Stochastic conservation laws: weak-in-time formulation and strong entropy condition

TL;DR

This paper revisits stochastic conservation laws, proposing a new entropy formulation that is weak in both time and space, addressing limitations of previous strong-in-time formulations.

Contribution

It introduces a novel weak-in-time and weak-in-space entropy formulation for stochastic conservation laws, improving upon existing strong-in-time approaches.

Findings

Proposes a new entropy inequality formulation.

Addresses limitations of previous formulations.

Enhances theoretical understanding of stochastic conservation laws.

Abstract

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng $&$ Nualarthave developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of {\it strong entropy condition}. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a-priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.