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G. Dhariwal, F. Huber, A. Jüngel, C. Kuehn, A. Neamtu:
"Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method";
in: "ASC Report 27/2019", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2019, ISBN: 978-3-902627-12-4, 1 - 33.

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 Brzezniak and coworkers.Three-species Maxwell-Stefan systems and n-species biofilm models are examples that satisfy the general assumptions.

Cross diffusion, martingale solutions, entropy method, tightness, Skorokhod- Jakubowski theorem, Maxwell-Stefan systems, biofilm model

http://www.asc.tuwien.ac.at/preprint/2019/asc27x2019.pdf