Keywords: sesquilinear forms, parabolic equation, maximal regularity, wave equation, Sobolev regularity, Besov regularity. Mots-clés formes sesquilinéaire, équation parabolique, régularité maximale, équation des ondes, régularité de Sobolev, régularité de Besov Preparatory lemmas . . . . . . . . . . . .

Cette thèse est dédiée a l'étude de certaines propriétés des équations d'évolutions non-autonomes u (t) + A(t)u(t) = f (t), u(0) = x. Il s'agit précisément de la propriété de la régularité maximale L p : étant donnée f ∈ L p (0, τ ; H), montrer l'existence et l'unicité de la solution u ∈ W 1,p (0, τ ; H). Ce problème a été intensivement étudié dans le cas autonome, i.e., A(t) = A pour tout t. Dans le cas non-autonome, le problème a été considéré par J.L.Lions en 1960. Nous montrons divers résultats qui étendent tout ce qui est connu sur ce problème. On suppose ici que la famille des opérateurs (A(t)) t∈[0,τ ] est associée à des formes quasi-coercives, non autonomes (a(t)) t∈[0,τ ] . Nous considérons également le problème de régularité maximale pour les équations d'ordre 2 (équations des ondes). Plusieurs exemples et applications sont considérés.

Introduction

Les méthodes des formes sesquilinéaire jouent un rôle important dans la théorie des équations d'évolution. Beaucoup d'équation parabolique s'écrivent comme un problème de Cauchy avec un opérateur associé á une forme sesquilinéaire. Sous des conditions classiques sur la forme, l'opérateur associé engendre un semi-groupe analytique et on obtient ainsi l'existence et l'unicité de la solution de l'équation parabolique. L'un des nombreux avantages de la méthode des formes est de pouvoir considérer des opérateurs elliptiques sous formes divergence et a coefficients non-réguliers. Elle permet aussi de considérer diverses conditions au bord telles que les conditions aux limites de Dirichlet, Neumann et Robin. En présence d'un terme source, les problèmes de Cauchy s'écrivent sous la forme    u (t) + Au(t) = f (t) u(0) = 0.

(CP)

Ces problèmes ont été profondément étudiés dans la théorie des semi-groupe sur les espaces de Banach. Par conséquent, il est trés naturel de prendre cette théorie comme point de départ pour la compréhension des problèmes non linéaires. Il est plus pratique d'écrire l'opérateur de la solution du problème linéaire L : u → u (t) + Au(t) comme la somme de deux opérateurs fermés B + A, ici B est l'opérateur de dérivation sur Z et A est l'opérateur de multiplication associé à A dans Z, où Z est l'espace des fonctions à valeurs dans l'espace de Banach X. Pour que L soit inversible, alors il faut que pour chaque f ∈ Z, il existe un unique u ∈ D(A) ∩ D(B) avec Bu + Au = f. Le cas le plus intéressant est quand Z = L p (0, ∞; X). On observe que si L est inversible alors pour f ∈ L p (0, ∞; X), on a les deux termes u et Au dans L p (0, ∞; X). En d'autres termes, cela signifie que les deux termes ont la même régularité que le côté droit f. Puisque c'est la meilleure régularité possible, on peut parler de la régularité maximale, ou plus précisement de la régularité maximale L p . Définition 0.0.1. On dit que A (ou le Problème (CP )) admet la régularité maximale L p (p ∈ (1, ∞)) si pour toute f ∈ L p (0, ∞; X), il existe une unique solution u du Problème (CP ) qui vérifie u ∈ W 1,p (0, ∞; X) ∩ L p (0, ∞; D(A)), où D(A) est le domaine de l'opérateur fermé A.

Beaucoup de travaux ont utilisé la régularité maximale pour l'étude de problème non linéaire de la forme    u (t) + Au(t) = F (u(t)) u(0) = 0, (NCP) où F (.) est un terme non linéaire. En effet, en choisissant un espace de Banach convenable et en fixant v ∈ W 1,p (0, ∞; X) ∩ L p (0, ∞; D(A)), on considère le problème linéaire    u (t) + Au(t) = F (v(t))

u(0) = 0.
La régularité maximale de ce problème permet d'avoir des estimations a priori. Celles-ci entraînent la continuité de l'opérateur S : v → u et on fait ensuite appel au théorème du point fixe. Il est bien connu qu'une condition nécessaire pour que A ait la régularité maximale est que -A soit le générateur d'un semi-groupe analytique. Notons que lorsque X = H un espace de Hilbert, de Simon [START_REF] De Simon | Un applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine[END_REF] a montré que tout générateur de semi-groupe analytique admet la propriété de la régularité maximale L p . Le problème formulé par H.Brézis en 1985 est de savoir si cette régularité maximale a lieu pour tout générateur d'un semi-groupe fortement continue et analytique dans X = L q (Ω). Kalton et Lancien [START_REF] Kalton | A solution to the problem of Lp-maximal regularity[END_REF] ont montré en 2000, en utilisant le semi-groupe de Poisson, que cela n'est vrai que dans l'espace de Hilbert, c'est à dire q = 2. En 2001, L.

Weis [START_REF] Weis | Operator-valued fourier multiplier theorems and maximal Lpregularity[END_REF] et aprés des travaux intensifs d'autre, a donné une caractérisation en terme de R-bornitude des résolvantes de l'opérateur ait la régularité maximale (Voir le premier Chapitre pour plus de détails et références).

L'objet de cette thèse est l'étude du problème de la régularité maximale dans les espaces de Hilbert des équations d'évolution non autonomes du premier ordre

   u (t) + B(t)u(t) = f (t) u(0) = x (P)
et du deuxième ordre Introduction maximale, c-à-d l'existence et l'unicité d'une solution pour les problèmes (P ) et (P ). Pour (P ) on cherche une solution u ∈ W 1,p (0, τ ; X) et pour (P ) une solution u ∈ W 2,p (0, τ ; X). Soit τ > 0 et V, H deux espaces de Hilbert tel que V → d H, c-à-d V s'injecte de façon continue et dense dans H. On dénote V l'espace (anti-) dual de V. On considère une famille de formes sesquilinéaires b(t), t ∈ [0, τ ] telle que

• [H1]: D(b(t)) = V (le domaine constant). • [H2]: |b(t, u, v)| ≤ M u V v V (la bornitude uniforme). • [H3]: Re b(t, u, u) + ν u 2 ≥ δ u 2 V (∀u ∈ V) pour certains δ > 0 et ν ∈ R (la quasi-coercivité uniforme).
Pour chaque forme b(t), on peut associer deux opérateurs B(t), B(t) dans H et V respectivement. L'opérateur B(t) est la part de B(t) dans H. Un résultat bien connu par J.L.Lions [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] affirme que le problème de Cauchy (P ) admet la régularité maximale L 2 dans V , c-à-d pour toute f ∈ L 2 (0, τ ; V ) et x ∈ H, il existe un unique u ∈ H 1 (0, τ ; V )∩L 2 (0, τ ; V) vérifiant (P ). La régularité maximale dans H est cependant plus intéressante car, quand il s'agit d'un problème aux limites on ne peut pas identifier les conditions aux limites si le problème de Cauchy est considéré dans V . La régularité maximale dans H est plus difficile à prouver. J.L.Lions [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] a prouvé que c'est le cas pour u 0 ∈ D(B(0)) sous une condition de régularité assez restrictive, à savoir t → b(t) est C 2 (ou C 1 si u 0 = 0). Lions a ensuite posé le problème de savoir si la régularité maximale L 2 dans H a lieu sans l'hypothèse de régularité C 2 ou C 1 des formes. Bardos [START_REF] Bardos | A regularity theorem for parabolic equations[END_REF] a amélioré le résultat de Lions, dans le sens que l'on peut prendre toute donnée initiale u 0 ∈ V. Bardos suppose que les formes satisfont la propriété de racine carrée de Kato (voir Définition 3.3.4) et que B(.) 1 2 est continument différentiable avec des valeurs dans L(V, V ). Beaucoup de progrés ont été fait ces derniéres années sur ce problème. Il a été prouvé par Ouhabaz et Spina [START_REF] Ouhabaz | Maximal regularity for non-autonomous Schrödinger type equations[END_REF] qu'on a la régularité maximale L p dans H, si t → b(t) est C α pour un α > 1 2 . Ce résultat est cependant prouvé pour le cas u 0 = 0. La preuve dans [START_REF] Ouhabaz | Maximal regularity for non-autonomous Schrödinger type equations[END_REF], est utilise le résultat de Hieber-Monniaux [START_REF] Hieber | Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations[END_REF] sur les équations d'évolutions non-autonomes satisfont la condition d'Aquistapace-Terreni. Haak et Ouhabaz [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] ont prouvé que pour u 0 ∈ (H, D(B(0))) 1-1 p ,p et |b(t; g, h) -b(s; g, h)| ≤ w(|t -s|) h V g V , pour certaine fonction croissante w : [0, τ ] → [0, ∞), telle que

τ 0 w(t) t 3 2
dt < ∞, alors le problème de Cauchy (P) admet la régularité maximale L p dans H. Dier [START_REF] Dier | Non-Autonomous Cauchy Problems Governed by Forms[END_REF] a observé que la réponse au problème de Lions est négative en général. Son exemple est basé sur des formes non symétriques pour lesquelles la propriété de la racine carrée uniforme de Kato n'est pas satisfaite. Dier [START_REF] Dier | Non-autonomous maximal regularity for forms of bounded variation[END_REF] a prouvé aussi qu'on a la régularité maximale L 2 pour des formes symétriques telle que t → b(.) est à variation bornée. Fackler [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF] a montré que α > 1 2 dans [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] est optimal dans le sens où il existe b(.) une forme symétrique et C 1 2 en temps pour lequel la régularité maximale dans H n'est pas satisfaite. Dier et Zacher [START_REF] Dier | Non-autonomous maximal regularity in Hilbert spaces[END_REF] ont prouvé que si t → B(t) est dans l'espace de Sobolev fractionnaire H 1 2 + (0, τ ; L(V, V )), pour un > 0 alors on a la régularité maximale L 2 . Pour une version d'espace de Banach de ce résultat, voir Fackler [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF]. Notons que l'exemple dans [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF] n'est pas un opérateur différentiel. Pour les opérateurs elliptiques sous forme divergence sur R n , Auscher et Egert [START_REF] Auscher | On non-autonomous maximal regularity for elliptic operators in divergence form[END_REF] ont montré la régularité maximale L 2 si les coefficients satisfont une certaine condition BM O -H 1 2 . La régularité maximale du problème d'ordre 2 a été aussi étudiée dans la littérature a la fois dans V et H. Comme pour le problème d'ordre 1, la régularité maximale dans H est plus intéressante dans les applications. La régularité maximale L p de (P ) dans H consiste à trouver u ∈ W 2,p (0, τ ; H) à condition que f ∈ L p (0, τ ; H). La première réponse à cette question dans le cas non-autonome a été donnée dans Batty, Chill et Srivastava [START_REF] Batty | Maximal regularity for second order non-autonomous Cauchy problems[END_REF] sous l'hypothèse que B(.) = kA(.) pour une constante k et que A(.) a la régularité maximale dans H. Dier et Ouhabaz [START_REF] Dier | Maximal regularity for non-autonomous second order Cauchy problems[END_REF] ont prouvé un premier résultat sans l'hypothèse assez forte B(.) = kA(.), mais que A(.) et B(.) associés avec des formes V-bornées quasi coercives a(.), b(.) respectivement et t → a(t, v, w); b(t, v, w) sont lipschitziennes pour tout v, w ∈ V. Notons aussi que Lions [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] a montré la régularité maximale dans V .

Dans cette thèse, nous nous proposons d'étudier le problème de régularité maximale de (P) et (P'). Nos résultats étendent ceux cités plus haut dans plusieurs directions.

Nous allons décrire nos contributions dans ce qui suit. Dans le deuxiéme chapitre nous étudions la régularité maximale L p pour les problèmes de Cauchy non autonomes perturbés

   u (t) + B(t)A(t)u(t) + P (t)u(t) = f (t) u(0) = x (P1) et    u (t) + A(t)B(t)u(t) + P (t)u(t) = f (t) u(0) = x, (P2)
où chaque opérateur A(t) associé à une forme V-bornée quasi-coercive, B(t) et P (t) sont deux opérateurs bornés tels que Re (B(t) -1 g, g) ≥ δ g 2 pour un δ > 0. Les deux problèmes sont motivés par des applications aux équations dt < ∞, alors (P1) admet la régularité maximale L p dans H, pour u 0 ∈ (H, D(A(0))) 1-1 p ,p . On montre aussi que si w(t) ≤ ct pour un > 0 et D(A(t)

1 2 ) = V pour tout t ∈ [0, τ ], alors la solution u ∈ C([0, τ ]; V) et s → A(s) 1 2 u(s) ∈ C([0, τ ]; H).
Concernant (P2), on suppose que t → B(t) est lipschitzienne continue sur [0, τ ] avec des valeurs dans L(H) et avec les mêmes hypothèses comme avant on obtient la régularité maximale L p dans H pour tout p ∈ (1, ∞). Notons que ces résultats généralisent ceux de Haak et Ouhabaz [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] et Augner, Jacob et Laasri [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF]. Ces derniers ont considéré les perturbations multiplicatives en supposant que a(t) est symétrique et t → a(t, u, v) ∈ C 1 , pour tout u, v ∈ V et ont étudié que la régularité maximale L 2 .

Dans le troisième chapitre nous étudions la régularité maximale L 2 pour le Problème (P). Notre résultat principal montre que pour des formes qui satisfont la propriété de la racine carrée de Kato uniforme et que t → A(.) est dans l'espace de Sobolev H 1 2 (0, τ ; L(V, V )) par morceaux alors la régularité maximale L 2 est satisfaite dans H. Dans le cas où A(t) -A(s) ∈ L(V, ([H, V] γ ) ) avec un γ ∈ (0, 1) il suffit alors de supposer que t → A(.) ∈ H γ 2 (0, τ ; L(V, V ). Il s'agit ici du meilleur résultat sur le problème de Lions et il est optimal, en utilisant le contre-exemple dans Arendt, Dier et Fackler [START_REF] Arendt | Lions' problem on maximal regularity[END_REF]. Les résultats précédents s'appliquent pour des opérateurs elliptiques dans R n de même que dans un domaine lipschitzien Ω, avec des conditions aux limites de Dirichlet, Neumann ou Robin. Nous considérons par exemple l'équation de la chaleur avec des conditions aux limites de Robin dépendantes du temps (voir Section 3.6).

Dans le chapitre 4 nous poursuivons notre étude sur la régularité maximale L p dans H avec p > 2. Il a été montré dans Fackler [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] que la famille des opérateurs (A(t)) t∈[0,τ ] vérifie la propriété de la régularité maximale L p si t → A(t) ∈ W 1 2 + ,p (0, τ ; L(V, V )) pour un > 0. Nous montrons la régularité maximale L p lorsque t ∈ A(t) est dans l'espace de Besov B 1 2 ,p 2 (0, τ ; L(V, V )). Notre résultat améliore ceux de [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] car

W 1 2 + ,p (0, τ ; L(V, V )) ⊂ B 1 2 ,p 2 (0, τ ; L(V, V ))
et il est optimal en utilisant toujours le contre-exemple dans [START_REF] Arendt | Lions' problem on maximal regularity[END_REF]. On prouve que

(voir Théorème 4.2.3) si A(.) ∈ B 1 2 ,p 2 (0, τ ; L(V, V )), P (.) ∈ L p (0, τ ; L(V, H)) et pour tout f ∈ L p (0, τ ; H), u 0 ∈ (H, D(A(0))) 1-1 p ,p il existe une unique solution du problème    u (t) + A(t)u(t) + P (t)u(t) = f (t) u(0) = u 0 , tel que u ∈ W 1,p (0, τ ; H) ∩ B 1 2 ,p 2 (0, τ ; V). De plus, u(t) ∈ (H, D(A(t))) 1-1 p ,p pour tout t ∈ [0, τ ].
Dans le chapitre 5 nous étudions la régularité maximale L p dans V et H pour le problème d'ordre 2 (P'). On prouve la régularité maximale et d'autres propriétés de la solution sachant que A(t) est un opérateur borné de V dans V et B(t) associé à une forme V-bornée quasi-coercive b(t). Dans ce chapitre aussi nous généralisons le résultat sur la régularité maximale pour le problème de Cauchy d'ordre 1 au problème de Cauchy d'ordre N avec N ∈ N * . En utilisant le contre-exemple dans Fackler [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF], nous obtenons que la régularité maximale dans H pour le problème (P') n'est pas satisfaite si B(t) = A(t) + I, où A(t) est un opérateur associé à une forme V-bornée quasi-coercive a(t) et t → a(t) ∈ C Introduction • Chapitre 5: Ce dernier chapitre, contient nos résultats sur la régularité maximale non-autonome pour le problème d'ordre 2, notamment l'équations des ondes amorties.

Chapter 1

Preliminaries

In this chapter we present some basic results on sesquilinear forms, semigroups and known results on maximal regularity.

Forms and their operators

In this section we recall some known results on forms, operators and semigroups which are frequently used in this thesis. For more details, see the monograph [START_REF] Ouhabaz | Analysis of Heat Equations on Domains[END_REF]. Let (H, (•, •), • ) be a separable Hilbert space over R or C. We consider another separable Hilbert space V which is densely and continuously embedded into H. We denote by V the (anti-) dual space of V so that

V → d H → d V .
Hence there exists a constant C > 0 such that

u ≤ C H u V u ∈ V, where • V denotes the norm of V. Similarly, ψ V ≤ C H ψ ψ ∈ H.
We denote by , the duality V -V and note that ψ

, v = (ψ, v) if ψ, v ∈ H. We consider a form a : V × V → C be sesquilinear and V-bounded, i.e. |a(u, v) ≤ M u V v V , (u, v ∈ V)
for some constant M > 0. The form a is called quasi-coercive if there exist constants ν ∈ R and δ > 0 such that Re a(u, u) + ν u 2 H ≥ δ u 2 V , (u ∈ V). If ν = 0 we say that the form a is coercive.

Preliminaries

Definition 1.1.1. Let a be a sesquilinear form V-bounded and quasi-coercive. The adjoint form of a is the sesquilinear form a * defined by

a * (u, v) = a(v, u), u, v ∈ V.
The symmetric part of a is defined by

b := 1 2 (a + a * ).
We say that a is a symmetric form if b = a (or a = a * ), that is

a(u, v) = a(v, u), u, v ∈ V. Definition 1.1.2. A sesquilinear form a : V × V → C, is called sectorial if there exists a non-negative constant C, such that |Im a(u, u)| ≤ C|Re a(u, u)|, u ∈ V. (1.1.1)
The numerical range of a is the set

N(a) = {a(u, u), u ∈ V, u = 1}.
Clearly, a satisfies (1.

N(a + νI) = {z ∈ C * , | arg z| ≤ arctan ( M δ )}.
Proof. Let u ∈ V, we have that

|Im (a + νI)(u, u)| ≤ |a(u, u)| ≤ M u 2 V ≤ M δ [Re a(u, u) + ν u 2 H ].
This proves the proposition.

Let a be a sesquilinear form V-bounded and quasi-coercive. The operator A ∈ L(V, V ) associated with a is defined by

Au, v = a(u, v), (u, v ∈ V).
Seen as an unbounded operator on V with domain D(A) = V. One can define also an unbounded operator A on H, it is the part of A on H, i.e.

D(A)

:= {v ∈ V : Av ∈ H} Av := Av.
Observe also that D(A) is the set of vectors u ∈ V for which the map v → a(u, v) is continuous on V with respect to the norm of H. The operators A and A are called the operators associated with a.

Forms and their operators

Proposition 1.1.4. Denote by A the operator associated with a sesquilinear V-bounded and quasi-coercive form a. Then A is densely defined and for every λ > ν, the operator λ + A is invertible (from D(A) into H) and its inverse (λ + A) -1 is a bounded operator on H.

Definition 1.1.5. A scalar λ ∈ C is in the resolvent set of A if λ -A is invertible (from D(A) into H ) and its inverse (λ -A) -1 is a bounded operator on H. For such λ, the operator (λ -A) -1 is called the resolvent of A at λ. The set ρ(A) := {λ ∈ C, λ -A is invertible and (λ -A) -1 ∈ L(H)} is called the resolvent set of A. The complement of ρ(A) in C is the spectrum of A.
Let θ ∈ (0, π) we define the sector

Σ θ = {z ∈ C * , | arg z| < θ} = {re iα , r > 0, |α| < θ}.
Definition 1.1.6. A semigroup on a Banach space E is a family of bounded linear operators (T (t)) t≥0 acting on H such that

T (0) = I and T (s + t) = T (s)T (t) for all s, t ≥ 0.
We say that a semigroup (T (t)) t≥0 is strongly continuous if for every x ∈ E, we have lim t→0

T (t)x = x.
Let (T (t)) t≥0 be a strongly continuous semigroup on E. The generator of (T (t)) t≥0 is the operator B defined by

D(B) := {x ∈ E, s.t lim t→0 T (t)x -x t exists}. Bx := lim t→0 T (t)x -x t .
(T (t)) t≥0 is called a bounded holomorphic semigroup on the sector Σ θ if (T (t)) t≥0 admits a holomorphic extension (T (z)) z∈Σ θ such that for each ψ ∈ (0, θ), (T (z)) z∈Σ ψ is uniformly bounded and strongly continuous at 0. Note that a holomorphic semigroup on the sector Σ θ satisfies

T (z + z ) = T (z)T (z ), for all z, z ∈ Σ θ .
Theorem 1.1.7. Let B be a densely defined operator on a complex Banach space E. Then B generates a semigroup which is bounded holomorphic on Σ θ if and only if Σ θ+ π 2 ⊂ ρ(B) and for every ψ ∈ (0, θ), one has sup

λ∈Σ ψ+ π 2 λ(λ -B) -1 L(E) < ∞.

Preliminaries

Proposition 1.1.8. Let a be sesquilinear form V-bounded and coercive. Denote by A the operator associated with a. Let θ = arctan M δ . Then Σ π-θ ⊂ ρ(-A) and there exists constants C θ , C θ > 0 depending on θ, such that

1-(λ + A) -1 L(H) ≤ C θ |λ| . 2-(λ + A) -1 L(H,V) ≤ C θ √ |λ| . Proof. Let u ∈ D(A), λ ∈ C. We get (λ -A)u u ≥ |((λ -A)u, u)| = |λ - (Au, u) u 2 | u 2 = |λ - a(u, u) u 2 | u 2 = |λ -a( u u , u u )| u 2 . Therefore (λ -A)u ≥ dist(λ, Σ arctan M δ ) u .
This implies that λ -A is injective and has closed range for λ / ∈ Σ arctan M δ . In order to prove that λ -A is invertible it remains to prove that it has dense range. By duality, one has to prove that the adjoint is injective and this true by the same argument as before. Therefore

(λ -A) -1 L(H) ≤ 1 dist(λ, Σ arctan M δ ) for all λ / ∈ Σ arctan M δ . Now we set θ = arctan M δ , then there exists a constant C θ such that (λ -A) -1 L(H) ≤ C θ |λ| .
In other words, λ + A is invertible for λ ∈ Σ π-θ and

(λ + A) -1 L(H) ≤ C θ |λ| . Now let x ∈ H and λ ∈ Σ π-θ . We have δ (λ + A) -1 x 2 V ≤Re (A(λ + A) -1 x, (λ + A) -1 x) ≤ A(λ + A) -1 x (λ + A) -1 x ≤ (1 + C θ ) C θ |λ| x 2 .
1.1. Forms and their operators

Therefore (λ + A) -1 L(H,V) ≤ C θ |λ| .
Proposition 1.1.9. Let A as in the previous proposition. Then -A is a generator of a bounded holomorphic contraction semigroup on H and we have 1-For all t ∈ (0, ∞), n ∈ N, there exists a constant C > 0 such that

A n e -tA L(H) ≤ C t n .
2-For all t ∈ (0, ∞),

e -tA L(H,V) ≤ C √ t .
Proof. Since Σπ A) and

2 +( π 2 -arctan M δ ) ⊂ ρ(-
(λ + A) -1 L(H) ≤ C θ |λ| , then by Theorem 1.1.7 -A is the generator of a bounded holomorphic semi- group on Σ ( π 2 -arctan M δ ) and for z ∈ Σ ( π 2 -arctan M δ )
we have e -zA x ∈ D(A) where x ∈ H. So for all x ∈ H we obtain ∂ ∂z e -zA x 2 = -2Re (Ae -zA x, e -zA x) < 0.

Therefore e -zA L(H) ≤ 1. For 1, we use the Cauchy's integral formula (Cauchy's differentiation formula). For all x ∈ H and t > 0 we get δ e -tA x 2

V ≤ Re a(e -tA x, e -tA x) = Re (Ae -tA x, e -tA x)

≤ Ae -tA x e -tA x ≤ C t .
This shows 2.

D(∆ D ) := {u ∈ H 1 0 (Ω) : ∆u ∈ L 2 (Ω)} ∆ D u = ∆u := d i=1 ∂ 2 u ∂x 2 i .

Preliminaries

We have that ∆ D is selfadjoint and generates a bounded holomorphic semigroup on L 2 (Ω). In fact, define a :

H 1 0 (Ω) × H 1 0 (Ω) → R by a(u, v) = Ω ∇u∇v.
Then clearly a is H 1 0 (Ω) bounded and coercive. Let A be the operator associated with a. We show that A = -∆ D . In fact, let u ∈ D(A) and we write f = Au.

Then Ω ∇u∇v = Ω f v for all v ∈ H 1 0 (Ω). Taking in partic- ular v ∈ C ∞ c (Ω), we get -∆u = f. Conversely, let u ∈ H 1 0 (Ω) be such that f := -∆u ∈ L 2 (Ω). Then Ω ∇u∇v = Ω f v = a(u, v) for all v ∈ C ∞ c (Ω)
. This is just the definition of the weak partial derivatives in

H 1 (Ω). Since C ∞ c (Ω) is dense in H 1 0 (Ω), it follows that Ω f v = a(u, v) for all v ∈ H 1 0 (Ω). Thus u ∈ D(A) and Au = f.

Fractional Powers

The fractional power A α with 0 < α < 1, can be defined by

A α = - sin πα π ∞ 0 µ α (µ + A) -1 dµ. (A1)
Let A be the operator associated with a V-bounded coercive sesquilinear form a. We consider 0 < α < 1 and the complex interpolation space [H, V] α .

Proposition 1.2.1. We have

(1)-V → D(A 1
2 ) if and only if D(A * 1 2 ) → V.

(

)-If A = A * , we get D(A 1 2 ) = D(A * 1 2 ) = V and √ δ u V ≤ A 1 2 u ≤ √ M u V . 2
(

)-D(A α ) = [H, V] 2α for all α < 1 2 . ( 3
)-D(A 1-α ) → V for all α < 1 2 . Proof. Let u ∈ D(A * ). If V → D(A 4
2 ) we get

u 2 V ≤ 1 δ Re (A 1 2 u, A * 1 2 u) ≤ 1 δ A 1 2 u A * 1 2 u ≤ C u V A * 1 2 u .
Then by the density of D(A * ) on D(A * 1 2 ) we obtain

u V ≤ C A * 1 2 u 1.2. Fractional Powers for all u ∈ D(A * 1 2 ). Then D(A * 1 2 ) → V. Now, we assume that D(A * 1 2 ) → V. It follows that A * -1 2 ∈ L(H, V). Let x ∈ H and we write A * 1 2 x = A * A * -1 2
x. Then we get

A * 1 2 x V ≤ A * L(V,V ) A * -1 2 x V ≤ M A * -1 2 L(H,V) x .
The boundedness implies A * 1 2 ∈ L(H, V ) and by duality we have A

1 2 ∈ L(V, H). Then V ⊆ D(A 1
2 ) and we get for all x ∈ V

x 2 D(A 1 2 ) = x 2 + A 1 2 x 2 H ≤ (C 2 H + A 1 2 2 L(V,H) ) x 2 V . Thus, V → D(A 1
2 ). This shows (1). We assume that A = A * . By the density of D(A) in V and D(A

1 2 ), we get for all u ∈ V δ u 2 V ≤ Re a(u, u) = A 1 2 u 2 ≤ M u 2 V .
This shows [START_REF] Achache | Lions' maximal regularity problem with H 1 2regularity in time[END_REF]. For (3), we refer to [START_REF] Kato | Fractional powers of dissipative operators[END_REF] (Theorem 3.1). Let α < 1 2 and u ∈ D(A). We have

u 2 V ≤ 1 δ A 1-α u A * α u ≤ 1 δ A 1-α u u [H,V] 2α ≤ C(α) δ A 1-α u u 2α V u 1-2α ,
where C(α) > 0 depending on α. Thus, for all u ∈ D(A 1-α ) we get

u V ≤ C 1-2α H C(α) δ A 1-α u .
This shows (4).

Remark 1.2.2. As a consequence from the previous lemma if

D(A * 1 2 ) = D(A 1 2 ), we have D(A 1 2 ) \ V is not empty or D(A * 1
2 ) \ V as well, where

D(A 1 2 ) \ V = {x ∈ D(A 1 2 ) s.t x / ∈ V}
and the same for D(A * 1 2 ) \ V.

