Eigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory

摘要

The Dafermos regularization of a system of n conservation laws in one space dimension admits smooth self-similar solutions of the form u = u(X/T). In particular, there are such solutions near a Riemann solution consisting of n possibly large Lax shocks. In Lin and Schecter (2004, SIAM. J. Math. Anal. 35, 884-921), eigenvalues and eigenfunctions of the linearized Dafermos operator at such a solution were studied using asymptotic expansions. Here we show that the asymptotic expansions correspond to true eigenvalue-ei-genfunction pairs. The proofs use geometric singular perturbation theory, in particular an extension of the Exchange Lemma.