Advertisement

manuscripta mathematica

, Volume 28, Issue 1–3, pp 89–99 | Cite as

On the Korteweg-de Vries equation

  • Tosio Kato
Article

Abstract

Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial function ϕ in the Sobolev space of order s>3/2 and the solution u(t) staying in the same space, s=∞ being included For the proper KdV equation, existence of global solutions follows if s≥2. The proof is based on the theory of abstract quasilinear evolution equations developed elsewhere.

Keywords

Initial Data Evolution Equation Sobolev Space Number Theory Algebraic Geometry 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Proceedings of the Symposium at Dundee, 1974, Lecture Notes in Mathematics, Springer 1975, pp. 25–70Google Scholar
  2. [2]
    J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London, Ser.A278 (1975), 555–601Google Scholar
  3. [3]
    J. L. Bona and R. Scott, Solutions of the Kortewegde Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), 87–99Google Scholar
  4. [4]
    J. C. Saut and R. Temam, Remarks on the Kortewegde Vries equation, Israel J. Math. 24 (1976), 78–87Google Scholar
  5. [5]
    A. Cohen Murray, Solutions of the Korteweg-de Vries equation evolving from irregular data, Duke Math. J. 45 (1978), 149–181Google Scholar
  6. [6]
    G. Darmois, Evolution equation in a Banach space, Thesis, University of California, 1974Google Scholar
  7. [7]
    G. Da Prato and M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case, Comm. Partial Differential Equations 1 (1976), 585–608Google Scholar
  8. [8]
    K. Kobayashi, On a theorem for linear evolution equations of hyperbolic type, preprint 1978Google Scholar
  9. [9]
    T. Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo, Sec. I, 17 (1970), 241–258Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Tosio Kato
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations