Oana Ivanovici
email: oana.ivanovici@math.u-psud.fr

Fabrice Planchon

On the energy critical Schrödinger equation in 3D non-trapping domains

We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for H 1 0 (Ω) data on any smooth, non-trapping domain Ω ⊂ R 3 . The key ingredient is a smoothing effect in L 5

x (L 2 t ) for the linear equation. We also derive scattering results for the whole range of defocusing subquintic Schrödinger equations outside a star-shaped domain.

Introduction

The Cauchy problem for the semilinear Schrödinger equation in R 3 is by now relatively well-understood: after seminal results by Ginibre-Velo [START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF] in the energy class for energy subcritical equations, the issue of local well-posedness in the critical Sobolev spaces ( Ḣ 3 2 -2 p-1 ) was settled in [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]. Scattering for large time was proved in [START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF] for energy subcritical defocusing equations, while the energy critical (quintic) defocusing equation was only recently successfully tackled in [START_REF] Colliander | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3[END_REF]. The local well-posedness relies on Strichartz estimates, while scattering results combine these local results with suitable non concentration arguments based on Morawetz type estimates. On domains, the same set of problems remains 1 INTRODUCTION 2 an elusive target, due to the difficulty in obtaining Strichartz estimates in such a setting. In [START_REF] Burq | On nonlinear Schrödinger equations in exterior domains[END_REF], the authors proved Strichartz estimates with an half-derivative loss on non trapping domains: the non trapping assumption is crucial in order to rely on the local smoothing estimates. However, the loss resulted in well-posedness results for strictly less than cubic nonlinearities; this was later improved to cubic nonlinearities in [START_REF] Anton | Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains[END_REF] (combining local smoothing and semiclassical Strichartz near the boundary) and in [START_REF] Ivanovici | Precise smoothing effect in the exterior of balls[END_REF] (on the exterior of a ball, through precised smoothing effects near the boundary). Recently there were two significant improvements, following different strategies:

• in [START_REF] Planchon | Bilinear virial identities and applications[END_REF], Luis Vega and the second author obtain an L 4 t,x Strichartz estimate which is scale invariant. However, one barely misses L 4 t (L ∞ (Ω)) control for H 1 0 data, and therefore local wellposedness in the energy space was improved to all subcritical (less than quintic) nonlinearities, but combining this Strichartz estimate with local smoothing close to the boundary and the full set of Strichartz estimates in R 3 away from it. Scattering was also obtained for the cubic defocusing equation, but the lack of a good local wellposedness theory at the scale invariant level ( Ḣ

2 ) led to a rather intricate incremental argument, from scattering in Ḣ 1 4 to scattering in H 1 0 ;

• in [START_REF] Ivanovici | On the Schrodinger equation outside strictly convex obstacles[END_REF], the first author proved the full set of Strichartz estimates (except for the endpoint) outside stricly convex obstacles, by following the strategy pioneered in [START_REF] Smith | On the critical semilinear wave equation outside convex obstacles[END_REF] for the wave equation, and relying on the Melrose-Taylor parametrix. In the case of the Schrödinger equation, one obtains Strichartz estimates on a semiclassical time scale (taking advantage of a finite speed of propagation principle at this scale), and then upgrading to large time results from combining them with the smoothing effect (see [START_REF] Burq | Estimations de Strichartz pour des perturbations à longue portée de l'opérateur de Schrödinger[END_REF] for a nice presentation of such an argument, already implicit in [START_REF] Staffilani | Strichartz estimates for a Schrödinger operator with nonsmooth coefficients[END_REF]). Therefore, one obtains the exact same local wellposedness theory as in the R 3 case, including the quintic nonlinearity, and scattering holds for all subquintic defocusing nonlinearities, taking advantage of the a priori estimates from [START_REF] Planchon | Bilinear virial identities and applications[END_REF].

In the present work, we aim at providing a local wellposedness theory for the quintic nonlinearity outside non trapping obstacles, a case which is not covered by [START_REF] Ivanovici | On the Schrodinger equation outside strictly convex obstacles[END_REF]. From explicit computations with gallery modes ( [START_REF] Ivanovici | Counter example to Strichartz estimates for the wave equation in domains[END_REF]), one knows that the full set of optimal Strichartz estimates does not hold for the Schrödinger equation on a domain whose boundary has at least one geodesically convex point; while this does not preclude a scale invariant Strichartz estimate with a loss (like the L 4 t (L ∞ x ) estimate in R 3 which is enough to solve the quintic NLS), it suggests to bypass the issue and use a different set of estimates, which we call smoothing estimates: in R 3 , these estimates may be stated as follows,

exp(it∆)f L 4 x (L 2 t ) f Ḣ-1 4 , (1.1)
from which one can infer various estimates by using Sobolev in time and/or in space. Formally, (1.1) is an immediate consequence of the Stein-Tomas restriction theorem in R 3 (or, more accurately, its dual version, on the extension): let τ > 0 be a fixed radius, one sees f (ξ) as a function on |ξ| = √ τ , and applies the extension estimate, with δ the Dirac function and F the space Fourier transform

F -1 (δ(τ -|ξ| 2 ) f(ξ)) L 4 x f (ξ) L 2 (|ξ|= √ τ ) .
Summing over τ yields the L 2 norm of f on the RHS, while on the left we use Plancherel in time and Minkowski to get (1.1). A similar estimate holds for the wave equation, replacing √ τ = |ξ| by τ = ±|ξ|, and usually goes under the denomination of square function (in time) estimates. In a compact setting (e.g. compact manifolds) a substitute for the Stein-Tomas theorem is provided by L p eigenfunction estimates, or better yet, spectral cluster estimates. In the context of a compact manifold with boundaries, such spectral cluster estimates were recently obtained by Smith and Sogge in [START_REF] Smith | On the L p norm of spectral clusters for compact manifolds with boundary[END_REF], and provided a key tool for solving the critical wave equation on domains, see [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF][START_REF] Burq | Global existence for energy critical waves in 3-d domains : Neumann boundary conditions[END_REF]. In this paper, we apply the same strategy to the Schrödinger equation:

• we derive an L 5 (Ω; L 2 I ) smoothing estimate for spectrally localized data on compact manifolds with boundaries, from the spectral cluster L 5 (Ω) estimate; here I is a time interval whose size is such that |I|| √ -∆ D | ∼ 1;

• we decompose the solution to the linear Schrödinger equation on a non trapping domain into two main regions: close to the boundary, where we can view the region as embedded into a 3D punctured torus, to which the previous semi-classical estimate may be applied, and then sumed up using the local smoothing effect; and far away from the boundary where the R 3 estimates hold.

• Finally, we patch together all estimates to obtain an estimate which is valid on the whole exterior domain. Local wellposedness in the critical Sobolev space

Ḣ 3 2 -2 p-1
immediatly follows for 3 + 2/5 < p ≤ 5, and together with the a priori estimates from [START_REF] Planchon | Bilinear virial identities and applications[END_REF], this implies scattering for the defocusing equation for 3 + 2/5 < p < 5. The remaining range 3 ≤ p ≤ 3 + 2/5 is sufficiently close to 3 that, as alluded to in [START_REF] Planchon | Bilinear virial identities and applications[END_REF], a suitable modification of the arguments from [START_REF] Planchon | Bilinear virial identities and applications[END_REF] yields scattering as well.

Remark 1.1. Clearly, such smoothing estimates are better suited to large values of p: the restriction 3 + 2/5 < p for the critical wellposedness is directly linked to the exponent 5 in the spectral cluster estimates; in R 3 , where the correct (and optimal !) exponent is 4, one may solve down to p = 3 by this method, while the Strichartz estimates allow to solve at scaling level all the way to the L 2 critical value p = 1 + 4/3.

Statement of results

Let Θ be a compact, non-trapping obstacle in R 3 and set Ω = R 3 \ Θ. By ∆ D we denote the Laplace operator with constants coefficients on Ω. For s ∈ R, p, q ∈ [1, ∞] we denote by Ḃs,q p (Ω) = Ḃs,q p the Besov spaces on Ω, where the spectral localization in their definition is meant to be with respect to ∆ D . We write L p x = L p (Ω) and Ḣσ = Ḃs,2 2 for the Lebesgue and Sobolev spaces on Ω. It will be useful to introduce the Banach-valued Besov spaces Ḃs,q p (L r t ), and we refer to the Appendix for their definition. Whenever L p t is replaced by L p T , it is meant that the time integration is restricted to the interval (-T, T ). We aim at studying wellposedness for the energy critical equation on Ω × R, with Dirichlet boundary condition,

i∂ t u + ∆ D u = ±|u| 4 u, u |∂Ω = 0, u |t=0 = u 0 (2.1)
and more generally

i∂ t u + ∆ D u = ±|u| p-1 u, u |∂Ω = 0, u |t=0 = u 0 (2.2)
with p < 5. T ). Moreover the solution is global in time and scatters in Ḣsp if the data is small. Remark 2.1. We elected to state both theorems for Dirichlet boundary conditions mostly for sake of simplicity. Indeed, both results hold with Neuman boundary conditions: the key ingredients for our linear estimates are known to hold for Neuman, see [START_REF] Smith | On the L p norm of spectral clusters for compact manifolds with boundary[END_REF][START_REF] Burq | On nonlinear Schrödinger equations in exterior domains[END_REF], while the nonlinear mappings from our appendix rely on [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF] (where all relevant estimates can be proved to hold in the Neuman case).

Finally, we consider the long time asymptotics for (2.2) in the defocusing case, namely the + sign on the left; in this situation, we are indeed restricted to the Dirichlet boundary conditions, as we rely on a priori estimates from [START_REF] Planchon | Bilinear virial identities and applications[END_REF].

