Global well-posedness for generalized fractional Hartree equations with rough initial data in all dimensions

TL;DR

This paper establishes the global well-posedness of fractional Hartree equations for rough initial data across all dimensions, expanding understanding of solution existence without smallness constraints.

Contribution

It introduces a novel approach using fractional Schr"odinger semigroup-based interpolation spaces and adapts a splitting method to prove global solutions in modulation spaces.

Findings

Global existence of solutions for fractional Hartree equations.

No smallness condition required on initial data.

Extension to all dimensions with rough initial data.

Abstract

We prove the global existence of the solution for fractional Hartree equations with initial data in certain real interpolation spaces between $L^{2}$ and some kinds of new function spaces defined by fractional Schr\"odinger semigroup, which could imply the global well-posedness of the equation in modulation spaces $M_{p,p'}^{s_{p}}$ for $p$ close to 2 with no smallness condition on initial data, where $s_{p}=(m-2)(1/2-1/p)$. The proof adapts a splitting method inspired by the work of Hyakuna-Tsutsumi, Chaichenets et al. to the modulation spaces and exploits polynomial growth of the fractional Schr\"odinger semi-group on modulation spaces $M_{p,p'}$ with loss of regularity $s_{p}$.