Advertisement

Fritz John pp 335-352 | Cite as

Algebraic Conditions for Hyperbolicity of Systems of Partial Differential Equations

  • Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

Partial differential equations or systems of such equations are classified according to “type”, such as “elliptic”, “parabolic”, “hyperbolic”, etc. Hyperbolic systems (with respect to the initial plane t = 0) are those for which the initial value problem is well posed in the sense of Hadamard (see [8]). General conditions that are both necessary and sufficient for hyperbolicity are complicated (see [10], [11], [2], [3], [14], [15], [16], [17]). The situation, however, is rather simple for an m-th order linear homogeneous system of equations with constant coefficients:
$$ P\left( {\frac{\partial }{{\partial t}},\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}} \right)u = 0 $$
(1)
for a vector u = u(t, x 1 • • •, x n ) = u(t, x) with N components. Here P is an N x N square matrix whose elements p ik are m-th degree forms in their n + 1 arguments. We associate with (1) the matrix (the “symbol” of (1))
$$ P(\lambda ,\xi ) = P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = (Pik(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n})) $$
(2)
and the characteristic form
$$ Q(\lambda ,\xi ) = Q(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = \det P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) $$
(3)
of degree mN in its arguments.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    Lax, P. D., Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24, 1957, pp. 627–646.CrossRefGoogle Scholar
  2. [2]
    Hörmander, L., Linear Partial Differential Operators, Academic Press, New York (third revised edition ), 1969.CrossRefGoogle Scholar
  3. [3]
    Chaillou, J., Les Polynômes Différentiels Hyperboliques et Leurs Perturbations Singulières, Gauthier-Villars, Paris, 1973.Google Scholar
  4. [4]
    Nuij, Wim, A note on hyperbolic polynomials, Math. Scand. 23, 1968, pp. 69–72.Google Scholar
  5. [5]
    Lancaster, P., Lambda-Matrices and Vibrating Systems, Pergamon Press, New York, 1966.Google Scholar
  6. [6]
    Gantmacher, F. R., The Theory of Matrices, Chelsea Publishing, New York, 1959.Google Scholar
  7. [7]
    Hopf, E., Lectures on Differential Geometry in the Large, Applied Mathematics and Statistics Laboratory, Stanford University, pp. 38–41.Google Scholar
  8. [8]
    Courant-Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience Publishers, New York, 1953.Google Scholar
  9. [9]
    John, F., Partial Differential Equations, Springer-Verlag, New York, 1978 (to appear).CrossRefGoogle Scholar
  10. [10]
    Gårding, L., Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85, pp. 1–62.Google Scholar
  11. [11]
    Lax, A., On Cauchy’s problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math. 9, 1956, pp. 135–170.CrossRefGoogle Scholar
  12. [12]
    John, F., Restrictions on the coefficients of hyperbolic systems of partial differential equations, Proc. Nat’l. Acad. Sci. USA 74, 1977.Google Scholar
  13. [13]
    Mizohata, S., Some remarks on the Cauchy problem, J. Math. Kyoto Univ. I, 1962/3, pp. 109–127.Google Scholar
  14. [14]
    Mizohata, S., and Ohya, Y., Sur la condition d’hyperbolicité pour les équations à charactéristiques multiples, II, Japan J. Math. 40, 1971, pp. 63–104.Google Scholar
  15. [15]
    Mizohata, S., and Ohya, Y., Sur la condition de E. E. Levi concernant des équations hyperboliques, Proc. Int’l. Conf. on Functional Analysis and Related Topics, Tokyo, 1969, pp. 177–185;Google Scholar
  16. [15]
    also Publ. Res. Inst. Math. Sei. Ser. A, 4, 1968/9, pp. 511–526.Google Scholar
  17. [16]
    Flaschka, H., and Strang, C., The correctness of the Cauchy problem, Advances in Math. 6, 1971, pp. 347–379.CrossRefGoogle Scholar
  18. [17]
    De Paris, J.-C., Problème de Cauchy oscillatoire par un opérateur différentiel à charactéristiques multiples; line avec Vhyperbolicité, J. Math. Pure Appl. 9, 1972, p. 51.Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Fritz John

There are no affiliations available

Personalised recommendations