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

TL;DR

This paper establishes sub-exponential stability times for certain infinite-dimensional Hamiltonian PDEs using ultra-differentiable normal form techniques, extending and optimizing previous stability results.

Contribution

It introduces a normal form lemma for ultra-differentiable Hamiltonian systems, proving optimal stability times for a broad class of PDEs, including Schrödinger and beam equations.

Findings

Proves sub-exponential stability times for Hamiltonian PDEs.

Extends stability results to ultra-differential classes.

Achieves Bourgain's predicted optimal bounds under certain conditions.

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 Schr\"odinger equation with convolution potentials, fractional-order Schr\"odinger 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.