Spectral stability of shock waves associated with not genuinely nonlinear modes

Zitieren

Dateien zu dieser Ressource

Dateien Größe Format Anzeige

Zu diesem Dokument gibt es keine Dateien.

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.

Das Dokument erscheint in:

KOPS Suche


Stöbern

Mein Benutzerkonto