
New files
An energy method for averaging lemmas,
with Diogo Arsénio,
file on arXiv,
This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approachbased upon the classical energy methodprovides a powerful duality principle in kinetic transport equations which allows for a natural extension of classical averaging lemmas to previously unknown cases where the density and the source term belong to dual spaces. More generally, this kinetic duality principle produces regularity results where one can trade a loss of regularity or integrability somewhere in the kinetic transport equation for a suitable opposite gain elsewhere. Also, it looks simpler and more robust to rely on proving inequalities instead of constructing exact parametrices. The results in this article are introduced from a functional analytic point of view and they are motivated by the abstract regularity theory of kinetic transport equations.
Carleman Inequalities: an Introduction and More,
a new book, published in 2019
in the SpringerVerlag Series
Grundlehren der Mathematischen Wissenschaften
.
This is a 576page book, dealing with various aspects of Carleman inequalities.
On that
link, you will find a short description of the contents of the book as well as some previews opportunities.
On the antiWick symbol as a GelfandShilov generalized function,
with Laurent Amour and Jean Nourrigat,
file on arXiv,
to appear in the
Proceedings of the American Mathematical Society.
The purpose of this article is to prove that the antiWick symbol of an operator mapping
S(ℝ^{n}) into
S'(ℝ^{n}),
which is generally not a tempered distribution, can still be defined as a GelfandShilov generalized function.
This result relies on test function spaces embeddings involving the Schwartz and GelfandShilov spaces. An
additional embedding concerning Schwartz and Gevrey spaces is also given.
On integrals over a convex set of the Wigner distribution,
with Bérangère Delourme and Thomas Duyckaerts,
Journal of Fourier Analysis and Applications,
volume 26, February 2020.
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.
Mehler's formula and functional calculus,
Science China Mathematics, 62 (2019), no. 6, 11431166.
We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.
Unique continuation through transversal characteristic hypersurfaces,
Journal d'Analyse Mathématique
,138, (2019), no.1, 135156.
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.
