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Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow

Dario Bambusi, Roberto Feola, Beatrice Langella, Francesco Monzani

Abstract

In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (including flat tori), homogeneous spaces and rotational invariant surfaces. As applications of the abstract result we prove almost global existence for a nonlinear Schrödinger equation with a convolution potential and for a nonlinear beam equation. We also prove $H^s$ stability of the ground state in NLS equation. The proof is based on a normal form procedure.

Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow

Abstract

In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (including flat tori), homogeneous spaces and rotational invariant surfaces. As applications of the abstract result we prove almost global existence for a nonlinear Schrödinger equation with a convolution potential and for a nonlinear beam equation. We also prove stability of the ground state in NLS equation. The proof is based on a normal form procedure.
Paper Structure (32 sections, 46 theorems, 326 equations)

This paper contains 32 sections, 46 theorems, 326 equations.

Table of Contents

  1. Introduction
  2. Statement of the Abstract Theorem
  3. Assumptions on the linear system
  4. Assumption on the nonlinearity and statement
  5. Remark on the abstract theorem
  6. Applications
  7. Manifolds
  8. Models
  9. Schrödinger equation with spectral multiplier
  10. Sobolev stability of ground state for NLS equation
  11. Beam equation
  12. Functional setting
  13. Phase space
  14. Hamiltonian structure
  15. Polynomials with localized coefficients
  16. ...and 17 more sections

Key Result

Lemma 2.6

There exists a sequence of intervals $\Sigma_n=[a_n,b_n]$, $n\in\mathbb N, \mathbb{N}\ge1$ and a positive constant $C$, with the following properties: $\bullet$$a_n<b_n<a_{n+1}< 3n$; $\bullet$$\Sigma\subset[0,b_0]\cup \bigcup_{n}\Sigma_n$; $\bullet$$\left|b_n-a_n\right|\equiv \left|\Sigma_n\right|\l

Theorems & Definitions (108)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Globally integrable quantum system
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10: Steepness
  • ...and 98 more