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Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition. (English) Zbl 1427.35373

Summary: We consider the free boundary problem for the flow of an incompressible inviscid elastic fluid. The columns of the deformation gradient are tangent and the pressure vanishes along the free interface. We prove the local existence of a unique smooth solution to the free boundary problem under the mixed type stability condition, provided that the Rayleigh-Taylor sign condition is satisfied at all the points of the initial interface where the non-collinearity condition for the deformation gradient (among three columns of the deformation gradient there are two non-collinear vectors) fails. In particular, we solve an open problem proposed by Y. Trakhinin [J. Differ. Equations 264, No. 3, 1661–1715 (2018; Zbl 1432.76211)] under the incompressible setting.

MSC:

35R35 Free boundary problems for PDEs
35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76W05 Magnetohydrodynamics and electrohydrodynamics

Citations:

Zbl 1432.76211
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References:

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