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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 0157 8849 9
Author(s):
Benzoni-Gavage, Sylvie; Serre, Denis; Zumbrun, Kevin:
Shorttitle:
Alternate Evans functions and viscous shock waves
Source:
SIAM J. Math. Anal. 32, No. 5, 929 - 962 (2001)
Classification:
34L15Estimation of eigenvalues, upper and lower bounds
35K45Initial value problems for parabolic systems
35L67Shocks and singularities
Primary Classification:
Secondary Classification:
Keywords:
traveling waves; asymptotic stability; viscous conservation laws
Review:

The spectrum of a finite matrix are zeros of characteristic
polynomial. For differential operators studied in the context of
asymptotic stability analysis of traveling waves, the role of this
polynomial is taken over by the so called Evans function D. The
paper starts in fact by a concise review of its possible
definition(s), with emphasis on the application in the study of
the viscous shock waves.

In the context of various applications of Evans functions, the
paper is a more or less immediate continuation of the work by
Gardner and Zumbrun from 1998, comparing the merits and
shortcomings related to different definitions of the Evans
functions in practical computations, and preferring the use of the
homotopy to the original rescaling approach. Authors emphasize the
useful role the so called ``dual" and ``mixed" type of the
definition of D. There are two directions of the new development
of its applications, viz., the improvement of the stability
analysis (especially for the so called Lax shock) and an extension
of the formalism to the general system of size n > 2 (giving, in
fact, a proof of the missing lemma in general theory).
Remarks to the editors:

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