Preprint 2002-019

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

Xiao-Biao Lin and Stephen Schecter

Abstract: In contrast to a viscous regularization of a system of $n$ 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 {\em Riemann-Dafermos solutions}. After a change of coordinates, 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 spectrum of the linearized system in a weighted function space, that stability is determined by eigenvalues only. We then use asymptotic methods to study the eigenvalues and eigenfunctions. We find there are fast eigenvalues of order \( \frac{1}{\epsilon} \) and slow eigenvalues of order one. The fast eigenvalues correspond to fast convergence of initial data to traveling wave solutions in singular layers, while the slow eigenvalues correspond to convection in regular layers connected by traveling waves in singular layers. For an example from gas dynamics, we show that all the slow eigenvalues are stable.

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Xiao-Biao Lin, <>
Stephen Schecter, <>
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<> April 4 2002.

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