Existence of Dafermos profiles for singular shocks
type: Item
instance of: scholarly article
title: Existence of Dafermos profiles for singular shocks
zbMATH Open document ID: 1077.35095
author: Stephen Schecter
published in: Journal of Differential Equations
publication date: 2004-11-22T00:00:00Z
review text: This paper considers the Riemann problem for a strictly hyperbolic, nonlinear system of conservation law \[ \begin{aligned} u_{1t}+(u_1^2-u_2)_x&=0, \quad u_1(x,0)=u_{1L}\text{ if }x<0, \quad u_1(x,0)=u_{1R}\text{ if }x>0, \\ u_{2t}+(\frac{1}{3}u_1^3-u_1)_x&=0, \quad u_2(x,0)=u_{2L}\text{ if }x<0, \quad u_2(x,0)=u_{2R}\text{ if }x>0, \end{aligned} \] where \(u_{iL}\) and \(u_{iR}\in \mathbb{R}\). \textit{B. L. Keyfitz} and \textit{H. C. Kranzer} [Lect. Notes Math. 1402, 185--197 (1989; Zbl 0704.35094), IMA Vol. Math. Appl. 27, 107--125 (1990; Zbl 0718.76071)] produced a unique solution containing {singular shocks}, i.e. allowing discontinuities made of a Heavyside function plus a \(\delta\)-function. They conjectured that such solutions were limits as \(\varepsilon\) goes to zero of self-similar solutions \(u_i^{\varepsilon}(\frac{x}{t})\) of a Dafermos regularization \[ \begin{aligned} u_{1t}+(u_1^2-u_2)_x&=\varepsilon tu_{1xx}, \\ u_{2t}+(\frac{1}{3}u_1^3-u_1)_x&=\varepsilon tu_{2xx}. \end{aligned} \] The main result of the paper gives conditions which ensure existence of such a singular shock which is the limit of solutions of Dafermos regularizations. The proof uses the blowing up approach to geometric singular perturbation problems that lack normal hyperbolicity. It also applies to some generalized Keyfitz-Kranzer system.
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 2117240
zbMATH Keywords: Dafermos regularization
zbMATH Keywords: geometric singular perturbation theory
zbMATH Keywords: blow up
zbMATH Keywords: Riemann problem
zbMATH Keywords: generalized Keyfitz-Kranzer system
reviewed by: Patrick Habets
MaRDI profile type: Publication
full work available at URL: j.jde.2004.06.013
OpenAlex ID: W2150279676
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DOI: 10.1016/J.JDE.2004.06.013
description: scientific article; zbMATH DE number 2117240