Theorem 2.3. Assume the domain Ω to be the exterior of a star-shaped compact obstacle (which implies Ω is non trapping). Let 3 ≤ p < 5, and u 0 ∈ H 1 0 (Ω). There exists a unique global in time solution u, which is in the energy class, C(R, H 1 0 (Ω)), to the nonlinear equation (2.2) in the defocusing case (+ sign in (2.2)). Moreover, this solution scatters for large times: there exists two scattering states u ± ∈ H 1 0 (Ω) such that

lim t←±∞ u(x, t) -e it∆ D u ± H 1 0 (Ω) = 0.
As mentioned in the introduction, the (global) existence part was dealt with in [START_REF] Planchon | Bilinear virial identities and applications[END_REF]; for the scattering part, the p = 3 case was also dealt with in [START_REF] Planchon | Bilinear virial identities and applications[END_REF]. In the setting of Theorem 2.2, one may adapt the usual argument from the R n case, combining a priori estimates and a good Cauchy theory at the critical regularity; this provides a very short argument in the range 3 + 2/5 < p < 5. In the remaining range, namely 3 < p ≤ 3 + 2/5, one unfortunately needs to adapt the intricate proof from [START_REF] Planchon | Bilinear virial identities and applications[END_REF], and this leads to a much lenghtier proof; we provide it mostly for the sake of completeness. This type of argument may however be of relevance in other contexts.

Smoothing type estimates

We start with definitions and notations. Let ψ(ξ 2 ) ∈ C ∞ 0 (R \ {0}) and ψ j (ξ 2 ) = ψ(2 -2j ξ 2 ). On the domain Ω, one has the spectral resolution of the Dirichlet Laplacian, and we may define smooth spectral projections ∆ j = ψ j (-∆ D ) as continuous operators on L 2 . Moreover, these operators are continuous on L p for all p, and if f is Hilbert-valued and such that f H L p (Ω) < +∞, then the operators ∆ j are continuous as well on L p (H). We refer to [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF] for an extensive discussion and references. We simply point out that if H = L 2 t , then ∆ j is continuous on all L p x L q t by interpolation with the obvious L p t (L p x ) bound and duality.

In this section we concentrate on estimates for the linear Schrödinger equation on Ω×R with Dirichlet boundary conditions,

i∂ t u L + ∆ D u L = 0, u L|∂Ω = 0, u L|t=0 = u 0 (3.1)
Theorem 3.1. The following local smoothing estimate holds for the homogeneous linear equation

(3.1), ∆ j u L L 5 x L 2 t 2 -j 10 ∆ j u 0 L 2 x . (3.2)
Moreover, let 2 ≤ q ≤ ∞, then

∆ j u L L 5 x L q t 2 -j( 2 q -9 10 ) ∆ j u 0 L 2 x . (3.3)
Consider now the inhomogeneous equation,

i∂ t v + ∆ D v = F, v |∂Ω = 0, v |t=0 = 0. (3.4)
From Theorem 3.1, we will obtain the following set of estimates:

Theorem 3.2. Let 2 ≤ q < r ≤ +∞, then ∆ j v Ct(L 2 x ) + 2 j( 2 q -9 10 ) ∆ j v L 5 x L q t 2 -j( 4 r -9 5 ) ∆ j F L 5 4 x L r ′ t , (3.5
)

with 1/r + 1/r ′ = 1.
Combining the previous theorems with the results from [START_REF] Planchon | Bilinear virial identities and applications[END_REF], we finally state the set of estimates which will be used later for

i∂ t u + ∆ D u = F 1 + F 2 , u |∂Ω = 0, v |t=0 = u 0 . (3.6) Theorem 3.3. Let 2 < r ≤ +∞, then ∆ j u Ct(L 2 x ) + 2 j 10 ∆ j u L 5 x L 2 t + 2 -3 4 j ∆ j u L 4 t,x ∆ j u 0 L 2 x + 2 -j( 4 r -9 5 ) ∆ j F 1 L 5 4 x L r ′ t + 2 -1 4 j ∆ j F 2 L 4 3 t,x , (3.7)
with 1/r + 1/r ′ = 1.

Proof of Theorem 3.1

Let ψ ∈ C ∞ 0 (R \ {0}) be such that ψ = 1 on the support of ψ: hence, if ∆j denotes the corresponding localization operator, ∆j ∆ j = ∆ j . We now split the solution of the linear equation ∆ j u L = ∆j ∆ j u L as a sum of two terms ∆j χ∆ j u L + ∆j (1χ)∆ j u L , where χ ∈ C ∞ 0 (R 3 ) is compactly supported and it is equal to 1 near the boundary ∂Ω.

Far from the boundary:

∆j (1 -χ)∆ j u L Set w h (t, x) = (1 -χ)∆ j e it∆ D u 0 (x). Then w h satisfies i∂ t w h + ∆ D w h = -[∆ D , χ]∆ j u L , w h | t=0 = (1 -χ)∆ j u 0 . (3.8)
Since χ is equal to 1 near the boundary ∂Ω, we can view the solution to (3.8) as the solution of a problem in the whole space R 3 . Consequently, the Duhamel formula writes

w h (t, x) = e it∆ 0 (1 -χ)∆ j u 0 - t 0 e i(t-s)∆ 0 [∆ D , χ]∆ j u L (s)ds, (3.9)
where ∆ 0 is the free Laplacian on R 3 and therefore the contribution of e it∆ 0 (1χ)∆ j u 0 satisfies the usual Strichartz estimates. We have thus reduced the problem to the study of the second term in the right hand-side of (3.9). Ideally, one would like to remove the time restriction s < t and use a variant of the Christ-Kiselev lemma. However, this would miss the endpoint case q = 2. Instead, we recall the following lemma:

Lemma 3.1 (Staffilani-Tataru [START_REF] Staffilani | Strichartz estimates for a Schrödinger operator with nonsmooth coefficients[END_REF]). Let x ∈ R n , n ≥ 3 and let f (x, t) be compactly supported in space, such that f ∈ L 2 t (H -1 2 ). Then the solution w to (i∂ t + ∆ 0 )w = f with w |t=0 = 0, is such that w

L 2 t (L 2n n-2 x ) f L 2 t (H - 1
2 ) .

(3.10)

In fact, one may shift regularity in (3.10) without difficulty. Now, the proof in [START_REF] Staffilani | Strichartz estimates for a Schrödinger operator with nonsmooth coefficients[END_REF] relies on a decomposition into traveling waves, to which homogeneous estimates are then applied. We can therefore use the L 4

x (L 2 t ) smoothing estimate, Sobolev in space, and extend the conclusion of Lemma 3.1 to

w L 5 x (L 2 t ) f L 2 t (H -1 2 -1 10 ) , (3.11)
where we chose to conveniently shift the regularity to the right handside. We now take

f = -[∆ D , χ]∆ j u L ∈ L 2 t H -1/2-1/10 comp (Ω) and [∆ D , χ]∆ j u L L 2 H -1/2-1/10 comp ∆ j u L L 2 Ḣ1/2-1/10 (Ω) ∆ j u 0 Ḣ1/10 (Ω) ,
from which the smoothing estimates follow

(1 -χ)∆ j u L L 5 (R 3 )L 2 t (1 -χ)∆ j u 0 Ḣ-1 10 (R 3 ) + [∆ D , χ]∆ j u L L 2 H -1/2-1/10 comp ∆ j u 0 Ḣ-1 10 (Ω) . (3.12)
We conclude using the continuity properties of ∆j which were recalled at the beginning of Section 3 (e.g. see [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF]Cor.2.5]). In fact, using (3.12), we get

∆j (1 -χ)∆ j u L L 5 x L 2 t (1 -χ)∆ j u L L 5 x L 2 t 2 -j 10 ∆ j u 0 L 2 (Ω) ,
where we have used the spectral localization ∆ j to estimate

∆ j u 0 Ḣσ (Ω) ≃ 2 σj ∆ j u 0 L 2 (Ω) .
3.1.2 Close to the boundary: ∆j χ∆ j u L For l ∈ Z let ϕ l ∈ C ∞ 0 (((l -1/2)π, (l + 1)π)) equal to 1 on [lπ, (l + 1/2)π]. We set v j = ∆j χ∆ j u L and for l ∈ Z we set v j,l = ϕ l (2 j t)v j . We have

v j 2 L 5 (Ω)L 2 (R) = l∈Z v j,l 2 L 5 x L 2 t ≃ l∈Z v j,l 2 L 2 t L 5/2 x l∈Z v j,l 2
L 2 t L 5/2 x ≤ l∈Z v j,l 2 L 5 x L 2 t , (3.13)
where for the first inequality we used the fact that the supports in time of ϕ l are almost orthogonal. In order to estimate v j 2 L 5

x L 2 t it will be thus sufficient to estimate each v j,l

2 L 5 x L 2 t . The equation satisfied by ṽj,l := ϕ l (2 j t)χ∆ j u L is i∂ t ṽj,l + ∆ D ṽj,l = -(ϕ l (2 j t)[∆ D , χ]∆ j u L -i2 j ϕ ′ l (2 j t)χ∆ j u L ), (3.14)
where we stress that ṽj,l vanishes outside the time interval (2 -j (l -1/2)π, 2 -j (l + 1)π). We denote V j,l the right hand side in (3.14), namely

V j,l := -ϕ l (2 j t)[∆ D , χ]∆ j u L + i2 j ϕ ′ l (2 j t)χ∆ j u L . (3.15)
Let Q ⊂ R 3 be an open cube sufficiently large such that ∂Ω is contained in the interior of Q. We denote by S the punctured torus obtained from removing the obstacle Θ (recall that Ω = R 3 \ Θ) in the compact manifold obtained from Q with periodic boundary conditions on ∂Q. Notice that defined in this way S coincides with the Sinaï billiard. Let also ∆ S := 3 j=1 ∂ 2 j denote the Laplace operator on the compact domain S. On S, we may define a spectral localization operator using eigenvalues λ k and eigenvectors

e k of ∆ S : if f = k c k e k , then ∆ S j f = ψ(2 -2j ∆ S )f = k ψ(2 -2j λ 2 k )c k e k . (3.16)
Remark 3.1. Notice that in a neighborhood of the boundary, the domains of ∆ S and

∆ D coincide, thus if χ ∈ C ∞ 0 (R 3 ) is supported near ∂Ω then ∆ S χ = ∆ D χ.
In order to apply estimates on the manifold S, we will need to relocalize close to the obstacle. Consider χ 1 ∈ C ∞ 0 (R 3 ) supported near the boundary and equal to 1 on the support of χ, we will write

χ 1 ∆j χ = χ 1 ∆S j χ + χ 1 ( ∆j -∆S j ) χ, (3.17)
with the expectation that the difference term is smoothing.

