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A Survey of Hyperbolic Systems of Conservation Laws in Two Dependent Variables

  • J. A. Smoller

Abstract

This paper is concerned with quasi-linear systems of equations in conservation form of the type
$$u_t + f\left( {u,\upsilon } \right)_x = 0,\,\,\,\,\,\,\,\upsilon _t + g\left( {u,\upsilon } \right)_x = 0,$$
(1)
where f and g are smooth functions of two real variables u and v, and u and v are functions of x and t, -∞ < x <∞, t ≥ 0. We assume that the system (1) is hyperbolic, i. e., if we denote by F, the vector function (f,g), then dF(u,v) has real and distinct eigenvalues λ1(u, v) < λ2(u, v), for all values of the argument (u, v) in question. We consider the Cauchy problem for the system (1).

Keywords

Initial Data Cauchy Problem Existence Theorem Uniqueness Theorem Hyperbolic System 
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References

  1. 1.
    Cesari, L.: Sulle Funzioni a Variazione Limita. Ann. Scuola Norm. Pisa 5 (2), 299–313 (1936).MathSciNetMATHGoogle Scholar
  2. 2.
    Conway, E., and J. Smoller: Global Solutions of the Cauchy Problem for Quasi-linear First Order Equations In Several Space Variables. Comm. Pur. Appl. Math. 19, 95–105 (1966).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Glimm, J.: Solutions in the Large for Nonlinear Hyperbolic Systems of Equations. Comm. Pur. Appl. Math. 18, 697–715 (1965).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Glimm, J. and P. D. Lax: Decay of Solutions of Systems of Hyperbolic Conservations Laws, (to appear).Google Scholar
  5. 5.
    Yu-Fa, Gua, and Zhang Tong: A class of Initial-value Problems for Systems of Aerodynamic Equations. Acta. Math. Sinica, 386–396 (1965). English translation in Chin. Math. 7, 90–101 (1965).Google Scholar
  6. 6.
    Johnson, J. L., and J. A. Smoller: Global Solutions for Certain Systems of Quasi-linear Hyperbolic Equations. J. Math. Mech. 17, 561–576 (1967).MathSciNetMATHGoogle Scholar
  7. 7.
    Johnson, J. L., and J. A. Smoller: Global Solutions for an Extended Class of Hyperbolic Systems of Conservation Laws. Arch. Rat. Mech. Math, (to appear).Google Scholar
  8. 8.
    Krickeberg, K.: Distributionen, Funktionen Beschränkter Variation und Lebesguescher Inhalt Nicht Parametrischer Flächen. Ann. Mat. Pura Appl., Ser. 4, 44, 105–133 (1957).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Landau, L. D., and E. M. Lifshitz: Fluid Mechanics. Reading, Mass.: Addison Wesley 1959.Google Scholar
  10. 10.
    Lax, F. D.: Hyperbolic Systems of Conservation Laws, II. Comm. Pur. Appl. Math. 10, 537–566 (1957).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Oleinik, O. A.: On the Uniqueness of the Generalized Solution of the Cauchy Problem for a Non-linear System of Equations Occurring in Mechanics. Usp. Mat. Nank. 78, 169–176 (1957).MathSciNetGoogle Scholar
  12. 12.
    Rozdestvenskii, B.: Discontinuous Solutions of Hyperbolic Systems of Quasi-linear Equations. Ups. Mat. Nank. 15, 59–117 (1960). English translation in Russ. Math. Surv. 15, 55–111 (1960).MathSciNetGoogle Scholar
  13. 13.
    Smoller, J. A.: On the Solution of the Riemann Problem with General Step Data For an Extended Class of Hyperbolic Systems. Michigan Math. J. (to appear).Google Scholar
  14. 14.
    Smoller, J. A.: A Uniqueness Theorem for Riemann Problems. Arch. Rat. Math. Mech. (to appear).Google Scholar

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© Springer-Verlag Berlin · Heidelberg 1970

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  • J. A. Smoller

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