High-Field Limit from a Stochastic BGK Model to a Scalar Conservation Law with Stochastic Forcing

TL;DR

This paper derives a scalar conservation law with stochastic forcing from a stochastic BGK model under high-field scaling, proving convergence to a kinetic formulation with a modified Maxwellian and establishing weak solutions satisfying entropy conditions.

Contribution

It introduces a new kinetic formulation with a modified Maxwellian and proves the existence of weak solutions for the stochastic conservation law.

Findings

Convergence of stochastic BGK model to a kinetic formulation

Existence of weak solutions satisfying entropy relations

Introduction of a modified Maxwellian in the kinetic limit

Abstract

We study the derivation of a scalar conservation law with stochastic forcing starting from a stochastic BGK model with a high-field scaling. We prove the convergence to a new kinetic formulation where appears a modified Maxwellian. We deduce from it the existence of a weak solution to the scalar conservation law with stochastic forcing. We establish that this solution satisfies some Krushkov-like entropy relations.