# Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

## TL;DR

This paper establishes well-posedness for scalar conservation laws on Riemannian manifolds driven by multiplicative Gaussian noise, proving existence, uniqueness, and stability of solutions using kinetic formulation and vanishing viscosity methods.

## Contribution

It extends the well-posedness theory of stochastic conservation laws to Riemannian manifolds, including existence, uniqueness, and a kinetic formulation for solutions.

## Key findings

- Existence of generalized kinetic solutions via vanishing viscosity.
- Uniqueness of solutions through an $L^1$ contraction principle.
- Equivalence of generalized and kinetic solutions on manifolds.

## Abstract

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ($L^1$ contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch (2007), who worked with Kruzkov-DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle (2010).