Almost sure well-posedness of the cubic nonlinear Schr\"odinger equation below L^2(T)

TL;DR

This paper proves almost sure local and global well-posedness for the one-dimensional cubic NLS with initial data below L^2, leveraging nonlinear smoothing effects from randomized initial data.

Contribution

It establishes almost sure well-posedness for initial data in H^s(T) with s > -1/3 locally and s > -1/12 globally, below the L^2 threshold.

Findings

Almost sure local well-posedness for s > -1/3

Almost sure global well-posedness for s > -1/12

Demonstrates nonlinear smoothing effects with randomized data

Abstract

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of NLS almost surely for the initial data in the support of the canonical Gaussian measures on H^s(T) for each s > -1/3, and global well-posedness for each s > -1/12.