Journal of Dynamics and Differential Equations

, Volume 13, Issue 3, pp 523–588

Classification of Codimension-One Riemann Solutions

  • Stephen Schecter
  • Bradley J. Plohr
  • Dan Marchesin
Article

DOI: 10.1023/A:1016634307145

Cite this article as:
Schecter, S., Plohr, B.J. & Marchesin, D. Journal of Dynamics and Differential Equations (2001) 13: 523. doi:10.1023/A:1016634307145

Abstract

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data. In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.

conservation law Riemann problem viscous profile 

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Stephen Schecter
    • 1
  • Bradley J. Plohr
    • 2
  • Dan Marchesin
    • 2
  1. 1.Mathematics DepartmentNorth Carolina State UniversityRaleigh
  2. 2.Instituto de Matemática Pura e AplicadaRio de Janeiro, RJBrazil

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