In what follows let χ ∈ C ∞ 0 (R 3 ) be equal to 1 on the support of χ and be supported in a neighborhood of ∂Ω such that on its support the operator -∆ D coincide with -∆ S . From their respective definition, ṽj,l = χṽ j,l , V j,l = χV j,l , consequently ṽj,l will also solve the following equation on the compact manifold S i∂ t ṽj,l + ∆ S ṽj,l = V j,l , ṽj,l | t<h(l-1/2)π = 0, ṽj,l | t>h(l+1)π = 0.

(3.18)

Therefore we can write the Duhamel formula either for the last equation (3.18) on S, or for the equation (3.14) on Ω. We now apply ∆j and use that v j.l = ∆j ṽj,l , χṽ j,l = ṽj,l and

∆j χ = χ 1 ∆S j χ + (1 -χ 1 ) ∆j χ + χ 1 ( ∆j -∆S j )χ, which yields v j,l (t, x) = χ 1 t h(l-1/2)π e i(t-s)∆ S ∆S j V j,l (s, x)ds + (1 -χ 1 ) t h(l-1/2)π e i(t-s)∆ D ∆j V j,l (s, x)ds + χ 1 ( ∆j -∆S j )ṽ j,l , (3.19)
where we conveniently chose to write Duhamel on S for the first term and Duhamel on Ω for the second one, which allows to commute the flow under the time integral. Denote by v j,l,m the first term in the second line of (3.19) by v j,l,f the second one and v j,l,s the last one. We deal with them separately. To estimate the L 5

x L 2 t norm of the v j,l,f we notice that its support is far from the boundary: as such, estimates on the L 5

x L 2 t norm will follow from Section 3.1.1. Indeed, we get

(1 -χ 1 ) ∆j e i(t-s)∆ D V j,l L 5 x L 2 t ∆j V j,l Ḣ-1/10 (Ω) ≃ 2 -j 10 ∆j V j,l L 2 (Ω) .
(3.20)

We then apply the Minkowski inequality to deduce

(1 -χ 1 ) t h(l-1/2)π ∆j e i(t-s)∆ D V j,l (s, x)ds L 5 x L 2 t ≤ 2 -j/2 ( I j,l (1 -χ 1 ) ∆j e i(t-s)∆ D V j,l (s, .) 2 L 5 (Ω)L 2 (I j,l ) ds) 1/2 , (3.21)
where we denoted I j,l = [2 -j (l -1/2)π, 2 -j (l + 1)π] and we used the Cauchy-Schwartz inequality. Using (3.20) we finally get

v j,l,f L 5 (Ω)L 2 (I j,l ) ≤ 2 -j(1/2+1/10) ∆j V j,l L 2 (I j,l )L 2 (Ω) . (3.22)
To estimate the L 5 x L 2 t norm of the main contribution v j,l,m we need the following:

Proposition 3.1. Let j ≥ 0, I j = (-π2 -j , π2 -j ), χ ∈ C ∞ 0 (R 3
) be supported near ∂Ω and V 0 ∈ L 2 (Ω). Then there exists C > 0 independent of j such that for the solution e it∆ S ∆S j χV 0 of the linear Schrödinger equation on S with initial data ∆S j χV 0 we have

e it∆ S ∆S j χV 0 L 5 (S)L 2 t (I j ) ≤ C2 -j 10 ∆S j χV 0 L 2 (S) . (3.23)
We postpone the proof of Proposition 3.1 to Subsection 3.3.

Using the fact that v j,l is supported in time in

I j,l = [2 -j (l -1/2)π, 2 -j (l + 1)π],
the Minkowski inequality, Proposition 3.1 with χ = 1 on the support of χ and with V 0 = V j,l , and since χ1 v j,l,m = v j,l,m for any χ1 ∈ C ∞ (R 3 ) with χ1 = 1 on the support of χ 1 , we obtain

v j,l,m L 5 (Ω)L 2 (I j,l ) = χ1 v j,l,m L 5 (Ω)L 2 (I j,l ) = v j,l,m L 5 (S)L 2 (I j,l ) ≤ 2 -j (l+1)π 2 -j (l-1)π e i(t-s)∆ S ∆S j V j,l (s, .) L 5 (S)L 2 (I j,l ) ds ≤2 -j 10 I j,l ∆S j V j,l (s) L 2 (S) ds ≤2 -j 10 I j,l χV j,l (s) L 2 (S) ds ≤2 -j 10 I j,l χV j,l (s) L 2 (Ω) ds (3.24)
where we used again V j,l = χV j,l to switch S and Ω and continuity of ∆ S j on L 2 (S). Using the Cauchy-Schwartz inequality in (3.24) yields

v j,l,m L 5 (Ω)L 2 (I j,l ) 2 -j(1/2+1/10) V j,l L 2 (I j,l )L 2 (Ω) (3.25)
We deal with the right handside in (3.25). Using the explicit expression of V j,l given in (3.15),

V j,l (s) L 2 (I j,l )L 2 (Ω) ( ϕ l (2 j t)[∆ D , χ]∆ j u L L 2 (I j,l )L 2 (Ω) + 2 j ϕ ′ l (2 j t)χ∆ j u L L 2 (I j,l )L 2 (Ω) ). (3.26) As [∆ D , χ] is bounded from H 1 0 to L 2 , we get ∆j V j,l L 2 (I j,l )L 2 (Ω) χ 1 ∆ j u L L 2 (I j,l )H 1 0 (Ω) + 2 j χ∆ j u L L 2 (I j,l )L 2 (Ω) (3.27)
Let us recall the following local smoothing result on a non trapping domain:

Lemma 3.2. (Burq, Gérard, Tzvetkov [2, Prop.2.7]) Assume that Ω = R 3 \Θ, where Θ = ∅ is a non-trapping obstacle. Then, for every χ ∈ C ∞ 0 (R 3 ), and σ ∈ [-1/2, 1], χ∆ j u L L 2 (R, Ḣσ+1/2 (Ω)) ≤ C ∆ j u 0 H σ (Ω) , (3.28)
where, as usual, u L (t, x) = e -it∆ D u 0 (x).

We now turn to the difference term v j,l,s and prove a smoothing lemma.

Lemma 3.3. Let χ 1 ∈ C ∞ 0 (R n
) be equal to 1 on a fixed neighborhood of the support of χ. Then we have for all N ∈ N,

v j,l,s L 5 (Ω)L 2 (I j,l ) ≤ C N 2 -N j V j,l (x, s) L 2 (I j,l ,L 2 (Ω)) .
(3.29)

In order to prove the lemma, one would like to rewrite ∆j = ψ(2 -2j ∆ D ) as a solution of the wave equation, using h = 2 -j as a time. Then the finite speed of propagation would let us switch ∆ D and ∆ S . However the inverse Fourier transform (in |ξ|) of Ψ(|ξ|) = ψ(|ξ| 2 ) is only Schwartz class, rather than compactly supported. The tails will eventually account for the right handside of (3.29). We now turn to the details: let ϕ 0 , ϕ(y) be even, compactly supported (ϕ(y) away from zero) and such that

ϕ 0 (y) + k≥1 ϕ(2 -k y) = 1.
We decompose Ψ(y) using this resolution of the identity, and set with obvious notations

Ψ(|ξ|) = k∈N φ k (|ξ|),
where the φ k have good bounds, say φ0 ∈ L ∞ and for k ≥ 1

∀N ∈ N, φk ∞ = Ψ(y)ϕ(2 -k y) ∞ ≤ C N 2 -kN .
(3.30)

At fixed k, we write (abusing notation and letting ∆ be either ∆ D or ∆ S )

φ k (h √ -∆) χṽ j,l = 1 2π e iyh √ -∆ χ(x)ṽ j,l (x) φk (y) dy. Notice that φ k (y) is compactly supported, in fact its support is roughly |y| ∈ [2 k-1 , 2 k+1 ].
As such the y integral is a time average of half-wave operators, which have finite speed of propagation. Therefore if the time |yh| ≤ 1, we can add another cut-off function χ 1 which is equal to one on the domain of dependency of χ on this time scale, and such that χ 1 is indifferently defined on S or Ω: namely, for k j,

φ k (h -∆ S ) χṽ j,l = χ 1 (x)φ k (h -∆ S ) χṽ j,l = χ 1 (x) 1 2π e iyh √ -∆ χ(x)ṽ j,l (x) φk (y) dy, φ k (2 -j -∆ S ) χṽ j,l = χ 1 (x)φ k (2 -j -∆ D ) χṽ j,l . (3.31)
From this identity, we obtain

v j,l,s = χ 1 (x) j k (φ k (2 -j -∆ D ) -φ k (2 -j -∆ S )) χ(x)ṽ j,l . (3.32)
At this point the difference in (3.32) is irrelevant and we estimate both terms using Sobolev embedding and energy estimates. Abusing notations, with ∆ ∈ {∆ D , ∆ S }, we have

χ 1 φ k (2 -j √ -∆) χṽ j,l L 5 (Ω)L 2 t (I j,l ) ≤ χ 1 φ k (2 -j √ -∆) χṽ j,l L 2 t (I j,l )L 5 (Ω) ≤2 -j 2 χ 1 φ k (2 -j √ -∆) χṽ j,l L ∞ t (I j,l )L 5 (Ω) 2 -j 2 φ k (2 -j √ -∆) χṽ j,l L ∞ t (I j,l )H 1 2 (Ω) C N 2 -j 2 -kN χṽ j,l L ∞ t (I j,l )H 1 2 (Ω)
where we used Minkowski, Hölder, (non sharp !) Sobolev and (3.30). Finally, by the dual estimate of (3.28), ṽj,l

L ∞ t (I j,l )H 1 2 (Ω) V j,l L 2 t (I j,l ,L 2 (Ω))
. Summing in k and relabeling N, we have

v j,l,s L 5 (Ω)L 2 t (I j,l ) ≤ C N 2 -jN V j,l L 2 t (I j,l ,L 2 (Ω)) , (3.33)
which concludes the proof of the lemma. Using this lemma and (3.27), we get for v j,l,s an estimate which matches (3.25): picking N = 1 is enough. From there, using (3.13), (3.22), (3.25), we write

∆j χ∆ j u L 2 L 5 (Ω)L 2 t 2 -2j( 1 2 + 1 10 ) l∈Z ∆j V j,l (s) 2 L 2 (I j,l )L 2 (Ω) 2 -2j( 1 2 + 1 10 ) l∈Z ( χ∆ j u L 2 L 2 (I j,l )H 1 0 (Ω) + 2 2j χ∆ j u L 2 L 2 (I j,l )L 2 (Ω) ) 2 -2j 10 (2 -j ∆j u 0 2 Ḣ 1 2 (Ω) + 2 j ∆j u 0 2 Ḣ-1 2 (Ω) ) 2 -2j 10 ( ∆j u 0 2 L 2 (Ω) ,
which is the desired result.