Preliminaries

For the next two results we refer to [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF] (Theorem 4.3.5 and Proposition 5.1.1) Proposition 1.2.3. Let a be V-bounded coercive form and let A be the associated operator to a. Then the imaginary powers A it , t ∈ R, are bounded operators and

A it L(H) ≤ e π |t| 2 , t ∈ R.
Let E be a Banach space. In the next proposition we suppose that B is a generator of holomorphic semigroup. We denote by (E, D(B)) θ,p the classical real interpolation space.

Proposition 1.2.4. For 0 < θ < 1, 1 ≤ p ≤ ∞, we have (E, D(B)) θ,p = {x ∈ E : φ(t) = t 1-θ Be tB x E ∈ L p (0, ∞; dt t )} with norm x p (E,D(B)) θ,p = x p E + ∞ 0 φ(t) p E dt t .
Holomorphic semigroups and interpolation spaces play an important role in the theory of evolution equations. In particular, if the semigroup generated by B is holomorphic, then the problem

   u (t) = Bu(t) u(0) = x (1.2.1)
have a unique solution u ∈ W 1,p (0, τ ; E) ∩ L p (0, τ ; D(B)) for every initial data x ∈ (E, D(B)) 1-1 p ,p . In fact, it is very known that the solution of the Problem (1.2.1) is giving by u(t) = e tB x and u (t) = Be tB x. Therefore

u p L p (0,τ ;E) = τ 0 Be tB x p E dt ≤ x p (E,D(B)) 1-1 p ,p .

Maximal Regularity for autonomous problem in Hilbert space

Let H be an Hilbert space and A be a closed (unbounded) operator with domain D(A) dense in H. Let f : [0, ∞[→ H be a measurable function and x ∈ H. We consider the problem of existence and regularity of solution to the following equation

   u (t) + Au(t) = f (t) u(0) = x.
(1.3.1)

Maximal Regularity for autonomous problem in Hilbert space

We define the maximal regularity space

M R(p, H) = W 1,p (0, ∞; H) ∩ L p (0, ∞; D(A))
endowed with norm

u M R(p,H) = u W 1,p (0,∞;H) + Au L p (0,∞;H) .
We define the associated trace space by

T R(p, H) := {u(0) : u ∈ M R(p, H)},
with norm

x T R(p,H) = inf{ u M R(p,H) : u ∈ M R(p, H), u(0) = x}.
u M R(p,H) ≤ C[ x T R(p,H) + f L p (0,∞;H) ].
Proposition 1.3.2. If A has the maximal regularity property, then -A generates a bounded holomorphic semigroup on H.

Proof. The proof is taken from [START_REF] Monniaux | Maximal regularity and applications to PDEs[END_REF]. Suppose x = 0. Let z ∈ C with Re (z) > 0.

Define f z ∈ L p (0, ∞; C) by

f z (t) = e zt , t ∈ [0, 1 Re (z) ]. 0, t > 1 Re (z) .
Let y ∈ H and denote by u z the solution of the Cauchy problem (1.3.1) with f = f z ⊗ y. Define then

R z y = Re (z) ∞ 0 e -zt u z (t)dt.
Then the following estimates hold

f L p (0,∞;H) = ( 1 Re z ) 1 p y H and R z y H ≤ Re (z) u z L p (0,∞;H) e -z.
L p (0,∞)

≤ C (e p -1)

R z y = Re (z) z ∞ 0 e -zt u z (t)dt.
The same arguments as before give the estimate

R z y H ≤ 1 |z| C (e p -1) 1 p p 1 p y H .
Therefore, we get

R z y H ≤ M |z| + 1 y H . (1.3.2)
with M = C (e p -1)

1 p p 1 p
. Let y ∈ D(A). We have

R z (z + A)y = zR z y + R z Ay = Re (z) ∞ 0 e -zt u z (t)dt + Re (z) ∞ 0 e -zt Au z (t)dt = Re (z) ∞ 0 e -zt f z (t)dty = y.
The equality R z (z +A)y = y for all y ∈ D(A) together with (1.3.2) ensure that R z is the resolvent of -A in z. Therefore, the spectrum of σ(A) ⊆ C + = {z ∈ C, Re (z) ≥ 0} and there exists M > 0 such that for all z with Re (z) > 0, we have (1.3.2). This implies that -A generates a bounded analytic semigroup in H.

The next theorem shows maximal L p -regularity holds on the setting of Hilbert space. The theorem is due to de Simon [START_REF] De Simon | Un applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine[END_REF] and [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF].

Proof. First we prove that

T R(p, H) = (H, D(A)) 1-1 p ,p . Let x ∈ (H, D(A)) 1-1 p ,p
and we set u(t) = e -tA x, so

u (t) = -Ae -tA x. Since x ∈ (H, D(A)) 1-1 p ,p it follows that u ∈ M R(p, H). Therefore (H, D(A)) 1-1 p ,p ⊆ T R(p, H). Let x ∈ T R(p, H) so there is u ∈ M R(p, H) such that u(0) = x. We write x = u(t) - t 0 u (s)ds, with t ∈ (0, ∞). It follows that Ae -tA x = Ae -tA u(t) -Ae -tA t 0 u (s)ds = e -tA Au(t) -Ae -tA t 0 u (s)ds = (R 1 u)(t) + (R 2 u)(t).

Maximal Regularity for autonomous problem in Hilbert space

Using that the semigroup (e -sA ) s≥0 generated by -A is bounded holomorphic to obtain (R 1 u)(t) ≤ C Au(t) . Thus, R 1 u L p (0,∞;H) ≤ C u L p (0,∞;D(A)) . For R 2 , by the analyticity of the semigroups (e -sA ) s≥0 we obtain

(R 2 u)(t) ≤ Ae -tA L(H) t 0 u (s)ds ≤ C t t 0 u (s)ds .
By the Hardy's inequality we have

R 2 u L p (0,∞;H) ≤ C 1 u L p (0,∞;H) . Then ∞ 0 Ae -tA x p dt ≤ C u p M R(p,H) .
Thus, x ∈ (H, D(A)) 1-1 p ,p and so T R(p, H) ⊆ (H, D(A)) 1-1 p ,p . Let u is the solution of the Problem (1.3.1) if it exists and fix 0 ≤ s ≤ t ≤ τ. We set v(s) = e -(t-s)A u(s), since u(t) = v(t) and v(0) = e -tA u(0), then

u(t) = e -tA x + t 0 e -(t-s)A f (s)ds (1.3.3) = u 1 (t) + u 2 (t). (1.3.4)
So on order to prove the maximal regularity it is enough to prove that u 1 , u 2 ∈ M R(p, H). Since x ∈ (H, D(A)) 1-1 p ,p = T R(p, H) we have u 1 ∈ M R(p, H). Next we prove u 2 ∈ M R(p, H). First we prove the result for p = 2. In fact, let f ∈ L 2 (0, ∞; H) and we extend f by 0 in (-∞, 0) and let f be the extended function of f. Let F f be the Fourier transform of f and F -1 f be the inverse of the Fourier transform of f . We write

Au 2 (t) = A t -∞ e -(t-s)A f (s)ds = A t -∞ e -(t-s)A F -1 F f (s)ds = 1 2π A t -∞ e -(t-s)A R e iξs F f (ξ)dξds = 1 2π A R ( t -∞ e -(t-s)(iξ+A) ds)e itξ F f (ξ)dξ = 1 2π R e itξ A(iξ + A) -1 F f (ξ)dξ = F -1 (A(i. + A) -1 F f ).
Since -A is a generator of a bounded holomorphic semigroup on H, we get

A(iξ + A) -1 L(H) ≤ C, 1.
Preliminaries with ξ ∈ R and C > 0. Then by Plancherel's theorem we obtain

Au 2 L 2 (0,∞;H) = A(i. + A) -1 F f L 2 (R;H) ≤ C F f L 2 (R;H) ≤ C f L 2 (R;H) = C f L 2 (0,∞;H) .
This proves that A has the maximal L 2 -regularity property. Now, we set

(Lf )(t) = Au 2 (t) = A t 0 e -(t-s)A f (s)ds.
The operator L is a singular integral operator with operator-valued kernel

K(t, s) = I 0≤s≤t Ae -(t-s)A ,
where I denotes the indicator function. As we have L ∈ L(L 2 (0, ∞; H)) we prove that both L and L * are of weak type (1, 1) operators and we conclude by the Marcinkiewicz interpolation theorem that L ∈ L(L p (0, ∞; H)) for all p ∈ (1, ∞). It is known (see e.g. [START_REF] Rubio De Francia | Calderón-Zygmund theory for operator-valued kernels[END_REF]

K(t, s) -K(t; s ) L(H) dt ≤ C
for some constant C independent of s, s . In fact

|t-s|≥2|s -s| K(s, t) -K(s , t) L(H) dt ≤ |t-s|≥2|s -s| Ae -(s-t)A -Ae -(s -t)A L(H) dt ≤ |t-s|≥2|s -s| s-t s -t A 2 e -lA L(H) dldt ≤ |t-s|≥2|s -s| | s-t s -t C l 2 dl|dt ≤ C |t-s|≥2|s -s| | 1 t -s - 1 t -s |dt ≤ C |r|≥2|s -s| | 1 r - 1 r -(s -s) |dr ≤ C ln(2).
The last inequality comes from the exact integration of the integral

|r|≥2|s| | 1 r - 1 r -s |dr,
which gives ln 2 if s > 0, 0 if s = 0 and ln 3 2 if s < 0. Therefore, L ∈ L(L p (0, ∞; H)) for all p ∈ (1, ∞) and we have the maximal L p -regularity.

Maximal regularity for autonomous Cauchy problems in Banach spaces

Maximal regularity for autonomous Cauchy problems in Banach spaces

The question now arises, whether every negative generator of bounded analytic semigroup in any Banach space E has the property of maximal L pregularity. The answer is no in general. First we define the notion of U M D-space. The Hilbert transform Hf of a measurable function f is whenever it exists the limit as → 0 + and τ → +∞ of

H ,τ f (t) = ≤|s|≤τ f (t -s) s ds, t ∈ R.
) k∈N ⊂ C, such that x = ∞ k=1 a k x k .
It is called an unconditional basis if the series converges unconditionally. It is interesting to remark that many classical Banach spaces have an unconditional basis. It is, for instance the case of finite dimensional spaces, p spaces and L p (1 < p < ∞). Theorem 1.4.6 (Kalton, Lancien [START_REF] Kalton | A solution to the problem of Lp-maximal regularity[END_REF]). Let X be a Banach space with an unconditional basis. Assume that each negative generator of an analytic semigroup on X has the maximal L p -regularity property. Then X is isomorphic to 2 .

Definition 1.4.7. Let X, Y be two Banach spaces. A set T ⊂ L(X, Y ) is called R-bounded if there is a constant C > 0 such that for all n ∈ N and T 1 , ..., T n ∈ T and x 1 , ..., x n ∈ X

1 0 n j=1 r j (s)T j x j Y ds ≤ C 1 0 n j=1 r j (s)x j X ds,
where (r j ) j=1,...,n is a sequence of independent {-1, 1}-valued random variables on [0, 1].

Preliminaries

It turns out that on a Banach space one needs extra conditions in order to have the maximal regularity.

Theorem 1.4.8 (L. [START_REF] Weis | Operator-valued fourier multiplier theorems and maximal Lpregularity[END_REF]. Let X be a U M D-Banach space and A be the negative generator of an analytic semigroup on X. Then A has the maximal L p -regularity if and only if the set {iσ(iσ

+ A) -1 , σ ∈ R} is R-bounded.
References for the proof. This theorem can be found in [START_REF] Weis | Operator-valued fourier multiplier theorems and maximal Lpregularity[END_REF](Theorem 3.4, Corollary 4.4).

Maximal regularity for non-autonomous problems in V

We consider a family of sesquilinear forms

a : [0, τ ] × V × V → C.
We assume the following usual properties.

• [H1]: D(a(t)) = V (constant form domain), • [H2]: |a(t, u, v)| ≤ M u V v V (uniform boundedness),
• [H3]: Re a(t, u, u) + ν u 2 ≥ δ u 2 V (∀u ∈ V) for some δ > 0 and some ν ∈ R (uniform quasi-coercivity).

We suppose that t → a(t, u, u) is measurable for all u ∈ V. We denote by A(t), A(t) the usual associated operators with a(t) as operators on H and V , respectively. In particular, A(t) : V → V as a bounded operator and

a(t, u, v) = A(t)u, v , for all u, v ∈ V.
The operator A(t) is the part of A(t) on H. Theorem 1.5.1 (Lions' theorem). For every f ∈ L 2 (0, τ ; V ) and u 0 ∈ H there exists a unique u ∈ M R(V, V ) = H 1 (0, τ ; V ) ∩ L 2 (0, τ ; V) which solves the equation

u (t) + A(t) u(t) = f (t), t ∈ (0, τ ] u(0) = u 0 . (P')
Lions' proof is based on the following representation result Proposition 1.5.2 (Lions' representation theorem). Let H be a Hilbert space, V a pre-Hilbert space such that V → H. Let E : H × V → C be a sesquilinear form such that 1.5. Maximal regularity for non-autonomous problems in V

• For all v ∈ V, E(., v) is a continuous linear functional on H.

• |E(v, v)| ≥ α v 2
V for all v ∈ V and some α > 0. Let L ∈ V . Then there exists u ∈ H such that

Lv = E(u, v) for all v ∈ V.
The previous proposition is proved in [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] (p. 61).

Lemma 1.5.3. Let τ > 0 and u ∈ M R(V, V ) := H 1 (0, τ ; V ) ∩ L 2 (0, τ ; V). We have u ∈ C([0, τ ]; H) ∩ H 1 2 (0, τ ; H) and 2Re t 0 u (s), u(s) ds = u(t) 2 -u(0) 2 , with t ∈ [0, τ ].
Proof. By [18] (Theorem 1, p. 473) we obtain M R(V, V ) → C([0, τ ]; H) and for all u ∈ M R(V, V ) we have

2Re t 0 u (s), u(s) ds = u(t) 2 -u(0) 2 .
By [18] (Lemma 2, p. 473) there exists a continuous extension operator

P : M R(V, V ) → H 1 (R; V ) ∩ L 2 (R; V). Now, let u ∈ M R 2 (V, V ) we get P u 2 H 1 2 (R;H) = P u 2 L 2 (R;H) + R |ξ|FP u(ξ) 2 dξ = P u 2 L 2 (R;H) + R |ξ|FP u(ξ), FP u(ξ) dξ ≤ P u 2 L 2 (R;H) + P u H 1 (R;V ) P u L 2 (R;V) ≤ 2 P u H 1 (R;V )∩L 2 (R;V) ≤ 2C u M R(V,V ) . Since u 2 H 1 2 (0,τ ;H) ≤ P u 2 H 1 2 (R;H) , then M R(V, V ) → H 1 2 (0, τ ; H). Proof of Theorem 1.5.1. Let c ≥ ν. Since u is a solution of (P ) if and only if v(.) = e -c. u is a solution of    v (t) + (A(t) + c)v(t) = e -ct f (t) v(0) = u 0 1. Preliminaries we may assume that a(t) is coercive. Let H = L 2 (0, τ ; V) endowed with norm g 2 H = τ 0 g(t) 2 V dt and V = {L 2 (0, τ ; V) ∩ H 1 (0, τ ; V ) s.t v(τ ) = 0}
with norm

v 2 V = τ 0 v(t) 2 V dt + v(0) 2 .
Further we define the sesquilinear form E :

H × V → C by E(u, v) = τ 0 a(t, u(t), v(t)) -u(t), v(t) dt and for u 0 ∈ H, f ∈ L 2 (0, τ ; V ) we define L : V → C by L(v) = τ 0 f (t), v(t) dt + (u 0 , v(0)). For all v ∈ V, is clear that the form u → E(u, v) is continuous on H. For v ∈ V, we have Re E(v, v) = τ 0 a(t, v(t), v(t))dt - τ 0 ∂ ∂t v(t) 2 dt (1.5.1) ≥ min{δ, 1}( τ 0 v(t) 2 V dt + v(0) 2 ). (1.5.2)
Finally, it easy to show that v → L(v) is continuous on V. Therefore by applying Theorem 1.5.1, there exists u ∈ H such that E(u, v) = Lv for all v ∈ V.

Let φ ∈ C ∞ c (0, τ ) and h ∈ V, then for w(.) = φ(.)h the identity E(u, w) = Lw implies

- τ 0 u, h φ dt = τ 0 f -Au, h φdt.

Thus by definition we have

u ∈ H 1 (0, τ ; V ) and u = f -Au ∈ L 2 (0, τ ; V ). It remains to show u(0) = u 0 . Let φ ∈ C ∞ (0, τ ) with φ(0) = 1, φ(τ ) = 0 and h ∈ V.
By integration by parts we have

- τ 0 u, h φ dt = (u(0), h)φ(0) + τ 0 u , h φdt.
On other hand, by the identity E(u, w) = Lw, where w(.) = φ(.)h we get

- τ 0 u, h φ dt = (u 0 , h)φ(0) + τ 0 u , h φdt.

Maximal regularity for non-autonomous problems in V

Then (u(0), h) = (u 0 , h), for all h ∈ V. Thus u(0) = u 0 and u is a solution for the Problem (P'). By Lemma 1.5.3, we get

u(t) 2 -u(0) 2 + 2δ τ 0 u(t) 2 V dt ≤ 2Re t 0 u(t), u(t) dt + 2Re t 0 a(t, u(t), u(t))dt = 2Re τ 0 f (t), u(t) dt ≤ 1 δ τ 0 f (t) 2 V dt + δ τ 0 u(t) 2 V dt.
Therefore

u(t) 2 + δ τ 0 u(t) 2 V dt ≤ u(0) 2 + 1 δ τ 0 f (t) 2 V dt.
So that, there exists a constant C > 0, such that

u C([0,τ ];H) + u L 2 (0,τ ;V) ≤ C( u(0) 2 + f L 2 (0,τ ;V ) ).
For the uniqueness we suppose there are two solutions u 1 , u 2 and we set w = u 1 -u 2 . So w is the solution of the Problem (P ) with f = u 0 = 0. Then by the previous estimate we have w = 0 and u 1 = u 2 .

Following [START_REF] Arendt | L p -maximal regularity for non-autonomous evolution equations[END_REF], we introduce the following definition Definition 1.5.4. Let (a(t)) t∈[0,τ ] be a family of V-bounded, sesquilinear forms. A function t → a(t) is called relatively continuous if for each t ∈ [0, τ ] and all > 0 there exists α > 0,

β ≥ 0 such that for all u, v ∈ V, s ∈ [0, τ ], |t -s| ≤ α implies that |a(t, u, v) -a(s, u, v)| ≤ ( u V + β u V ) v V . Example 1.5.5. Let (a(t)) t∈[0,τ ] be a family of V-bounded, sesquilinear forms such that t → a(t) is measurable. We suppose that |a(t, u, v) -a(s, u, v)| ≤ u (V ,V) α,2 v V
fore some α ∈ (0, 1). By the interpolation inequality (see [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF] Corollary 1.1.7) we get

u (V ,V) α,2 ≤ c u α V u 1-α V ≤ cγ α u α V 1 γ α u 1-α V ≤ αγ u V + (1 -α)( c γ α ) 1. Preliminaries
where c > 0 and γ > 0 is an arbitrary constant. Therefore

|a(t, u, v) -a(s, u, v)| ≤ αγ u V + (1 -α)( c γ α ) 1 1-α u V v V .
Then t → a(t) is relatively continuous.

We assume in the next proposition that ν = 0.

Proposition 1.5.6. Let p ∈ (1, ∞) and s ∈ [0, τ ]. Then for all f ∈ L p (0, τ ; V ) and u 0 ∈ (V , V) 1-1 p ,p , there exists a unique u ∈ M R p (V, V ) = W 1,p (0, τ ; V ) ∩ L p (0, τ ; V), be the solution of the autonomous problem    u (t) + A(s)u(t) = f (t) u(0) = u 0 (1.5.3)
and the solution is given by

u(t) = e -tA(s) u 0 + t 0 e -(t-l)A(s) f (l)dl.
Morever there exists a positive constant C independant of u 0 , f and τ such that

u M Rp(V,V ) ≤ C u (V ,V) 1-1 p ,p + f L p (0,τ ;V ) . Proof. Let u 0 ∈ (V , V) 1-1 p ,p , f ∈ L p (0, τ ; V ) and s ∈ [0, τ ]. Since A(s)
is a generator of an analytic semigroup on V ( [START_REF] Ouhabaz | Analysis of Heat Equations on Domains[END_REF] Theorem 1.55), we have by Theorem 1.3.3 that there exists a unique v ∈ W 1,p (0, τ ; V) which solves the following equation

   v (t) + A(s)v(t) = A(s) -1 f (t) v(0) = A(s) -1 u 0 . (1.5.4) We set u(t) = A(s)v(t), with t ∈ [0, τ ]. Then u ∈ M R p (V, V ) is the solution of the Problem (1.5.3).
For the non-autonomous maximal L p -regularity with p = 2 we have the following result Theorem 1.5.7. Let (a(t)) t∈[0,τ ] be a family of V-bounded, quasi coercive, sesquilinear forms and we suppose that t → a(t) is relatively continuous. Then for all f ∈ L p (0, τ ; V ), with p ∈ (1, ∞) and u 0 ∈ (V , V) 1-1 p ,p there exists a unique u ∈ M R p (V, V ), which solves the equation (P ).

Maximal regularity for non-autonomous problems in V

Proof. First we prove the unicity. Given c ∈ R, (P ) has maximal L p -regularity if and only if the Cauchy problem

   v (t) + (A(t) + c)v(t) = e -ct f (t) v(0) = u 0 (1.5.5) has the maximal L p -regularity. The reason is that v(t) = e -ct u(t) and u ∈ M R p (V, V ) if and only if v ∈ M R p (V, V ).
Therefore, by adding a large constant c we may assume [H3] holds with ν = 0. We suppose there are two solutions

v 1 , v 2 , then v = v 1 -v 2 is a solution of the problem    v (t) + A(t)v(t) = 0 v(0) = 0. (1.5.6)
Therefore, for t > 0 we get

2Re t 0 v (s), v(s) ds + 2Re t 0 A(s)v(s), v(s) ds = 0. Then by Lemma 1.5.3 we obtain v(t) 2 + 2δ t 0 v(s) 2
V ds = 0. Thus, for all t ∈ [0, τ ] we get v(t) = 0 and so v 1 (t) = v 2 (t). Now, for the existence we apply Proposition 1.5.6 and [7] (Theorem 2.7) to get the diserd result.

Chapter 2

Non-autonomous right and left multiplicative perturbations and maximal regularity

Les résultats présentés dans ce chapitre ont fait l'objet de l'article [START_REF] Achache | Non-autonomous right and left multiplicative perturbations and maximal regularity[END_REF] en collaboration avec El Maati Ouhabaz.

Introduction

The present paper deals with maximal L p -regularity for non-autonomous evolution equations in the setting of Hilbert spaces. Before explaining our results we introduce some notations and assumptions. Let (H, (•, •), • ) be a Hilbert space over R or C. We consider another Hilbert space V which is densely and continuously embedded into H. We denote by V the (anti-) dual space of V so that

V → d H → d V .
We denote by , the duality V-V and note that ψ

, v = (ψ, v) if ψ, v ∈ H. Given τ ∈ (0, ∞) and consider a family of sesquilinear forms a : [0, τ ] × V × V → C such that • [H1]: D(a(t)) = V (constant form domain), • [H2]: |a(t, u, v)| ≤ M u V v V (uniform boundedness), • [H3]: Re a(t, u, u) + ν u 2 ≥ δ u 2 V (∀u ∈ V)
for some δ > 0 and some ν ∈ R (uniform quasi-coercivity).

Introduction

Here and throughout this paper, • V denotes the norm of V.

To each form a(t) we can associate two operators A(t) and A(t) on H and V , respectively. Recall that u ∈ H is in the domain D(A(t)) if there exists h ∈ H such that for all v ∈ V: a(t, u, v) = (h, v). We then set A(t)u := h. The operator A(t) is a bounded operator from V into V such that A(t)u = a(t, u, •). The operator A(t) is the part of A(t) on H. It is a classical fact that -A(t) and -A(t) are both generators of holomorphic semigroups (e -rA(t) ) r≥0 and (e -rA(t) ) r≥0 on H and V , respectively. The semigroup e -rA(t) is the restriction of e -rA(t) to H. In addition, e -rA(t) induces a holomorphic semigroup on V (see, e.g., Ouhabaz [START_REF] Ouhabaz | Analysis of Heat Equations on Domains[END_REF]Chapter 1]).

A well known result by J.L. Lions asserts that the Cauchy problem

u (t) + A(t)u(t) = f (t), u(0) = u 0 ∈ H (2.1.1) has maximal L 2 -regularity in V , that is, for every f ∈ L 2 (0, τ ; V ) there exists a unique u ∈ W 1 2 (0, τ ; V ) ∩ L 2 (0, τ ; V) which satisfies (2.1.1
) in the L 2 -sense. The maximal regularity in H is however more interesting since when dealing with boundary value problems one cannot identify the boundary conditions if the Cauchy problem is considered in V . The maximal regularity in H is more difficult to prove. J.L. Lions has proved that this is the case for initial data u 0 ∈ D(A(0)) under a quite restrictive regularity condition, namely t → a(t, g, h) is C 2 (or C 1 if u 0 = 0). It was a question by him in 1961 (see [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] p. 68) whether maximal L 2 -regularity holds in general in H.

A lot of progress has been made in recent years on this problem. It was proved by Ouhabaz and Spina [START_REF] Ouhabaz | Maximal regularity for non-autonomous Schrödinger type equations[END_REF] that maximal L p -regularity holds in H if t → a(t, g, h) is C α for some α > 1/2 (for all g, h ∈ V). This result is however proved for the case u 0 = 0 only. In Haak and Ouhabaz [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF], it is proved that for u 0 ∈ (H, D(A(0))) 1-1 p ,p and

|a(t, g, h) -a(s, g, h)| ≤ ω(|t -s|) h V g V (2.1.2)
for some non-decreasing function ω such that

τ 0 ω(t) t 3 2 dt < ∞ and τ 0 ω(t) t p dt < ∞, (2.1.3)
then the Cauchy problem (2.1.1) has maximal L p -regularity in H. The condition (2.1.3) can be improved if (2.1.2) holds with norms in some complex interpolation spaces (see Arendt and Monniaux [START_REF] Arendt | Maximal regularity for non-autonomous Robin boundary conditions[END_REF] and Ouhabaz [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF]). It was observed by Dier [20] that the answer to Lions' problem is negative in general. His example is based on a non-symmetric form for which the Kato square root property D(A(t)) 1/2 ) = V is not satisfied. Recently, Fackler [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF] gave a negative answer to the maximal regularity problem for forms which are C α for any α ≤ 1/2 (even symmetric ones). Let us also mention a recent positive result of

Non-autonomous right and left multiplicative perturbations and maximal regularity

Dier and Zacher [START_REF] Dier | Non-autonomous maximal regularity in Hilbert spaces[END_REF] on maximal L 2 -regularity in which the condition (2.1.3) is replaced by a norm in a Sobolev space of order > 1 2 . For forms associated with divergence form elliptic operators, Auscher and Egert [START_REF] Auscher | On non-autonomous maximal regularity for elliptic operators in divergence form[END_REF] proved maximal L 2 -regularity under a BMO-H 1/2 condition on the forms. More recently, Fackler [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] proved maximal L p -regularity under fractional Sobolev regularity.

One of the aims of the present paper is to study the same problem for multiplicative perturbations. More precisely, we study maximal L p -regularity for

u (t) + B(t)A(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 (2.1.4)
and also for

u (t) + A(t)B(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 , (2.1.5)
where B(t) and P (t) are bounded operators on H such that Re (B(t) -1 g, g) ≥ δ g 2 for some δ > 0 and all g ∈ H. The left perturbation problem (2.1.4) was already considered by Arendt et al. [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF] and the right perturbation one (2.1.5) by Augner et al. [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF]. The two problems are motivated by applications to semi-linear evolution equations and boundary value problems. We extend the results in [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF] and [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF] in three directions. The first one is to consider general forms which may not satisfy the Kato square root property, a condition which was used in an essential way in the previous two papers. The second direction is to deal with maximal L p -regularity, whereas in the mentioned papers only the maximal L 2 -regularity is considered. The third direction, which is our main motivation, is to assume less regularity on the forms a(t) with respect to t. In both papers [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF] and [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF] it is assumed that t → a(t, g, h) is Lipschitz continuous on [0, τ ]. In applications to elliptic operators with time dependent coefficients, the regularity assumption on the forms reflects the regularity needed for coefficients with respect to t.

Our main results can be summarized as follows (see Theorems 2.3.6 and 2.5.1 for more general and precise statements). Suppose that for some β, γ ∈

[0, 1], |a(t, g, h) -a(s, g, h)| ≤ ω(|t -s|) g [H,V] β h [H,V] γ , u, v ∈ V where ω : [0, τ ] → [0, ∞) is a non-decreasing function such that τ 0 ω(t) t 1+ γ 2 dt < ∞. Suppose also that t → B(t) is continuous on [0, τ ] with values in L(H). Then the Cauchy problem (2.1.4) has maximal L p -regularity in H for all p ∈ (1, ∞) when u 0 = 0. If in addition, τ 0 ω(t) p t 1 2 (β+pγ) dt < ∞ (2.1.6) 2.1. Introduction then (2.1.4) has maximal L p -regularity in H provided u 0 ∈ (H, D(A(0))) 1-1 p ,p . We also prove that if ω(t) ≤ Ct ε for some ε > 0 and D(A(t) 1/2 ) = V for all t ∈ [0, τ ], then the solution u ∈ C([0, τ ]; V) and s → A(s) 1/2 u(s) ∈ C([0, τ ]; H).
Concerning (2.1.5), we assume as in [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF] that t → B(t) is Lipschitz continuous on [0, τ ] with values in L(H). The assumptions on a(t) are the same as above. The maximal L p -regularity results we prove are the same as previously. We could also consider both left and right perturbations, see the end of Section 3.5.

