Strong ill-posedness for fractional Hartree and cubic NLS Equations

TL;DR

This paper demonstrates that fractional Hartree and cubic NLS equations exhibit norm inflation at all initial data in various Fourier-based spaces, indicating severe ill-posedness especially for large dispersion indices.

Contribution

It introduces a unified Fourier analytic approach to establish norm inflation phenomena for fractional Hartree and NLS equations across multiple function spaces.

Findings

Norm inflation occurs at all initial data in Fourier amalgam spaces with negative regularity.

The phenomena include infinite loss of regularity, indicating very strong ill-posedness.

The approach applies to both Hartree and power-type NLS equations, and potentially more nonlinear equations.

Abstract

We consider fractional Hartree and cubic nonlinear Schr\"odinger equations on Euclidean space $\mathbb R^d$ and on torus $\mathbb T^d$. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data in Fourier amalgam spaces with negative regularity. In particular, these spaces include Fourier-Lebesgue, modulation and Sobolev spaces. We further show that this can be even worse by exhibiting norm inflation with an infinite loss of regularity. To establish these phenomena, we employ a Fourier analytic approach and introduce new resonant sets corresponding to the fractional dispersion $(-\Delta)^{\alpha/2}$. In particular, when dispersion index $\alpha$ is large enough, we obtain norm inflation {above} scaling critical regularity in some of these spaces. It turns out that our approach could treat both equations (Hartree and power-type NLS) in a unified manner. The method should also work for a broader range of nonlinear equations with Hartree-type nonlinearity.