End of the proof of Theorem 3.1

Until now we have prove Theorem 3.1 only for q = 2. We shall use the Gagliardo-Nirenberg inequality in order to deduce (3.3) for every q ≥ 2. We have

∆ j u L L ∞ t ∆ j u L 1/2 L 2 t ∆ j ∂ t u L 1/2 L 2 t .
which gives, taking the L 5 x norms and using the Cauchy-Schwartz inequality

∆ j u L 5 L 5 x L ∞ t ∆ j u L 5/2 L 5 L 2 t ∆ j ∂ t u L 5/2 L 5 x L 2 t . (3.34) It remains to estimate ∆ j ∂ t u L L 5 x L 2 t : notice that since u L = e -it∆ D u 0 ∆ j ∂ t u L = -i∆ D ∆ j u L = i2 2j ∆j u L ,
where ∆j is defined with

ψ 1 (x) = xψ(x) ∈ C ∞ 0 (R \ {0}). Therefore ∆ j ∂ t u L L 5 x L ∞ t ≤ C2 j(2-1/10) ∆j u 0 L 2 (Ω) , (3.35)
consequently ∆ j ∂ t u L L 5 x L q t ≤ C2 -j(2/q-9/10) ∆ j u 0 L 2 (Ω) and Theorem 3.1 is proved.

Proof of Theorems and 3.3

We recall a lemma due to Christ and Kiselev [START_REF] Christ | Maximal functions associated to filtrations[END_REF]. We state the corollary we will use, with only the time variable: we refer to [START_REF] Burq | Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications[END_REF] for a simple direct proof of all the different cases we use, with Banach-valued L p t (B) spaces or B(L p t ). Its use in the context of reversed norms L q

x (L p t ) goes back to [START_REF] Planchon | Dispersive estimates and the 2D cubic NLS equation[END_REF] and it greatly simplifies obtaining inhomogeneous estimates from homogeneous ones. Lemma 3.4. (Christ and Kiselev [START_REF] Christ | Maximal functions associated to filtrations[END_REF]) Consider a bounded operator

T : L r (R) → L q (R)
given by a locally integrable kernel K(t, s). Suppose that r < q. Then the restricted operator

T R f (t) = s<t K(t, s)f (s)ds is bounded from L r (R) to L q (R) and T R L r (R)→L q (R) ≤ C(1 -2 -(1/q-1/r) ) -1 T L r (R)→L q (R) .
From the lemma, the proof of the inhomogeneous set of estimates in Theorem 3.2 is routine from the homogeneous estimates in Theorem 3.1 and the Duhamel formula. Combining both homogeneous and inhomogeneous estimates yields Theorem 3.3.

Proof of Proposition 3.1

Let S denote the compact domain defined above. Recall (e n ) n is the eigenbasis of L 2 (S) consisting of eigenfunctions of -∆ S associated to the eigenvalues λ 2 n . Following [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF], we define an abstract self adjoint operator on L 2 (S) as follows

A h (e n ) := -[hλ 2 n ]e n ,
where [λ] is the integer part of λ. Notice that in some sense A h = "[h∆ S ]". We first need to establish estimates for the linear Schrödinger equation on the compact domain S with spectrally localized initial data. We now set h = 2 -j and state estimates on the evolution equation where h∆ S is replaced by A h .

Lemma 3.5. Let 0 < h ≤ 1, q ≥ 2, I h = (-πh, πh), χ ∈ C ∞ 0 (R 3
) be supported near ∂Ω and V 0 ∈ L 2 (Ω). There exists C > 0 independent of h such that

e i t h A h ∆S j χV 0 L 5 (S)L q (I h ) ≤ Ch 2/q-9/10 ∆S j χV 0 L 2 (S) . (3.36)
We postpone the proof of Lemma 3.5 and proceed with the proof of Proposition 3.1. Denote by V h (t, x) := e it∆ S ∆S j χV 0 (x), then

(ih∂ t + A h )V h = (ih∂ t + h∆ S )V h + (A h -h∆ S )V h = (A h -h∆ S )e it∆ S ∆S j χV 0 .
Writing Duhamel formula for V h yields

V h (t, x) = e i t h A h ∆S j χV 0 (x) - i h t 0 e i (t-s) h A h (A h -h∆ S )e is∆ S ∆S j χV 0 (x)ds. (3.37)
Using (3.36) with q = 2, (3.37), the Minkowski inequality and boundedness of the operator

e i t h A h ∆S j L 2 (S)→L 5 (S)L 2 (I h ) 2 -j 10 ∼ h 1/10
(which follows from the proof of Lemma 3.5), we obtain

e it∆ S ∆S j χV 0 L 5 (S)L 2 (I h ) h 1 10 ∆S j χV 0 L 2 (S) + 1 h (A h -h∆ S )e is∆ S ∆S j χV 0 L 1 (-hπ,hπ)L 2 (S) , (3.38)
where to estimate the second term in the right hand side of (3.37) we used the fact that A h commutes with the spectral localization ∆S j . Changing variables s = hτ in the second term in the right hand side of (3.38) yields

1 h (A h -h∆ S )e is∆ S ∆S j χV 0 L 1 (-hπ,hπ)L 2 (S) = π -π (A h -h∆ S )e iτ h∆ S ∆S j χV 0 L 2 (S) dτ 2π ∆S j χV 0 L 2 (S) , (3.39)
where we used the fact that the operator (A h -h∆ S ) is bounded on L 2 (S) and the mass conservation of the linear Schrödinger flow. If follows from (3.38) and (3.39) that

e it∆ S ∆S j χV 0 L 5 (S)L 2 (I h ) h 1/10 ∆S j χV 0 L 2 (S) ,
which ends the proof of Proposition 3.1.

We now return to Lemma 3.5 for the rest of this section. Writing ∆S j V 0 = n ψ(h 2 λ 2 n )V λn e n , we decompose (for 0 < h ≤ 1/4)

e i t h A h ∆S j V 0 (t, x) = k∈N e i t h k v k (x) with v k (x) = ((k+1)2 j ) 1/2 -1 λ=(k2 j ) 1/2 λn∈[λ,λ+1) Ψ(h 2 λ 2 n )V λn e n = ((k+1)2 j ) 1/2 -1 λ=(k2 j ) 1/2 Π λ ( ∆S j V 0 ),
where Π λ denotes the spectral projector λ+1) . Let us estimate the L 5 (S)L q (I h ) norm of e i t h A h ∆S j V 0 :

Π λ = 1 √ -∆ S ∈[λ,
e i t h A h ∆S j V 0 2 L 5 (S)L q (I h ) h 2/q e isA h ∆S j V 0 2 L q s (-π,π) L 5/2 (S) h 2/q e isA h ∆S j V 0 2 H 1/2-1/q (s∈(-π,π)) L 5/2 (S) h 2/q k∈N (1 + k) 2( 1 2 -1 q ) e isk v k (x) 2 L 2 s (-π,π) L 5/2 (S) h 2/q k∈N (1 + k) 1-2/q e isk v k (x) 2 L 5 (S)L 2 (-π,π) h 2/q k∈N (1 + k) 1-2/q e isk v k (x) 2 L 2 (-π,π)L 5 (S) ,
where we used Sobolev injection in the time variable H 1/2-1/q ⊂ L q and Plancherel in time.

We recall a result of [START_REF] Smith | On the L p norm of spectral clusters for compact manifolds with boundary[END_REF] of Smith and Sogge on the spectral projector Π λ :

Theorem 3.4. (Smith and Sogge [START_REF] Smith | On the L p norm of spectral clusters for compact manifolds with boundary[END_REF]) Let S be a compact manifold of dimension 3, then

Π λ L 2 (S)→L 5 (S) ≤ λ 2/5 .
Using Theorem 3.4 we have

e i t h A h ∆S j V 0 2 L 5 (S)L q (I h ) h 2/q 1/4h-1≤k≤4/h (1 + k) 1-2/q+4/5 ∆S j V 0 2 L 2 (S) hk∈[1/4,4] k 1-4/q+4/5 ∆S j V 0 2 L 2 (S)
∆S j V 0 2 Ḣ2/q-9/10 (S) , since for hk > 4 or h(k + 1) < 1/4 and λ n ∈ [(k2 j ) 1/2 , ((k + 1)2 j ) 1/2 ) we have Ψ(h 2 λ 2 n ) = 0 and on the other hand for these values of k we have

k/ √ 2 ≤ (k2 j ) 1/2 ≤ λ n ≤ ((k + 1)2 j ) 1/2 ≤ √ 2(k + 1), h ≤ 5(k + 1) -1 .
This completes the proof of Lemma 3.5.

Local existence

In this section we prove Theorem 2.1.

Definition 4.1. Let u ∈ S ′ (R × Ω) and let ∆ j = ψ(-2 -2j ∆ D
) be a spectral localization with respect to the Dirichlet Laplacian ∆ D in the x variable, such that j ∆ j = Id and let S j = k<j ∆ j . We introduce the "Banach valued" Besov space Ḃs,q p (L r t ) as follows: we say that u ∈ Ḃs,q p (L r t ) if

2 js ∆ j u L p x L r t ∈ l q ,
and j ∆ j f converges to f in S ′ . If L r t is replaced by L r T , the time integration is meant to be over (-T, T ). Moreover, when s < 0, ∆ j may be replaced by S j in the norm and both norms are equivalent.

