Global Regularity of the Vlasov-Poisson-Boltzmann System Near Maxwellian Without Angular Cutoff for Soft Potential

TL;DR

This paper proves the global regularity of the non-cutoff Vlasov-Poisson-Boltzmann system with soft potential in three dimensions, showing solutions become smooth for any positive time, extending previous results for hard potentials.

Contribution

It establishes the global regularity for the soft potential case of the VPB system without angular cutoff, completing the understanding of smoothing effects for all potential types.

Findings

Solutions are smooth for all positive times.

The proof uses time-weighted energy methods and pseudo-differential calculus.

Completes the theory for soft potential cases in VPB systems.

Abstract

We consider the non-cutoff Vlasov-Poisson-Boltzmann (VPB) system of two species with soft potential in the whole space $\mathbb{R}^3$ when an initial data is near Maxwellian. Continuing the work Deng [Comm. Math. Phys. 387, 1603-1654 (2021)] for hard potential case, we prove the global regularity of the Cauchy problem to VPB system for the case of soft potential in the whole space for the whole range $0<s<1$. This completes the smoothing effect to the Vlasov-Poisson-Boltzmann system, which shows that any classical solutions are smooth with respect to $(t,x,v)$ for any positive time $t>0$. The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.