Regularity estimates and open problems in kinetic equations

TL;DR

This paper surveys recent regularity estimates for the Boltzmann and Landau equations, discusses open problems, and explores ideas related to the global well-posedness of the Coulomb potential case.

Contribution

It introduces new a priori regularity estimates conditional on macroscopic boundedness and discusses open problems, including approaches to the Coulomb potential Landau equation.

Findings

New regularity estimates for Boltzmann and Landau equations

Identification of open problems in kinetic equations

Ideas for global well-posedness of Coulomb potentials

Abstract

We survey some new results regarding a priori regularity estimates for the Boltzmann and Landau equations conditional to the boundedness of the associated macroscopic quantities. We also discuss some open problems in the area. In particular, we describe some ideas related to the global well posedness problem for the space-homogeneous Landau equation for Coulomb potentials.