Consider u 0 ∈ Ḣ1 0 and u L the solution to the linear equation (3.1). Applying Theorem 3.1 with q = 2, 5 and taking s = 1 in the definition above we obtain

u L ∈ Ḃ1+ 1 10 ,2 5 (L 2 t ) ∩ Ḃ 1 2 ,2 5 (L 5 t ) and ∂ t u L ∈ Ḃ-3 2 ,2 5 (L 5 t ).
From this, by Gagliardo-Nirenberg in the time variable, one should have

u L ∈ Ḃ1,2 5 (L 20 9 t ) ∩ Ḃ3/20,2 5 (L 40 t ) ⊂ L 20/3 x L 40 t ,
and consequently

u 4 L ∈ L 5/3 x L 10 t as well as |u L | 4 u L ∈ Ḃ1,2 5 4
(L

20 11 t )
which should be enough to iterate. However, our spaces are Banach valued Besov spaces (if one sees time as a parametrer) and justifying Berstein-like inequalities and Sobolev embedding is not entirely trivial (but doable, using the estimates from [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF]). We choose an apparently complicated space in order to set up the fixed point, but the little gain in regularity from the smoothing estimate will turn out to be crucial for subcritical scattering.

Remark 4.1. By this choice, we only restrict the uniqueness class. It is likely that one may prove a better result, but there is no immediate benefit in the present setting, except proving additional estimates. We retained, however, the uniqueness class that would be provided by the argument above in the Theorems'statements. Another remark is that one may dispense with the use of Lemma 3.1, miss the endpoint q = 2 and still get the exact same nonlinear results, as there is room (due to the use of Sobolev embedding) in all mapping estimates. Moreover, as soon as we use an estimate with a (however small) gain in regularity, we do not need Lemma 4.11, as we could use a simpler embedding in a Besov space of negative regularity and play regularities against each other. In fact, in the same spirit as [START_REF] Planchon | Dispersive estimates and the 2D cubic NLS equation[END_REF] one could replace the critical Sobolev norm by a Besov norm Ḃsp,∞ 2 .

For T > 0 let

X T := {u | u ∈ Ḃ1+ 1 10 ,2 5 (L 2 T ) ∩ Ḃ 1 2 ,2 5 (L 5 T ) and ∂ t u ∈ Ḃ-3 2 ,2 5 (L 5 T )}. ( 4
φ(u)(t) := s<t e i(t-s)∆ D F (u(s))ds. Then φ(u) C T ( Ḣ1 0 ) + φ(u) X T F (u) Ḃ1,2 5/4 (L 20/11 T ) u 5 X T , (4.2)
and

φ(u) -φ(v) X T F (u) -F (v) Ḃ1,2 5/4 (L 20/11 T ) u -v X T ( u X T + v X T ) 4 . (4.
3)

The estimate for the inhomogeneous problem writes

e -is∆ D F L 2 x ≤ C F Ḃ0,2 5/4 (L 20/11 t ) ,
Shifting the regularity to s = 1 and using the Christ-Kiselev lemma provides the first step of both estimates 4.2 and 4.3. Now, Lemma 4.10 in the Appendix provides the nonlinear part of both estimates (note however that, as p = 5 is an integer, one could prove directly the nonlinear mappings by product rules).

One may now set up the usual fixed point argument in X T if T is sufficiently small of if the data is small. This concludes the proof of Theorem 2.1 (scattering for small data follows the usual way from the global in time space-time estimates).

We now consider local wellposedness for p < 5, e.g. Theorem 2.2. The critical Sobolev exponent w.r.t. scaling is s p = 3/2 -2/(p -1). We aim at setting up a contraction argument in a small ball of

X T := {u | u ∈ Ḃsp+ 1 10 ,2 5 (L 2 T ) ∩ Ḃsp-1 4 ,2 4 (L 4 T ) and ∂ t u ∈ Ḃsp-1 4 -2,2 4 (L 4 T )}. (4.4)
The important fact (if we were to ignore issues with Banach valued Besov spaces) would be that

X T ⊂ Ḃsp,2 5 (L 20/9 T ) ∩ L 5(p-1)/3 x L 10(p-1) T .
Remark 4.2. Some numerology is in order: if one were only to have the L 5 x L 2 t smoothing estimate and use Sobolev (in time and in space), it would require 5(p -1)/3 ≥ 5, namely p ≥ 4. However, we have the Strichartz estimate from [START_REF] Planchon | Bilinear virial identities and applications[END_REF], which allows 5(p -1)/3 ≥ 4, or p ≥ 3 + 2/5.

Again from the Appendix, the nonlinear mapping verifies

F (u) -F (v) Ḃsp,2 5/4 (L 20/11 T ) u -v X T ( u p-1 X T + v p-1 X T )
and existence and uniqueness follow by fixed point again.

4.1 Scattering for 3 + 2/5 < p < 5

We now deal with scattering in the same range of p ∈ (3 + 2/5, 5): from [START_REF] Planchon | Bilinear virial identities and applications[END_REF], we have an a priori bound S j u 4

L 4 t L 4 x u 4 L 4 t L 4 x u 0 3 L 2 x sup t u H 1 0 ≤ M 3 2 E 1 2 ,
where M and E are the conserved charge and hamiltonian,

M = Ω |u| 2 dx and E = Ω |∇u| 2 + 2 p + 1 |u| p+1 dx. (4.5)
Notice how this estimate is below the critical scaling s p , as the RHS regularity is s = 1/4. From the energy a priori bound and Sobolev embedding, one has on the other hand

S j u L ∞ t,x 2 j 2 sup t u H 1 0 2 j 2 E 1 2 .
Interpolating between the two bounds to get the right scaling yields,

S j u L q t,x C(M, E)2 j( 1 2 -5-p 3(p-1) ) , (4.6)
where 1/q = (5p)/6(p -1). In order to proceed with the usual scattering argument, we need to revisit the fixed point, or more precisely the nonlinear estimate on F (u): indeed, if we wish to use (4.6), even at a power ε, we cannot afford to use the same regularity on both sides of the Duhamel formula. Fortunately, we have off diagonal inhomogeneous estimates, e.g.

e i(t-s)∆ D F Ḃsp,2 5 (L 20/9 t )∩ Ḃsp-3/4,2 4 (L 4 t ) ≤ C F (u) Ḃsp-1 10 ,2 5/4 (L 2 t )
.

In order to evaluate F (u), one needs to place the S j u factors in such a way that

(S j u) p-1 L 5/3 x L 20 t 2 j 10
.

However, we have from (4.6)

(∆ j u) p-1 L 6 5-p t,x
C(M, E)2 j( 5p-13 6) , (4.7) and 6/(5p) > 5/3. As such, one may interpolate with

∆ j u L 4 x L 4 t 2 -j(sp-1 4 ) ,
to get (after Sobolev embedding)

(∆ j u) p-1 L 5 3 x L 20 t 2 j 10 .
Suming over low frequencies recovers the desired bound. Notice that scaling dictates the exponents (hence there is no need to compute explicitely the interpolation θ).

Scattering for 3 ≤ p ≤ 3 + 2/5

In this part we the remaining case, e.g. nonlinearities which are close to 3 and for which our main results do not provide a scale-invariant local Cauchy theory. As mentioned before, this case will be dealt with using the approach from [START_REF] Planchon | Bilinear virial identities and applications[END_REF]. As such, this entire Subsection is somewhat disconnected from the rest of the paper; the combination of several technical difficulties makes it lenghty and cumbersome, but we hope the underlying strategy is clear. We have two a priori bounds on the nonlinear equation at our disposal: local smoothing, which is at the scale of Ḣ 1 2 regularity for the data, and an L 4 t,x space-time bound, which is at the scale of Ḣ 1 4 regularity for the data. Both are below the scale of critical H s regularity, which is

s p = 3 2 -2 (p-1)
. Interpolation with the energy bound provides bounds at the critical level, but the lack of flexible scale-invariant estimates on the inhomogeneous problem make them seemingly useless. As such, one has to improve both the local smoothing bound and the L 4 t,x space-time bounds obtained in [START_REF] Planchon | Bilinear virial identities and applications[END_REF], to reach critical scaling and beyond. This is accomplished through several steps, which we informally summarize as follows:

• improve the space-time bounds by using the equations far and close to the boundary.

As the resulting commutator source term can only be handle at H 1 2 regularity, this will improve estimates from Ḣ 1 4 regularity to Ḣ 1 2 -ε regularity, which is still below scale invariance;

• combine this improved estimates with the energy bound to obtain yet again better space-time bounds through the equation (but splitting the source terms in close and far away terms). As an added bonus we also improve our local smoothing estimate; moreover we now go beyond scale-invariance;

• turn the crank a few more times, going back and forth between estimates on the split equations and estimates on the equation with split source terms, until we reach the correct set of estimates to prove scattering at the H 1 0 regularity. It is worth noticing that the numerology gets worse with p > 3 + 2/5, and that the forthcoming argument would probably break down before even reaching p = 4.

We start by stating a few linear estimates which will be needed in the proof and are simple consequences of our Theorem 3.3 by summing over dyadic frequencies. (Ω)

+ w L 4 t Ẇ s,4 f L 4 3 t Ẇ s+ 1 2 , 4 3
.

(4.9)

The next lemma is just the Christ-Kiselev lemma again, stated in a form which is convenient for later use. Lemma 4.2. (see [START_REF] Planchon | Bilinear virial identities and applications[END_REF]Lemma 5.6]) Let U(t) be a one parameter group of operators, 1 ≤ r < q ≤ ∞, H an Hilbert space and B r and B q two Banach spaces. Suppose that

U(t)ϕ L q t (Bq) ϕ H and s U(-s)g(s)ds H g L r t (Br ) , then s<t U(t -s)g(s)ds L q t (Bq) g L r t (Br ) .
finally, we recall that we have Lemma 3.1 at our disposal, should we need the endpoint Strichartz on the left handside in Lemma 4.2, provided that we used a (dual) local smoothing norm on the right handside.

