Classical solutions to a model of two-phase incompressible flows with variable density

Abstract

We study the global well-posedness of three-dimensions Allen-Cahn-Navier-Stokes equations, a diffuse-interface model for two-phase incompressible flows with different densities. We first prove the local-in-time existence of classical solutions with finite initial energy. The key point is to carefully design an energy norm. The derivative f′(ϕ) of the physical relevant energy density f(ϕ) produces a damping effect near equilibrium ϕ=±1. We thereby establish the unique global classical solution near equilibrium under small size of initial energy.