The mixed fractional Hartree equations in Fourier amalgam and modulation spaces

TL;DR

This paper establishes local and global well-posedness for mixed fractional Hartree equations with low regularity data in Fourier amalgam and modulation spaces, extending previous results and addressing Hartree-Fock equations.

Contribution

It introduces new well-posedness results in Fourier amalgam and modulation spaces for fractional Hartree and Hartree-Fock equations, broadening the functional framework.

Findings

Proves well-posedness in Fourier amalgam and modulation spaces.

Extends previous results to all p,q in [1, ∞] for these spaces.

Addresses Hartree-Fock equations with multiple particles.

Abstract

We prove local and global well-posedness for mixed fractional Hartree equation and with low regularity Cauchy data in Fourier amalgam $\F W(L^p,\ell^q)$ and modulation $M^{p,q}$ spaces. Similar results also hold for the Hartree equation with harmonic potential in some modulation spaces. Our approach also addresses Hartree-Fock equations of finitely many (but arbitrary large) particles. A key ingredient of our method is to establish trilinear estimates for Hartree non-linearity and the use of Strichartz estimates. As a consequence, we could gain $\F W(L^p,\ell^q)$ and $M^{p,q}-$regularity for all $p,q\in [1, \infty].$ In particular, we extend result of Bhimani-Grillakis-Okoudju \cite{bhimani2020hartree} in $M^{p,q}$ for all $p,q$ and complement known results in Sobolev spaces.