In what follows we shall write p = 3 + 2η, with η ∈ [0, 1/5]. All the nonlinear mappings which we use can be proved using the appendix and we will no longer refer to it. We recall all a priori bounds at our disposal: the first two are uniform in time bounds for the L 2 (Ω) and H 1 0 (Ω) norms of the solution to the defocusing NLS, irrespective of the power p, and were already stated in the previous section, see 4.5. The next two were obtained in [START_REF] Planchon | Bilinear virial identities and applications[END_REF], again in the defocusing case and irrespective of p: a space-time norm estimate

u L 4 t (L 4 (Ω)) ≤ E 1 8 M 3 8 , (4.10)
which has the same scaling as Ḣ 1 4 for the data; and a local smoothing norm estimate

∇u L 2 t (L 2 (K)) ≤ C(K)E 1 4 M 1 4 , (4.11)
which has the same scaling as Ḣ 1 2 for the data; here K is meant to be a compact set which includes the obstacle, and (4.11) holds only under the star-shaped condition on obstacle, while proving (4.10) makes an essential use of (4.11).

We start with proving Proposition 4.2. Let u be a solution to the nonlinear problem (2.2). Let χ ∈ C 2 0 (R 3 ) be a smooth function equal to 1 near ∂Ω. Then

χu ∈ L 4 t Ḃ1/4-η,2 4 (Ω) and (1 -χ)u ∈ L 2 t Ḃ1/2-η,2 6 (Ω). (4.12)
Remark 4.3. Notice that our cut χ is only C 2 rather thant C ∞ , and this will remain so for the rest of the section. This is in no way a difficulty, and it allows to conveniently take χ = χ p 1 or χ = χ p-1 1 , where χ 1 ∈ C 2 0 as an admissible cut if we need, as p -1 > 2. This is particulary convenient for nonlinear mappings where all factors can be considered equal. Alternatively, one may retain C ∞ 0 cuts and play with at least 3 overlapping ones, as was done in [START_REF] Planchon | Bilinear virial identities and applications[END_REF], at the expense of desymetrizing various nonlinear estimates. These are (mildly ennoying) considerations that the reader should ignore at first read.

Proof. In order to prove the Proposition, we split the equation (2.2), treating differently the neighborhood of the boundary (using local smoothing type arguments) and spatial infinity (where the full range of sharp Stricharz estimates holds).

Consider the equation satisfied by χu,

(i∂ t + ∆ D )(χu) = χ|u| 2+2η u -[χ, ∆ D ]u. (4.13)
We need to show that the nonlinear term belongs to L 2 t H -η comp (Ω). The commutator term is controlled by χu L 2 t H 1 comp for some χ ∈ C 2 0 (R 3 ) equal to 1 on the support of χ and it belongs to

L 2 t L 2 comp (Ω) ⊂ L 2 t H -η comp (Ω)
. We now deal with the nonlinear term: let q be such that Ḃ1,2 q (Ω) ⊂ H -η (Ω), hence 1

-3 q = -η -3 2 . Then 1 q = 1 2 + 2(1+η) 6 and χ|u| 2(1+η) u L 2 t H -η comp0 (Ω) χ|u| 2(1+η) u L 2 t Ḃ1,2 q (Ω) χ 1 u L 2 t H 1 0 (Ω) (χ 1 u) 1+η L ∞ t L 6 1+η (Ω)
, where χ p 1 = χ and we used u ∈ L ∞ t H 1 0 (Ω) ⊂ L ∞ t L 6 (Ω) on two factors and u ∈ L 2 t H 1 comp (Ω) on one factor. Hence the right hand side in (4.13) is in L 2 t H -η comp (Ω) and we can apply Lemma 4.2 with L q (B q ) := L 4 t Ẇ 1/4-η,4 (Ω), H := H 1/2-η (Ω) and L r (B r ) := L 2 t H -η comp (Ω). This gives the first assertion in (4.12). Let us deal now with (1χ)u which is solution to

(i∂ t + ∆ D )((1 -χ)u) = (1 -χ)|u| 2+2η u + [χ, ∆]u, (4.14)
where ∆ denotes the free Laplacian (notice that we can consider (4.14) in the whole space R 3 since both source terms vanish near the boundary ∂Ω). The commutator term is dealt with exactly as in the previous part and is therefore in

L 2 t L 2 comp (Ω). Let v := (1 -χ 1 )u for some χ 1 ∈ C 2 0 (R 3 ) such that (1 -χ 1 ) p = 1 -χ.
In order to prove (4.12) we only need to prove

|v| 2+2η v ∈ L 2 t Ḃ1/2-η,2 6/5
(Ω), since then we may apply the dual end-point Strichartz estimates (from the R 3 case) on the nonlinear term. Using the embedding Ḃ1-η,2

1 (Ω) ⊂ Ḃ1/2-η,2 6/5
(Ω), it suffices to get

|v| 2+2η v ∈ L 2 t Ḃ1-η,2 1
(Ω). When evaluating the product |v| 2+2η v we will use for one factor v the energy bound and Sobolev embedding,

L ∞ t H 1 0 (Ω) ⊂ L ∞ t Ḃ1-η,2 q (Ω) with 1 q = 1 2 -η 3 .
On the other hand, from our a priori bound from [START_REF] Planchon | Bilinear virial identities and applications[END_REF], we have

v ∈ L 4 t L 4 (Ω), while v ∈ L ∞ t H 1 0 (Ω) ⊂ L ∞ t L 6 (Ω) and hence v 1+η ∈ L 4/(1+η) t L 4/(1+η) (Ω) ∩ L ∞ t L 6/(1+η) (Ω). Interpolation with weights 1/(1 + η) and η/(1 + η) gives v 1+η ∈ L 4 t L 12/(3+2η) (Ω). Consequently, |v| 2+2η v L 2 t Ḃ1/2-η,2 6/5
(Ω)

|v| 2+2η v L 2 t Ḃ1-η,2 1 (Ω) v L ∞ t Ḃ1-η,2 q (Ω) |v| 1+η 2 L 4 t L 12/(3+2η)) (Ω) .
This achieves the proof of Proposition 4.2.

Remark 4.4. One should point out that the proof of this last estimate is slightly incorrect, as it conveniently ignores the situation where low frequencies are on the v factor and high frequencies are on |v| 2+2η . This can be easily fixed by revisiting the proof of Lemma 4.9 and 4.10 in the Appendix, noticing that we may suppose that factors f there are in several different L r spaces and distribute them when using Hölder on the low frequencies in the proofs. The same situation occurs several times in the present proof and we leave details to the reader.

The next iterative step will be the following lemma:

Proposition 4.3. Let u be a solution to the nonlinear problem (2.2). Then

u ∈ L 4 t Ẇ 1/4+η,4 (Ω) ∩ L 2 t H 1+η comp (Ω). (4.15)
Proof. The split of the equation into equations for χu and (1χ)u is no longer of any use: the resulting commutator source term is no better than [χ, ∆]u ∈ L 2 t L 2 comp (Ω). However we now have estimates from Proposition 4.2 which turn out to be good enough that splitting the nonlinear term in (2.2) in two parts, using the partition χ + (1χ) = 1 will allow us to use the somewhat restricted set of inhomogeneous estimates we have for the equation on a domain. Setting g 1 := χ|u| 2+2η u, g 2 := (1χ)|u| 2+2η u and using Duhamel formula, we have u(t, x) = e it∆ D u 0 + t 0 e i(t-s)∆ D g 1 (s)ds + t 0 e i(t-s)∆ D g 2 (s)ds ; (4.16) the idea is then that one may use (4.9) on the g 1 Duhamel term, while the g 2 term may be handled in L 1 t ( Ḣs ) for a suitable s.

Lemma 4.3. Let v := (1 -χ 1 )u, where χ 1 ∈ C 2 0 (R 3 ) is such that (1 -χ 1 ) p = 1 -χ. We have g 2 ∈ L 2 t Ḃ1/2,
g 2 L 1 t ( Ḣ 1 2 +η (Ω)) . (4.18)
Proof. From Proposition 4.2, the energy and mass bound, and interpolation, we have

v ∈ L 2 t Ẇ 1/2-η,6 (Ω) ∩ L ∞ t ( Ḣ 1 2 -η (Ω) ⊂ L 4 t L q (Ω) for 1 q = 1 6 + η 3 , hence |v| 1+η ∈ L 4/(1+η) t
L q/(1+η) (Ω) ∩ L ∞ t L 6/(1+η) (Ω). We now interpolate again and obtain |v| 1+η ∈ L 4 t L r (Ω), where 2 r = 1 3 + η. Therefore, the nonlinear term

g 2 = |v| 2+2η v belongs to L 2 t Ḃ1-3η,2 6/5 (Ω). Indeed, let 1 m = 1 2 + 2 r = 5 6 + η, then g 2 L 2 t Ḃ1-3η,2 6/5
(Ω) In fact, the commutator term [χ 1 , ∆]u is in L 2 t L 2 (Ω) and, consequently, it also belongs to

g 2 L 2 t Ḃ1,2 m (Ω) v L ∞ t Ḣ1 0 (Ω) |v| 1+η 2 L 4 t L r (Ω) . ( 4
L 2 t H 1/2-3η (Ω) since in this case 1/2 -3η < 0, while (1 -χ 1 )|v| 2+2η v ∈ L 2 t Ḃ1-3η,2 6/5
(Ω) as shown before. Therefore, with 1 -3η -

3/r = 2(1 -3η) -1, v|v| ∈ L 1 t Ḃ1-3η,2 r (Ω) ⊂ L 1 t Ḃ1-6η,2 ∞ (Ω).
(4.21)

In order to estimate g 2 we use (4.21) for a factor v|v|, while for the remaining factor |v| 1+2η we use v ∈ L ∞ t H 1 0 (Ω), which yields (Ω). Using Lemma 3.1, we also obtain

|v| 1+2η ⊂ L ∞ t Ḃ1,2 λ (Ω) ⊂ L ∞ t H 1-η (Ω) for 1 λ = 1 2 + η 3 . ( 4
t 0 e i(t-s)∆ [χ 1 , ∆]u(s)ds L 2 t Ḃ1/2,2 6 (Ω) [χ 1 , ∆]u L 2 t L 2 comp (Ω) .
Finally, the linear evolution

e it∆ R 3 (1 -χ 1 )u 0 is evidently in L 2 t Ḃ1/2,2 6
(Ω) and we obtain (4.17).

