 goes to 0 and thus we may replace t 0 by 0 in this inequality. Adittionaly, letting t 1 goes to t, by * -weak continuity, A R (t) ≤ lim t 1 →t A R (t 1 ), and so we can replace t 1 by t ∈ (0, T). Therefore, for every t ∈ (0, T) :

|u(t, x)| 2 + |b(t, x)| 2 2 φ R dx + t 0 (|∇u| 2 + |∇b| 2 ) φ R ds dx ≤ |u 0 (x)| 2 + |b 0 (x)| 2 2 φ R dx + t 0 |u| 2 + |b| 2 2 ∆φ R ds dx + 3 ∑ i=1 t 0 [( |u| 2 2 + |b| 2 2 )v i + pu i ]∂ i φ R dx ds + 3 ∑ i=1 t 0 [(u • b)c i + qb i ]∂ i φ R dx ds -∑ 1≤i,j≤3 ( t 0 F i,j u j ∂ i φ R dx ds + t 0 F i,j ∂ i u j φ R dx ds).
(6.15)

To estimate the second term in the right-hand side, as R ≥ 1 we write

1 R 2 (|u| 2 + |b| 2 )∆φ R dx ≤ C R 4 B(0,R) (|u| 2 + |b| 2 ) dx ≤ C( u 2 B 2 + b 2 B 2 ).
To treat third and fourth terms, we consider first the expressions where the pressure terms p and q do not appear. By Hölder inequalities and Sobolev embeddings we obtain:

3 ∑ i=1 (u • b) 2 (b i * θ )∂ i φ R dx ≤ u L 12 5 (B(0,R)) b L 12 5 (B(0,R)) b * θ L 6 (B(0,R)) ∇φ R L ∞ ≤ C R u 3/4 L 2 (B(0,R)) u 1/4 L 6 (B(0,R)) b 3/4 L 2 (B(0,R)) b 5/4 L 6 (B(0,R+1)) ≤ C R b 3/4 L 2 (B(0,R)) u 3/4 L 2 (B(0,R)) U 1/4 B 5/4 ,
where we have denoted

U = |φ 2R ∇u| 2 dx 1/2 + |x|≤2R |u| 2 dx 1/2 and B = |φ 2(R+1) ∇b| 2 dx 1/2 + |x|≤2(R+1) |b| 2 dx 1/2 .
Then, by the Young's inequalities for products with 1

= 1 8 + 1 8 + 1 8 + 5 8 , 1 R 2 3 ∑ i=1 (u • b) 2 (b i * θ )∂ i φ R dx ≤ C( u L 2 (B(0,R)) R ) 3/4 ( b L 2 (B(0,R)) R ) 3/4 ( U R ) 1/4 ( B R ) 5/4 ≤ C (u, b) 6 B 2 + C (u, b) 2 B 2 + C 0 R 2 |φ 2R ∇u| 2 + |φ 2(R+1) ∇b| 2 dx,
where C 0 > 0 is an arbitrarily small constant to be fixed later. Now, we use the following technical lemma which will be proved below, in order to estimate the expressions where the terms p and q appear. Lemma 6.6. Under the hypothesis of Theorem 15, p and q belong to L 3/2 loc and there exist an arbitrarily small constant C 0 > 0 and a constant C = C(C 0 ) > 0, which do not depend on : T, u, b, u 0 , b 0 , F nor ; such that for all R ≥ 1 and for all 0 ≤ t ≤ T,

1 R 2 3 ∑ i=1 t 0 (pu i + qb i ) ∂ i φ R ds dx ≤C F 2 B 2 L 2 (0,t) + C t 0 (u, b)(s) 2 B 2 + (u, b)(s) 6 B 2 + C 0 R 2 t 0 |ϕ 2(5R+1) ∇u| 2 + |ϕ 2(5R+1) ∇b| 2 dx.
Finally, for the fifth term in the right side of (6.15) we have 1

R 2 ∑ 1≤i,j≤3 t 0 F i,j (∂ i u j )φ R dx ds ≤ C F 2 B 2 L 2 (0,t) + C 0 R 2 t 0 |x|<R |∇u| 2 dx ds, and 1 R 2 ∑ 1≤i,j≤3 t 0 F i,j u i ∂ j (φ R ) dx ds ≤ C F 2 B 2 L 2 (0,t) + C t 0 u(s) 2 B 2 ds,
where C 0 > 0 always denote a small enough constant.

Once we dispose of all these estimates, we conclude that

( |u(t, x)| 2 2 + |b(t, x)| 2 2 )φ R dx + t 0 (|∇u| 2 + |∇b| 2 ) φ R ds dx ≤ ( |u(0, x)| 2 2 + |b(0, x)| 2 2 )φ R dx + C F 2 B 2 L 2 (0,t) ds + C t 0 (u, b)(s, •) 2 B 2 + (u, b)(s, •) 6 B 2 ds + C 0 R 2 t 0 |ϕ 2(5R+1) ∇u| 2 + |ϕ 2(5R+1) ∇b| 2 dx,
and the desired control (6.12) follows. To finish this proof, (6.13) follows directly from (6.12) and the Lemma at the end of this section.

Proof of Lemma 6.6. As in the proof of the theorem above, we consider the case (v, c) = (u * θ ε , b * θ ε ). We focus only on the expression which involves the pressure p, because the computations for the other expression, where q appears, are very similar.

We have

1 R 2 3 ∑ k=1 t 0 |x|≤R |pu k || ∂ k φ R | dx ds ≤ c R 3 3 ∑ k=1 t 0 |x|≤R |pu k | dx ds,
then, recalling the formula

p = ∑ 1≤i,j≤3 R i R j ((u i * θ ε )u j -(b i * θ ε )b j -F i,j ), we find 1 R 2 3 ∑ k=1 t 0 |x|≤R |pu k || ∂ k φ R | dx ds ≤ c R 3 3 ∑ k=1 t 0 |x|≤R |u k 3 ∑ i,j=1 R i R j ((u i * θ ε )u j )| dx ds + c R 3 3 ∑ k=1 t 0 |x|≤R |u k 3 ∑ i,j=1 R i R j ((b i * θ ε )b j -F i,j )| dx ds.
In view that we have the same information on u and b, it is enough to analyse the last term above. For R ≥ 1, we define :

p 1 = ∑ i,j R i R j (1 |y|<5R (θ * b i )b j ), p 2 = -∑ i,j R i R j (1 |y|≥5R (θ * b i )b j ), and
p 3 = -∑ i,j R i R j (1 |y|<5R F i,j ), p 4 = ∑ i,j R i R j (1 |y|≥5R F i,j ).
By the Young's inequalities we get,

c R 3 3 ∑ k=1 t 0 |x|≤R |u k 3 ∑ i,j=1 R i R j ((b i * θ ε )b j -F i,j )| dx ds ≤ C R 3 t 0 |x|≤R (|p 1 | 3/2 + |p 2 | 3/2 + |u| 3 + |p 3 | 2 + |p 4 | 2 + |u| 2 )dx ds,
and next we will study each term.

