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

www.elsevier.com/locate/anihpc

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

Oana Ivanovici

, Fabrice Planchon

aUniversité Paris-Sud, Orsay, Mathématiques, Bat. 430, 91405 Orsay Cedex, France

bLaboratoire Analyse, Géométrie & Applications, UMR 7539 du CNRS, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France

Received 21 July 2009; accepted 16 March 2010 Available online 22 April 2010

Abstract

We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed forH₀¹(Ω)data on any smooth, non-trapping domainΩ⊂R³. The key ingredient is a smoothing effect inL⁵_x(L²_t)for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.

Published by Elsevier Masson SAS.

1. Introduction

The Cauchy problem for the semilinear Schrödinger equation in R³ is by now relatively well understood: after seminal results by Ginibre and Velo [10] in the energy class for energy subcritical equations, the issue of local well- posedness in the critical Sobolev spaces(H˙^32−p−21)was settled in [7]. Scattering for large time was proved in [10]

for energy subcritical defocusing equations, while the energy critical (quintic) defocusing equation was only recently successfully tackled in [9]. 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 an elusive target, due to the difficulty in obtaining Strichartz estimates in such a setting.

In [2], the authors proved Strichartz estimates with a 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 non-linearities; this was later improved to cubic non-linearities in [1] (combining local smoothing and semiclassical Strichartz near the boundary) and in [11] (on the exterior of a ball, through pre- cised smoothing effects near the boundary). Recently there were two significant improvements, following different strategies:

* Corresponding author.

E-mail addresses:oana.ivanovici@math.u-psud.fr (O. Ivanovici), fab@math.univ-paris13.fr (F. Planchon).

1 The author was partially supported by grant A.N.R.-07-BLAN-0250.

2 The author was partially supported by A.N.R. grant ONDE NON LIN.

0294-1449/$ – see front matter Published by Elsevier Masson SAS.

doi:10.1016/j.anihpc.2010.04.001

• in [16], Luis Vega and the second author obtain anL⁴_t,x Strichartz estimate which is scale invariant. However, one barely missesL⁴_t(L^∞(Ω))control forH₀¹data, and therefore local well-posedness in the energy space was improved to all subcritical (less than quintic) non-linearities, but combining this Strichartz estimate with local smoothing close to the boundary and the full set of Strichartz estimates inR³away from it. Scattering was also obtained for the cubic defocusing equation, but the lack of a good local well-posedness theory at the scale invariant level (H˙¹²) led to a rather intricate incremental argument, from scattering inH˙¹⁴ to scattering inH₀¹;

• in [13], the first author proved the full set of Strichartz estimates (except for the endpoint) outside strictly convex obstacles, by following the strategy pioneered in [17] 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 upgrades them to large time results by combining them with the smoothing effect (see [3] for a nice presentation of such an argument, already implicit in [19]). Therefore, one obtains the exact same local well-posedness theory as in theR³case, including the quintic non-linearity, and scattering holds for all sub quintic defocusing non-linearities, taking advantage of the a priori estimates from [16].

In the present work, we aim at providing a local well-posedness theory for the quintic non-linearity outside non- trapping obstacles, a case which is not covered by [13]. From explicit computations with gallery modes [12], 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 theL⁴_t(L^∞_x )estimate inR³ 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: inR³, these estimates may be stated as follows,

exp(it )f

L⁴_x(L²_t)f_˙

H⁻¹⁴, (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 inR³(or, more accurately, its dual version, on the extension): let τ >0 be a fixed radius, one seesf (ξ )ˆ as a function on|ξ| =√

τ, and applies the extension estimate, withδthe Dirac function andF the space Fourier transform

F⁻¹ δ

τ− |ξ|²f (ξ )ˆ

L⁴_x f (ξ )ˆ

L²(|ξ|=√ τ ).

