A kinetic Nash inequality and precise boundary behavior of the kinetic Fokker-Planck equation

TL;DR

This paper establishes a kinetic Nash inequality and uses it to analyze the boundary behavior of solutions to the kinetic Fokker-Planck equation, achieving sharp regularity results at the boundary.

Contribution

It introduces a new kinetic Nash inequality and applies it to determine precise boundary regularity for the kinetic Fokker-Planck equation.

Findings

Proved a kinetic Nash inequality for functions in kinetic Sobolev spaces.

Derived a new functional inequality with absorbing boundary conditions.

Established sharp regularity of solutions at the boundary and grazing set.

Abstract

In this paper, we prove a kinetic Nash type inequality and adapt it to a new functional inequality for functions in a kinetic Sobolev space with absorbing boundary conditions on the half-space. As an application, we address the boundary behavior of the kinetic Fokker-Planck equations in the half-space. Our main result is the sharp regularity of the solution at the absorbing boundary and grazing set.