Almost sure local well-posedness for the supercritical quintic NLS

TL;DR

This paper proves almost sure local well-posedness for the supercritical quintic nonlinear Schrödinger equation on  with randomized initial data below the critical regularity, extending techniques from previous cubic problem studies.

Contribution

It establishes almost sure local well-posedness for supercritical quintic NLS with randomized data below critical regularity, building on recent probabilistic methods.

Findings

Almost sure local well-posedness in  for s in ((d-2)/2, (d-1)/2)

Extension of techniques from cubic to quintic NLS

Conditions for almost sure global well-posedness

Abstract

This paper studies the quintic nonlinear Schr\"odinger equation on $\mathbb{R}^d$ with randomized initial data below the critical regularity $H^{\frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s \in (\frac{d-2}{2}, \frac{d-1}{2})$. The argument further develops the techniques introduced in the work of \'A. B\'enyi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.