Summing over τ yields the L² norm off on the right-hand side, 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|byτ = ±|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 [18], and provided a key tool for solving the critical wave equation on domains, see [4,6]. In this paper, we apply the same strategy to the Schrödinger equation:

• we derive anL⁵(Ω;L²_I)“smoothing” estimate for spectrally localized data on compact manifolds with bound- aries, from the spectral clusterL⁵(Ω)estimate; hereI 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 3Dpunctured torus, to which the previous semiclassical estimate may be applied, and then summed up using the local smoothing effect; and far away from the boundary where theR³estimates hold.

• Finally, we patch together all estimates to obtain an estimate which is valid on the whole exterior domain. Local well-posedness in the critical Sobolev spaceH˙^32−p−21 immediately follows for 3+2/5< p5, and together with the a priori estimates from [16], this implies scattering for the defocusing equation for 3+2/5< p <5. The remaining range 3p3+2/5 is sufficiently close to 3 that, as alluded to in [16], a suitable modification of the arguments from [16] yields scattering as well.

Remark 1.1.Clearly, such smoothing estimates are better suited to “large” values ofp: the restriction 3+2/5< p for the critical well-posedness is directly linked to the exponent 5 in the spectral cluster estimates; inR³, where the correct (and optimal!) exponent is 4, one may solve down top=3 by this method, while the Strichartz estimates allow to solve at scaling level all the way to theL²critical valuep=1+4/3.

2. Statement of results

LetΘbe a compact, non-trapping obstacle inR³and setΩ=R³\Θ. By_Dwe denote the Laplace operator with constants coefficients onΩ. We writeL^p_x=L^p(Ω)andH˙^σ for the Lebesgue and Sobolev spaces onΩ. Fors∈R, p, q∈ [1,∞]we denote byB˙_p^s,q(Ω)= ˙B_p^s,q Besov spaces onΩ, where the spectral localization in their definition is meant to be with respect to_D; it will be useful to introduce Banach-valued Besov spacesB˙_p^s,q(L^r_t), and we refer to Section 4 (Definition 4.1) for relevant definitions. WheneverL^p_t is replaced byL^p_T, it means that the time integration is restricted to the interval(−T , T ). Notice thatH˙^s= ˙B₂^s,2(using the spectral decomposition).

Remark 2.1.A reader who is well acquainted with Besov spaces inRⁿmay think our spaces are indeed equivalent to the usual ones, which are defined on domains via extensions or with finite differences when the regularitysis strictly positive. While most likely true, these equivalences are non-trivial to prove and we elected to just define the spaces we need in a way which is convenient for our purposes. We however have a (small) price to pay in Appendix A where several non-linear mappings are proved within our framework.

We aim at studying well-posedness for the energy critical equation onΩ×R, with Dirichlet boundary condition,

i∂tu+Du= ±|u|⁴u, u_|∂Ω=0, u_|t=0=u0 (2.1)

and more generally

i∂tu+Du= ±|u|^p−1u, u_|∂Ω=0, u_|t=0=u0 (2.2)

withp <5.

Theorem 2.1(Well-posedness for the quintic Schrödinger equation). Letu₀∈H₀¹(Ω). There existsT (u₀)such that the quintic non-linear equation(2.1)admits a solutionuinC([−T , T], H₀¹(Ω));moreover,uis unique inB˙₅^1,2(L

20 11

T )∩ L

20

x3 L⁴⁰_T . If the data is small, then the solution is global in time and scatters inH₀¹. The previous theorem extends to the following subcritical range:

Theorem 2.2.Let3+²₅ < p <5,s_p=³₂−_p²₋₁ andu₀∈ ˙H^sp. There existsT (u₀)such that the non-linear equa- tion(2.2)admits a solutionuinC([−T , T],H˙^sp), andu is unique inB˙₅^sp,2(L

20 11

T )∩L

5(p−1)

x 3 L^10(p_T ⁻¹⁾. Moreover the solution is global in time and scatters inH˙^spif the data is small.

Remark 2.2.We elected to state both theorems for Dirichlet boundary conditions mostly for sake of simplicity. Indeed, both results hold with Neumann boundary conditions, at least forp4: the key ingredients for the required linear estimates are known to hold for Neumann, see [18,2], while the non-linear mappings from our appendix rely on [14]

(where all relevant estimates can be proved to hold in the Neumann 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 [16].

