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.
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Schecter, S., Plohr, B.J. & Marchesin, D. Classification of Codimension-One Riemann Solutions. Journal of Dynamics and Differential Equations 13, 523–588 (2001). https://doi.org/10.1023/A:1016634307145
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DOI: https://doi.org/10.1023/A:1016634307145