Normal form approach to the one-dimensional periodic cubic nonlinear Schr\xf6dinger equation in almost critical Fourier-Lebesgue spaces

Tadahiro Oh,Yuzhao Wang

doi:10.1007/s11854-021-0168-1

Abstract

In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp({cal F}{L^p}(mathbb{T})), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp({cal F}{L^p}(mathbb{T})) for 1 leq p leq {3 over 2}.

Highlights

We study the periodic cubic nonlinear Schrodinger equation (NLS) (1.1) in the Fourier–Lebesgue spaces FLp(T) defined via the norm: f FLp(T) :=
We apply an infinite iteration of normal form reductions and transform the cubic NLS into the so-called normal form equation
3In [22], we only proved well-posedness of the cubic NLS (1.1) in the sense of weak solutions in the extended sense

Summary

Introduction

Our main goal in this paper is (i) to develop further the normal form approach to study the (renormalized) cubic NLS, introduced in [22], and provide the solution theory for (1.2) in almost critical Fourier–Lebesgue spaces (Theorem 1.4) in the sense of Definitions 1.2 and 1.3 without using any auxiliary function spaces, in particular, without using the Fourier restriction norm method as in [6, 19] and (ii) to prove unconditional uniqueness of the (renormalized) cubic NLS in FLp(T) for In proving these results, we apply an infinite iteration of normal form reductions and transform the (renormalized) cubic NLS into the so-called normal form equation. Relevant multilinear estimates were studied based on the idea introduced in [29], namely, successive applications of basic trilinear estimates (called localized modulation estimates)

Proof of the main results

Sensible weak solutions

Normal form reduction

General step