Theorem 2.3.Assume the domainΩ to be the exterior of a star-shaped compact obstacle(which impliesΩ is non- trapping). Let3p <5, andu₀∈H₀¹(Ω). There exists a unique global in time solutionu, which is in the energy

class,C(R, H₀¹(Ω)), to the non-linear equation(2.2)in the defocusing case(+sign in(2.2)). Moreover, this solution scatters for large times:there exist two scattering statesu^±∈H₀¹(Ω)such that

t←±∞lim u(x, t )−e^{it D}u^±

H₀¹(Ω)=0.

As mentioned in the introduction, the (global) existence part was dealt with in [16]; for the scattering part, the p=3 case was also dealt with in [16]. In the setting of Theorem 2.2, one may adapt the usual argument from the Rⁿ 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< p3+2/5, one unfortunately needs to adapt the intricate proof from [16], and this leads to a much lengthier proof; we provide it mostly for the sake of completeness. This type of argument may however be of relevance in other contexts.

3. Smoothing type estimates

We start with definitions and notations. Letψ (ξ²)∈C₀^∞(R\{0})andψ_j(ξ²)=ψ (2^−2jξ²). On the domainΩ, one has the spectral resolution of the Dirichlet Laplacian, and we may define a smooth spectral projection_j=ψ_j(−_D) as a bounded operator onL². Moreover, this operator is bounded onL^pfor allp, and iff is Hilbert-valued and such thatfHL^p(Ω)<+∞, thenj is bounded as well onL^p(H ). We refer to [14] for an extensive discussion and references. We simply point out that ifH=L²_t, then_j is continuous on allL^p_xL^q_t by interpolation with the obvious L^p_t(L^p_x)bound and duality.

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

i∂tuL+DuL=0, uL|∂Ω=0, uL|t=0=u0. (3.1)

Theorem 3.1.The following local smoothing estimate holds for the homogeneous linear equation(3.1), juL_L⁵

xL²_t +2^−2j∂tjuL_L⁵

xL²_t 2^−10jju0L²_x. (3.2)

Moreover, let2q∞, then

_ju_LL⁵_xL^q_t +2^−2j∂_tju_LL⁵_xL^q_t 2^{−j (q2−109)}_ju₀L²_x. (3.3) Consider now the inhomogeneous equation,

i∂_tv+_Dv=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.Let2q+∞,2< r+∞, then

jvCt(L²_x)+2^{j (2q−109)}jv_L⁵_xL^q_t +2^{j (2q−2910)}∂tjv_L⁵_xL^q_t 2^{−j (4r−95)}jF

L

5 x4L^r_t

, (3.5)

with1/r+1/r=1andvsolution to(3.4).

Combining the previous theorems with the results from [16], we finally state the set of estimates which will be used later for

i∂tu+Du=F1+F2, u_|∂Ω=0, v_|t=0=u0. (3.6)

Theorem 3.3.Let2q+∞,2< r+∞, then _juC_t(L²_x)+2^{j (2q−109)}_ju_L⁵

xL^q_t +2^{j (q2−2910)}∂_tju_L⁵

xL^q_t +2^−34j_ju_L⁴

t,x+2^−114j∂_tju_L⁴

t,x

ju0L²_x +2^{−j (4r−95)}jF1

L

5

x4L^r_t+2^−14jjF2

L

4 3 t,x

, (3.7)

with1/r+1/r=1andusolution to(3.6).

3.1. Proof of Theorem 3.1

Let P_j =_j−1+_j +_j+1, then P_jj =_j because of support properties. We now split the solution of the linear equation_ju_L=P_jju_L as a sum of two termsP_jχ _ju_L+P_j(1−χ )_ju_L, whereχ∈C₀^∞(R³)is compactly supported andχ=1 near the boundary∂Ω.

