Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. APPLIED DYNAMICAL SYSTEMS c  2009 Society for Industrial and Applied Mathematics Vol. 8, No. 3, pp. 822–853 Persistence of Rarefactions under Dafermos Regularization: Blow-Up and an Exchange Lemma for Gain-of-Stability Turning Points ∗ Stephen Schecter † and Peter Szmolyan ‡ Abstract. We construct self-similar solutions of the Dafermos regularization of a system of conservation laws near structurally stable Riemann solutions composed of Lax shocks and rarefactions, with all waves possibly large. The construction requires blowing up a manifold of gain-of-stability turning points in a geometric singular perturbation problem as well as a new exchange lemma to deal with the remaining hyperbolic directions. Key words. conservation laws, Riemann problem, geometric singular perturbation theory, loss of normal hy- perbolicity, blow-up AMS subject classifications. 35L65, 34E15 DOI. 10.1137/080715305 1. Introduction. This paper is the last in a series of three; the others are [22] and [23]. An introduction to the series is in [22]. We construct self-similar solutions of the Dafermos regularization of a system of conservation laws near structurally stable Riemann solutions composed of Lax shocks and rarefactions, with all waves possibly large. The construction requires blowing up a manifold of gain-of-stability turning points in a geometric singular per- turbation problem. In addition, it requires a new exchange lemma to deal with the remaining hyperbolic directions. The latter is a consequence of the general exchange lemma from [23]. In this introduction, we briefly describe the conservation law background, and we describe some solutions near gain-of-stability turning points in order to help the reader’s intuition. A system of conservation laws in one space dimension is a partial differential equation of the form (1.1) u T + f (u) X =0, with X ∈ R, u ∈ R n , and f : R n → R n a smooth function. For background on this class of equations, see, for example, [26]. An important initial value problem is the Riemann problem, which has piecewise constant initial conditions: (1.2) u(X, 0) =  u L for X< 0, u R for X> 0. ∗ Received by the editors February 7, 2008; accepted for publication (in revised form) by C. Wayne May 4, 2009; published electronically July 10, 2009. http://www.siam.org/journals/siads/8-3/71530.html † Mathematics Department, North Carolina State University, Box 8205, Raleigh, NC 27695 (schecter@math.ncsu. edu). This author’s research was supported by the National Science Foundation under grant DMS-0406016. ‡ Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 6–10, A-1040 Vienna, Austria (szmolyan@tuwien.ac.at). This author’s research was supported by the Austrian Science Foundation under grant Y 42-MAT. 822 Downloaded 02/26/13 to 152.1.252.106. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php