A stochastic conservation law with nonhomogeneous Dirichlet boundary conditions

TL;DR

This paper develops a kinetic formulation approach to establish existence and uniqueness for a stochastic conservation law with nonhomogeneous Dirichlet boundary conditions, using stochastic parabolic approximations.

Contribution

It introduces a novel kinetic formulation with truncated boundary defect measures for stochastic conservation laws with boundary conditions.

Findings

Proves existence and uniqueness of solutions.

Shows solutions are limits of stochastic parabolic approximations.

Provides a framework for boundary value problems with noise.

Abstract

This paper discusses the initial-boundary value problem (with a nonhomogeneous boundary condition) for a multi-dimensional scalar first-order conservation law with a multiplicative noise. One introduces a notion of kinetic formulations in which the kinetic defect measures on the boundary of a domain are truncated. In such a kinetic formulation one obtains a result of uniqueness and existence. The unique solution is the limit of the solution of the stochastic parabolic approximation.