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Singular solutions of a fully nonlinear $2×2$ system of conservation laws. (English) Zbl 06101591

Authors abstract : Existence and admissibilityof $\delta -$shock solutions is discussed for the non-convex srictly hyperbolic system of equations

$\begin{array}{cc}\hfill {\partial }_{t}u+{\partial }_{x}\left(\frac{1}{2}\left({u}^{2}+{v}^{2}\right)\right)& =0\hfill \\ \hfill {\partial }_{t}v+{\partial }_{x}\left(v\left(u-1\right)\right)& =0\hfill \end{array}$

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 Danilov and Shelkovich. In deed, in this context, we can show that every 2$×$2 system of conservation laws admits $\delta -$ shock solutions.

MSC:
 35L65 Conservation laws 35L67 Shocks and singularities 76W05 Magnetohydrodynamics and electrohydrodynamics