×

New developments of delta shock waves and its applications in systems of conservation laws. (English) Zbl 1248.35127

The authors develop the theory of delta shock type solutions to the \(2\times 2\) system of conservation laws \(u_t+(\phi(r)u)_x=v_t+(\phi(r)v)_x=0\), where \(r=au+bv\) with some constants \(a,b\). Both state variables \(u,v\) may contain delta shocks. The generalized Rankine-Hugoniot relation and the entropy condition are proposed, the existence and uniqueness of delta shock type solutions of the Riemann problem are established. The authors also prove the existence and convergence of viscous approximations. Some applications to known systems are given, numerical simulations are presented.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
76L05 Shock waves and blast waves in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bouchut, F., On zero-pressure gas dynamics, advances in kinetic theory and computing, (Ser. Adv. Math. Appl. Sci., vol. 22 (1994), World Scientific: World Scientific River Edge, NJ), 171-190 · Zbl 0863.76068
[2] Brenier, Y., Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas dynamics, J. Math. Fluid Mech., 7, S326-S331 (2005) · Zbl 1085.35097
[3] Chen, G.; Liu, H., Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations, SIAM J. Math. Anal., 34, 925-938 (2003) · Zbl 1038.35035
[4] Chen, G.; Liu, H., Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D, 189, 141-165 (2004) · Zbl 1098.76603
[5] Cheng, H.; Liu, W.; Yang, H., Two-dimensional Riemann problems for zero-pressure gas dynamics with three constant states, J. Math. Anal. Appl., 343, 127-140 (2008) · Zbl 1139.35073
[6] Cheng, H.; Yang, H., Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl., 381, 1, 17-26 (2011) · Zbl 1220.35126
[7] Cheng, H.; Yang, H., Delta shock waves in chromatography equations, J. Math. Anal. Appl., 380, 475-485 (2011) · Zbl 1217.35120
[8] 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
[9] Dafermos, C. M.; Diperna, R. J., The Riemann problem for certain classes if hyperbolic systems of conservation laws, J. Differential Equations, 20, 90-114 (1976) · Zbl 0323.35050
[10] Danilov, V. G.; Mitrovic, D., Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations, 245, 3704-3734 (2008) · Zbl 1192.35120
[11] Danilov, V. G.; Shelkovich, V. M., Dynamics of propagation and interaction of delta-shock waves in conservation law systems, J. Differential Equations, 211, 333-381 (2005) · Zbl 1072.35121
[12] Danilov, V. G.; Shelkovich, V. M., Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math., 63, 3, 401-427 (2005)
[13] Ding, X.; Wang, Z., Existence and uniqueness of discontinuous solutions defined by Lebesgue-Stieltjes integral, Sci. China Ser. A, 39, 807-819 (1996) · Zbl 0866.35065
[14] Ercole, G., Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math., LVIII, 1, 177-199 (2000) · Zbl 1157.35430
[15] Forester, A.; Le Floch, P., Multivalued solutions to some nonlinear and nonstrictly hyperbolic systems, Japan J. Indust. Appl. Math., 9, 1-23 (1992) · Zbl 0768.35058
[16] Glimm, J., Solutions in the large for nonlinear systems of equations, Comm. Pure Appl. Math., 18, 697-715 (1965) · Zbl 0141.28902
[17] Guo, L.; Sheng, W.; Zhang, T., The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9, 431-458 (2010) · Zbl 1197.35164
[18] Hu, J., A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws, Quart. Appl. Math., 55, 361-373 (1997) · Zbl 0877.35076
[19] F. Huang, Existence and uniqueness of discontinuous solutions for a class nonstrictly differential equations and related areas, in: Advances in Nonlinear Partial Differential Equations and Related Areas, Beijing, 1997, pp. 187-208.; F. Huang, Existence and uniqueness of discontinuous solutions for a class nonstrictly differential equations and related areas, in: Advances in Nonlinear Partial Differential Equations and Related Areas, Beijing, 1997, pp. 187-208. · Zbl 0933.35126
[20] Jiang, G.; Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19, 1892-1917 (1998) · Zbl 0914.65095
[21] Joseph, K. T., A Riemann problem whose viscosity solution contain \(δ\)-measures, Asymptot. Anal., 7, 105-120 (1993) · Zbl 0791.35077
[22] Keyfitz, B. L.; Kranzer, H. C., A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal., 72, 219-241 (1980) · Zbl 0434.73019
[23] Keyfitz, B. L.; Kranzer, H. C., A viscosity approximation to a system of conservation laws with no classical Riemann solution, (Nonlinear Hyperbolic Problems. Nonlinear Hyperbolic Problems, Lecture Notes in Math., vol. 1402 (1989)), 185-197 · Zbl 0704.35094
[24] Keyfitz, B. L.; Kranzer, H. C., Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118, 420-451 (1995) · Zbl 0821.35096
[25] D.J. Korchinski, Solution of a Riemann problem for a \(2 \times 2\); D.J. Korchinski, Solution of a Riemann problem for a \(2 \times 2\)
[26] Kranzer, H. C.; Keyfitz, B. L., A strictly hyperbolic system of conservation laws admitting singular shock, (Nonlinear Evolution Equations that Change Type. Nonlinear Evolution Equations that Change Type, IMA Vol. Math. Appl., vol. 27 (1990), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0718.76071
[27] Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973), SIAM: SIAM Philadelphia · Zbl 0268.35062
[28] Le Floch, P., An existence and uniqueness result for two nonstrictly hyperbolic systems, (Nonlinear Evolution Equations that Change Type. Nonlinear Evolution Equations that Change Type, IMA Vol. Math. Appl., vol. 27 (1990), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0727.35083
