
New files
Unique continuation through transversal characteristic hypersurfaces,
file on arXiv,
to appear in the
Journal d'Analyse Mathématique.
We prove a unique continuation result for an illposed characteristic problem.
A model problem of this type occurs in A.D. Ionescu & S. Klainerman article
(Theorem 1.1 in [MR2470908]) and we extend their modelresult using only geometric assumptions.
The main tools are Carleman estimates and Hörmander's pseudoconvexity conditions.
Energy decay for a locally undamped wave equation,
with Matthieu Léautaud,
file on arXiv,
to appear in the
Annales de la Faculté des Sciences de Toulouse.
We study the decay rate for the energy of solutions of a damped wave equation in a situation where the
Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions.
We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.
Our method relies on a refined microlocal analysis linked to a second microlocalization procedure
to cut the phase space into tiny regions respecting the uncertainty principle
but way too small to enter a standard semiclassical analysis localization.
Using a multiplier method, we obtain the energy estimates in each region
and we then patch the microlocal estimates together.
The onset of instability in firstorder systems,
with Toan T. Nguyen and Benjamin Texier,
file on arXiv,
to appear
in the
Journal of the European Mathematical Society.
We study in this paper the Cauchy problem for firstorder quasilinear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., when hyperbolicity is violated at initial time, then the Cauchy problem is strongly unstable, in the sense of Hadamard. This phenomenon, which extends the linear LaxMizohata theorem, was explained by G. Métivier in
Remarks on the wellposedness of the nonlinear Cauchy problem,
(Contemp. Math. 2005). In this article,
we are interested in the transition from hyperbolicity to nonhyperbolicity, that is the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under such an hypothesis, we generalize a recent work by
N. Lerner, Y. Morimoto and C.J. Xu,
Instability of the CauchyKovalevskaya solution for a class of nonlinear systems,
(American J. Math. 2010), on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and KleinGordonZakharov systems. Our analysis relies on an approximation result for pseudodifferential flows, introduced by B. Texier in
Approximations of pseudodifferential flows,
file on arXiv.
GelfandShilov and Gevrey smoothing effect for the spatially inhomogeneous noncutoff Kac equation,
with Yoshinori Morimoto, Karel PravdaStarov and ChaoJiang Xu,
published by the
Journal of Functional Analysis,
Volume 269, Issue 2, 15 July 2015, pages 459535,
link to the file.
We consider the spatially inhomogeneous noncutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys GelfandShilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
GelfandShilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff,
with Yoshinori Morimoto, Karel PravdaStarov and ChaoJiang Xu,
Journal of Differential Equations,
256 (2014), no. 2, 797831.
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous noncutoff Boltzmann equation with Maxwellian molecules enjoys the same GelfandShilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.
Carleman estimates for anisotropic elliptic operators with jumps at an interface,
file on arxiv,
with Jérôme Le Rousseau,
Analysis & PDE
67 (2013), 16011648.
We consider a secondorder selfadjoint elliptic operator with an
anisotropic diffusion matrix having a jump across a smooth
hypersurface. We prove the existence of a weightfunction such that
a Carleman estimate holds true. We moreover prove that the
conditions imposed on the weight function are necessary.
Fonctions classiques, Cours de master 1.
Eléments d'analyse pour le M2, Cours introductif de master 2.
