Classical solutions of the periodic Camassa-Holm equation. (English) Zbl 1158.37311

Summary: We study the periodic Cauchy problem for the Camassa-Holm equation \[ \partial_t u - \partial_t\partial_x^2 u + 3u\partial_x u - 2\partial_x u\partial_x^2 u - u\partial_x^3 u \equiv 0 \tag{CH} \]
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 \(H^s\) 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.


37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
35B10 Periodic solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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