Projektdetails
Abstract
1) Wider research context
Biological membranes can be mathematically treated as deformable inextensible fluidic surfaces governed by bending energies. The interest in the modeling of two-phase biomembranes has been continuously growing. The fluidic behavior of the membrane, curvature elasticity and phase separation phenomena have been modeled and an increasing number of works concerning theoretical and numerical results has been produced. Among these, phase-field models have already proved successful. Nevertheless, their Mathematical Analysis still presents many unsolved issues.
2) Research questions
This project aims at advancing the rigorous mathematical understanding of biomembrane modeling. It will connect the world of classical phase-field models on flat domains with geometrical notions about compact surfaces. The first goal is to study the well-posedness of weak and strong solutions to phase-field hydrodynamic models on compact 2D surfaces evolving with prescribed motion, also establishing a suitable analytical framework for the analysis of the longtime behavior of the solutions. The second goal is to study the well-posedness of suitably defined solutions to some free boundary value problems modeling biomembranes, where the evolution of the membrane surface is determined by the model itself.
3)Methods
This project is meant as a bridge between nonlinear PDE analysis, differential geometry and geometric measure theory. The main tools are the geometry of evolving surfaces modeling the biomembranes, connected to the study of evolving Banach spaces. In the free boundary value problems one deals with functions of bounded variation and sets of finite perimeter. Being forced by the low regularity of solutions ensured by the energy estimates, it is necessary to introduce generalized notions of solutions, namely varifolds solutions: the surface is substituted by the more general concept of varifolds. In addition, to analyze the longtime behavior of solutions, one needs to settle the corresponding nonautonomous dynamical system in the framework of processes and pullback attractors.
4)Level of innovation
The project focuses on frontier mathematical issues with application to cell biology. The AGG phase-field model with logarithmic potential is proposed in the novel setting of evolving two-dimensional compact surfaces. The study of the longtime behavior of solutions is an application of the quite recent theory of nonautonomous dynamical systems in the new setting of evolving surfaces. Moreover, the mathematical analysis of free boundary value problems for phase-field models on a lipid bilayer membrane immersed in a fluid is new in the literature both from the point of view of global generalized solutions and local strong solutions.
5)Primary researchers involved
Andrea Poiatti (PI), Ulisse Stefanelli (mentor), Helmut Abels, Harald Garcke, Charles M. Elliott and Benoît Perthame (collaborators)
Biological membranes can be mathematically treated as deformable inextensible fluidic surfaces governed by bending energies. The interest in the modeling of two-phase biomembranes has been continuously growing. The fluidic behavior of the membrane, curvature elasticity and phase separation phenomena have been modeled and an increasing number of works concerning theoretical and numerical results has been produced. Among these, phase-field models have already proved successful. Nevertheless, their Mathematical Analysis still presents many unsolved issues.
2) Research questions
This project aims at advancing the rigorous mathematical understanding of biomembrane modeling. It will connect the world of classical phase-field models on flat domains with geometrical notions about compact surfaces. The first goal is to study the well-posedness of weak and strong solutions to phase-field hydrodynamic models on compact 2D surfaces evolving with prescribed motion, also establishing a suitable analytical framework for the analysis of the longtime behavior of the solutions. The second goal is to study the well-posedness of suitably defined solutions to some free boundary value problems modeling biomembranes, where the evolution of the membrane surface is determined by the model itself.
3)Methods
This project is meant as a bridge between nonlinear PDE analysis, differential geometry and geometric measure theory. The main tools are the geometry of evolving surfaces modeling the biomembranes, connected to the study of evolving Banach spaces. In the free boundary value problems one deals with functions of bounded variation and sets of finite perimeter. Being forced by the low regularity of solutions ensured by the energy estimates, it is necessary to introduce generalized notions of solutions, namely varifolds solutions: the surface is substituted by the more general concept of varifolds. In addition, to analyze the longtime behavior of solutions, one needs to settle the corresponding nonautonomous dynamical system in the framework of processes and pullback attractors.
4)Level of innovation
The project focuses on frontier mathematical issues with application to cell biology. The AGG phase-field model with logarithmic potential is proposed in the novel setting of evolving two-dimensional compact surfaces. The study of the longtime behavior of solutions is an application of the quite recent theory of nonautonomous dynamical systems in the new setting of evolving surfaces. Moreover, the mathematical analysis of free boundary value problems for phase-field models on a lipid bilayer membrane immersed in a fluid is new in the literature both from the point of view of global generalized solutions and local strong solutions.
5)Primary researchers involved
Andrea Poiatti (PI), Ulisse Stefanelli (mentor), Helmut Abels, Harald Garcke, Charles M. Elliott and Benoît Perthame (collaborators)
| Kurztitel | Elastisch-fluidische Biomembrane |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 7/03/24 → 6/03/27 |