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

2024/12/07 by Zaidni, Azeddine, Benjelloun, Saad, Boukharfane, Radouan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2412.05757

Abstract

We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain Ω⊂ ℝ^d. This work extends previous studies on the isotropic case by incorporating anisotropic surface energy, represented by \mathfrakF= ∫_Ω \fracε2 Γ²(∇ ϕ) . The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in [3,5,7]. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two and three dimensions (d=2,3). A crucial element in extending the existence of approximate solutions from local to global lies in the application of Bihari's inequality and a fixed-point argument.

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