Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three

TL;DR

This paper proves the invariance of the Gibbs measure and the existence of global strong solutions for the defocusing Hartree NLS in three dimensions with potentials decaying like |k|^{-eta}, extending previous results to lower decay rates.

Contribution

It introduces a new method based on random averaging operators to establish invariance and global solutions for lower decay rates of the potential, improving upon Bourgain's earlier work.

Findings

Gibbs measure invariance for  < eta > eta_0

Global strong solutions exist for  < eta > eta_0

Extension of Bourgain's result to  < eta  1

Abstract

In this paper we consider the defocusing Hartree nonlinear Schr\"odinger equations on $\mathbb T^3$ with real valued and even potential $V$ and Fourier multiplier decaying like $|k|^{-\beta}$. By relying on the method of random averaging operators in arXiv:1910.08492, we show that there exists $\frac{1}{2} \ll \beta_0 <1 $ such that for $ \beta > \beta_0 $ we have invariance of the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way we extend Bourgain's seminal result [7] which requires $\beta >2$ in this case.