Communications in Mathematical Sciences

Volume 18 (2020)

Number 8

On the existence of weak solutions to non-local Cahn–Hilliard/Navier–Stokes equations and its local asymptotics

Pages: 2121 – 2147



Zhilei Liang (School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China)


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.


weak solutions, Cahn–Hilliard/Navier–Stokes, non-local model, local asymptotics

2010 Mathematics Subject Classification

35K25, 45K05, 76D05, 76D10

Received 9 November 2019

Accepted 21 May 2020

Published 22 December 2020