On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions $d=2,3$
Yilin Song, Ruixiao Zhang, Jiqiang Zheng
TL;DR
The paper proves polynomial-in-time growth bounds for higher-order Sobolev norms of cubic NLS with a harmonic potential in dimensions 2 and 3. It introduces sharp bilinear Strichartz estimates for the harmonic oscillator, and adapts the upside-down I-method to the harmonic-oscillator setting, enabling control over high-frequency interactions via explicit eigenfunction products. A key innovation is a precise frequency-interaction formula for products of eigenfunctions, which, combined with a spectral multiplier theorem, yields almost-conserved modified energy and thus the polynomial growth bounds. The results extend the 2D bound to 3D and establish a robust framework for growth control in trapped NLS models, with potential extensions to other geometries like S^3.
Abstract
In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-Δ+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three.