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October  2004, 10(4): 885-924. doi: 10.3934/dcds.2004.10.885

An Evans function approach to spectral stability of small-amplitude shock profiles

Ramon Plaza 1, and  K. Zumbrun 2,

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, United States
2. 

Mathematics Department, Indiana University, Bloomington, IN 47405

Received  September 2002 Revised  December 2003 Published  March 2004

In recent work, the second author and various collaborators have shown using Evans function/refined semigroup techniques that, under very general circumstances, the problems of determining one- or multi-dimensional nonlinear stability of a smooth shock profile may be reduced to that of determining spectral stability of the corresponding linearized operator about the wave. It is expected that this condition should in general be analytically verifiable in the case of small amplitude profiles, but this has so far been shown only on a case-by-case basis using clever (and difficult to generalize) energy estimates. Here, we describe how the same set of Evans function tools that were used to accomplish the original reduction can be used to show also small-amplitude spectral stability by a direct and readily generalizable procedure. This approach both recovers the results obtained by energy methods, and yields new results not previously obtainable. In particular, we establish one-dimensional stability of small amplitude relaxation profiles, completing the Evans function program set out in Mascia&Zumbrun [MZ.1]. Multidimensional stability of small amplitude viscous profiles will be addressed in a companion paper [PZ], completing the program of Zumbrun [Z.3].
Keywords: traveling waves, Evans function, Spectral stability.
Mathematics Subject Classification: 35K5.
Citation: Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
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