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Spectral stability of shock waves associated with not genuinely nonlinear modes

Spectral stability of shock waves associated with not genuinely nonlinear modes

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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} }

Spectral stability of shock waves associated with not genuinely nonlinear modes 2014 eng 2015-02-27T17:52:22Z Freistühler, Heinrich Wächtler, Johannes Szmolyan, Peter Szmolyan, Peter 2015-02-27T17:52:22Z Wächtler, Johannes Freistühler, Heinrich We study viscous shock waves that are associated with a simple mode (λ,r) of a system u<sub>t</sub>+f(u)<sub>x</sub>=u<sub>xx</sub> 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⋅∇)<sup>2</sup>λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law u<sub>t</sub>+(u<sup>3</sup>)<sub>x</sub>=u<sub>xx</sub>, 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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