Geometry and phases on elastic-fluidic biomembranes

The interest in the modeling of two-phase biomembranes has been continuously growing in the last years, attracting many researchers of different scientific areas. Biomembranes provide the fundamental separation structure in eukaryotic cells, namely they form the structure separating the cell interior from its exterior. Their peculiar behavior combines the mechanics of solids (curvature or elasticity) with that of fluids. This means that biological membranes can be mathematically treated as deformable inextensible fluidic surfaces governed by bending energies, which involve the curvature of the membrane. The fluidic behavior of biomembranes is at the basis of the lipid-raft formation phenomenon, involving the separation of the lipids on the membrane into two immiscible liquid phases, and leading to the formation of liquid-ordered phase platforms, which are called rafts and are believed to play an important role in a large number of biological processes. Coming back to mathematical modeling, when these lipid rafts are present, i.e., there are at least two liquid phases on the membrane, the bending energies associated to the elastic cell membrane will depend on the individual phases, and the local shape of the membrane will vary according to the corresponding phase. Concerning the mathematical study of biological membranes, an increasing number of works concerning new models and numerical results has been produced. Still, their Mathematical Analysis has not kept pace and many issues remain unsolved. Therefore, this project aims at regaining ground by advancing some solid theoretical framework to the arising numerical literature on biomembranes modeling. In particular, it will connect the world of classical phase-field models on flat domains with geometrical notions related to compact surfaces. The key objectives of this three-year project are then the following. First, we will address the well-posedness of weak and strong solutions to phase-field hydrodynamic models on two-dimensional surfaces evolving with a priori prescribed motion. We will also compare the analytical results obtained for various modeling choices. Secondly, we aim at establishing a suitable analytical framework for the analysis of the longtime behavior of the solutions to the same problems. Namely, we will tackle the study of the convergence to equilibria and the existence of properly defined attractors. In conclusion, we will face the existence and, possibly, uniqueness of solutions to some free boundary value problems for phase-field models on a biomembrane immersed in a fluid. In this case, the evolution of the membrane surface is not a priori determined, but rather imposed by the model itself.