Refined probabilistic global well-posedness for the weakly dispersive NLS

TL;DR

This paper advances the understanding of the cubic fractional nonlinear Schrödinger equation with very weak dispersion by constructing strong solutions for a broader range of dispersion parameters, improving previous results and employing novel analytical techniques.

Contribution

It extends the well-posedness results for the fractional NLS with weaker dispersion, using new resolution methods and adapting recent ideas to handle singular frequency interactions.

Findings

Constructed strong solutions for ta > 1.124, below previous thresholds.

Improved previous well-posedness results for weakly dispersive NLS.

Utilized novel resolution ansatz and frequency interaction analysis.

Abstract

We continue our study of the cubic fractional NLS with very weak dispersion $\alpha>1$ and data distributed according to the Gibbs measure. We construct the natural strong solutions for $\alpha>\alpha_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann \cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular we adapt to our situation the new resolution ansatz in \cite{Deng2} which captures the most singular frequency interaction parts in the $X^{s,b}$ type space. To overcome the difficulties caused by the weakly dispersive effect, our specific strategy is to benefit from the "almost" transport effect of these singular parts and to exploit their $L^{\infty}$ as well as the Fourier-Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions.