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

In this paper, we study the anisotropic, incompressible Cahn–Hilliard–Navier–Stokes system with variable density in a bounded smooth domain [Formula: see text]. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by [Formula: see text]. The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in [D. M. Anderson, G. B. McFadden and G. B. Wheeler, A phase-field model of solidification with convection, Physica D 135 (2000) 175–194; J. E. Taylor and J. W. Cahn, Diffuse interfaces with sharp corners and facets: Phase field models with strongly anisotropic surfaces, Physica D 112 (1998) 194–249; A. Zaidni, P. J. Morrison and S. Benjelloun, Thermodynamically consistent Cahn-Hilliard-Navier-Stokes equations using the metriplectic dynamics formalism, Physica D 468 (2024) 134–303]. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two and three dimensions [Formula: see text]. 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.