To study p 1 , we use the continuity of R i on L 3 2 (R 3 ) and the fact that supp(θ ) ⊂ B(0, 1) to obtain

|x|≤R |p 1 | 3/2 dx ≤ C |p 1 | 3/2 dx ≤ C |(1 |x|<5R (θ * b) ⊗ b)| 3/2 dx ≤ C |1 |x|<5R (θ * b)| 3 dx 1/2 |1 |y|<5R b| 3 dx 1/2 ≤ C |x|≤5R |x-z|≤1 θ (x -z)|b(z)| 3 dzdx 1/2 |(1 |y|<5R b)| 3 dx 1/2 ≤ C |x|≤5R |z|≤5R+1 θ (x -z)|b(z)| 3 dzdx 1/2 |(1 |y|<5R b)| 3 dx 1/2 ≤ C |z|≤5R+1 |b| 3 dz, so we see that |x|≤R |u| 3 + |p 1 | 3/2 dx ≤ C |x|≤5R+1 |u| 3 + |b| 3 dx.
Using a Sobolev embedding, we find

C R 3 |x|≤5R+1 |u| 3 dx ≤ C R 3 u 3/2 L 2 (B(0,5R+1)) u 3/2 L 6 (B(0,5R+1)) ≤ C R 3/2 u 3/2 L 2 (B(0,5R+1)) 1 R 2 |φ 2(5R+1) ∇u| 2 dx 1 2 + 1 R 2 |x|≤2(5R+1) |u| 2 dx 1 2 3 2 ≤ C u 6 B 2 + C 0 u 2 B 2 + C 0 R 2 |φ 2(5R+1) ∇u| 2 dx,
where C 0 > 0 is a arbitrarily small constant. Similar estimates works for b.

To study p 2 . Remark first that there exist a constant C > 0, which does not depend on R ≥ 1, such that for all |x| ≤ R and all |y| ≥ 5R, the kernel of the operator R i R j named K i,j verifies |K i,j (xy)| ≤ C |y| 3 (we refer to (Grafakos, 2009) for a proof), and then we may write:

|x|≤R |p 2 | 3/2 dx 2/3 ≤ C ∑ i,j |x|≤R |K i,j (x -y)| |(θ * b i )(y)b j (y)| 1 |y|≥5R dy 3/2 dx 2/3 ≤ C |x|≤R |y|≥5R 1 |y| 3 |(θ * b) ⊗ b| dy 3/2 dx 2/3 ≤ CR 2 |y|≥5R 1 |y| 3 |(θ * b) ⊗ b|dy ≤ CR 2 |y|≥5R 1 |y| 3 |θ * b| 2 dy 1/2 |y|≥5R 1 |y| 3 |b| 2 dy 1/2 ≤ CR 2 |y|≥5R 1 |y| 3 |y-z|<1 θ (y -z)|b(z)| 2 dz dy 1/2 |y|≥5R 1 |y| 3 |b| 2 dy 1/2 ≤ CR 2 |y|≥5R |z|≥5R-1 1 |z| 3 θ (y -z)|b(z)| 2 dz dy 1/2 |y|≥5R 1 |y| 3 |b| 2 dy 1/2 ≤ CR 2 |z|≥5R-1 1 |z| 3 |b| 2 dz. As B 2 (R 3 ) ⊂ L 2 w 3 (R 3 ), we finally get C R 3 |y|≤R |p 2 | 3/2 dx ≤ C 1 (1 + |z|) 3 |b| 2 3/2 ≤ C b 3 B 2 .
Now, we treat the terms p 3 and p 4 which involve the tensor F. For p 3 , continuity of the Riesz transform R i on

L 2 gives c R 3 t 0 |x|≤R |p 3 | 2 dx ds ≤ C R 3 ∑ i,j t 0 |x|<5R |F i,j | 2 dx ds ≤ C F 2 B 2 L 2 (0,t) .
For the term p 4 , observe that

|x|≤R |p 4 | 2 dx 1/2 C ≤ ∑ i,j |x|≤R |y|≥5R |K i,j (x -y)F i,j | dy 2 dx 1/2 ≤ C ∑ i,j |x|≤R |y|≥5R 1 |y| 3 |F i,j | dy 2 dx 1/2 ≤ C ∑ i,j R 3/2 |y|≥5R 1 |y| 3 |F i,j | dy,
then, taking 0 < δ < 1, by the Hölder inequalities we find

C R 3 t 0 |x|≤R |p 4 | 2 dx ds ≤ C ∑ i,j t 0 1 (1 + |x|) 3 |F i,j | dx 2 ds ≤ C ∑ i,j t 0 1 (1 + |x|) 2+δ |F i,j | 2 dx ds ≤ C ∑ i,j 1 (1 + |x|) 2+δ t 0 |F i,j | 2 ds dx ≤ C F 2 B 2 L 2 (0,t)
and the proof is finished.

Before to finish this section we present the following lemma, used in the proof of the theorem above, which is a variant of Lemma 3.5. Lemma 6.7. Let α be a non-negative bounded measurable function on [0, T) which satisfies, for three constants A, B > 0 and b > 1,

α(t) ≤ A + B t 0 1 + α(s) b ds. If T 0 > 0 and T 1 = min(T, T 0 , 1 4(b-1)B(A+BT 0 ) b-1 ), we have, for every t ∈ [0, T 1 ], α(t) ≤ √ 2(A + BT 0 ).
Proof. We define

Φ(t) = A + BT 0 + B t 0 α b ds and Ψ(t) = A + BT 0 + B t 0 Φ(s) b ds,
so that for all t ∈ [0, T 1 ], α ≤ Φ ≤ Ψ, and then

Ψ (t) = BΦ(t) b ≤ BΨ(t) b so 1 Ψ(0) b-1 - 1 Ψ(t) b-1 ≤ (b -1)Bt,
and we conclude

Ψ(t) b-1 ≤ Ψ(0) b-1 1 -(b -1)BΨ(0) b-1 t ≤ 2Ψ(0) b-1 .

A stability result

Theorem 16. Let 0 < T < +∞. Let u 0,n , b 0,n be divergence-free vector fields belonging to B 2 . Let F n be tensors belonging to B 2 L 2 (0, T). Suppose that (u n , b n , p n , q n ) is a solution of the (MHD n ) problem, with n → 0 + , or a solution of the (MHD) problem (in which case q n = 0) :

             ∂ t u n = ∆u n -(v n • ∇)u n + (c n • ∇)b n -∇p n + ∇ • F n , ∂ t b n = ∆b n -(v n • ∇)b n + (c n • ∇)u n -∇q n , ∇ • u n = 0, ∇ • b n = 0, u n (0, •) = u 0,n , b n (0, •) = b 0,n .
which verifies the same hypothesis of Theorem 15.

