Local Well-posedness of Vlasov-Poisson-Boltzmann Equation with Generalized Diffuse Boundary Condition

TL;DR

This paper establishes local well-posedness for the Vlasov-Poisson-Boltzmann equation in convex domains with generalized Cercignani-Lampis boundary conditions, using novel estimates and an iteration scheme.

Contribution

It introduces a new iteration scheme and boundary integral decomposition to prove local existence and uniqueness of solutions under generalized diffuse boundary conditions.

Findings

Established local existence and uniqueness of solutions.

Developed a new iteration scheme along characteristics.

Created an intrinsic boundary integral decomposition.

Abstract

The Vlasov-Poisson-Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We consider the system in a convex domain with the Cercignani-Lampis boundary condition. We construct a uniqueness local-in-time solution based on an $L^\infty$-estimate and $W^{1,p}$-estimate. In particular, we develop a new iteration scheme along the characteristic with the Cercignani-Lampis boundary for the $L^\infty$-estimate, and an intrinsic decomposition of boundary integral for $W^{1,p}$-estimate.