Global strong solutions to the Vlasov-Poisson-Boltzmann system with soft potential in a bounded domain

TL;DR

This paper proves the existence and regularity of strong solutions to the Vlasov-Poisson-Boltzmann system with soft potential in a bounded domain, using new energy estimates and bootstrap methods.

Contribution

It introduces a novel weighted energy estimate framework in specific function spaces for soft potential cases, extending local solutions to global in time.

Findings

Existence of strong solutions in bounded convex domains.

Extension of solutions from small to large time scales.

New weighted energy estimates for soft potential.

Abstract

Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study the existence and regularity of solutions to the VPB system with soft potential in a bounded convex domain with in-flow boundary condition. We establish the existence of strong solutions in the time interval $[0,T]$ for an arbitrary given $T>0$ when the initial distribution function is near an absolute Maxwellian. Our contribution is based on a new weighted energy estimate in some $W^{1,p}$ space and $L_x^3 L_v^{1+}$ space for soft potential. By using the classical $L^2$--$L^\infty$ method and bootstrap argument, we extend the local solutions from small time scale to large time scale.