Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model

In this paper, we consider a model describing the dynamics of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated with a bending energy plus a penalization related to the area conservation (volume is exactly conserved). This problem has a dissipative in time free energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon's result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.