Nonlinear Schr\"odinger equation with Ornstein-Uhlenbeck operator

TL;DR

This paper introduces and analyzes nonlinear Schr"odinger equations with anisotropic dispersion involving Ornstein-Uhlenbeck operators, establishing key estimates and results like blow-up and scattering in specific models, capturing waveguide-like behavior.

Contribution

It is the first rigorous analysis of NLS with Ornstein-Uhlenbeck operators in both divergence and non-divergence forms, providing new estimates and well-posedness results.

Findings

Established Strichartz estimates for both models

Proved finite-time blow-up for divergence form model

Proved global well-posedness and scattering for non-divergence form model

Abstract

In this work, we introduce and study nonlinear Schr\"odinger equations (NLS) with anisotropic dispersion, where the standard Laplacian acts on the Euclidean variable \(x \in \mathbb{R}^d\), and an Ornstein-Uhlenbeck ($\mathcal{OU}$) operator governs the confined direction \(\alpha \in \mathbb{R}\). We consider models with two natural variants of $\mathcal{OU}$-induced confinement: (Model Div) based on the divergence form \(\nabla_\alpha \cdot (e^{-\frac{\alpha^2}{2}} \nabla_\alpha)\), and (Model Non-Div) based on the non-divergence form \(\Delta_\alpha - \alpha \cdot \nabla_\alpha\). For both models, we establish the Strichartz estimates and Gaussian-weighted Morawetz estimates. In addition, for (Model Div), we prove a virial-type finite-time blow-up result; for (Model Non-Div), we establish global well-posedness and small data scattering in the 2D quintic and 3D cubic cases. The primary motivation of this work is to capture waveguide-type dispersive behavior in a Euclidean setting. To the best of our knowledge, this is the first rigorous analysis of NLS with $\mathcal{OU}$ operators in both divergence and non-divergence forms.