  It was recently proven by De Lellis, Kappeler, and Topalov that the periodic
Cauchy problem for the Camassa-Holm equations is locally well-posed in the
space Lip (T) endowed with the topology of H^1 (T). We prove here that the
Lagrangian flow of these solutions are analytic with respect to time and smooth
with respect to the initial data. These results can be adapted to the
higher-order Camassa-Holm equations describing the exponential curves of the
manifold of orientation preserving diffeomorphisms of T using the Riemannian
structure induced by the Sobolev inner product H^l (T), for l in N, l > 1 (the
classical Camassa-Holm equation corresponds to the case l=1): the periodic
Cauchy problem is locally well-posed in the space W^{2l-1,infty} (T) endowed
with the topology of H^{2l-1} (T) and the Lagrangian flows of these solutions
are analytic with respect to time with values in W^{2l-1,infty} (T) and smooth
with respect to the initial data. These results extend some earlier results
which dealt with more regular solutions. In particular our results cover the
case of peakons, up to the first collision.
