Non-existence of solutions for the periodic cubic NLS below $L^2$

Abstract

We prove non-existence of solutions for the cubic nonlinear Schr\"odinger equation (NLS) on the circle if initial data belong to $H^s(\mathbb{T}) \setminus L^2(\mathbb{T})$ for some $s \in (-\frac18, 0)$. The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.