Spectral stability of shock waves associated with not genuinely nonlinear modes

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2014
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Journal of Differential Equations. 2014, 257(1), pp. 185-206. ISSN 0022-0396. eISSN 1090-2732. Available under: doi: 10.1016/j.jde.2014.03.018
Zusammenfassung

We study viscous shock waves that are associated with a simple mode (λ,r) of a system ut+f(u)x=uxx 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 ut+(u3)x=uxx, 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.

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510 Mathematik
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Viscous shock waves, Spectral stability, Evans function, Geometric singular perturbation theory
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ISO 690FREISTÜHLER, Heinrich, Peter SZMOLYAN, Johannes WÄCHTLER, 2014. Spectral stability of shock waves associated with not genuinely nonlinear modes. In: Journal of Differential Equations. 2014, 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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