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New preprint: Global regularity estimates for the Boltzmann equation

I wrote with Luis Silvestre (University of Chicago) a series of articles about the Boltzmann equation without cut-off in the inhomogeneous case. The goal was to prove C∞ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. The final paper is … Lire la suite

New preprint: partial regularity in time for the Landau-Coulomb equation

In collaboration with F. Golse, M. P. Gualdani and A. F. Vasseur. It can be found in the hal and also in arxiv. Abstract: We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), … Lire la suite

New preprint: Gaussian lower bounds for the Boltzmann equation

Together with C. Mouhot and L. Silvestre, we just uploaded on hal and arxiv a new preprint entitled Gaussian lower bounds for the Boltzmann equation. Abstract: The study of positivity of solutions to the Boltzmann equation goes back to Carleman (1933), and the initial argument of Carleman was developed byPulvirenti-Wennberg (1997), the second author and … Lire la suite

New preprint: The Schauder estimate for kinetic integral equations

A new preprint entitled « The Schauder estimate for kinetic integral equations » is available on arxiv and hal. This is a joint work with Luis Silvestre. Abstract: We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. We study equations of the form ft+v⋅∇xf=Lvf+c, where Lv is an integro-differential diffusion operator of order 2s acting … Lire la suite

New preprint: Decay estimates for large velocities in the Boltzmann equation without cut-off

C. Mouhot, L. Silvestre and myself just uploaded on HAL and arXiv a new preprint entitled Decay estimates for large velocities in the Boltzmann equation without cut-off. Abstract: We establish pointwise large velocity decay rates for the solution of the inhomogeneous Boltzmann equation without cutoff, under the assumption that three hydrodynamic quantities (mass, energy, entropy … Lire la suite

New preprint: A toy model in kinetic theory

I just uploaded on HAL and arxiv a new preprint entitled « A nonlinear toy model from kinetic theory », in collaboration with C. Mouhot (Cambridge). Abstract: This note is concerned with the study of a toy nonlinear model in kinetic theory. It consists in a non-linear kinetic Fokker-Planck equation whose diffusion in the velocity variable is … Lire la suite

New preprint: flux-limited solutions & classical viscosity solutions

We uploaded on hal with Guy Barles, Ariela Briani and Emmanuel Chasseigne a preprint entitled Flux-limited and classical viscosity solutions for regional control problems. Abstract: The aim of this paper is to compare two different approaches for regional control problems: the first one is the classical approach, using a standard notion of viscosity solutions, which … Lire la suite

New preprint about singular/degenerate parabolic equations

With Tianling Jin and Luis Silvestre, we uploaded on arxiv and hal a preprint entitled Hölder gradient estimates for a class of singular or degenerate parabolic equations. Abstract: We prove interior Hölder estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation ut=|∇u|κdiv(|∇u|{p−2}∇u), where p∈(1,∞) and κ∈(1−p,∞). This includes … Lire la suite

New preprint about the Boltzmann equation

We wrote with Luis Silvestre a paper entitled « Weak Harnack inequality for the Boltzmann equation without cut-off ». See arxiv and hal. Abstract: « In this paper, we obtain the weak Harnack inequality and H\ »older estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cut-off can be written in this … Lire la suite

New preprint about kinetic Fokker-Planck equations

I recently uploaded a new preprint on HAL and arxiv entitled « Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation ». It is written in collaboration with François Golse, Clément Mouhot and Alexis F. Vasseur. Abstract: We extend the De~Giorgi–Nash–Moser theory to a class of kinetic Fokker-Planck equations and deduce … Lire la suite