# zbMATH — the first resource for mathematics

The Rankine-Hugoniot conditions and balance laws for $$\delta$$ -shocks. (English) Zbl 1151.35400
J. Math. Sci., New York 151, No. 1, 2781-2792 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 213-229 (2006).
Summary: New definitions of $$\delta$$-shock-wave-type solutions are introduced for two (one-dimensional) types of hyperbolic systems of conservation laws. The corresponding Rankine-Hugoniot conditions for $$\delta$$-shocks are derived and their geometrical interpretation is given. Balance laws connected with “area” mass and momentum transportation for $$\delta$$-shocks are derived.

##### MSC:
 35L65 Hyperbolic conservation laws 35L67 Shocks and singularities for hyperbolic equations
Full Text:
##### References:
 [1] F. Bouchut, ”On zero pressure gas dynamics,” Adv. Math. Sci. Appl., 22, 171–190 (1994). · Zbl 0863.76068 [2] V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, ”Algebras of the singularities of singular solutions of first-order quasilinear strictly hyperbolic systems,” Theor. Math. Phys., 114, No. 1, 1–42 (1998). · Zbl 0946.35049 · doi:10.1007/BF02557106 [3] V. G. Danilov, G. A. Omel’yanov, and V. M. Shelkovich, ”Weak asymptotics method and interaction of nonlinear waves,” in: M. Karasev, ed., Asymptotic Methods for Wave and Quantum Problems, Transl. Amer. Math. Soc., Ser. 2, Vol. 208, Amer. Math. Soc. (2003), pp. 33–165. · Zbl 1140.35382 [4] V. G. Danilov and V. M. Shelkovich, ”Propagation and interaction of nonlinear waves to quasilinear equations,” in: Hyperbolic problems: Theory, Numerics, Applications (8th Int. Conf. in Magdeburg, February/March 2000, Vol. I), Int. Series of Numerical Math., Vol. 140, Birkhäuser, Basel (2001), pp. 267–276. [5] V. G. Danilov and V. M. Shelkovich, ”Propagation and interaction of shock waves of quasilinear equations,” Nonlinear Stud., 8, No. 1, 135–169 (2001). · Zbl 1008.35041 [6] V. G. Danilov and V. M. Shelkovich, ”Propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws,” in: T. Y. Hou and E. Tadmor, eds., Hyperbolic Problems: Theory, Numerics, Applications, Proc. 9th Int. Conf. on Hyperbolic Problems held in CalTech, Pasadena, March 25–29, 2002, Springer-Verlag (2003), pp. 483–492. · Zbl 1098.35106 [7] V. G. Danilov and V. M. Shelkovich, ”Propagation and interaction of $$\delta$$-shock waves to hyperbolic systems of conservation laws,” Russ. Acad. Sci. Dokl. Math., 69, No. 1, 4–8 (2004). · Zbl 1129.35440 [8] V. G. Danilov and V. M. Shelkovich, ”Delta-shock-wave-type solution of hyperbolic systems of conservation laws,” Quart. Appl. Math., 63, No. 3, 401–427 (2005). [9] V. G. Danilov and V. M. Shelkovich, ”Dynamics of propagation and interaction of delta-shock waves in conservation law systems,” J. Differential Equations, 211, 333–381 (2005). · Zbl 1072.35121 · doi:10.1016/j.jde.2004.12.011 [10] G. Ercole, ”Delta-shock waves as self-similar viscosity limits,” Quart. Appl. Math., 58, No. 1, 177–199 (2000). · Zbl 1157.35430 [11] K. T. Joseph, ”A Riemann problem whose viscosity solutions contain $$\delta$$-measures,” Asympt. Anal., 7, 105–120 (1993). · Zbl 0791.35077 [12] F. Huang, ”Existence and uniqueness of discontinuous solutions for a class nonstrictly hyperbolic systems,” in: G.-Q. Chen et al., eds., Advances in Nonlinear Partial Differential Equations and Related Areas, Proc. of Conf. Dedicated to Prof. Xiaqi Ding, China (1997), pp. 187–208. · Zbl 0933.35126 [13] B. L. Keyfitz, ”Conservation laws, delta-shocks and singular shocks,” in: M. Grosser, G. Hormann, and M. Oberguggenberger, eds., Nonlinear Theory of Generalized Functions, Chapman & Hall/CRC (1999), pp. 99–112. · Zbl 0933.35134 [14] B. L. Keyfitz and H. C. Kranzer, ”Spaces of weighted measures for conservation laws with singular shock solutions,” J. Differential Equations, 118, 420–451 (1995). · Zbl 0821.35096 · doi:10.1006/jdeq.1995.1080 [15] P. Le Floch, ”An existence and uniqueness result for two nonstrictly hyperbolic systems,” in: Nonlinear Evolution Equations That Change Type, Springer-Verlag (1990), pp. 126–138. [16] T.-P. Liu and Zh. Xin, ”Overcompressive shock waves,” in: Nonlinear Evolution Equations That Change Type, Springer-Verlag (1990), pp. 139–145. [17] V. P. Maslov, ”Three algebras corresponding to nonsmooth solutions of systems of quasilinear hyperbolic equations,” Usp. Mat. Nauk, 35, No. 2, 252–253 (1980). [18] V. P. Maslov, ”Non-standard characteristics in asymptotical problems,” in: Proc. Int. Congr. Math., August 16–24, 1983, Warszawa, Vol. I, North-Holland, Amsterdam (1984), pp. 139–185. [19] S. F. Shandarin and Ya. B. Zeldovich, ”The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium,” Rev. Modern Phys., 61, 185–220 (1989). · doi:10.1103/RevModPhys.61.185 [20] V. M. Shelkovich, ”Delta-shock waves of a class of hyperbolic systems of conservation laws,” in: A. Abramian, S. Vakulenko and V. Volpert, eds., Patterns and Waves, AkademPrint, St. Petersburg (2003), pp. 155–168. [21] V. M. Shelkovich, A Specific Hyperbolic System of Conservation Laws Admitting Delta-Shock-Wave-Type Solutions, Preprint 2003-059, http://www.math.ntnu.no/conservation/2003/059.html . [22] W. Sheng and T. Zhang, eds., The Riemann Problem for the Transportaion Equations in Gas Dynamics, Mem. Amer. Math. Soc., Vol. 654, Amer. Math. Soc. (1999). [23] D. Tan, T. Zhang, and Y. Zheng, ”Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws,” J. Differential Equations, 112, 1–32 (1994). · Zbl 0804.35077 · doi:10.1006/jdeq.1994.1093 [24] A. I. Volpert, ”The space BV and quasilinear equations,” Math. USSR Sb., 2, 225–267 (1967). · Zbl 0168.07402 · doi:10.1070/SM1967v002n02ABEH002340 [25] E. Weinan, Yu. Rykov, and Ya. G. Sinai, ”Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics,” Commun. Math. Phys., 177, 349–380 (1996). · Zbl 0852.35097 · doi:10.1007/BF02101897 [26] G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). · Zbl 0373.76001 [27] H. Yang, ”Riemann problems for a class of coupled hyperbolic systems of conservation laws,” J. Differential Equations, 159, 447–484 (1999). · Zbl 0948.35079 · doi:10.1006/jdeq.1999.3629 [28] Ya. B. Zeldovich, ”Gravitational instability: An approximate theory for large density perturbations,” Astronom. Astrophys., 5, 84–89 (1970).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.