Well-posedness of the free boundary problem in compressible elastodynamics

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. 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-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.

Original languageEnglish
Pages (from-to)1661-1715
Number of pages55
JournalJournal of Differential Equations
Volume264
Issue number3
DOIs
Publication statusPublished - 5 Feb 2018

Keywords

  • Compressible elastodynamics
  • Free boundary problem
  • Rayleigh–Taylor instability
  • Symmetric hyperbolic system
  • Well-posedness
  • CURRENT-VORTEX SHEETS
  • EXISTENCE
  • VACUUM INTERFACE PROBLEM
  • VISCOELASTIC FLOWS
  • HYPERBOLIC SYSTEMS
  • SOBOLEV SPACES
  • Rayleigh-Taylor instability
  • MOTION
  • FREE-SURFACE BOUNDARY
  • WATER-WAVE PROBLEM
  • EULER EQUATIONS

Fingerprint Dive into the research topics of 'Well-posedness of the free boundary problem in compressible elastodynamics'. Together they form a unique fingerprint.

Cite this