Abstract
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.
Highlights
We consider the Cauchy problem for the Camassa-Holm equations:∂tu − ∂txxu + 3u∂xu − 2∂xu∂xxu − u∂xxxu = 0, u(0, x) = u0(x), (1.1)where x runs over the line R or the one-dimensional torus T
Fokas and Fuchssteiner [18] where it appears as a member of a whole family of bi-hamiltonian equations generated by the method of recursion operator
The aim of this paper is to prove that the flow map of these solutions benefits from extra smoothness properties
Summary
We consider the Cauchy problem for the Camassa-Holm equations:. where x runs over the line R or the one-dimensional torus T. Let us first recall that the Camassa-Holm equation describes the exponential curves of the manifold of orientation preserving diffeomorphisms of T using the Riemannian structure induced by the Sobolev inner product H1(T) This can be seen as a counterpart of the celebrated papers [1] and [16] where classical solutions of the incompressible Euler equations are interpreted as geodesics of a Riemannian manifold of infinite dimension. In this spirit some higher-order Camassa-Holm equations can be considered using the Sobolev space Hl(T), for l ∈ N, l 2, instead of H1(T), see for instance [10, 11]. Let us mention here the papers [19] and [30] which deal with the issues of control and of stabilization of the Camassa-Holm equation
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