3.1.1. “Far” from the boundary:P_j(1−χ )_ju_L

In this case the spectral localization provided byP_j is useless until the very last step and we therefore drop it. Set w_j(t, x)=(1−χ )_je^{it D}u₀(x). Thenw_j satisfies

i∂_tw_j+_Dw_j= −[_D, χ]_ju_L,

wj|t=0=(1−χ )ju0. (3.8)

Sinceχ=1 near the boundary∂Ω, we can view the solution to (3.8) as a solution to the Schrödinger equation inR³. Consequently, the Duhamel formula writes

wj(t, x)=e^{it 0}(1−χ )ju0− t 0

e^i(t−s)0[D, χ]juL(s) ds, (3.9) where₀is the free Laplacian onR³. Hence, fore^{it 0}(1−χ )_ju₀, usual Strichartz estimates hold. We now have to deal with the second term in the right-hand side of (3.9). Ideally, one would like to remove the time restrictions < t and use a variant of the Christ–Kiselev lemma. However, this would miss the endpoint caseq=2. Instead, we recall the following lemma:

Lemma 3.1.(See Staffilani and Tataru [19].) Letx∈Rⁿ,n3and letf (x, t )be compactly supported in space, such thatf ∈L²_t(H⁻¹²). Then the solutionwto(i∂t+0)w=f withw_|t=0=0, is such that

w

L²_t(L

2n n−2

x )

f

L²_t(H⁻¹²). (3.10)

In fact, one may shift regularity in (3.10) without difficulty. Now, the proof in [19] relies on a decomposition into traveling waves, to which homogeneous estimates are then applied. We can therefore use theL⁴_x(L²_t)smoothing estimate, Sobolev in space, and extend the conclusion of Lemma 3.1 to

w_L5

x(L²_t)f

L²_t(H^−12−101), (3.11)

where we chose to conveniently shift the regularity to the right hand-side.

We now takef = −[D, χ]juL∈L²_tH_comp^{−1/2−1/10}(Ω)and [_D, χ]_ju_L

L²H_comp^{−1/2−1/10}_ju_LL2H˙^1/2−1/10(Ω)_ju_0H˙−1/10(Ω), from which the smoothing estimate for solutions to (3.8) follows

(1−χ )juL

L⁵(R³)L²_t (1−χ )ju0

˙

H⁻¹⁰¹(R³)+[D, χ]juL

L²Hcomp^{−1/2−1/10}

ju0_˙

H⁻¹⁰¹ (Ω). (3.12)

We conclude using the continuity properties ofP_j which were recalled at the beginning of Section 3 (e.g. see [14, Corollary 2.5]). In fact, using (3.12), we get

P_j(1−χ )_ju_L

L⁵_xL²_t (1−χ )_ju_L

L⁵_xL²_t

2^−10j_ju₀L²(Ω), where we have used the spectral localization_j to estimate

_ju₀H˙^σ(Ω)2^σj_ju₀L²(Ω).

3.1.2. “Close” to the boundary:P_jχ _ju_L

Forl∈Zletϕ_l∈C₀^∞(((l−1/2)π, (l+1)π ))equal to 1 on[lπ, (l+1/2)π]. We setv_j=P_jχ _ju_Land forl∈Z we setv_j,l=ϕ_l(2^jt )v_j. We have

v_j²_L5 xL²_t =

l∈Z

v_j,l ²

L⁵_xL²_t

l∈Z

v_j,l ²

L²_t

L^5/2_x

l∈Z

vj,l²_L2 t

L^5/2_x

l∈Z

vj,l²_L5

xL²_t, (3.13)

where for the first inequality rests upon almost orthogonality in time of the(ϕ_l)_l. In order to estimatev_j²_L5

xL²_t it will be thus sufficient to estimate eachv_j,l²_L5

xL²_t. We derive an equation forv˜_j,l^def=ϕ_l(2^jt )χ _ju_L: i∂_tv˜_j,l+_Dv˜_j,l= −

ϕ_l 2^jt

[_D, χ]_ju_L−i2^jϕ_l 2^jt

χ _ju_L

, (3.14)

and we stress thatv˜_j,lvanishes outside the time interval(2^−j(l−1/2)π,2^−j(l+1)π ). LetV_j,lbe the right-hand side in (3.14), namely

