Cahn–Hilliard/Navier–Stokes system is the combination of the Cahn–Hilliard equation with the Navier–Stokes equations. It describes the motion of unsteady mixing fluids and has a wide range of applications ranging from turbulent two-phase flows to microfluidics. In this paper we consider the non-local Cahn–Hilliard equation (the gradient term of the order parameter in the free energy is replaced with its spatial convolution) coupled with the Navier–Stokes equations. Assuming that the densities of the incompressible fluids are constant and the double-well potential is singular, we establish the existence of global weak solutions to the non-local system in three dimensional torus. In addition, we show that, under suitable initial assumptions, the solutions are asymptotic to those of the local Cahn–Hilliard/Navier–Stokes equations.

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关于非局部Cahn-Hilliard / Navier-Stokes方程的弱解的存在性及其局部渐近性

Cahn–Hilliard / Navier–Stokes系统是Cahn–Hilliard方程与Navier–Stokes方程的组合。它描述了不稳定混合流体的运动，并具有广泛的应用范围，从湍流两相流到微流体。在本文中，我们考虑了非局部Cahn–Hilliard方程（自由能中的阶跃参数的梯度项被其空间卷积代替）和Navier–Stokes方程。假设不可压缩流体的密度是恒定的，并且双井势是奇异的，我们建立了三维环面中非局部系统整体弱解的存在性。此外，我们表明，在适当的初始假设下，解与局部Cahn–Hilliard / Navier–Stokes方程的解是渐近的。