Well-Posedness of the Cauchy Problem for a Shallow Water Equation on the Circle

In this paper we consider the periodic Cauchy problem for a fifth order modification of the Camassa–Holm equation. We prove local well-posedness in appropriate Bourgain spaces for initial data in a Sobolev space Hs(T), s>1/2. We also prove global well-posedness for data in H1(T) and of arbitrary size. The proofs are based on a priori estimates using Fourier analysis techniques, microlocalization in phase space, an interpolation argument and a fixed point theorem.