V_j,l^def= −ϕ_l 2^jt

[_D, χ]_ju_L+i2^jϕ_l 2^jt

χ _ju_L. (3.15)

Let Q⊂R³ be an open cube which is sufficiently large so that∂Ω is contained in the interior ofQ. We now viewQas a compact, boundary less, manifold with periodic boundary conditions on∂Q, and we denote byS the punctured torusQ\Θ(recall thatΩ=R³\Θ). Let also_S^def=₃

j=1∂_j²denote the Laplace operator on the compact domainS.

On S, we may define a spectral localization operator using eigenvalues λ_k and eigenvectors e_k of _S: if f =

kc_ke_k, then ^S_jf =ψ

2^−2jS

f=

k

ψ 2^−2jλ²_k

ckek. (3.16)

We also defineP_j^S=^S_j−1+^S_j+^S_j+1.

Remark 3.1.Notice that in a neighborhood of the boundary, the domains ofSandDcoincide, thus ifχ˜∈C₀^∞(R³) is supported near∂Ω then

_Sχ˜=_Dχ .˜

In order to apply estimates on the manifoldS, we will need to re-localize close to the obstacle. Considerχ₁∈C₀^∞(R³) supported near the boundary and equal to 1 on the support ofχ, we will write˜

χ₁P_jχ˜=χ₁P_j^Sχ˜+χ₁

P_j−P_j^S

˜

χ , (3.17)

and our expectation will be that the difference term is smoothing.

In what follows letχ˜∈C₀^∞(R³)be equal to 1 on the support ofχand be supported in a neighborhood of∂Ωsuch that on its support the operator−_Dcoincide with−_S. From their support properties in space, we havev˜_j,l= ˜χv˜_j,l andV_j,l= ˜χ V_j,l. Consequentlyv˜_j,lwill also solve the following equation on our compact manifoldS

i∂_tv˜_j,l+_Sv˜_j,l=V_j,l,

˜

vj,l|t <h(l−1/2)π=0, v˜j,l|t >h(l+1)π=0. (3.18)

Therefore we can write the Duhamel formula either for the last equation (3.18) onS, or for Eq. (3.14) onΩ. We now applyP_j and use thatv_{j ,l}=P_jv˜_j,l,χ˜v˜_j,l= ˜v_j,l andP_jχ˜=χ₁P_j^Sχ˜+(1−χ₁)P_jχ˜+χ₁(P_j−P_j^S)χ, which yields˜

v_j,l(t, x)=χ1

t h(l−1/2)π

e^i(t−s)SP_j^SV_j,l(s, x) ds

+(1−χ1) t h(l−1/2)π

e^i(t−s)DP_jV_j,l(s, x) ds

+χ₁

P_j−P_j^S

˜

v_j,l, (3.19)

where we conveniently chose to write Duhamel onSfor the first term and Duhamel onΩ for the second one, which allows to commute the flow under the time integral. Denote byvj,l,mthe first term in the right-hand side of (3.19) by v_j,l,f the second one andv_j,l,sthe last one. We deal with them separately. To estimate theL⁵_xL²_t norm of thev_j,l,f we notice that its support is far from the boundary: as such, estimates on theL⁵_xL²_t norm will follow from Section 3.1.1.

Indeed, we get

(1−χ1)Pje^i(t−s)DVj,l

L⁵_xL²_t PjVj,lH˙^−1/10(Ω)2^−10jPjVj,lL²(Ω). (3.20) We then apply the Minkowski inequality to deduce

(1−χ₁) t h(l−1/2)π

P_je^i(t−s)DV_j,l(s, x) ds L⁵_xL²_t

2^−j/2

Ij,l

(1−χ₁)P_je^i(t−s)DV_j,l(s, .)²

L⁵(Ω)L²(I_j,l)ds 1/2

, (3.21)

where we denotedIj,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,fL⁵(Ω)L²(Ij,l)2^{−j (1/2+1/10)}P_jV_j,lL²(Ij,l)L²(Ω). (3.22) To estimate theL⁵_xL²_t norm of the main contributionv_j,l,mwe need the following:

