Regularity of the Vlasov-Poisson-Boltzmann System without angular cutoff

TL;DR

This paper proves that solutions to the non-cutoff Vlasov-Poisson-Boltzmann system with hard potential become smooth over time, establishing regularity and decay properties for plasma particles in three-dimensional space.

Contribution

It demonstrates the global existence and smoothing effect for the system with hard potential, a previously open problem, using advanced energy methods and pseudo-differential calculus.

Findings

Solutions become smooth for any positive time

Establishment of decay rates for solutions

Extension of regularity results to hard potential case

Abstract

In this paper we study the regularity of the non-cutoff Vlasov-Poisson-Boltzmann system for plasma particles of two species in the whole space $\mathbb{R}^3$ with hard potential. The existence of global-in-time nearby Maxwellian solutions is known for soft potential from [15]. However the smoothing effect of these solutions has been a challenging open problem. We establish the global existence and regularizing effect to the Cauchy problem for hard potential with large time decay. Hence, the solutions are smooth with respect to $(t,x,v)$ for any positive time $t>0$. This gives the regularity to Vlasov-Poisson-Boltzmann system, which enjoys a similar smoothing effect as Boltzmann equation. The proof is based on the time-weighted energy method building also upon the pseudo-differential calculus.