Étiquette : Boltzmann

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Preprint

Survey: Regularity for the Boltzmann equation conditional to macroscopic bounds

Together with Luis Silvestre (University of Chicago), we wrote a survey article about a series of works we recently completed about the Boltzmann equation. See hal and arxiv.

Two systems of coordinates to represent Boltzmann’s collision kernel. On the right hand side, the one introduced by Carleman in 1933.

Abstract

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.

Catégories
Preprint

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 entitled Global regularity estimates for the Boltzmann equation without cut-off.

Ludwig Boltzmann

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.

Catégories
Preprint

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 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).