Stability of periodic solutions of conservation laws with viscosity: analysis of the Evans function (Q1566395)

The paper studies the stability of periodic traveling-wave solutions of conservation laws with viscosity. The paper deals with the traveling-wave solutions of conservation laws with viscosity \[ u_t+f(u)_x=(B(u)u_x)_x\,, \] where \(u,f\in \mathbb R^n\), \(B\in \mathbb R^{n\times n}\), modeling flow in compressible media. Periodic solutions do not occur for classical systems possessing a convex entropy. However, they arise in a natural way in certain nonclassical systems, modeling media with multiple phases. The stability of periodic traveling waves within the abstract Evans-function is investigated. The main task is to derive stability index analogous to that developed by \textit{R. A. Gardner} [J. Math. Pures Appl. 72, 415--439 (1993; Zbl 0831.35077)]. The paper is well supported with several numerical experiments.