Refined probabilistic global well-posedness for the weakly dispersive NLS

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 root natural strong solutions for alpha > alpha(0) = 31 root 233/14 approximate to 1.124 which is strictly smaller than 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 and Tzvetkov (2020). We rely on recent ideas of Bringmann (2021) and Deng et al. (2019). In particular we adapt to our situation the new resolution ansatz in Deng et al. (2019) which captures the most singular frequency interaction parts in the X-s,X-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-infinity as well as the Fourier-Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The study focuses on cubic fractional NLS with weak dispersion and data distributed according to the Gibbs measure. By constructing root natural strong solutions, the research improves upon previous results and relies on new ideas to overcome difficulties caused by weakly dispersive effect.