We assume that (u 0,n , b 0,n ) is strongly convergent to (u 0,∞ , b 0,∞ ) in B 2 , and the sequence F n is strongly convergent to F ∞ in B 2 L 2 (0, T). Then, there exist (u ∞ , b ∞ , p ∞ ) and an increasing sequence (n k ) k∈N with values in N such that:

• (u n k , b n k ) converges *-weakly to (u ∞ , b ∞ ) in L ∞ ((0, T), B 2 ), (∇u n k , ∇b n k ) converges weakly to (∇u ∞ , ∇b ∞ ) in B 2 L 2 (0, T). • (u n k , b n k ) converges strongly to (u ∞ , b ∞ ) in L 2 loc ([0, T) × R 3 ). • For 2 < γ < 5/2, the sequence (p n k , q n k )converges weakly to (p ∞ , 0) in L 3 ((0, T), L 6/5 w 6γ 5 ) + L 2 ((0, T), L 2 w γ ). Moreover, (u ∞ , b ∞ , p ∞ ) is a solution of the problem (MHD):              ∂ t u ∞ = ∆u ∞ -(u ∞ • ∇)u ∞ + (b ∞ • ∇)b ∞ -∇p ∞ + ∇ • F ∞ , ∂ t b ∞ = ∆b ∞ -(u ∞ • ∇)b ∞ + (b ∞ • ∇)u ∞ ∇ • u ∞ = 0, ∇ • b ∞ = 0, u ∞ (0, •) = u 0,∞ , b ∞ (0, •) = b 0,∞ ,
and verifies all the hypothesis of Theorem 15.

Proof. We will verify that the sequence (u n , b n ) satisfy the hypothesis of the Rellich lemma. Remark first that: since for 2 < γ we have that u n , b n is bounded in L ∞ ((0, T), B 2 ) ⊂ L ∞ ((0, T), L 2 w γ ) and moreover, since we have that ∇u n , ∇b n is bounded in B 2 L 2 (0, T) ⊂ L 2 ((0, T), L 2 w γ ), then for all ϕ ∈ D(R 3 ) we have that (ϕu n , ϕb n ) are bounded in L 2 ((0, T), H 1 ). On the other hand, for the pressure p n and the term q n we write p n = p n,1 + p n,2 with

p n,1 = 3 ∑ i=1 3 ∑ j=1 R i R j (v n,i u n,j -c n,i b n,j ), p n,2 = - 3 ∑ i=1 3 ∑ j=1 R i R j (F n,i,j ), and
q n = 3 ∑ i=1 3 ∑ j=1 R i R j (v n,i b n,j -c n,i u n,j ).
From now on, we fix γ ∈ (2, 5

2 ). Interpolation inequalities and the continuity of the Riesz transforms in the Lebesgue weighted spaces permit to conclude that the sequence (p n,1 , q n,1 ) is bounded in L 3 ((0, T), L 6/5 w 6γ

5

). Indeed, to treat the term p n,1 recall that for 0 < γ < 5/2 the weight w 6γ/5 belongs to the Muckenhoupt class

A p (R 3 ), with p ∈ (1, +∞), then ∑ i,j R i R j (u n,i u n,j )w γ L 6/5 ≤ (u n ⊗ u n )w γ L 6/5 ≤ √ w γ u n 3 2 L 2 √ w γ u n 1 2 L 6 ≤ √ w γ u 3 2 L 2 ( √ w γ u L 2 + √ w γ ∇u L 2 ) 1 2 .
Similar estimates holds for the term q n,1 . Of course, the sequences q n,2 and p n,2 are bounded in L 2 ((0, T), L 2 w γ ). Then, we obtain that the sequence (ϕ∂

t u n , ϕ∂ t b n ) is bounded in L 2 L 2 + L 2 W -1,6/5 + L 2 H -1 ⊂ L 2 ((0, T), H -2
). Thus, we can apply the Rellich lemma, there exists an increasing sequence (n k ) k∈N in N, and there exists a couple of functions

(u ∞ , b ∞ ) such that (u n k , b n k ) converges strongly to (u ∞ , b ∞ ) in L 2 loc ([0, T) × R 3 ). We also know that (v n k , c n k ) = (u n k * θ n k , b n k * θ n k ) converges strongly to (u ∞ , b ∞ ) in L 2 loc ([0, T) × R 3 ). As the sequence (u n , b n ) is bounded in L ∞ ((0, T), L 2 w γ ) and (∇u n , ∇b n ) is bounded in L 2 ((0, T), L 2 w γ ), we get (u n k , b n k ) converges *-weakly to (u ∞ , b ∞ ) in L ∞ ((0, T), L 2 w γ ), and (∇u n k , ∇b n k ) converges weakly to (∇u ∞ , ∇b ∞ ) in L 2 ((0, T), L 2 w γ ).
Additionally, by the Sobolev embeddings and the interpolation inequalities we have (u

n k , b n k ) converges weakly to (u ∞ , b ∞ ) in L 3 ((0, T), L 3 w 3γ/2 ). Moreover, we find that (v n k , c n k ) = (v n k * θ n k , c n k * θ n k ) converges weakly to (u ∞ , b ∞ ) in L 3 ((0, T), L 3 w 3γ/2 ) as well, since it is bounded in L 3 ((0, T), L 3 w 3γ/2 ). Thus, the terms v n k ,i u n k ,j , c n k ,i b n k ,j , v n k ,i b n k ,j
and c n k ,i u n k ,j are weakly convergent in (L 6/5 L 6/5 ) loc and hence in D ((0, T) × R 3 ).

Those terms are bounded in L 3 ((0, T), L 6/5 w 6γ 5

), then they are weakly convergent in

L 3 ((0, T), L 6/5 w 6γ 5 ). Therefore, defining p ∞ = p ∞,1 + p ∞,2 with p ∞,1 = 3 ∑ i=1 3 ∑ j=1 R i R j (v ∞,i u ∞,j -c ∞,i b ∞,j ), p 2 = - 3 ∑ i=1 3 ∑ j=1 R i R j (F ∞,i,j ), and q ∞ = 3 ∑ i=1 3 ∑ j=1 R i R j (v ∞,i b ∞,j -c ∞,i u ∞,j ), we conclude that (p n k ,1 , q n k ,1 ) is weakly convergent in L 3 ((0, T), L 6/5 w 6γ 5 ) to (p ∞,1 , q ∞,1 ), and p n k ,2 is strongly convergent in L 2 ((0, T), L 2 w γ ) to p ∞,2 . As v n k = θ n k * (u n k -u) + θ n k * u, then we verify that v n k is convergent to u in L α loc ([0, T 0 ) × R 3 ). Thus, we find (v ∞ , c ∞ ) = (u ∞ , b ∞ ) and then q ∞ = 0 and (u ∞ , b ∞ , p ∞ ) verify the three first equations in the system (MHD) in D ((0, T) × R 3 ). It remains to verify the initial conditions. Since (∂ t u ∞ , ∂ t b ∞ ) are locally in L 2 H -2 the distribution (u ∞ , b ∞ ) has a representative such that t → (u ∞ (t, .), b ∞ (t, .)) is continuous from [0, T) to D (R 3 ) (hence *-weakly continuous from [0, T) to B 2 ) and additionally, they coincide with u ∞ (0, .) + t 0 ∂ t u ∞ ds and b ∞ (0, .) + t 0 ∂ t b ∞ ds. Then, we get in D ((0, T) × R 3 ), u ∞ (0, .) + t 0 ∂ t u ∞ ds = u ∞ = lim n k →+∞ u n k = lim n k →+∞ u n k ,0 + t 0 ∂ t u n k ds = u ∞,0 + t 0 ∂ t u ∞ ds, hence u ∞ (0, .) = u ∞,0 . Similarly b ∞ (0, .) = b ∞,0 . Thus (u ∞ , b ∞ , p ∞ ) is a solution of the (MHD * ) system.
Now, we study the local energy balance. We define

