The Existence and Stability of Shock Fronts in Several Space Variables

  • A. Majda
Part of the Applied Mathematical Sciences book series (AMS, volume 53)


The phenomenon of shock wave formation described in the previous chapter indicates the necessity for studying discontinuous weak solutions of systems of conservation laws. Here we turn to the construction of discontinuous solutions of conservation laws in several space variables with an emphasis on the physical equations of compressible fluid flow. We concentrate on the rigorous short-time existence and structural stability of shock fronts in several space variables -- these are the simplest multi-D nonlinear progressing wave patterns, and in Section 4.1 we introduce the basic preliminary facts for this problem. We restrict our discussion for two reasons: 1) shock fronts are the most important nonlinear wave patterns in compressible fluid flow and other systems of conservation laws; 2) these special wave patterns are the only inherently discontinuous waves in multi-D for m × m systems where a rigorous theory has been developed ([15], [16]).


Weak Solution Shock Front Mixed Problem Uniform Stability Linear Hyperbolic Equation 
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Bibliography for Chapter 4

  1. [1]
    Agemi, R.: “Mixed problems for the linearized shallow water equations”, Comm. Partial Differential Equations 5 (1980), 645–681.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Agmon, S.: “Problèmes mixtes pour les equations hyperboliques d’ordre superieure”, Colloques Internationaux du C.R.N.S. (1963), 13–18.Google Scholar
  3. [3]
    Ben-Dor, G., and I. Glass: “Domains and boundaries of non-stationary oblique shock-wave reflexions, I. Diatomic gas”, J. Fluid Mech. 92 (1979), 459–496.ADSCrossRefGoogle Scholar
  4. [4]
    Courant, R., and K. O. Friedrichs: Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1949.Google Scholar
  5. [5]
    DiPerna, R.: “Uniqueness of solutions of hyperbolic conservation laws”, Indiana Univ. Math. J. 28 (1979), 137–187.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Duff, G. F. D.: “Hyperbolic differential equations and waves” in Boundary Value Problems for Evaluation Partial Differential Equations, edited by H. Garnir, Reidel, Boston, pp. 27–155.Google Scholar
  7. [7]
    Hersh, R.: “Mixed problems in several variables”, J. Math. Mech. 12 (1963), 317–334.MathSciNetzbMATHGoogle Scholar
  8. *[8]
    Kreiss, H. O.: “Initial boundary value problems for hyperbolic systems”, Comm. Pure Appl. Math. 23 (1970), 277–298.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Lax, P. D.: “Hyperbolic systems of conservation laws, II”, Comm. Pure Appl. Math. 10 (1957), 537–567.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Appl. Math. #11 (1973), Philadelphia.Google Scholar
  11. *[11]
    Lax, P.D., and R. S. Phillips: “Local Boundary conditions for dissipative symmetric linear differential operators”, Comm. Pure Appl. Math. 13 (1960), 427–456.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Li Da-Qian and Yu Wen-Ci: “Some existence theorems for quasi-linear hyperbolic systems of partial differential equations in two independent variables, II”, Sci. Sinica 13 (1964), 551–564.MathSciNetGoogle Scholar
  13. [13]
    Li Da-Qian and Yu Wen-Ci: “The local solvability of boundary value problems for quasi linear hyperbolic systems”, Sci. Sinica 23 (1980), 1357–1367.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Liu, T. P.: “The Riemann problem for general systems of conservation laws”, J. Differential Equations 28 (1975), 218–234.CrossRefGoogle Scholar
  15. [15]
    Majda, A.: “The stability of multi-dimensional shock fronts — a new problem for linear hyperbolic equations”, Mem. Amer. Math. Soc. #275, 1983.Google Scholar
  16. [16]
    Majda, A.: “The existence of multi-dimensional shock fronts”, Mem. Amer. Math. Soc. #281, 1983.Google Scholar
  17. [17]
    Majda, A., and R. Rosales: “A theory for the spontaneous formation of Mach stems in reacting shock fronts; I, the basic perturbation analysis” (to appear in 1983 in SIAM J. Appl. Math.). 156Google Scholar
  18. [18]
    Majda, A., and S. Osher: “Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary”, Comm. Pure Appl. Math. 28 (1975), 607–676.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Ralston, J. V.: “Note on a paper of Kreiss”, Comm. Pure Appl. Math. 24 (1971), 759–762.MathSciNetCrossRefGoogle Scholar
  20. *[20]
    Rauch, J.: “L2 is a continuable initial condition for Kreiss’ mixed problems”, Comm. Pure Appl. Math. 15 (1972), 265–285.MathSciNetCrossRefGoogle Scholar
  21. *[21]
    Rauch, J., and F. Massey: “Differentiability of solutions to hyperbolic initial boundary value problems”, Trans. Amer. Math. Soc. 189 (1974), 303–318.MathSciNetzbMATHGoogle Scholar
  22. *[22]
    Sakamoto, R.: “Mixed problems for hyperbolic equations, I, II”, J. Math. Kyoto Univ. 10 (1970), 349–373 and 403-417.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Wendroff, B.: “The Riemann problem for materials with nonconvex equations of states. II. General flow”, J. Math. Anal. Appl. 38 (1972), 640–658.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Courant, R., and D. Hilbert: Methods of Mathematical Physics, Vol. II, Wiley, New York, 1962.zbMATHGoogle Scholar
  25. [25]
    Majda, A. and E. Thomann, “Multi-dimensional shock fronts for 2nd order wave equations” (in preparation).Google Scholar

Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • A. Majda
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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