Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ³

Árpád Bényi,Tadahiro Oh,Oana Pocovnicu

doi:10.1090/btran/29

Abstract

We consider the cubic nonlinear Schrödinger equation (NLS) on R 3 \mathbb {R}^3 with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

Highlights

By introducing a modified iterative approach, we prove almost sure local wellposedness of (1.1) in an almost optimal range with respect to the original iterative procedure (Theorem 1.8)
In proving Theorem A, we studied the perturbed nonlinear Schrodinger equation (NLS) (1.8) for v = u − z1
One novelty of this work is that we introduced an infinite sequence {z2res−1} ∈N of stochastic (2 − 1)-linear objects and considered the following expansion of an infinite order:

Summary

Introduction

We discuss an iterative approach to lower this regularity threshold by studying further expansions in terms of the random linear solution z1. 1.8 based on the modified iterative approach (1.33), we can prove almost sure local well-posedness of the cubic NLS (1.16) with the random initial data of the form v0 + φω for the same range of s. Do not pursue this direction in this paper since (i) our main purpose is to present the iterative procedures in their simplest forms and (ii) estimating the higher order stochastic terms by exploiting randomness at the multilinear level would require a significant amount of additional work, which would blur the main focus of this paper.

Strichartz estimates and function spaces

VΔ2 L2

On the second order term z3

On the higher order terms