Probabilistic well-posedness of generalized cubic nonlinear Schr\"odinger equations with strong dispersion using higher order expansions

TL;DR

This paper establishes probabilistic local well-posedness for generalized cubic nonlinear Schrödinger equations with strong dispersion, using higher order expansions and random initial data in negative Sobolev spaces.

Contribution

It introduces a novel framework combining multilinear expansions and probabilistic estimates to prove well-posedness in low regularity settings for dispersive PDEs.

Findings

Solutions exist almost surely with initial data in negative Sobolev spaces.

Developed directional space-time norms for controlling multilinear expansions.

Improved bilinear probabilistic-deterministic Strichartz estimates.

Abstract

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit scale of a given function $f$ and $\mathcal{L}$ being an operator of degree $\sigma\geq 2$. In particular, we prove that a solution exists almost-surely locally in time provided $f\in H^{S}_{x}(\mathbb{R}^{d})$ with $S>\frac{2-\sigma}{4}$ for $d\leq \frac{3\sigma}{2}$, i.e. even if the initial datum is taken in certain negative order Sobolev spaces. The solutions are constructed as a sum of an explicit multilinear expansion of the flow in terms of the random initial data and of an additional smoother remainder term with deterministically subcritical regularity. We develop the framework of directional space-time norms to control the (probabilistic) multilinear expansion and the (deterministic) remainder term and to obtain improved bilinear probabilistic-deterministic Strichartz estimates.