Global Existence of Weak Solutions to a Nonlocal Cahn-Hilliard-Navier-Stokes System

TL;DR

This paper proves the global existence of weak solutions for a nonlocal variant of the coupled Navier-Stokes and Cahn-Hilliard system, extending the understanding of binary-fluid mixture models with nonlocal interactions.

Contribution

It establishes the existence of weak solutions for a nonlocal Cahn-Hilliard-Navier-Stokes system, including energy identities and dissipative estimates in two dimensions.

Findings

Global weak solutions exist for the nonlocal model.

Energy identity holds in 2D under certain conditions.

Dissipative estimates are established for 2D solutions.

Abstract

A well-known diffuse interface model consists of the Navier-Stokes equations nonlinearly coupled with a convective Cahn-Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary-fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn-Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter phi, while the potential F may have any polynomial growth. Therefore the coupling with the Navier-Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of phi. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition.