Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition

2018/03/30 by Xumin Gu, Fan Wang, Gu, Xumin +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #math.AP

paper · pdf · doi:10.48550/arxiv.1804.00972

arXiv admin note: substantial text overlap with arXiv:1609.07013, arXiv:1712.02152 and text overlap with arXiv:1705.11120 by other authors

arxiv created 2018/03/30 · openalex publication_date 2018/03/30 · arxiv updated 2018/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the free boundary problem for the incompressible elastodynamics equations. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local existence of a unique smooth solution of the free boundary problem, under the mixed type stability condition that some regions of the initial free boundary satisfy the Rayleigh-Taylor sign condition, while the remaining boundary satisfy the non-collinearity condition. In particular, we solve an open problem proposed by Y. Trakhinin in the paper \citeTrak₁8.

Citations

• Well-posedness for the motion of an incompressible liquid with free surface boundary
• Global solutions for the gravity water waves equation in dimension 3
• Well-posedness of the plasma-vacuum interface problem for ideal incompressible MHD
• Well-posedness of the free boundary problem in incompressible elastodynamics
• Well-posedness in Sobolev spaces of the full water wave problem in 2-D
• On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations
• Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the Rayleigh-Taylor sign condition
• Well-posedness of the free boundary problem in compressible elastodynamics

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