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Comptes Rendus Mathématique
Volume 355, n° 2
pages 161-165 (février 2017)
Doi : 10.1016/j.crma.2017.01.005
Received : 16 September 2016 ;  accepted : 5 January 2017
Dispersive estimates for the wave equation inside cylindrical convex domains: A model case
Estimation de dispersion pour les ondes dans un convexe : le cas modèle
 

Len Meas
 Laboratoire Jean-Alexandre-Dieudonné, UMR CNRS 7351, Université de Nice, parc Valrose, 06108 Nice Cedex 02, France 

Abstract

In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain   with a smooth boundary  . Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge [[1], [2]]. Better estimates in strictly convex domains have been obtained in [[4]]. Our case of cylindrical domains is an extension of the result of [[4]] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.

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Résumé

Dans ce travail, nous allons établir des estimations de dispersion locales en temps pour les solutions de l'équation des ondes dans un domaine cylindrique convexe   à bord    . Les estimations de dispersion sont classiquement utilisées pour prouver les estimations de Strichartz. Dans un domaine Ω général, des estimations de Strichartz non optimales ont été démontrées par Blair–Smith–Sogge [[1], [2]]. De meilleures estimations ont été prouvées dans [[4]] lorsque Ω est strictement convexe. Le cas des domaines cylindriques que nous considérons ici généralise les resultats de [[4]] dans le cas où la courbure ≥0 dépend de l'angle d'incidence et s'annule dans certaines directions.

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Version française abrégée

Soit  . Nous établissons des estimations de dispersion en temps petit pour l'équation des ondes dans Ω avec condition de Dirichlet.
Pu=0,u|t=0=δa,∂tu|t=0=0,u|x=0=0 avec  ,  , et pour  ,   . Le problème est local près de chaque point du bord. Les phénomènes nouveaux apparaissent pour   petit. Nous utilisons les notations  ,  ,  ,  .

Résultats

On note   la fonction de Green associée à ((1)). La fonction χ appartient à   et est égale à 1 sur l'intervalle  . Nous prouvons le théorème suivant.

Théorème 0.1

Il existe   tel que, pour tout  ,  ,  , on a
‖χ(hDt)Ga(t,x,y,z)‖L∞≤Ch−3min⁡(1,(h|t|)1/2γ(t,h,sup⁡(x,a))), avec γ(t,h,b)={(h|t|)1/2+b1/8h1/4sib≥h2/3−ϵ, pour tout ϵ>0 petit,h1/4+(h|t|)1/3sib≤h1/2.

Introduction

We will study the following model case. Let  . Our goal is to establish local in time dispersive estimates for solutions to the linear Dirichlet wave equation inside Ω with smooth boundary ∂Ω:
(1)Pu=0,u|t=0=δa,∂tu|t=0=0,u|x=0=0 with  ,  , and for  ,   (the Dirac distribution). The problem is local near any points on the boundary. We are interested only in highly reflected waves and near points where the order of tangency is infinity whose source points are located at a small distance   to the boundary. This gives us interesting phenomena such as caustics near the boundary for such domains. We will use the notation  ,  ,  ,  . We recall that the dispersive estimates for the free wave in   read as follows:‖χ(hDt)e±itΔ(δa)‖L∞≤Ch−3min⁡(1,h|t|).

Main results

In the following, we prove dispersive estimates in cylindrical domains. The main results are obtained for appropriately localized Green function in each region corresponding to different values of η and we prove that the estimates are worse than that of the free case. To be more precise, the following theorem holds true.

Theorem 1.1

There exists C such that for every  , every  , every  , the following holds:
(2)‖χ(hDt)Ga(t,x,y,z)‖L∞≤Ch−3min⁡(1,(h|t|)1/2γ(t,h,sup⁡(x,a))), with γ(t,h,b)={(h|t|)1/2+b1/8h1/4ifb≥h2/3−ϵ, for any ϵ>0 small,h1/4+(h|t|)1/3ifb≤h1/2.

Idea of the proof

The main idea to prove the theorem is to construct a suitable local parametrix together with the Airy–Poisson summation formula. In this way, the parametrix is represented as a sum over eigenmodes, which is used to prove the estimates for  . On the other hand, the parametrix is represented as a sum over multiple reflections, which is used to prove the estimates for  .

Main ingredients

(1)
Airy function. Let  . The Airy function is defined by
Ai(−z)=12π∫Rei(s3/3−sz)ds. It satisfies the Airy equation  . We rewrite as(3)Ai(−z)=e−iπ/3Ai(e−iπ/3z)+eiπ/3Ai(eiπ/3z)=A+(z)+A−(z), where we have defined  . Notice that  . Moreover, we haveA−(z)A+(z)=ie−43iz3/2eiB(z3/2), where   for   and  .
(2)
Green function. Let
ek(x,η)=fk|η|1/3k1/6Ai(|η|2/3x−ωk), where   are constants such that   for every  . We have that   is represented microlocally near tangential directions by(4)χ(hDt)Ga(t,x,y,z)=14π2h2∑k≥1∫eih(yη+zζ)eith(η2+ζ2+ωkh2/3|η|4/3)1/2ek(x,η/h)ek(a,η/h)×χ0(η2+ζ2)χ1(ωkh2/3|η|4/3)dηdζ. Here  ,  ,   is supported in the neighborhood of 1 and  ,  ,   is supported in  ,   on  .
(3)
Airy–Poisson summation formula. For  , set
L(ω)=π+ilog⁡(A−(ω)A+(ω)). The function L is analytic, strictly increasing and from the asymptotic expansion of   we have the basic properties of the function L . It satisfiesL(0)=π/3,limω→−∞⁡L(ω)=0,L(ω)∼ω→+∞43ω3/2, and for all  , we haveL(ωk)=2πk⇔Ai(−ωk)=0,L′(ωk)=2π∫0∞Ai2(x−ωk)dx.
Lemma 2.1

The following equality holds true in  ,
∑N∈Ze−iNL(ω)=2π∑k∈N⁎1L′(ωk)δω=ωk.


