Uniqueness for stochastic scalar conservation laws on Riemannian manifolds revisited

Abstract

We revise a uniqueness question for the scalar conservation law with stochastic forcing$$d u +\mathrm{div}_g {\mathfrak f}(\mx,u) dt= \Phi(\mx,u) dW_t, \ \ {\bf x} \in M, \ \ t\geq 0$$ on a smooth compact Riemannian manifold $(M,g)$ where $W_t$ is the Wiener process and ${\bf x}\mapsto {\mathfrak f}(\mx,\xi)$ is a vector field on $M$ for each $\xi\in \R$. We introduce admissibility conditions, derive the kinetic formulation and use it to prove uniqueness in a more straight-forward way than in the existing literature.