Local well-posedness for Gross-Pitaevskii hierarchies

TL;DR

This paper proves local well-posedness for the Gross-Pitaevskii hierarchy on  space using a novel (F)-norm in Sobolev spaces, providing existence, uniqueness, stability, and explicit estimates.

Contribution

Introduces a new (F)-norm framework for analyzing the Gross-Pitaevskii hierarchy, enabling local well-posedness results with explicit estimates.

Findings

Established local existence and uniqueness of solutions.

Derived explicit space-time estimates for solutions.

Demonstrated compatibility of the (F)-norm with Sobolev norms for factorized data.

Abstract

We consider the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n.$ By introducing a (F)-norm in certain Sobolev type spaces of sequences of marginal density matrices, we establish local existence, uniqueness and stability of solutions. Explicit space-time type estimates for the solutions are obtained as well. In particular, this (F)-norm is compatible with the usual Sobolev space norm whenever the initial data is factorized.