In this paper, we consider a stochastic version of a nonlinear system which consists of the incompressible Navier–Stokes equations with shear dependent viscosity controlled by a power \(p>2\), coupled with a convective nonlocal Cahn–Hilliard-equations. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluid having the same density. We prove the existence of a weak martingale solutions when \(p\in [11/5,12/5)\), and their exponential decay when the time goes to infinity.

中文翻译：

---

具有剪切相关粘度的3D随机非局部Cahn-Hilliard-Navier-Stokes系统的ting解决方案的全球存在和长时间行为

在本文中，我们考虑了一个非线性系统的随机版本，该非线性系统由不可压缩的Navier–Stokes方程组成，其剪切强度由幂\（p> 2 \）控制，并具有对流非局部Cahn-Hilliard方程。这是一个扩散接口模型，描述了具有相同密度的两种（部分）不混溶流体的不可压缩等温混合物的运动。我们证明了当\（p \ in [11 / 5,12 / 5）\）时，存在mar弱解的存在，并且当时间到无穷远时，它们的指数衰减。