A n k = -∂ t ( |u n k | 2 + |b n k | 2 2 ) + ∆( |u n k | 2 + |b n k | 2 2 ) -∇ • ( |u n k | 2 2 + |b n k | 2 2 )v n k -∇ • (p n k u n k ) -∇ • (q n k b n k ) + ∇ • ((u n k • b n k )c n k ) + u n k • (∇ • F n k ). By interpolation (u n , b n ) is bounded in L 10/3 ((0, T), L 10/3 w 5γ/3 ), then (u n k , b n k ) are locally bounded in L 10/3 t L 10/3
x and locally strongly convergent in L 2 t L 2 x . Therefore, (u n k , b n k ) converges strongly in (L 3 t L 3 x ) loc . By Lemma 6.6 we know that (p n k , q n k ) are locally bounded in L 3/2 t L 3/2 x . Thus the quantity A n k converges in the distributional sense to

A ∞ = -∂ t ( |u ∞ | 2 + |b ∞ | 2 2 ) + ∆( |u ∞ | 2 + |b ∞ | 2 2 ) -∇ • ( |u ∞ | 2 2 + |b ∞ | 2 2 )u ∞ -∇ • (p ∞ u ∞ ) + ∇ • ((u ∞ • b ∞ )b ∞ ) + u ∞ • (∇ • F ∞ ).
By hypothesis, there exist µ n k a non-negative locally finite measure on (0, T) × R 3 such that

∂ t ( |u n k | 2 + |b n k | 2 2 ) = ∆( |u n k | 2 + |b n k | 2 2 ) -|∇u n k | 2 -|∇b n k | 2 -∇ • ( |u n k | 2 2 + |b n k | 2 2 )v n k -∇ • (p n k u n k ) -∇ • (q n k b n k ) + ∇ • ((u n k • b n k )c n k ) + u n k • (∇ • F n k ) -µ n k , or equivalently, A n k = |∇u n k | 2 + |∇b n k | 2 + µ n k .
Hence, we have

A ∞ = lim n k →+∞ |∇u n k | 2 + |∇b n k | 2 + µ n k .
Let us take Φ ∈ D((0, T) × R 3 ) be a non-negative function. As √ Φ(∇u n k , ∇b n k ) is weakly convergent to √ Φ(∇u ∞ , ∇b ∞ ) in L 2 t L 2 x , we find

A ∞ Φ dx ds = lim n k →+∞ A n k Φ dx ds ≥ lim sup n k →+∞ (|∇u n k | 2 + |∇b n k | 2 )Φ dx ds ≥ (|∇u ∞ | 2 + |∇b ∞ | 2 )Φ dx ds.
Thus, there exists a non-negative locally finite measure µ ∞ on (0,

T) × R 3 such that A ∞ = (|∇u ∞ | 2 + |∇b ∞ | 2 ) + µ ∞ .
We have obtained the desired local energy balance:

∂ t ( |u ∞ | 2 + |b ∞ | 2 2 ) = ∆( |u ∞ | 2 + |b ∞ | 2 2 ) -|∇u ∞ | 2 -|∇b ∞ | 2 -∇ • ( |u ∞ | 2 2 + |b ∞ | 2 2 )u ∞ -∇ • (p ∞ u ∞ ) + ∇ • ((u ∞ • b ∞ )b ∞ ) + u ∞ • (∇ • F ∞ ) -µ ∞ .
To finish this proof, we must to prove the convergence to the initial data (u 0,∞ , b 0,∞ ).

+ 3 ∑ i=1 t 0 [( |u ∞ | 2 2 + |b ∞ | 2 2 )u ∞,i + p ∞ u ∞,i ]∂ i φ R dx ds + 3 ∑ i=1 t 0 [(u ∞ • b ∞ )b ∞,i ]∂ i φ R dx ds -∑ 1≤i,j≤3 ( t 0 F ∞,i,j u ∞,j ∂ i φ R dx ds + t 0 F ∞,i,j ∂ i u ∞,j φ R dx ds).
As u n k = u 0,n k + t 0 ∂ t u n k ds, we observe that u n k (t, .) converges to u ∞ (t, .) in D (R 3 ), thus, it converges weakly in L 2 loc (R 3 ) and we find:

|u ∞ (t
+ 3 ∑ i=1 t 0 [( |u ∞ | 2 2 + |b ∞ | 2 2 )u ∞,i + p ∞ u ∞,i ]∂ i φ R dx ds + 3 ∑ i=1 t 0 [(u ∞ • b ∞ )b ∞,i ]∂ i φ R dx ds -∑ 1≤i,j≤3 ( t 0 F ∞,i,j u ∞,j ∂ i φ R dx ds + t 0 F ∞,i,j ∂ i u ∞,j φ R dx ds).
Letting t goes to 0,

lim sup t→0 (u ∞ , b ∞ )(t, .) 2 L 2 (φ R (x)dx) ≤ (u 0,∞ , b 0,∞ ) 2 L 2 (φ R (x)dx) .
Reciprocally, by weak convergence

(u 0,∞ , b 0,∞ ) 2 L 2 (φ R (x)dx) ≤ lim inf t→0 (u ∞ , b ∞ )(t, .) 2 L 2 (φ R (x)dx) .
and ∇( ˜un , ˜bn ) 2 B 2 L 2 (0,λ -2n T T) ≤ C(1 + ( ũ0,n T , b0,n T ) 2 B 2 + F n T 2 B 2 L 2 (0,λ -2n T T) ).

These estimates gives uniforms controls for u n and b n , as ( ˜un , ˜bn )(t) 2 B 2 ≥ λ -n T (u n , b n )(λ 2n T t, .) 2 B 2 , and ∇( ˜un , ˜bn ) 2 B 2 L 2 (0,λ -2n T T) ≥λ -n T ∇(u n , b n ) 2 B 2 L 2 (0,T) .

To finish, we observe that we have controlled uniformly u n , b n on (0, T) for n ≥ n T . Thus, we are able to apply Theorem 9 and we get a solution on (0, T). As T > 0 is an arbitrary time, a diagonal permits to obtain a solution (u, b, p) on (0, +∞). The control in the statement of the theorem for the solution (u, b, p) is given by Theorem 15, and thus the proof is finished.

