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Singular solutions of a fully nonlinear 2 × 2 system of conservation laws

Part of: Hyperbolic equations and systems Magnetohydrodynamics and electrohydrodynamics

Published online by Cambridge University Press:  30 August 2012

Henrik Kalisch  and
Darko Mitrović
Henrik Kalisch
Affiliation:
Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen, Norway ([email protected])
Darko Mitrović
Affiliation:
Department of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Montenegro ([email protected])
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Abstract

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Existence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyperbolic system of equations

The system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive δ-shock solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued, and the solutions can be seen to fit into the framework of weak singular solutions defined by Danilov and Shelkovich. Indeed, in this context, we can show that every 2 × 2 system of conservation laws admits δ-shock solutions.

Keywords

conservation lawsRiemann problemsingular solutionsweak asymptoticsmagnetohydrodynamics

MSC classification

Secondary: 35L65: Conservation laws 35L67: Shocks and singularities 76W05: Magnetohydrodynamics and electrohydrodynamics
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 711 - 729
Copyright
Copyright © Edinburgh Mathematical Society 2012

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