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Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator

Xueying Yu, Haitian Yue, Zehua Zhao

TL;DR

This work introduces nonlinear Schrödinger equations with an Ornstein-Uhlenbeck drift acting as a confining direction, analyzed in two formulations: divergence-form OU and non-divergence-form OU. The authors establish Strichartz estimates and Gaussian-weighted Morawetz inequalities in this mixed-geometry setting, proving a virial-type blow-up result for the divergence form and global well-posedness with small-data scattering for the non-divergence form in key 2D and 3D regimes. The analysis leverages spectral decompositions via OU eigenfunctions and Keel–Tao machinery, capturing waveguide-like dispersive behavior in a non-product geometry and revealing how drift confinement mimics compactness. The results provide foundational tools for NLS under soft confinement and open avenues for stochastic extensions, broader confinement mechanisms, and long-time dynamics in mixed-geometry dispersive PDEs.

Abstract

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.

Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator

TL;DR

This work introduces nonlinear Schrödinger equations with an Ornstein-Uhlenbeck drift acting as a confining direction, analyzed in two formulations: divergence-form OU and non-divergence-form OU. The authors establish Strichartz estimates and Gaussian-weighted Morawetz inequalities in this mixed-geometry setting, proving a virial-type blow-up result for the divergence form and global well-posedness with small-data scattering for the non-divergence form in key 2D and 3D regimes. The analysis leverages spectral decompositions via OU eigenfunctions and Keel–Tao machinery, capturing waveguide-like dispersive behavior in a non-product geometry and revealing how drift confinement mimics compactness. The results provide foundational tools for NLS under soft confinement and open avenues for stochastic extensions, broader confinement mechanisms, and long-time dynamics in mixed-geometry dispersive PDEs.

Abstract

In this work, we introduce and study nonlinear Schrödinger equations (NLS) with anisotropic dispersion, where the standard Laplacian acts on the Euclidean variable , and an Ornstein-Uhlenbeck () operator governs the confined direction . We consider models with two natural variants of -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 . 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 operators in both divergence and non-divergence forms.
Paper Structure (26 sections, 13 theorems, 121 equations, 2 tables)

This paper contains 26 sections, 13 theorems, 121 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Divergence form vs non-divergence form
  4. Connection to Gaussian measure
  5. NLS models confined with $\mathcal{OU}$ operators
  6. The statement of the main results
  7. Comparisons with models on waveguides and models with compact-like behaviors
  8. Organization of the rest of this paper
  9. Preliminarily
  10. Notations
  11. Strichartz estimates
  12. Conservation Laws
  13. Strichartz Estimates
  14. Strichartz Estimates for \ref{['maineq1']}
  15. Strichartz Estimates for \ref{['maineq2']}
  16. ...and 11 more sections

Key Result

Lemma 2.2

Let $(q,r)$ be an admissible Strichartz pair in $\mathbb{R}^d$. Then we have the bound Also, for any two Strichartz pairs $(q_1, r_1)$ and $(q_2,r_2)$, we have where $q_2'$ and $r_2'$ are conjugates of $q_2$ and $r_2$.

Theorems & Definitions (40)

  • Definition 2.1
  • Lemma 2.2: Strichartz estimate
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Proposition \ref{['prop: Conserv']}
  • proof : Proof of Theorem \ref{['mainthm1']}
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4: Strichartz estimate
  • ...and 30 more