Proof of Remark 6.1. We detail the computations for u 0,n and F n . First, we observe that Navier-Stokes equations in B 2 2 have been recently discussed by Bradshaw, Kukavica and Tsai [START_REF] Bradshaw | Existence of global weak solutions to the Navier-Stokes equations in weighted spaces[END_REF].

u 0,n 2 B 2 λ n = sup R≥1 1 λ n R 2 |x|≤R |λ n u 0 (λ n x)| 2 dx = sup R≥1 1 (λ n R) 2 |x|≤λ n R |u 0 (x)| 2 dx,
The case d = 2 is more intricate. Indeed, while the Leray projection operator is bounded on B 2 2 (R 3 ), this is no longer the fact on B 2 2 (R 2 ), in which case one must be careful in the handling of the pressure. Basson treat this case to find local solutions of Navier-Stokes equations in his Ph. D. thesis in 2006 [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF].

Solutions in local Morrey spaces in dimension 2

We indicate here how to treat the pressure and moreover, we give a sketch of the proof for the local and global existence of weak suitable solutions of the (MHD) system. It is worth remark that for the (NS) equations, A. Basson obtained in his Ph. D. thesis [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF] the local existence of weak solutions with initial data in B 2 (R 2 ). Thus, our main contribution is the study of global weak solutions in the generalized setting of the (MHD) system.

The main idea in [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF] consists in giving a useful decomposition for the pressure. First, we fix some notation. We denote R the vector field of the Riesz transforms and we write H i,j the kernel of the operator R i R j and H = (H i,j ). Consider ϕ ∈ D(R 2 ) a non negative function supported on B(0, 2) such that ϕ = 1 on B(0, 1). For each k ∈ N, we define the functions ψ k (x) = ϕ(2 -k-1 x)ϕ(2 -k x) and χ k = ϕ(2 -k-3 x)ϕ(2 -k+2 x). Then, ψ k (x) = 1 for all 2 k-1 ≤ |x| ≤ 2 k+3 , sup(χ k ) ⊂ {x ∈ R 2 : 2 k-2 ≤ |x| ≤ 2 k+4 }, and sup(ψ k ) ⊂ {x ∈ R 2 : 2 k ≤ |x| ≤ 2 k+2 }.

For an index-family A, consider (u α ) α∈A and (b α ) α∈A two families of time dependent vector fields defined on [0, T) × R 2 , and consider a family (F α ) α∈A of tensors defined on [0, T) × R 2 . We denote for each α ∈ A, As usual, we will consider an approximated solutions (u n , b n , p n ). After, using the local energy balance where we split the term p n as in the expressions above, we will find an uniform bound on the approximated solutions. Passing to the limit we will able to exhibit a solution (u, b, p) of the (MHD) system.

A α = u α ⊗ u α -b α ⊗ b α -F α , ( 6
Theorem 17 (Local and global solutions). Let 0 < T < +∞. Let u 0 , b 0 ∈ B 2 (R 2 ) be divergence-free vector fields. Let F be a tensor belonging to B 2 L 2 (0, T). Then, there exists a time 0 < T 0 < T such that the system (MHD) has a solution (u, b, p) which satisfies :

• u, b belong to L ∞ ((0, T 0 ), B 2 ) and ∇u, ∇b belong to B 2 L 2 (0, T 0 ).

• The pressure p is related to u, b and F as follows. Let φ ∈ D(R 2 ) be a test function such that φ(x) = 1 on a neighborhood of the origin. We define

Φ i,j, φ = (1 -φ)∂ i ∂ j G 2 .
where G 2 = 1 2π ln( 1 |x| ) is a fundamental solution of the operator -∆ (we have -∆G 2 = δ 0 ). Then p and can be defined by : p φ(t, x) = ∑ i,j ( φ∂ i ∂ j G 2 ) * (u i u jb i b j -F i,j )(t, x) + ∑ i,j (Φ i,j, φ(xy) -Φ i,j, φ(-y)) u i (t, y)u j (t, y) -b i (t, y)b j (t, y) -F i,j (t, y) dy (6.19)

• The map t ∈ [0, T) → (u(t, •), b(t, •)) is * -weakly continuous from [0, T) to B 2 (R 2 ), and for all compact set K ⊂ R 2 we have: • The solution (u, b, p) is suitable : there exists a non-negative locally finite measure µ on (0, T) × R 2 such that:

∂ t ( |u| 2 + |b| 2 2 ) =∆( |u| 2 + |b| 2 2 ) -|∇u| 2 -|∇b| 2 -∇ • [ |u| 2 2 + |b| 2 2 + p]u + ∇ • [(u • b)b] + u • (∇ • F) -µ.
In particular we have the following global control on the solution: for all 0 ≤ t ≤ T 0 ,

max{ (u, b)(t) 2 B 2 , ∇(u, b) 2 B 2 L 2 (0,T 0 ) } ≤ C (u 0 , b 0 ) 2 B 2 + C F 2

Sketch of the proof.

The key point is to provide a priori controls for the following approximated solutions. Let φ R as in (6.14). Basson proves in [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF]) that one may take a sequence R n → +∞ such that P(φ R n u 0 ) converges *-weakly to u 0 and P(φ R n b 0 ) converges *-weakly to b 0 in B 2 . We define u 0,n = P(φ R n u 0 ), b 0,n = P(φ R n b 0 ), F n = φ R n F. Then, there exists a unique solution (u n , b n ) of the approximated system (MHD n ):

(MHD n )              ∂ t u n = ∆u n -(u n • ∇)u n + (b n • ∇)b n -∇p n + ∇ • F n , ∂ t b n = ∆b n -(u n • ∇)b n + (b n • ∇)u n , ∇ • u n = 0, ∇ • b n = 0, u n (0, •) = u 0,n , b n (0, •) = b 0,n ,
which belongs to L ∞ ((0, +∞), L 2 (R 2 )) ∩ L 2 ((0, +∞), Ḣ1 (R 2 )), and even, belongs to C([0, +∞), L 2 (R 2 )).

Next, we denote ∇p n = ∇p n,1 + ∇p n,2 , where p n,1 and ∇p n,2 are given by (6.17) and (6.18).

We make use of a technical lemma which we will prove later.

Lemma 6.8. Let F = (F i,j ) 1≤i,j≤2 ∈ L 1 loc be a tensor. Then we have We get back to (6.16) (and we set α = (n, i, j) ∈ N × {1, 2} × {1, 2}), using this lemma we find

∇[(1 -χ k )R i R j (ψ k F i,j )] L ∞ (R 2 ) ≤ C ψ k (
∇[(1 -χ k )R i R j (ψ k A n,i,j )] L ∞ (R 2 ) ≤ C ψ k (|u n | 2 + |b n | 2 ) (1 + |y|) 3 dy + C ψ k |F n | (1 + |y|) 3 dy ≈ 2 -3k ψ k (|u n | 2 + |b n | 2 )dy + 2 -3k ψ k |F n |dy.
After summation over k and using the Hölder inequality in the term with the forcing tensor, we get for 2 < γ 0 < 4, (6.22) We consider the solutions (u n , b n , p n ) of the approximated system (MHD n ), but, for simplicity, we will get rid of the index n and we shall just write (u, b, p). This part of the proof seems to the case of dimension 3, for this reason we only detail the main computations.

