Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

Abstract

We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity ($i{\partial }_{t}u-(-\Delta {)}^{\frac{\alpha }{2}}u\pm (K\ast |u{|}^2)u=0$) with Cauchy data in the modulation spaces $M^{p,q}({\mathbb{R}}^d).$ For $K(x)=|x{|}^{-\gamma }$ $(0<\gamma <\text{min}\{\alpha ,d/2\})$, we establish global well-posedness results in $M^{p,q}({\mathbb{R}}^d)(1\le p\le 2,1\le q<2d/(d+\gamma ))$ when $\alpha =2,d\ge 1$, and with radial Cauchy data when $d\ge 2,\frac{2d}{2d-1}<\alpha <2.$ Similar results are proven in Fourier algebra $\mathcal{F}{L}^1({\mathbb{R}}^d)\cap L^2({\mathbb{R}}^d).$

arXiv comment15 pages