TY - JOUR

T1 - Structural stability of shock waves in 2D compressible elastodynamics

AU - Morando, Alessandro

AU - Trakhinin, Yuri

AU - Trebeschi, Paola

N1 - Publisher Copyright: © 2019, Springer-Verlag GmbH Germany, part of Springer Nature. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

AB - We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

KW - 35Q35

KW - 35L67

KW - 35L04

KW - 35L05

KW - 76L05

KW - BOUNDARY-VALUE-PROBLEMS

KW - VORTEX SHEETS

KW - MIXED PROBLEM

KW - GASDYNAMICS

KW - EQUATIONS

KW - SYSTEMS

UR - http://www.scopus.com/inward/record.url?scp=85074582719&partnerID=8YFLogxK

U2 - 10.1007/s00208-019-01920-6

DO - 10.1007/s00208-019-01920-6

M3 - Article

AN - SCOPUS:85074582719

VL - 378

SP - 1471

EP - 1504

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 3-4

ER -