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On Existence and Uniqueness of Singular Solutions for Systems of Conservation Laws Kalisch, Henrik


Existence and admissibility of delta-shock solutions is discussed for hyperbolic systems of conservation laws. One of the systems discussed is fully nonlinear, and does not admit a classical Lax-admissible solution to certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive delta-shock wave solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued and the resulting distributions fit into a generalized concept of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of delta-shock waves in hyperbolic systems, J. Differential Equations 211 (2005), 333-381]. In this framework, it can be shown that every 2x2 system of conservation laws admits delta-shock solutions. As an example, it is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally leads to singular solutions featuring Dirac delta distributions in the context of shallow-water flows.

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