# Well-posedness of the non-local conservation law by stochastic perturbation

## TL;DR

This paper establishes the existence and uniqueness of strong solutions for stochastic non-local conservation laws with discontinuous flux functions, extending previous linear results to nonlinear cases using probabilistic methods.

## Contribution

It provides the first nonlinear extension of linear stochastic conservation laws, proving well-posedness in an $L^{1}igcap L^{2}$ setting with constructive proofs.

## Key findings

- Proved existence and uniqueness of weak solutions.
- Solutions are strong in the probabilistic sense.
- Extended linear results to nonlinear stochastic conservation laws.

## Abstract

Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an $L^{1}\cap L^{2}$ setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove existence and uniqueness of weak solutions. The solution is strong solution in the probabilistic sense. The proofs are constructive and based on the method of characteristics (in the presence of noise), It\^o-Wentzell-Kunita formula and commutators. Our results are new , to the best of our knowledge, and are the first nonlinear extension of the seminar paper [20] where the linear case was addressed.