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

Abstract

In this paper we consider the defocusing Hartree nonlinear Schrödinger 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.

arXiv comment46 pages

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