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

TL;DR

This paper proves the existence of infinitely many conserved quantities for the Degasperis-Procesi equation near the origin, which control Sobolev norms, leading to global well-posedness for small data and insights into its Birkhoff normal form.

Contribution

It establishes the analyticity of conserved quantities controlling Sobolev norms and demonstrates the action-preserving nature of the Birkhoff normal form for the equation.

Findings

Infinitely many conserved quantities control Sobolev norms.

Global well-posedness for small initial data is proved.

Birkhoff normal form is action-preserving at any order.

Abstract

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.