+ C k 0 +1 ∑ k=1 |x|≤2R |χ k R ⊗ R(ψ k (A n ))| 3/2 dx ≤C |x|≤2R |ϕ(u n ⊗ u n + b n ⊗ b n )| 3/2 dx + C k 0 +1 ∑ k=1 |x|≤2R |ψ k (u n ⊗ u n + b n ⊗ b n )| 3/2 dx + C |ϕF n | 3/2 + C k 0 +1 ∑ k=1 |ψ k F n | 3/2 ≤C |x|≤2 5 R |u n | 3 + |b n | 3 dx + C |x|≤2 5 R |F n | 3/2 dx.
We define α η,t 0 ,t 1 (t) as in (5.7), and φ R as in (6.14

+ 2 ∑ i=1 (u • b)b i α η,t 0 ,t 1 ∂ i φ R dx ds -∑ 1≤i,j≤2
F i,j u j α η,t 0 ,t 1 ∂ i φ R dx ds -∑ 1≤i,j≤2 F i,j ∂ i u j α η,t 0 ,t 1 φ R dx ds.

We divide this expression by R 2 , we use ∇φ R L ∞ ≤ c/R and ∆φ R L ∞ ≤ c/R 2 , and for the term involving F, we apply the Cauchy-Schwarz inequalities and the Young inequalities. Then, there exist two constants C 0 > 0 and C = C(C 0 ) > 0, where C 0 is arbitrarily small, such that To control the term (a) we use the decomposition ∇p = ∇p 1 + ∇p 2 where p 1 and ∇p 2 are always given by the equations (6.17) and (6.18). Recalling that div(u) = 0, we may write

φ R u • ∇p dx = -p 1 u • ∇φ R dx + φ R u • ∇p 2 dx.
The following fact is now useful : by the interpolation inequalities we have (6.24) Using (6.21) and (6.22), and (6.24), we obtain for 2 < γ 0 < 4 and 2 < γ 1 < 10/3, we obtain

|x|≤2 5 R |u| 3 dx ≤ ϕ 2 5 R u 3 L 3 ≤ C ϕ 2 5 R u 2 L 2 ∇(ϕ 2 5 R u) L 2 ≤ C ϕ 2 5 R u 3 L 2 + C ϕ 2 5 R u 2 L 2 ϕ 2 5 R ∇u L 2 .
1 R 2 φ R u • ∇p dx ≤ C R 3 |x|≤2R (|u| 3 + |p 1 | 3/2 )dx + C ∇p 2 ∞ R φ R |u| 2 dx 1/2 ≤ C R 3 |x|≤2 5 R (|u| 3 + |b| 3 )dx + C |F| 3/2 (1 + |x|) 3 dx + C R ( u 2 B 2 + b 2 B 2 ) φ R u L 2 + C R 2 φ R u 2 L 2 + C |F| 2 (1 + |x|) γ 0 dx ≤ C R 3 ϕ 2 5 R u 3 L 2 + C R 3 ϕ 2 5 R u 2 L 2 ϕ 2 5 R ∇u L 2 + C |F| 2 (1 + |x|) γ 1 dx 3/4 + C R 3 ϕ 2 5 R b 3 L 2 + C R 3 ϕ 2 5 R b 2 L 2 ϕ 2 5 R ∇b L 2 + C R ( u 2 B 2 + b 2 B 2 ) |x|≤2R |u| 2 dx 1/2 + C R 2 φ R u 2 L 2 dx + C |F| 2 (1 + |x|) γ 0 dx.
We can inject this estimate on term (a), to (6.23). Then, we find

+ C R 3 ( ϕ 2 5 R u 3 L 2 + ϕ 2 5 R b 3 L 2 )α η,t 0 ,t 1 ds + C ( u 3 B 2 + b 3 B 2 )α η,
+ C 0 R 2 t 1 t 0 |x|≤2 5 R
|∇u| 2 + |∇b| 2 dx ds.

In this inequality, the continuity at 0 of the map t ∈ [0, T) → (u, b)(t) ∈ L 2 loc (R 2 ) permit to let t 0 goes to zero. Additionally, by the *-weak continuity of this map we can let t 1 goes to t ∈ (0, T). We thus conclude that for all t ∈ (0, T),

1 R 2 ( |u(t)| 2 + |b(t)| 2 2 )φ R dx + 1 R 2 t 0 (|∇u| 2 + |∇b| 2 ) φ R dx ≤C( u 0 2 B 2 + b 0 2 B 2 ) + C F 2 B 2 L 2 (0,t) + C G 2 B 2 L 2 (0,t) + C t 1 t 0 1 + u 4 B 2 + b 4 B 2 ds + C 0 R 2 t 0 |x|≤2 5 R (|∇u| 2 + |∇b| 2 ) dx ds.
From this estimate and using the Young inequalities, we obtain the global energy control (6.20) for the approximated solutions.

With these uniform controls, we may follow the ideas in [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF] to obtain a subsequence (u n k , b n k , p n k ) which converges in the sense of distributions to a local solution (u, b, p) of the (MHD) system on [0, T 0 ], where

T 0 ≈ 1 1 + (u 0 , b 0 ) 2 B 2 + F 2 B 2 L 2 (0,+∞)
.

We observe that the pressure term is given by the formula ∇p = ∇p 1 + ∇p 2 , with p 1 = lim k→∞ p n k ,1 and ∇p 2 = lim k→∞ ∇p n k ,2 . We obtain p 1 and ∇p 2 satisfy (6.17) and (6.18) (with u α = u). Additionally, by Theorem 3 (with d = 2) we get that p can be written as in (6.19).

It remains to study the existence of global solutions. This part is obtained in the same way as for dimension 3, see Section 6.4.4. The only difference in the case of dimension 2 is that, we use the Lemma 6.7, with b = 2.

Proof of Lemma 6.8

We proceed as in [START_REF] Basson | Solutions spatialement homogènes adaptées des équations de Navier-Stokes[END_REF]. In dimension 2, for the kernel H we have By the localization properties of ϕ we have that (1ϕ(x/8))ϕ(y) = 0 and ∇ϕ(x/8)ϕ(y) = 0, so the second estimate in the statement of this lemma follows in the same way.