Proposition 3.1.Letj0,Ij=(−π2^−j, π2^−j),χ˜ ∈C^∞₀ (R³)be supported near∂Ω andV0∈L²(Ω). Then there existsC >0independent ofj such that for the solutione^{it S}P_j^Sχ V˜ 0 of the linear Schrödinger equation onS with initial dataP_j^Sχ V˜ 0we have

e^{it S}P_j^Sχ V˜ ₀

L⁵(S)L²_t(I_j)C2^−10j P_j^Sχ V˜ ₀

L²(S). (3.23)

We postpone the proof of Proposition 3.1 to Section 3.4.

Using the fact that v_j,l is supported in time inI_j,l = [2^−j(l−1/2)π,2^−j(l+1)π], the Minkowski inequality, Proposition 3.1 withχ˜=1 on the support ofχ and withV₀=V_j,l, and sinceχ˜₁v_j,l,m=v_j,l,mfor anyχ˜₁∈C^∞(R³) withχ˜₁=1 on the support ofχ₁, we obtain

v_j,l,mL⁵_xL²(I_j,l)= ˜χ1v_j,l,mL⁵_xL²(I_j,l)= v_j,l,mL⁵(S)L²(I_j,l)

2^−j(l+1)π 2^−j(l−1)π

e^i(t−s)SP_j^SV_j,l(s, .)

L⁵(S)L²(Ij,l)ds

2^−10j

Ij,l

P_j^SV_j,l(s)

L²(S)ds

2^−10j

I_j,l

χ V˜ j,l(s)

L²(S)ds

2^−10j

Ij,l

χ V˜ _j,l(s)

L²(Ω)ds (3.24)

where we used again V_j,l = ˜χ V_j,l to switch S andΩ as well as continuity of^S_j on L²(S). Using the Cauchy–

Schwartz inequality in (3.24) yields

v_j,l,mL⁵_xL²(I_j,l)2^{−j (1/2+1/10)}V_j,lL²(I_j,l)L²(Ω). (3.25) We postpone for a moment further treatment of the right-hand side of this inequality, and now turn to the difference termvj,l,s. We rely on the following smoothing lemma, whose proof is postponed to Section 3.2.

Lemma 3.2.Letχ₁∈C₀^∞(Rⁿ)be equal to1on a fixed neighborhood of the support ofχ˜. Then we have for allN∈N, v_j,l,sL⁵(Ω)L²(Ij,l)C_N2^−NjV_j,l(x, s)

L²(Ij,l,L²(Ω)). (3.26)

Using this lemma, 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

P_jχ _ju_L²_L5

xL²_t 2^{−2j (12+101)}

l∈Z

P_jV_j,l(s)²

L²(Ij,l)L²(Ω) (3.27)

and we are left with the right-hand side in (3.27). Using the explicit expression ofV_j,l given in (3.15), we have V_j,l(s)

L²(I_j,l)L²(Ω)ϕ_l 2^jt

[_D, χ]_ju_L

L²(I_j,l)L²(Ω)

+2^jϕ_l 2^jt

χ juL

L²(Ij,l)L²(Ω)

. (3.28)

As[_D, χ]is bounded fromH₀¹toL², we get PjVj,lL²(Ij,l)L²(Ω)χ1juL_L²_(Ij,l)H¹

0(Ω)+2^jχ juLL²(Ij,l)L²(Ω). (3.29) Now, let us recall the following local smoothing result on a non-trapping domain:

Lemma 3.3. (See Burq, Gérard, Tzvetkov [2, Proposition 2.7].) Assume thatΩ=R³\Θ, whereΘ= ∅is a non- trapping obstacle. Then, for everyχ˜∈C₀^∞(R³), andσ∈ [−1/2,1],

˜χ _ju_LL²(R,H˙^σ+1/2(Ω))C_ju₀H^σ(Ω), (3.30)

where, as usual,u_L(t, x)=e^{−it D}u₀(x).

