Global solutions for 3D nonlocal Gross-Pitaevskii equations with rough data

TL;DR

This paper establishes global well-posedness for 3D nonlocal Gross-Pitaevskii equations with rough initial data, even when the energy is not positive definite, using a modified I-method.

Contribution

It extends global existence results to rough data and non-positive energy cases for nonlocal Gross-Pitaevskii equations in three dimensions.

Findings

Proves global well-posedness for rough data with nonlocal interactions

Handles cases where the energy functional is not positive definite

Uses a modified I-method for the analysis

Abstract

We study the Cauchy problem for the Gross-Pitaevskii equation with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and its Fourier transform decays sufficiently rapidly the problem is shown to be globally well-posed for large rough data which not necessarily have finite energy and also in a situation where the energy functional is not positive definite. The proof uses a suitable modification of the I-method.