Long-Time Behavior of a Cahn-Hilliard-Navier-Stokes Vesicle-Fluid Interaction Model

Abstract

A model about the dynamic of vesicle membranes in incompressible viscous fluids is introduced. The system consists of the Navier-Stokes equations with an extra stress depending on the membrane, coupled with a Cahn-Hilliard phase-field equation in 3D domains. This problem has a time dissipative energy which leads, in particular, to the existence of global in time weak solutions. By using some extra regular estimates, we prove that every weak solution is strong and unique for sufficiently large times. Moreover, the asymptotic behavior of these solutions is analyzed. We prove that the w-limit set is a subset of the set of equilibrium points. By using a Lojasiewic-Simon type inequality and a continuity result with respect to the initial values, we demonstrate the convergence of the whole trajectory to a single equilibrium.