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
function u such
that its Wigner distribution W(u,u) has an integral >1 on the
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
along with a precise numerical analysis of its
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 ill-posed 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 model-result using only geometric assumptions.
The main tools are Carleman estimates and Hörmander's pseudo-convexity conditions.
Some natural subspaces and quotient spaces of L1,
with Gilles Godefroy,
Advances in Operator Theory,
3 (2018), no.1, 73-86,
The onset of instability in first-order systems,
with Toan T. Nguyen and Benjamin Texier,
Journal of the European Mathematical Society,
20, 6, 1303-1373, (2018),
paper on the JEMS website.
We study in this paper the Cauchy problem for first-order quasi-linear 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 Lax-Mizohata theorem, was explained by G. Métivier in
Remarks on the well-posedness of the nonlinear Cauchy problem,
(Contemp. Math. 2005). In this article,
we are interested in the transition from hyperbolicity to non-hyperbolicity, 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 Cauchy-Kovalevskaya solution for a class of non-linear 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 Klein-Gordon-Zakharov systems. Our analysis relies on an approximation result for pseudo-differential flows, introduced by B. Texier in
Approximations of pseudo-differential 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. 157-205,
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 semi-classical analysis localization.
Using a multiplier method, we obtain the energy estimates in each region
and we then patch the microlocal estimates together.