Entropy dissipation estimates for the relativistic Landau equation, and applications

TL;DR

This paper analyzes the relativistic Landau equation with Coulomb interactions, providing entropy dissipation estimates, moment propagation, and existence results for weak solutions, advancing mathematical understanding of this complex physical model.

Contribution

It offers a detailed decomposition of the relativistic collision operator and establishes foundational analytical results for the equation's solutions.

Findings

Proved global entropy dissipation estimate.

Established propagation of polynomial moments.

Proved existence of weak solutions for broad initial data.

Abstract

In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.