From two species Vlasov-Maxwell-Boltzmann system to magnetohydrodynamics system

TL;DR

This paper rigorously derives the magnetohydrodynamics system from the two-species Vlasov-Maxwell-Boltzmann equations, using hypocoercivity techniques and handling singular Lorentz terms to establish uniform estimates and the limiting process.

Contribution

It provides a formal derivation of magnetohydrodynamics from kinetic equations with new analytical techniques for singular terms.

Findings

Uniform estimates of solutions with respect to the Knudsen number

Successful derivation of MHD system from kinetic equations

Handling of singular Lorentz terms in the derivation process

Abstract

In this work, the magnetohydrodynamics system is formally derived from two species Vlasov-Maxwell-Boltzmann system. By employing the hypocoercivity of the linear Boltzmann operator and overcoming the difficulties resulting from the singular Lorentz term, we first obtain the uniform estimates of solutions with respect to the Knudsen number and then derive the magnetohydrodynamics system from the dimensionless Valsov-Maxwell-Boltzmann system.