Generalized characteristic analysis and Guckenheimer structure
type: Item
instance of: scholarly article
title: Generalized characteristic analysis and Guckenheimer structure
zbMATH Open document ID: 0924.35079
author: Peng Zhang
author: Tong Zhang
published in: Journal of Differential Equations
publication date: 1999-11-04T00:00:00Z
review text: The authors study the two-dimensional scalar conservation law \(u_t+f(u)_x+g(u)_y=0\) and by transformation the Riemann problem corresponding to \(-\xi u_{\xi}-\eta u_{\eta}+f(u)_{\xi}+g(u)_{\eta}=0\) with \(\lim_{r \rightarrow \infty}u(\xi,\eta)=u_0(\theta)\), where \((r,\theta)\) are the polar coordinates of \((\xi,\eta)\). The authors develop a generalized characteristic analysis method and study the intersections of elementary waves and construct the Riemann solutions for the conservation law. Finally, necessary and sufficient conditions for the appearance of Guckenheimer structure is given. This structure was shown by Guckenheimer to arise in the example \(f(u)=u^2/2\), \(g(u)=u^3/3\) and \(u_0\) is \(1,-1\) or \(0\), and is similar to Mach reflection in gas dynamics.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35R05
zbMATH DE Number: 1283670
zbMATH Keywords: Riemann problem
zbMATH Keywords: intersections of elementary waves
zbMATH Keywords: Mach reflection in gas dynamics
reviewed by: Emmanuel N. Barron
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3526
OpenAlex ID: W2080336596
cites work: Q4040818
cites work: Structure of Riemann solutions for 2-dimensional scalar conservation laws
cites work: Global solutions of the cauchy problem for quasi‐linear first‐order equations in several space variables
cites work: Shocks and rarefactions in two space dimensions
cites work: FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES
cites work: The Scalar Riemann Problem in Two Spatial Dimensions: Piecewise Smoothness of Solutions and Its Breakdown
cites work: The Riemann Problem in Two Space Dimensions for a Single Conservation Law
cites work: Two-Dimensional Riemann Problem for a Single Conservation Law
cites work: On two-dimensional Riemann problem for pressure-gradient equations of the Euler system
DOI: 10.1006/JDEQ.1998.3526
description: scientific article; zbMATH DE number 1283670