[29] Li, J.; Zhang, T., Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations, (Advances in Nonlinear Partial Differential Equations and Related Areas (1998), World Scientific: World Scientific Singapore), 219-232 · Zbl 0929.35092
[30] Li, J.; Yang, S.; Zhang, T., The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monogr. Surv. Pure Appl. Math., vol. 98 (1998), Longman · Zbl 0935.76002
[31] Li, J.; Yang, H., Delta-shocks as limit of solutions of multidimensional zero-pressure gas dynamics, Quart. Appl. Math., 59, 315-342 (2001) · Zbl 1019.76040
[32] Liu, T.; Wang, C., On a nonstrictly hyperbolic system of conservation laws, J. Differential Equations, 57, 1-14 (1980) · Zbl 0576.35075
[33] Mazzotti, M., Nonclassical composition fronts in nonlinear chromatography: delta-shock, Ind. Eng. Chem. Res., 48, 7733-7752 (2009)
[34] Mazzotti, M.; Tarafder, A.; Cornel, J.; Gritti, F.; Guiochon, G., Experimental evidence of a delta-shock in nonlinear chromatography, J. Chromatogr. A, 1217, 2002-2012 (2010)
[35] Nedeljkov, M.; Oberguggenberger, M., Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl., 344, 1143-1157 (2008) · Zbl 1155.35059
[36] Oberguggenberger, M., Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Res. Notes Math. Ser., vol. 259 (1992), Longman: Longman Harlow · Zbl 0818.46036
[37] Oberguggenberger, M.; Wang, Y. G., Generalized solutions to conservation laws, Z. Anal. Anwend., 13, 7-18 (1994) · Zbl 0794.35102
[38] Panov, E. Y.; Shelkovich, V. M., \( \delta^\prime \)-shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations, 228, 49-86 (2006) · Zbl 1108.35116
[39] Serre, D., Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191, 539-577 (2009) · Zbl 1161.76025
[40] Shandarin, S. F.; Zeldovich, Y. B., The large-scale structure of the universe: turbulence, intermittency, structure in a self-gravitating medium, Rev. Modern Phys., 61, 185-220 (1989)
[41] Shelkovich, V. M., Delta-shock waves for a class of hyperbolic systems of conservation laws, (Abramian, A.; Vakulenko, S.; Volpert, V., Patterns and Waves (2003), Akadem Print: Akadem Print St. Petersburg), 155-168
[42] Shelkovich, V. M., The Riemann problem admitting \(δ-, \delta^\prime \)-shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231, 459-500 (2006) · Zbl 1108.35117
[43] V.M. Shelkovich, Delta-shock waves in nonlinear chromatography, in: 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, June 15-19, 2010, Xijiao Hotel, Beijing, China, Program and Abstracts, Beijing, China, 2010, pp. 103-106.; V.M. Shelkovich, Delta-shock waves in nonlinear chromatography, in: 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, June 15-19, 2010, Xijiao Hotel, Beijing, China, Program and Abstracts, Beijing, China, 2010, pp. 103-106.
[44] V.M. Shelkovich, One class of systems of conservation laws admitting delta-shocks, in: Proceedings of 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Beijing, China, 2011, in press.; V.M. Shelkovich, One class of systems of conservation laws admitting delta-shocks, in: Proceedings of 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Beijing, China, 2011, in press. · Zbl 1262.35166
[45] Shen, C.; Sun, M., Interactions of delta shock waves for the transport equations with split delta functions, J. Math. Anal. Appl., 351, 747-755 (2009) · Zbl 1159.35042
[46] Sheng, W.; Zhang, T., The Riemann problem for transportation equation in gas dynamics, Mem. Amer. Math. Soc., 137, 654 (1999) · Zbl 0913.35082
[47] Slemrod, M., A limiting viscosity approach to the Riemann problem for materials exhibiting change of phase, Arch. Ration. Mech. Anal., 41, 327-366 (1989) · Zbl 0701.35101
[48] Slemrod, M.; Tzavaras, A. E., A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Indiana Univ. Math. J., 4, 1047-1074 (1989) · Zbl 0675.76073
[49] Tan, D.; Zhang, T., Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws, I. Four-\(J\) cases, II. Initial data involving some rarefaction waves, J. Differential Equations, 111, 203-253 (1994), 253-283
[50] Tan, D.; Zhang, T.; Zheng, Y., Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conversation laws, J. Differential Equations, 112, 1-32 (1994) · Zbl 0804.35077
[51] Temple, B., Global solution of the Cauchy problem for a class of \(2 \times 2\) nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3, 335-375 (1982) · Zbl 0508.76107
[52] Temple, B., Systems of conservations laws with invariant submanifolds, Trans. Amer. Math. Soc., 280, 781-795 (1983) · Zbl 0559.35046
[53] Tupciev, V. A., On the method of introducing viscosity in the study of problems involving decay of a discontinuity, Dokl. Akad. Nauk SSR. Dokl. Akad. Nauk SSR, Soviet Math. Dokl., 14, 978-982 (1973), transl. in: · Zbl 0294.35065
[54] Volʼpert, A. I., The space BV and quasilinear equations, Math. USSR Sb., 2, 225-267 (1967) · Zbl 0168.07402
[55] Weinan, E.; Rykov, Yu. G.; Sinai, Ya. G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in ashesion particle dynamics, Comm. Math. Phys., 177, 349-380 (1996) · Zbl 0852.35097
[56] Yang, H., Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159, 447-484 (1999) · Zbl 0948.35079
[57] Yang, H., Generalized plane delta-shock waves for \(n\)-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260, 18-35 (2001) · Zbl 0985.35044
[58] Yang, H.; Sun, W., The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws, Nonlinear Anal., 67, 3041-3049 (2007) · Zbl 1120.35067
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.