Available online xxxx Communicated by J. Bourgain MSC: primary 35Q53 secondary 37K10 Keywords: Ovsyannikov theorem for nonlocal equations Well-posedness of Cauchy problem in analytic spaces Integrable Camassa-Holm equations Continuity of solution mapThis work presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution. Then, using this theorem it studies the Cauchy problem for Camassa-Holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line, which is the main goal of this paper. Finally, it studies the continuity of the data-to-solution map in spaces of analytic functions.

We consider complex vector fields L on the two-torus. We regard L as an operator acting on smooth functions and study conditions for L to have a closed range. We also give conditions for the range of L to have finite codimension. Our results involve condition (P) of Nirenberg and Treves. One-dimensional orbits diffeomorphic to the unit circle are allowed.

A necessary and sufficient condition is given for a sum of squares operator to be globally hypoelliptic on an N-dimensional torus. This condition is expressed in terms of Diophantine approximation properties of the coefficients. The proof of the Theorem is based on L 2 -estimates and microlocalization. Academic Press

Let (x, t) ∈ R m × R and u ∈ C 2 (R m × R). We study the Gevrey micro-regularity of solutions u of the nonlinear equationis a Gevrey function of order s > 1 and holomorphic in (ζ 0 , ζ ). We show that the Gevrey wave-front set of any C 2 solution u is contained in the characteristic set of the linearized operatorTo achieve this, we study the notion of Gevrey approximate solutions, a concept which we believe is of independent interest and could be applied to much more general situations.

Using the multilinear estimates, which were derived for proving well-posedness of the generalized Korteweg-de Vries (gKdV) equation, it is shown that if the initial data belongs to Gevrey space G σ , σ 1, in the space variable then the solution to the corresponding Cauchy problem for gKdV belongs also to G σ in the space variable. Moreover, the solution is not necessarily G σ in the time variable. However, it belongs to G 3σ near 0. When σ = 1 these are analytic regularity results for gKdV.