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Preprint 2011-001

Singular solutions for nonlinear hyperbolic systems

Henrik Kalisch and Darko Mitrovic

Abstract: An extension of the method of weak asymptotics is presented which allows the construction of singular solutions of Riemann problems for systems of hyperbolic conservation laws. The method is based on using complex-valued approximations which become real-valued in the distributional limit. It is shown how this approach can be used to construct solutions containing combinations of classical hyperbolic shock waves and Dirac delta distributions. The method is applied to two particular systems of conservation laws in one spatial variable. First, existence of solutions for the shallow-water system is obtained for a class of initial data which includes delta distributions. Uniqueness is obtained in a smaller class of distributions which satisfy a condition of Oleinik type. As a second example, a hyperbolic system appearing in the study of magneto-hydrodynamics is studied. This system was investigated in [22], and it was noticed that there exists no classical Lax-admissible solution for a particular Riemann problem. It was conjectured that this initial configuration might lead to singular solutions featuring combinations of Dirac delta distributions and shock waves. By introducing the concept of vanishing complex-valued correction in the weak asymptotic method, we are able to settle this question.

[22] B. Hayes, P. G. LeFloch, Measure-solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), 1547–1563. MR1419460.
Available as PDF (217 Kbytes).
Henrik Kalisch,
Darko Mitrovic,
Submitted by:
; 2011-01-20.