Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

Abstract

Abstract. The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u (X/T ). In [19] it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant.In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in . We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.

L² semigroup and linear stability for Riemann solutions of conservation laws

“…We will also show that non-zero eigenvalues are the zeros of the determinant of the "SLEP" matrix as defined in [14,10].…”

“…Moreover, this condition is equivalent to that s is the root of the SLEP function (determinant of the corresponding SLEP matrix [15]) as defined in [10].…”

“…In contrast to the "fast eigenvalues" of (1.6) which are of O( 1 ) and come from the dynamics of the singular layers near the shocks of the conservation laws, the critical eigenvalues are so called "slow eigenvalues" in [14,16,10] which are of O(1) and are the eigenvalues of the inviscid system (1.7).…”

Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws

“…The method of factorization can be used to convert a second order singularly perturbed equation to a first order system of which the fast and slow variables naturally split [4,16,25,37].…”