Obstacle problem for a stochastic conservation law and Lewy Stampacchia inequality

Abstract

Our interest in this paper evolves around rigorous mathematical formalisation of obstacle problem for stochastic conservation laws. We formulate a stochastic entropy solution framework for a bilateral obstacle problem and establish well-posedness. Our method primarily relies on a combination of techniques involving penalization and regularisation by viscous/parabolic perturbation. In tandem with Kruzkhov's doubling of the variables method, this combination manifests itself via Lagrange multipliers and we are able to obtain existence and uniqueness of entropy solution.