Refined probabilistic global well-posedness for the weakly dispersive NLS

Abstract

We continue our study of the cubic fractional NLS with very weak dispersion α>1 and data distributed according to the Gibbs measure. We construct the natural strong solutions for α>α0=31−23314≈1.124 which is strictly smaller than 87, 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 Xs,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∞ as well as the Fourier–Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions.