Unconditional global well-posedness for the 3D Gross-Pitaevskii equation for data without finite energy

TL;DR

This paper proves the global well-posedness of the 3D Gross-Pitaevskii equation for a broad class of initial data without finite energy, using an advanced method that overcomes the lack of energy conservation.

Contribution

It establishes unconditional global well-posedness for data in 1 + H^s with 5/6 < s < 1, extending previous results to less regular data without finite energy.

Findings

Global solution exists for broader initial data classes.

The I-method effectively handles the absence of energy conservation.

Improves upon previous results by Bethuel-Saut and Gerard.

Abstract

The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof uses the I-method which is complicated by the fact that no L^2 -conservation law holds. This improves former results of Bethuel-Saut and Gerard.