Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L2(T)

Abstract

We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation 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 the NLS equation almost surely for the initial data in the support of the canonical Gaussian measures on $H^s\left(\mathbb{T}\right)$ for each $s>-\frac{1}{3}$, and global well-posedness for each $s>-\frac{1}{12}$.