| Abstract: |
| Diffuse interface models play nowadays an important role in Fluid dynamics as analytical and numerical methods to describe the behavior of multi-component or multi-phase fluids flows. The Cahn-Hilliard-Brinkman (CHB) system belongs to a class of diffuse interface models for the description of the interaction between incompressible and viscuos binary fluids. In particular, it is employed for phase separation phenomena in porous media and constitutes a relaxation of the well-known Cahn-Hilliard-Navier-Stokes and Cahn-Hilliard-Hele-Shaw systems. The CHB model couples a modified Darcy`s law which rules the volume-averaged fluid velocity with a convective Cahn-Hilliard equation for the difference of the fluid concentrations. I will present in this talk a fairly complete mathematical theory for the CHB model with unmatched viscosities and logarithmic potential in three dimensions. More precisely, I will discuss about uniqueness of weak solutions, global well-posedness of strong solutions and validity of the separation property. This analysis validates teh CHB system as a robust diffuse interface model for the description of three dimensional two-component flows. |
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