Global well-posedness and long-time behavior of the fractional NLS

Abstract

In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension $\ge 2$. In the high regularity result, an {\it Inviscid - Infinite dimensional (IID) limit} is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an independent approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss the adaptation to the periodic settings, in any dimension, for smooth regularities.

arXiv comment39 pages

References in corpus (17)

• Fractional Quantum Mechanics
• On the continuum limit for discrete NLS with long-range lattice interactions
• Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces
• Energy cascades for NLS on the torus
• Invariant measures for passive scalars in the small noise inviscid limit
• Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two
• Existence and Uniqueness theory for the fractional Schrödinger equation on the torus
• Invariant measures and global well posedness for SQG equation
• Invariant measure and long time behavior of regular solutions of the Benjamin-Ono equation
• Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit
• Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces
• Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations
• Global well-posedness for the cubic fractional NLS on the unit disk
• Models for energy and charge transport and storage in biomolecules
• Continuum limit for discrete NLS with memory effect
• Almost sure global well-posedness for the energy supercritical NLS on the unit ball of ${\mathbb{R}}^3$
• Almost sure global well-posedness for the energy supercritical Schrödinger equations

Cited by in corpus (1)

• Singular Levy processes and dispersive effects of generalized Schrödinger equations