Stochastic fractional conservation laws

TL;DR

This paper studies stochastic fractional conservation laws, establishing well-posedness, long-term behavior, and invariant measures for solutions driven by multiplicative noise, using kinetic solutions and the vanishing viscosity method.

Contribution

It introduces a framework for analyzing stochastic fractional conservation laws, proving existence, uniqueness, and ergodicity of solutions with new techniques.

Findings

Existence of kinetic solutions via vanishing viscosity

Existence of invariant measures under certain conditions

Uniqueness and ergodicity of the invariant measure

Abstract

In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws driven by a multiplicative noise. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.