Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Zihua Guo, Tadahiro Oh

Type: Article

Publication Date: 2016-10-26

Citations: 37

DOI: https://doi.org/10.1093/imrn/rnw271

Abstract

We prove non-existence of solutions for the cubic nonlinear Schrödinger 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.

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