We point out that condition (2.1.6) is slightly better than the second condition in (2.1.3) which was assumed in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] and [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF] (for the unperturbed problem). In the natural case ω(t) ∼ t α , one sees immediately that for large p, (2.1.3) requires larger α (and then more regularity) than (2.1.6).

In order to prove our results we follow similar ideas as in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] and [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF]. However, several modifications are needed in order to deal with multiplicative perturbations. Also, at several places we appeal to classical tools from harmonic analysis such as square function estimates or Hörmander type conditions for singular integral operators with vector-valued kernels.

Our results on maximal L p -regularity could be applied to boundary value problems as well as to some semi-linear evolution equations such as

       u (t) = m(t, x, u(t), ∇u(t))∆u(t) + f (t) u(0) = u 0 ∈ H 1 (Ω) ∂u(t) ∂n + β(t, .
)u(t) = 0 on ∂Ω on a bounded Lipschitz domain Ω. This and many other applications have been already considered in [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF] and [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF]. The gain here is that we are able to assume less regularity with respect to the variable t. We shall not write these applications explicitly in this paper since the ideas are the same as in [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF] and [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF], one has just to insert our new results on maximal regularity. The reader interested in applications of non-autonomous maximal regularity is referred to the previous articles and the references therein.

Non-autonomous right and left multiplicative perturbations and maximal regularity

Maximal regularity for the unperturbed problem

Let H and V be as in the introduction. We consider a family of sesquilinear forms a(t) :

V × V → C, t ∈ [0, τ ]
which satisfy the classical assumptions [H1]-[H3]. We denote again by A(t) and A(t) the operators associated with a(t) on H and V , respectively. Note that by adding a positive constant to A(t) we may assume that [H3] holds with ν = 0. Therefore, there exists

w 0 ∈ [0, π 2 ) such that a(t, u, u) ∈ Σ(w 0 ), ∀t ∈ [0, τ ], u ∈ V. (2.2.1)
Here

Σ(w 0 ) := {z ∈ C * , | arg(z)| ≤ w 0 }.
In (2.2.1) we take w 0 to be the smallest possible value for which the inclusion holds.

Definition 2.2.1. Fix u 0 ∈ H. We say that the problem

u (t) + A(t)u(t) = f (t) (t ∈ [0, τ ]), u(0) = u 0 (2.2.2)
has maximal L p -regularity in H if for each f ∈ L p (0, τ ; H), there exists a unique u ∈ W 1 p (0, τ ; H) such that u(t) ∈ D(A(t)) for almost all t and satisfies (4.1.1) in the L p -sense.

We denote by V β := [H, V] β the classical complex interpolation space. Its usual norm is denoted • V β . We start with the following result on maximal L p -regularity of (4.1.1). Theorem 2.2.2. Suppose that the forms (a(t)) t∈[0,τ ] satisfy the standing hypotheses [H1]-[H3]. Suppose that for some β, γ ∈ [0, 1]

|a(t, u, v) -a(s, u, v)| ≤ ω(|t -s|) u V β v Vγ , u, v ∈ V, (2.2.3)
where

ω : [0, τ ] → [0, ∞) is a non-decreasing function such that τ 0 w(t) t 1+ γ 2 dt < ∞.
u W 1 p (0,τ ;H) + Au Lp(0,τ ;H) ≤ C f Lp(0,τ ;H) + u 0 (H,D(A(0))) 1-1 p ,p .
Here, (H, D(A(0))) 1-1 p ,p denotes the classical real-interpolation space and the constant C is independent of f and u 0 .

The first part of the theorem (i.e., the case u 0 = 0) was proved in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] when

β = γ = 1 (and hence [H, V] β = [H, V] γ = V).
The case with different values β and γ was proved in [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF]. See also [START_REF] Arendt | Maximal regularity for non-autonomous Robin boundary conditions[END_REF] for a related result. In order to treat the case of a non-trivial initial data u 0 ∈ (H, D(A(0))) 1-1 p ,p , the assumption required on ω in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF]

is τ 0 ω(t) t p dt < ∞, (2.2.5)
and in [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF],

τ 0 ω(t) t β+γ 2 p dt < ∞. (2.2.6)
In the previous theorem we replace these conditions by the weaker condition (2.2.4). The important example ω(t) = t α shows that (2.2.5) and (2.2.6) require a large α (and hence more regularity) in the case p > 2, whereas (2.2.4) does not require any additional regularity than α > γ 2 which is already needed for the first condition

τ 0 w(t) t 1+ γ 2 dt < ∞.
Proof. As explained above the sole novelty here is the treatment of the case u 0 ∈ (H, D(A(0))) 1-1 p ,p under the condition (2.2.4). Following [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF], Lemma 12 and [37], Lemma 2.3 we have to prove that

t → A(t)e -tA(t) u 0 ∈ L p (0, τ ; H).
(2.2.7)

Since we can assume without loss of generality that A(0) is invertible, then

u 0 ∈ (H, D(A(0))) 1-1 p ,p is equivalent to (see [46, Theorem 1.14]) t → A(0)e -tA(0) u 0 ∈ L p (0, τ ; H). (2.2.8)
Let g ∈ H and choose a contour Γ = ∂Σ(θ) for appropriate θ ∈ (0, π 2 ) we write

Non-autonomous right and left multiplicative perturbations and maximal regularity

by the holomorphic functional calculus (A(t)e -tA(t) u 0 -A(0)e -tA(0) u 0 , g)

= 1 2πi Γ (ze -tz (zI -A(t)) -1 -(zI -A(0)) -1 u 0 , g) dz = 1 2πi Γ (ze -tz A(t) -A(0) (zI -A(0)) -1 u 0 , (zI -A(t) * ) -1 g)dz = 1 2πi Γ ze -tz a(t, (zI -A(0)) -1 u 0 , (zI -A(t) * ) -1 g)- a(0, (zI -A(0)) -1 u 0 , (zI -A(t) * ) -1 g) dz.
Hence by (2.2.3), the modulus is bounded by

Cω(t) ∞ 0 |z|e -ct|z| (zI -A(0)) -1 u 0 V β (zI -A(t) * ) -1 g Vγ d|z|.
Note that by interpolation (see e.g. [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF])

(zI -A(t) * ) -1 L(H,Vγ ) ≤ C |z| 1-γ 2 .
(2.2.9)

On the other hand for f ∈ D(A(0)),

δ (zI -A(0)) -1 f 2 V ≤ Re (A(0)(zI -A(0)) -1 f, (zI -A(0)) -1 f ) ≤ (zI -A(0)) -1 A(0)f (zI -A(0)) -1 f ≤ C |z| A(0)f (zI -A(0)) -1 f V .
The embedding V → V β gives

(zI -A(0)) -1 L(D(A(0)),V β ) ≤ C |z| .
Hence, by (2.2.9) and interpolation

(zI -A(0)) -1 L((H,D(A(0))) 1-1 p ,p ,V β ) ≤ C |z| 1-β 2p . (2.2.10)
Using these estimates we obtain

|(A(t)e -tA(t) u 0 -A(0)e -tA(0) u 0 , g)| ≤ Cω(t) ∞ 0 e -ct|z| |z| 1-1 2 (γ+ β p ) d|z| g u 0 (H,D(A(0))) 1-1 p ,p ≤ C ω(t) t 1 2 (γ+ β p ) g u 0 (H,D(A(0))) 1-1 p ,p . Hence, t → A(t)e -tA(t) u 0 ∈ L p (0, τ, H) for u 0 ∈ (H, D(A(0))) 1-1 p ,p if ω(t) satisfies (2.2.4).

Maximal regularity for left perturbations

Maximal regularity for left perturbations

This section is devoted to the main subject of this paper in which we are interested in maximal regularity for operators B(t)A(t) for a wide class of operators B(t) and A(t). We will consider in another section the same problem for right multiplicative perturbations A(t)B(t).

Single left multiplicative perturbation-Resolvent estimates

Let H and V be as above. We denote again by • and • V their associated norms, respectively. Let a : V × V → C be a closed, coercive and continuous sesquilinear form. We denote by A and A its associated operators on H and V , respectively. Let b : H × H → C be a bounded sesquilinear form. We assume that b is coercive, that is there exists a constant δ > 0 such that Re b(u, u) ≥ δ u 2 , u ∈ H.

(

There exists a unique bounded operator associated with b. We denote temporarily this operator by C. Note that by coercivity, it is obvious that C is invertible on H. Now we introduce another operator A b which we call the operator associated with a with respect to b. It is defined as follows

D(A b ) = {u ∈ V, ∃v H : a(u, φ) = b(v, φ) ∀φ ∈ V}, A b u := v.
The difference with A is that we take the form b instead of the scalar product of H in the equality a(u, φ) = b(v, φ). The operator A b is well defined. Indeed, if b(v 1 , φ) = b(v 2 , φ) for all φ ∈ V then by density this equality holds for all φ ∈ H. Therefore, taking φ = v 2 -v 1 and using (2.3.1), we obtain

v 2 = v 1 . Proposition 2.3.1. Let B := C -1 . Then A b = BA with domain D(A b ) = D(A). Proof. Let u ∈ D(A b ) and v = A b u. Then a(u, φ) = b(v, φ) = (Cv, φ) ∀φ ∈ V. Thus, u ∈ D(A) and Au = Cv = B -1 v. This gives, u ∈ D(A) and A b u = v = BAu.
For the converse, we write for u ∈ D(A) and φ ∈ V

a(u, φ) = (Au, φ) = (CBAu, φ) = b(BAu, φ).
This gives u ∈ D(A b ) and BAu = A b u.

Non-autonomous right and left multiplicative perturbations and maximal regularity

It is obvious that BA is a closed operator on H. In order to continue we assume that a is coercive (i.e., it satisfies [H3] with ν = 0) and define w 0 and w 1 to be the angles of the numerical ranges of A and B, respectively. That is

(Au, u) ∈ Σ(w 0 ) := {z ∈ C * , | arg(z)| ≤ w 0 } and b(u, u) = (B -1 u, u) ∈ Σ(w 1 )
where w 0 and w 1 are the smallest possible values for which these two properties hold for all u ∈ V. Note that w 0 , w 1 ∈ [0, π 2 ) because of the coercivity property.

Proposition 2.3.2. For all λ / ∈ Σ(w 0 + w 1 ), the operator λI -BA is invertible on H and

(λI -BA) -1 L(H) ≤ δ -1 B -1 L(H)
dist(λ, Σ(w 0 + w 1 )) .

Proof. Let u ∈ D(A). We write

(λI -BA)u u = B(λB -1 -A)u u ≥ 1 B -1 L(H) (λB -1 I -A)u u ≥ 1 B -1 L(H) |(λB -1 u -Au, u)| = |(B -1 u, u)| B -1 L(H) |λ - (Au, u) (B -1 u, u) |. Since (Au,u) (B -1 u,u) = a(u,u) b(u,u) ∈ Σ(w 0 + w 1 ) it follows that |(λ - (Au, u) (B -1 u, u) | ≥ dist(λ, Σ(w 0 + w 1 )).
On the other hand, by (2.3.1), |(B -1 u, u)| ≥ δ u 2 and so

(λI -BA)u u ≥ δ B -1 L(H) u 2 dist(λ, Σ(w 0 + w 1 )).
Hence,

(λI -BA)u ≥ δ B -1 L(H) u dist(λ, Σ(w 0 + w 1 )) ∀u ∈ D(A). (2.3.2)
This implies that λI -BA is injective and has closed range for λ ∈ Σ(w 0 + w 1 ).

In order to prove that λI -BA is invertible it remains to prove that it has 2.3. Maximal regularity for left perturbations dense range. By duality, one has to prove that the adjoint is injective. The adjoint operator is λI -A * B * . We write

λI -A * B * = (λB * -1 -A * )B * .
The previous arguments show that λB -1 -A is injective. This also applies to λB * -1 -A * . Since B * is invertible, we obtain λI -A * B * is injective and hence λI -BA is invertible . Now (2.3.2) gives

(λI -BA) -1 ≤ B -1 L(H) δ dist(λ, Σ(w 0 + w 1 ))
for all λ ∈ Σ(w 0 + w 1 ).

Corollary 2.3.3. Suppose that w 0 + w 1 < π 2 . Then -BA is the generator of a bounded holomorphic semigroup on H.

Proof. By Proposition 2.3.2, (λI -BA) -1 ≤ c |λ| , ∀λ ∈ Σ(w 0 + w 1 ).
In other words, λI + BA is invertible for λ ∈ Σ(π -(w 0 + w 1 )) and

(λI + BA) -1 ≤ c |λ| , ∀λ ∈ Σ(π -(w 0 + w 1 )).
It is a classical fact that the latter estimate implies that -BA generates a bounded holomorphic semigroup of angle π 2 -(w 0 + w 1 ). Obviously, one cannot remove the assumption w 0 + w 1 < π 2 in the previous result. Indeed, let A = -e i π 3 ∆ on L 2 (R d ) and B be the multiplication by e i π 3 . Then -BA = e i 2π 3 ∆ is not a generator of a C 0 -semigroup.

Single perturbation-Maximal regularity

Let (a(t)) t∈[0,τ ] , A(t), A(t) and b be as in the previous sections. We assume that [H3] holds with ν = 0. In particular, (2.2.1) holds. We also have

b(u, u) ∈ Σ(w 1 ) (2.3.3)
for some w 1 ∈ [0, π 2 ) by coercivity of b. We make the assumption for λ ∈ ρ(-BA(t)).

w 0 + w 1 < π 2 .

Non-autonomous right and left multiplicative perturbations and maximal regularity

Proposition 2.3.4. Assume that

w 0 + w 1 < π 2 . Then 1-(λB -1 + A(t)) -1 L(H) ≤ C |λ|+1 , λ ∈ Σ(π -(w 0 + w 1 )), 2-R(λ, BA(t))B L(V ,H) ≤ C (|λ|+1) 1 2 , λ ∈ Σ(π -(w 0 + w 1 )), 3-e -(t-s)BA(t) B L(V ,H) ≤ C (t-s) 1 2 , 4-e -(t-s)BA(t) B L(V ,V) ≤ C (t-s)
. The constant C is independent of t and λ.

Proof. We have (λB -1 + A(t)) -1 = (λ + BA(t)) -1 B, then we obtain assertion 1-from Proposition 2.3.2. Note that

(λB -1 + A(t)) -1 = (λ + A(t)) -1 + (λB -1 + A(t)) -1 (λ(-B -1 + I))(λ + A(t)) -1 . (2.3.4) Then R(λ, BA(t))B L(V ,H) = (λB -1 + A(t)) -1 L(V ,H) ≤ (λI + A(t)) -1 L(V ,H) + (λB -1 + A(t)) -1 (λ(-B -1 + I)) L(H) (λ + A(t)) -1 L(V ,H) . Since (λI + A(t)) -1 L(V ,H) ≤ C (|λ| + 1) 1 2
(see e.g. [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF], Proposition 6), we obtain

R(λ, BA(t))B L(V ,H) ≤ C (|λ| + 1) 1 2
, which proves assertion 2. Now we choose an appropriate contour Γ = ∂Σ(θ) with θ < π 2 and write by the functional calculus

e -(t-s)BA(t) B = 1 2πi Γ e -(t-s)λ (λ -BA(t)) -1 Bdλ. Then e -(t-s)BA(t) B L(V ,H) ≤ 1 2π Γ e -(t-s)Re λ (λ -BA(t)) -1 B L(V ,H) d|λ| ≤ C Γ e -(t-s)Re λ 1 (|λ| + 1) 1 2 d|λ| ≤ C (t -s) 1 2
.

Maximal regularity for left perturbations

In order to prove assertion 4-we write

e -(t-s)BA(t) B L(V ,V) ≤ e -(t-s) 2 BA(t) BB -1 L(H,V) e -(t-s) 2 BA(t) B L(V ,H) and e -(t-s) 2 BA(t) BB -1 L(H,V) ≤ B -1 L(H) e -(t-s) 2 BA(t) B L(H,V)
. We use the equality

(λB -1 + A(t)) -1 = (λI + A(t)) -1 + (λI + A(t)) -1 λ(I -B -1 )(λB -1 + A(t)) -1
in place of (2.3.4) to estimate R(λ, BA(t))B L(H,V) and then argue as previously. Now, let P (t) ∈ L(H) be such that t → P (t) is strongly measurable and

P (t) L(H) ≤ M, t ∈ [0, τ ] (2.3.5)
for some constant M . We consider the Cauchy problem

u (t) + BA(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 . (2.3.6)
Recall that B -1 is the operator associated with b. We are interested in maximal regularity of (2.3.6). As explained at the beginning of the proof of the next proposition, we may assume without loss of generality that the forms a(t) are coercive and hence (2.2.1) is satisfied for some w 0 ∈ [0, π 2 ). Proposition 2.3.5. Suppose that the forms (a(t)) t∈[0,τ ] satisfy [H1]-[H3], the form b satisfies (2.3.1) and w 0 + w 1 < π 2 . Suppose that for some β, γ

∈ [0, 1] |a(t, u, v) -a(s, u, v)| ≤ ω(|t -s|) u V β v Vγ , u, v ∈ V where ω : [0, τ ] → [0, ∞) is a non-decreasing function such that : τ 0 w(t) t 1+ γ 2 dt < ∞. Then the Cauchy problem (2.3.6) with u 0 = 0 has maximal L p -regularity in H for all p ∈ (1, ∞). If in addition, τ 0 ω(t) p t 1 2 (β+pγ) dt < ∞ (2.3.7) then (2.3.6) has maximal L p -regularity for all u 0 ∈ (H, D(A(0))) 1-1
p ,p . Moreover there exists a positive constant C such that

u W 1 p (0,τ ;H) + Au Lp(0,τ ;H) ≤ C f Lp(0,τ ;H) + u 0 (H,D(A(0))) 1-1 p ,p
.

Here C depends on the constants in [H1]-[H3], (4.1.2), B L(H) , B -1 L(H) , p, β and γ and it is independent of f and u 0 .

Non-autonomous right and left multiplicative perturbations and maximal regularity

Proof. Firstly, we note that for c ∈ R, (2.3.6) has maximal L p -regularity if and only if the Cauchy problem

v (t) + (BA(t) + P (t) + cI)v(t) = e -ct f (t), v(0) = u 0 has maximal L p -regularity. The reason is that v(t) = u(t)e -ct
and it is clear that u ∈ W 1 p (0, τ ; H) if and only if v ∈ W 1 p (0, τ ; H). Thus, by adding a large constant c we may assume that [H3] holds with ν = 0 and BA(t

) + P (t) is invertible for each t ∈ [0, τ ].
Note that BA(t) = A(t) b is the operator associated with the form a(t) with respect to b (see Section 2.3.1). This allows us to use the same strategy of proof as for Theorem 2.2.2 (cf. [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF] or [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] in the case

β = γ = 1). Set v(s) := e -(t-s)BA(t) u(s). Writing v(t) -v(0) = t 0 v (s)ds we obtain A(t)u(t) = A(t)e -tBA(t) u 0 + A(t) t 0 e -(t-l)BA(t) B(A(t) -A(l))u(l)dl + A(t) t 0 e -(t-l)BA(t) (-P (l))u(l)dl + A(t) t 0 e -(t-l)BA(t) f (l)dl.
Note that by Proposition 2.3.4, the term e -(t-l)BA(t) B(A(t) -A(l))u(l) is well defined.

We first prove the proposition in the case u 0 = 0. We define

(Lf )(t) := A(t) t 0
e -(t-l)BA(t) f (l)dl.

Following [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF], by extending the forms to R we can view the operator L as a pseudo-differential operator with the vector-valued symbol σ(t, ξ) given by

σ(t, ξ) :=    A(0)(iξ + BA(0)) -1 if t < 0 A(t)(iξ + BA(t)) -1 if 0 ≤ t ≤ τ A(τ )(iξ + BA(τ )) -1 if t > τ.
Then we use Proposition 2.3.2 and argue as in the proof of Lemmas 10 and 11 in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] to prove the boundedness on L p (0, τ ; H), 1 < p < ∞, of the operator L.

We continue as in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] and [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF]. We set

(Sg)(t) := A(t) t 0
e -(t-l)BA(t) P (l)A(l) -1 g(l)dl.

By the boundedness of the operator L on L p (0, τ ; H), Sg Lp(0,τ ;H) ≤ C A -1 g Lp(0,τ ;H) .

Maximal regularity for left perturbations

We define

(Qg)(t) := A(t) t 0 e -(t-l)BA(t) B(A(t) -A(l))A(l) -1 g(l)dl.
Then, arguing as in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] or [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF] we obtain easily from Proposition 2.3.4

(Qg)(t) ≤ t 0 w(|t -l|) (t -l) 1+ γ 2 A -1 (l)g(l) V dl.
Thus,

Qg Lp(0,τ ;H) ≤ C τ 0 w(t) t 1+ γ 2 dt A -1 g Lp(0,τ ;V) .
From these estimates, we see that by replacing A(t) by A(t) + cI for c large enough we obtain

S L(Lp(0,τ ;H)) < 1 4 and Q L(Lp(0,τ ;H)) < 1 4 .
In particular, I -

(S + Q) is invertible. Since (Au)(t) = (I -(S + Q)) -1 (L(f ))(t)
we obtain Au ∈ L p (0, τ ; H) and hence u ∈ W 1 p (0, τ ; H). This proves maximal L p -regularity.

In order to treat the case u 0 = 0 we need to estimate the difference of the resolvents, i.e., R(λ, A(t) b ) -R(λ, A(s) b ) L(H) in terms of ω(|t -s|). Let f, g ∈ H and λ ∈ Σ(π -(w 0 + w 1 )). We write

([R(λ, A(t) b ) -R(λ, A(s) b )]f, g) = ([R(λ, A(t) b )B(A(s) -A(t))R(λ, A(s) b )]f, g).
Note that the RHS is well defined since R(λ, BA(t))B is a bounded operator from V to V (see the proof of Proposition 2.3.4, assertion 2). Therefore,

([R(λ, A(t) b ) -R(λ, A(s) b )]f, g) = (A(s) -A(t))R(λ, A(s) b )f, B * R(λ, A(t) b ) * g = a(s, R(λ, A(s) b )f, (λB * -1 + A(t) * ) -1 g) -a(t, R(λ, A(s) b )f, (λB * -1 + A(t) * ) -1 g).
Hence the modulus is bounded by

ω(|t -s|) R(λ, A(s) b )f V β (λB * -1 + A(t) * ) -1 g Vγ .

Non-autonomous right and left multiplicative perturbations and maximal regularity

Let w 0 be the common angle for the numerical range of a(t). By Proposition 2.3.2 we have for all λ / ∈ Σ(w 0 + w 1 )

δ R(λ, A(s) b )f 2 V ≤ Re a(s, R(λ, A(s) b )f, R(λ, A(s) b )f ) = Re (A(s)R(λ, A(s) b )f, R(λ, A(s) b )f ) = Re (BA(s)R(λ, A(s) b )f, (B -1 ) * R(λ, A(s) b )f ) ≤ C |λ| f 2 .
Hence, by interpolation

R(λ, A(s) b )f V β ≤ C |λ| 1-β 2 f . (2.3.8)
Putting together the previous estimates yields

[R(λ, A(t) b ) -R(λ, A(s) b )]f, g) ≤ C ω(|t -s|) |λ| 2-β+γ 2 f g . This shows R(λ, A(t) b ) -R(λ, A(s) b ) L(H) ≤ C ω(|t -s|) |λ| 2-β+γ 2 .
This is the estimate we need in order to obtain the proposition when u 0 ∈ (H, D(A(0)) 1-1 p ,p (see [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] or [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF] for the details).

Time dependent perturbations-Maximal regularity

Let a(t), A(t), V and H be as above and suppose again that the standard assumptions [H1]-[H3] are satisfied. Let (B(t)) t∈[0,τ ] be a family of bounded invertible operators on H. We assume that there exist constants δ > 0 and M > 0 independent of t such that Re (B(t

) -1 u, u) ≥ δ u 2 H ∀u ∈ H, (2.3.9)
and

B(t) -1 L(H) ≤ M. (2.3.10)
Let (P (t)) t∈[0,τ ] be a family of bounded operators on H. We assume that

P (t) L(H) ≤ M. (2.3.11)
As a consequence of (2.3.9) and (2.3.10) the numerical range of B(t) -1 is contained in a sector of angle w 1 for some w 1 ∈ [0, π 2 ), independent of t. Note that (2.3.9) implies that B(t) -1 u ≥ δ u We denote as previously by w 0 the common angle of the numerical range of forms a(t), t ∈ [0, τ ]. We assume again that

w 0 + w 1 < π 2 . (2.3.13)
The following is our main result.

|a(t, u, v) -a(s, u, v)| ≤ ω(|t -s|) u V β v Vγ , u, v ∈ V (2.3.14)
where ω : [0, τ ] → [0, ∞) is a non-decreasing function such that :

τ 0 ω(t) t 1+ γ 2 dt < ∞. (2.3.15)
Then the Cauchy problem

u (t) + B(t)A(t)u(t) + P (t)u(t) = f (t), u(0) = 0 (2.3.16
)

has maximal L p -regularity in H for all p ∈ (1, ∞). If in addition, τ 0 ω(t) p t 1 2 (β+pγ) dt < ∞ (2.3.17) then u (t) + B(t)A(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 (2.3.18)
has maximal L p -regularity in H provided u 0 ∈ (H, D(A(0))) 1-1 p ,p . Moreover there exists a positive constant C such that :

u W 1 p (0,τ ;H) + BAu Lp(0,τ ;H) ≤ C f Lp(0,τ ;H) + u 0 (H,D(A(0))) 1-1 p ,p .
(2.3.19) The constant C depends on the constants in [H1]-[H3], δ, M in (2.3.9)-(2.3.11) as well as β, γ and p and it is independent of f and u 0 .

Remark. As we shall see in the proof, the regularity assumption on B(t) can be weakened considerably. Indeed, continuity at finite number of appropriate points is sufficient.

Non-autonomous right and left multiplicative perturbations and maximal regularity

Before starting the proof, let us define the maximal regularity space M R(p, H) := {u ∈ W 1 p (0, τ ; H) : u(t) ∈ D(A(t)) a.e., A(.)u(.) ∈ L p (0, τ ; H)}.

It is a Banach space for the norm

u M R(p,H) := u W 1 p (0,τ ;H) + Au Lp(0,τ ;H) . Proof. Let f ∈ L p (0, τ ; H) and u 0 ∈ (H, D(A(0)) 1-1 p ,p
. By Proposition 2.3.5, there exists a unique u ∈ M R(p, H) such that

u (t) + B(0)A(t)u(t) + P (t)u(t) = f (t) u(0) = u 0 .
Hence, for a given v ∈ M R(p, H), there exists a unique u ∈ M R(p, H) such that

u (t) + B(0)A(t)u(t) + P (t)u(t) = f (t) + (B(0) -B(t))A(t)v(t) u(0) = u 0 . (2.3.20)
We define

S : M R(p, H) → M R(p, H) v → u.
For v 1 , v 2 ∈ M R(p, H) we set u 1 := Sv 1 and u 2 := Sv 2 . Obviously, u := u 1 -u 2 satisfies

u (t) + B(0)A(t)u(t) + P (t)u = (B(0) -B(t))A(t)(v 1 -v 2 ) u(0) = 0.
Thus, by Proposition 2.3.5, there exists a constant C such that

u 1 -u 2 M R(p,H) ≤ C (B(0) -B(•))A(•)(v 1 -v 2 ) Lp(0,τ ;H) ≤ C sup t∈[0,τ ] B(0) -B(t) L(H) v 1 -v 2 M R(p,H) .
By continuity at 0, for > 0 there exists t 0 > 0 such that for t ∈ [0, t 0 ] (B(0) -B(t)) L(H) < .

Hence for τ = t 0 small enough, the operator S is a contraction on M R(p, H) and so it has a fixed point u ∈ M R(p, H).

i (t i-1 ) = u i-1 (t i-1
). The forgoing arguments prove existence and uniqueness of a solution on

[t i-1 , t i ] with maximal L p -regularity provided u i-1 (t i-1 ) ∈ (H, D(A(t i-1 ))) 1-1 p ,p .
Once we do this we glue these solutions and obtain a unique solution u of (2.3.16) with maximal L p -regularity on [0, τ ] for all u 0 ∈ (H, D(A(0))) 1-1 p ,p . Thus, our task now is to prove that u i-1 (t i-1 ) ∈ (H, D(A(t i-1 ))) 1-1 p ,p . In order to make the notation simpler, we work on [0, τ ] (with τ small enough) instead of [t i-1 , t i ] and set A := A(τ ). We have to prove that the solution u to (2.3.16) satisfies u(τ ) ∈ (H, D(A)) 1-1 p ,p . This means that (remember we always assume w.l.o.g. that the operators A(t) are invertible, i.e. ν = 0 in [H3])

t → Ae -tA u(τ ) ∈ L p (0, τ ; H). (2.3.22)
We start with an expression for u(τ ). Set v(s) := e -(τ -s)A u(s), 0 ≤ s ≤ τ.

