Stochastic perturbation and zero noise limit for scalar conservation laws

TL;DR

This paper introduces a new stochastic perturbation method for scalar conservation laws, demonstrating that as noise diminishes, solutions converge uniquely to the entropy solution, thus providing a noise-based selection criterion.

Contribution

It presents the first rigorous proof that stochastic perturbations can select the entropy solution in nonlinear hyperbolic conservation laws.

Findings

Proved well-posedness of the stochastically perturbed equation.

Established convergence of solutions as noise tends to zero.

Demonstrated the noise acts as a selection mechanism for entropy solutions.

Abstract

Scalar conservation laws sit at the intersection between being simple enough to study analytically, while being complex enough to exhibit a wide range of nonlinear phenomena. We introduce a novel stochastic perturbation of scalar conservation laws, inspired by mean field games. We prove well-posedness of the stochastically perturbed equation; prove that it converges as the noise parameter is sent to $0$; and that the limit is the unique entropy solution of the conservation law. Thus, the noise acts as a selection criterion for (deterministic) conservation laws. This is the first such result for nonlinear hyperbolic conservation laws.