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

TL;DR

This paper proves the diffusive limit and global existence of solutions for the Vlasov-Poisson-Boltzmann system without angular cutoff, covering the full range of non-cutoff potentials, and establishes the hydrodynamic limit to a two-fluid Navier-Stokes-Fourier-Poisson system.

Contribution

It introduces a weighted energy method with a novel weight function to handle the full range of non-cutoff potentials, solving an open problem in the field.

Findings

Established uniform estimates with respect to the Knudsen number

Proved global existence of solutions for the full range of non-cutoff potentials

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

Abstract

Diffusive limit of the Vlasov-Poisson-Boltzmann system without angular cutoff in the framework of perturbation around global Maxwellian still remains open. By employing the weighted energy method with a newly introduced weight function $w_l(\alpha,\beta)$ and some novel treatments, we solve this problem for the full range of non-cutoff potentials $\gamma>-3$ and $0<s<1$. 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 Vlasov-Poisson-Boltzmann system without angular cutoff for the full range of non-cutoff potentials and hydrodynamic limit to the two-fluid incompressible Navier-Stokes-Fourier-Poisson system with Ohm's law. As a byproduct, this approach also extends the global existence results of previous studies on the Vlasov-Poisson-Boltzmann system without angular cutoff to the full range of non-cutoff potentials $\gamma>-3$ and $0<s<1$.