The Well - Posedness And Asymptotic Behavior Of Navier - Stokes Equations In The Biofilm Phase Of Incompressible Fluid

In this paper, we study a Mathematical Model describing the deformation of vesicle membranes in incompressible viscous fluids. This model consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. In the three dimensional case, we prove the existence/uniqueness of global solutions of Dirichlet problem for arbitrary initial data under the large viscosity assumption. We study the long-time properties of this system as well, and prove its regularity and asymptotic behavior when time goes to infinite.