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Acta Mathematica Scientia

, Volume 39, Issue 1, pp 57–82 | Cite as

Global Solutions of the Perturbed Riemann Problem for the Chromatography Equations

  • Ting Zhang (张婷)
  • Wancheng Sheng (盛万成)Email author
Article

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 

2010 MR Subject Classification

35L65 35L45 

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Notes

Acknowledgements

The frist author would like to express her gratitude to all those who helped her during the writing of this thesis.

References

  1. [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, 2001Google Scholar
  2. [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, 2001Google Scholar
  3. [3]
    Aris R, Amundson N. Mathematical Methods in Chemical Engineering, Vol 2. Prentice-Hall, Englewood Cliffs, NJ, 1966Google Scholar
  4. [4]
    Rhee H K, Aris R, Amundson N R. On the theory of multicomponent chromatography. Philos Trans R Soc London, 1970, A267: 419–455CrossRefzbMATHGoogle Scholar
  5. [5]
    Helfferich F, Klein G. Multicomponent Chromatography. New York: Marcel Dekker, 1970Google Scholar
  6. [6]
    Temple B. Systems of conservation laws with invariant submanifolds. Trans Amer Math Soc, 1983, 280: 781–795MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Ostrov D N. Asymptotic behavior of two interacting chemicals in a chromatography reactor. SIAM J Math Anal, 1996, 27: 1559–1596MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Mazzotti M. Local equilibrium theory for the binary chromatography of species subject to a generalized Langmuir isotherm. Ind Eng Chem Res, 2006, 45: 5332–5350CrossRefGoogle Scholar
  9. [9]
    Mazzotti M. Non-classical composition fronts in nonlinear chromatography: delta-shock. Ind Eng Chem Res, 2009, 48: 7733–7752CrossRefGoogle Scholar
  10. [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): 2002–2012CrossRefGoogle Scholar
  11. [11]
    Shen C. Wave interactions and stability of the Riemann solutions for the chromatography equations. J Math Anal Appl, 2010, 365: 609–618MathSciNetCrossRefzbMATHGoogle Scholar
  12. [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: 1090–1920MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Sun M. Delta shock waves for the chromatography equations as self-similar viscosity limits. Q Appl Math, 2011, 69: 425–443MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Bressan A, Shen W. Uniqueness if discintinuous ODE and conservation laws. Nonlinear Anal, 1998, 34: 637–652MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tsikkou C. Singular shocks in a chromatography model. J Math Anal Appl, 2016, 439: 766–797MathSciNetCrossRefzbMATHGoogle Scholar
  16. [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: 185–197Google Scholar
  17. [17]
    Keyfitz B L, Kranzer H C. Spaces of weighted measures for conservation laws with singular shock solutions. J Differential Equations, 1995, 118(2): 420–451MathSciNetCrossRefzbMATHGoogle Scholar
  18. [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: 107–125Google Scholar
  19. [19]
    Wang L, Bertozzi A L. Shock solutions for high concentration particle-laden thin films. SIAM J Appl Math, 2014, 74(2): 322–344MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Mavromoustaki A, Bertozzi A L. Hyperbolic systems of conservation laws in gravity-driven, particles-laden thin-film flows. J Engrg Math, 2014, 88: 29–48MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proc Edinb Math Soc, 2012, 55(3): 711–729MathSciNetCrossRefzbMATHGoogle Scholar
  22. [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): 683–730MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Nedeljkov M. Singular shock waves in interactions. Quarterly of Applied Mathematics, 2006, 66(2): 112–118MathSciNetGoogle Scholar
  24. [24]
    Canon E. On some hyperbolic systems of temple class. Nonlinear Anal TMA, 2012, 75: 4241–4250MathSciNetCrossRefzbMATHGoogle Scholar
  25. [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: 28–63MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Barti P, Bressan A. The semigroup generated by a Temple class system with large data. Differential Integral Equations, 1997, 10: 401–418MathSciNetzbMATHGoogle Scholar
  27. [27]
    Bianchini S. Stability of L solutions for hyperbolic systems with coinciding shocks and rarefactions. SIAM J Math Anal, 2001, 33: 959–981MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Bressan A, Goatin P. Stability of L solutions of temple class systems. Differential Integral Equations. 2000, 13: 1503–1528MathSciNetzbMATHGoogle Scholar
  29. [29]
    Liu T P, Yang T. L 1 stability of conservation laws with coinciding Hugoniot and characteristic curves. Indiana Univ Math J, 1999, 48: 237–247MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Li T T. Global Classical Solutions for Quasilinear Hyperbolic Systems. New York: John Wiley and Sons, 1994zbMATHGoogle Scholar
  31. [31]
    Shen C, Sun M. Interactions of delta shock waves for the transport equations with split delta functions. J Math Anal Appl, 2009, 351: 747–755MathSciNetCrossRefzbMATHGoogle Scholar
  32. [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: 190–201MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Qu A, Wang Z. Stability of the Riemann solutions for a Chaplygin gas. J Math Anal Appl, 2014, 409: 347–361MathSciNetCrossRefzbMATHGoogle Scholar
  34. [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): 825–841MathSciNetzbMATHGoogle Scholar
  35. [35]
    Sun M. Interactions of elementary waves for Aw-Rascle model. SIAM J Appl Math, 2009, 69: 1542–1558MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Guo L, Pan L, Yin G. The perturbed Riemann problem and delta contact discontinuity in chromatography equations. Nonlinear Analysis, TMA, 2014, 106: 110–123MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    Shen C. Wave interactions and stability of the Riemann solutions for the chromatography equations. J Math Anal Appl, 2010, 365: 609–618MathSciNetCrossRefzbMATHGoogle Scholar
  38. [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: 76–101MathSciNetCrossRefGoogle Scholar
  39. [39]
    Sun M. Interactions of delta shock waves for the chromatography equations. Appl Math Lett, 2013, 26: 631–637MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Sun M, Sheng W. The ignition problem for a scalar nonconvex combustion model. J Differential Equations, 2006, 231: 673–692MathSciNetCrossRefzbMATHGoogle Scholar
  41. [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, 1989Google Scholar
  42. [42]
    Dafermos C M. Hyperbolic Conversation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenchaften. Berlin, Heidelberg, New York: Springer, 2000CrossRefzbMATHGoogle Scholar
  43. [43]
    Serre D. Systems of Conversation Laws 1/2. Cambridge: Cambridge University Press, 1999/2000Google Scholar
  44. [44]
    Sekhar T R, Sharma V D. Riemann problem and elementary wave interactions in isentropic magnetogas-dynamics. Nonlinear Analysis: Real World Applications, 2010, 11: 619–636MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    Dafermos C. Generalized characteristics in hyperbolic systems of conservation laws. Arch Rational Mech Anal, 1989, 107: 127–155MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer, 1994CrossRefzbMATHGoogle Scholar
  47. [47]
    Bressan A. Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford Lecture Ser Math Appl, Vol 20. Oxford: Oxford University Press, 2000Google Scholar
  48. [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): 391–411CrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Ting Zhang (张婷)
    • 1
  • Wancheng Sheng (盛万成)
    • 1
    Email author
  1. 1.Department of Mathematics, College of ScienceShanghai UniversityShanghaiChina

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