加权空间中的NLS全局理论I：有限伪共形能量
Global theory for NLS in the weighted spaces I: finite pseudo conformal energy
浏览论文内容
中文总结 AI 辅助
本文证明散焦非线性薛定谔方程在有限伪共形能量下解全局存在且散射，将Bourgain结果推广至$s_c<0$情形，利用超临界界替代次临界控制。
中文摘要 AI 辅助
我们旨在证明对于散焦非线性薛定谔方程，具有有限伪共形能量的解必定是全局存在且散射的。这一事实首先由Bourgain在$0<s_c<1$情形下证明：方程在$H_x^{s_c}$中局部适定，若进一步假设初值满足$xu_0\in L_x^2$，则解全局存在且散射。$s_c<0$的情形尚未被研究，目前最佳结果由Beceanu、Deng、Soffer和Wu给出：若初值属于$H_x^{s_c}$、径向且具有紧支撑，则解全局适定。本文中，我们将Bourgain的结果推广到$s_c<0$情形。我们证明：若初值满足$|x|^{-s_c}u_0\in L_x^2$且$xu_0\in L_x^2$，或径向初值满足$u_0\in \dot{H}_x^{s_c}$且$xu_0\in L_x^2$，则解全局存在且散射。Bourgain的结果基于$L_x^{p+2}$上的次临界先验控制，而我们的论证依赖于$\mathcal{F}\dot H_x^1$上的超临界界。
英文摘要
We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when $0<s_c<1$: the equation is locally well-posed in $H_x^{s_c}$, and if one further assumes that the initial data satisfies $xu_0\in L_x^2$, then the solution is global and scatters. The $s_c<0$ case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in $H_x^{s_c}$, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain's result to the $s_c<0$ case. We prove that if the initial data satisfies $|x|^{-s_c}u_0\in L_x^2$ and $xu_0\in L_x^2$, or if the radial initial data satisfies $u_0\in \dot{H}_x^{s_c}$ and $xu_0\in L_x^2$, then the solution is global and scatters. Bourgain's result is based on the subcritical a priori control on $L_x^{p+2}$, while our argument relies on the supercritical bound on $\mathcal{F}\dot H_x^1$.
发表机构
- Center for Applied Mathematics Tianjin University（天津大学应用数学中心）
- School of Mathematical Sciences and LPMC Nankai University（南开大学数学科学学院及低维物理和材料计算重点实验室）
- Tianjin University（天津大学）
机构由 AI 辅助整理，请以论文原文为准。
