Well-posedness for stochastic scalar conservation laws on Riemannian manifolds

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