A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies

TL;DR

This paper introduces a novel proof demonstrating the existence of solutions for cubic and quintic Gross-Pitaevskii hierarchies in multiple dimensions, avoiding the need for prior spacetime norm bounds used in earlier methods.

Contribution

It provides a new proof technique for existence of solutions to GP hierarchies that bypasses the a priori spacetime norm bounds required in previous approaches.

Findings

Existence of solutions established without spacetime norm bounds.

Applicable to both focusing and defocusing interactions.

Valid in dimensions d ≥ 1.

Abstract

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d\geq1$ dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, \cite{klma}, and used in our earlier work, \cite{chpa2}.