Benzoni-Gavage, Sylvie; Serre, Denis; Zumbrun, Kevin Alternate Evans functions and viscous shock waves. (English) Zbl 0985.34075 SIAM J. Math. Anal. 32, No. 5, 929-962 (2001). The spectrum of a finite matrix consists of the zeros of its 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 an 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 R. A. Gardner and K. Zumbrun [Commun. Pure Appl. Math. 51, No. 7, 797-855 (1998; Zbl 0933.35136)], 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. The 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 the general theory). Reviewer: Miloslav Znojil (Řež) Cited in 29 Documents MSC: 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 76L05 Shock waves and blast waves in fluid mechanics 35L67 Shocks and singularities for hyperbolic equations 35K45 Initial value problems for second-order parabolic systems 35Q35 PDEs in connection with fluid mechanics Keywords:traveling waves; asymptotic stability; viscous conservation laws; viscous shock waves; Evans function D Citations:Zbl 0933.35136 × Cite Format Result Cite Review PDF Full Text: DOI