Hyperbolic Conservation Laws: An Illustrated Tutorial

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2062)

Abstract

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: (1) Meaning of a conservation equation and definition of weak solutions. (2) Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. (3) Shock waves: Rankine–Hugoniot equations and admissibility conditions. (4) Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. (5) Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. (6) Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. (7) Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. (8) Vanishing viscosity approximations. (9) Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections.Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.

Keywords

Weak Solution Cauchy Problem Hyperbolic System Riemann Problem Piecewise Constant Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    D. Amadori, L. Gosse, G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Ration. Mech. Anal. 162, 327–366 (2002)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    F. Ancona, S. Bianchini, Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary, in “WASCOM 2005”–13th Conference on Waves and Stability in Continuous Media (World Scientific, Hackensack, 2006), pp. 13–21Google Scholar
  3. 3.
    F. Ancona, A. Marson, Existence theory by front tracking for general nonlinear hyperbolic systems. Arch. Ration. Mech. Anal. 185, 287–340 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    F. Ancona, A. Marson, A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems. Arch. Ration. Mech. Anal. 196, 455–487 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    P. Baiti, H.K. Jenssen, On the front tracking algorithm. J. Math. Anal. Appl. 217, 395–404 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    S. Bianchini, On the Riemann problem for non-conservative hyperbolic systems. Arch. Ration. Mech. Anal. 166, 1–26 (2003)MathSciNetMATHGoogle Scholar
  7. 7.
    S. Bianchini, A. Bressan, Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann. Math. 161, 223–342 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Bressan, Contractive metrics for nonlinear hyperbolic systems. Indiana Univ. J. Math. 37, 409–421 (1988)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    A. Bressan, Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170, 414–432 (1992)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    A. Bressan, The unique limit of the Glimm scheme. Arch. Ration. Mech. Anal. 130, 205–230 (1995)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem (Oxford University Press, Oxford, 2000)Google Scholar
  12. 12.
    A. Bressan, BV solutions to systems of conservation laws by vanishing viscosity, C.I.M.E. course in Cetraro, 2003, in Springer Lecture Notes in Mathematics, vol. 1911, ed. by P. Marcati (Springer-Verlag, Berlin, 2007), pp. 1–78Google Scholar
  13. 13.
    A. Bressan, A tutorial on the Center Manifold Theorem, C.I.M.E. course in Cetraro, 2003, in Springer Lecture Notes in Mathematics, vol. 1911, ed. by P. Marcati (Springer-Verlag, Berlin, 2007), pp. 327–344Google Scholar
  14. 14.
    A. Bressan, R.M. Colombo, The semigroup generated by 2 ×2 conservation laws. Arch. Ration. Mech. Anal. 133, 1–75 (1995)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    A. Bressan, P. Goatin, Oleinik type estimates and uniqueness for n ×n conservation laws. J. Differ. Equat. 156, 26–49 (1999)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A. Bressan, P. LeFloch, Uniqueness of weak solutions to hyperbolic systems of conservation laws. Arch. Ration. Mech. Anal. 140, 301–317 (1997)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    A. Bressan, M. Lewicka, A uniqueness condition for hyperbolic systems of conservation laws. Discrete. Cont. Dyn. Syst. 6, 673–682 (2000)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    A. Bressan, A. Marson, Error bounds for a deterministic version of the Glimm scheme. Arch. Ration. Mech. Anal. 142, 155–176 (1998)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    A. Bressan, T. Yang, On the convergence rate of vanishing viscosity approximations. Comm. Pure Appl. Math. 57, 1075–1109 (2004)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    A. Bressan, T.P. Liu, T. Yang, L 1 stability estimates for n ×n conservation laws. Arch. Ration. Mech. Anal. 149, 1–22 (1999)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    A. Bressan, G. Crasta, B. Piccoli, Well posedness of the Cauchy problem for n ×n systems of conservation laws. Am. Math. Soc. Mem. 694 (2000)Google Scholar
  22. 22.
    A. Bressan, K. Jenssen, P. Baiti, An instability of the Godunov scheme. Comm. Pure Appl. Math. 59, 1604–1638 (2006)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    C. Cheverry, Systèmes de lois de conservation et stabilité BV [Systems of conservation laws and BV stability]. Mém. Soc. Math. Fr. 75 (1998) (in French)Google Scholar
  24. 24.
