Eigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory |
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Authors: | Stephen Schecter |
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Affiliation: | (1) Mathematics Department, North Carolina State University, Box 8205, Raleigh, NC 27695, USA |
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Abstract: | 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–eigenfunction pairs. The proofs use geometric singular perturbation theory, in particular an extension of the Exchange Lemma. |
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Keywords: | Conservation law Dafermos regularization eigenfunction eigenvalue geometric singular perturbation theory Riemann problem stability |
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