On the sharp interface limit of a phase field model for near spherical two phase biomembranes. (English) Zbl 1489.35287
Summary: We consider sharp interface asymptotics for a phase field model motivated by lipid raft formation on near spherical biomembranes involving a coupling between the local mean curvature and the local composition. A reduced diffuse interface energy depending only on the membrane composition is introduced and a \(\Gamma \)-limit is derived. It is shown that the Euler-Lagrange equations for the limiting functional and the sharp interface energy coincide. Finally, we consider a system of gradient flow equations with conserved Allen-Cahn dynamics for the phase field model. Performing a formal asymptotic analysis, we obtain a system of gradient flow equations for the sharp interface energy coupling geodesic curvature flow for the phase interface to a fourth order PDE free boundary problem for the surface deformation.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35Q35 | PDEs in connection with fluid mechanics |
| 92C10 | Biomechanics |
| 76Z05 | Physiological flows |
| 74L15 | Biomechanical solid mechanics |
| 74K15 | Membranes |
| 35A15 | Variational methods applied to PDEs |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 35R35 | Free boundary problems for PDEs |
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