Obstacle problem for a stochastic conservation law and Lewy Stampacchia inequality

Lewy–Stampacchia’s inequality for a stochastic T-monotone obstacle problem

“…Concerning the well-posedness questions, many authors have been interested in the topic, without seeking to be exhaustive, let us mention [9] about the reflected solutions of parabolic SPDEs driven by a space time white noise on the spatial interval [0, 1], quasilinear stochastic PDEs with obstacle in [8] and [12] for stochastic variational inequalities. Besides the well posedness, a regularity properties of the reflected stochastic measure was studied in [27], where stochastic Lewy-Stampacchia's inequalities where proved in the case of stochastic T-monotone obstacle problems and then [4,24] for stochastic scalar conservation laws with constraint and pseudomonotone parabolic obstacle problems.…”