The incompressible Navier-Stokes-Fourier-Maxwell system limits of the Vlasov-Maxwell-Boltzmann system for soft potentials: the noncutoff cases and cutoff cases

TL;DR

This paper establishes uniform energy estimates for the scaled Vlasov-Maxwell-Boltzmann system with soft potentials, both cutoff and non-cutoff, and justifies the incompressible Navier-Stokes-Fourier-Maxwell equations with Ohm's law as a limit.

Contribution

It provides the first rigorous derivation of the incompressible Navier-Stokes-Fourier-Maxwell equations from the Vlasov-Maxwell-Boltzmann system for soft potentials, including non-cutoff cases.

Findings

Uniform energy estimates for the system with soft potentials.

Justification of the incompressible Navier-Stokes-Fourier-Maxwell equations as a limit.

Introduction of new weight functions and energy constructions for cutoff cases.

Abstract

We obtain the global-in-time and uniform in Knudsen number $\epsilon$ energy estimate for the cutoff and non-cutoff scaled Vlasov-Maxwell-Boltzmann system for the soft potential. For the non-cutoff soft potential cases, our analysis relies heavily on additional dissipative mechanisms with respect to velocity, which are brought about by the strong angular singularity hypothesis, i.e. $\frac12\leq s<1$. In the case of cutoff cases, our proof relies on two new kinds of weight functions and complex construction of energy functions, and here we ask $\gamma\geq-1$. As a consequence, we justify the incompressible Navier-Stokes-Fourier-Maxwell equations with Ohm's law limit.