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Singular solutions of a fully nonlinear \(2 \times 2\) system of conservation laws. (English) Zbl 1286.35162

Authors abstract: Existence and admissibility of \(\delta\)-shock solutions is discussed for the non-convex strictly hyperbolic system of equations \[ \begin{aligned} \partial _{t}u+\partial _{x}(\frac{1}{2}(u^{2}+v^{2})) &=0, \\ \partial _{t}v+\partial _{x}(v(u-1)) &=0. \end{aligned} \] The system is fully nonlinear, i.e. it is nonlinear with respect to both unknows, 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 \(\delta\)-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 V. G. Danilov and V. M. Shelkovich [J. Differ. Equations 211, No. 2, 333–381 (2005; Zbl 1072.35121)]. Indeed, in this context, we can show that every 2\(\times \)2 system of conservation laws admits \(\delta\)-shock solutions.

MSC:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76W05 Magnetohydrodynamics and electrohydrodynamics

Citations:

Zbl 1072.35121
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