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

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.