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

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.