Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Growth of Sobolev norms for 2D cubic nonlinear Schrödinger equation with partial harmonic potential

Mingming Deng, Xiaoyan Su, Jiqiang Zheng

TL;DR

This work analyzes the 2D cubic NLS with a partial harmonic potential, formulating the problem in Sobolev spaces adapted to the operator A = -∂_x^2 - ∂_y^2 + y^2. It establishes local well-posedness in Bourgain spaces tailored to A by proving key bilinear and trilinear estimates and leveraging local Strichartz bounds for the linear flow. The authors then develop a Planchon–Tzvetkov–Visciglia–style energy framework using modified energies to obtain polynomial-in-time bounds for higher-order Sobolev norms, extending existing results from the Laplacian and the full harmonic oscillator to the partial harmonic setting. The results provide insight into weak turbulence phenomena under partial confinement and furnish robust analytic tools for spectral-localized analysis of NLS with mixed spectra.

Abstract

In this paper, we study the $2$D cubic nonlinear Schrödinger equation (NLS) with the partial harmonic potential. First, we prove the local well-posedness in Bourgain spaces by establishing a key bilinear estimate associated with the partial harmonic oscillator. Then, we give the polynomial bound of the Sobolev norms for the solutions using the method of the Planchon, Tzvetkov, and Visciglia.

Growth of Sobolev norms for 2D cubic nonlinear Schrödinger equation with partial harmonic potential

TL;DR

This work analyzes the 2D cubic NLS with a partial harmonic potential, formulating the problem in Sobolev spaces adapted to the operator A = -∂_x^2 - ∂_y^2 + y^2. It establishes local well-posedness in Bourgain spaces tailored to A by proving key bilinear and trilinear estimates and leveraging local Strichartz bounds for the linear flow. The authors then develop a Planchon–Tzvetkov–Visciglia–style energy framework using modified energies to obtain polynomial-in-time bounds for higher-order Sobolev norms, extending existing results from the Laplacian and the full harmonic oscillator to the partial harmonic setting. The results provide insight into weak turbulence phenomena under partial confinement and furnish robust analytic tools for spectral-localized analysis of NLS with mixed spectra.

Abstract

In this paper, we study the D cubic nonlinear Schrödinger equation (NLS) with the partial harmonic potential. First, we prove the local well-posedness in Bourgain spaces by establishing a key bilinear estimate associated with the partial harmonic oscillator. Then, we give the polynomial bound of the Sobolev norms for the solutions using the method of the Planchon, Tzvetkov, and Visciglia.
Paper Structure (12 sections, 15 theorems, 140 equations)

This paper contains 12 sections, 15 theorems, 140 equations.

Table of Contents

  1. introduction
  2. preliminaries
  3. Hermite functions (See Than93book)
  4. Sobolev spaces adapted to $A$ (See SWX22')
  5. Littlewood--Paley theory for $A$
  6. The Bourgain space $X^{s,b}$ adapted to $A$
  7. Local Strichartz estimates (See AC20)
  8. Local well-posedness in Bourgain spaces
  9. Bilinear estimates
  10. Trilinear estimate
  11. Local well-posedness in Bourgain spaces
  12. Growth of Sobolev norms

Key Result

Proposition 1.1

Let $\delta_0\in(0,\frac{1}{2}]$ and assume that $1 \leqslant M \leqslant N$ are dyadic integers. Then there exists $0<b<\frac{1}{2}$ and $C>0$ such that for every $\delta\in(0,\delta_0]$ where $\Delta_N$ is given in Subsection lwd and $X^{s, b}({\mathbb{R}}\times{\mathbb{R}}^2)$ is the Bourgain space defined in Section subsection of Bourgain space.

Theorems & Definitions (32)

  • Proposition 1.1
  • Theorem 1.1: Growth of Sobolev norms
  • Remark 1.1
  • Lemma 2.1: See BTT13
  • Remark 2.1
  • Lemma 2.2: Multiplier theorem for $A$
  • Lemma 2.3: Bernstein-type estimates
  • proof
  • Remark 2.2
  • Lemma 2.4
  • ...and 22 more