Geometric & Functional Analysis GAFA

, Volume 12, Issue 5, pp 1080–1104

Classical solutions of the periodic Camassa—Holm equation

  • G. Misiolek

DOI: 10.1007/PL00012648

Cite this article as:
Misiolek, G. GAFA, Geom. funct. anal. (2002) 12: 1080. doi:10.1007/PL00012648


We study the periodic Cauchy problem for the Camassa—Holm equation and prove that it is locally well-posed in the space of continuously differentiable functions on the circle. The approach we use consists in rewriting the equation and deriving suitable estimates which permit application of o.d.e. techniques in Banach spaces. We also describe results in fractional Sobolev Hs spaces and in Appendices provide a direct well-posedness proof for arbitrary real s > 3/2 based on commutator estimates of Kato and Ponce as well as include a derivation of the equation on the diffeomorphism group of the circle together with related curvature computations.

Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • G. Misiolek
    • 1
  1. 1.Department of Mathematics, University of Notre Dame, IN 46556, USA, e-mail: misiolek.1@nd.eduUS