Stochastic Scalar Conservation Laws on Moving Hypersurfaces

Abstract

We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an It\^{o} formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.