We have

v (s) = Ae -(τ -s)A u(s) + e -(τ -s)A (-B(s)A(s)u(s) -P (s)u(s) + f (s)).
Hence

u(τ ) = e -τ A u 0 + τ 0 e -(τ -s)A (A(τ ) -A(s))u(s)ds (2.3.23) + τ 0 e -(τ -s)A [(I -B(s))A(s)u(s) -P (s)u(s) + f (s)] ds. Since Ae -(t+τ )A L(H) ≤ C t + τ it is clear that t → Ae -tA e -τ A u 0 ∈ L p (0, τ ; H).

Non-autonomous right and left multiplicative perturbations and maximal regularity

For the second term we have

A τ 0 e -(t+τ -s)A (A(τ ) -A(s))u(s)ds ≤ τ 0 C t + τ -s e -1 2 (t+τ -s)A (A(τ ) -A(s))u(s)ds ≤ C τ 0 ω(τ -s) (t + τ -s) 1+ γ 2 u(s) V β ds ≤ C τ 0 ω(t + τ -s) (t + τ -s) 1+ γ 2 u(s) V β ds,
where ω(r) = ω(r) for r ∈ [0, τ ] and = ω(τ ) for r > τ . Note that the second estimate uses Proposition 2.3.4, assertion 3 and assumption (2.3.14). Therefore, using the assumption (2.3.15) on ω and Young's inequality we obtain

τ 0 A τ 0 e -(t+τ -s)A (A(τ ) -A(s))u(s)ds p dt ≤ C u p Lp(0,τ ;V) . (2.3.24)
Now we consider the last term in (2.3.23). We start with the case p = 2. We have for every g ∈ L 2 (0, τ ; H)

τ 0 Ae -tA τ 0 e -(τ -s)A g(s)ds 2 dt 1/2 = τ 0 A 1/2 e -tA τ 0 A 1/2 e -(τ -s)A g(s)ds 2 dt 1/2 ≤ C τ 0 A 1/2 e -(τ -s)A g(s)ds .
In the last inequality we use the boundedness of the square function, namely

∞ 0 A(t) 1/2 e -rA(t) x 2 dr ≤ C x 2 (2.3.25)
for all x ∈ H. This estimate is a consequence of the fact that A(t) has a bounded holomorphic functional calculus as an accretive operator, see [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF].

We repeat the previous argument but since g is not necessarily constant in s we cannot use directly the square function estimate. We argue by duality. For

x ∈ H we have

|( τ 0 A 1/2 e -(τ -s)A g(s)ds, x)| = | τ 0 (g(s), A * 1/2 e -(τ -s)A * x)| ≤ ( τ 0 g(s) 2 ds) 1/2 ( τ 0 A * 1/2 e -(τ -s)A * x 2 ds) 1/2 ≤ C x ( τ 0 g(s) 2 ds) 1/2 .

Maximal regularity for left perturbations

Since this is true for all x ∈ H, we obtain

τ 0 Ae -tA τ 0 e -(τ -s)A g(s)ds 2 dt 1/2 ≤ C( τ 0 g(s) 2 ds) 1/2 .
We define the operator T by

T g(t) = τ 0
Ae -(τ +t-s)A g(s)ds.

We have proved that T : L 2 (0, τ ; H) → L 2 (0, τ ; H) is bounded. We extend this operator to L p (0, τ ; H) for all p ∈ (1, ∞). Indeed, note that T is a singular integral operator with kernel

K(t, s) = Ae -(τ +t-s)A
and we use Hörmander's integral condition for K(t, s) and K(s, t) (see, e.g. [START_REF] Rubio De Francia | Calderón-Zygmund theory for operator-valued kernels[END_REF], Theorems III 1.2 and III 1.3). A similar argument was used in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF]. We have to prove that |t-s|≥2|s -s|

K(t, s) -K(t, s ) L(H) dt ≤ C (2.3.26)
for some constant C independent of s, s ∈ (0, τ ). Note that we use here Hörmander's condition on an interval as a space of homogeneous type, see [START_REF] Rubio De Francia | Calderón-Zygmund theory for operator-valued kernels[END_REF], p. 15.

Assume for example that s ≤ s . Since the semigroup generated by -A is bounded holomorphic we have for some constant C |t-s|≥2|s -s|

K(t, s) -K(t, s ) L(H) dt ≤ τ 2s -s Ae -(τ +t-s)A -Ae -(τ +t-s )A L(H) dt = τ 2s -s s s A 2 e -(τ +t-r)A dr L(H) dt ≤ C τ 2s -s s s 1 (τ + t -r) 2 drdt = C τ 2s -s 1 τ + t -s - 1 τ + t -s dt = C log τ + t -s τ + t -s t=τ t=2s -s ≤ C log 2.
This proves (2.3.26). The same arguments apply for the kernel of the adjoint T * . Hence T : L p (0, τ ; H) → L p (0, τ ; H)

u W 1 p (t i ,t i+1 ;H) + A(•)u(•) Lp(t i ,t i+1 ;H) ≤ C f Lp(t i ,t i+1 ;H) + u(t i ) (H;D(A(t i ))) 1-1 p ;p . ( 2
A(t) + cI) -1 L(H,V) ≤ c 0 (c + 1) 1 2
it follows that

u(t) V ≤ (A(t) + cI) -1 L(H,V) (A(t) + cI)u(t) H ≤ c 0 (c + 1) 1 2
(A(t) + cI)u(t) H .

Summing over i in (2.3.28) and taking c large enough we see that for some constant

C 1 u W 1 p (0,τ ;H) + A(•)u(•) Lp(0,τ ;H) ≤ C 1 f Lp(0,τ ;H) + u 0 (H,D(A(0))) 1-1 p ,p .
This proves the desired a priori estimate since the operators B(t) are uniformly bounded with respect to t. This finishes the proof of the theorem.

Further regularity results

Further regularity results

We continue our investigations on the solution of the problem (4.2.7). We work with the same assumptions as in Theorem 2.3.6. For f ∈ L 2 (0, τ ; H) we know that the solution u is in W 1 2 (0, τ ; H) and hence u ∈ C([0, τ ]; H) by the Sobolev embedding. It is interesting to know whether u is also continuous for the norm of V. This is indeed the case if the forms a(t) are symmetric (or perturbations of symmetric forms) and t → a(t, x, y) is Lipschitz continuous on [0, τ ] for all x, y ∈ V. This is proved in [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF]. Continuity in V was also proved in [START_REF] Arendt | Maximal regularity for non-autonomous Robin boundary conditions[END_REF] for the unperturbed problem (i.e., without multiplicative and additive perturbations) when γ in (2.3.14) is < 1. This is a rather restrictive condition but turns out to be satisfied in some cases such as time-dependent Robin boundary conditions. Here we make no restriction on γ and β and we assume less regularity for t → a(t, x, y) than what was previously known. The continuity of the solution with respect to the norm of V is used in [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF] in applications to some semi-linear PDE's. In this section, we look again at this question in the setting of Theorem 2.3.6 in which we assume less regularity (than Lipschitz continuous) on t → a(t, g, h).

In the statements below we shall need the following square root property (called Kato's square root property)

D(A(t) 1/2 ) = V and c 1 A(t) 1/2 v ≤ v V ≤ c 2 A(t) 1/2 v (2.4.1)
for all v ∈ V and t ∈ [0, τ ], where the positive constants c 1 and c 2 are independent of t. Note that this assumption is always true for symmetric forms when ν = 0 in [H3]. It is also valid for uniformly elliptic operator on R n , see [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF].

We start with the following lemma which will be used later.

Lemma 2.4.1. Suppose (2.4.1). Then for all f ∈ L 2 (0, τ ; H),

0 ≤ s ≤ t ≤ τ , t s e -(t-r)A(t) f (r)dr V ≤ C f L 2 (s,t;H) . Proof. By (2.4.1), t s e -(t-r)A(t) f (r)dr V ≤ c 2 t s A(t) 1/2 e -(t-r)A(t) f (r)dr = c 2 sup x =1 | t s (f (r), A(t) * 1/2 e -(t-r)A(t) * x)dr| ≤ c 2 sup x =1 t s A(t) * 1/2 e -(t-r)A(t) * x 2 dr 1 2 × f L 2 (s,t;H) ≤ C f L 2 (s,t;H) .

Non-autonomous right and left multiplicative perturbations and maximal regularity

Note that in the last inequality we use again the square function estimate for A(t) * (see (2.3.25)). This proves the lemma.

In the next result we prove continuity of the solution to (4.2.7) as a function with values in V. Note that if D(A(0) 1/2 ) = V, then (D(A(0), H)1

2 ,2 = D(A(0) 1/2 ) = V. See [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF], Corollaries 4.37 and 4.30.

Theorem 2.4.2. Suppose (2.4.1) and that the assumptions of Theorem 2.3.6 are satisfied. Suppose also that ω(t) ≤ ct ε for some ε > 0. Let f ∈ L 2 (0, τ ; H) and u 0 ∈ V. Then the solution u to the problem (4.2.7) satisfies u ∈ C([0, τ ]; V).

Proof. We fix s and t in [0, τ ] such that s < t. We first derive a formula similar to (2.3.23). Define v(r) := e -(t-r)A(t) u(r) for r ∈ [s, t]. After differentiation and integration from s to t we obtain

u(t) = e -(t-s)A(t) u(s) + t s e -(t-r)A(t) (A(t) -B(r)A(r))u(r)dr + t s e -(t-r)A(t) [-P (r)u(r) + f (r)]dr.
Hence u(t) = e -(t-s)A(t) u(s) + and so

u(t) -u(s) V ≤ t s e -(t-r)A(t) (A(t) -A(r))u(r)dr V + t s e -(t-r)A(t) [(I -B(r))A(r)u(r) -P (r)u(r) + f (r)]dr V + [e -(t-s)A(t) -I]u(s) V .
We

Further regularity results

Hence

u(t) -u(s) V ≤ C( t s ω(t -r) t -r u(r) V dr + f L 2 (s,t;H) + Au L 2 (s,t;H) + u L 2 (s,t;H) ) + [e -(t-s)A(t) -I]u(s) V ≤ C( t-s 0 ω(r) r dr u L∞(0,τ ;V) + f L 2 (s,t;H) + Au L 2 (s,t;H) + u L 2 (s,t;H) ) + [e -(t-s)A(t) -I]u(s) V .
Note that M R(2, H) is continuously embedded into L ∞ (0, τ ; V) by Proposition 4.5 in [START_REF] Monniaux | The incompressible Navier-Stokes system with time-dependent Robin-type boundary conditions[END_REF]. This proposition is proved for forms which are symmetric but it remains true under the assumption (2.4.1). Thus, by maximal regularity result (Theorem 2.3.6), u ∈ M R(2, H) and hence u ∈ L ∞ (0, τ ; V). Next, since ω(t) ≤ ct ε by assumptions then

τ 0 ω(r)
r dr < ∞ and since u, Au ∈ L 2 (0, τ ; H) we see that the first four terms in the RHS converge to 0 as t → s (or as s → t). It remains to prove that [e -(t-s)A(t) -I]u(s) V → 0 as t → s (or as s → t).

(

We first prove (2.4.3) when t → s (for fixed s). We write [e -(t-s)A(t) -I]u(s) V ≤ [e -(t-s)A(t) -e -(t-s)A(s) ]u(s) V

+ [e -(t-s)A(s) -I]u(s) V .

We estimate the first term on the RHS. We write again by the functional calculus

e -(t-s)A(t) -e -(t-s)A(s) = 1 2π Γ e -(t-s)λ (λ-A(t)) -1 (A(t)-A(s))(λ-A(s)) -1 dλ where Γ = ∂Σ(θ) for appropriate θ ∈ (0, π 2 ). Note that for v ∈ V (λI -A(t)) -1 v V ≤ C A(t)(λI -A(t)) -1 v V = C v -λ(λI -A(t)) -1 v V and hence (λI -A(t)) -1 v V ≤ C θ v V
for all λ ∈ Γ. On the other hand, the usual resolvent estimates hold on V (for the part of the operator A(t) on V) as in V (this can also be seen by duality).

Non-autonomous right and left multiplicative perturbations and maximal regularity

Therefore,

[e -(t-s)A(t) -e -(t-s)A(s) ]u(s)

V = 1 2π Γ e -(t-s)λ (λ -A(t)) -1 (A(t) -A(s))(λ -A(s)) -1 u(s)dλ V ≤ cω(t -s) Γ e -(t-s)|λ| cos θ (1 + |λ|) -1 d|λ| u(s) V ≤ c ω(t -s) (t -s) ε u(s) V .
Here we use u(s) ∈ D(A(s)

2 ) (see the proof of Theorem 2.3.6) and (2.4.1). We have proved

[e -(t-s)A(t) -e -(t-s)A(s) ]u(s) V ≤ c ω(t -s) (t -s) ε u(s) V (2.4.4)
for appropriate small ε > 0. Now the fact that ω(t) ≤ ct ε for some ε > 0 and the strong continuity of the semigroup e -tA(s) on V imply that [e -(t-s)A(t) -I]u(s) V → 0 as t → s. This proves that u is right continuous for the norm of V.

It remains to prove left continuity of u. We need a formula similar to (2.4.2) but with u(s) expressed in terms of u(t). Fix 0 ≤ s < t ≤ τ and set v(r) := e -(r-s)A(s) u(r) for r ∈ [s, t]. Then v (r) = -e -(r-s)A(s) (A(s) + B(r)A(r) + P (r))u(r) + e -(r-s)A(s) f (r), and hence u(s) = e -(t-s)A(s) u(t) + t s e -(r-s)A(s) (A(s) + B(r)A(r))u(r)dr (2.4.5)

- t s e -(r-s)A(s) [f (r) -P (r)u(r)]dr. Therefore u(s) -u(t) = [e -(t-s)A(s) u(t) -u(t)] + t s e -(r-s)A(s) (A(s) -A(r))u(r)dr + t s e -(r-s)A(s) ((B(r) + I)A(r)u(r))dr - t s
e -(r-s)A(s) [f (r) -P (r)u(r)]dr =: I 1 (s, t) + I 2 (s, t) + I 3 (s, t) + I 4 (s, t).

By Lemma 2.4.1,

I 4 (s, t) V ≤ C[ u L 2 (s,t;H) + f L 2 (s,t;H) ].

Further regularity results

By Lemma 2.4.1

I 3 (s, t) V ≤ C A(•)u(•) L 2 (s,t;H) .
For I 2 (s, t) we have immediately,

I 2 (s, t) V ≤ C t s
w(r -s) r -s dr u L∞(0;τ,V) .

For I 1 (s, t) we write

[e -(t-s)A(s) u(t) -u(t)] = [e -(t-s)A(s) u(t) -e -(t-s)A(t) u(t)] + [e -(t-s)A(t) u(t) -u(t)]
and use again the functional calculus to obtain as in (2.4.4)

[e -(t-s)A(s) u(t) -e -(t-s)A(t) u(t)] V ≤ c ω(t -s) (t -s) ε u(t) V .
The remaining term e -(t-s)A(t) u(t) -u(t) V converges to 0 as s → t by strong continuity of the semigroup on V. We have proved that u is left continuous in V and finally u ∈ C([0, τ ]; V).

Proposition 2.4.3. Suppose that the assumptions of the previous theorem are satisfied. Let f ∈ L 2 (0, τ ; H), u 0 ∈ V and u be the solution of (4.2.7). Then

s → A(s) 1 2 u(s) ∈ C([0, τ ]; H).
Proof. We use again (2.4.2) and write

A(t) 1 2 u(t) -A(s) 1 2 u(s) = A(t)
1 2 e -(t-s)A(t) u(s) -A(s)

1 2 u(s) + A(t) 1 2
t s e -(t-r)A(t) (A(t) -A(r))u(r)dr

+ A(t) 1 2
t s e -(t-r)A(t) [(-B(r) + I)A(r)u(r) -P (r)u(r) + f (r)]dr.

By (2.4.1), the norms in H of the last two terms are equivalent to the norms in V of the same terms but without A(t)

1 2
. We have seen in the proof of Theorem 2.4.2 that these norms in V converge to 0 as t → s or as s → t. It remains to consider the term A(t) 1 2 e -(t-s)A(t) u(s)-A(s) 1 2 u(s). We use again the functional calculus to write

A(t) 1 2 e -(t-s)A(t) u(s) -A(s) 1 2 u(s) = A(t) 1 2 e -(t-s)A(t) u(s) -A(s) 1 2 e -(t-s)A(s) u(s) + A(s) 1 2 e -(t-s)A(s) u(s) -A(s) 1 2 u(s) = 1 2πi Γ λ 1 2 e -(t-s)λ [(λ -A(t)) -1 -(λ -A(s)) -1 ]u(s)dλ + A(s)
1 2 e -(t-s)A(s) u(s) -A(s) 1 2 u(s).

Non-autonomous right and left multiplicative perturbations and maximal regularity

By the resolvent equation

(λ -A(t)) -1 -(λ -A(s)) -1 = (λ -A(t)) -1 (A(t) -A(s))(λ -A(s)) -1
and Proposition 2.3.4 we have

(λ -A(t)) -1 -(λ -A(s)) -1 L(V,H) ≤ (λ -A(t)) -1 L(V ,H) (A(s) -A(t)) L(V,V ) (λ -A(s)) -1 L(V) ≤ C|λ| -1/2 ω(|t -s|) 1 1 + |λ| .
Therefore for any ε > 0,

A(t) 1 2 e -(t-s)A(t) u(s) -A(s) 1 2 e -(t-s)A(s) u(s) ≤ Cω(|t -s|) ∞ 0 1 1 + r e -(t-s)r dr u(s) V ≤ C ε ω(|t -s|) |t -s| ε u(s) V .
Remember that u ∈ L ∞ (0, τ ; V) by Theorem 2.4.2. Using the assumption on ω, the latest term converges to 0 as |t -s| → 0. The term A(s)

1 2 e -(t-s)A(s) u(s) -A(s) 1 2 u(s) converges to 0 as t → s by the strong continuity of the semigroup on V. This proves the right continuity of s → A(s) 1 2 u(s). The left continuity is proved similarly, we use (2.4.5) instead of (2.4.2).

Right perturbations-Maximal regularity

Let B(t) and P (t) (t ∈ [0, τ ]) be bounded operators on H. We investigate the maximal L p -regularity property for right multiplicative perturbations A(t)B(t). As mentioned in the introduction, this problem has been considered in [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF] and was motivated there by several applications. We will extend the results from [START_REF] Augner | On the right multiplicative perturbation of non-autonomous L p -maximal regularity[END_REF] in the sense that we require much less regularity for t → a(t).

Let a(t) be a family of sesquilinear forms satisfying again [H1]-[H3] and denote as before A(t) the corresponding associate operators. Under the assumptions of Theorem 2.3.6, for each t, the operator -B(t)A(t) generates a holomorphic semigroup on H. The same is also true for -B(t) * A(t) * since the adjoint operators B(t) * and A(t) * satisfy the same properties as B(t) and A(t). Hence by duality, -A(t)B(t) generates a holomorphic semigroup on H. Here, the domain of A(t)B(t) is given by

D(A(t)B(t)) = {x ∈ H, B(t)x ∈ D(A(t))}.
For right perturbations, we say that the Cauchy problem u (t) + A(t)B(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 2.5. Right perturbations-Maximal regularity has maximal L p -regularity if for every f ∈ L p (0, τ, H) there exists a unique u ∈ W 1 p (0, τ ; H), B(t)u(t) ∈ D(A(t)) a.e. and u satisfies the Cauchy problem in the L p -sense.

Our main result in this section is the following.

[0, 1] |a(t, u, v) -a(s, u, v)| ≤ ω(|t -s|) u V β v Vγ , u, v ∈ V
where ω : [0, τ ] → [0, ∞) is a non-decreasing function such that :

τ 0 ω(t) t 1+ γ 2 dt < ∞.
Then the Cauchy problem

u (t) + A(t)B(t)u(t) + P (t)u(t) = f (t), u(0) = 0 (2.5.1) has maximal L p -regularity in H for all p ∈ (1, ∞). If in addition, τ 0 ω(t) p t 1 2 (β+pγ) dt < ∞ (2.5.2) then u (t) + A(t)B(t)u(t) + P (t)u(t) = f (t), u(0) = u 0 (2.5.3)
has maximal L p regularity in H provided u 0 ∈ B(0) -1 (H, D(A(0))) 1-1 p ,p . Moreover there exists a positive constant C such that :

u W 1 p (0,τ ;H) + A(•)B(•)u(•) Lp(0,τ ;H) (2.5.4) ≤ C f Lp(0,τ ;H) + B(0)u 0 (H,D(A(0))) 1-1 p ,p .
We start with the following lemma.

Lemma 2.5.2. Under the above assumptions on B(t), t → B(t) -1 x is differentiable on (0, τ ) with values in L(H) and

d dt B(t) -1 x = -B(t) -1 B (t)B(t) -1 x
for all x ∈ H.

Proof. We write

B(t + h) -1 -B(t) -1 = -B(t + h) -1 (B(t + h) -B(t))B(t) -1 (2.5.5)
and since B(t + h) -1 has norm bounded with respect to h it follows that B(t + h) -1 converges uniformly to B(t) -1 . Using this and the fact that t → B(t)x is Lipschitz we obtain the lemma.

Lions' maximal regularity problem with

H 1 2 -regularity in time • [H2]: |a(t, u, v)| ≤ M u V v V for t ∈ [0, τ ],
u, v ∈ V and some constant M > 0 (uniform boundedness),

• [H3]: Re a(t, u, u)+ν u 2 ≥ δ u 2 V for u ∈ V and some δ > 0 and ν ∈ R (uniform quasi-coercivity).

We denote by A(t), A(t) the usual associated operators with a(t) as operators on H and V , respectively. In particular, A(t) : V → V as a bounded operator and a(t, u, v) = A(t)u, v for all u, v ∈ V.

The operator A(t) is the part of A(t) on H. We consider the non-homogeneous Cauchy problem

u (t) + A(t) u(t) = f (t), t ∈ (0, τ ] u(0) = u 0 . (P) Definition 3.1.1.
The Cauchy problem (P) has maximal L 2 -regularity in H if for every f ∈ L 2 (0, τ ; H), there exists a unique u ∈ H 1 (0, τ ; H) with u(t) ∈ D(A(t)) for a.e. t ∈ [0, τ ] and u is a solution of (P) in the L 2 -sense.

By a very well known result of J.L. Lions, maximal L 2 -regularity always holds in the space V . That is, for every f ∈ L 2 (0, τ ; V ) and u 0 ∈ H there exists a unique u ∈ H 1 (0, τ ; V ) ∩ L 2 (0, τ ; V) which solves the equation

u (t) + A(t) u(t) = f (t), t ∈ (0, τ ] u(0) = u 0 . (P')
In applications one needs however maximal regularity in H (for example for elliptic boundary value problems one has to work on H rather than V in order to identify the boundary conditions). Maximal regularity in H differs considerably from the same property in V . Before we recall known results and explain our main contribution in this paper, we recall that one of the reasons why maximal regularity (both in the autonomous and non-autonomous cases) was intensively studied is due to the fact that it is a very useful tool to prove existence results for non-linear evolution equations.

For symmetric forms a(t), Lions [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] (IV Sec. 6, Théorème 6.1]) proved that if t → a(t, u, v) is C 1 and u 0 = 0, then maximal L 2 -regularity in H is satisfied. For general u 0 ∈ D(A(0)), Lions imposes the stronger regularity property that t → a(t, u, v) is C 2 . Bardos [START_REF] Bardos | A regularity theorem for parabolic equations[END_REF] improves the latter result for forms satisfying the uniform Kato square root property (see Definition 3.3.4 below) by assuming that A(.) 1 2 is continuously differentiable with values in L(V, V ) and u 0 ∈ V. Ouhabaz and Spina [START_REF] Ouhabaz | Maximal regularity for non-autonomous Schrödinger type equations[END_REF] proved maximal regularity in H if t → a(t, u, v) is C α for some α > 1 2 when u 0 = 0. This result was extended in Haak and Ouhabaz [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] who prove maximal L p -regularity under a slightly better regularity condition and allowing u 0 ∈ D(A(0) 1 2 ). Dier [START_REF] Dier | Non-autonomous maximal regularity for forms of bounded variation[END_REF]

Main results

proved maximal L 2 -regularity for symmetric forms such that t → a(t, u, v) is of bounded variations. Fackler [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF] proved that the order α > 1 2 in [START_REF] Ouhabaz | Maximal regularity for non-autonomous Schrödinger type equations[END_REF] or [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] is optimal in the sense that there exist a(.) symmetric and C 1 2 for which maximal regularity in H fails. A counter-example already appeared in Dier [START_REF] Dier | Non-Autonomous Cauchy Problems Governed by Forms[END_REF] and it is based on a form which does not satisfy the Kato square root property. Dier and Zacher [START_REF] Dier | Non-autonomous maximal regularity in Hilbert spaces[END_REF] proved that if t → A(t) is in the fractional Sobolev space H 1 2 +δ (0, τ ; L(V, V )) for some δ > 0 then maximal L 2 -regularity in H holds. For a Banach space version of this result, see Fackler [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF]. The example in [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF] is not a differential operator. For elliptic operators in divergence form on R n , Auscher and Egert [START_REF] Auscher | On non-autonomous maximal regularity for elliptic operators in divergence form[END_REF] proved maximal regularity if the coefficients satisfy a certain BMO-H 1 2 condition. The example from [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] also shows that A(.) ∈ W 1 2 ,p (0, τ ; L(V, V )) for p > 2 is not enough to obtain maximal regularity. The example in [START_REF] Dier | Non-autonomous maximal regularity for forms of bounded variation[END_REF] shows that A(.) ∈ W 1 2 ,p (0, τ ; L(V, V )) for p < 2 does not imply maximal regularity, at least for form which does not satisfy Kato's square root property. For a discussion on these negative results, see the review paper of Arendt, Dier and Fackler [START_REF] Arendt | Lions' problem on maximal regularity[END_REF]. As pointed in [START_REF] Arendt | Lions' problem on maximal regularity[END_REF], the remaining problem is the case of fractional regularity H 1 2 . We solve this problem in the present paper. Our main result shows that for forms satisfying the uniform Kato square root property and an integrability condition (see ( 3

.2.1) below), if t → A(t) is piecewise in the Sobolev space H 1 2 (0, τ ; L(V, V )) then maximal L 2 -regularity in H is satisfied. The initial data u 0 is arbitrary in V.
This result is optimal. The required Soblev regularity cannot be smaller than

1 2 since C 1 2 ⊂ H α for α < 1 2 .
In the case where A(t) -A(s) maps into the dual space of [H, V] γ we allow the fractional Sobolev regularity to be γ 2 . This extends related results in Ouhabaz [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF] and Arendt and Monniaux [START_REF] Arendt | Maximal regularity for non-autonomous Robin boundary conditions[END_REF].

We give the precise statements of the main results in the next section. In Sections 3.3 and 3.4 we prove several key estimates and develop the necessary tools for the proofs of the main results. Some of these tools are quadratic estimates and L ∞ (0, τ ; V)-estimates for the solution of the Cauchy problem. The main results are proved in Section 3.5 and several examples are given in Section 3.6.

Main results

In this section we state explicitly our main results. For clarity of exposition we consider separately the cases γ = 1, γ ∈ (0, 1) and γ = 0.

We start by recalling the definition of vector-valued fractional Sobolev spaces. Definition 3.2.1. Let X be a Banach space, α ∈ (0, 1) and I an open subset

1 2 -regularity in time of R. A function f ∈ L 2 (I; X) is in the fractional Sobolev space H α (I; X) if f 2 H α (I;X) := f 2 L 2 (I;X) + I×I f (t) -f (s) 2 X |t -s| 2α+1 dsdt < ∞.
We say that f is in the homogeneous Sobolev space Ḣα (I; X) if

f 2 Ḣα (I;X) := I×I f (t) -f (s) 2 X |t -s| 2α+1 dsdt < ∞.
We shall say that f is piecewise in H α (I; X) (resp. Ḣα (I; X)) if there exists t 0 < t 1 < ... < t n such that I = ∪ i [t i , t i+1 ] and the restriction of f to each sub-interval

(t i , t i+1 ) is in H α (t i , t i+1 ; X) (resp. Ḣα (t i , t i+1 ; X)).
Let a(t) : V × V → C for 0 ≤ t ≤ τ be a family of forms satisfying [H1]-[H3] and let A(t) and A(t) be the associated operators on H and V , respectively. We shall need the following property.

Given ε > 0, there exists

τ 0 = 0 < τ 1 < ... < τ n = τ such that sup t∈(τ i-1 ,τ i ) τ i τ i-1 A(t) -A(s) 2 L(V,V ) |t -s| ds < ε. (3.2.1)
Note that this assumption is satisfied in many cases. Suppose for example that t → a(t, u, v) is C α for some α > 0 in the sense that

|a(t, u, v) -a(s, u, v)| ≤ M |t -s| α u V v V (3.2.2)
for some positive constant M and all u, v ∈ V. Then clearly

A(t) -A(s) L(V,V ) ≤ M |t -s| α
and this implies (3.2.1). More generally, if ω i denotes the modulus of continuity of A on the interval (τ i-1 , τ i ) then (3.2.1) is satisfied if V,V )) and satisfies (3.2.1) then (P) has maximal L 2 -regularity in H for all u 0 ∈ V. In addition, there exists a positive constant C independent of u 0 and f such that

|r|≤τ i -τ i-1 ω i (r) 2 r dr < ε. ( 3
A(.)u(.) L 2 (0,τ ;H) + u H 1 (0,τ ;H) ≤ C u 0 V + f L 2 (0,τ ;H) . (3.2.4)
We refer to the next section for the definition of the uniform Kato square root property and for few more details on such property.