  Leray P(w) de w défini par Pw = w -∇p ϕ où ∇p ϕ est l'unique solution de -∆∇p = -∇(∇ • w) telle que lim τ→+∞ e τ∆ ∇p = 0, est un champ des vecteurs à divergence nulle.

  est défini comme le sous-espace de toutes les fonctions u ∈ B p γ telles que

  L p (0, T) is the subspace of all functions u ∈ B p γ L p (0, T) such that lim , x)| p dx dt = 0. The following result highlights how B p γ is strongly linked with the weighted spaces L p w γ = L p (w γ dx), with w γ = (1 + |x|) -γ , which are considered in (Fernández-Dalgo and Jarrín, 2021b; Fernández-Dalgo and Lemarié-Rieusset, 2020b). Lemma 6.5. Let γ ≥ 0 and γ < δ < +∞. We have the continuous embedding for 0 < T ≤ +∞, L p ((0, T), L p w γ ) ⊂ B p γ,0 L p (0, T) ⊂ B p γ L p (0, T) ⊂ L p ((0, T), L p w δ ).

  + |x|) γ dx dt, by dominated convergence we can take the limit when n → +∞ and conclude.Thereafter, we have the following result involving the interpolation theory of Banach spaces: Theorem 13. B p γ can be found by interpolation: for all 0

Corollary 6. 1 .

 1 If δ ∈ (0, d) and p ∈ (1, +∞), then :• The Riesz transforms R j are bounded on the space B

  x)| 2 dx ds = 0.

  .16) and we define the terms p α,1 and ∇p α,2 by the formulasp α,1 = ϕ(x/8)R ⊗ R(ϕ(A α )) + +∞ ∑ k=1 χ k R ⊗ R(ψ k (A α )), (6.17) ∇p α,2 =∇[(1ϕ(x/8))R ⊗ R(ϕ(A α ))] χ k )R ⊗ R(ψ k (A α ))]. (6.18)

  t, •)u 0 , b(t, •)b 0 ) L 2 (K) = 0.

  x)| 2 + |b 0 (x)| 2 dx = 0, , x)| 2 dx dt = 0,then we get a global solution (u, b, p).

  y)|F(y)| (1 + |y|) 3 dy ≈ C2 -3k ψ k (y)|F(y)| dy. and ∇[(1ϕ(x/8))R i R j (ϕF i,j )] L ∞ (R 2 ) ≤ C ϕ(y)|F(y)| dy

  ∇p n,2 L ∞ < C |u n | 2 + |b n | 2 (1 + |x|) 3 dy + C |F n | 2 (1 + |x|) ∇p n,2 L ∞ is uniformly bounded.Now, we study the term p n,1 . Let R ≥ 1 fix, and we take k 0 ∈ N such that 2 k 0 -1 ≤ 2R ≤ 2 k 0 . Then, by the localization of the function χ k ,|x|≤2R |p n,1 | 3/2 dx ≤C |x|≤2R |ϕ(x/8)R ⊗ R(ϕ(A n ))| 3/2 dx

  |H(x)| ≤ C |x| 2 and |∇H(x)| ≤ C |x| 3 . By the localization properties of the functions ψ k and χ k , we obtain for all k ∈ N, for |x -y| < 1+|y| 16 ,(1χ k )(x)ψ k (y) = 0 and ∇χ k (x)ψ k (y) = 0.Chapter 6. The incompressible magneto-hydrodynamics equationsThen we findR i R j (ψ k F i,j )(x) = |x-y|> 1+|y| 16 H i,j (xy)ψ k (y)F i,j dy,and|∇[(1χ k )R i R j (ψ k F i,j )](x)| ≤|∇(1χ k )(x) |x-y|> 1+|y| 16 H i,j (xy)ψ k (y)F i,j dy | + |(1χ k )(x) |x-y|> 1+|y| 16 ∇H i,j (xy)ψ k (y)F i,j dy | ≤C2 -k ψ k (y)|F(y)| (1 + |y|) 2 dy + C ψ k (y)|F(y)| (1 + |y|) 3 dy ≈C ψ k (y)|F(y)| (1 + |y|) 3 dy ≈ C2 -3k ψ k (y)|F(y)| dy.

  dz = +∞, as for |k -x| ≥ |k| 2 we have |k -x| p ≥ c k |k -x|, for some constant c k > 0, we deduce |k-x|≥ |k| dz = +∞, as the condition |k -x| < |k| 2 implies |x| > |k| -|x -k| > |x -k|, we deduce |k-x|≥ |k|

	|k-x| (1+|z|) 2 not belong to L If |k-x|≥ |k| 2 p w d (R d ) which is a contradiction. |k-x| p (1+|z|) d dz = +∞ and then y = k -x does
	|k-x| (1+|z|) 2 w d (R d ) which is a contradiction. If |k-x|< |k| 2 L 1	|x| (1+|z|) d dz = +∞ and then x does not belong to
	1 w d (R d )) and u belongs to L 1 ((0, T), L that this polynomial is identically equal to a constant k = (k 1 , ..., k d ) ∈ R d and be-p w d (R d )), we have longs to L 1 p w d (R d ), and now we prove that k is zero. w d (R d ) + L We suppose that k is non zero. We have k = x + y, with x ∈ L 1 w d (R d ) and y ∈ L p w d (R d ). As x ∈ L 1 w d (R d ) and k does not belong to L 1 w d (R d ) we have y = k -x does not
	belong to L 1 w d (R d ), thus we have two cases : |k-x|≥ |k| 2 |k-x|< |k| 2 |k-x| (1+|z|) d dz = +∞.	|k-x| (1+|z|) d dz = +∞ or

d d

  Using the local energy balance, we obtain:|u n (t, x)| 2 + |b n (t, x)| 2 2 φ R dx + t 0 (|∇u n | 2 + |∇b n | 2 ) φ R dx ds ≤ |u 0,n (x)| 2 + |b 0,n (x)| 2 2 φ R dx + + p n u n,i ]∂ i φ R dx ds • b n )c n,i + q n b n,i ]∂ i φ R dx ds (|∇u n k | 2 + |∇b n k | 2 ) φ R dx ds

					0	t	|u n | 2 + |b n | 2 2	∆φ R dx ds
	+ )v n,i + 3 ∑ i=1 t 0 [( |u n | 2 2 + |b n | 2 2 3 t ∑ i=1 0 [(u n -∑ (
		1≤i,j≤3
	Therefore,
	lim sup n k →+∞	|u n k (t, x)| 2 + |b n k (t, x)| 2 2	φ R dx +		0	t
	≤	|u 0 (x)| 2 + |b 0 (x)| 2 2	φ R dx +	0	t	|u ∞ | 2 + |b ∞ | 2 2	∆φ R dx ds

t 0 F n,i,j u n,j ∂ i φ R dx ds + t 0 F n,i,j ∂ i u n,j φ R dx ds).

