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

Logarithmically Improved Blow-up Criteria for a Phase Field Navier-Stokes Vesicle-Fluid Interaction Model