We have the following corollary which recovers the maximal regularity result proved [START_REF] Dier | Non-autonomous maximal regularity in Hilbert spaces[END_REF] under the assumption that t → A(t) is in H 1 2 +δ (0, τ ; L(V, V )).

Main results

Corollary 3.2.3. Suppose that [H1]-[H3] and the uniform Kato square property are satisfied. Suppose that t → A(t) is piecewise in H 1 2 +δ (0, τ ; L(V, V )) for some δ > 0. Then (P) has maximal L 2 -regularity in H for all u 0 ∈ V. In addition, there exists a positive constant C independent of u 0 and f such that

A(.)u(.) L 2 (0,τ ;H) + u H 1 (0,τ ;H) ≤ C u 0 V + f L 2 (0,τ ;H) .
(3.2.5)

Proof. It follows from [START_REF] Simon | Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval[END_REF], Corollary 26 that H

1 2 +δ (τ i-1 , τ i ; L(V, V )) is contin- uously embedded into C δ (τ i-1 , τ i ; L(V, V )).
As explained above this implies condition (3.2.1). We then apply Theorem 3.2.2.

Let γ ∈ [0, 1] and V γ := [H, V] γ be the usual complex interpolation space between H and V. We denote by V γ := (V γ ) its (anti-) dual. In some situations, one may have A(t) -A(s) : V → V γ for some γ ∈ [0, 1) (see Section 3.3 for some additional details). For example, this happens for forms a(t) associated with differential operators such that the difference a(t) -a(s) has only terms of smaller order or boundary terms. In this case the required regularity in the previous theorem can be improved. Before we state the results we introduce the following assumption Ḣ γ 2 (0, τ ; L(V, V γ )), then (P) has maximal L 2 -regularity in H for all u 0 ∈ V. In addition, there exists a positive constant C independent of u 0 and f such that

• [H4]: A(t) -A(s) L(V,V γ ) ≤ M γ
A(.)u(.) L 2 (0,τ ;H) + u H 1 (0,τ ;H) ≤ C u 0 V + f L 2 (0,τ ;H) . (3.2.6) Note that if [H4] holds with γ = 0 then | A(t)u -A(s)u, v | ≤ M 0 u V v
for all u, v ∈ V. In that case A(t) -A(s) defines a bounded operator from V into H. This implies in particular that the operators A(t) have the same domain D(A(t)) = D(A(0)). For operators satisfying the later property maximal regularity of (P) holds under the assumption that t → A(t)v is relatively continuous for all v ∈ D(A(0)), see [START_REF] Arendt | L p -maximal regularity for non-autonomous evolution equations[END_REF], Theorem 3.3. See also [START_REF] Gallarati | Maximal regularity for non-autonomous equations with measurable dependence on time[END_REF] where the later regularity assumption is replaced by a certain commutation property. We prove here that maximal regularity holds without requiring any property on the operators (or the forms). More precisely, we have

2 ) = V for some t 0 ∈ [0, τ ]. Then (P) has maximal L 2 -regularity in H for all u 0 ∈ V. In addition, there exists a positive constant C independent of u 0 and f such that

A(.)u(.) L 2 (0,τ ;H) + u H 1 (0,τ ;H) ≤ C u 0 V + f L 2 (0,τ ;H) .
(3.2.7)

Preparatory lemmas

In this section we prove several estimates which will play an important role in the proofs of the main results. We emphasize that one of the important points here is to prove estimates with constants which are independent of t.

Before we start let us point out that we may assume without loss of generality that assumption [H3] is satisfied with ν = 0, that is the forms are coercive with constant δ > 0 independent of t. The reason is that the maximal regularity of (P ) is equivalent to the same property for

v (t) + (A(t) + ν)v(t) = g(t), v(0) = u 0 . (3.3.1)
This can be seen by observing that for g(t) := f (t)e -νt , then v(t) = u(t)e -νt and clearly v ∈ H 1 (0, τ ; H) if and only if u ∈ H 1 (0, τ ; H) (and obviously f ∈ L 2 (0, τ ; H) if and only if g ∈ L 2 (0, τ ; H)).

When [H3] holds with ν = 0 then clearly the operators A(t) are invertible on H. In addition, one has the resolvent estimate

(µ + A(t)) -1 L(H) ≤ C 1 + µ (3.3.2)
for all µ ≥ 0. The constant is independent of t ∈ [0, τ ] (see e.g., Proposition 1.1.8 or [START_REF] Arendt | Maximal regularity for evolution equations governed by non-autonomous forms[END_REF], Proposition 2.1). The same estimate holds for

A(t) on V . Recall that V γ = [H, V] γ (for γ ∈ [0, 1]
) is the complex interpolation space between H and V and V γ := (V γ ) denotes its (anti-) dual space. Lemma 3.3.1. Suppose that [H1]-[H3] are satisfied with ν = 0. Then there exists a constant C > 0 such that the following estimates hold for all µ ≥ 0, r > 0 and all t ∈ [0, τ ].

1-(µ + A(t)) -1 L(V γ ,V) ≤ C (µ+1) 1-γ 2 , 2-(µ + A(t)) -1 L(V γ ,H) ≤ C (µ+1) 1-γ 2 , 3-e -rA(t) L(V γ ,H) ≤ C r γ 2
.

Preparatory lemmas

Proof. The arguments are classical but we write them here for clarity of the exposition.

We have for w ∈ V

δ (µ + A(t)) -1 w 2 V ≤ Re A(t)(µ + A(t)) -1 w, (µ + A(t)) -1 w = Re w -µ(µ + A(t)) -1 w, (µ + A(t)) -1 w ≤ w V (µ + A(t)) -1 w V + C (µ + A(t)) -1 w V , which gives (µ + A(t)) -1 L(V ,V) ≤ C . A similar argument gives the estimate (µ + A(t)) -1 L(H,V) ≤ C (µ+1) 1 2
. The first assertion follows by interpolation.

For the second one we use

(µ + A(t)) -1 L(V γ ,H) ≤ (µ + A(t)) -1 γ L(V ,H) (µ + A(t)) -1 1-γ L(H) ≤ C (µ + 1) 1-γ 2 .
We have e rA (t) L(H) ≤ 1 and e -rA(t) L(V ,H) ≤ C √ r (see e.g., [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF], Proposition 6). Since V γ = [V , H] 1-γ we use interpolation and obtain the third estimate.

We make some comments on property [H4]. Let γ ∈ [0, 1] and consider for fixed t, s

∈ [0, τ ] A(t) -A(s) ∈ L(V, V γ ). ( 3
|a(t, u, v) -a(s, u, v)| ≤ C t,s u V v Vγ (3.3.4)
for some positive constant C t,s and all u, v ∈ V. Morover, one can take C t,s = A(t) -A(s) L(V,V γ ) . In order to see this, one writes for u, v ∈ V

a(t, u, v) -a(s, u, v) = A(t)u -A(s)u, v , (3.3.5)
and obtains immediately that (3.3.3) implies (3.3.4). For the converse, we note that by (3.3.5), v → A(t)u -A(s)u, v extends to a (anti-) linear continuous functional on V γ (for fixed u ∈ V). The rest of the claim is easy to check. Similarly to the previous remark, A(t) -A(s) extends to a bounded operator from V γ to V if and only if

|a(t, u, v) -a(s, u, v)| ≤ C t,s u Vγ v V (3.3.6)
for al u, v ∈ V.

Our next lemma shows stability of the Kato square root property if (3.3.4) or (3.3.6) holds for some γ ∈ [0, 1).

Preparatory lemmas

This gives u ∈ D(A(t) 1 2 ) and

A(t) 1 2 u ≤ CC t,s u V + A(s) 1 2 u
which proves the lemma. Note that if we assume (3.3.6) then we argue by duality and prove the lemma for A(t) * 1 2 . It is well known that the equality

D(A(t) * 1 2 ) = V is equivalent to D(A(t) 1
2 ) = V.

with a constant C independent of t ∈ [0, τ ] (i.e., C t,s ≤ C for all t). If D(A(s) 1 2 ) = V, then D(A(t) 1
2 ) = V for all t ∈ [0, τ ] and there exist positive constants C 1 , C 2 such that

C 1 u V ≤ A(t) 1 2 u ≤ C 2 u V for all u ∈ V and all t ∈ [0, τ ]. (3.3.7)
Proof. All the details are already given in the proof of the previous lemma.

Definition 3.3.4. We say that (A(t)) (or the corresponding forms a(t)) satisfy the uniform Kato square root property if D(A(t)

1 2 ) = V for all t ∈ [0, τ ] and there exist constants C 1 , C 2 > 0 such that (3.3.7) is satisfied.
The uniform Kato square root property is obviously satisfied for symmetric forms. It is also satisfied for uniformly elliptic operators (not necessarily symmetric)

A(t) = - d k,l=1 ∂ k (a kl (t, x)∂ l ) on L 2 (R d ) since ∇u 2 is equivalent to A(t)
1 2 u 2 with constants depending only on the dimension and the ellipticity constants. See [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF]. The previous proposition says that in order to have the uniform Kato square root property one needs only to check that D(A(s)

1 2 ) = V for one s ∈ [0, τ ] provided (3.3.4) or (3.3.6) holds for some γ ∈ [0, 1).
In the next lemma we show a quadratic estimate for A(t) with constant independent of t. Here we assume the uniform Kato square root property and give a short proof for the quadratic estimate. It is possible to prove the same result without the uniform Kato square root property by proving that the holomorphic functional calculus of A(t) has uniform estimate with respect to t (this is the case since the resolvent have uniform estimates). It is well known that quadratic estimates in H are intimately related to the holomorphic functional calculus, see [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF]. Quadratic estimates are an important tool in harmonic analysis and we will use them at several places in the proofs of maximal regularity.

Lions' maximal regularity problem with H

t ∈ [0, τ ] τ 0 A(t) 1 2 e -sA(t) x 2 ds ≤ C x 2 (3.3.8)
for all x ∈ H.

Proof. By the uniform Kato square root property, we have

τ 0 A(t) 1 2 e -sA(t) x 2 ds ≤ C 2 τ 0 e -sA(t) x 2 V ds ≤ C τ 0
Re a(t, e -sA (t) x, e -sA(t) x) ds t) x, e -sA(t) x) ds

= C τ 0 Re (A(t)e -sA(
= - C 2 τ 0 d ds e -sA(t) x 2 ds = C 2 [ x 2 -e -τ A x 2 ] ≤ C 2 x 2 .
This proves the lemma.

We note that the constant C is also independent of τ . We could formulate the lemma with ∞ 0

A(t)

1 2 e -sA(t) x 2 ds. Let us also mention the following L p -version. Lemma 3.3.6. Suppose the assumptions of the previous lemma. Let p ≥ 2. Then there exists a constant C p such that

τ 0 A(t) 1/p e -sA(t) x p ds ≤ C p x p (3.3.9)
for all x ∈ H.

Proof. We fix t ∈ [0, τ ] and s > 0. We define

F (z)x := A(t) z/2 e -sA(t) x.
It is a classical fact that F is a holomorphic function on C + . In addition, each operator A(t), as an accretive operator on H, has bounded imaginary powers. That is A is L(H) ≤ e π |s| 2 for all s ∈ R. See Proposition 1.2.3 or [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF]. Using this one obtains immediately that

F (is)x L ∞ (0,τ ;H) ≤ e π |s| 2 x .

Key estimates

On the other hand, by Lemma 3.3.5 and again uniform boundedness of imaginary powers on H we obtain

F (1 + is)x L 2 (0,τ ;H) ≤ Ce π |s| 2 x ,
where C is a positive constant. We apply Stein's complex interpolation theorem to obtain that for all p ≥ 2

F ( 2 p )x L p (0,τ ;H) ≤ C p x
and we obtain the lemma.

Let u be the solution of (P') by Lions' theorem. Lions also proved that

u ∈ C([0, τ ]; H). Since u ∈ L 2 (0, τ ; V) we have u(t) ∈ V for a.e. t ∈ [0, τ ].
It is very useful to know whether u(t) ∈ V for every t ∈ [0, τ ]. We prove this in the following lemma under an additional assumption that u ∈ L ∞ (0, τ ; V). We shall see later that this property holds when we assume that (3.2.1) is satisfied. Lemma 3.3.7. Suppose [H1]-[H3] and suppose in addition that the solution u belongs to L ∞ (0, τ ; V). Then u(t) ∈ V for every t ∈ [0, τ ].

Proof. Let Γ = {t ∈ [0, τ ] s.t. u(t) / ∈ V}. Since Γ has measure zero, [0, τ ] \ Γ is dense in [0, τ ]. Suppose that t ∈ Γ. There exists a sequence (t n ) n ∈ [0, τ ] \ Γ such that t n → t as n → ∞. Since the sequence (u(t n ))
is bounded in V we can exact a subsequence u(t n k ) which converges weakly to some v in V. By continuity of u in H, u(t n k ) converges (in H) to u(t). This gives u(t) = v ∈ V. Hence Γ is empty.

Key estimates

In this section we state and prove some other estimates which we will need in the proofs of the main results.

L 0 (f )(t) := t 0 e -(t-s)A(t) f (s) ds. Then L 0 : L 2 (0, τ ; H) → L ∞ (0, τ ; V) is a bounded operator.
Proof. By the uniform Kato square root property,

L 0 (f )(t) V ≤ C 2 A(t) 1 2
t 0 e -(t-s)A(t) f (s) ds .

Lions' maximal regularity problem with H 1 2 -regularity in time

On the other hand, for x ∈ H |(A(t)

1 2 t 0 e -(t-s)A(t) f (s) ds, x)| = t 0 (f (s), A(t) * 1 2 e -(t-s)A(t) * x) ds ≤ f L 2 (0,τ ;H) t 0 A(t) * 1 2 e -(t-s)A(t) * x 2 ds 1 2 ≤ C f L 2 (0,τ ;H) x .
The latest inequality follows from Lemma 3.3.5 applied to the adjoint operator A(t) * (note that A(t) * satisfies the same properties as A(t)). The constant C is independent of t. Therefore,

L 0 (f )(t) V ≤ C 2 C f L 2 (0,τ ;H) . (3.4.1)
This implies immediately that L 0 :

L 2 (0, τ ; H) → L ∞ (0, τ ; V) is bounded.
Now we study boundedness on L 2 (0, τ ; H) of the operator

L(f )(t) := t 0 A(t)e -(t-s)A(t) f (s) ds. (3.4.2)
It is proved in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] that L is bounded on L p (0, τ ; H) for all p ∈ (1, ∞) provided t → a(t, ., .) is C for some > 0 (or similarly, t → A(t) is C on [0, τ ] with values in L(V, V )). The proof for the case p = 2 is based on vector-valued pseudo-differential operators. The extension from p = 2 to p ∈ (1, ∞) uses Hörmander's almost L 1 -condition for singular integral operators. Here we give a direct proof for the case p = 2 which does not appeal to pseudo-differential operators. It is essentially based on the quadratic estimate of Lemma 3.

sup s∈[0,τ ] τ s A(t) -A(s) 2 L(V,V γ ) |t -s| γ dt < ∞, (3.4.3)
then the operator L is bounded on L 2 (0, τ ; H).

Key estimates

Proof. Fix γ ∈ [0, 1]. Take g ∈ L 2 (0, τ ; H). We have

| τ 0 t 0 (A(t)e -(t-s)A(t) f (s), g(t)) dsdt| = | τ 0 t 0 A(t) 1 2 e -(t-s) 2 A(t) f (s), A(t) * 1 2 e -(t-s) 2 A(t) * g(t) dsdt| ≤ τ 0 t 0 A(t) 1 2 e -(t-s) 2 A(t) f (s) A(t) * 1 2 e -(t-s) 2 A(t) * g(t) dsdt ≤ τ 0 t 0 A(t) 1 2 e -(t-s) 2 A(t) f (s) 2 ds 1 2 t 0 A(t) * 1 2 e -(t-s) 2 A(t) * g(t) 2 ds 1 2 dt ≤ C τ 0 t 0 A(t) 1 2 e -(t-s) 2 A(t) f (s) 2 ds 1 2 g(t) dt ≤ C τ 0 t 0 A(t) 1 2 e -(t-s) 2 A(t) f (s) 2 dsdt 1 2 g L 2 (0,τ ;H) .
Here we use the quadratic estimate of Lemma 3.3.5 for the adjoint operator

A(t) * . It follows that for all f ∈ L 2 (0, τ ; H) L(f ) L 2 (0,τ ;H) ≤ C τ 0 t 0 A(t) 1 2 e -(t-s) 2 A(t) f (s) 2 dsdt 1 2 . (3.4.4)
We use again Lemma 3.3.5 and obtain

τ 0 t 0 A(t) 1 2 e -(t-s)A(t) f (s) 2 dsdt ≤ 2 τ 0 t 0 A(t) 1 2 e -(t-s)A(t) f (s) -A(s) 1 2 e -(t-s)A(s) f (s) 2 dsdt + 2 τ 0 τ s A(s) 1 2 e -(t-s)A(s) f (s) 2 dtds ≤ 2 τ 0 t 0 A(t) 1 2 e -(t-s)A(t) f (s) -A(s) 1 2 e -(t-s)A(s) f (s) 2 dsdt + 2C τ 0 f (s) 2 ds.
We choose a contour Γ in the positive half-plane and we write by the holomorphic functional calculus

A(t) 1 2 e -(t-s)A(t) f (s) -A(s) 1 2 e -(t-s)A(s) f (s) = 1 2πi Γ λ 1 2 e -(t-s)λ [(λ -A(t)) -1 -(λ -A(s)) -1 ] dλ f (s) = 1 2πi Γ λ 1 2 e -(t-s)λ (λ -A(t)) -1 (A(t) -A(s))(λ -A(s)) -1 f (s) dλ.

Lions' maximal regularity problem with

H 1 2 -

regularity in time

We estimate the norm in H of the latest term. For λ = |λ|e iθ we apply Lemma 3.3.1 and obtain

Γ λ 1 2 e -(t-s)λ (λ -A(t)) -1 (A(t) -A(s))(λ -A(s)) -1 f (s) dλ ≤ Γ |λ| 1 2 e -(t-s)|λ| cos θ (λ -A(t)) -1 L(V γ ,H) A(t) -A(s) L(V,V γ ) × (λ -A(s)) -1 L(H,V) |dλ| f (s) ≤ C Γ |λ| 1 2 e -(t-s)|λ| cos θ 1 (1 + |λ|) 1-γ/2 1 (1 + |λ|) 1 2 A(t) -A(s) L(V,V γ ) |dλ| f (s) ≤ C A(t) -A(s) L(V,V γ ) |t -s| γ/2 f (s) .
Here and at other places we use the estimate

∞ 0 e -r(t-s) (1 + r) 1-γ 2 dr ≤ C (t -s) γ 2 (3.4.5)
for some constant C and all s < t. This is seen by making the change of the variable v := r(t -s) in the LHS which then coincides with

1 (t -s) γ 2 ∞ 0 e -v (t -s + v) 1-γ 2 dv.
The latter term is bounded by

1 (t -s) γ 2 ∞ 0 e -v v 1-γ 2 dv = C (t -s) γ 2 .
The previous estimates give

τ 0 t 0 A(t) 1 2 e -(t-s)A(t) f (s) -A(s) 1 2 e -(t-s)A(s) f (s) 2 dsdt ≤ C τ 0 τ s A(t) -A(s) 2 L(V,V γ ) |t -s| γ dt f (s) 2 ds ≤ C sup s∈[0,τ ] τ s A(t) -A(s) 2 L(V,V γ ) |t -s| γ dt τ 0 f (s) 2 ds. (3.4.6)
This proves that L is bounded on L 2 (0, τ ; H).

Remark 3.4.3. One may argue as in the proof of Lemma 11 in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] and obtain boundedness of L on L p (0, τ ; H) at least for p ∈ (1, 2).

Corollary 3.4.4. 1) If A satisfies (3.2.1) then L is bounded on L 2 (0, τ ; H). 2) If [H4]
is satisfied for some γ ∈ (0, 1) then L is bounded on L 2 (0, τ ; H).

Key estimates

Proof. Assertion 1) follows directly from Proposition 3.4.2 by noticing that (3.2.1) implies (3.4.3) with γ = 1. For assertion 2) one uses [H4] to obtain

τ s A(t) -A(s) 2 L(V,V γ ) |t -s| γ dt ≤ M 2 γ τ s |t -s| -γ dt ≤ cM 2 γ
for some constant c > 0. The result follows from Proposition 3.4.2.

Note that if γ ∈ (0, 1) we do not require any regularity property for A in assertion 2) of the previous proposition. Proposition 3.4.5. Suppose [H1]-[H3] (with ν = 0) and the uniform Kato square root property. Let f ∈ L 2 (0, τ ; H), u 0 ∈ V and let u be the Lions' solution to the problem (P'). 1) If A satisfies (3.2.1) then u ∈ L ∞ (0, τ ; V) and there exists a constant C independent of u 0 and f such that

u L ∞ (0,τ ;V) ≤ C u 0 V + f L 2 (0,τ ;H) . (3.4.7)
2) Let γ ∈ [0, 1) and suppose that [H4] is satisfied. Then u ∈ L ∞ (0, τ ; V) and

Proof. As we already mentioned above, by Lions' theorem there exists a unique solution u to the problem (P') such that u ∈ H 1 (0, τ ; V ) ∩ L 2 (0, τ ; V). For 0 ≤ s ≤ t ≤ τ , we define v(s) := e -(t-s)A(t) u(s). We write v(t) = v(0) + t 0 v (s) ds and obtain as in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] (Lemma 8)

u(t) = e -tA(t) u 0 + t 0 e -(t-s)A(t) (A(t) -A(s))u(s) ds + t 0 e -(t-s)A(t) f (s) ds (3.4.8)
=: R 0 u 0 (t) + S 0 u(t) + L 0 f (t).

Clearly there exists a constant C > 0 such that for all u 0 ∈ V,

R 0 u 0 (t) V = e -tA(t) u 0 V ≤ C u 0 V . (3.4.9) By Lemma 3.4.1, L 0 f (t) V ≤ C f L 2 (0,τ ;H) . (3.4.10)
Next we prove that S 0 ∈ L(L ∞ (0, τ ; V)). Let g ∈ L ∞ (0, τ ; V). We have by the uniform Kato square root property

S 0 g(t) V ≤ C 2 t 0 A(t)
1 2 e -(t-s)A(t) (A(t) -A(s))g(s) ds .

Lions' maximal regularity problem with

H 1 2 -

regularity in time

In order to estimate the term on the RHS we argue as in the proof of Lemma 3.4.1 and use Lemma 3.3.1. For x ∈ H and γ ∈ [0, 1], we have

|( t 0 A(t) 1 2 e -(t-s)A(t) (A(t) -A(s))g(s) ds, x)| = | t 0 (e -(t-s) 2 A(t) (A(t) -A(s))g(s), A(t) * 1 2 e -(t-s) 2 A(t) * x) ds| ≤ t 0 e -(t-s) 2 A(t) (A(t) -A(s))g(s) 2 ds 1 2 t 0 A(t) * 1 2 e -(t-s) 2 A(t) * x 2 ds 1 2 ≤ C x t 0 e -(t-s) 2 A(t) 2 L(V γ ,H) A(t) -A(s) 2 L(V,V γ ) ds 1 2 g L ∞ (0,t;V) ≤ C x t 0 A(t) -A(s) 2 L(V,V γ ) (t -s) γ ds 1 2 g L ∞ (0,t;V) .
Therefore,

S 0 g(t) V ≤ CC 2 t 0 A(t) -A(s) 2 L(V,V γ ) (t -s) γ ds 1 2 g L ∞ (0,τ ;V) . (3.4.11)
Suppose γ = 1. It follows from the assumption (3.2.1) that S 0 is a bounded operator on L ∞ (0, τ ; V) with

S 0 L(L ∞ (0,τ ;V)) ≤ CC 2 sup t∈[0,τ ] t 0 A(t) -A(s) 2 L(V,V ) t -s ds 1 2
.

(3.4.12)

In order to continue we wish to take the inverse of I -S 0 . Let ε > 0 and τ 1 be as (3.2.1). We work on the interval [0, τ 1 ]. We have

sup t∈[0,τ 1 ] τ 1 0 A(t) -A(s) 2 L(V,V ) t -s ds < ε. It follows from (3.4.12) that S 0 L(L ∞ (0,τ 1 ;V)) < 1.
Therefore, u = (I -S 0 ) -1 (R 0 u 0 + L 0 f ) and we obtain from (3.4.9) and (3.4.10) that u ∈ L ∞ (0, τ 1 ; V) and (3.4.7) is satisfied on [0, τ 1 ]. Now repeat the same strategy. We use (3.2.1), we work on [τ i-1 , τ i ] and argue exactly as before. We obtain (3.4.7) on each sub-intervals [τ i-1 , τ i ]. This implies (3.4.7) on [0, τ ] for arbitrary τ > 0 and finishes the proof of assertion 1). In order to prove assertion 2), we use [H4] and (3.4.11). We obtain

S 0 g(t) V ≤ CC 2 M γ t 0 1 (t -s) γ ds 1 2 g L ∞ (0,τ ;V) ≤ C τ 1-γ 2 g L ∞ (0,τ ;V) .
We see that S 0 L(L ∞ (0,τ ;V)) < 1 for small τ > 0. We split [0, τ ] into a finite number of intervals with small sizes and then argue as previously.

Key estimates

Proposition 3.4.6. Suppose the assumptions of the previous proposition. Then u(t) ∈ V for every t ∈ [0, τ ].

Proof. This is an application of Lemma 3.3.7 and Proposition 3.4.5. Note that u(t) is well defined for every t since u ∈ C([0, τ ], H) by Lions' theorem.

The following lemma was first proved in [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] under the assumption that A(.) ∈ C α (0, τ ; L(V, V )) for some α > 1 2 . See also [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF]. We prove it here in the case where A satisfies (3.2.1) and for arbitrary A if [H4] is satisfied for some γ ∈ (0, 1).

for u 0 ∈ V and t ∈ [0, τ ]. 1) Suppose (3.2.1). Then R is bounded from V into L 2 (0, τ ; H). 2) If [H4] is satisfied for some γ ∈ (0, 1) then R is bounded from V into L 2 (0, τ ; H).
Proof. We write Ru 0 (t) = [A(t)e -tA(t) -A(0)e -tA(0) ]u 0 + A(0)e -tA(0) u 0 =: R 1 u 0 (t) + R 2 u 0 (t).

We use Lemma 3.3.5 to obtain

R 2 u 0 2 L 2 (0,τ ;H) = τ 0 A(0)
1 2 e -tA(0) A(0)

1 2 u 0 2 dt ≤ C A(0) 1 2 u 0 2 ≤ C u 0 2 V .
We estimate R 1 u 0 . We argue as in the proof of Proposition 3.4.2. By the holomorphic functional calculus for the sectorial operators A(t) and A(0) we have

R 1 u 0 (t) = 1 2πi Γ λe -λt (λ -A(t)) -1 (A(t) -A(0))(λ -A(0)) -1 u 0 dλ.
Now taking the norm in H we have

R 1 u 0 (t) ≤ 1 2π Γ |λ|e -tRe λ (λ -A(t)) -1 L(V γ ,H) × A(0) -A(t) L(V,V γ ) (λ -A(0)) -1 L(V) |dλ| u 0 V ≤ C A(0) -A(t) L(V,V γ ) t γ 2 u 0 V .

Proofs of the main results

together with the properties of R and L we just mentioned above and obtain A(.)u(.) ∈ L 2 (0, τ ; H) with A(.)u(.) L 2 (0,τ ;H) ≤ Ru 0 L 2 (0,τ ;H) + Su L 2 (0,τ ;H) + Lf L 2 (0,τ ;H)

≤ C u 0 V + u L ∞ (0,τ ;V) + f L 2 (0,τ ;H) ≤ C u 0 V + f L 2 (0,τ ;H) .
This proves the two theorems in the case where A ∈ Ḣ γ 2 (0, τ ; L(V, V γ )) for some γ ∈ (0, 1]. Now we prove (3.5.1). We have

Su(t) = A(t) t 0 e -(t-s)A(t) (A(t) -A(s))u(s) ds ≤ sup x∈H, x =1 t 0 |(A(t) 1 2 e -(t-s) 2 A(t) (A(t) -A(s))u(s), A(t) * 1 2 e -(t-s) 2 A(t) * x)| ds ≤ C t 0 A(t) 1 2 e -(t-s) 2 A(t) (A(t) -A(s))u(s) 2 ds 1 2
.

Here we use again the quadratic estimate of Lemma 3.3.5. By analyticity of the semigroup together with Lemma 3.3.1 we have

A(t) 1 2 e -(t-s) 2 A(t) (A(t) -A(s))u(s) 2 ≤ C t -s e -(t-s) 4 A(t) (A(t) -A(s))u(s) 2 ≤ C A(t) -A(s) 2 L(V,V γ ) |t -s| 1+γ u(s) 2 V .
Therefore,

Su(t) ≤ C u L ∞ (0,τ ;V) t 0 A(t) -A(s) 2 L(V,V γ ) |t -s| 1+γ ds 1 2
.

This gives

Su L 2 (0,τ ;H) ≤ C u L ∞ (0,τ ;V) A Ḣ γ 2 (0,τ ;L(V,V γ ))
and finishes the proof of (3.5.1).

