Hilbert Expansion for Coulomb Collisional Kinetic Models

2022/06/30 by Ouyang, Zhimeng, Wu, Lei, Xiao, Qinghua
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.00126

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.

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• Hilbert expansion for kinetic equations with non-relativistic Coulomb collision
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• Hilbert Expansion for the Relativistic Landau Equation
• Classical limit of the relativistic Vlasov-Maxwell-Landau system
• Hilbert’s 6th problem: the relativistic version