Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator
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| Format: | Preprint |
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2025
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| _version_ | 1866915563425497088 |
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| author | Yu, Xueying Yue, Haitian Zhao, Zehua |
| author_facet | Yu, Xueying Yue, Haitian Zhao, Zehua |
| contents | 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 ($\mathcal{OU}$) operator governs the confined direction \(α\in \mathbb{R}\). We consider models with two natural variants of $\mathcal{OU}$-induced confinement: (Model Div) based on the divergence form \(\nabla_α\cdot (e^{-\frac{α^2}{2}} \nabla_α)\), and (Model Non-Div) based on the non-divergence form \(Δ_α- α\cdot \nabla_α\). 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator Yu, Xueying Yue, Haitian Zhao, Zehua Analysis of PDEs 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 ($\mathcal{OU}$) operator governs the confined direction \(α\in \mathbb{R}\). We consider models with two natural variants of $\mathcal{OU}$-induced confinement: (Model Div) based on the divergence form \(\nabla_α\cdot (e^{-\frac{α^2}{2}} \nabla_α)\), and (Model Non-Div) based on the non-divergence form \(Δ_α- α\cdot \nabla_α\). 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. |
| title | Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.17178 |