A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations
type: Item
instance of: scholarly article
title: A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations
zbMATH Open document ID: 1247.37082
author: Jian-Jun Liu
author: Xiaoping Yuan
published in: Communications in Mathematical Physics
publication date: 2011-11-23T00:00:00Z
review text: The authors prove a KAM theorem in infinite dimension for parameter-depending Hamiltonian maps of the form \(H= N_\xi+ P_\xi\), where \[ N= \sum_{1\leq j\leq n} \omega_j(\xi) y_j+{1\over 2} \sum_{j\geq 1} \Omega_j(\xi)(u^2_j+ v^2_j), \] and \(P_\xi\) is a small perturbation. Here \(x_j\in\mathbb{T}= (\mathbb{R}/2\pi\mathbb{Z})\), \(y_j, u_j, v_j\in \mathbb{R}\), the parameter \(\xi\) is in \(\mathbb{R}^n\), and the symplectic form is \[ \sum_{1\leq j\leq n} dy_i\wedge dx_j+ \sum_{j\geq 1} du_j\wedge dv_j. \] \(P_\xi\) is an unbounded perturbation, inducing a loss of regularity of order \(\widetilde d\): the Hamiltonian vector-field \(X_P\) sends \(\mathbb{T}^n\times \mathbb{R}^n\times (l^{a,p})^2\) to \(\mathbb{R}^n\times \mathbb{R}^n\times (l^{a,p-\widetilde d})^2\), where \(l^{a,p}\) is the set of the sequences \((w_j)_{j\geq 1}\) of real numbers for which \[ \sum_{j\geq 1} e^{2aj} j^{2p}|w_j|^2< \infty. \] For all \(\xi\), the torus defined by \(y_j= 0\), \(u_j= v_j= 0\) is invariant for the Hamiltonian vector-field \(X_N\), and the flow on this torus is linear, of frequency vector \(\omega(\xi)= (\omega_1(\xi),\dots, \omega_n(\xi))\). The authors assume that the map \(\xi\mapsto\omega(\xi)\) satisfies a usual nondegeneracy condition and that, roughly, \(\Omega_j(\xi)\sim j^d\). In particular, the normal frequencies \(\Omega_j(\xi)\) are all distinct. Under some additional assumptions on \(N_\xi\), implying the Melnikov non-resonance conditions for almost all \(\xi\), the authors prove that if \(P_\xi\) is a small enough real analytic perturbation such that the order \(\widetilde d\) of the Hamiltonian field \(X_P\) is not greater than \(d-1\), then, for most values of the parameter \(\xi\), the invariant torus is preserved through the perturbation, the perturbed torus having a frequency vector close to \(\Omega(\xi)\). This extends a, result of \textit{S. B. Kuksin} [Analysis of Hamiltonian PDE's. Oxford: Oxford University Press (2000; Zbl 0960.35001)], which wa…
reviewed by: Philippe Bolle
Mathematics Subject Classification ID: 37K55
zbMATH DE Number: 5977583
zbMATH Keywords: Hamiltonian PDE
zbMATH Keywords: KAM theorem
zbMATH Keywords: unbounded perturbation
MaRDI profile type: Publication
full work available at URL: s00220-011-1353-3
OpenAlex ID: W2041219148
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DOI: 10.1007/S00220-011-1353-3
description: scientific article; zbMATH DE number 5977583