On $C^2$ solution of the free-transport equation in a disk

TL;DR

This paper investigates the regularity of solutions to the free transport equation in a disk with specular boundary conditions, establishing compatibility conditions and regularity estimates for solutions with high smoothness.

Contribution

It provides the first detailed analysis of $C^2$ regularity for solutions to the free transport equation in a bounded domain with boundary conditions, including compatibility conditions and estimates.

Findings

Derived initial-boundary compatibility conditions for $C^1$ and $C^2$ solutions.

Established regularity estimates for solutions with high smoothness.

Demonstrated the feasibility of obtaining $C^2$ regularity in this setting.

Abstract

The free transport operator of probability density function $f(t,x,v)$ is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular boundary condition. We obtain initial-boundary compatibility conditions for $C^1_{t,x,v}$ and $C^2_{t,x,v}$ regularity of the solution. We also provide regularity estimates.