Diffusive Limit of the Vlasov-Poisson-Boltzmann System for the Full Range of Cutoff Potentials

Abstract

Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff soft potentials $-3<\gamma <0$ in the perturbative framework around global Maxwellian still remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2,\infty }$ approach with time decay, we solve this problem for the full range of cutoff potentials $-3<\gamma \le 1$. The core of this approach lies in the interplay between the velocity weighted $H_{x,v}^2$ energy estimate with time decay and the time-velocity weighted $W_{x,v}^{2,\infty }$ estimate with time decay for the Vlasov-Poisson-Boltzmann system, which leads to the uniform estimate with respect to the Knudsen number $\epsilon \in (0,1]$ globally in time. As a result, global strong solution is constructed and incompressible Navier-Stokes-Fourier-Poisson limit is rigorously justified for both hard and soft potentials. Meanwhile, this uniform estimate with respect to $\epsilon \in (0,1]$ also yields optimal $L^2$ time decay rate and $L^{\infty }$ time decay rate for the Vlasov-Poisson-Boltzmann system and its incompressible Navier-Stokes-Fourier-Poisson limit. This newly introduced weighted $H_{x,v}^2$-$W_{x,v}^{2,\infty }$ approach with time decay is flexible and robust, as it can deal with both optimal time decay problems and hydrodynamic limit problems in a unified framework for the Boltzmann equation as well as the Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials. It is also expected to shed some light on the more challenging hydrodynamic limit of the Landau equation and the Vlasov-Poisson-Landau system.

59 pages