  , x)| 2 2 φ R dx ≤ lim sup of course, a similar estimates hold for b ∞ . Then, we get |u ∞ (t, x)| 2 + |b ∞ (t, x)| 2 2 φ R dx + t 0 (|∇u ∞ | 2 + |∇b ∞ | 2 ) φ R dx ds ≤ |u 0 (x)| 2 + |b 0 (x)| 2 2 φ R dx +

				0	t	|u ∞ | 2 + |b ∞ | 2 2	∆φ R dx ds
						n k →+∞			|u n k (t, x)| 2 2	φ R dx.
	Moreover, by weakly convergence we know that
	0	t	|∇u ∞ (s, x)| 2 2	φ R dx ds ≤ lim sup n k →+∞	0	t	|∇u n k (s, x)| 2 2	φ R dx ds,

  ). The local energy balance gives∂ t ( |u| 2 + |b| 2 2 )α η,t 0 ,t 1 φ R dx ds + |∇u| 2 + |∇b| 2 α η,t 0 ,t 1 φ R dx ds )u i α η,t 0 ,t 1 ∂ i φ R dx ds + u • ∇pφ R dx ds

	≤		|u| 2 + |b| 2 2	α η,t 0 ,t 1 ∆φ R dx ds
	+	2 ∑ i=1	(	|u| 2 + |b| 2 2

  + |∇b| 2 ) α η,t 0 ,t 1 φ R dx ds + |∇b| 2 ) α η,t 0 ,t 1 dx ds.

	1 R 2 (|∇u| 2 ≤ ∂ t ( |u| 2 + |b| 2 2 )α η,t 0 ,t 1 φ R dx ds + 1 R 2 C R 2 |x|≤2R (|u| 2 + |b| 2 )α η,t 0 ,t 1 dx ds + 1 R 3 |x|≤2R ( |u| 3 + |b| 2 |u| 2	)α η,t 0 ,t 1 dx ds
	+	2 R 2	φ R (u • ∇p)α η,t 0 ,t 1 dx ds
				(a)
	+	C R 2 |x|≤2R	|F| 2 α η,t 0 ,t 1 dx ds
	+	C 0 R 2 |x|≤2R	(|∇u| 2 (6.23)

1

  R 2 ∂ t ( |u| 2 + |b| 2 2 )α η,t 0 ,t 1 φ R dx ds + 1 R 2 (|∇u| 2 + |∇b| 2 ) α η,t 0 ,t 1 φ R dx ds ≤ C R 2 |x|≤2R (|u| 2 + |b| 2 )α η,t 0 ,t 1 dx ds

  t 0 ,t 1 ds |x|) γ 1 α η,t 0 ,t 1 dx ds + C |F| 2 (1 + |x|) γ 0 α η,t 0 ,t 1 dx ds + C R 2 |x|≤2R |F| 2 α η,t 0 ,t 1 ds ds + C 0 R 2 |x|≤2R (|∇u| 2 + |∇b| 2 ) α η,t 0 ,t 1 dx ds.At this moment, we let η goes to 0. If we take t 0 and t 1 two Lebesgue points of thefunction A R (s) = (|u(s, x)| 2 + |b(s, x)| 2 )φ R (x) dx, we get -|u| 2 + |b| 2 2 ∂ t α η,t 0 ,t 1 φ R dx ds = -1 2 ∂ t α η,t 0 ,t 1 A R (s) ds, R (t 1 ) -A R (t 0 )),

	+ + + C C R 4 C R 4 (1 + and ϕ 2 5 R u 4 L 2 α η,t 0 ,t 1 ds + ϕ 2 5 R b 4 L 2 α η,t 0 ,t 1 ds + 1 + |F| 2 lim η→0 -|u| 2 + |b| 2 2 ∂ t α η,t 0 ,t 1 φ R dx ds = C 0 R 2 ϕ 2 5 R ∇u 2 L 2 α η,t 0 ,t 1 ds C 0 R 2 ϕ 2 5 R ∇b 2 L 2 α η,t 0 ,t 1 ds 1 2 1 R 2 ( |u(t 1 )| 2 + |b(t 1 )| 2 2 -|u(t 0 )| 2 + |b(t 0 )| 2 2 )φ R dx + 1 R 2 t 1 t 0 (|∇u| 2 + |∇b| 2 φ R ) dx ds ≤C t 1 t 0 1 + u 4 B 2 + b 4 B 2 ds (A so that + C F 2 B 2 L 2 (t 0 ,t 1 )

Thus, in the Hilbert space L 2 (φ R (x)dx), we obtain strong convergence to the initial data.

B 2 L 2 (0,T 0 )

Remerciements

gratitude à mon tuteur Pierre Gilles Lemarié-Rieusset pour sa générosité, sa patience et aussi pour son soutien constant surtout en cette période difficile de crise sanitaire.

Cela a été un privilège de soutenir cette thèse devant les membres du jury dont les noms figurent sur la couverture de cette thèse, j'apprécie particulièrement les suggestions pour

Proof of Theorem 14

Local in time existence

Consider φ R (x) = φ( x R ) given in (6.14), we define u 0,R = P(φ R u 0 ), b 0,R = P(φ R b 0 ), F R = φ R F. We denote (MHD R, ) the following approximated problem

Additionally, this solution belongs to C([0, +∞), L 2 ) and fulfills the hypothesis of Theorem 15. We apply this theorem (for the case (v, c) = (u * θ , b * θ )) to obtain uniform controls. More precisely, there exists a constant C > 0 such that for T 0 satisfying

Now, in the setting of Theorem 9, we denote (u 0,n , b 0

Letting R n → +∞ and n → 0 we thus obtain a local solution of the (MHD) system satisfying the properties stated in Theorem 14.

Global in time existence

Let λ > 1. For each n ∈ N we consider the (MHD) system with initial value

and forcing tensor

By the local in time existence, there exists a solution ( ũn , bn ) on (0, T n ), where

Now, we use the scaling of the (MHD) system :

where (u n , b n ) is a solution of the (MHD) on (0, λ 2n T n ) associated with the initial value (u 0 , b 0 ) and forcing tensor F. Now, we use the following simple remark, which will be proved at the end of this section.

Therefore, we have lim n→+∞ λ 2n T n = +∞.

Consider T > 0 arbitrary. Let n T such that for all n ≥ n T , λ 2n T n > T. Thus, (u n , b n ) is a solution of the (MHD) equations on (0, T).

We denote ( ˜un (t, x), ˜bn (t, x)) = (λ n T u n (λ 2n T t, λ n T x), λ n T b n (λ 2n T t, λ n T x)). Observe that for n ≥ n T , ( ˜un , ˜bn ) is a solution of the (MHD) system on (0, λ -2n T T) with initial value ( ũ0,n T , b0,n T ) and forcing tensor F n T . Then, since we have λ -2n T T ≤ T n T , we find

and by Theorem 15 we obtain: Abstract : In this thesis, energy estimates for the Navier-Stokes equations are studied, in a sufficiently robust context to be applied to the construction of discretely self-similar solutions for initial data satisfying the weak condition to be locally square integrable. These ideas also are applied to construct regular axially symmetrical solutions without swirl, for initial data which together with his gradient belong to a weighted L 2 space, where the weight allows to consider functions which tend to infinity in a piece of the space R 3 . More specifically an example is given by consider an axisymmetric initial velocity without swirl such that both the initial velocity and its vorticity belong to L 2 ((1 + r 2 ) -γ/2 dx), with r = x 2 1 + x 2 2 and γ ∈ (0, 2).

Universit é Paris Saclay

91190 Saint-Aubin, France