  In this paper, we introduce a model describing the dynamic 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
to a bending energy plus a penalization term related to the area conservation.
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.