(4)
Oscillatory integrals. To evaluate various oscillatory integrals, we use the following lemma.
Lemma 2.2

Let  , and let   be a classical symbol of degree 0 in   with   for  . Let  ,   and   be a phase function such that
∑2≤j≤k|Φ(j)(ξ)|≥c0,ξ∈K. Then there exists C such that |∫eiλΦ(ξ)a(ξ,λ)dξ|≤Cλ−1/k,∀λ≥1. Moreover, the constant C depends only on   and on an upper bound of a finite number of derivatives of order ≥2 of Φ, a in a neighborhood of K. In particular case  , the usual stationary phase holds [see [[3]]].


Outline of the proof

These ingredients play a crucial role in obtaining dispersive estimates ((2)). Our local parametrix is represented as a sum of Fourier Integral Operators in two different ways. On the one hand, it is given as in ((4)) as the sum of gallery modes. On the other hand, applying Lemma 2.1 to ((4)), we can write the parametrix as a sum over N corresponding to the number of reflections on the boundary as follows:
(5)Ga=∑N∈ZGa,N=∑N∈Z(−i)Na2(2π)5h4∫eihΦa,N,h|η|3χ0χ1dsdσdωdζdη, with the phase functionΦa,N,h(t,x,y,z;s,σ,ω,ζ,η)=yη+|η|zζ+|η|t(1+ζ2+aω)1/2+a3/2|η|(s33+s(X−ω)+σ33+σ(1−ω)−43Nω3/2+ha3/2|η|NB(ω3/2a3/2|η|/h)) which generates a Lagrangian submanifold parametrized by  . Then we can apply stationary phase method for ζ -integration and decompose  , where   is defined by introducing a cutoff function  ,  ,   on   in the integral ((5)). This   corresponds to the regime of swallowtails. Its counterpart   is defined by introducing   in ((5)). Then the following results hold.

Proposition 2.3

Let   and   be fixed. There exists C such that for all  , all  , all  , all  , all  , all  , the following holds:
|∑2≤N≤C0a−1/2Ga,N,1(t,x,y,z;h)|≤Ch−3(ht)1/2h1/3.

Proposition 2.4

Let   and   be fixed. There exists C such that for all  , all  , all  , all  , all  , all  , the following holds:
|∑2≤N≤C0a−1/2Ga,N,2(t,x,y,z;h)|≤Ch−3(ht)1/2a1/8h1/4.

Proposition 2.5

Let   and   be fixed. There exists C such that for all  , all  , all  , all  , all  , all  , the following holds:
|Ga,1(t,x,y,z;h)|≤Ch−3(ht)1/2((ht)1/2+a1/8h1/4).

Putting together above results yields the following theorem which implies Theorem 1.1.

Theorem 2.6

Let   and   be fixed. There exists C such that for all  , all  , all  , all  , all  , all  , the following holds:
|∑0≤N≤C0a−1/2Ga,N(t,x,y,z;h)|≤Ch−3(ht)1/2((ht)1/2+a1/8h1/4).

Next, we use ((4)) to obtain the dispersive estimates for  . To do that we first apply the stationary phase method for ζ -integration then for some values of k with   for appropriate choice of L , we use the following lemma.

Lemma 2.7

(Lemma 3.5 [[4]]) There exists   such that for  , the following holds:
supb∈R⁡(∑1≤k≤Lk−1/3Ai2(b−ωk))≤C0L1/3.

For the values of k with  , we use ((3)) together with the asymptotic expansion of   to obtain oscillatory integrals to which we apply Lemma 2.2. We have the following result.

Proposition 2.8

For small ε, there exists a constant C independent of  ,  ,  ,   and   such that the following holds true:
|∫eiλψk±,±σk±,±dη|≤C(hk)−2/3λ−1/3 where hλψk±,±(t,x,y,z,η)=yη+|η|t1−z˜2(1+γ)1/2±23|η|(γ−x)3/2±23|η|(γ−a)3/2, and   are symbols obtained from the Airy expansion. Here   with   and  . For  , the integral is   by integration by parts.


 This work was supported by the ERC project SCAPDE.

References

Blair M.D., Smith H.F., Sogge C.D. On Strichartz estimates for Schrödinger operators in compact manifolds with boundary Proc. Amer. Math. Soc. 2008 ;  130 : 247-256
Blair M.D., Smith H.F., Sogge C.D. Strichartz estimates for the wave equation on manifolds with boundary Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2009 ;  26 : 1817-1829 [cross-ref]
Hörmander L. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis  New York: Springer-Verlag (2003). 
Ivanovici O., Lebeau G., Planchon F. Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case Ann. Math. (2) 2014 ;  180 : 323-380 [cross-ref]
Ivanovici O., Lascar R., Lebeau G., Planchon F. Dispersion for the wave equation inside strictly convex domains II: the general casearXiv:1605.08800v12016



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