Global well-posedness for generalized fractional Hartree equations with rough initial data in all dimensions
Yufeng Lu
TL;DR
This work addresses global well-posedness for the fractional Hartree equation in all dimensions with rough initial data. The authors develop a real-interpolation–based splitting method, combining an $L^2$-based flow with a linear evolution of a rough component defined via the fractional Schrödinger semigroup $S_m(t)$, to obtain global solutions without small data assumptions for $p$ near $2$. Central contributions include constructing real-interpolation spaces $(L^2,X_{q,r}^{s,m})_{\theta,\infty}$ and the associated evolution framework, proving local and global well-posedness for data in these spaces, and deriving global well-posedness in modulation spaces $M_{p,p'}^{s_p}$ with $s_p=(m-2)(1/2-1/p)$, under subcritical mass conditions and parameter relations. The approach relies on polynomial growth estimates for $S_m(t)$ on modulation spaces and yields a broad, robust mechanism for rough-data well-posedness in fractional dispersive settings.
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 Schrödinger 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 Schrödinger semi-group on modulation spaces $M_{p,p'}$ with loss of regularity $s_{p}$.