Almost global existence for Ultra-differential Hamiltonian in L² space

2025/12/18 by Yu, Bingqi, Yong, Li
Engineering · Mathematics · Physics and Astronomy · #37K45 #37K55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.16332

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

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.

Citations

• A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori
• On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori
• Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori
• Sub-exponential stability for the Beam equation
• An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS
• Long time behavior of the solutions of NLW on the d-dimensional torus
• A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential
• Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrodinger Equation
• Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
• Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians
• On invariant tori of full dimension for 1D periodic NLS
• Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds
• Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations

Related

• Nekhoroshev type stability for non-local semilinear Schrödinger equations
• Hamiltonian perturbation theory for ultra-differentiable functions
• Generic super-exponential stability of invariant tori in Hamiltonian systems
• Sub-exponential stability for the Beam equation
• Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians