Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system

TL;DR

This paper proves the global existence of weak solutions for an anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in 2D and 3D, extending previous isotropic results.

Contribution

It introduces anisotropic surface energy into the system and establishes global weak solutions using Galerkin approximation, Bihari's inequality, and fixed-point methods.

Findings

Proved existence of global weak solutions in 2D and 3D.

Extended isotropic results to anisotropic surface energy case.

Used Galerkin scheme and fixed-point argument for proof.

Abstract

We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain $\Omega \subset \mathbb{R}^d$. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by $\mathfrak{F}= \int_{\Omega} \frac{\epsilon}{2}\, \Gamma^2(\nabla \phi) $. The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in \cite{anderson2000phase,taylor-cahn98, zaidni2024}. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions $(d=2,3)$. A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument.