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  • Proceedings of the Edinburgh Mathematical Society, Volume 55, Issue 3
  • October 2012, pp. 711-729

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

  • Henrik Kalisch (a1) and Darko Mitrović (a2)
  • DOI: http://dx.doi.org/10.1017/S0013091512000065
  • Published online: 30 August 2012
Abstract
Abstract

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.

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