Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method

TL;DR

This paper proves the existence of global bounded martingale solutions for a class of stochastic cross-diffusion systems with entropy structure, using a boundedness-by-entropy method and stochastic Galerkin approximations.

Contribution

It introduces a novel approach combining the boundedness-by-entropy method with stochastic Galerkin techniques to establish solutions for complex SPDEs with multiplicative noise.

Findings

Existence of positive lower and upper bounds for solutions.

Applicability to Maxwell--Stefan and biofilm models.

Method handles systems with volume-filling effects.

Abstract

The existence of global-in-time bounded martingale solutions to a general class of cross-diffusion systems with multiplicative Stratonovich noise is proved. The equations describe multicomponent systems from physics or biology with volume-filling effects and possess a formal gradient-flow or entropy structure. This structure allows for the derivation of almost surely positive lower and upper bounds for the stochastic processes. The existence result holds under some assumptions on the interplay between the entropy density and the multiplicative noise terms. The proof is based on a stochastic Galerkin method, a Wong--Zakai type approximation of the Wiener process, the boundedness-by-entropy method, and the tightness criterion of Brze\'{z}niak and coworkers. Three-species Maxwell--Stefan systems and $n$-species biofilm models are examples that satisfy the general assumptions.