Remark 4.5. For the last part of the proof of Lemma 4.3 we shall use less information than that, precisely we only need the fact that for ǫ > 0 small enough we have

v ∈ L 2 t Ḃ1/2-ǫ,2 6 (Ω) ⊂ L 2 t (L 3 ǫ (Ω)) ⊂ L 2 t Ḃ-ǫ,∞ ∞ (Ω), (4.24
)

and |v| ∈ L 3 ǫ (Ω) ⊂ L 2 t Ḃ-ǫ,∞ ∞ (Ω) as well.
We refine our knowledge on g 2 = v|v|v 1+2η : using the previous remark, we now have

v|v| ∈ L 1 t Ḃ-2ǫ,∞ ∞ (Ω). From (4.22) we also have |v| 1+2η ∈ L ∞ t Ḃ1,2 λ (Ω) if λ = 6
3+2η . Thus, the source term g 2 can be estimated as follows

g 2 L 1 t H 1-η-2ǫ (Ω) g 2 L 1 t Ḃ1-2ǫ,2 λ (Ω) v|v| L 1 t Ḃ-2ǫ,∞ ∞ (Ω) |v| 1+2η L ∞ t Ḃ1,2 λ (Ω) .
(4.25)

Using again Lemma 4.1, this time with (Ω) u 1-θ

L q (B q ) := L 4 t Ḃ3/4-η-2ǫ,2 4 (Ω), H := H 1-η-2ǫ (Ω) and L r (B r ) := L 1 t H 1-η-2ǫ (Ω),
L 4 t,x g 2 L 1 t H 1-η-2ǫ (Ω) + u L 4 t,x ; (4.26)
where for the first (interpolation) inequality in (4.26) we used that 3/4η -2ǫ > 1/4 + η if ǫ is sufficiently small (take 0 < ǫ ≤ 1/20 for example).

that T = ∞ is allowed. For this, we interpolate between L

4 (Ω) ≤ C 3 (E, M) + C 4 (E, M) χu γ L 2 t H 1 comp (Ω) u ρ L 4 T Ḃσ,2 4 (Ω) , (4.34)
where ρ, γ > 0. The coefficients are uniformly bounded, and a splitting time argument performed on the L At this point of the proof, we could establish scattering in the scale-invariant Sobolev space; however we want to reach H 1 0 . Recall that we may write

u(t, x) -e it∆ D (u 0 + +∞ 0 e -is∆ D |u| p-1 u(s)ds) H 1 0 = +∞ t e i(t-s)∆ D |u| p-1 u(s)ds H 1 0 ,
from which we wish to use Duhamel to get

+∞ t e i(t-s)∆ D |u| p-1 u(s)ds H 1 0 g 1 L 4/3 (t,+∞; Ḃ5/4,2 4/3 (Ω)) + g 2 L 1 (t,+∞;H 1 0 (Ω)) , (4.35)
from which scattering easily follows (the same argument applies at t = -∞ as well). Therefore we focus on the right handside and start with the easiest part, which is g 2 .

Lemma 4.5. We have g

2 = (1 -χ)u p ∈ L 1 t H 1 0 (Ω).
Proof. We start by proving that (Ω). Interpolating again between this bound and the energy bound u ∈ L ∞ t H 1 0 (Ω), followed by Sobolev embedding yields (4.39). Now we write

v = (1 -χ 1 )u ∈ L 2(1+η) t L ∞ (Ω). ( 4
v|v| 2+2η L 1 t H 1 0 (Ω) ≤ |v| 2(1+η) L 1 t L ∞ (Ω) v L ∞ t H 1 0 (Ω) . ( 4
g 1 L 4/3 t Ḃ5/4,2 4/3 (Ω) χu L 2 t H 5/4 comp (Ω) u 2+2η L 4 t L 4 (Ω) , (4.40)
and also by the Duhamel formula and the local smoothing estimate on the domain,

u L 2 t H 5/4 comp (Ω) ≤ u 0 H 3/4 (Ω) + g 1 L 4/3 t Ḃ1,2 4/3 (Ω) + g 2 L 1 t H 3/4 (Ω) . (4.41)
Certainly, using Lemma 4.5, the g 2 term is bounded. For g 1 , we may write

g 1 L 4/3 t Ḃ1,2 4/3 (Ω) χu L 2 t H 1 comp (Ω) u 2+2η L 4 t L 4 (Ω) ; ( 4
(I -S j )f j ) p 2 -ks k<j 2 -s(j-k) η j ,
which by an l 1l q convolution provides the result.

Lemma 4.9. Consider α = 1 or α ≥ 2, f ∈ Ḃs,q p and g ∈ L r , with 0 < s < 2,

1 m = α r + 1 p : let T α g f = j (S j g) α ∆ j f.
Then T α g f ∈ Ḃs,q m . We split the paraproduct T α g f :

T α g f = j S j ((S j g) α ∆ j f ) + j (I -S j )((S j g) α ∆ j f );
the first part is easily dealt with by Lemma 4.7. For the second one, K g f , taking once again advantage of the spectral supports

∆ k K g f = ∆ k j<k (I -S j )((S j g) α ∆ j f ).
Notice the situation is close to the one in Lemma 4.8, but we don't have a negative regularity for summing. We therefore derive

∆ D K g f = j<k (I -S j )∆ D ((S j g) α ∆ j f ) = j<k (I -S j ) ∆ D (S j g) α ∆ j f + (∆ D ∆ j f )(S j g) α + 2α(S j g) α-1 ∇S j g • ∇∆ j f = j<k (I -S j ) α∆ D S j g(S j g) α-1 ∆ j f + α(α -1)|∇S j g| 2 (S j g) α-2 ∆ j f + (∆ D ∆ j f )(S j g) α + 2α(S j g) α-1 ∇S j g • ∇∆ j f .
The first two pieces are again easily dealt with with Lemma 4.8, and the resulting function is in Ḃs-2,q m . The remaining cross term is handled with some help from [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF]:

∇∆ j f = ∇ exp(4 -j ∆ D ) ∆j f,
where the new dyadic block ∆j is built on the function ψ(ξ) = exp(|ξ| 2 )ψ(ξ). From the continuity properties of √ s∇ exp(s∆ D ) on L p , 1 < p < +∞, we immediatly deduce

∇∆ j f p 2 j ∆j f p , (4.43)
and we can easily sum and conclude. This will be enough to deal with the critical case, but for differences of nonlinear power-like mappings, we need Lemma 4.10.

Consider α ≥ 3, f, g ∈ X = Ḃs,q p ∩ L r , with 0 < s < 2, 1 m = α-1 r + 1 p : Then, if F (x) = |x| α-1 x or F (x) = |x| α , F (u) -F (v) Ḃs,q m u -v X ( u α-1 X + v α-1 X ).
In order to obtain a factor uv, we write where on the third line we wrote the worst case, namely 2 ≤ β < 3 (otherwise the power of ∆ j w in the third bound will be replaced by |∆ j w|(|S j w| β-3 + |S j+1 w| β-3 )). By integrating, applying Hölder and using (4.43) to eliminate the ∇ operator, we obtain as an intermediary result F ′ (w) ∈ Ḃs,q λ , with

F (u) -F (v) = (u -v) 1 0 F ′ (θu + (1 -θ)v)dθ. ( 4
1 λ = α -2 r + 1 p .
We may now go back to the difference F (u) -F (v) as expressed in (4.44) and perform a simple paraproduct decomposition in two terms to which Lemma 4.9 may be applied.

Observe that there is no difficulty in estimating F ′ (w) in L m/(α-1) , and that the integration in θ is irrelevant. This completes the proof. We now go back to the first nonlinear estimate, namely (4.2). We write a telescopic series for the product five factors u 1 , u 2 , u 3 , u 4 , u 5 ∈ X T , u 1 u 2 u 3 u 4 u 5 = j S j+1 u 1 S j+1 u 2 S j+1 u 3 S j+1 u 4 S j+1 u 5 -S j u 1 S j u 2 S j u 3 S j u 4 S j u 5 and we are reduced to studying five sums of the same type, of which the following is generic S 1 = j ∆ j u 1 S j u 2 S j u 3 S j u 4 S j u 5 , and we intend to apply Lemma 4.9, which is trivially extended to a product of several factors. In principle, u k ∈ Ḃ1,2 5 (L 20 11

T ) ∩ L 20 3

x L 40 T is enough, using the first space of the ∆ j factor and the second one for all remaining S j factors, except for the use of (4.43) in the proof. Consider, from u ∈ X T , 2 11 10 j ∆ j u L 5 x L 2 T + 2 -3 2 j ∂ t ∆ j u L 5 T L 5

x = µ 0 j ∈ l 2 j .

We will have, using [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF],

2 11 10 j ∇∆ j u L 5 x L 2 T + 2 -3 2 j ∂ t ∇∆ j u L 5 T L 5 x = µ 1 j ∈ l 2 j , with µ 1 l 2 µ 0 l 2 .
By Gagliardo-Nirenberg in time, we have the correct estimate for ∆ j u, for k = 0, 1

2 (1-k)j ∇ k ∆ j u L 5 x L 20 11 T µ k j .
We proceed with the low frequencies by proving a suitable Sobolev embedding. Notice that the estimate with a gradient is much easier: just interpolate between (4.45) and (4.46) with k = 1 to obtain 2 -j ∇∆ j u L 20 3

x L 40 T µ 7 j , which we can now sum over k < j to obtain control of S j u. The case p < 5 is handled in an similar way, and we leave the details to the reader, sparing him the complete set of exponents (depending on p !) that would appear in the proof. For scaling reasons there is actually no need to perform the computation: the previous one on the critical case simply illustrates that we can sidestep issues related to the usual Littlewood-Paley theory by using direct arguments.

