Abstract: This article deals with kinetic Fokker-Planck equations with essentially bounded coefficients. A weak Harnack inequality for non-negative super-solutions is derived by considering their Log-transform and following S. N. Kruzhkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.
The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an outstanding open problem. In the present article, we review a program focused on the type of particle interactions known as non-cutoff. It is dedicated to the derivation of a priori estimates in C∞, depending only on physically meaningful conditions. We prove that the solution will stay uniformly smooth provided that its mass, energy and entropy densities remain bounded, and away from vacuum.
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 entitled Global regularity estimates for the Boltzmann equation without cut-off.
It relies on the L∞ estimate derived by Luis Silvestre (see this paper), the local Hölder estimate derived in this paper, the Schauder estimate for kinetic equations with integral diffusion derived in this paper and the pointwise decay estimates for large velocities derived in this paper with Clément Mouhot and Luis Silvestre.
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), 273-307] has Hausdorff dimension at most 1/2.
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 Briant (2015). The appearance of a lower bound with Gaussian decay had however remained an open question for long-range interactions (the so-called non-cutoff collision kernels). We answer this question and establish such Gaussian lower bound for solutions to the Boltzmann equation without cutoff, in the case of hard and moderately soft potentials, with spatial periodic conditions, and under the sole assumption that hydrodynamic quantities (local mass, local energy and local entropy density) remain bounded. The paper is mostly self-contained, apart from the uniform upper bound on the solution established by the third author (2016).
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 in the v-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of Lv, we obtain an a priori estimate for f in a properly scaled Hölder space.
Le projet de réforme des retraites et la loi pluriannuelle de programmation de la recherche (LPPR) sont largement rejetées par la communauté de l’enseignement supérieur et de la recherche.
Les personnels et étudiant.es de l’ENS sont mobilisés, notamment via un comité de mobilisation.