Publications – ANADEL

My publications are available also here : ORCID, HAL, arXiv

New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain, with G. Lebeau and F. Planchon,

Journal de l’Ecole Polytechnique, to appear, (2021)

Abstract : We prove that the range of Strichartz estimates on a model 2D convex domain may be further restricted compared to the known counterexamples. Our new family ofcounterexamples is built on the parametrix construction from our recent paper “Dispersion for the wave equation inside strictly convex domains”. Interestingly enough, it is sharp in at least some regions of phase space.

Strichartz estimates for the wave equation on a 2D model convex domain, with G. Lebeau and F. Planchon, Journal of Differential Equations, Vol. 300 November (2021), 830-880

Abstract : We prove sharper Strichartz estimates than expected from the optimal dispersion bounds. This follows from taking full advantage of the space-time localization of caustics. Several improvements on the parametrix construction from our previous works are obtained along theway and are of independent interest.

Dispersive estimates for the semi-classical Schrödinger equation inside a strictly convex domain, Annales de l’IHP Analyse Non-Linéaire, to appear

Abstract : We consider a model case for a strictly convex domain of dimension d >1 with smooth boundary and we describe dispersion for the semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the optimal fixed time decay rate for the linear semi-classical flow : a loss of ( h/t )^{1/4} occurs with respect to the boundary less case due to repeated swallowtail type singularities. The result is optimal and implies corresponding Strichartz estimates.

Dispersive estimates inside the Friedlander model domain for the Klein-Gordon equation and the wave equation in large time,

Discrete and Continuous Dynamical Systems, to appear (2021)

Abstract : We prove global in time dispersion estimates for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay that the wave, unlike in the flat space.

Dispersion for the wave equation inside strictly convex domains II, with Richard Lascar, Gilles Lebeau and Fabrice Planchon

Annals of PDE (accepted)

Abstract : In this work, we consider the wave equation on a generic strictly convex domain of dimension at least two with smooth non empty boundary and with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a tˆ¼ loss with respect to the boundary less case. Moreover, we precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set,proving that the resulting decay is optimal. The new methods in this paper (compared to the Friedlander model case) provide a much deeper understanding of the complex propagation pattern near the boundary and extends the parametrix construction to the largest possible phase space region. This may have far reaching consequences, beyond pointwise bounds, as the parametrix will prove to bea powerful tool to prove sharp propagation of singularitiesresults which were out of reach until now.

Dispersion estimates for the wave and the Schrödinger equations outside a ball and counterexamples, preprint, with Gilles Lebeau

Abstract : We consider the wave and the Schrödinger equations with Dirichlet boundary conditions in the exterior of a ball in R^d. In dimension d=3 we construct a sharp, global in time parametrix and then proceed to obtain sharp dispersive estimates, matching the R^3 case, for all frequencies (low and high). If d ≥4 we provide an explicit solution to the wave equation localized at large frequency 1/h with data a Dirac mass at a point Q_0 at large distance s from the center of the ball : taking s~h^{-1/3}, the decay rate of that solution exhibits a (t/h)^{(d-3)/4} loss with respect to the boundary less case, that occurs at t~ 2s with an observation point being symmetric to Q_0 with respect to the center of the ball (at the Poisson Arago spot). A similar counterexample is also obtained for the Schrödinger flow.

The question about whether or not dispersion did hold outside general strictly convex obstacles was raised more than 20 years ago : in this work we give sharp answers which highlight the importance of diffractive effects, especially in higher dimensions where we provide unexpectedcounterexamples.

Dispersion for the wave flow outside a ball in R^d, with Gilles Lebeau

Comptes Rendus – Mathématique, 355 (2017), pages 774-779

Abstract: In this paper we prove dispersive estimates for the wave equation outside a ball in R^d. If d = 3, we show that the linear flow satisfies the dispersive estimates as in the free case. In higher dimensions d >3, we show that losses in dispersion do appear and this happens at the Poisson spot.

Square function and heat flow estimates on domains, with Fabrice Planchon

Comm. PDE (2017), 42(9): 1447-1466

Abstract: The first purpose of this note is to provide a proof of the usual square function estimate on L^p(M) (which follows from a generic Mikhlin multiplier theorem). We also relate such bounds to a weaker version of the square function estimate which is enough in most instances involving dispersive PDEs and relies on Gaussian bounds on the heat kernel. Moreover, we obtain several useful L^p(M;H) bounds for (the derivatives of) the heat flow with values in a Hilbert space H.

Estimations de Strichartz pour l’équation des ondes dans un domaine strictement convexe : le cas général, with F. Planchon and G. Lebeau,

Publications de la SMF, 30 (2017), pages 69-79

In this paper we establish Strichartz estimates with 1/6 loss for the wave equation inside a generic strictly convex domain of R^3. To do that, we need to focus on the space-time localisation of caustics and show that they appear at exceptional times so that, averaging by integration in time, can produce better Strichartz estimates (recall that dispersion holds with 1/4 loss).

Note : this is a chapter of the book “PDE’s, Dispersion, Scattering Theory and Control Theory”, ed. by K.Ammari and G.Lebeau.

Dispersion for the wave equation inside strictly convex domains I : the Friedlander model case, with Gilles Lebeau and Fabrice Planchon,

Annals of Mathematics, vol.180 issue 1 (2014), pages 323-380

We consider a model case for a strictly convex domain Ω ⊂ R^d of dimension d>1 with smooth, nonempty boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a t^(1/4) loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.

Counter-examples to the Strichartz estimates for the wave equation in general domains with boundary,

Journal of the European Math. Soc. (JEMS), vol. 14, issue 5 (2012), pages 1357-1388

A detailed version (59 pages) is available here.

Counter-examples to the Strichartz estimates for the wave equation in domains,

Mathematische Annalen vol 347 issue 3 (2010), pages 627-672

Abstract : in these two papers (dealing first with a model case, then with a general convex domain) we have shown that the (local in time) Strichartz estimates for the wave equation suffer losses when compared to the usual flat case R^d, at least for a subset of the usual range of admissible indices (q,r). This result was striking as the whispering gallery modes, which seemed to have the maximum amount of concentration and easily rule out the spectral projectors estimates with a bound like in the flat case, do NOT rule out the Strichartz estimates : so one could think that the latter should hold for all admissible exponents.

On the energy critical Schrödinger equation in non-trapping domains, with Fabrice Planchon,

Annales de l’IHP Analyse Non-Linéaire vol. 27. no.5 (2010)

Abstract : In this work we prove a local well-posedness theory for the solution to the quintic nonlinear Schrödinger equation outside non-trapping obstacles, bypassing the absence of suitable known Strichartz estimates.

On the Schrödinger equation outside strictly convex domains,

Analysis and PDE, vol. 3, no.3 (2010), pages 261-293

Abstract : Outside a strictly convex obstacle, we obtain the full set of Strichartz estimates (except for the end-points) for solutions to the Schrödinger equation, an open question since H.Smith and Ch.Sogge had obtained similar results for the wave equation ( in 1995 ).