
New files
On integrals over a convex set of the Wigner distribution,
with Bérangère Delourme and Thomas Duyckaerts,
file on arXiv.
We provide an example of a normalized
L^{2}(ℝ)
function u such
that its Wigner distribution W(u,u) has an integral >1 on the
square [0,a]×[0,a]
for a suitable choice of a. This provides a
negative answer to a question raised by P. Flandrin in 1988. Our arguments are
based upon the study of the Weyl quantization of the indicatrix of
ℝ_{+}×ℝ_{+}
along with a precise numerical analysis of its
discretization.
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.
Some natural subspaces and quotient spaces of L^{1},
with Gilles Godefroy,
Advances in Operator Theory,
3 (2018), no.1, 7386,
paper online.
The onset of instability in firstorder systems,
with Toan T. Nguyen and Benjamin Texier,
Journal of the European Mathematical Society,
20, 6, 13031373, (2018),
paper on the JEMS website.
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.
Energy decay for a locally undamped wave equation,
with Matthieu Léautaud,
Annales de la Faculté des Sciences de Toulouse,
Sér. 6, 26 no. 1 (2017), p. 157205,
paper online.
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.