Suppose now that A is piecewise in

Ḣ γ 2 (0, τ ; L(V, V γ )). Then [0, τ ] = ∪ n i=1 [τ i-1 , τ i ]
and the restriction of A to each sub-interval is in Ḣ γ 2 . We apply the previous proof to each sub-interval and obtain a solution u i in the subinterval [τ i-1 , τ i ] which has maximal regularity and satisfies apriori estimates. By Proposition 3.4.6, u i (τ i ) ∈ V and hence we can solve u i+1 (t)+A(t)u i+1 (t) = f (t) on [τ i , τ i+1 ] with u i+1 (τ i ) = u i (τ i ) and u i+1 has maximal regularity and

Applications

This shows that for CM 0 √ τ < 1 the operator K is a contraction. Hence there exists a unique u ∈ E such that K(u) = u. This gives that u satisfies (P) on [0, τ ] for τ < 1 (CM 0 ) 2 and it follows from (3.5.2) that u satisfies the apriori estimate

u E ≤ C [ u 0 V + f L 2 (0,τ ;H) ].
Finally, for arbitrary τ > 0, we split [0, τ ] into a finite number of sub-intervals with small sizes and proceed exactly as in the previous proof.

Applications

In this section we give some applications of the previous results to concrete differential operators.

-Elliptic operators on R n . Let H = L 2 (R n ) and V = H 1 (R n ) and define the sesquilinear forms

a(t, u, v) = n k,l=1 R n c kl (t, x)∂ k u∂ l v dx, u, v ∈ V.
We assume that the matrix C(t, x) = (c kl (t, x)) 1≤k,l≤n satisfies the usual ellipticity condition, that is, there exists α, M > 0 such that α|ξ| 2 ≤ Re (C(t, x)ξ. ξ) and |C(t, x)ξ.ν| ≤ M |ξ||ν| for all ξ, ν ∈ C n and a.e t ∈ [0, τ ], x ∈ R n . The forms a(t) satisfy the assumptions [H1]-[H3]. For each t, the corresponding operator is formally given by

A(t) = -n k,l=1 ∂ l (c kl (t, x)∂ k ). Next we assume that C ∈ H 1 2 (0, τ ; L ∞ (C n 2 )). We note that A(t) -A(s) L(V,V ) ≤ M C(t, .) -C(s, .) L ∞ (C n 2 )
for some constant M . This implies that A ∈ H 1 2 (0, τ ; L(V, V )). We assume in addition that each c kl is Hölder continuous of order α for some α > 0 with

|c kl (t, x) -c kl (s, x)| ≤ c|t -s| α
for a.e. x ∈ R n . This assumption implies in particular (3.2.1). We could also weaken this assumption by formulating it in terms of the modulus of continuity, see (3.2.3). We are now allowed to apply Theorem 3.2.2. We obtain maximal L 2 -regularity and apriori estimate for the parabolic problem

u (t) + A(t)u(t) = f (t) u(0) = u 0 ∈ H 1 (R n ).

Lions' maximal regularity problem with

H 1 2 -

regularity in time

That is, for every f ∈ L 2 (0, τ ; L 2 (R n )) and u 0 ∈ H 1 (R n ) there is unique solution u ∈ H 1 (0, τ ; L 2 (R n )). Note that we also have from Proposition 3.4.5 that u ∈ L ∞ (0, τ ; H 1 (R n )). As we already mentioned before, the uniform Kato square root property required in Theorem 3.2.2 is satisfied in this setting, see [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF]. As we mentioned in the introduction, maximal L 2 -regularity for these elliptic operators was proved recently in [START_REF] Auscher | On non-autonomous maximal regularity for elliptic operators in divergence form[END_REF] under the slightly stronger assumption that the coefficients satisfy a BMO-H Here ∂u ∂n denotes the normal derivative in the weak sense. Note that for any ε > 0

|a(t; u, v) -a(s; u, v)| = | ∂Ω [β(t, .) -β(s, .)]Tr(u)Tr(v) dσ| ≤ β(t, .) -β(s, .) L ∞ (∂Ω) u H 1 2 +ε (Ω) v H 1 2 +ε (Ω) ,
where we used the fat that the trace operator is bounded from H 1 2 +ε (Ω) into L 2 (∂Ω). Now assumption (3.6.1) allows us to apply Theorem 3.2.4 with γ = 1 2 + ε and obtain maximal L 2 -regularity for the corresponding evolution equation with initial data u 0 ∈ H 1 (Ω). The forms considered here are symmetric and therefore the uniform Kato square root property can be checked easily.

Applications

Maximal L 2 -regularity for time dependent Robin boundary condition with β(., x) ∈ C α for some α > 1 4 was previously proved in [START_REF] Arendt | Maximal regularity for non-autonomous Robin boundary conditions[END_REF] and [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF]. In [START_REF] Ouhabaz | Maximal regularity for non-autonomous evolution equations governed by forms having less regularity[END_REF] maximal L p -regularity is proved for all p ∈ (1, ∞) is proved.

-Operators with terms of lower order. Let Ω be a domain of R n and let b k , m : [0, τ ] × Ω → R be a bounded measurable function for each k = 1, • • • , n. We define the forms

a(t, u, v) = Ω ∇u.∇vdx + n k=1 Ω b k (t, x)∂ k uvdx + Ω m(t, x)uvdx, with domain V, a closed subset of H 1 (Ω) which contains H 1 0 (Ω). It is clear that |a(t, u, v) -a(s, u, v)| ≤ M 0 u V v 2
for some constant M 0 . This means that assumption [H4] is satisfied with γ = 0. We apply Proposition 3.2.5 and obtain maximal L 2 -regularity for the correspond evolution equation.

As we mentioned in the introduction, the domains of the corresponding operators are independent of t and one may apply the results from [START_REF] Arendt | L p -maximal regularity for non-autonomous evolution equations[END_REF] to obtain maximal regularity. Doing so, one needs to assume some regularity with respect to t for the coefficients b k (t, x) and m(t, x) whereas the result we obtain from Proposition 3.2.5 does not require any regularity.

Chapter 4

Non-autonomous maximal regularity under Besov regularity in time

The main purposes of this chapter is to study the maximal L p -regularity for p > 2. It was shown in [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] that the family of the operators (A(t)) t∈[0,τ ] satisfy the maximal L p -regularity if

(i) A(.) ∈ Ḣ 1 2 + (0, τ ; L(V, V )) for p ≤ 2 (ii) A(.) ∈ Ẇ 1 2 + ,p (0, τ ; L(V, V )) for p > 2,
where is an arbitrary positive constant and (A(t)) t∈[0,τ ] is associated with Vbounded quasi-coercive forms (a(t)) t∈[0,τ ] . In this chapter we extend the results in [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] and [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] in case p > 2 by assuming less regularity as much possible.

Our main result shows that for forms satisfying the uniform Kato square root property and if t → A(t) is piecewise in the Besov space Ḃ 1 2 ,p 2 (0, τ ; L(V, V )) then the maximal L p -regularity in H is satisfied. The initial data u 0 is arbitrary in the interpolation space (H, D(A(0))) 1-1 p ,p . We note that for all p > 2 and

> 0 Ẇ 1 2 + ,p (0, τ ; L(V, V ) → Ḃ 1 2 ,p 2 (0, τ ; L(V, V )).
Our result is optimal, by using an example in [START_REF] Arendt | Lions' problem on maximal regularity[END_REF] (Proposition 10.1).

Autonomous first order Cauchy problems

Definition 4.1.1. Let X be a Banach space, p, q ∈ [1, ∞), and α ∈ (0, 1). The Bochner measurable function f : [0, τ ] → X belongs to the homogeneous Besov space Ḃα,p q (0, τ ; X) if

f q Ḃα,p q (0,τ ;X) = τ 0 1 αq τ f (t) -f (t -l) p X dt q p d < ∞,

Autonomous first order Cauchy problems

and the classical Besov space is defined as B α,p q (0, τ ; X) = {f ∈ L p (0, τ ; X), s.t f q Ḃα,p q (0,τ ;X) < ∞} endowed with norm f B α,p q (0,τ ;X) = f L p (0,τ ;X) + f Ḃα,p q (0,τ ;X) .

It is well known that B α,p q (0, τ ; X) = L p (0, τ ; X), W 1,p (0, τ ; X) α,q , where (., .) α,q is the real interpolation space. For 0 < α < 1, we define the Hölder space

C α (0, τ ; X) = {f ∈ L ∞ (0, τ ; X) : f Ċα (0,τ ;X) < ∞},
where f is endowed with the norm in Ċα (0, τ ; X) as

f Ċα (0,τ ;X) = sup s,t∈[0,τ ] f (t) -f (s) X |t -s| α .
For more details and references, we let the readers to see Section 4.5.

We introduce the subspace

W 1,p 0 (0, τ ; X) := {u ∈ W 1,p (0, τ ; X), s.t u(0) = 0}.
We equip this subspace with the norm u → u L p (0,τ ;X) . Let X and Y be two Banach spaces. Then X ∩ Y is a Banach space and we equip this subspace with the norm x X∩Y = max{ x X , x Y }. p (0, τ ; E), with the estimate

g Ċ 1 2 -1 p (0,τ ;E) ≤ 72 1 2 - 1 p g Ḃ 1 2 ,p 2 (0,τ ;E) . (c) B 1 2 ,p 2 (0, τ ; E) → C(0, τ ; E), with the estimate g C(0,τ ;E) ≤ 1 1 2 - 1 p g Ḃ 1 2
,p 2 (0,τ ;E)

+ (2p + ( 2 τ ) 1 p ) g L p (0,τ ;E) .
(d) B α,p 2 (0, τ ; E) → W α,p (0, τ ; E), for all p ≥ 2, 1 > α > 0, with constant of injection depending only on p and α.

Proof. For assertions (a), (b) and (c), we refer to [START_REF] Simon | Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval[END_REF] (Corollaries 15 and 26). For (d), we use [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF] (Proposition 1.1.3).

Non-autonomous maximal regularity under Besov regularity in time

As we seen in the previous chapters, we may suppose without loss of generality that ν = 0. Proposition 4.1.3. For all f ∈ W 1,p (0, τ ; V ) with f (0) ∈ (V , V) 1-1 p ,p , there exists a unique u ∈ W 1,p (0, τ ; V) be the solution of the problem In addition,

   u (t) + A(0)u(t) = f (t) u(0) = 0.
u W 1,p (0,τ ;V) ≤ C(δ, M, p) f (0) (V ,V) 1-1 p ,p + f W 1,p (0,τ ;V )
, where the positive constant C(δ, M, p) depending only on δ, p, M.

Proof. According to Proposition 1.5.6, there exists a unique solution u of the Problem (4.1.1) and for all f ∈ L p (0, τ ; V ) the solution is given by

u(t) = t 0 e -(t-s)A(0) f (s) ds.
We have also u ∈ W 1,p (0, τ ; V ) ∩ L p (0, τ ; V) and

u W 1,p (0,τ ;V )∩L p (0,τ ;V) ≤ C(δ, M, p) f L p (0,τ ;V ) . (4.1.2)
For t ∈ [0, τ ] and f ∈ W 1,p (0, τ ; V ) with f (0) ∈ (V , V) 1-1 p ,p , we have by integration by parts

A(0)u(t) = A(0) t 0 e -(t-s)A(0) f (s) ds = f (t) -e -tA(0) f (0) - t 0 e -(t-s)A(0) f (s) ds = u (t) + A(0)u(t) -e -tA(0) f (0) - t 0 e -(t-s)A(0) f (s) ds. Hence u (t) = e -tA(0) f (0) + t 0 e -(t-s)A(0) f (s)ds. Since f (0) ∈ (V , V) 1-1 p ,p , we have by Proposition 1.2.4 τ 0 e -tA(0) f (0) p V dt ≤ A(0) -1 p L(V ,V) τ 0 A(0)e -tA(0) f (0) p V dt ≤ 1 δ p τ 0 A(0)e -tA(0) f (0) p V dt ≤ 1 δ p f (0) p (V ,V) 1-1 p ,p .

Autonomous first order Cauchy problems

As a consequence, we obtain the final estimate

u L p (0,τ ;V) ≤ C(δ, M, p) f (0) (V ,V) 1-1 p ,p + f W 1,p (
0,τ ;V ) . Then we get the desired result. Lemma 4.1.4. Suppose that D(A(0)

1 2 ) = V. Then D(A(0) 1 2 ) = H and D(A(0) * 1 2 ) = V.
Proof. For any x ∈ H, we write A(0)

1 2 x = A(0)A(0) -1 2
x. Then we get

A(0) 1 2 x V ≤ A(0) L(V,V ) A(0) -1 2 x V ≤ M A(0) -1 2 L(H,V) x .
The boundedness implies A(0)

1 2 ∈ L(H, V
) and by duality we have 2 ). Now, we use the same argument to the adjoint and we apply Proposition 1.2.1(1) to get the desired result.

A(0) * 1 2 ∈ L(V, H). It follows that V → D(A(0) * 1 2 ) and H → D(A(0) * 1
u(t) = t 0 e -(t-s)A(0) f (s)ds = t 0 e -(t-s)A(0) (f (s) -f (t))ds + A(0) -1 [f (t) -e -tA(0) f (t)].
On the other hand, by Lemma 3.3.5 we have the quadratic estimate

τ 0 A(0) 1 2 e -tA(0) x 2 dt ≤ C x 2 , (4.1.3)
where C be a positive constant independent of τ.

We have the following estimate

u(t) V ≤ sup x V =1,x∈V t 0 e -(t-s)A(0) (f (s) -f (t)) ds, x + A(0) -1 f (t) -e -tA(0) f (t) V = A + B.

Non-autonomous maximal regularity under Besov regularity in time

We get

A = sup x V =1,x∈V t 0 e -(t-s) 2 A(0) (f (s) -f (t)), A(0) 1 2 * e -(t-s) 2 A(0) * A(0) * -1 2 x ds ≤ t 0 e -(t-s) 2 A(0) (f (s) -f (t)) 2 ds 1 2 × sup x V =1,x∈V t 0 A(0) * 1 2 e -(t-s) 2 A(0) * A(0) * -1 2 x 2 ds 1 2 ≤ C t 0 e -(t-s) 2 A(0) 2 L(V ,H) f (s) -f (t) 2 V ds 1 2 sup x V =1,x∈V A(0) * -1 2 x ≤ C t 0 e -(t-s) 2 A(0) 2 L(V ,H) f (s) -f (t) 2 V ds 1 2 .
The last inequality holds by Lemma 4.1.4. For the second inequality we have used (4.1.3).

Concerning B, we have

B ≤ A(0) -1 L(V ,V) f (t) -e -tA(0) f (t) V ≤ C 2 f (t) V .
Finally, we obtain

u(t) V ≤ A + B ≤ C 1 t 0 f (t) -f (s) 2 V t -s ds 1 2 + C 2 f (t) V ≤ C 1 τ p-2 2p t 0 f (t) -f (s) p V (t -s) p 2 ds 1 p + C 2 f (t) V ≤ C 3 τ p-2 2p f Ẇ 1 2 ,p (0,τ ;,V ) + C 2 f (t) V .
Note that the last inequality is derived from Lemma 4.2.1. Here C, C 1 , C 2 and C 3 are positive constants independent of τ and f . Combining with Proposition 4.1.2 (d) the result is following.

We define the space

G = {f ∈ W 1,p (0, τ ; V ), s.t f (0) ∈ (V , V) 1-1 p ,p }. Let us consider the operator L : f → u,
where f ∈ G and u is the unique solution of the Cauchy problem We start with the following known result (see [START_REF] Prüss | Existence of analytic solutions for the classical Stefan problem[END_REF] p. 745, (6.8)).

   u (t) + A(0)u(t) = f (t) u(0) = 0. ( 4
Lemma 4.2.1. For α > 1 p and any f ∈ Ẇ α,p (0, τ ; L(V, V )) with f (0) = 0, we have

τ 0 f (s) p L(V,V ) s αp ds ≤ 1 + α -1 p α -1 p f p Ẇ α,p (0,τ ;L(V,V )) . Proposition 4.2.2. Suppose that A(.) ∈ B 1 2 ,p 2 (0, τ ; L(V, V )). Let p > 2 and we set f (t) = (A(t) -A(0))v(t), where v ∈ B 1 2 ,p 2 (0, τ ; V), t ∈ [0, τ ]. Then we get f ∈ B 1 2 ,p 2 (0, τ ; V ). Moreover, f B 1 2 ,p 2 (0,τ ;V ) ≤ C(p)(1 + τ p-2 2p ) A Ḃ 1 2 ,p 2 (0,τ ;L(V,V )) v B 1 2 ,p 2 (0,τ ;V)∩L ∞ (0,τ ;V) , (4.2.1)
where C(p) is a positive constant independent of f and τ.

Proof. We write

f B 1 2 ,p 2 (0,τ ;V ) = (A(.) -A(0))v(.) L p (0,τ ;V ) + τ 0 1 ( τ f (t) -f (t -) p V dt) 2 p d 1 2 = D + E.
By using Proposition 4.1.2 (b) the terms D and E can be estimated as follows

D ≤ A(.) -A(0) L ∞ (0,τ ;L(V,V )) v L p (0,τ ;V) ≤ τ p-2 2p A Ċ 1 2 -1 p (0,τ ;L(V,V )) v L p (0,τ ;V) ≤ c(p)τ p-2 2p A Ḃ 1 2 ,p 2 (0,τ ;L(V,V )) v L p (0,τ ;V) and E ≤ A(.) -A(0) L ∞ (0,τ ;L(V,V )) v Ḃ 1 2 ,p 2 (0,τ ;V) + A Ḃ 1 2 ,p 2 (0,τ ;L(V,V )) v L ∞ (0,τ ;V) ≤ C 1 (p)(1 + τ p-2 2p ) A Ḃ 1 2 ,p 2 (0,τ ;L(V,V )) v B 1 2 ,p 2 (0,τ ;V)∩L ∞ (0,τ ;V)
.

As a result, we obtain finally

f B 1 2 ,p 2 (0,τ ;V ) ≤ C(p)(1 + τ p-2 2p ) A Ḃ 1 2 ,p 2 (0,τ ;L(V,V )) v B 1 2 ,p 2 (0,τ ;V)∩L ∞ (0,τ ;V)
.

Here C(p), C 2 (p), c(p) are positive constants independent of f and τ.

Non-autonomous maximal regularity under Besov regularity in time

Similarly

B(t) = A( -t), t ∈ [0, ]. A τ τ -l (t -) , t ∈ [ , τ ].
In the case where = τ , we set v(t) = u(τ -t) and B(t

) = A(τ -t) for t ∈ [0, τ ]. So we have v ∈ W 1,p (0, τ ; H) ∩ B 1 2 ,p 2 (0, τ ; V) with B(.)v(.) ∈ L p (0, τ ; H). We write u( ) = v(0) = v( ) -0 v(s)ds. Note that v(0) ∈ (H, D(B(0))) 1-1 p ,p if and only if r → B(0)e -rB(0) v(0) ∈ L p (0, τ ; H) (see Proposition 1.2.4). We write B(0)e -rB(0) v(0) = B(0)e -rB(0) -B(r)e -rB(r) v(0) + e -rB(r) B(r)v(r) -B(r)e -rB(r) r 0 v(s) ds = I 1 (r) + I 2 (r) + I 3 (r).
By the holomorphic functional calculus for the operators B(0), B(r) we can write

I 1 (r) = 1 2πi Γ λe -rλ (λ -B(0)) -1 (B(0) -B(r))(λ -B(r)) -1 dλ v(0),
where Γ is the boundary of an appropriate sector S θ for appropriate θ ∈ (0, π 2 ). Now, taking the norm in H and we apply Proposition 1.1.8 to deduce

I 1 (r) ≤ ∞ 0 |λ|e -r cos θ|λ| (λ -B(0)) -1 L(V ,H) (λ -B(r)) -1 L(V) |dλ| × B(0) -B(r) L(V,V ) v(0) V ≤ C B(0) -B(r) L(V,V ) r 1 2 v(0) V .
For the term I 3 , by the analyticity of the semigroups (e -rB(r) ) r≥0 it follows that

I 3 (r) ≤ B(r)e -rB(r) L(H) r 0 v (s) ds ≤ C r r 0 v (
s) ds.

Maximal regularity for the Damped Wave equation

This is an abstract damped non-autonomous wave equation and our aim is to prove well-posedness and the maximal L p -regularity for p ∈ (1, ∞) in V and in H.

Definition 5.0.1. Let X = H or V . We say that Problem (5.0.1) has the maximal L p -regularity in X, if for all f ∈ L p (0, τ ; X) and all (u 0 , u 1 ) in the trace space (see Sections 5.1 and 5.2 for more details) there exists a unique u ∈ W 2,p (0, τ ; X) ∩ W 1,p (0, τ ; V) wich satisfies (5.0.1) in the L p -sense.

The maximal regularity in V in the case where p = 2 was first considered by Lions [START_REF] Lions | Équations Différentielles Opérationnelles et Problèmes aux Limites[END_REF] (p. 151). He assumes that A(t) is associated with a sesquilinear forms a(t) wich satisfies the same properties as b(t) and by assuming an additional L 2 -regularity assumption on the forms t → a(t, u, v) and t → b(t, u, v) for every fixed u, v ∈ V. Dautray-Lions [18] (p.667) proved the maximal regularity in V without the regularity assumption as before by taking f ∈ L 2 (0, τ ; H) and considering mainly symmetric forms. Recently, Batty, Chill, Strivastava [START_REF] Batty | Maximal regularity for second order non-autonomous Cauchy problems[END_REF] proved the maximal L p regularity for general forms B(.) and A(.) for the case u 0 = u 1 = 0 and h ∈ L p (0, τ ) by reducing the problem to a first order non-autonomous Cauchy problem. Dier-Ouhabaz [START_REF] Dier | Maximal regularity for non-autonomous second order Cauchy problems[END_REF] proved the maximal L 2 -regularity in V for u 0 ∈ V, u 1 ∈ H and A(t) is also associated with a V-bounded quasi-coercive non-autonomous form a(t). We improve the result in [START_REF] Batty | Maximal regularity for second order non-autonomous Cauchy problems[END_REF], where we prove the maximal L p -regularity in V for u 0 and u 1 are different to 0 and t → t 1-1 p h(t) ∈ L p (0, τ ). Our proof is based on the result of the first order problem as in [START_REF] Batty | Maximal regularity for second order non-autonomous Cauchy problems[END_REF], but the main difference being that we use a fixed point argument.

More interesting is the question of second order maximal regularity in H, i.e. whether the solution u of (5.0.1) is in H 2 (0, τ ; H) provided that f ∈ L 2 (0, τ ; H). A first answer of this question was giving by Batty, Chill, Strivastava [START_REF] Batty | Maximal regularity for second order non-autonomous Cauchy problems[END_REF] in the particular case B(.) = kA(.) for some constant k and that A(.) has the maximal regularity in H. By using the form method, Dier and Ouhabaz [START_REF] Dier | Maximal regularity for non-autonomous second order Cauchy problems[END_REF], proved the maximal L 2regularity in H without the rather strong assumption B(.) = kA(.), but A(t) is also associated with V-bounded quasi-coercive form a(t) and t → a(t, u, v), b(t, u, v) are symmetric, Lipschitz continuous for all u, v ∈ V. We extend the results in [START_REF] Dier | Maximal regularity for non-autonomous second order Cauchy problems[END_REF] in three directions. The first one is to consider general forms which may not be symmetric. The second direction is to deal with maximal L p -regularity, for all p ∈ (1, ∞). The third direction, which is our main motivation, is to assume less regularity on the operators A(t), B(t) with respect to t.

Our main results can be summarized as follows (see Theorems 5.2.6 and 5.2.7 for more general and precise statements). For p ∈ (1, ∞) we assume the following

-|b(t, u, v) -b(s, u, v)| ≤ w(|t -s|) u V v V , for all u, v ∈ V.

Maximal regularity for the damped wave equation in

V -A(t) -A(s) L(V,V ) ≤ w(|t -s|). Such that τ 0 w(t) t 3 2 dt < ∞, ( 5
2 ) → V, we assume τ 0 w(t) 2 t 1+ dt < ∞, 1
for some > 0.

Here w : [0, τ ] → [0, ∞) is a non-decreasing function. We prove that for f ∈ L p (0, τ ; H) and one of the following conditions holds

1-for p ≥ 2, u 0 ∈ (V, D(A(0))) 1-1 p ,p and u 1 ∈ (H, D(B(0))) 1-1 p ,p , 2-for p < 2, u 0 ∈ V and u 1 ∈ (H, D(B(0))) 1-1
p ,p , then (5.0.1) has the maximal L p -regularity in H. Assume in addition that D(B(t) 1 2 ) = V for all t ∈ [0, τ ] and w(t) ≤ Ct , for some > 0, then for all f ∈ L 2 (0, τ ; H) and u 0 , u 1 ∈ V, we prove that the solution u ∈ H 2 (0, τ ; H) ∩ C 1 (0, τ ; V). By induction, we observe that our approach allows to consider Cauchy problem of order N for any N ≥ 3. Finally, using similar ideas as in [START_REF] Fackler | Lions' problem concerning maximal regularity of equations governed by non-autonomous forms[END_REF] and [START_REF] Arendt | Lions' problem on maximal regularity[END_REF], we give counterexamples where the maximal regularity may fails. Notation. We denote by C, C or c... all inessential positive constants. Their values may change from line to line.

Maximal regularity for the damped wave equation in V

In this section we prove the maximal regularity in V for the Problem (5.0.1), we start by recalling a well-known result on maximal regularity for the first order non autonomous problem (see Theorems 1.5.7, 1.5.1).

Theorem 5.1.1. Let p ∈ (1, ∞). We assume one of the following conditions -for p = 2, t → b(t) is measurable .

Maximal regularity for the damped wave equation in H

We deduce that we may replace B(t) by B(t) + γ. Therefore, by adding a large constant c from the previous proposition we may suppose without loss of generality that [H3] holds with ν = 0. We note that for γ > 0 is big enough (γ > max{ M δ , |ν|}) and for t ∈ [0, τ ], we have that C(t) = A(t) + γB(t) + γ 2 is associated with V-bounded coercive form c(t). In fact, let u ∈ V. We get

Re c(t, u, u) = Re A(t)u, u + γRe b(t, u, u) + γ 2 u 2 ≥ -M u 2 V + γδ u 2 V + (γ 2 -ν) u 2 ≥ (γδ -M ) u 2
V . We denote by S θ the open sector S θ = {z ∈ C * : |arg(z)| < θ} with vertex 0. Lemma 5.2.2. For any t ∈ [0, τ ], the operators -B(t) and -B(t) generate strongly continuous analytic semigroups of angle γ = π 2 -arctan( M α ) on H and V ,respectively. In addition, there exist constants C and C θ , independent of t, such that 1e -zB(t) L(H) ≤ 1 and e -zB(t)

L(V ) ≤ C for all z ∈ S γ . 2-B(t)e -sB(t) L(H) ≤ C s and B(t)e -sB(t) L(V ) ≤ C s for all s ∈ R. 3-e -sB(t) L(H,V) ≤ C √ s . 4-(z -B(t)) -1 L(H,V) ≤ C θ √ |z| and (z -B(t)) -1 L(V ,H) ≤ C θ √ |z| for all z / ∈ S θ with fixed θ > γ. 5-(z -B(t)) -1 L((H,V) β,p ;V) ≤ C θ (1+|z|) 1+β 2
for all β ∈ [0, 1], z / ∈ S θ and p ∈

(1, ∞).

6-All the previous estimates hold for B(t) + α with constants independent of α for α > 0.

Proof. For assertions 1, 2, 3 and 4, 6 we refer to [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] (Proposition 2.1). For 5 we use the interpolation.

For p ∈ (1, ∞) and f ∈ L p (0, τ ; H), we define the operator L by

L(f )(t) = B(t) t 0 e -(t-s)B(t) f (s)ds, with t ∈ [0, τ ]. Proposition 5.2.3. Let p ∈ (1, ∞). We suppose that B(t) -B(s) L(V,V ) ≤ w(t -s), where w : [0, τ ] → [0, ∞) is a non-decreasing function such that τ 0 w(t) t dt < ∞.
Then the operator L is bounded on L p (0, τ ; H).

Maximal regularity for the damped wave equation in H

Proof. First, we note that in case p < 2 we have (H, D(B(0

))) 1-1 p ,p = (H, V) 2(1-1 p ),p
(see [START_REF] Fackler | Non-autonomous maximal L p -regularity under fractional Sobolev regularity in time[END_REF] (p. 5)). Then by Lemma 5.2.2

(λ -B(0)) -1 L((H,D(B(0))) 1-1 p ,p ;V) ≤ C |λ| 3 2 -1 p .
In case p > 2, the embedding (H, D(B(0))) 1-1 p ,p → V holds. In fact, we use the inclusion properties of the real interpolation spaces ( [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF], Proposition 1.1.4) to obtain

(H, D(B(0))) 1-1 p ,p = (H, D(B(0)))1 2 +[ 1 2 -1 p ],p → (H, D(B(0)))1 2 +[ 1 2 -1 p ]-,2 = D(B(0) 1-( 1 p + ) ), with < [ 1 2 -1 p ]. Since by Proposition 1.2.1 D(B(0) 1-( 1 p + ) ) → V, it follows that (λ -B(0)) -1 L((H,D(B(0))) 1-1 p ,p ;V) ≤ C |λ| . Now, if D(B(0) 1 2 ) → V we get (λ -B(0)) -1 u 1 V ≤ C 1 B(0) -1 2
L(H,D(B(0) For the other case (D(B(0)

1 2 ) → V), since D(B(0) 1+
2 ) → V for all > 0 (see Proposition 1.2.1) then

(λ -B(0)) -1 u 1 V ≤ C 1 B(0) -1+ 2 L(H,D(B(0) 1+ 2 )) B(0) 2 (λ -B(0)) -1 B(0) 1 2 u 1 ≤ C 1 |λ| 1-2 u 1 D(B(0) 1 2
) .

