GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS

doi:10.1007/s10473-019-0106-9

Abstract

Abstract: The Riemann problem for the chromatography equations in a conservative form is considered. The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states. The wave interaction problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann problem. In addition, it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.

Key words: Riemann problem, chromatography equation, wave interactions, Temple class, hyperbolic conservation laws

Ting ZHANG, Wancheng SHENG. GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS[J].Acta mathematica scientia,Series B, 2019, 39(1): 57-82.

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References

[1] Rhee H K, Aris R, Amundson N R. First-Order Partial Differential Equations, Vol 1:Theory and Application of Single Equations. New York:Dover Publications, 2001
[2] Rhee H K, Aris R, Amundson N R. First-Order Partial Differential Equations, Vol 2:Theory and Application of Hyperbolic Systems of Quasilinear Equations. New York:Dover Publications, 2001
[3] Aris R, Amundson N. Mathematical Methods in Chemical Engineering, Vol 2. Prentice-Hall, Englewood Cliffs, NJ, 1966
[4] Rhee H K, Aris R, Amundson N R. On the theory of multicomponent chromatography. Philos Trans R Soc London, 1970, A267:419-455
[5] Helfferich F, Klein G. Multicomponent Chromatography. New York:Marcel Dekker, 1970
[6] Temple B. Systems of conservation laws with invariant submanifolds. Trans Amer Math Soc, 1983, 280:781-795
[7] Ostrov D N. Asymptotic behavior of two interacting chemicals in a chromatography reactor. SIAM J Math Anal, 1996, 27:1559-1596
[8] Mazzotti M. Local equilibrium theory for the binary chromatography of species subject to a generalized Langmuir isotherm. Ind Eng Chem Res, 2006, 45:5332-5350
[9] Mazzotti M. Non-classical composition fronts in nonlinear chromatography:delta-shock. Ind Eng Chem Res, 2009, 48:7733-7752
[10] Mazzotti M, Tarafder A, Cornel J, Gritti F, Guiochon G. Experimental evidence of a delta-shock in nonlinear chromatography. J Chromatogr A, 2010, 1217(13):2002-2012
[11] Shen C. Wave interactions and stability of the Riemann solutions for the chromatography equations. J Math Anal Appl, 2010, 365:609-618
[12] Ambrosio L, Crippa G, Fifalli A, Spinolo L A. Some new well-posedness results for continuity and transport equations and applications to the chromatography system. SIAM J Math Anal, 2009, 41:1090-1920
[13] Sun M. Delta shock waves for the chromatography equations as self-similar viscosity limits. Q Appl Math, 2011, 69:425-443
[14] Bressan A, Shen W. Uniqueness if discintinuous ODE and conservation laws. Nonlinear Anal, 1998, 34:637-652
[15] Tsikkou C. Singular shocks in a chromatography model. J Math Anal Appl, 2016, 439:766-797
[16] Keyfitz B L, Kranzer H C. A viscosity approximation to a system of conservation laws with no classical Riemann solution//Nonlinear Hyperbolic Problems, Bordeaux 1988. Lecture Notes in Math, Vol 1402. Berlin:Springer, 1989:185-197
[17] Keyfitz B L, Kranzer H C. Spaces of weighted measures for conservation laws with singular shock solutions. J Differential Equations, 1995, 118(2):420-451
[18] Kranzer H C, Keyfitz B L. A strictly hyperbolic system of conservation laws admitting singular shocks//Nonlinear Evolution Equations that Change Type. IMA Vol Math Appl, Vol 27. New York:Springer, 1990:107-125
[19] Wang L, Bertozzi A L. Shock solutions for high concentration particle-laden thin films. SIAM J Appl Math, 2014, 74(2):322-344
[20] Mavromoustaki A, Bertozzi A L. Hyperbolic systems of conservation laws in gravity-driven, particles-laden thin-film flows. J Engrg Math, 2014, 88:29-48
