A Short Introduction to One-Dimensional Conservation Laws

  • Raluca Eftimie
Part of the Lecture Notes in Mathematics book series (LNM, volume 2232)


The one-equation advection models that are being used to describe the movement of various animal populations have been extensively investigated over the last decades. Since the theory behind these equations is well known (and can be found in any textbook on hyperbolic conservation laws), our goal here is to give the reader a brief review of this theory (while leaving behind most technical details). This approach will help the reader understand the analytical results presented in the upcoming chapters regarding the existence and the types of patterns displayed by various hyperbolic models for populations dynamics that exist in the literature.


  1. 1.
    P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, Philadelphia, 1973)CrossRefGoogle Scholar
  2. 2.
    D. Serre, Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves (Cambridge University Press, Cambridge, 1999)Google Scholar
  3. 3.
    D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems (Cambridge University Press, Cambridge, 2000)Google Scholar
  4. 4.
    A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem (Oxford University Press, Oxford, 2000)Google Scholar
  5. 5.
    A. Bressan, Lecture Notes on Functional Analysis. With Applications to Linear Partial Differential Equations (American Mathematical Society, Rhodes Island, 2013)Google Scholar
  6. 6.
    A. Bressan, Hyperbolic Conservation Laws: An Illustrated Tutorial (Springer, Berlin, 2013), pp. 157–245Google Scholar
  7. 7.
    C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenschaften, vol. 325 (Springer, Berlin, 2000)Google Scholar
  8. 8.
    V. Sharma, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves (CRC Press, Boca Raton, 2010)CrossRefGoogle Scholar
  9. 9.
    M. Fey, R. Jeltsch (eds.), Hyperbolic Problems: Theory, Numerics, Applications (Birkhäuser Verlag, Basel, 1999)zbMATHGoogle Scholar
  10. 10.
    B. Perthame, Kinetic Formulation of Conservation Laws (Oxford University Press, Oxford, 2002)zbMATHGoogle Scholar
  11. 11.
    A. Bressan, D. Serre, M. Williams, K. Zumbrun, Hyperbolic Systems of Balance Laws (Springer, Berlin, 2007)Google Scholar
  12. 12.
    R. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1992)CrossRefGoogle Scholar
  13. 13.
    M. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications (Springer, Berlin, 2013)CrossRefGoogle Scholar
  14. 14.
    S. Goldstein, Quart. J. Mech. Appl. Math. 4, 129 (1951)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Bressan, Hyperbolic Systems of Balance Laws (Springer, Berlin, 2007), pp. 1–78CrossRefGoogle Scholar
  16. 16.
    J. Glimm, Commun. Pure Appl. Math. 18, 697 (1965)CrossRefGoogle Scholar
  17. 17.
    S. Bianchini, A. Bressan, Ann. Math. 161, 223 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    T. Liu, Commun. Math. Phys. 57, 135 (1977)CrossRefGoogle Scholar
  19. 19.
    C. Dafermos, J. Math. Anal. Appl. 38, 33 (1972)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Bressan, J. Math. Anal. Appl. 170, 414 (1992)MathSciNetCrossRefGoogle Scholar
  21. 21.
    O. Oleinik, Amer. Math. Soc. Transl. 26, 95 (1957)Google Scholar
  22. 22.
    S. Kruz̆kov, Math. USSR Sb. 42, 217 (1970)Google Scholar
  23. 23.
    H. Holden, N. Risebro, Front Tracking for Hyperbolic Conservation Laws (Springer, Berlin, 2011)CrossRefGoogle Scholar
  24. 24.
    G. Sod, J. Comput. Phys. 27, 1 (1978)MathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Liu, J. Math. Anal. Appl. 53, 78 (1976)MathSciNetCrossRefGoogle Scholar
  26. 26.
    P. Lax, Commun. Pure Appl. Math. 10, 537 (1957)CrossRefGoogle Scholar
  27. 27.
    L. Evans, Partial Differential Equations (American Mathematical Society, Rhodes Island, 1997)Google Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Raluca Eftimie
    • 1
  1. 1.Division of MathematicsUniversity of DundeeDundeeUK

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