Average Regularity of the Solution to an Equation with the Relativistic-free Transport Operator

TL;DR

This paper investigates the regularity properties of the velocity-averaged solution to a relativistic transport equation, demonstrating fractional Sobolev space regularity which aids in solving the relativistic Boltzmann equation.

Contribution

It extends the regularity analysis of velocity averages to the relativistic case using methods from Golse et al., providing new insights for relativistic kinetic equations.

Findings

Established fractional Sobolev regularity of velocity averages

Applied methods to relativistic transport operator

Facilitated existence proofs for relativistic Boltzmann solutions

Abstract

Let $u=u(t,{\bf x},{\bf p})$ satisfy the transport equation $\frac {\partial u}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial u}{\partial{\bf x}}=f$, where $f=f(t,\bf x,\bf p)$ belongs to $ L^{p}((0,T)\times {\bf R}^{3}\times {\bf R}^{3})$ for $1<p<\infty$ and $\frac {\partial}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial}{\partial{\bf x}}$ is the relativistic-free transport operator. We show the regularity of $\int_{{\bf R}^{3}}u(t, {\bf x}, {\bf p})d{\bf p}$ using the same method as given by Golse, Lions, Perthame and Sentis. This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation.