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)
Wave interactions and stability of the Riemann solutions for the chromatography equations. (English) Zbl 1184.35207
Summary: We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case.
MSC:
35L67Shocks and singularities
35L65Conservation laws
35B35Stability of solutions of PDE
References:
[1]Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields, Invent. math. 158, 227-260 (2004) · Zbl 1075.35087 · doi:10.1007/s00222-004-0367-2
[2]Ambrosio, L.; Crippa, G.; Figalli, A.; Spinolo, L. A.: 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
[3]Ancona, F.; Goatin, P.: Uniqueness and stability of L solutions for temple class systems with boundary and properties of the attenaible sets, SIAM J. Math. anal. 34, 28-63 (2002) · Zbl 1020.35050 · doi:10.1137/S0036141001383424
[4]Barti, P.; Bressan, A.: The semigroup generated by a temple class system with large data, Differential integral equations 10, 401-418 (1997) · Zbl 0890.35083
[5]Bianchini, S.: Stability of L solutions for hyperbolic systems with coinciding shocks and rarefactions, SIAM J. Math. anal. 33, 959-981 (2001) · Zbl 1009.35052 · doi:10.1137/S0036141000377900
[6]Bressan, A.: Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem, Oxford lecture ser. Math. appl. 20 (2000) · Zbl 0997.35002
[7]Bressan, A.; Goatin, P.: Stability of L solutions of temple class systems, Differential integral equations 13, 1503-1528 (2000) · Zbl 1047.35095
[8]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
[9]Butto, A.; Storti, G.; Mazzotti, M.: Shock formation in binary systems with nonlinear characteristic curves, Chem. eng. Sci. 63, 4159-4170 (2008)
[10]Chang, T.; Hsiao, L.: The Riemann problem and interaction of waves in gas dynamics, Pitman monogr. Surv. pure appl. Math. 41 (1989) · Zbl 0698.76078
[11]Dafermos, C. M.: Hyperbolic conservation laws in continuum physics, Grundlehren math. Wiss. (2000)
[12]Liu, T. P.; Yang, T.: L1 stability of conservation laws with coinciding hugoniot and characteristic curves, Indiana univ. Math. J. 48, 237-247 (1999) · Zbl 0935.35090 · doi:10.1512/iumj.1999.48.1601 · doi:http://www.iumj.indiana.edu/TOC/991.htm
[13]Mazzotti, M.: Local equilibrium theory for the binary chromatography of species subject to a generalized Langmuir isotherm, Ind. eng. Chem. res. 45, 5332-5350 (2006)
[14]Ostrov, D. N.: Asymptotic behavior of two interreacting chemicals in a chromatography reactor, SIAM J. Math. anal. 27, 1559-1596 (1996) · Zbl 0865.35086 · doi:10.1137/S0036141094275701
[15]Rhee, H. K.; Aris, R.; Amundson, N. R.: First-order partial differential equations, vol. 1: theory and application of single equations, (2001)
[16]Rhee, H. K.; Aris, R.; Amundson, N. R.: First-order partial differential equations, vol. 2: theory and application of hyperbolic systems of quasilinear equations, (2001)
[17]Serre, D.: Systems of conservation laws I/II, (1999/2000)
[18]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 · doi:10.1016/j.jmaa.2008.11.005
[19]Smoller, J.: Shock waves and reaction – diffusion equations, (1994)
[20]Strohlein, G.; Morbidelli, M.; Rhee, H. K.; Mazzotti, M.: Modeling of modifier – solute peak interactions in chromatography, Aiche J. 52, 565-573 (2006)
[21]Sun, M.: Interactions of elementary waves for aw – rascle model, SIAM J. Appl. math. 69, 1542-1558 (2009) · Zbl 1184.35208 · doi:10.1137/080731402
[22]Sun, M.; Sheng, W.: The ignition problem for a scalar nonconvex combustion model, J. differential equations 231, 673-692 (2006) · Zbl 1113.35120 · doi:10.1016/j.jde.2006.08.014
[23]Temple, B.: Systems of conservation laws with invariant submanifolds, Trans. amer. Math. soc. 280, 781-795 (1983) · Zbl 0559.35046 · doi:10.2307/1999646