On a stochastic Cahn-Hilliard-Brinkman model
Z. Brzeźniak, A. Ndongmo Ngana, T. Tachim Medjo
TL;DR
The paper studies a stochastic Cahn-Hilliard-Brinkman system with dynamic bulk-surface boundary coupling on a bounded smooth domain, incorporating multiplicative Wiener noise in both bulk and boundary CH equations. It develops a double-approximation framework using Faedo-Galerkin discretization and Yosida regularization, derives sharp energy estimates via Itô calculus, and employs stochastic compactness to pass to the limit, yielding a global weak martingale solution for K>0 in dimensions two and three. While the model is not a pure gradient flow due to velocity coupling, it remains thermodynamically consistent with respect to the energy $E$, and the coupling is implemented through a Robin-type boundary condition with parameter $K$. The results extend deterministic existence theory for CHB with dynamic boundary conditions to the stochastic setting and lay the groundwork for future treatment of singular potentials and more general boundary dynamics.
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.