Norm-inflation for periodic NLS equations in negative Sobolev spaces

TL;DR

This paper demonstrates that certain nonlinear Schrödinger equations on periodic domains are strongly ill-posed in negative Sobolev spaces, with solutions exhibiting norm-inflation and loss of regularity, especially for higher-order nonlinearities.

Contribution

It proves norm-inflation and ill-posedness in negative Sobolev spaces for a class of periodic NLS equations, including the cubic case and higher odd-order nonlinearities.

Findings

Strong ill-posedness for σd ≥ 2 in H^s for any s < 0.

Norm-inflation with infinite regularity loss in these spaces.

Specific results for the one-dimensional cubic NLS and its renormalized version for s < -2/3.

Abstract

In this paper we consider Schr{\"o}dinger equations with nonlinearities of odd order 2$\sigma$ + 1 on T^d. We prove that for $\sigma$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{\"o}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3.