zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. (English) Zbl 1192.35120
Summary: We describe δ-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:
35L67Shocks and singularities
35L65Conservation laws
References:
[1]Arnold, V. I.: Obyknovennyje differencial’nyje uravnenija, (1971)
[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, No. 4, 925-938 (2003) · Zbl 1038.35035 · doi:10.1137/S0036141001399350
[3]Dafermos, C. M.: Hyperbolic conservation laws in continuum physics, (2000)
[4]Danilov, V. G.: Generalized solution describing singularity interaction, Int. J. Math. math. Sci. 29, No. 22, 481-494 (February 2002) · Zbl 1011.35093 · doi:10.1155/S0161171202012206 · doi:http://www.hindawi.com/journals/ijmms/volume-29/S0161171202012206.html
[5]Danilov, V. G.: On singularities of conservation equation solution, Nonlinear anal. (2007)
[6]Danilov, V. G.; Shelkovich, V. M.: Propagation and interaction of nonlinear waves to quasilinear equations, , 326-328 (2000)
[7]Danilov, V. G.; Shelkovich, V. M.: Propagation and interaction of shock waves of quasilinear equations, Nonlinear stud. 8, No. 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 · doi:10.1016/j.jde.2004.12.011
[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 · doi:10.1016/j.na.2005.01.034
[11]V.G. Danilov, D. Mitrovic, Smooth approximations of global in time solutions to scalar conservation law, preprint · Zbl 1172.35450 · doi:10.1155/2009/350762
[12]Danilov, V. G.; Omelianov, G. A.; Shelkovich, V. M.: Weak asymptotic method and interaction of nonlinear waves, Amer. math. Soc. transl. Ser. 208, 33-165 (2003) · Zbl 1140.35382
[13]Ding, X.; Wang, Z.: Existence and uniqueness of discontinuous solution defined by Lebesgue – Stieltjes integral, Sci. China ser. A 39, No. 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 (1998) · Zbl 0933.35126
[15]Huang, F.: Weak solution to pressureless type system, Comm. partial differential equations 30, No. 1 – 3, 283-304 (2005) · Zbl 1074.35021 · doi:10.1081/PDE-200050026
[16]Huang, F.: Existence and uniqueness of discontinuous solutions for a hyperbolic system, Proc. roy. Soc. Edinburgh sect. A 127, No. 6, 1193-1205 (1997) · Zbl 0887.35093 · doi:10.1017/S0308210500027013
[17]Ilin, A. M.: Matching of asymptotic expansions of solutions of boundary value problems, (1989) · Zbl 0671.35002
[18]Ercole, G.: Delta-shock waves as self-similar viscosity limits, Quart. appl. Math. 58, No. 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) · Zbl 0768.35058 · doi:10.1007/BF03167192
[20]Hayes, B.; Lefloch, P. G.: Measure-solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9, 1547-1563 (1996) · Zbl 0908.35075 · doi:10.1088/0951-7715/9/6/009
[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 · doi:10.1006/jdeq.1995.1080
[23]Lefloch, P. G.: An existence and uniqueness result for two nonstrictly hyperbolic systems, IMA vol. Math. appl. 27, 126-138 (1990) · Zbl 0727.35083
[24]Liu, Y. -P.; Xin, Z.: Overcompressive shock waves, Nonlinear evolution equations that change type, 145-149 (1990)
[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, No. 114, 1-22 (2007) · Zbl 1138.35367 · doi:emis:journals/EJDE/Volumes/2007/114/abstr.html
[26]Mitrovic, D.; Nedeljkov, M.: Delta shock waves as a limit of shock waves, J. hyperbolic differ. Equ. 4, No. 4, 629-653 (2007) · Zbl 1145.35086 · doi:10.1142/S021989160700129X
[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 · doi:10.1002/mma.480
[29]Panov, E. Yu.; Shelkovich, V. M.: δ ' -shock waves as a new type of solutions to systems of conservation laws, J. differential equations 228, No. 1, 49-86 (2006) · Zbl 1108.35116 · doi:10.1016/j.jde.2006.04.004
[30]Shelkovich, V. M.: The Riemann problem admitting δ-,δ ' -shocks, and vacuum states (the vanishing viscosity approach), J. differential equations 231, No. 2, 459-500 (2006) · Zbl 1108.35117 · doi:10.1016/j.jde.2006.08.003
[31]Sheng, W.; Zhang, T.: The Riemann problem for transportation equations in gas dynamics, Mem. amer. Math. soc. 137, No. 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 · doi:10.1006/jdeq.1994.1093
[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 · doi:10.1006/jdeq.1999.3629