# American Institute of Mathematical Sciences

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May  2019, 18(3): 1117-1138. doi: 10.3934/cpaa.2019054

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

 Department of Mathematics and Satistics, Florida International University, MMC, Miami, Florida 33199, USA

Received  January 2018 Revised  June 2018 Published  November 2018

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.

Citation: T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054
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