Obstacle problem for a stochastic conservation law and Lewy Stampacchia inequality

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. (c) 2023 Elsevier Inc. All rights reserved.