Stability of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws

In contrast to a viscous regularization of a systemof n conservation laws, a Dafermos regularization admits many self-similar solutions of the form u = u( X T ). In particular, it is known in many cases that Riemann solutions of a system of conservation laws have nearby self-similar smooth solutions of an associated Dafermos regularization. We refer to these smooth solutions as Riemann-Dafermos solutions. In the coordinates x = X , t =l nT , Riemann-Dafermos solutions become stationary, and their time-asymptotic stability as solutions of the Dafermos regularization can be studied by linearization. We study the stability of Riemann-Dafermos solutions near Riemann solutions consisting of n Lax shock waves. We show, by studying the essential spectrumof the linearized systemin a weighted function space, that stability is determ ined by eigenvalues only. We then use asymptotic methods to study the eigenvalues and eigenfunctions. We find there are fast eigenvalues of order 1 and slow eigenvalues of order 1. The fast eigenvalues correspond to eigenvalues of the viscous profiles for the individual shock waves in the Riemann solution; these have been studied by other authors using Evans function methods. The slow eigenvalues are related to inviscid stability conditions that have been obtained by various authors for the underlying Riemann solution.

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