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

2024/02/01 by Dario Bambusi, Bambusi, Dario, Roberto Feola +4
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2402.00521

openalex publication_date 2024/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Almost global existence for Hamiltonian PDEs on compact manifolds
• Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori
• Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds
• Persistence of invariant manifolds for nonlinear PDEs
• Supercritical Nonlinear Schrödinger Equations II: Almost Global Existence