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Well-posedness of hydrodynamics on the moving elastic surface. (English) Zbl 1256.35182

Summary: The dynamics of a membrane are that of a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35R37 Moving boundary problems for PDEs
35Q35 PDEs in connection with fluid mechanics
74K15 Membranes
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