On a stochastic Cahn-Hilliard-Brinkman model

TL;DR

This paper introduces a stochastic version of the Cahn-Hilliard-Brinkman model for two-phase flows, proving the existence of weak solutions with multiplicative noise, and extends previous deterministic models to stochastic settings.

Contribution

It is the first to generalize the deterministic Cahn-Hilliard-Brinkman model to include stochastic effects with multiplicative Wiener noise.

Findings

Proved existence of weak solutions in stochastic setting

Developed a solution construction using Faedo-Galerkin and Yosida approximations

Extended deterministic models to stochastic frameworks

Abstract

In this paper, we consider a stochastic version of the Cahn-Hilliard-Brinkman model in a smooth two- or three-dimensional domain with dynamical boundary conditions. The system describes creeping two-phase flows and is basically a coupling of the Brinkman equation for the velocity field that governs the flow through the porous media coupled with convective Cahn-Hilliard equations for the phase field, both with two independent sources of randomness given by general multiplicative-type Wiener noises in the Cahn-Hilliard equations. The existence of a weak solution, both in the probabilistic and PDEs sense, is proved. Our construction of a solution is based on the classical Faedo-Galerkin approximation, the Yosida approximation and uses a compactness method. Our paper is the first attempt to generalize the paper \cite{Colli+Knopf+Schimperna+Signor_2024} to a stochastic setting.