| Abstract: |
| In this talk I would like to present a model concerning lipid rafts formation on cell membranes (lipid bilayers). This phenomenon consists in the separation of lipids composing the cell membrane into two immiscible liquid phases, leading to the formation of heterogeneous liquid-ordered phase platforms (rafts). These rafts are believed to play important roles in the biology of the cell.
I will focus on a diffuse interface model for incompressible viscous two-phase fluids with different densities, known as Abels-Garcke-Gr\{u}n model, over an evolving surface. After briefly showing the derivation of the model, I will introduce a suitable framework of evolving families of Banach and Hilbert spaces, and explain some recent results concerning the well-posedness of strong solutions to the problem, when the surface evolution is a priori prescribed. Namely I will first focus on the existence of a local strong solution, which is separated from pure phases, and then on how to extend this solution to a global-in-time unique separated strong solution. I will also present some new techniques for obtaining the validity of the strict separation property from pure phases on two-dimensional surfaces, under very weak assumptions on the behavior of the singular potential. |
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