Hydrodynamic Limit of the Incompressible Navier–Stokes–Fourier–Maxwell System with Ohm’s Law from the Vlasov–Maxwell–Boltzmann System: Hilbert Expansion Approach

Abstract

We obtain the global-in-time and uniform in Knudsen number ε 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. 1 2 ≤ 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 γ ≥ −1. As a consequence, we justify the incompressible Navier-Stokes-Fourier-Maxwell equations with Ohm's law limit.