During the program related to kinetic theory that I co-organize in Berkeley, I have been appointed as a Chancellor’s visiting professor by the department of Mathematics of the University of California in Berkeley.
I give lectures about regularity à la De Giorgi for elliptic, parabolic and kinetic equations. See this page.
I am one of the organizers of the semester-long program Kinetic Theory: Novel Statistical, Stochastic and Analytical Methods at SLmath in Berkeley.
Overview.
The focus of the proposed program is on so-called kinetic equations, describing the evolution of the of many-particle interacting systems. These models have the form of statistical flows, with their solutions being either a single or multiple point probability density functions or measures, supported in a space of attributes. The attributes are problem-dependent and can be molecular velocity, energy, opinion, wealth, and many others. The flow then predicts the evolution of the probability measure in time, position in space, and the interchanging of the particles’ states by the transition probability.
Probably the most classical kinetic equation is the Boltzmann equation which describes the evolutions of the phase-space density function for a dilute gas under binary molecular collisions. Other well-known classical kinetic models include the Landau equation, Vlasov equation for plasmas or other systems, Fokker-Planck equations or kinetic formulations of various macroscopic or hyperbolic systems.
In recent years, the successes of kinetic theory gave rise to an rapidly increasing variety of mathematical models beyond physics to applications in life sciences, social sciences, economy. Even more recently fascinating connections between kinetic theory and some aspects of data science have emerged.
Kinetic theory has strong and fascinating interactions with a large variety of other fields, including statistical mechanics, stochastic processes, dynamical systems…
The program will strive to give an overview of the novel mathematical tools used in kinetic theory through a broad range of classical and more recent applications.
I presented the result obtained with Luis Silvestre and Cédric Villani about the monotonicity of the Fisher information for the space-homogeneous Boltzmann equation during the Journées EDP (2025).
I wrote a short note (14 pages) related to this presentation. See HAL and arxiv.
With Amélie Loher, we wrote and uploaded to hal and arxiv a preprint entitled Conditional appearance of decay for the non-cutoff Boltzmann equation in a bounded domain.
Abstract: This work is concerned with the appearance of decay bounds in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff posed in a domain in the case of hard and moderately soft potentials. Such bounds are derived for general non-negative suitable weak subsolutions. These estimates hold true as long as mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The proof relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced by F. Golse, L. Silvestre and the first author (2023). To the best of our knowledge, the generation of arbitrary polynomial decay in the velocity variable for the Boltzmann equation without cutoff is new in the case of soft potentials, even for classical solutions.
With J. Guerand and C. Mouhot, we uploaded on HAL and arxiv a new preprint entitled Gehring’s Lemma for kinetic Fokker-Planck equations.
Abstract: In this article, we establish a « Gehring lemma » for a real function satisfying a reverse Hölder inequality on all « kinetic cylinders » contained in a large one: it asserts that the integrability degree of the function improves under such an assumption. The kinetic cylinders are derived from the non-commutative group of invariances of the Kolmogorov equation. Our contributions here are (1) the extension of Gehring’s Lemma to this kinetic (hypoelliptic) scaling used to generate the cylinders, (2) the localisation of the lemma in this hypoelliptic context (using ideas from the elliptic theory), (3) the streamlining of a short and quantitative proof. We then use this lemma to establish that the velocity gradient of weak solutions to linear kinetic equations of Fokker-Planck type with rough coefficients have Lebesgue integrability strictly greater than two, while the natural energy estimate merely ensures that it is square integrable. Our argument here is new but relies on Poincaré-type inequalities established in previous works.
With Luis Silvestre and Cédric Villani, we proved that the Fisher information decreases along the flow of the space-homogeneous Boltzmann equation for a large class of collision kernels including the most classical ones.
With F. Golse and Luis Silvestre, we uploaded a new paper entitled: Partial regularity in time for the space-homogeneous Boltzmann equation with very soft potentials. HAL | arxiv
Summary: We prove that the set of singular times for weak solutions of the homogeneous Boltzmann equation with very soft potentials constructed as in Villani (1998) has Hausdorff dimension at most |γ+2s|/2s with γ∈[−4s,−2s) and s∈(0,1).
With Nicolas Forcadel et Régis Monneau, we recently uploaded on HAL and arxiv two new preprints about Hamilton-Jacobi equations.
The first one is interested in the understanding of viscosity solutions for HJ equations posed on a domain. This theory allows one to easily construct weak solutions for boundary value problems. These weak viscosity solutions can be studied thanks to the relaxation operator introduced by J. Guerand. In this new work, a link with classical Godunov fluxes is exhibited. As an application, the classical Neumann and Dirichlet problems are discussed.
The second one is interested in uniqueness of viscosity solutions. We elaborate on a method introduced by P.-L. Lions and P. Souganidis to prove new comparison principles for HJ equations posed on a domain. We follow them by performing a blow analysis and by reducing to a 1D problem but we depart from their method by blowing up the sub-solution and the super-solution simultaneously.
With F. Golse and A. F. Vasseur, we uploaded a new preprint about local regularity estimates for the Landau equation with very soft potentials.
Reference. F. Golse, C. Imbert and A. F. Vasseur. Local regularity for the space homogeneous Landau equation with very soft potentials. hal | arxiv
We just posted online with Jessica Guerand a new preprint entitled Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations. It is available on HAL and on arxiv.
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.