Research output: Contribution to journal › Article › peer-review
Well-posedness of the free boundary problem in compressible elastodynamics. / Trakhinin, Yuri.
In: Journal of Differential Equations, Vol. 264, No. 3, 05.02.2018, p. 1661-1715.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 9160283