Stochastic non-isotropic degenerate parabolic-hyperbolic equations

TL;DR

This paper introduces a new framework for analyzing degenerate parabolic-hyperbolic equations with stochastic elements, proving well-posedness, convergence to spatial average, and regularization effects, extending existing theories.

Contribution

It develops the concept of pathwise entropy solutions for a broad class of stochastic degenerate equations with non-isotropic nonlinearities and rough time dependence, including new regularization results.

Findings

Proved well-posedness of pathwise entropy solutions.

Showed convergence to spatial average for Brownian noise cases.

Established a new regularization result akin to averaging lemmata.

Abstract

We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of Brownian noise and periodic boundary conditions, we prove that the pathwise entropy solutions converge to their spatial average and provide an estimate on the rate of convergence. The third main result of the paper is a new regularization result in the spirit of averaging lemmata. This work extends both the framework of pathwise entropy solutions for stochastic scalar conservation laws introduced by Lions, Perthame and Souganidis and the analysis of the long time behavior of stochastic scalar conservation laws by the authors to a new class of equations.