Strong ill-posedness for fractional Hartree and cubic NLS Equations

Abstract

We consider fractional Hartree and cubic nonlinear Schrödinger 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.

arXiv comment34 pages, 4 figures, to appear in Journal of Functional Analysis