Scalar conservation laws with stochastic forcing

TL;DR

This paper proves the well-posedness of stochastic scalar conservation laws with periodic boundary conditions, establishing existence, uniqueness, and solution characterization via kinetic formulation, and connecting it to stochastic parabolic approximations.

Contribution

It introduces a kinetic formulation approach for stochastic conservation laws and demonstrates their well-posedness through approximation by stochastic parabolic equations.

Findings

Unique solutions exist for the stochastic conservation law.

Solutions can be characterized by a kinetic formulation.

The kinetic formulation is obtained as a limit of stochastic parabolic approximations.

Abstract

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.