Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws

TL;DR

This paper investigates the long-term behavior of stochastic scalar conservation laws, showing solutions converge to a unique invariant measure and establishing new convergence rates, including regularization effects induced by noise.

Contribution

It introduces a new regularization result for stochastic scalar conservation laws, demonstrating convergence to invariant measures and novel rates even in higher dimensions.

Findings

Solutions converge to their spatial average

Established a rate of convergence for the solutions

Proved regularization effects due to noise

Abstract

We study the long-time behavior and the regularity of pathwise entropy solutions to stochastic scalar conservation laws with random in time spatially homogeneous fluxes and periodic initial data. We prove that the solutions converge to their spatial average, which is the unique invariant measure, and provide a rate of convergence, the latter being new even in the deterministic case for dimensions higher than two. The main tool is a new regularization result in the spirit of averaging lemmata for scalar conservation laws, which, in particular, implies a regularization by noise-type result for pathwise quasi-solutions.