## Delta shock waves with Dirac delta function in both components for systems of conservation laws.(English)Zbl 1304.35422

Summary: We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine-Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine-Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.

### MSC:

 35L67 Shocks and singularities for hyperbolic equations 35L65 Hyperbolic conservation laws 78A05 Geometric optics
Full Text:

### References:

 [1] Engquist, B.; Runborg, O., Multiphase computations in geometrical optics, J. Comput. Appl. Math., 74, 175-192 (1996) · Zbl 0947.78001 [2] Engquist, B., Multiscale and multiphase methods for wave propagation (1998), Royal Institute of Technology: Royal Institute of Technology Stockholm, Sweden, Doctoral dissertation [3] Dafermos, C. M., Solutions of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Ration. Mech. Anal., 52, 1-9 (1973) · Zbl 0262.35034 [4] Slemrod, M.; Tzavaras, A., A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Indiana Univ. Math. J., 38, 1047-1074 (1989) · Zbl 0675.76073 [5] Sheng, W.; Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137, 564 (1999) · Zbl 0913.35082 [6] Shu, C., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, 7-65 (1998), ICASE Report · Zbl 0927.65111 [7] Yang, H.; Cheng, H., Riemann problem for a geometrical optics system, Acta Math. Sin., Engl. Series (2014), to appear · Zbl 1307.35167 [8] Yang, H.; Zhang, Y., New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252, 5951-5993 (2012) · Zbl 1248.35127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.