Theorem 2 . 1 .(L 20 11 T

 2111 (Well-posedness for the quintic Schrödinger equation) Let u 0 ∈ H 1 0 (Ω). There exists T (u 0 ) such that the quintic nonlinear equation (2.1) admits a unique solutionu ∈ C([-T, T ], H 1 0 (Ω)) ∩ Ḃ1,25). Moreover, the solution is global in time and scatters in H 1 0 if the data is small.The previous theorem extends to the following subcritical range:Theorem 2.2. Let 3+ 2 5 < p < 5, s p = 3 2 -2 p-1 and u 0 ∈ Ḣsp . There exists T (u 0 ) such that the nonlinear equation (2.2) admits a unique solution u ∈ C([-T, T ], Ḣsp ) ∩ Ḃsp,2 5 (L 20 11

. 1 )

 1 and for u ∈ X T set F (u) := |u| 4 u. Proposition 4.1. Define a nonlinear map φ as follows,

Lemma 4 . 1 . 4 t 8 )

 4148 (see[16, Lemma 5.4]) Let Ω be a non trapping domain and denote u L = e it∆ D the linear flow for the Schrödinger equation on Ω with Dirichlet boundary conditions. Thene it∆ D u 0 LDenote by w the solution of the inhomogeneous equation, e.g. w = t 0 e i(t-s)∆ D f (s)ds, then w

=S 1 + S 2 . 2 ∆ 2 .

 1222 .44)We need to efficiently split this difference into two paraproducts involving uv and F ′ (w) with w = θu + (1θ)v, and this requires an estimate on F ′ (w): write another telescopic seriesF ′ (w) = j F ′ (S j+1 w) -F ′ (S j w) = j S j (F ′ (S j+1 w) -F ′ (S j w)) + j (I -S j )(F ′ (S j+1 w) -F ′ (S j w))Exactly as before, the first sum S 1 is easily disposed of with Lemma 4.7, as|F ′ (S j+1 w) -F ′ (S j w)| |∆ j w|(|S j+1 w| α-2 + |S j w| α-2 ).The second sum S 2 requires again a trick; to avoid uncessary cluttering, we set F (x) = x α , ignoring the sign issue (recall that α ≥ 3, hence F ′′′ (x) is well-defined as a function): we apply ∆ D , letβ = α -1 ≥ D S 2 = j (I -S j )∆ D ((S j+1 w) α-1 -(S j w) α-1 ) = j (I -S j ) β(S j+1 w) β-1 ∆ D S j+1 wβ(S j w) β-1 ∆ D S j w + β(β -1)(S j+1 w) β-2 (∇S j+1 w) 2β(β -1)(S j w) β-2 (∇S j w)We now apply Lemma 4.8 after inserting the right factors: we have four types of differences,|((S j+1 w) β-1 -(S j w) β-1 )∆ D S j+1 w| C β |∆ j w||∆ D S j+1 |(|S j+1 w| β-2 + |S j w| β-2 ) |(S j+1 w) β-1 ∆ D ∆ j w| ≤ |∆ D ∆ j w||S j+1 w| β-2 |((S j+1 w) β-2 -(S j w) β-2 )(∇S j+1 w) 2 | Cβ |∆ j w| β-2 |∇S j+1w| 2 |(S j+1 w) β-2 ((∇S j w) 2 -(∇S j+1 w) 2 )| ≤ |∇∆ j w|(|∇S j w| + |∇S j+1 w||S j+1 w| β-2

Lemma 4 . 11 . 1 2 , 5 5 (L 5 T 3 2 ,5 5 (L 5 T 5 T L 5 x 5 T 2 -j 10 ∆ j u L 5 T L ∞ x µ 5 j 2 |∆ j u| + 2 -J 6 sup j 2 j 6 |∆ j u| |u| 4 sup j 2 -j 2 |∆ j u| sup j 2 j 6

 4111553555552105522262226 Let u ∈ Ḃ ) and ∂ t u ∈ Ḃ-). Then u ∈ L )j ∇ k ∆ j u L 5 x L 5 T + 2 -(k+ 3 2 )j ∂ t ∇ k ∆ j u L = µ k j ∈ l 5j , notice we can easily switch time and space Lebesgue norms. Using Gagliardo-Nirenberg in time, we have 2 ( 1 6 -k)j ∇ k ∆ j u L 5 x L 30 -Nirenberg in space, we also have2 -j 10 ∆ j u L ∞ x Land the same thing for 2 -2j ∂ t ∆ j u (or with an additional 2 j ∇). Now another Gagliardo-Nirenberg in time provides 2-(k+ 1 2 )j ∇ k ∆ j u L ∞ T,x µ 6 j . (4.46)Finally, we take advantage of a discrete embedding between l 1 and weighted l ∞ sequences:

  we get by interpolation

	t				t
	0	e i(t-s)∆ g 2 (s)ds L 4 t	Ḃ1/4+η,2 4	(Ω)	0	e i(t-s)∆ g 2 (s)ds θ L 4 t B 4 3/4-η-2ǫ,2

  Lemma 4.3 we consequently also have a uniform bound on the Duhamel part coming from g 2 , see (4.18). Finally, using (4.29) for g 1 and the uniform bounds we already have for the linear part and the g 2 part,

								4 t	Ḃ1/4-η,2 4	(Ω) and L 4 T	Ḃ1/4+η,2 4	(Ω)
	with interpolation exponent θ = η 2(1+η) to obtain an estimate on the L 4 T	Ḃσ,2 4 (Ω) norm,
	where σ = 1/4 + η/(1 + η):
				u L 4 T	Ḃσ,2 4 (Ω) ≤ u θ L 4 t	Ḃ1/4-η,2 4	(Ω) u 1-θ L 4 T Ḃ1/4+η,2 4	(Ω)	.	(4.31)
	Recall that from Proposition 4.2 we have now a uniform bound,
						u L 4 t	Ḃ1/4-η,2 4	(Ω)	C(E, M),	(4.32)
	and from u L 4 T	Ḃ1/4+η,2 4	(Ω)	C 1 (E, M) + C 2 (E, M) χu	1/2-η L 2 t H 1 comp (Ω) u	2(1+η) L 4 T Ḃσ,2 4 (Ω) .	(4.33)
	Plugging (4.32), (4.33) in (4.31) yields
		u L 4 T	Ḃσ,2

  Lemma 4.6. We have g 1 = χu p ∈ L

					.37)
	We proceed with (4.36). From Lemma 4.3 we know that g 2 ∈ L 1 t H 1-η (Ω) and [χ, ∆ D ]u ∈ t H η L 2 comp (Ω), so using again the equation for (1 -χ)u and Lemma 4.2,
	(1 -χ)u ∈ L 2 t	Ḃ1-η,2 6	(Ω) ∩L ∞ t H 1 0 (Ω) .	(4.38)
	Recall that from Lemma 4.3 we also have v ∈ L 2 t follows by interpolation and the Gagliardo-Nirenberg inequality (a similar key step exists Ḃ1/2,2 6 t H 1/2 (Ω). The Lemma now ∩ L ∞
	in [16]).
	4/3 t	Ḃ5/4,2 4/3 (Ω).
	Proof. We first prove
	u ∈ L 8(1+η) t	L 8(1+η) (Ω).	(4.39)
	Indeed, from Propositions 4.2, 4.3 and interpolation, we get u ∈ L 4 t	Ḃ1/4+η/2,2 4

* The first author was partially supported by grant A.N.R.-07-BLAN-0250 † The second author was partially supported by A.N.R. grant ONDE NON LIN 1

On the other hand, by Lemma 4.2 again, t 0 e i(t-s)∆ g 2 (s)ds L 2 t H 1+η comp (Ω)

which finally achieves the proof of Lemma 4.3.

It remains now to deal with the Duhamel term coming from g 1 in (4.16).

Lemma 4. [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF]. Suppose that we know moreover that

(Ω) and

Taking the lemma for granted, we can complete the proof of Proposition 4.3: using Lemmas 4.3, 4.4, the fact that the linear flow is in

comp (Ω) and Duhamel formula (4.16), estimate (4.15) follows immediately.

Proof. (of Lemma 4.4):

The a-priori information (4.28) gives

and consequently u 2(1+η) ∈ L 2/(1+η) t L 3/(1-η) (Ω). On the other hand, interpolating between

where

It remains to notice that for M defined above, the embedding Ḃ1,2

(Ω) holds (indeed, 1 > 3/4 + η and 1 -3/M = 3/4 + η -9/4) and to use again Lemmas 4.2, 3.1. Another application of Lemma 4.2 with

(Ω) achieves the proof of (4.29) and Lemma 4.4.

End of the proof of Proposition 4.3:

In order to complete the proof of Proposition 4.3 it remains to prove that (4.28) holds indeed, since we have used it to deduce (4.15). Let 0 < T < ∞ be small enough, so that by the local existence theory (see [START_REF] Planchon | Bilinear virial identities and applications[END_REF]) the L 4 T Ḃσ,2 4 (Ω) norm of u is finite; in fact, the same can be said with σ replaced by η + 1 4 . We shall prove

Appendix

In order to perform the various product estimates, we need a couple of useful lemma.

Observe that with the spectral localization one cannot take advantage of convolution of Fourier supports. As a first step and in order to avoid cumbersome notations, we only consider functions and Besov spaces which do not depend on time. We will then explain how to re-instate the time dependance in the nonlinear estimates.

It is worth noting at this stage, however, that both ∆ j and S j operators are well-defined on L p t L q x and L q x L p t for all the pairs (p, q) to be considered: this follows from [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF] for the case L p t L q x where the time norm is harmless. In the case L q x L 2 t , the arguments from [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF] apply as well (heat estimates are proved for data in L p

x (H) where H is an abstract Hilbert space, and when H = L 2 t , the heat kernel is diagonal and therefore Gaussian as well). By interpolation and duality we recover all pairs (p, q). Remark 4.8. In R n , one may perform product estimates in an easier way because of the convolution of Fourier supports. However, when dealing with non integer power-like nonlinearities, one cannot proceed so easily: the usual route is to use a characterization of Besov spaces via finite differences; here, because of the Banach valued Besov spaces, we perform a direct argument which is directly inspired by computations in [START_REF] Planchon | Dispersive estimates and the 2D cubic NLS equation[END_REF], where the same sort of time-valued Besov spaces were unavoidable. Lemma 4.7. Let f j be such that S j f j = f j , and f j L p 2 -js η j , with s > 0 and (η j ) j ∈ l q . Then g = j f j ∈ Ḃs,q p .

We have, by support conditions,

which by an l 1l q convolution provides the result.

Lemma 4.8. Let f j be such that (I -S j )f j = f j , and f j L p 2 -js η j , with s < 0 and (η j ) j ∈ l q . Then g = j f j ∈ Ḃs,q p .

We have, by support conditions,