Structural stability of shock waves in 2D compressible elastodynamics
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Mathematische Annalen
Structural stability of shock waves in 2D compressible elastodynamics Alessandro Morando1 · Yuri Trakhinin2,3
· Paola Trebeschi1
Received: 20 March 2019 / Revised: 22 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract 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. Mathematics Subject Classification 35Q35 · 35L67 · 35L04 · 35L05 · 76L05
Communicated by Y. Giga.
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Yuri Trakhinin [email protected] Alessandro Morando [email protected] Paola Trebeschi [email protected]
1
DICATAM, Sezione di Matematica, Università di Brescia, Via Valotti, 9, 25133 Brescia, Italy
2
Sobolev Institute of Mathematics, Koptyug av. 4, 630090 Novosibirsk, Russia
3
Novosibirsk State University, Pirogova str. 1, 630090 Novosibirsk, Russia
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A. Morando et al.
1 Introduction We consider the equations of elastodynamics [14,20,21] governing the motion of compressible isentropic inviscid elastic materials. We restrict ourself to two-dimensional (2D) elastic flows. Then, the elastodynamics equations read ⎧ ⎨ ∂t ρ + div (ρv) = 0, ∂ (ρv) + div (ρv ⊗ v) + ∇ p − div (ρ F F ) = 0, ⎩ t ∂t (ρ F j ) + div (ρ F j ⊗ v − v ⊗ ρ F j ) = 0, j = 1, 2,
(1)
where ρ is the density, v ∈ R2 is the velocity, F ∈ M(2, 2) is the deformation gradient, F1 = (F11 , F21 ) and F2 = (F12 , F22 ) are the columns of F, and the pressure p = p(ρ) is a smooth function of ρ. Moreover, system (1) is supplemented by the identity div (ρ F ) = 0 which is the set of the two divergence constraints div (ρ F j ) = 0 ( j = 1, 2)
(2)
on initial data, i.e., one can show that if equations (2) are satisfied initially, then they hold for all t > 0. We note that system (1) arises as the inviscid limit of the equations of compressible viscoelasticity [14,20,21] of Oldroyd type [35,36]. Taking into account the divergence constrain
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