The Schauder estimate in kinetic theory with application to a toy nonlinear model

TL;DR

This paper establishes Schauder estimates for hypoelliptic kinetic Fokker-Planck equations with H"older coefficients and applies these results to prove the global well-posedness of a nonlinear kinetic model with Maxwellian steady states.

Contribution

It provides the first Schauder estimate for linear hypoelliptic kinetic equations with H"older coefficients and uses it to analyze a nonlinear kinetic model.

Findings

Proved Schauder estimates for kinetic Fokker-Planck equations.

Established global well-posedness of a nonlinear kinetic model.

Identified Maxwellian steady states in the nonlinear model.

Abstract

This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with H\"older continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker-Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.