We write R 1 u 1 (t) = (B(t)e -tB(t) -B(0)e -tB(0) )u 1 + B(0)e -tB(0) u 1 .

Choose a contour Γ in the positive half-plane we write by the holomorphic functional calculus B(t)e -tB(t) -B(0)e -tB(0) = 1 2πi Γ λe -tλ (λ-B(t)) -1 (B(t)-B(0))(λ-B(0)) -1 dλ.

Maximal regularity for the damped wave equation in H

Therefore R 2 u 0 L p (0,τ ;H) ≤ C u 0 V . Now for p ≥ 2, we write R 2 u 0 (t) = B(t)e -tB(t) B(t) -1 C(t)u 0 = B(t)e -tB(t) B(t) -1 (C(t) -C(0))u 0 + B(t)e -tB(t) (B(t) -1 -B(0) -1 )C(0)u 0 + B(t)e -tB(t) B(0) -1 C(0)u 0 = I 1 (t) + I 2 (t) + I 3 (t).

For i = 1 or i = 2, we have the following estimate

I i (t) ≤ Cw(t) e -tB(t) L(V ,H) u 0 V ≤ C w(t) √ t u 0 V ,
where C, C are two positive constants. Then where C(0) | (V ,H)α,p is the part of C(0) on (V , H) α,p . Thus, B(0) -1 C(0) ∈ L((V, D(C(0))) 1-1 p ,p ; (V, D(B(0)) 1-1 p ,p ). Observing that for t ∈ [0, τ ], I 3 (t) = R 1 B(0) -1 C(0)u 0 (t). Using the first part of the proposition, we obtain I 3 L p (0,τ ;H) ≤ C u 0 (V,D(C(0))) 1-1 p ,p . This shows that t → R 2 u 0 (t) ∈ L p (0, τ ; H).

The Main Result

Our main result in this chapter is the following Theorem 5.2.6. Let f ∈ L p (0, τ ; H) and u 1 ∈ (H, D(B(0))) 1-1 p ,p , u 0 ∈ X p , with p ∈ (1, ∞). Then the problem p ,p + u 0 X p + f L p (0,τ ;H) .

Proof. Let f ∈ L p (0, τ ; H) and (u 0 , u 1 ) ∈ (V × (V , V) 1-1 p ,p ). By Theorem 5.1.5 there exists a unique solution v ∈ W 2,p (0, τ ; V ) ∩ W 1,p (0, τ ; V) of the Problem (5.2.6). Then by Proposition 5.2.1, there exists a unique solution u ∈ W 2,p (0, τ ; V ) ∩ W 1,p (0, τ ; V) of the Problem (5.2.2). To simplify the writing, we replace B(t) + γ by B(t). Fix 0 ≤ t ≤ τ. We get from the equation in (5.2.2) e -(t-s)B(t) g(s)ds.

B(t)
(5.2.7)

By performing an integration by parts, we have We shall now prove the boundedness of Q, W 1 , W 1 , L 1 on L p (0, τ ; H) for p ∈ (1, ∞). In fact, take h ∈ L p (0, τ ; H). We get h(s) dsα -1 2 .

B(t)
(
Therefore, by using Young's inequality we obtain

Q L(L p (0,τ ;H)) ≤ C τ 0 w(s) s 3 2 dsα -1 2 .
Using the assumption on w and taking α large enough makes Q strictly contractive, so that (I -Q) -1 is bounded on L p (0, τ ; H) by the Neumann series.

For W i with i = 1, 2 we have ds u W 1,p (0,τ ;V) .

Maximal regularity for the Damped Wave equation

Proof. First we set w 1 (t) = w 0 (t)+γw(t). It is clear that |c(t, u, v)-c(s, u, v)| ≤ w 1 (|t -s|) u V v V and w 1 satisfies the same conditions with w 0 . From (5.2.11), we have (I -Q)(B(.)u + C(.)u)(t) = B(t)e -tB(t) u 1 + e -tB(t) (C(t)u 0 )

-B(t) We follow the same proof as in Theorem 5.2.6. The difference is to prove that t → R 2 u 0 (t) = e -tB(t) C(t)u 0 ∈ L 2 (0, τ ; H). Observing that R 2 u 0 (t) = e -tB(t) C(t)u 0 = e -tB(t) (C(t) -C(0))u 0 + e -tB(t) C(0)u 0 .

(5.2.15)

For the first term in (5. We write e -tB(t) C(0)u 0 = e -tB(t) C(0)u 0 -e -tB(0) C(0)u 0 + e -tB(0) C(0)u 0 .

The functional calculus for the sectorial operators B(t), B(0) gives e -tB(t) C(0)u 0 -e -tB(0) C(0)u 0 ≤ c w(t) t 1 2 u 0 V .

We write again e -tB(0) C(0)u 0 = B(0)

1 2 e -tB(0) B(0) -1 2 C(0)u 0 .

Maximal regularity for the damped wave equation in H

The quadratic estimate of Lemma 3. Then R 2 u 0 L 2 (0,τ ;H) ≤ C u 0 V and the maximal L 2 -regularity holds in H. Fix 0 ≤ t ≤ τ and using (5.2.10), we obtain We shall now prove that u ∈ L ∞ (0, τ ; V). We define the operator K in L ∞ (0, τ ; V) by ds < c, with 0 < c < 1. Thus, K(h) L ∞ (0,τ ;V) ≤ c h L ∞ (0,τ ;V) and hence I -K is invertible on L ∞ (0, τ ; V). Using (5.2.16), we get (I -K)(u )(t) = -B(t) -1 C(t)u(t) + e -tB(t) B(t) -1 C(t)u 0 + e -(t)B(t) u 1 + t 0 e -(t-s)B(t) (C(t) -C(s))u(s)ds + t 0 e -(t-s)B(t) B(t) -1 C(t)u (s)ds + t 0 e -(t-s)B(t) g(s)ds.

u (t) = -B(t

Maximal regularity for the Damped Wave equation

Therefore for τ small enough and by using Lemma 3.4.1, we have u L ∞ (0,τ ;V) ≤ C[ u L ∞ (0,τ ;V) + u 0 V + u 1 V + g L 2 (0,τ ;H) + u L 2 (0,τ ;V) + t 0 w 1 (l) l dl u L ∞ (0,τ ;V) ].

Since u ∈ L ∞ (0, τ ; V) ∩ C([0, τ ]; H), by Lemma 3.3.7 we have u (t) ∈ V for all t ∈ [0, τ ]. Now, for τ arbitrary we split (0, τ ) into small intervals and proceed exactly as in the previous proof. In order to obtain a solution u ∈ W 1,∞ (0, τ ; V), we glue the solutions of each-interval. For 0 ≤ s ≤ t ≤ τ, as in (5.2.16) we note that u (t) = e -(t-s)B(t) u (s) + Hence, by performing an integration by parts as before we obtain u (t) -u (s) = [-B(t) -1 C(t)u(t) + e -(t-s)B(t) B(t) -1 C(t)u(s)]

+ [e -(t-s)B(t) u (s) -u (s)] + e -(t-l)B(t) g(l)dl = K 1 (t, s) + K 2 (t, s) + K 3 (t, s) + K 4 (t, s) + K 5 (t, s) + K 6 (t, s).

Observing that e -(t-s)B(t) B(t) -1 C(s)u(s) -B(t) -1 C(t)u(t)

= [e -(t-s)B(t) -e -(t-s)B( (5.2.17)

We write by the functional calculus e -(t-s)B(t) -e -(t-s)B(s) = 1 2πi Γ e -(t-s)λ (λ-B(t)) -1 (B(t)-B(s))(λ-B(s)) -1 dλ, where Γ = ∂Σ(θ) for appropriate θ ∈ (0, π 2 ). Then we get I 1 (s, t) V = [e -(t-s)B(t) -e -(t-s)B(s) ]B(t) -1 C(s)u(s) V where is an appropriate small positive constant and c > 0 depending on . For i = 2, 3, we have immediately

I i (s, t) V ≤ C B(t) -B(s) L(V,V ) u L ∞ (s,t;V) .
For the last terms in 5.2.17 we get Since C(t)u(t) ∈ C([0, τ ]; V ) and by the strong continuity of the semigroup e -sB(t) on V we get e -(t-s)B(t) B(t) -1 C(s)u(s) -B(t) -1 C(t)u(t) → 0 as t → s. Therefore K 1 (t, s) → 0 as t → s.

Taking the norm in V, we obtain Hence, u (t) -u (s) V → 0 as t → s and u is right continuous on V.

K 3 (t,
We shall now prove the left continuous of u . Fix 0 ≤ s ≤ t ≤ τ. We integrate the equation (5.2.6) from s to t to obtain Using the same steps and arguments as in the proof of the right continuity of u, we obtain that for s → t u (t) -u (s) V → 0.

Therefore, u ∈ C 1 ([0, τ ]; V) and so v ∈ C 1 ([0, τ ]; V).

For higher equations we have Theorem 5.2.8. Let (A i (t)) t∈[0,τ ],i∈ [1,N ] , N ∈ N * such that A i (t) ∈ L(V, V ) for all i ∈ [1, N ] and A i (t) L(V,V ) ≤ M. We suppose that A N (t) is associated with V-bounded quasi-coercive forms and for all i ∈ [1, N ]

A i (t) -A i (s) L(V,V ) ≤ K|t -s| α for some K > 0 and α > 1 2 . We assume that (A N (t) + ν) t∈[0,τ ] satisfies the uniform Kato property (5.2.14). Then for all f ∈ L 2 (0, τ ; H) and u 0 , ..., u N -1 ∈

1 2 -

 2 regularity in time 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Preparatory lemmas . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Key estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Proofs of the main results . . . . . . . . . . . . . . . . . . . . . 3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Example 1 . 1 . 10 .

 1110 (Dirichlet Laplacian) Let Ω ⊂ R d be a bounded open set. Let H = L 2 (Ω) and define the operator ∆ D on H, by

Theorem 1 . 3 . 3 .

 133 If -A generates a bounded holomorphic semigroup on H then A has the maximal regularity property and T R(p, H) = (H, D(A)) 1-1 p ,p for all p ∈ (1, ∞).

Notation.

  We denote by L(E, F ) (or L(E)) the space of bounded linear operators from E to F (from E to E). The spaces L p (a, b; E) and W 1 p (a, b; E) denote respectively the Lebesgue and Sobolev spaces of function on (a, b) with values in E. Recall that the norms of H and V are denoted by • and • V . The scalar product of H is (•, •). Finally, we denote by C, C or c... all inessential constants. Their values may change from line to line.

  t-r)A(t) (A(t) -A(r))u(r)dr + t s e -(t-r)A(t) [(-B(r) + I)A(r)u(r) -P (r)u(r) + f (r)]dr, (2.4.2)

1 2

 1 for some positive constant M γ and all t, s ∈ [0, τ ]. Theorem 3.2.4. Suppose [H1]-[H3] and that D(A(t 0 ) ) = V for some t 0 ∈ [0, τ ]. Let γ ∈ (0, 1) and suppose [H4]. If t → A(t) is in the homogeneous Sobolev space

3. Lions' maximal regularity problem with H 1 2 -regularity in time Proposition 3 . 2 . 5 .

 1325 Suppose [H1]-[H3]. Suppose that [H4] holds for γ = 0 and that D(A(t 0 )

Proposition 3 . 3 . 3 .

 333 Suppose that [H1]-[H3] are satisfied with ν = 0. Fix s ∈ [0, τ ] and suppose that either (3.3.4) or (3.3.6) holds for some γ ∈ [0, 1)

1 2 -

 2 regularity in time Lemma 3.3.5. Suppose in addition to [H1]-[H3] (with ν = 0) that the uniform Kato square root property is satisfied. Then there exists a constant C such that for every

Lemma 3 . 4 . 1 .

 341 Suppose in addition to [H1]-[H3] (with ν = 0) that the uniform Kato square root property is satisfied. Define

3 . 5 . 3 . 4 . 2 .

 35342 Proposition Suppose [H1]-[H3] (with ν = 0) and the uniform Kato square root property. Let γ ∈ (0, 1]. If

Lemma 3 . 4 . 7 .

 347 Suppose [H1]-[H3] (with ν = 0) and the uniform Kato square root property. Define the operator Ru 0 (t) := A(t)e -tA(t) u 0

1 2 4 .

 14 regularity with respect to t. The maximal L 2 -regularity we proved here holds also in the case of elliptic operators on Lipschitz domains with Dirichlet or Neumann boundary conditions. The arguments are the same. One define the previous forms a(t) with domain V = H 1 0 (Ω) (for Dirichlet boundary conditions) or V = H 1 (Ω) (for Neumann boundary conditions). -Robin boundary conditions. Let Ω be a bounded domain of R d with Lipschitz boundary ∂Ω. We denote by Tr the classical trace operator. Let β : [0, τ ] × ∂Ω → [0, ∞) be bounded and such that In particular, β(., x) ∈ H α . We define the forms a(t, u, v) := Ω ∇u.∇v dx + ∂Ω β(t, .)Tr(u)Tr(v) dσ, for all u, v ∈ V := H 1 (Ω). Formally, the associated operator A(t) is (minus) the Laplacian with the time dependent Robin boundary condition ∂u ∂n + β(t, .)u = 0 on ∂Ω.

Proposition 4 . 1 . 2 . 2 + 1 2

 41221 Let E be a Banach space. For all > 0 and p > 2 we have(a) W 1 ,p (0, τ ; E) ⊂ B 1 2 ,p 2 (0, τ ; E) ⊂ W ,p (0, τ ; E).

Lemma 4 . 1 . 5 . 1 2

 4151 Let p > 2 and we suppose that D(A(0) ) = V. Then for all f ∈ B 1 2 ,p 2 (0, τ ; V ), the solution of the Problem (4.1.1) belongs to the space L ∞ (0, τ ; V). Moreover, there exist a positive constant C(δ, M, p) > 0 depending only on δ, M and p such that u L ∞ (0,τ ;V) ≤ C(δ, M, p) τ ;V )∩L ∞ (0,τ ;V ) . Proof. From the representation of the solution for the Problem (4.1.1), we obtain

.0. 2 )--

 2 For p = 2 or p = 2 with D(B(0) In the case p = 2 but D(B(0)

I

  i L p (0,τ ;H) ≤ C ( 0) | (V ,H)α,p ) = (V, D(C(0))) α,p ,

v

  (t) + B(t)v (t) + A(t)v(t) = f (t) v(0) = u 0 , v (0) = u 1 + γu 0 (5.2.6)5. Maximal regularity for the Damped Wave equationhas the maximal L p -regularity in H. In addition, there exists a positive constant C such that v M R(p,H) ≤ C u 1 (H,D(B(0))) 1-1

t 0 e

 0 -(t-s)B(t) u (s)ds + B(t)t 0 e -(t-s)B(t) B(s)u (s)ds + B(t) t 0 e -(t-s)B(t) C(s)u(s)ds = B(t) t 0 e -(t-s)B(t) g(s)ds. Here g(s) = e -γs f (s) for all s ∈ [0, τ ]. Hence B(t) t 0 e -(t-s)B(t) u (s)ds + B(t) t 0 e -(t-s)B(t) B(s)u (s)ds + B(t)t 0 e -(t-s)B(t) (C(s) -C(t))u(s)ds + B(t) t 0 e -(t-s)B(t) C(t)u(s)ds = B(t) t 0

t 0 e 5 .

 05 -(t-s)B(t) (B(t) -B(s))B(s) -1 (B(s)u (s) + C(s)u(s))ds-B(t) t 0 e -(t-s)B(t) (B(t) -B(s))B(s) -1 C(s)u(s)ds + t 0 e -(t-s)B(t) C(t)u (s)ds + B(t) t 0 e -(t-s)B(t) (C(t) -C(s))u(s)ds + B(t) t 0e -(t-s)B(t) g(s)ds.This allows us to write(I -Q)(B(.)u + C(.)u)(t) = + B(t)e -tB(t) u 1 + e -tB(t) C(t)u 0 -B(t) t 0 e -(t-s)B(t) (B(t) -B(s))B(s) -1 C(s)u(s)ds + B(t) t 0 e -(t-s)B(t) (C(t) -C(s))u(s)ds + t 0 e -(t-s)B(t) C(s)u (s)ds + B(t) t-s)B(t) (B(t) -B(s))B(s) -1 h(s)ds. Then, if I -Q is invertible on L p (0, τ ; H) we obtain B(t)u (t) + C(t)u(t) = (I -Q) -1 [R 1 u 1 + R 2 u 0 + W 1 (u) + L(f ) + L 2 (u) + W 2 (u)](t), (5.2.12) Maximal regularity for the Damped Wave equation here R 1 and R 2 are as in Proposition 5.2.4 and L as in Proposition 5.2.3 andW 1 (u)(t) := -B(t) t 0 e -(t-s)B(t) (B(t) -B(s))B(s) -1 (C(s)u(s))ds. W 2 (u)(t) := B(t)t 0 e -(t-s)B(t) (C(t) -C(s))u(s)ds. L 2 u(t) := t 0 e -(t-s)B(t) C(t)u (s)ds.

W

  i (u) L p (0,τ ;H) ≤ C 1 ds u L ∞ (0,τ ;V) .By the Sobolev embedding we haveW i (u) L p (0,τ ;H) ≤ C

t 0 e 0 e

 00 -(t-s)B(t) (B(t) -B(s))B(s) -1 C(s)u(s)ds + B(t) t 0 e -(t-s)B(t) (C(s) -C(t))u(s)ds + t -(t-s)B(t) C(t)u (s)ds + B(t) t 0 e -(t-s)B(t) g(s)ds, such that (Qh)(t) = B(t) t 0e -(t-s)B(t) (B(t) -B(s))B(s) -1 h(s)ds.

2 . 15 ), we get τ 0 edt τ 0 e

 21500 -tB(t) (C(t) -C(0))u 0 2 -tB(t) 2 L(V ,H) (C(t) -C(0)) 2 L(V,V )

3 .5 gives τ 0 e

 30 -tB(0) C(0)u 0 2 dt ≤ c B(0)

  t-s)B(t) (B(t) -B(s))h(s)ds, where t ∈ [0, τ ]. Taking the norm in V, it follows thatK(h)(t) V ≤ t 0 e -(t-s)B(t) L(V ,V) B(t) -B(s) L(V,V ) h(s) V ds ≤ C t 0 w(t -s) t -s ds h L ∞ (0,t;V) ds h L ∞ (0,t;V) ,where C is a positive constant independent of t. Now, we take τ small enough such that C √ t

  t-l)B(t) (B(t) -B(l))u (l)dl t s e -(t-l)B(t) C(l)u(l)dl + t se -(t-l)B(t) g(l)dl.

  t-l)B(t) (B(t) -B(l))u (l)dl + t s e -(t-l)B(t) (C(t) -C(l))u(l)dl + t s e -(t-l)B(t) B(t) -1 C(t)u (l)dl + t s

  s) ]B(t) -1 C(s)u(s) + e -(t-s)B(s) [B(t) -1 -B(s) -1 ]C(s)u(s) -[B(t) -1 -B(s) -1 ]C(t)u(t) + B(s) -1 [C(s)u(s) -C(t)u(t)] + B(s) -1 [e -(t-s)B(s) -I]C(s)u(s) = I 1 (s, t) + I 2 (s, t) + I 3 (s, t) + I 4 (s, t) + I 5 (s, t).

  t-s)λ (λ -B(t)) -1 (B(t) -B(s))(λ -B(s)) -1 B(t) -1 C(s)u(s)dλ V ≤ cw(t -s) Γ e -(t-s)|λ| cos θ (1 + |λ|) -1 d|λ| u(s) V ≤ c w(t -s) (t -s) u(s) V ,

I 4

 4 (s, t) V ≤ C 1 C(s)u(s) -C(t)u(t) V . I 5 (s, t) V ≤ C 2 [e -(t-s)B(s) -I]C(s)u(s) V .

5 .

 5 l-s)B(s) u (l)dl + t s e -(l-s)B(s) B(l)u (l)dl + t s e -(l-s)B(s) C(l)u(l)dl = t s e -(l-s)B(s) g(l)dl. Maximal regularity for the Damped Wave equation Now, by integration by parts we getu (s) = e -(t-s)B(s) u (t) -B(s) -1 e -(t-s)B(s) C(t)u(t) + B(s) -1 C(t)u(s) l-s)B(s) (B(l)u (l) + C(l)u(l))dl + t s e -(l-s)B(s) (B(s) -B(l))u (l)dl t s e -(l-s)B(s) (C(l) -C(t))u(l)dl t s e -(l-s)B(s) B(s) -1 C(t)u (l)dl -t s e -(l-s)B(s) g(l)dl. Hence u (s) -u (t) = [e -(t-s)B(s) u (t) -u (t)] -B(s) -1 e -(t-s)B(s) C(t)u(t) + B(s) -1 C(t)u(s) l-s)B(s) (B(l)u (l) + C(l)u(l))dl + t s e -(l-s)B(s) (B(s) -B(l))u (l)dlt s e -(l-s)B(s) (C(l) -C(t))u(l)dl t s e -(l-s)B(s) B(s) -1 C(t)u (l)dl -t s e -(l-s)B(s) g(l)dl.

  Further regularity results . . . . . . . . . . . . . . . . . . . . . . 2.5 Right perturbations-Maximal regularity . . . . . . . . . . . . . .

	3 Lions' maximal regularity problem with H

Introduction 1 Preliminaries 1.1 Forms and their operators . . . . . . . . . . . . . . . . . . . . . 1.2 Fractional Powers . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Maximal Regularity for autonomous problem in Hilbert space . 1.4 Maximal regularity for autonomous Cauchy problems in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Maximal regularity for non-autonomous problems in V . . . . . 2 Non-autonomous right and left multiplicative perturbations and maximal regularity 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Maximal regularity for the unperturbed problem . . . . . . . . . 2.3 Maximal regularity for left perturbations . . . . . . . . . . . . . 2.3.1 Single left multiplicative perturbation-Resolvent estimates 2.3.2 Single perturbation-Maximal regularity . . . . . . . . . 2.3.3 Time dependent perturbations-Maximal regularity . . . 2.4

  où Ω est un domaine lipschitzien borné. Nos principaux résultats peuvent être résumés comme suit (voir les Théorèmes 2.3.6, 2.5.1 et 2.4.2 pour des énoncés plus généraux et précises). On suppose que pour certains β, γ ∈ [0, 1],|b(t; g, h) -b(s; g, h)| ≤ w(|t -s|) h [H;V] β g [H;V]γ ; g, h ∈ V, où w : [0, τ ] → [0, ∞) estune fonction croissante telle que V] β désigne l'espace d'interpolation complexe entre H et V. On suppose que t → B(t) est continue dans [0, τ ] avec des valeurs dans L(H). On montre alors que le problème de Cauchy (P1) admet la rǵularité maximale L p dans H pour tout p ∈ (1, ∞), quand u 0 = 0. Si on suppose de plus que

	Introduction
	d'évolutions semi-linéaires et aux limites, comme par exemple
				   	u (t) = m(t, x, u(t), ∇u(t))∆u(t) + f (t) u(0) = u 0 ∈ H 1 (Ω)
				  	∂u(t) ∂n	+ β(t, .)u(t) = 0 on ∂Ω,
					τ 0	w(t) t 1+ γ 2	dt < ∞
	et [H, τ w(t)
	0	t	β 2 +	pγ 2

  Definition 1.4.1. A Banach space E is said to be class U M D if the Hilbert transform H is bounded in L p (R; E) for all (or equivalently for one) p ∈]1, ∞[.

	Remark 1.4.2. There are other definitions of U M D-space. For example E
	is a U M D space if is ξ-convex i.e, there exists a symmetric biconvex function
	ξ on E × E such that ξ(0, 0) > 0 and
	∀x, y ∈ E, x E ≥ 1, ξ(x, y) ≤ x + y E .
	Example 1.4.3. A Hilbert space is a U M D space.
	Theorem 1.4.4 (Coulhon, Lamberton 1986). If -A generates the Poisson
	semigroup on X = L 2 (Y ) (i.e. e -tA where Y is a Banach space ) then A has
	the maximal L p -regularity property on X if and only if X is U M D.
	Definition 1.4.5. Let X be a Banach space. A sequence (x k ) k∈N ⊂ X is called
	Shauder basis if for every x ∈ X there exists a unique sequence (a k

  Maximal regularity for the unperturbed problem then (4.1.1) has maximal L p -regularity in H for all u 0 ∈ (H, D(A(0))) 1-1 p ,p . Moreover there exists a positive constant C such that

Then the Cauchy problem (4.1.1) with u 0 = 0 has maximal L p -regularity in H for all p ∈ (1, ∞). If in addition, τ 0 w(t) p t 1 2 (β+pγ) dt < ∞ (2.2.4) 2.2.

  By Corollary 2.3.3, for each t ∈ [0, τ ], the operator -BA(t) generates a holomorphic semigroup (e -sBA(t) ) s≥0 on H.Our aim in this section is to prove maximal regularity in H for the Cauchy problem associated with BA(t), t ∈ [0, τ ]. The definition of maximal L pregularity in this context is the same as in Definition 2.2.1.

Set R(λ, BA(t)) := (λI + BA(t)) -1

  Theorem 2.3.6. Suppose that (a(t)) t satisfies [H1]-[H3]. Let B(t) and P (t) be bounded operators which satisfy (2.3.9)-(2.3.11) and (2.3.13). Suppose in

addition that t → B(t) is continuous on [0, τ ] with values in L(H). Suppose that for some β, γ ∈ [0, 1]

  The uniqueness of u follows from the uniqueness 2.3. Maximal regularity for left perturbations of the fixed point of S. The a priori estimate of Proposition 2.3.5 and the fact that v = u on [0, t 0 ] give Next, we divide the interval [0, τ ] into ∪ i∈1,....,N [t i-1 , t i ] with t i -t i-1 small enough. On each interval [t i-1 , t i ], we search for a solution u i to (2.3.16) with initial data u

	u M R(p,H) ≤ C f Lp(0,t 0 ;H) + u 0 (H,D(A(0))) 1-1 p ,p	.	(2.3.21)

Clearly, u is a solution of the Cauchy problem (2.3.16) on [0, t 0 ].

  2. Non-autonomous right and left multiplicative perturbations and maximal regularityis a bounded operator for all p ∈ (1, ∞). We obtain and note that for f ∈ L p (0, τ ; H) we have A(•)u(•) ∈ L p (0, τ ; H) since we have proved maximal L p -regularity for small τ . Hence g ∈ L p (0, τ ; H). This finishes the proof of u(τ ) ∈ (H, D(A(τ )) 1-1 p ,p and we obtain maximal L p -regularity of (2.3.16) on [0, τ ] for every τ > 0. The uniqueness of the solution on [0, τ ] follows from the uniqueness on each small sub-interval [t i , t i+1 ]. It remains to prove the a priori estimate (2.3.19). On each small subinterval [t i ; t i+1 ] we have the a priori estimate (2.3.21). That is

	τ	τ	1/p	τ
	Ae -tA	e -(τ -s)A g(s)ds p dt	≤ C(	g(s) p ds) 1/p	(2.3.27)
	0	0	0
	for all g ∈ L p (0, τ ; H).
	Set
	g(s) := (I -B(s)A(s)u(s) -P (s)u(s) + f (s)

  5.2. Maximal regularity for the damped wave equation in HCombining (5.2.8), (5.2.9) and (5.2.7) we obtain B(t)u (t) -B(t)e -tB(t) u 1 -B(t)

			t
			e -(t-s)B(t) (B(t) -B(s))u (s)ds
			0
		t
	+ B(t)	e -(t-s)B(t) (C(s) -C(t))u(s)ds
	0
		t
	+ C(t)u(t) -e -tB(t) C(t)u 0 -	e -(t-s)B(t) C(t)u (s)ds
		0
		t
	= B(t)	e -(t-s)B(t) g(s)ds.		(5.2.10)
	0
			t
			e -(t-s)B(t) B(t)u (s)ds	(5.2.8)
			0
	and
	t
	B(t)	e -(t-s)B(t) C(t)u(s)ds = C(t)u(t) -e -tB(t) C(t)u(0)
	0
		-

t 0 e -(t-s)B(t) u (s)ds = B(t)u (t) -B(t)e -tB(t) u (0) -B(t) t 0 e -(t-s)B(t) C(t)u (s)ds.

