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# Spectral stability of shock waves associated with not genuinely nonlinear modes

Type of Publication: | Journal article |

Author: | Freistühler, Heinrich; Szmolyan, Peter; Wächtler, Johannes |

Year of publication: | 2014 |

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

DOI (citable link): | https://dx.doi.org/10.1016/j.jde.2014.03.018 |

Summary: |
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⋅∇)^{2}λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law u_{t}+(u^{3})_{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 |

Bibliography of Konstanz: | Yes |

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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

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