The Strong Trace Property and the Neumann Problem for Stochastic Conservation Laws

TL;DR

This paper proves the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise, introducing a new strong trace property and analyzing stochastic parabolic problems for the first time.

Contribution

It establishes the strong trace property for stochastic conservation laws and proves existence and uniqueness of solutions using vanishing viscosity and detailed stochastic analysis.

Findings

Well-posedness of the Neumann problem established

New strong trace property for stochastic conservation laws

Existence of kinetic solutions proved via vanishing viscosity

Abstract

We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a major step for establishing the uniqueness of the kinetic solution to the referred problem we establish the new strong trace property for stochastic conservation laws. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.