TY - JOUR

T1 - Well-posedness of the free boundary problem in compressible elastodynamics

AU - Trakhinin, Yuri

PY - 2018/2/5

Y1 - 2018/2/5

N2 - 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.

AB - 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.

KW - Compressible elastodynamics

KW - Free boundary problem

KW - Rayleigh–Taylor instability

KW - Symmetric hyperbolic system

KW - Well-posedness

KW - CURRENT-VORTEX SHEETS

KW - EXISTENCE

KW - VACUUM INTERFACE PROBLEM

KW - VISCOELASTIC FLOWS

KW - HYPERBOLIC SYSTEMS

KW - SOBOLEV SPACES

KW - Rayleigh-Taylor instability

KW - MOTION

KW - FREE-SURFACE BOUNDARY

KW - WATER-WAVE PROBLEM

KW - EULER EQUATIONS

UR - http://www.scopus.com/inward/record.url?scp=85030774179&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2017.10.005

DO - 10.1016/j.jde.2017.10.005

M3 - Article

AN - SCOPUS:85030774179

VL - 264

SP - 1661

EP - 1715

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 3

ER -