On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

TL;DR

This paper demonstrates that the periodic cubic Schrödinger equation is ill-posed in negative Sobolev spaces, showing the flow-map's discontinuity and providing new insights into the ill-posedness phenomena.

Contribution

It establishes the ill-posedness in $H^s(\mathbb{T})$ for $s<0$ with a novel proof that clarifies the underlying ill-posedness mechanisms.

Findings

Flow-map is discontinuous in $H^s(\mathbb{T})$ for $s<0$.

The proof introduces a new approach based on the weak topology of $L^2(\mathbb{T})$.

Results strengthen previous work by Christ-Colliander-Tao.

Abstract

We prove the ill-posedness in $ H^s(\T) $, $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from $H^s(\T) $ into itself for any fixed $ t\neq 0 $. This result is slightly stronger than the one obtained by Christ-Colliander-Tao where the discontinuity of the solution map is established. Moreover our proof is different and clarifies the ill-posedness phenomena. Our approach relies on a new result on the behavior of the associated flow-map with respect to the weak topology of $ L^2(\T) $.