Well-posedness for stochastic scalar conservation laws on Riemannian manifolds

We consider the scalar conservation law with stochastic forcing $$\partial_t u +\mathrm{div}_g {\mathfrak f}(\mx,u)= \Phi(\mx,u) dW, \ \ {\bf x} \in M, \ \ t\geq 0$$ on a smooth compact Riemannian manifold $(M,g)$ where $W$ is the Wiener process and ${\bf x}\mapsto {\mathfrak f}(\mx,\xi)$ is a vector field on $M$ for each $\xi\in {\bf R}$. We introduce admissibility conditions, derive the kinetic formulation and use it to prove well posedness.