From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions

TL;DR

This paper rigorously justifies the convergence of classical solutions of the two-species Vlasov-Maxwell-Boltzmann system to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law, establishing uniform estimates and global existence.

Contribution

It provides the first classical solution framework for the convergence from VMB to NSFM systems under the specified scaling, complementing recent results on dissipative solutions.

Findings

Established uniform estimates for fluctuations with respect to Knudsen number

Proved existence of global classical solutions for all small epsilon

Rigorously justified the convergence to the NSFM system with Ohm's law

Abstract

For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigorously justified. This limit was justified in the recent breakthrough of Ars\'enio and Saint-Raymond \cite{Arsenio-SRM-2016} from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the corresponding scaling. In this sense, our result gives a classical solution analogue of the corresponding limit in \cite{Arsenio-SRM-2016}.