An Abstract Framework for Nonlinear Stochastic Parabolic Systems with Application to a Chemotaxis-Fluid Model

Abstract

The purpose of the paper is twofold. Firstly, we want to present an abstract existence Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic partial differential equations. The idea is first to split the system by rewriting the nonlinear part in a linear part acting on a given process. This is done in such a way that the fixed point with respect to the given process would be the solution of the original Stochastic Partial Differential Equation (SPDE). However, it may be necessary to show the well-posedness of the linearized system using a cut-off argument, which may impose additional constraints. We work out under which conditions on the underlying spaces and the cut-off argument one gets a martingale solution of the original SPDE. A similar strategy was applied by Hausenblas and Tolle in [E. Hausenblas and J. Tolle. The stochastic Klausmeier system and a schauder tychonoff type theorem. Potential Analusis, 310:506-554, 2022] to investigate the stochastic Klausmeier system. Secondly, we want to verify the full applicability of the existence Theorem by showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamics.

arXiv comment53 pages