FUNDING

Research interests and publications
 Microlocal
analysis, nonlinear PDE, hyperbolic equations on manifolds with/without
boundary, dispersion, long time behavior of solutions, propagation et
reflection of singularities, diffraction phenomenon.
Preprints
 preprint ; with Fabrice
Planchon, Gilles Lebeau and Richard Lascar
 In this very recent work (started in 2012), 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
t^{1/4} 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 be a powerful tool to prove sharp propagation of singularities results which were out of reach until now.
 Dispersion estimates for wave and Schrödinger equations outside a strictly convex obstacles and counterexamples
 preprint ; in collaboration with
Gilles Lebeau
 Abstract: We consider the linear wave equation and the linear Schrödinger equation
outside a compact, strictly convex obstacle in R^d with smooth boundary. In dimension
d = 3 we show that the linear flow satisfies the dispersive estimates as in R3. For
d >3 and if the obstacle is a ball, we show that there exists points (near the PoissonArago
spot) where the dispersive estimates fail for both wave and Schrödinger quations.
 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 unexpected counterexamples.
 Strichartz estimates for the wave equation in
strictly convex domains : d=2
 preprint in collaboration with
Gilles Lebeau and Fabrice Planchon
 Abstract: We prove sharper Strichartz estimates than expected from the
optimal dispersion bounds. This follows from reducing the analysis to gliding
wave packets and taking
full advantage of the spacetime localization of caustics. Several
improvements on the parametrix construction are obtained along the way
and are of independent interest. Moreover, we extend the range of known
counterexamples by propagating carefully constructed Gaussian beams,
proving that our Strichartz estimates are sharp, up to logarithmic
losses, in some regions of phase space.
 Strichartz estimates for the wave equation in
strictly convex domains : d=3,4
 in collaboration with
Gilles Lebeau and Fabrice Planchon
Publications
 in collaboration with Gilles Lebeau
 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.
 to appear in Communications and PDE (2017)
 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
 in collaboration with Fabrice Planchon et Gilles Lebeau
 Publications de la SMF, Séminaires et Congrès, 30 (2017) pages 6979
 Abstract: In this note we establish Strichartz estimates with 1/6 loss for the wave equation inside a generic
stritcly convex domain of R^3. To do that, we need to focus on the
spacetime localization of caustics and show that they appear at
exceptional times so that, averaging by integration in time, can
produce better Strichartz estimates (recall that we have obtained optimal dispersive estimates with a loss of 1/4).
 Abstract: In this paper we show that inside a strictly convex model domain there is a loss compared to the dispersive estimates in the free case and this is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. This result is optimal.
 Journal of the European
Math.Soc. (JEMS),
volume 14, issue 5 (2012), pages 13571388
 a detailed version (59 pages) is available here.
 Mathematische Annalen vol. 347 issue 3 (2010),
pages 627672
 Abstract: In these two
papers (dealing first with a model case, then with a general convex
domain) I showed that the (local in time) Strichartz estimates for the
wave equation suffer losses
when compared to the usual case R^d, at least for a subset of the usual range of admissible
indices (q; r). This result was striking since the whispering gallery modes, which seemed
to have the maximum amount of concentration and easily rule out the spectral projectors
estimate with a bound like in the flat case, do NOT rule out the Strichartz estimates : so
people felt that the latter should hold for all admissible exponents.
 Annales de l'IHP Analyse NonLinéaire vol.27.
no.5 (2010)
 Abstract: In this work we
prove a local wellposedness theory for the solution to the quintic
nonlinear Schrödinger equation outside nontrapping obstacles,
bypassing the absence of suitable
known Strichartz estimates (back then).
 Analysis and PDE vol.3 no.3 (2010), pages
261293
 Abstract: Outside a strictly convex obstacle, I obtained the full set of Strichartz estimates (except for
the endpoints) for solutions to the Schrödinger equation, an open question since Smith and
Sogge had obtained similar results for the wave in 1995.
 Asymptotic Analysis vol. 53 no. 4 (2007), pages
189208
 Abstract: We investigate
the smoothing effect properties of the solutions of the Schrödinger
equation outside one or more balls in R^3.
Proceedings
 Séminaire Laurent Schwartz (XEDP) (20132014)
 Oberwolfach Reports (2013)
PhD Thesis
 Thèse de Doctorat, Université Paris Sud,
Faculté des Sciences d'Orsay (2009)
 The final thesis report is available here.
