ABSTRACT

We consider the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system with shear-dependent viscosity on a bounded domain $\mathcal{M}\subset {\mathbb{R}}^d$, d = 2, 3, driven by a multiplicative noise of Lévy and Gaussian types. The velocity u is governed by a Navier–Stokes system with a shear-dependent viscosity controlled by a power p>2. This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn–Hilliard equation for the order parameter φ. The existence of a global weak martingale solution is proved. In the 2D case, we prove the pathwise uniqueness of the weak solution, when $11/5<p<3$.

中文翻译：

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具有剪切相关粘度的随机 3D 非局部 Cahn–Hilliard–Navier–Stokes 系统的弱解

摘要

我们考虑在有界域上具有依赖于剪切的粘度的随机非局部 Cahn-Hilliard-Navier-Stokes 系统$\mathcal{米}\subset {\mathbb{R}}^d$, d  = 2, 3，由 Lévy 和高斯类型的乘法噪声驱动。速度u由 Navier-Stokes 系统控制，该系统具有由功率p > 2控制的剪切相关粘度。该系统通过 Korteweg 力与阶参数φ的对流非局部 Cahn-Hilliard 方程非线性耦合。证明了全局弱鞅解的存在性。在二维情况下，我们证明了弱解的路径唯一性，当$11/\mathrm{5个}<p<\mathrm{3个}$.