×

zbMATH — the first resource for mathematics

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. (English) Zbl 1192.35120
Summary: We describe \(\delta \)-shock wave generation from continuous initial data in the case of triangular conservation law system arising from “generalized pressureless gas dynamics model”. We use smooth approximations in the weak sense that are more general than small viscosity approximations.

MSC:
35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arnold, V.I., Obyknovennyje differencial’nyje uravnenija, (1971), Nauka Moskva, (in Russian)
[2] Chen, G.-Q.; Liu, H., Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. math. anal., 34, 4, 925-938, (2003) · Zbl 1038.35035
[3] Dafermos, C.M., Hyperbolic conservation laws in continuum physics, (2000), Springer Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Singapore, Tokyo · Zbl 0940.35002
[4] Danilov, V.G., Generalized solution describing singularity interaction, Int. J. math. math. sci., 29, 22, 481-494, (February 2002)
[5] Danilov, V.G., On singularities of conservation equation solution, Nonlinear anal., (2007) · Zbl 1132.80002
[6] Danilov, V.G.; Shelkovich, V.M., Propagation and interaction of nonlinear waves to quasilinear equations, (), 326-328 · Zbl 1008.35041
[7] Danilov, V.G.; Shelkovich, V.M., Propagation and interaction of shock waves of quasilinear equations, Nonlinear stud., 8, 1, 135-169, (2001) · Zbl 1008.35041
[8] Danilov, V.G.; Shelkovich, V.M., Dynamics of propagation and interaction of δ-shock waves in conservation law system, J. differential equations, 211, 333-381, (2005) · Zbl 1072.35121
[9] Danilov, V.G.; Shelkovich, V.M., Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. appl. math., 63, 401-427, (2005)
[10] Danilov, V.G.; Mitrovic, D., Weak asymptotic of shock wave formation process, Nonlinear anal. TMA, 61, 613-635, (2005) · Zbl 1079.35067
[11] V.G. Danilov, D. Mitrovic, Smooth approximations of global in time solutions to scalar conservation law, preprint · Zbl 1172.35450
[12] Danilov, V.G.; Omelianov, G.A.; Shelkovich, V.M., Weak asymptotic method and interaction of nonlinear waves, (), 33-165 · Zbl 1140.35382
[13] Ding, X.; Wang, Z., Existence and uniqueness of discontinuous solution defined by lebesgue – stieltjes integral, Sci. China ser. A, 39, 8, 807-819, (1996) · Zbl 0866.35065
[14] Huang, F., Existence and uniqueness of discontinuous solutions for a class of non-strictly hyperbolic system, (), 187-208 · Zbl 0933.35126
[15] Huang, F., Weak solution to pressureless type system, Comm. partial differential equations, 30, 1-3, 283-304, (2005) · Zbl 1074.35021
[16] Huang, F., Existence and uniqueness of discontinuous solutions for a hyperbolic system, Proc. roy. soc. Edinburgh sect. A, 127, 6, 1193-1205, (1997) · Zbl 0887.35093
[17] Ilin, A.M., Matching of asymptotic expansions of solutions of boundary value problems, (1992), Amer. Math. Soc. RI, English transl.:
[18] Ercole, G., Delta-shock waves as self-similar viscosity limits, Quart. appl. math., LVIII, 1, 177-199, (2000) · Zbl 1157.35430
[19] Forestier, A.; LeFloch, P.G., Multivalued solutions to some nonlinear and nonstrictly hyperbolic systems, Japan J. indust. appl. math., 9, 1-23, (1992)
[20] Hayes, B.; LeFloch, P.G., Measure-solutions to a strictly hyperbolic system of conservation laws, Nonlinearity, 9, 1547-1563, (1996) · Zbl 0908.35075
[21] Joseph, K.T., A Riemann problem whose viscosity solution contains δ measures, Asymptot. anal., 7, 105-120, (1993) · Zbl 0791.35077
[22] Keyfitz, B.L.; Krantzer, H.C., Spaces of weighted measures for conservation laws with singular shock solutions, J. differential equations, 118, 420-451, (1995) · Zbl 0821.35096
[23] LeFloch, P.G., An existence and uniqueness result for two nonstrictly hyperbolic systems, (), 126-138
[24] Liu, Y.-P.; Xin, Z., Overcompressive shock waves, (), 145-149
[25] Mitrovic, D.; Susic, J., Global in time solution to Hopf equation and application to a non-strictly hyperbolic system of conservation laws, Electron. J. differential equations, 2007, 114, 1-22, (2007) · Zbl 1138.35367
[26] Mitrovic, D.; Nedeljkov, M., Delta shock waves as a limit of shock waves, J. hyperbolic differ. equ., 4, 4, 629-653, (2007) · Zbl 1145.35086
[27] Nedeljkov, M., Unbounded solutions to some systems of conservation laws – split delta shock waves, Mat. vesnik, 54, 145-149, (2002) · Zbl 1138.35368
[28] Nedeljkov, M., Delta and singular delta locus for one-dimensional systems of conservation laws, Math. methods appl. sci., 27, 931-955, (2004) · Zbl 1056.35115
[29] Panov, E.Yu.; Shelkovich, V.M., \(\delta^\prime\)-shock waves as a new type of solutions to systems of conservation laws, J. differential equations, 228, 1, 49-86, (2006) · Zbl 1108.35116
[30] Shelkovich, V.M., The Riemann problem admitting δ-, δ′-shocks, and vacuum states (the vanishing viscosity approach), J. differential equations, 231, 2, 459-500, (2006) · Zbl 1108.35117
[31] Sheng, W.; Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. amer. math. soc., 137, 645, 1-77, (1999) · Zbl 0913.35082
[32] Tan, D.; Zhang, T.; Zheng, Y., Delta shock waves as a limits of vanishing viscosity for a system of conservation laws, J. differential equations, 112, 1-32, (1994) · Zbl 0804.35077
[33] Volpert, A.I., The space BV and quasilinear equations, Math. USSR sb., 2, 225-267, (1967)
[34] Yang, H., Riemann problems for class of coupled hyperbolic system of conservation laws, J. differential equations, 159, 447-484, (1999) · Zbl 0948.35079
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.