On the Cauchy problem for Hartree equation in the Wiener algebra

TL;DR

This paper establishes global well-posedness for the mass-subcritical Hartree equation in the Wiener algebra and demonstrates ill-posedness in a broader Fourier integrable space, highlighting the importance of the kernel's properties.

Contribution

It proves global well-posedness in the Wiener algebra for the Hartree equation with homogeneous kernels and shows ill-posedness in the Fourier integrable space, extending results to non-homogeneous kernels.

Findings

Global well-posedness in Wiener algebra for mass-subcritical Hartree equation.

Ill-posedness in Fourier integrable space for the same equation.

Results extend to non-homogeneous kernels with Fourier transform in Lebesgue spaces.

Abstract

We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous, and is such that its Fourier transform belongs to some Lebesgue space.