  In this paper, we study a hydrodynamical system modeling the deformation of
vesicle membrane under external incompressible viscous flow fields. The system
is in the Eulerian formulation and is governed by the coupling of the
incompressible Navier-Stokes equations with a phase field equation. In the
three dimensional case, we establish two logarithmically improved blow-up
criteria for local smooth solutions of this system in terms of the vorticity
field only in the homogeneous Besov spaces.