Applying Lemma 3.3 to the right-hand side of (3.29), P_jχ _ju_L²_L5

xL²_t 2^{−2j (12+101)}

l∈Z

χ_1ju_L²_L2(I_j,l)H₀¹(Ω)+2^2jχ _ju_L²_L2(I_j,l)L²(Ω)

2^−2j10

2^−j_ju₀²

˙

H¹2(Ω)+2^j_ju₀²

˙ H⁻¹2(Ω)

2^−2j10_ju₀²_L2(Ω),

which is the first part of inequality (3.2). Therefore we have proved Theorem 3.1q=2, but without the time derivative term. We now use the Gagliardo–Nirenberg inequality in order to deduce (3.3) for everyq2. We have

_ju_LL^∞_{t j}u_L^1/2_L2

t _j∂_tu_L^1/2_L2 t

.

Taking theL⁵_xnorms and using Cauchy–Schwartz yields _ju_L⁵_L5

xL^∞_{t j}u_L^5/2_L5L2

t_j∂_tu_L^5/2_L5

xL²_t. (3.31)

It remains to estimatej∂tuL_L⁵

xL²_t: notice that sinceuL=e^{−it D}u0

j∂tuL= −iDjuL=i2^2jQjuL,

whereQj=ψ1(2^−2jD)andψ1(x)=xψ (x)∈C^∞₀ (R\ {0}). Therefore _j∂_tu_LL5

xL²_t C2^{j (2−1/10)} ˜_ju₀L²(Ω), (3.32)

consequently _j∂_tu_LL⁵

xL^q_t C2^{−j (2/q−9/10)}_ju₀L²(Ω)

and Theorem 3.1 is proved.

3.2. Proof of Lemma 3.2

In order to prove the lemma, one would like to rewriteP_j =ψ (2^−2j−2_D)+ψ (2^−2j_D)+ψ (2^−2j+2_D)as a solution of the wave equation, usingh=2^−j as a time. Then by finite speed of propagation we could switch_D and_S. However the inverse Fourier transform (in|ξ|) ofΨ (|ξ|)=ψ (|ξ|²/4)+ψ (|ξ|²)+ψ (4|ξ|²)is only Schwartz class, rather than compactly supported. The tails will eventually account for the right-hand side of (3.26). We now turn to the details: letϕ₀(y), ϕ(y)be even functions which are compactly supported (and away from zero forϕ(y)) and such that

ϕ₀(y)+

k1

ϕ 2^−ky

=1.

We decomposeΨ (y)ˆ using this resolution of the identity, and set with obvious notations Ψ

|ξ|

=

k∈N

φ_k

|ξ| ,

where, being Schwartz class, theφ_k have good bounds,φˆ₀∈L^∞and fork1

∀N∈N, ˆφ_k∞=Ψ (y)ϕˆ

2^−ky_∞C_N2^−kN. (3.33)

At fixedk, we write (abusing notation and lettingbe either_Dor_S) φ_k(h√

−)χ˜v˜_j,l= 1 2π

e^iyh√−χ (x)˜ v˜_j,l(x)φˆ_k(y) dy.

Notice thatφ_k(y)is compactly supported, in fact its support is roughly|y| ∈ [2^k−1,2^k+1]. As such they 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 functionχ1which is equal to one on the domain of dependency ofχ˜ on this time scale, and such thatχ1is indifferently defined onSorΩ: namely, forkj,

φ_k(h

−_S)χ˜v˜_j,l=χ₁(x)φ_k(h

−_S)χ˜v˜_j,l

=χ₁(x) 1 2π

e^iyh√−χ (x)˜ v˜_j,l(x)φˆ_k(y) dy, φ_k

2^−j

−_S

˜

χv˜_j,l=χ₁(x)φ_k 2^−j

−_D

˜

χv˜_j,l. (3.34)

From this identity, we obtain v_j,l,s=χ₁(x)

jk

φ_k 2^−j

−_D

−φ_k 2^−j

−_S

˜

χ (x)v˜_j,l. (3.35)