Long-Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

We study the long-time behavior and regularity of the 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 of the associated random dynamical system, and provide a rate of convergence, the latter being new even in the deterministic case for dimensions higher than 2. 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.(c) 2016 Wiley Periodicals, Inc.