Modified scattering for long-range Hartree equations of infinite rank near vacuum

TL;DR

This paper analyzes the long-term behavior of solutions near vacuum for the 3D Hartree equation with Coulomb potential, establishing decay estimates and global well-posedness without phase correction.

Contribution

It provides a direct method to derive decay estimates and proves global well-posedness for the long-range Hartree equation near vacuum, with explicit phase modification.

Findings

Established sharp dispersive decay estimates for the density function

Proved global in time well-posedness without phase correction

Derived explicit phase modification in terms of the density function

Abstract

We establish the asymptotic behavior and decay of solutions near vacuum to the Hartree equation with the Coulomb interaction potential in three dimensions. Our approach is direct, which consists of independently deriving the sharp dispersive decay estimates for the density function and establishing the boundedness of energy norms. The global in time well-posedness is done without introducing the phase correction, while the phase modification is explicit in terms of the density function.