(5.2.9) Therefore

B(t)u (t) + C(t)u(t) = B(t)e -tB(t) u 1 + e -tB(t) C(t)u 0 + B(t)

  Qh)(t) ≤

	t	B(t)e -(t-s) 2 B(t)	L(H) e -(t-s) 2 B(t)	L(V ,H)
	0
		3	B(s) -1	L(H,V) h(s) ds.	(5.2.13)
		2
				3
				2

× B(s) -B(t) L(V,V ) B(s) -1 L(H,V) h(s) ds ≤ C t 0 w(t -s) (t -s) Now,

we replace B(s) by α + B(s), with α > 0 is big enough (here we use Proposition 5.2.1). (5.2.13) is valid with a constant independent of α. Using the estimate (α + B(s)) -1 L(H,V) ≤ c √ α in (5.2.13) for α + B(s), we get (Qh)(t) ≤ C t 0 w(t -s) (t -s)

  ) -1 C(t)u(t) + e -(t)B(t) B(t) -1 C(t)u 0 + e -(t)B(t) u 1 +

t 0 e -(t-s)B(t) (B(t) -B(s))u (s)ds + t 0 e -(t-s)B(t) (C(t) -C(s))u(s)ds + t 0 e -(t-s)B(t) B(t) -1 C(t)u (s)ds + t 0 e -(t-s)B(t) g(s)ds.

(5.2.16)

  where C 1 , C 2 , C 3 are a positive constants independents of s and t. By using Lemma 3.4.1 we getK 6 (t, s) V ≤ C g L 2 (s,t;V) .

s) V ≤ C 1 t s B(t) -B(s) L(V,V ) t -s ds u L ∞ (s,t;V) . K 4 (t, s) V ≤ C 2 t s C(t) -C(s) L(V,V ) t -s ds u L ∞ (s,t;V) . K 5 (t, s) V ≤ C 3 (t -s) u L ∞ (s,t;V) ,

Remerciements

Non-autonomous right and left multiplicative perturbations and maximal regularity

Proof of Theorem 2.5.1. Let f ∈ L p (0, τ ; H) and initial data u 0 be such that u 0 ∈ B(0) -1 (H, D(A(0))) 1-1 p ,p . We consider the Cauchy problem with left multiplicative perturbations v (t) + B(t)A(t)v(t) -B (t)B(t) -1 v(t) + B(t)P (t)B(t) -1 v(t) = B(t)f (t) v(0) = B(0)u 0 .

(2.5.6) Note that B(•)f (.) ∈ L p (0, τ ; H) and B(0)u 0 ∈ (H, D(A(0))) 1-1 p ,p . Note also that t → -B (t)B(t) -1 +B(t)P (t) is strongly measurable with values in L(H). Thus, we can apply Theorem 2.3.6. We obtain existence and uniqueness of v ∈ W 1 p (0, τ ; H) such that v(t) ∈ D(A(t)) for a.e. t ∈ [0, τ ] which satisfies (2.5.6). We set u(t) := B(t) -1 v(t). Using Lemma 2.5.2 we check easily that u ∈ W 1 p (0, τ ; H), B(t)u(t) ∈ D(A(t)) for a.e. t and it is the unique solution of (2.5.1). Finally, (2.5.4) follows immediately from the a priori estimate of Theorem 2.3.6.

Note that we may consider both left and right multiplicative perturbations at the same time. Let B 0 (t) and B 1 (t) be bounded operators satisfying the same assumptions (2.3.9) and (2.3.10). We assume that t → B 0 (t) is continuous and t → B 1 (t) is Lipschitz continuous on [0, τ ]. We assume that the forms a(t) and P (t) are as in Theorem 2.5.1. We consider the Cauchy problem u (t) + B 0 (t)A(t)B 1 (t)u(t) + P (t)u(t) = f (t), u(0) = u 0 .

(2.5.7)

Then the maximal L p -regularity results of Theorem 2.5.1 hold for (2.5.7) for initial data u 0 ∈ B 1 (0) -1 (H, D(A(0))) 1-1 p ,p . The proof is very similar to the previous one. We consider the Cauchy problem with left perturbations

(2.5.8) We obtain the maximal L p -regularity for (2.5.8) by Theorem 2.3.6 and set as above u(t) = B 1 (t) -1 v(t).

Chapter 3

Lions' maximal regularity problem with H 1 2

-regularity in time

Les résultats présentés dans ce chapitre ont fait l'objet de l'article [START_REF] Achache | Lions' maximal regularity problem with H 1 2regularity in time[END_REF] en collaboration avec El Maati Ouhabaz.

Introduction

Let (H, (•, •), • ) be a separable Hilbert space over R or C. We consider another separable Hilbert space V which is densely and continuously embedded into H. We denote by V the (anti-) dual space of V so that

Hence there exists a constant C > 0 such that

where • V denotes the norm of V. Similarly,

We denote by , the duality V -V and note that ψ, v = (ψ, v) if ψ ∈ H and v ∈ V. We consider a family of sesquilinear forms

We assume throughout this paper the following usual properties.

• [H1]: D(a(t)) = V (constant form domain), 3. Lions' maximal regularity problem with H 1 2 -regularity in time Lemma 3.3.2. Suppose the assumptions of the previous lemma. Given t, s ∈ [0, τ ]. Suppose that there exists γ ∈ [0, 1) such that either (3.3.4) or (3.3.6) holds for t, s. If D(A(s)

Proof. Suppose that (3.3.4) is satisfied. Recall that

Hence

Using the previous lemma, we estimate the H -V norm of the term in the integral by

This implies that A(t) -1 2 -A(s) -1 2 is bounded from H to V and hence D(A(t)

). Then we write as before

We estimate the norm in H of the term inside the integral by

Lions' maximal regularity problem with

regularity in time

The last estimate follows as in the proof of Proposition 3.4.2 in which we also use (3.4.5). It is valid for γ ∈ (0, 1]. Therefore

For γ = 1 we use the assumption (3.2.1) and obtain

for some constant C > 0. This proves assertion 1) of the proposition. Assertion 2) follows directly from (3.4.13) when γ ∈ (0, 1).

Proofs of the main results

After the auxiliary results of the last two sections we are now ready to give the proofs of the main results of this paper. Note that we may assume without loss of generality that [H3] holds with ν = 0, see the beginning of Section 3.3.

Proof of Theorems 3.2.2 and 3.2.4. Let γ ∈ (0, 1]. We give the proof for the two theorems without considering separately the cases γ = 1 and γ ∈ (0, 1). If γ = 1 we assume (3.2.1).

Suppose first that A ∈ Ḣ γ 2 (0, τ ; L(V, V γ )). Let f ∈ L 2 (0, τ ; H) and u 0 ∈ V. Let u ∈ L 2 (0, τ ; V) ∩ H 1 (0, τ ; V ) be the solution of (P') given by Lions' theorem. Our aim is to prove that u ∈ H 1 (0, τ ; H) or equivalently that A(.)u(.) ∈ L 2 (0, τ ; H). Using (3.4.8) we have

By Lemma 3.4.7 the operator R is bounded from V into L 2 (0, τ ; H) and by Corollary 3.4.4 the operator L is bounded on L 2 (0, τ ; H). Concerning the operator S, we have

for some constant C independent of f and u 0 . Suppose for a moment that (3.5.1) is proved. Then we apply Proposition 3.4.5 ]. Now we "glue" the solutions u i and obtain a solution u of (P) on [0, τ ] such that u ∈ H 1 (0, τ ; H). The apriori estimate (3.2.5) on [0, τ ] follows by summing the corresponding apriori estimates on each sub-interval and by using Proposition 3.4.5. The uniqueness of the solution of (P) follows from this apriori estimate. Note that in Theorem 3.2.4 we assume merely that D(A(t 0 )

This assumption implies the uniform Kato square root property by Proposition 3.3.3.

In addition, there exists a constant C, independent of u 0 , f and v such that

Bounding u H 1 (0,τ ;H) by the term on the RHS follows from the classical maximal regularity for the operator A(0) in the Hilbert space H. For the bound of u L ∞ (0,τ ;V) by the same term we use either Proposition 3.4.5 or the classical embedding

).

Define the operator K on E by K(v) := u. We prove that for τ > 0 small, K is a contraction operator. Indeed, let

Hence by (3.5.2) and [H4] with γ = 0

Autonomous first order Cauchy problems

Then by Proposition 4.1.3 we obtain L ∈ L(G; W 1,p 0 (0, τ, V)) and

Since (V , V)1 2 ,2 = H, we get by interpolation (Proposition 4.5.1) that

In addition,

Here C(δ, M, p) is a positive constant depending only on δ, M, p.

For more details and references see Section 4.5.

Remark 4.1.6. We define the operator ∂ in L 2 (0, τ ; V ) by

with (∂v)(t) = v (t) and the operator of multiplication A

For t ∈ [0, τ ] we can rewrite the operator L as follows

2 (0, τ ; V) be the solution of the Problem (4.1.1) and there is a positive constant C(δ, M, p) depending only on δ, M, p such that

Proof. To prove the proposition we apply Lemma 4.1.5 and we use the fact that the operator L is bounded from

Maximal Regularity for non-autonomous first order Cauchy problems

Recall that (A(t)) t∈[0,τ ] satisfies the uniform Kato property that if D(A(t) 1 2 ) = V for all t ∈ [0, τ ] and there are c 1 , c 1 > 0 such that

Let us define the maximal regularity space

It is a Banach space for the norm

We assume that P (t) ∈ L(V, H) and P (t) L(V,H) ≤ h(t), such that h ∈ L p (0, τ ).

Our main result in this chapter is the following Theorem 4.2.3. Suppose (4.2.2). Let p > 2 and we assume that

Then for every f ∈ L p (0, τ ; H) there exists a unique solution u ∈ M R(p, H) of the non-autonomous problem

p ,p for all t ∈ [0, τ ] and there exists a positive constant C such that

-As we shall see in the proof, the regularity assumption on A(.) can be weakened considerably. Indeed, A(.) is piecewise in

) is sufficient and with this assumption the uniform Kato property is necessary by the example in [START_REF] Arendt | Lions' problem on maximal regularity[END_REF](Example 5.1).

0, τ ; L(V, V ) and we define the space

Non-autonomous maximal regularity under Besov regularity in time

Let u 1 be the solution of the problem

where v ∈ Z. We note that t → P (t)v(t) ∈ L p (0, τ ; H) and P (.

Since -A(0) is the generator of an analytic semigroup on H, we have the maximal L p -regularity for the previous problem (see Proposition 1.3.3). Thus,

Further, by the real interpolation formula for vector-valued Besov spaces (see [START_REF] Amann | Compact embeddings of vector-valued Sobolev and Besov spaces[END_REF] Corollary 4.3)

Indeed, we use the inclusion properties of the real interpolation spaces (see [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF]

Note that u 1 is giving by

By the quadratic estimate (4.1.3) and (4.2.2), (4.2.5) we obtain that there exists positive constants C(δ, M ), C 1 (δ, M, p) such that

). Therefore there exists a constant C(δ, M, p) depending only on δ, M, p such that

). Using Propositions 4.1.5 and 4.2.2, there exists u 2 ∈ Z be the solution of the problem

Maximal Regularity for non-autonomous first order Cauchy problems

(4.2.7)

By using Proposition 4.2.2, we get

Now, we define the operator

where u is the unique solution of the Problem (4.2.7) for the given v ∈ Z. For all v 1 , v 2 ∈ Z, we have by (4.2.8)

Let τ > 0 be small enough such that

By the contraction fixed point theorem we get that there exists a unique u ∈ Z such that Su = u. This gives the existence and uniqueness of the solution for the non-autonomous problem (4.2.3) and it follows from (4.2.8) that u satisfies the apriori estimate

). Now for τ arbitrary we split (0, τ ) into small intervals and we use the same procedure in each sub-interval. In order to obtain a solution u ∈ W 1,p (0, τ ; H) we "glue" the solutions of each-interval. What remains then to prove is that

Optimality of the Results

As a result, by Hardy's inequality and Lemma 4.2.1 we obtain finally

where C, C 1 , C 2 are arbitrary positive constants. Thus, u( ) ∈ (H, D(B(0)))

Then for every f ∈ L p (0, τ ; H) there exists a unique solution u ∈ M R(p, H) of the Problem (4.2.3).

Proof. It follows from [START_REF] Simon | Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval[END_REF], Theorem 10 that Ḣ1-

We then apply Theorem 4.2.3 to get the desired result. Theorem 4.2.6. Let p > 2 and α ∈ ( 1 p , 1 2 ]. We consider f ∈ L p (0, τ ; H) and we suppose that A ∈ B α,p 2 (0, τ ; L(V, V )). Then there exists a unique solution u ∈ W 1,p (0, τ ; (V , V) α,2 ) ∩ B α,p 2 (0, τ ; V) of the Problem (4.2.3).

Proof. The theorem is proved through the same steps as in the proof of Theorem 4.2.3.

Optimality of the Results

In this section we shall discuss the optimality of our result where we use an example in [START_REF] Arendt | Lions' problem on maximal regularity[END_REF] (Proposition 10.1). Proposition 4.3.1. Let p > 2 and q ∈]2, ∞[. There exists a Gelfand triple V → H → V and a family of forms a(.) coercive, symmetric and nonautonomous associated to A(.) ∈ B 1 2 ,p q (0, τ ; L(V, V )) and the correspond Cauchy problem (4.2.3) does not satisfy the maximal L p -regularity in H. [START_REF] Simon | Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval[END_REF] Theorem 11) and the inclusion is strict (see [START_REF] Triebel | Interpolation Theory, Function Spaces, Differential Operators[END_REF], 2.3.3). According to [START_REF] Arendt | Lions' problem on maximal regularity[END_REF], there exists a coercive, symmetric and non-autonomous 4. Non-autonomous maximal regularity under Besov regularity in time forms a(t) with A(.) ∈ B 1 2 ,p q (0, τ ; L(V, V )) that does not satisfy the maximal L 2 -regularity in H. The example in [START_REF] Arendt | Lions' problem on maximal regularity[END_REF] proves that there exists f ∈ L ∞ (0, τ ; H) such that the solution u of the Cauchy problem (4.2.3) belongs to

Therefore, we also have a counter-example for the maximal L p -regularity for all p ∈ (1, ∞). Remark 4.3.2. The example in [START_REF] Arendt | Lions' problem on maximal regularity[END_REF] proves also that if A(.) ∈ B s,p 2 (0, τ ; L(V, V )), with s < 1 2 then the maximal L p -regularity may fails in H for all p ∈ (1, ∞).

Applications

This section is devoted to some applications of the results given in the previous sections. We give examples illustrating the theory without seeking for generality. The first example is similar to the example given in 3.6. Here we study the maximal L p -regularity for p > 2.

-Elliptic operators on R n .

We define the operator P (t) ∈ L(V, H) by P (t)u = n j=1 b j (t)∂ j u, where u ∈ V and t ∈ [0, τ ]. We assume that the matrix C(t, x) = (c kl (t, x)) 1≤k,l≤n satisfies the usual ellipticity condition. Next we assume that C ∈ B 1 2 ,p 2 (0, τ ; L ∞ (C n 2 )) and b j ∈ L p (0, τ ; L ∞ (R n )) where j ∈ {1, ..., n}. We note that

for some constant M . This implies that A ∈ B 1 2 ,p 2 (0, τ ; L(V, V )).

We are now allowed to apply Theorem 4.2.3. We obtain the maximal L pregularity and apriori estimate for the parabolic problem

Applications

-Schrödinger operators with time-dependent potentials. Let 0 ≤ m 0 ∈ L 1 loc (R n ) and m : [0, τ ] × R n → R be a measurable function for which there exist positive constants α 1 , α 2 and M such that for a.e. x and all t ∈ [0, τ ]

We define the form

It is clear that V is a Hilbert space for the norm u V given by

In addition, a is V-bounded and coercive. Its associated operator on

with domain

Next we assume that t → m(t, .)m 0 (.

, with p > 2. We have

Then we get

, we apply Theorem 4.2.3 and obtain a unique

Non-autonomous maximal regularity under Besov regularity in time

Appendix

Let X be a Banach space and consider p ∈ [1, +∞). If D max is the differentiation operator on L p (0, τ ; X) with maximal domain i.e D(D max ) := W 1,p (0, τ ; X) D max u := u , then we have (X, D(D max )) θ,q = B θ,p q (0, τ ; X), (θ ∈ (0, 1), 1 ≤ q < ∞). In particular, (X, D(D max )) θ,p = B θ,p p (0, τ ; X) = W θ,p (0, τ ; X), (θ ∈ (0, 1)). Let D be the restriction of D max on L p (0, τ ; X) to the domain D(D)

for θ > 1 p . For θ < 1 p we get (X, D(D)) θ,q = (X, D(D max )) θ,q = B θ,p q (0, τ ; X). We recall that the operator D is sectorial of angle π 2 , while D max is not sectotrial. In fact σ(D max ) = C. For the case where D is the differentiation operator on C(0, τ ; X) with domain

For more details and references see [START_REF] Batty | Maximal regularity in interpolation spaces for second order Cauchy problems, Operator semigroups meet complex analysis[END_REF] (Section 2), [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF] (Example 1.9, Exercises 5 and 6, p.18) and [START_REF] Triebel | Theory of Function Spaces[END_REF] (Theorem, p.204). Proposition 4.5.1. Let X, Y be two Banach spaces which form an interpolation couple. Then for all p ∈ (1, ∞) and α ∈ (0, 1) we have a natural isometric isomorphism of spaces (L p (0, τ ; X), L p (0, τ ; Y )) α,p = L p (0, τ ; (X, Y ) α,p ).

(4.5.1)

Similarly,

Proof. For (4.5.1) see [START_REF] Hytönen | Analysis in Banach Spaces[END_REF] (Theorem 2.2.10). Since D : W 1,p 0 (0, τ ; Y ) → L p (0, τ ; Y ) and D : W 1,p 0 (0, τ ; X) → L p (0, τ ; X) is an isomorphism we get by interpolation and (4.5.1)

) is an isomorphism. On the other hand, since D is an isomorphism from W 1,p 0 (0, τ ; (X, Y ) α,p ) to L p (0, τ ; (X, Y ) α,p ), then (4.5.2) holds.

Chapter 5 Maximal regularity for the Damped Wave equation

In this chapter we consider maximal regularity for second order Cauchy problems. We focus on the damped wave equation. We consider a family of sesquilinear forms

• [H3]: Re b(t, u, u) + ν u 2 ≥ δ u 2 V (∀u ∈ V) for some δ > 0 and some ν ∈ R (uniform quasi-coercivity).

We denote by B(t), B(t) the usual associated operators with b(t)(as operators on H and V ). Let A(t) ∈ L(V, V ) such that there exists a function h :

We denote by A(t) the part of A(t) in H, with domain

Given a function f defined on [0, τ ] with values either in H or in V and consider the second order evolution equation

(5.0.1)

Maximal regularity for the Damped Wave equation

-for p = 2, t → b(t) is piecewise relatively continuous.

Then for all u 1 ∈ (V , V) 1-1 p ,p and f ∈ L p (0, τ ; V ) there exists a unique solution

(5.1.1)

In addition, there exists a positive constant C such that

Here,

From [3] (Theorem III 4.10.2) we have the following lemma

Let us consider the space

Let v ∈ M R p (V, V, V ) be a solution of (5.1.1). For u 0 ∈ V, we set w(t) = u 0 + t 0 v(s)ds. Then w (t) = v(t) and

(5.1.2)

Moreover, we have the following estimate

(5.1.3)

We note also that the solution of the Problem (5.1.2) is unique. Indeed, we suppose there are two solutions

(5.1.4)

Maximal regularity for the damped wave equation in V

Therefore, for t > 0 we get

Then by Lemma 1.5.3 we obtain v (t) 2 + 2δ

. By Gronwall's lemma we get that for all t ∈ [0, τ ] v(t) = 0 and so v 1 (t) = v 2 (t). Using Lemma 5.1.2 and the Sobolev embedding we have

We define the associated trace space to M R p (V, V, V ) by

p ,p is obtained by (5.1.5). For the second injection "← " let us take u 1 ∈ (V , V) 1-1 p ,p . Then by [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF] (Corollary 1.14) there exists w ∈ M R p (V, V ) such that w(0) = u 1 . We set u(t) = u 0 + t 0 w(s)ds, where u 0 ∈ V. Thus, u ∈ M R p (V, V, V ) and

We note that the trace space associated to M R p (V, V ) is isomorphic to the real interpolation space (V , V) 1-1 p ,p (see [START_REF] Lunardi | Analytic Semigroups and Optimal Regularity in Parabolic Problems[END_REF], Chapter 1). Then

From the previous proposition and (5.1.5), we can deduce that the operator

is well defined and bounded.

Our main result on maximal L p -regularity in V is the following Theorem 5.1.5. Let p ∈ (1, ∞). We assume one of the following conditions -for p = 2, t → b(t) is measurable.

-for p = 2, t → b(t) is piecewise relatively continuous.

Let

(5.1.6) Moreover, there exists positive constant C independent of u 0 , u 1 and f such that the following estimate holds

(5.1.7)

As mentioned in the introduction, this theorem was proved by Batty, Chill and Srivastava [START_REF] Batty | Maximal regularity for second order non-autonomous Cauchy problems[END_REF] but they consider only the case u 0 = u 1 = 0 and suppose t → A(t) L(V,V ) ∈ L p (0, τ ).

Proof. We introduce the subspace

We equip this subspace with the norm u → u L p (0,τ ;V ) + u L p (0,τ ;V) . To prove the existence and the uniqueness of the solution we use the contraction fixed point theorem and the existence of a solution in M R p 0 (V, V, V ) for the Problem (5.1.2). Indeed, let z ∈ M R p 0 (V, V, V ) and u ∈ M R p 0 (V, V, V ) be the solution of the problem

(5.1.9)

Maximal regularity for the damped wave equation in V

Therefore, by (5.1.3) we get

We choose τ small enough such that C τ 0 h(t) p t p-1 dt < 1. Thus, L is a contraction operator. So by the contraction fixed point theorem, there exists a unique solution of the problem

for all f ∈ L p (0, τ ; V ) and τ > 0 small enough and we get from the estimate (5.1.3)

This completes the proof when τ is small enough. We note that by Remark 5.1.4 (z, z ) ∈ C([0, τ ]; T R p (V, V )). For arbitrary τ > 0, we split [0, τ ] into a finite number of sub-intervals with small sizes and proceed exactly as in the previous proof. Finally, we stick the solutions and we get the desired result.

Maximal regularity for the Damped Wave equation

Maximal regularity for the damped wave equation in H

Let A(t) and B(t) be as before and we assume that

endowed with norms respectively

The maximal L p -regularity in H for the Problem (5.1.6) consists of proving existence and uniqueness of a solution u ∈ M R(p, H) provided f ∈ L p (0, τ ; H) and (u 0 , u 1 ) ∈ T r(p, H).

Preparatory lemmas

Proposition 5.2.1. The maximal regularity on H of the problem

is equivalent to the maximal regularity of the problem

for all γ ∈ C.

Proof. Let v be the solution of (5.2.1) and γ ∈ C. We set u(t) = e -γt v(t).

By a simple computation we obtain that u satisfies (5.2.2). In addition, f ∈ L p (0, τ ; H) if and only if t → e -γt f (t) ∈ L p (0, τ ; H) and it is clear that v ∈ W 2,p (0, τ ; H)∩W 1,p (0, τ ; V) if and only if u ∈ W 2,p (0, τ ; H)∩W 1,p (0, τ ; V).

Maximal regularity for the Damped Wave equation

Proposition 5.2.3 follows from [START_REF] Haak | Maximal regularity for non-autonomous evolution equations[END_REF] (Lemmas 2.5 and 2.6). Let p ∈ (1, ∞). We introduce the following assumptions -for p = 2 : t → b(t) is relatively continuous and for p = 2 : t → b(t) is measurable.

-

-

where w

dt < ∞.

(5.2.3)

We assume also that -for p = 2 (or p = 2 with D(B(0)

-In the case where p = 2, but D(B(0)

for arbitrary small > 0.

Let γ > 0 be big enough such that C(t) = A(t) + γB(t) + γ 2 is associated with a V-bounded coercive forms c(t). We denote by C(t) the part of C(t) on H.

We set X p = V for all p ∈ (1, 2[ and X p = (V, D(C(0))) 1-1 p ,p for p ≥ 2.

Lemma 5.2.4. Let u 1 ∈ (H, D(B(0))) 1-1 p ,p and u 0 ∈ X p , then the operators

are bounded from (H, D(B(0))) 1-1 p ,p and X p into L p (0, τ ; H), respectively.

Remark 5.2.5. We note that the operator R 1 is already studied in Theorem 2.2.2. Here we assume less regularity on the operators B(t) with respect to t compared with Theorem 2.2.2.

Maximal regularity for the Damped Wave equation

Therefore

Then -for p = 2 or p = 2 with D(B(0)

2 ) → V, we have

-for p = 2 and D(B(0)

) .

Using the definition of the interpolation space of the domain for a sectorial operator (see Proposition 1.2.4), we obtain ∞ 0 B(0)e -tB(0) u 1 p dt ≤ u 1 (H,D(B(0))) 1-1 p ,p .

Then

-for p = 2 (or p = 2 with D(B(0)

-if p = 2 and D(B(0)

) .

For R 2 with p < 2, we get

Maximal regularity for the damped wave equation in H

The following estimate holds for L 2

As a result, we obtain from (5.2.12), (5.1.7) and the previous estimates

Then u = g -C(.)u + B(.)u ∈ L p (0, τ ; H) and we have the maximal L pregularity in H for the Problems (5.2.2) and (5.2.6).

Recall that (B(t)) t∈[0,τ ] satisfies the uniform Kato property if D(B(t)

(5.2.14)

In the next result we prove the maximal L 2 -regularity where we improve the assumption on u 0 and we prove that the solution belongs to C 1 ([0, τ ], V). More precisely Theorem 5.2.7. We assume the uniform Kato property (5.2.14) and the following two conditions that for all s, t ∈ [0, τ ]

for an arbitrary > 0.

2-

Then for all f ∈ L 2 (0, τ ; H) and (u 0 , u 1 ) ∈ V × V, there exists a unique u ∈ M R(2, H) be the solution to the Problem (5.2.6). Moreover, v ∈ C 1 ([0, τ ]; V).

Maximal regularity for the damped wave equation in H

V there exists a unique u ∈ H N (0, τ ; H) ∩ C N -1 (0, τ ; V) be the solution of the problem

(5.2.18) In addition, there exists a positive constant C such that

Proof. We prove the theorem by induction. In case N = 1 the result follows from Theorem 2.4.2. The theorem holds for N = 2 by Theorem 5.2.7. Now we assume that the theorem is true at order N -1 where N is an arbitrary positive integer. By integration and following the same strategy of proof as in Theorem 5.1.5 we prove the maximal regularity in V for the Cauchy problem (5.2.18) and we get u ∈ H N (0, τ ; V ) ∩ H N -1 (0, τ ; V). Let γ > 0 and we set v(t) = e -γt u(t). By Leibniz's rule and using the equation (5.2.18) we get that v is the solution of the problem

, for all j ∈ [0, N -1] and C m j = m! j!(m-j)! is the binomial coefficient. We assume that γ > |ν| N , then C N -1 (t) = A N (t) + N γ is associated with Vbounded coercive form. By performing an integration by parts as (5.2.8), (5.2.9) we obtain

e -(t-s)C N -1 (t) e -γs f (s)ds.

Maximal regularity for the Damped Wave equation

Now, to prove the maximal regularity in H we follow the same steps and arguments as in the proof of Theorem 5.2.7.

The counter-examples

In this section we shall give some examples where the maximal regularity fails. Let c : V × V → C be bounded coercive form and let C is the operator associated to c in V and C = C | H . We introduce the space

Let us consider the space

We define the associated trace space by

The counter-examples

Next, we prove that D ⊆ T R c (2, H). Indeed, let (u 0 , u 1 ) ∈ D. Then by [START_REF] Lunardi | Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie[END_REF] (Corollary 1.14), there exist v ∈ L 2 (0, τ ; D(C))∩H 1 (0, τ ; H) s.t v(0) = u 0 +u 1 . We set u(t) = e -t u 0 + t 0 v(s)ds.

Then u(0) = u 0 , u (t) = -e -t u 0 + v(t) and u (0) = u 1 . We get u ∈ H 2 (0, τ ; H) ∩ H 1 (0, τ ; V) and u + u ∈ L 2 (0, τ ; D(C)).

Hence, D ⊆ T R c (2, H).

Dier's counter-example

The next counterexample is inspired from [START_REF] Arendt | Lions' problem on maximal regularity[END_REF] for the first order Cauchy problem.

According to [START_REF] Mcintosh | On the comparability of A 1 2 and A * 1 2[END_REF], there exist Hilbert spaces V, H with V → d H and a V-bounded coercive form b : V × V → C such that D(B 2 ) \ V is not empty. Let v(t) = -t 2 φ(1 -t), where t ∈ [0, 1]. We get v (t) = -2tφ(1 -t) + t 2 φ (1 -t) with v (0) = 0, v(0) = 0 and v(1) = 0, v (1) = u 1 . Note that (0, u 1 ) ∈ T R b (2, H) but (0, u 1 ) / ∈ T R c (2, H). We define the non-autonomous form a(t; ., .) = I [0,1] (t)b(., .) + I [1,2] (t)c(., .), where I is the indicator function and we denote by A(t) the associated operator to a(t).

Let on L 2 (0, 1; V ). Thus, w / ∈ H 2 (0, 1; H) and u / ∈ H 2 (0, 2; H). does not have the maximal L 2 -regularity in H, or equivalently, there exists f ∈ L 2 (0, τ ; H) such that u / ∈ H 1 (0, τ ; H). Now, we take b(t) = a(t) + I and we set v(t) = t 0 e -(t-s) u(s)ds. Then v(t) + v (t) = u(t). So v (t) + v (t) = u (t) and we get by Theorem 5.1.5 that v is the unique solution of the Problem (5. 3.5)

Fackler's counter-example