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

TL;DR

This paper proves global well-posedness for the fractional Hartree equation with cubic convolution nonlinearity in modulation spaces and Fourier algebra, extending results to various dimensions, fractional orders, and radial data.

Contribution

It establishes new global well-posedness results for the fractional Hartree equation in modulation spaces and Fourier algebra, covering a range of fractional orders and data types.

Findings

Global well-posedness in modulation spaces for specific parameters.

Results for fractional order b1=2 and b1a0da0 2d/(2d-1).

Extension to Fourier algebra L^1 and L^2 spaces.

Abstract

We study the Cauchy problem for fractional Schr\"odinger 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 $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+\gamma))$ when $\alpha =2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< \alpha < 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$