带斥性平方反比势的薛定谔方程的双线性估计及其应用

中文摘要 AI 辅助

本文针对带斥性平方反比势的线性薛定谔方程 $i\partial_tu+\La u=0$（其中 $\La=-\Delta+a|x|^{-2}$，$a\geq0$）建立了非径向数据的双线性Strichartz估计。我们的证明依赖于物理空间方法（相互作用Morawetz估计）。这类方法最初由Planchon-Vega [Ann. Sci. Éc. Norm. Supér. (4) {\bf 42} (2009), 261--290] 在平坦情形引入，并由Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam, \textbf{39}(2023), no. 4, 1405-1436] 用于调和振荡子。我们的双线性估计将Camps [Ann. Inst. H. Poincaré Anal. Non Linéaire {\bf 39} (2022), 1--58] 和Pusateri-Soffer [Mem. Amer. Math. Soc. {\bf 299} (2024), v+107 pp] 的结果推广到临界情形。作为应用，我们建立了带平方反比势的三次NLS方程 \begin{equation*} iu_{t} +\La u = -|u|^2u,\\ \\ (t,x)\in \mathbb{R}\times\mathbb{R}^{3}. \end{equation*} 对径向初值 $u_0\in H_a^s$（其中 $s>\frac{11}{13}$，$a>0$，$H_a^s$ 是适应于带平方反比势的薛定谔算子的Sobolev空间）的全局适定性。证明依赖于Bourgain的高-低频分解论证和我们的双线性估计。这是关于具有标度临界势的NLS低正则性适定性的首个结果。

英文摘要

In this paper, we establish the bilinear Strichartz estimates for the following linear Schrödinger equation with repulsive inverse-square potential $i\partial_tu+\La u=0$ where $\La=-Δ+a|x|^{-2}$ with $a\geq0$ for non-radial data. Our proof relies on the physical space method (interaction Morawetz estimate). Such type of method was first introduced in Planchon-Vega [Ann. Sci. Éc. Norm. Supér. (4) {\bf 42} (2009), 261--290] in flat case, and Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam, \textbf{39}(2023), no. 4, 1405-1436] for harmonic oscillator. Our bilinear estimate extends results obtained in Camps [Ann. Inst. H. Poincaré Anal. Non Linéaire {\bf 39} (2022), 1--58] and Pusateri-Soffer [Mem. Amer. Math. Soc. {\bf 299} (2024), v+107 pp] to the critical case. As an application, we establish the global well-posedness for cubic NLS with inverse-square potential \begin{equation*} iu_{t} +\La u = -|u|^2u,\ \ (t,x)\in \mathbb{R}\times\mathbb{R}^{3}. \end{equation*} for radial initial data $u_0\in H_a^s$ with $s>\frac{11}{13}$ and $a>0$ where $H_a^s$ is the Sobolev space adapted to the Schrödinger operator with inverse-square potential. The proof relies on Bourgain's high-low frequency decomposition argument and our bilinear estimates. This is the first result on the low regularity well-posedness for NLS with scaling-critical potential.

发表机构

• Zhengzhou University（郑州大学）
• Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
• Northeastern University（东北大学）

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