Hilbert Expansion for Coulomb Collisional Kinetic Models

TL;DR

This paper proves the validity of the Hilbert expansion for relativistic kinetic models, establishing their convergence to relativistic hydrodynamic limits as the Knudsen number approaches zero, thus solving a longstanding open problem.

Contribution

It introduces a new weighted energy method and rigorously confirms the hydrodynamic limits for relativistic Landau-type equations, a significant advancement in kinetic theory.

Findings

Established the relativistic Euler-Maxwell limit for r-VML system

Proved the relativistic Euler limit for r-LAN equation

Resolved the open problem of hydrodynamic limits for Landau-type equations

Abstract

The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic Landau equation (r-LAN) are fundamental models that describe the dynamics of an electron gas. In this paper, we introduce a novel weighted energy method and establish the validity of the Hilbert expansion for the r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This successfully resolves the long-standing open problem regarding the hydrodynamic limits of Landau-type equations.