Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential

TL;DR

This paper studies how solutions of the Vlasov-Maxwell-Landau system with Coulomb potential approach rarefaction waves in the hydrodynamic limit, overcoming dissipation challenges with new energy estimates.

Contribution

It establishes the convergence to rarefaction waves for the VML system with Coulomb potential as the Knudsen number tends to zero, introducing novel energy methods.

Findings

Solution converges to rarefaction wave as pproaches zero

Develops velocity weight and space-time scaling techniques

Addresses dissipation loss in electromagnetic interactions

Abstract

In this paper, we investigate the hydrodynamic limit of rarefaction wave for the two-species Vlasov-Maxwell-Landau(VML) system with Coulomb potential. We prove that for any given time interval, the solution of the Vlasov-Maxwell-Landau system with appropriate initial data converges to a rarefaction wave as the Knudsen number $\epsilon$ approaches zero. The main difficulty in the analysis lies in the loss of dissipation in the interaction between the electromagnetic field and the microscopic component, and the weak dissipation induced by the Lorentz force and the scaling with small parameter $\epsilon$. For this, we introduce a velocity weight function and a space-time scaling parameter together with suitable $\epsilon$-dependent energy estimates.