The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness

TL;DR

This paper proves the local existence and uniqueness of strong solutions for the Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems under stochastic electromagnetic fields, highlighting regularization effects and addressing stochastic transport complexities.

Contribution

It provides the first rigorous mathematical analysis of strong solutions to nonlinear stochastic kinetic equations with external electromagnetic forcing.

Findings

Solutions are unique and locally well-posed in Sobolev spaces.

Solutions to VPFP become instantly smooth due to hypoelliptic regularization.

The study addresses the complexity of stochastic transport in kinetic equations.

Abstract

In this paper, we construct unique, local-in-time strong solutions to the Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected to external, spatially regular, white-in-time electromagnetic fields in $\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^\sigma$ with $\sigma > d/2 + 1$ (in addition to polynomial velocity weights). We additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due to hypoelliptic regularization if the external force fields are smooth. The external forcing arises in the kinetic equation as a stochastic transport in velocity, which means, together with the anisotropy between $x$ and $v$ in the nonlinearity, that the local theory is a little more complicated than comparable fluid mechanics equations subjected to either additive stochastic forcing or stochastic transport. Although stochastic electromagnetic fields are often discussed in the plasma physics literature, to our knowledge, this is the first mathematical study of strong solutions to nonlinear stochastic kinetic equations.