Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

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

Henrik Kalischa1 and Darko Mitrovića2

a1 Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen, Norway (henrik.kalisch@math.uib.no)

a2 Department of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Montenegro (matematika@t-com.me)

Abstract

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

$$\begin{align*}\partial_tu+\partial_x(\tfrac12(u^2+v^2))&=0,\\\partial_tv+\partial_x(v(u-1))&=0.\end{align*}$$

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.

(Received May 23 2011)

Keywords

  • conservation laws;
  • Riemann problem;
  • singular solutions;
  • weak asymptotics;
  • magnetohydrodynamics

2010 Mathematics subject classification

  • Primary 35L65;
  • 35L67;
  • Secondary 76W05