Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator

In this work, we introduce and study nonlinear Schrödinger equations (NLS) with anisotropic dispersion, where the standard Laplacian acts on the Euclidean variable $x\in {\mathbb{R}}^d$, and an Ornstein-Uhlenbeck ($\mathcal{O}\mathcal{U}$) operator governs the confined direction $\alpha \in \mathbb{R}$. We consider models with two natural variants of $\mathcal{O}\mathcal{U}$-induced confinement: (Model Div) based on the divergence form ${\mathrm{\nabla }}_{\alpha }\cdot (e^{-\frac{{\alpha }^2}{2}}{\mathrm{\nabla }}_{\alpha })$, and (Model Non-Div) based on the non-divergence form ${\mathrm{\Delta }}_{\alpha }-\alpha \cdot {\mathrm{\nabla }}_{\alpha }$. 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 $\mathcal{O}\mathcal{U}$ operators in both divergence and non-divergence forms.