    C. Christoforou, Hyperbolic systems of balance laws via vanishing viscosity. J. Differ. Equat. 221, 470–541 (2006)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    M.G. Crandall, The semigroup approach to first order quasilinear equations in several space variables. Isr. J. Math. 12 108–132, (1972)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer, Berlin, 1999)Google Scholar
  28. 28.
    R.J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J. Differ. Equat. 20, 187–212 (1976)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    R. DiPerna, Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82, 27–70 (1983)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    C. Donadello, A. Marson, Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws. Nonlinear Differ. Equat. Appl. 14, 569-592 (2007)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, 1998)MATHGoogle Scholar
  32. 32.
    L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, Boca Raton, Fl, 1992)MATHGoogle Scholar
  33. 33.
    M. Garavello, B. Piccoli, in Traffic Flow on Networks. Conservation Laws Models (AIMS Series on Applied Mathematics, Springfield, 2006)Google Scholar
  34. 34.
    J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    J. Glimm, P. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am. Math. Soc. Mem. 101 (1970)Google Scholar
  36. 36.
    J. Goodman, Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 121, 235–265 (1992)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    H. Holden, N.H. Risebro, Front Tracking for Hyperbolic Systems of Conservation Laws (Springer, New York, 2002)CrossRefGoogle Scholar
  38. 38.
    H.K. Jenssen, Blowup for systems of conservation laws. SIAM J. Math. Anal. 31, 894–908 (2000)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    S. Kruzhkov, First-order quasilinear equations with several space variables. Mat. Sb. 123, 228–255 (1970). English translation in Math. USSR Sb. 10, 217–273 (1970)Google Scholar
  40. 40.
    P.D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566 (1957)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    M. Lewicka, Well-posedness for hyperbolic systems of conservation laws with large BV data. Arch. Ration. Mech. Anal. 173, 415–445 (2004)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems (Wiley, Chichester, 1994)MATHGoogle Scholar
  43. 43.
    M. Lighthill, G. Whitham, On kinematic waves, II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. A 229, 317–345 (1955)MathSciNetMATHGoogle Scholar
  44. 44.
    T.P. Liu, The Riemann problem for general systems of conservation laws. J. Differ. Equat. 18, 218–234 (1975)MATHCrossRefGoogle Scholar
  45. 45.
    T.P. Liu, The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53, 78–88 (1976)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    T.P. Liu, The deterministic version of the Glimm scheme. Comm. Math. Phys. 57, 135–148 (1977)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    T.P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Comm. Pure Appl. Math. 30, 767–796 (1977)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    T.-P. Liu, T. Yang, L 1 stability of conservation laws with coinciding Hugoniot and characteristic curves. Indiana Univ. Math. J. 48, 237–247 (1999)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    T.-P. Liu, T. Yang, L 1 stability of weak solutions for 2 ×2 systems of hyperbolic conservation laws. J. Am. Math. Soc. 12, 729–774 (1999)MATHCrossRefGoogle Scholar
  50. 50.
    Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method (Chapman & Hall/CRC, Boca Raton, 2003)MATHGoogle Scholar
  51. 51.
    O. Oleinik, Discontinuous solutions of nonlinear differential equations. Am. Math. Soc. Transl. 26, 95–172 (1963)MathSciNetGoogle Scholar
  52. 52.
    B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Göttingen Abh. Math. Cl. 8, 43–65 (1860)Google Scholar
  53. 53.
    F. Rousset, Viscous approximation of strong shocks of systems of conservation laws. SIAM J. Math. Anal. 35, 492–519 (2003)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    S. Schochet, Sufficient conditions for local existence via Glimm’s scheme for large BV data. J. Differ. Equat. 89, 317–354 (1991)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    S. Schochet, The essence of Glimm’s scheme, in Nonlinear Evolutionary Partial Differential Equations, ed. by X. Ding, T.P. Liu (American Mathematical Society/International Press, Providence, RI, 1997), pp. 355–362Google Scholar
  56. 56.
    D. Serre, Systems of Conservation Laws, vols. 1–2 (Cambridge University Press, Cambridge, 1999)Google Scholar
  57. 57.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1983)MATHCrossRefGoogle Scholar
  58. 58.
    S.H. Yu, Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Ration. Mech. Anal. 146, 275–370 (1999)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    A.I. Volpert, The spaces BV and quasilinear equations. Math. USSR Sbornik 2, 225–267 (1967)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA

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