Well-posedness for stochastic scalar conservation laws on Riemannian manifolds

TL;DR

This paper establishes well-posedness for stochastic scalar conservation laws on compact Riemannian manifolds by formulating kinetic solutions and deriving admissibility conditions, extending classical results to curved geometric settings.

Contribution

It introduces a kinetic formulation and admissibility conditions for stochastic conservation laws on manifolds, proving well-posedness in this geometric context.

Findings

Proved existence and uniqueness of solutions.

Extended kinetic formulation to Riemannian manifolds.

Established well-posedness under new admissibility conditions.

Abstract

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.