[21] Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proc Edinb Math Soc, 2012, 55(3):711-729
[22] Levine H A, Sleeman B D. A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J Appl Math, 1997, 57(3):683-730
[23] Nedeljkov M. Singular shock waves in interactions. Quarterly of Applied Mathematics, 2006, 66(2):112-118
[24] Canon E. On some hyperbolic systems of temple class. Nonlinear Anal TMA, 2012, 75:4241-4250
[25] 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, 2002, 34:28-63
[26] Barti P, Bressan A. The semigroup generated by a Temple class system with large data. Differential Integral Equations, 1997, 10:401-418
[27] Bianchini S. Stability of L^∞ solutions for hyperbolic systems with coinciding shocks and rarefactions. SIAM J Math Anal, 2001, 33:959-981
[28] Bressan A, Goatin P. Stability of L^∞ solutions of temple class systems. Differential Integral Equations. 2000, 13:1503-1528
[29] Liu T P, Yang T. L¹ stability of conservation laws with coinciding Hugoniot and characteristic curves. Indiana Univ Math J, 1999, 48:237-247
[30] Li T T. Global Classical Solutions for Quasilinear Hyperbolic Systems. New York:John Wiley and Sons, 1994
[31] Shen C, Sun M. Interactions of delta shock waves for the transport equations with split delta functions. J Math Anal Appl, 2009, 351:747-755
[32] Guo L, Zhang Y, Yin G. Interactions of delta shock waves for the Chaplygin gas equations with split delta functions. J Math Anal Appl, 2014, 410:190-201
[33] Qu A, Wang Z. Stability of the Riemann solutions for a Chaplygin gas. J Math Anal Appl, 2014, 409:347-361
[34] Wang Z, Zhang Q. The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Acta Math Sci, 2012, 32B(3):825-841
[35] Sun M. Interactions of elementary waves for Aw-Rascle model. SIAM J Appl Math, 2009, 69:1542-1558
[36] Guo L, Pan L, Yin G. The perturbed Riemann problem and delta contact discontinuity in chromatography equations. Nonlinear Analysis, TMA, 2014, 106:110-123
[37] Shen C. Wave interactions and stability of the Riemann solutions for the chromatography equations. J Math Anal Appl, 2010, 365:609-618
[38] Shen C. The asymptotic behaviors of solutions to the perturbed Riemann problem near the singular curve for the chromatography system. J Nonlin Math Phys, 2015, 22:76-101
[39] Sun M. Interactions of delta shock waves for the chromatography equations. Appl Math Lett, 2013, 26:631-637
[40] Sun M, Sheng W. The ignition problem for a scalar nonconvex combustion model. J Differential Equations, 2006, 231:673-692
[41] Chang T, Hsiao L. The Riemann problem and interaction of waves in gas dynamics//Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol 41. Longman Scientific and Technical, 1989
[42] Dafermos C M. Hyperbolic Conversation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenchaften. Berlin, Heidelberg, New York:Springer, 2000
[43] Serre D. Systems of Conversation Laws 1/2. Cambridge:Cambridge University Press, 1999/2000
[44] Sekhar T R, Sharma V D. Riemann problem and elementary wave interactions in isentropic magnetogasdynamics. Nonlinear Analysis:Real World Applications, 2010, 11:619-636
[45] Dafermos C. Generalized characteristics in hyperbolic systems of conservation laws. Arch Rational Mech Anal, 1989, 107:127-155
[46] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York:Springer, 1994
[47] Bressan A. Hyperbolic Systems of Conservation Laws:The One-Dimensional Cauchy Problem. Oxford Lecture Ser Math Appl, Vol 20. Oxford:Oxford University Press, 2000
[48] Shen C, Sheng W, Sun M. The asymptotic limits of Riemann solutions to the scaled Leroux system. Commun Pure Appl Anal, 2017, 17(2):391-411