具Hardy与库伦位势的色散方程的动力学行为研究

The nonlinear dispersion equations are very important physical models in quantum mechanics. The method of induction on energy pioneered by Fields Prize winner Bourgain opened the way for the study of the scattering theory of the critical dispersion equation. The concentrated compact method developed by Bocher Prize winner Kenig-Merle solved the scattering conjecture of the dispersion equation without potential. However, the models involved in a large number of specific physical problems are equations with potential (for example: Schrödinger equation with potential describes the propagation of light in nonlinear fibers and physical phenomena, such as Bose-Einstein condensation). The above classical research framework cannot be directly applied to the study of nonlinear dispersion equations with potential. This project is proposed to study the dynamic behavior of the nonlinear dispersion equation with potential, by establishing the harmonic analysis tool (such as: Sobolev space theory and Littlewood-Paley theory, etc.) corresponding to the Laplace operator with potential. These problems are one of the hot core researches in the field of partial differential equations in the past 30 years. They also involve some difficult problems that have not been solved for a long time. The solution of these problems has profound theoretical value in the fields of partial differential equations and harmonic analysis. And a wide range of application prospects.

非线性色散方程是量子力学中的重要物理模型。菲尔兹奖得主Bourgain开创的极小能量归纳法为研究临界色散方程的散射理论开辟了道路, Bocher奖得主Kenig-Merle发展的集中紧致方法解决了经典色散方程的散射猜想。然而大量具体物理问题所涉及的模型均是带位势的方程(例如: 带位势的薛定谔方程描述光在非线性光纤中的传播以及 Bose-Einstein 凝聚等物理现象)。经典的研究框架不能直接用于具位势的非线性色散方程的研究。本项目正是基于此，拟通过建立与具位势的拉普拉斯算子对应的调和分析工具(Sobolev 空间理论和 Littlewood-Paley 理论等)来研究具位势非线性色散方程的动力学行为。这些问题是偏微分方程领域近三十年来的热门核心研究之一,同时也涉及到一些长期尚未解决的难点问题, 这些问题的解决对偏微分方程和调和分析等领域均都具有深刻的理论价值和广泛的应用前景。

非线性色散方程是量子力学中的重要物理模型。菲尔兹奖得主Bourgain开创的极小能量归纳法为研究临界色散方程的散射理论开辟了道路, Bocher奖得主Kenig-Merle发展的集中紧致方法解决了经典色散方程的散射猜想。本项目(1)利用Guth的 k-broad-norm 估计提升了之前关于分数次薛定谔算子的局部光滑估计的结果; (2) 利用球调和级数展开和振荡积分技术解决锥奇性空间上波方程解对应的限制性估计;(3)通过开发调和分析方法中的交换子估计和bootstrap 方法克服非光滑非线性项带来的困难，高维质量临界Schrödinger 方程中log-log型爆破解在Hs-扰动下的稳定性; (4)通过构造具电磁场薛定谔算子谱测度与预解式估计建立了具电磁场位势色散算子的色散估计与一致预解式估计;(5)解决具Hardy位势与库伦位势色散方程的散射与爆破理论. 这些问题的解决对偏微分方程和调和分析等领域均都具有深刻的理论价值和广泛的应用前景。