Diffusive Limit of the Vlasov-Maxwell-Boltzmann System without Angular Cutoff

TL;DR

This paper proves the diffusive limit of the non-cutoff Vlasov-Maxwell-Boltzmann system, establishing global solutions and hydrodynamic limits for a broad range of potentials and angular singularities using novel analytical techniques.

Contribution

It introduces a new weight function and leverages anisotropic dissipation to handle non-cutoff potentials, extending the range of parameters for global solutions and hydrodynamic limits.

Findings

Established uniform estimates with respect to the Knudsen number.

Proved global existence of solutions for the non-cutoff Vlasov-Maxwell-Boltzmann system.

Derived the hydrodynamic limit to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system.

Abstract

Diffusive limit of the non-cutoff Vlasov-Maxwell-Boltzmann system in perturbation framework still remains open. By employing a new weight function and making full use of the anisotropic dissipation property of the non-cutoff linearized Boltzmann operator, we solve this problem with some novel treatments for non-cutoff potentials $\gamma > \max\{-3, -\frac{3}{2}-2s\}$, including both strong angular singularity $\frac{1}{2} \leq s <1$ and weak angular singularity $0 < s < \frac{1}{2}$. Uniform estimate with respect to the Knudsen number $\varepsilon\in (0,1]$ is established globally in time, which eventually leads to the global existence of solutions to the non-cutoff Vlasov-Maxwell-Boltzmann system as well as hydrodynamic limit to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law. The indicators $\gamma > \max\{-3, -\frac{3}{2}-2s\}$ and $0 < s <1$ in this paper cover all ranges that can be achieved by the previously established global solutions to the non-cutoff Vlasov-Maxwell-Boltzmann system in perturbation framework.