Existence, uniqueness of the solution and convergence of nite volume approximations for hyperbolic scalar conservation laws with multiplicative noise

We are interested here in multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a general time and space dependent ux-function. We address simultaneously theoretical and numerical issues. More precisely we establish existence and uniqueness of a stochastic entropy solution together with the convergence of a nite volume scheme. The results proposed in this paper suppose more general uxes than the ones considered in the literature and the main novelty here is the use of the numerical approximation to get both the existence and the uniqueness of the solution. It constitutes an original approach, which moreover is expected to be an important step towards the establishment of error estimates. We also provide a L ∞ stability result on the stochastic entropy solution to complete this study