New preprint: Hölder continuity of solutions to quasilinear hypoelliptic equations (w/ C. Imbert)

http://arxiv.org/abs/1505.04608

We prove that $L^2$ weak solutions to a quasilinear hypoelliptic equations with bounded measurable coefficients are H\”older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and regularity transfers developed in kinetic theory. The latter tool is used repeatedly: first in the proof of the local gain of integrability of sub-solutions, second in proving that the gradient with respect to the speed variable is $L^{2+\epsilon} _{loc}$, third, in the proof of an “hypoelliptic isoperimetric De Giorgi lemma”. To get such a lemma, we develop a new method which combines the classical isoperimetric inequality on the diffusive variable with the structure of the integral curves of the first-order part of the operator. It also uses that the gradient of solutions w.r.t. v is $L^{2+\epsilon} _{loc}$.

Théorème vivant et amortissement Landau

Le livre Théorème vivant de Cédric Villani vient de sortir, voir l’annonce sur son site internet.

Ce livre suit l’élaboration de notre théorème sur l’amortissement Landau. Pour plus d’informations mathématiques sur ce travail, on pourra consulter

Kac’s Program in Kinetic Theory (new preprint)

I have just uploaded on arXiv a joint work with Stéphane Mischler on “Kac’s Program in Kinetic Theory”. In this paper, we answer a set of questions raised by Mark Kac in his seminal paper Foundations of kinetic theory (Proc. Third Berkeley Symp. Math. Stat. & Prob., 1956) about the derivation of Boltzmann equations from many-particle jump processes.

UPDATE: We have just uploaded an announcement note, to appear shortly in the Comptes-rendus de l’Académie des Sciences de Paris.