zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 waves in chromatography equations. (English) Zbl 1217.35120
Summary: The previous investigations on delta shock waves were mostly focused on those with Dirac delta function in only one state variable. In this paper, we obtain another kind from the nonlinear chromatography equations, in which the Dirac delta functions develop simultaneously in both state variables. It is strictly proved to satisfy the system in the sense of distributions. The generalized Rankine-Hugoniot relation and entropy condition are clarified. The numerical results completely coinciding with the theoretical analysis are presented.
35L67Shocks and singularities
35L60Nonlinear first-order hyperbolic equations
35L65Conservation laws
[1]D. Korchinski, Solution of a Riemann problem for a 2×2 system of conservation laws possessing no classical weak solution, thesis, Adelphi University, 1977.
[2]Keyfitz, B.; Kranzer, H.: A viscosity approximation to a system of conservation laws with no classical Riemann solution, Lecture notes in math. 1402, 185-197 (1989) · Zbl 0704.35094
[3]Keyfitz, B.; Kranzer, H.: 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
[4]Lefloch, P.: An existence and uniqueness result for two nonstrictly hyperbolic systems, vol. 219, (1990)
[5]Joseph, K.: A Riemann problem whose viscosity solution contain δ-measures, Asymptot. anal. 7, 105-120 (1993) · Zbl 0791.35077
[6]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 · doi:10.1006/jdeq.1994.1093
[7]Sheng, W.; Zhang, T.: The Riemann problem for transportation equation in gas dynamics, Mem. amer. Math. soc. 137, No. 654 (1999)
[8]Li, J.; Zhang, T.: Generalized rankine-hugoniot relations of delta-shocks in solutions of transportation equations, Nonlinear PDE and related areas, 219-232 (1998) · Zbl 0929.35092
[9]Li, J.; Yang, S.; Zhang, T.: The two-dimensional Riemann problem in gas dynamics, Pitman monogr. Surv. pure appl. Math. 98 (1998) · Zbl 0935.76002
[10]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 · doi:10.1016/j.jmaa.2008.01.042
[11]Yang, H.: 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
[12]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 · doi:10.3934/cpaa.2010.9.431
[13]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 · doi:10.1137/S0036141001399350
[14]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 · doi:10.1016/j.physd.2003.09.039
[15]Danilov, V.; Shelkovich, V.: Dynamics of propagation and interaction of δ-shock waves in conservation laws systems, J. differential equations 221, 333-381 (2005) · Zbl 1072.35121 · doi:10.1016/j.jde.2004.12.011
[16]Temple, B.: Systems of conservations laws with invariant submanifolds, Trans. amer. Math. soc. 280, 781-795 (1983) · Zbl 0559.35046 · doi:10.2307/1999646
[17]Temple, B.: Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws, Adv. in appl. Math. 3, 335-375 (1982) · Zbl 0508.76107 · doi:10.1016/S0196-8858(82)80010-9
[18]Mazzotti, M.: Nonclassical composition fronts in nonlinear chromatography: delta-shock, Ind. eng. Chem. res. 48, 7733-7752 (2009)
[19]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)
[20]Ambrosio, L.; Crippa, G.; Figalli, A.; Spinolo, L.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system, SIAM J. Math. anal. 41, 1890-1920 (2009) · Zbl 1222.35060 · doi:10.1137/090754686
[21]Bressan, A.; Shen, W.: Uniqueness of discontinuous ODE and conservation laws, Nonlinear anal. 34, 637-652 (1998) · Zbl 0948.34006 · doi:10.1016/S0362-546X(97)00590-7
[22]Shen, C.: Wave interactions and stability of the Riemann solutions for the chromatography equations, J. math. Anal. appl. 365, 609-618 (2010) · Zbl 1184.35207 · doi:10.1016/j.jmaa.2009.11.037
[23]Nessyahu, H.; Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. Phys. 87, 408-463 (1990) · Zbl 0697.65068 · doi:10.1016/0021-9991(90)90260-8