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Israel Journal of Mathematics

, Volume 225, Issue 2, pp 721–741 | Cite as

Global weak solutions for the chromatography system

  • Yun-guang Lu
Article
  • 39 Downloads

Abstract

In this paper, by using a new technique from the compensated compactness method, we study the Cauchy problem of the chromatography system of two equations, and obtain the existence of the global weak solutions when the regular BV estimate is assumed for only one characteristic field.

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References

  1. [BB]
    S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals of Mathematics 161 (2005), 223–342.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BS1]
    S. Benzoni-Gavage and D. Serre, Compacité par compensation pour une classe de systémes hyperboliques de p ≥ 3 lois de conservation, Revista Matemática Iberoamericana 10 (1994), no. 3, 557–579.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BS2]
    S. Benzoni-Gavage and D. Serre, Existence of solutions for a class of hyperbolic systems of p conservation laws ( p ≥ 3), in Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects (Taormina, 1992), Notes on Numerical Fluid Mechanics, Vol. 43, Frieder. Vieweg, Braunschweig, 1993, pp. 56–61.CrossRefGoogle Scholar
  4. [BS3]
    S. Benzoni-Gavage and D. Serre, Compensated compactness for a class of hyperbolic systems of p conservation laws with p ≥ 3, in Progress in Partial Differential Equations: the Metz Surveys, 2 (1992), Pitman Research Notes in Mathematics Series, Vol. 296, Longman Sci. Tech., Harlow, 1993, pp. 3–11.MathSciNetzbMATHGoogle Scholar
  5. [Chen]
    G.-Q. Chen, Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics. III, Acta Mathematica Scientia 6 (1986), 75–120.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [DCL]
    X.-X. Ding, G.-Q. Chen and P.-Z. Luo, Convergence of the fractional step Lax–Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Communications in Mathematical Physics 121 (1989), 63–84.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Di1]
    R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Communications in Mathematical Physics 91 (1983), 1–30.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Di2]
    R. J. DiPerna, Convergence of approximate solutions to conservation laws, Archive for Rational Mechanics and Analysis 82 (1983), 27–70.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Gl]
    J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Communications on Pure and Applied Mathematics 18 (1965), 95–105.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [He]
    A. Heibig, Existence and uniqueneons for some hyperbolic systems of conservation laws, Archive for Rational Mechanics and Analysis 126 (1994), 79–101.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [HW]
    F. M. Huang and Z. Wang, Convergence of viscosity solutions for isothermal gas dynamics, SIAM Journal on Mathematical Analysis 34 (2003), 595–610.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Isa]
    E. Isaacson, Global solution of a Riemann problem for a non-strictly hyperbolic system of conservation laws arising in enhanced oil recovery, preprint, Rockefeller University, New York, 1981.Google Scholar
  13. [JPP]
    F. James, Y.-J. Peng and B. Perthame, Kinetic formulation for chromatography and some other hyperbolic systems, Journal de Mathématiques Pures et Appliquées 74 (1995), 367–385.MathSciNetzbMATHGoogle Scholar
  14. [KK]
    B. Keyfitz and H. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity, Archive for Rational Mechanics and Analysis 72 (1980), 219–241.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [KT]
    K. H. Karlsen and J. D. Towers, Convergence of the Lax–Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux, Chinese Annals of Mathematics, Series B. 25 (2004), 287–318.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Le]
    A. Y. le Roux, Numerical stability for some equations of gas dynamics, Mathematics of Computation, 37 (1981), 435–446.MathSciNetCrossRefGoogle Scholar
  17. [LPS]
    P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Communications on Pure and Applied Mathematics 49 (1996), 599–638.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [LPT]
    P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-system, Communications in Mathematical Physics 163 (1994), 415–431.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [LT]
    R. J. LeVeque and B. Temple, Stability of Godunov’s method for a class of 2 × 2 systems of conservation laws, Transactions of the American Mathematical Society 288 (1985), 115–123.MathSciNetzbMATHGoogle Scholar
  20. [Lu1]
    Y.-G. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 128, Chapman & Hall/CRC, Boca Raton, FL, 2003.zbMATHGoogle Scholar
  21. [Lu2]
    Y.-G. Lu, Existence of global bounded weak solutions to a non-symmetric system of Keyfitz–Kranzer type, Journal of Functional Analysis 261 (2011), 2797–2815.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Lu3]
    Y.-G. Lu, Existence of global entropy solutions to general system of Keyfitz–Kranzer type, Journal of Functional Analysis 264 (2013), 2457–2468.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Lu4]
    Y.-G. Lu, Existence of global weak entropy solutions to some nonstrictly hyperbolic systems, SIAM Journal on Mathematical Analysis 45 (2013), 3592–3610.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [LMR]
    Y.-G. Lu, I. Mantilla and L. Rendon, Convergence of approximated solutions to a nonstrictly hyperbolic system, Advanced Nonlinear Studies 1 (2001), 65–79.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Mu]
    F. Murat, Compacité par compensation, Annali della Scuola Normale Superiore di Pisa 5 (1978), 489–507.MathSciNetzbMATHGoogle Scholar
  26. [RAA]
    H. Rhee, R. Aris and N. Admundson, On the theory of multicomponent chromatography, Philosophical Transactions of the Royal Society of London A267 (1970), 419–455.Google Scholar
  27. [Se]
    D. Serre, Solutions `a variations bornées pour certains systèmes hyperboliques de lois de conservation, Journal of Differential Equations 68 (1987), 137–168.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Ta]
    T. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics, Heriot–Watt symposium, Vol. 4, Research Notes in Mathematics, Vol. 39, Pitman Press, London, 1979, pp. 136–212.MathSciNetzbMATHGoogle Scholar
  29. [Te]
    B. Temple, Systems of conservation laws with invariant submanifolds, Transactions of the American Mathematical Society 280 (1983), 781–795.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [Te1]
    B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws, Advances in Applied Mathematics 3 (1982), 335–375.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.K. K. Chen Institute for Advanced StudiesHangzhou Normal UniversityHangzhouChina
  2. 2.Escuela de MatemáticasUniversidad Industrial de SantanderSantanderColombia

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