On the integrability of Degasperis-Procesi equation: control of the Sobolev norms and Birkhoff resonances

We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-$\mathtt{c}$ u_{xxx}+4 $\mathtt{c}$ u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$with$$\mathtt{c}$\neq 0$. In \cite{Deg} the authors proved that this equation possesses infinitely many conserved quantities. We prove that, in a neighborhood of the origin, there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of some $H^s$ Sobolev space, both on $\mathbb{R}$ and $\mathbb{T}$. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis-Procesi at any order is action-preserving.