Spectral stability of shock waves associated with not genuinely nonlinear modes
In: Journal of Differential Equations, Jg. 257 (2014-07-01), S. 185-206
Online
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Zugriff:
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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Spectral stability of shock waves associated with not genuinely nonlinear modes
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Autor/in / Beteiligte Person: | Szmolyan, Peter ; Freistühler, Heinrich ; Wächtler, Johannes |
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Zeitschrift: | Journal of Differential Equations, Jg. 257 (2014-07-01), S. 185-206 |
Veröffentlichung: | Elsevier BV, 2014 |
Medientyp: | unknown |
ISSN: | 0022-0396 (print) |
DOI: | 10.1016/j.jde.2014.03.018 |
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