Stochastic Cahn-Hilliard-Navier-Stokes equations with the dynamic boundary: Martingale weak solution, Markov selection

TL;DR

This paper proves the existence of global martingale weak solutions for stochastic 2D and 3D Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions, incorporating multiplicative noise and establishing Markov selection.

Contribution

It introduces a novel three-level approximation scheme to handle nonlinearity, dynamic boundary conditions, and stochastic effects, and establishes Markov selection for the system.

Findings

Existence of global martingale weak solutions in 2D and 3D.

Development of a three-level approximation scheme.

Proof of Markov selection for the stochastic system.

Abstract

The existence of global martingale weak solution for the 2D and 3D stochastic Cahn-Hilliard-Navier-Stokes equations driven by multiplicative noise in a smooth bounded domain is established. In particular, the system is supplied with the dynamic boundary condition which accounts for the interaction between the fluid components and the rigid walls. The proof is completed by a three-level approximate scheme combining a fixed point argument and the stochastic compactness argument, overcoming challenges from strong nonlinearity, dynamic boundary and random effect. Then, we prove the existence of an almost surely Markov selection to the associated martingale problem following the abstract framework established by F. Flandoli and M. Romito.