The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

Abstract

In this article, we study the stability of weak solutions to a stochastic version of a coupled Cahn-Hilliard-Navier-Stokes model with multiplicative noise. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.

Existence of a solution to the stochastic nonlocal Cahn–Hilliard Navier–Stokes model via a splitting-up method

“…The model (1.1) is obtained by the suitable coupling between the nonlocal Cahn-Hilliard equation through the transport term u • ∇φ and the stochastic Navier-Stokes model through the capillarity term (or the Korteweg force) μ∇φ with a multiplicative infinite-dimensional Gaussian type noise. Adding a stochastic force in the equation for the relative concentration in (1.1) is possible and the corresponding local version has been studied by some authors such as [13,15,40]. But, the presence of this noise in (1.1) will involve tedious calculations and will increase significantly the size of the paper.…”