On weak stability of shock waves in 2D compressible elastodynamics

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2021-02-17 DOI:10.1142/s0219891622500035

Y. Trakhinin

引用次数: 0

Abstract

By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471–1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e. they are always either uniformly or weakly stable.

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二维可压缩弹性动力学中激波的弱稳定性研究

利用1-激波的一致Lopatinski条件的等价形式，证明了[A]中能量法得到的稳定性条件。Morando, Y. Trakhinin和P. Trebeschi，二维可压缩弹性动力学中激波的结构稳定性，数学。Ann. 378(2020) 1471-1504]对于可压缩等熵无粘弹性材料二维流动中直线激波的均匀稳定性不仅是充分的，而且是必要的(意味着相应弯曲激波的结构非线性稳定性)。我们光谱分析的重点是精细地研究均匀稳定和弱稳定之间的过渡。此外，我们还证明了直线激波从不剧烈不稳定，即它们总是均匀稳定或弱稳定的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Hyperbolic Differential Equations 数学-物理：数学物理

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

24 months

期刊介绍： This journal publishes original research papers on nonlinear hyperbolic problems and related topics, of mathematical and/or physical interest. Specifically, it invites papers on the theory and numerical analysis of hyperbolic conservation laws and of hyperbolic partial differential equations arising in mathematical physics. The Journal welcomes contributions in: Theory of nonlinear hyperbolic systems of conservation laws, addressing the issues of well-posedness and qualitative behavior of solutions, in one or several space dimensions. Hyperbolic differential equations of mathematical physics, such as the Einstein equations of general relativity, Dirac equations, Maxwell equations, relativistic fluid models, etc. Lorentzian geometry, particularly global geometric and causal theoretic aspects of spacetimes satisfying the Einstein equations. Nonlinear hyperbolic systems arising in continuum physics such as: hyperbolic models of fluid dynamics, mixed models of transonic flows, etc. General problems that are dominated (but not exclusively driven) by finite speed phenomena, such as dissipative and dispersive perturbations of hyperbolic systems, and models from statistical mechanics and other probabilistic models relevant to the derivation of fluid dynamical equations. Convergence analysis of numerical methods for hyperbolic equations: finite difference schemes, finite volumes schemes, etc.