Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nekhoroshev type stability for Ultra-differential Hamiltonian in $L^2$ space

Bingqi Yu, Li Yong

TL;DR

This work extends Nekhoroshev-type stability to infinite-dimensional Hamiltonian PDEs by developing a general framework based on ultra-differentiable regularity, high/low mode separation, and a truncation-aware normal form. Through an $N$-cutting normal form and carefully controlled homological equations, it establishes sub-exponential stability times that align with Bourgain’s optimal predictions under broad non-resonance conditions and applies the theory to Schrödinger equations with convolution, fractional Schrödinger equations, and beam equations. The results include explicit coefficient estimates, measure-based non-resonance theorems (often zero-measure resonant sets), and a unified approach that covers Gevrey and logarithmic ultra-differentiable classes. Overall, the paper significantly broadens the class of infinite-dimensional systems for which Bourgain-optimal Nekhoroshev-type stability can be proven, with concrete implications for long-time behavior of nonlinear dispersive PDEs.

Abstract

This paper combines the decay of high modes with the smallness introduced by high orders, leading to a normal form lemma for infinite-dimensional Hamiltonian systems under ultra-differentiable regularity. We prove the sub-exponential stability time of a wide class of Hamiltonian PDEs, including the Schrödinger equation with convolution potentials, fractional-order Schrödinger equations, and beam equations with metrics. When the conditions are equivalent to previous ones, the stability time we obtain reaches Bourgain's predicted optimal bound. Furthermore, we approach earlier results under lower conditions. These results are discussed within a general framework we propose, which applies to the ultra-differential class.

Nekhoroshev type stability for Ultra-differential Hamiltonian in $L^2$ space

TL;DR

This work extends Nekhoroshev-type stability to infinite-dimensional Hamiltonian PDEs by developing a general framework based on ultra-differentiable regularity, high/low mode separation, and a truncation-aware normal form. Through an -cutting normal form and carefully controlled homological equations, it establishes sub-exponential stability times that align with Bourgain’s optimal predictions under broad non-resonance conditions and applies the theory to Schrödinger equations with convolution, fractional Schrödinger equations, and beam equations. The results include explicit coefficient estimates, measure-based non-resonance theorems (often zero-measure resonant sets), and a unified approach that covers Gevrey and logarithmic ultra-differentiable classes. Overall, the paper significantly broadens the class of infinite-dimensional systems for which Bourgain-optimal Nekhoroshev-type stability can be proven, with concrete implications for long-time behavior of nonlinear dispersive PDEs.

Abstract

This paper combines the decay of high modes with the smallness introduced by high orders, leading to a normal form lemma for infinite-dimensional Hamiltonian systems under ultra-differentiable regularity. We prove the sub-exponential stability time of a wide class of Hamiltonian PDEs, including the Schrödinger equation with convolution potentials, fractional-order Schrödinger equations, and beam equations with metrics. When the conditions are equivalent to previous ones, the stability time we obtain reaches Bourgain's predicted optimal bound. Furthermore, we approach earlier results under lower conditions. These results are discussed within a general framework we propose, which applies to the ultra-differential class.

Paper Structure

This paper contains 15 sections, 27 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Classic Nekhoroshev stability
  3. Nekhoroshev type stability for infinite-dimensional Hamiltonian systems
  4. Main Contributions
  5. Setting
  6. Resonance and Normal Form
  7. Iteration lemma
  8. Normal Form Lemma
  9. Stability Time
  10. Applications
  11. Schrödinger Equations with Convolution Potentials
  12. Fractional Schrödinger equation
  13. Beam equation
  14. Constants
  15. Technical Lemmas

Key Result

Theorem 1

For Hamiltonian Hamiltonian with initial $u(0)=u_0$ in the $\theta$-Gevrey differentiable space $W_{s,\theta}^G$, assume the frequencies $\omega_{j}$ satisfy Assumption 2 with $\beta>1$. Then for sufficiently large $s$ , there exist a threshold $\varepsilon_0>0,$ and constants $C_{\mathtt{sta}},C_{\ then where

Theorems & Definitions (46)

  • Theorem 1: Result for Gevrey class case
  • Remark 1.1
  • Theorem 2: Result for logarithmic ultra-differential case
  • Remark 1.2
  • Definition 1
  • Definition 2: $N$-cutting normal form
  • Lemma 1: Homological equation
  • proof
  • Lemma 2: Iteration lemma
  • proof
  • ...and 36 more