Spectral stability of shock waves associated with not genuinely nonlinear modes

Freistühler, Heinrich; Szmolyan, Peter; Wächtler, Johannes

Spectral stability of shock waves associated with not genuinely nonlinear modes

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Date

2014

Authors

Freistühler, Heinrich

Szmolyan, Peter

Wächtler, Johannes

DOI (citable link)

10.1016/j.jde.2014.03.018

Collections

Mathematik und Statistik

Publication type

Journal article

Published in

Journal of Differential Equations ; 257 (2014), 1. - pp. 185-206. - ISSN 0022-0396. - eISSN 1090-2732

Abstract

We study viscous shock waves that are associated with a simple mode (λ,r) of a system u_t+f(u)_x=u_xx of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ in state space at whose points r⋅∇λ=0 and (r⋅∇)²λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law u_t+(u³)_x=u_xx, occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves.

Subject (DDC)

510 Mathematics

Keywords

Viscous shock waves, Spectral stability, Evans function, Geometric singular perturbation theory

Cite This

ISO 690

FREISTÜHLER, Heinrich, Peter SZMOLYAN, Johannes WÄCHTLER, 2014. Spectral stability of shock waves associated with not genuinely nonlinear modes. In: Journal of Differential Equations. 257(1), pp. 185-206. ISSN 0022-0396. eISSN 1090-2732. Available under: doi: 10.1016/j.jde.2014.03.018

BibTex

@article{Freistuhler2014Spect-30127,
  year={2014},
  doi={10.1016/j.jde.2014.03.018},
  title={Spectral stability of shock waves associated with not genuinely nonlinear modes},
  number={1},
  volume={257},
  issn={0022-0396},
  journal={Journal of Differential Equations},
  pages={185--206},
  author={Freistühler, Heinrich and Szmolyan, Peter and Wächtler, Johannes}
}

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    <dcterms:abstract xml:lang="eng">We study viscous shock waves that are associated with a simple mode (λ,r) of a system u&lt;sub&gt;t&lt;/sub&gt;+f(u)&lt;sub&gt;x&lt;/sub&gt;=u&lt;sub&gt;xx&lt;/sub&gt; of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ   in state space at whose points r⋅∇λ=0 and (r⋅∇)&lt;sup&gt;2&lt;/sup&gt;λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law u&lt;sub&gt;t&lt;/sub&gt;+(u&lt;sup&gt;3&lt;/sup&gt;)&lt;sub&gt;x&lt;/sub&gt;=u&lt;sub&gt;xx&lt;/sub&gt;, occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves.</dcterms:abstract>
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Yes