非线性薛定谔方程以及多参数、多线性乘子L^p估计的研究

In this project, we will investigate some important open problems in the study of the Cauchy problems for NLS equations and L^p estimates for multi-parameter multi-linear Fourier multipliers. Our research plan mainly include two aspects. On one hand, we plan to investigate the global well-posedness and scattering properties for the solutions to mass-supercritical NLS equations. More precisely, we will study the global well-posedness and scattering properties for the H^1 solutions to 3D focusing energy-critical NLS equations with general non-radial initial data below the ground state threshold; we will also study the "scattering conjecture", that is, for any 0<s<1, try to prove that the H^s solutions to the H^s-critical defocusing NLS equations are global well-posed and scatter in H^s. On the other hand, our project also includes the study of the L^p estimates for multi-parameter bilinear multipliers given by singular symbols having different singularity sets (0-dimensional or 1-dimensional, linear or nonlinear) with respect to different spatial variables, and we will try to derive some L^p estimates for the bilinear Hilbert transforms when p is equal to or less than 2/3.

本课题将对非线性Schrödinger（NLS）方程Cauchy问题和多参数、多线性Fourier乘子的L^p估计的研究中尚未得到解决的若干难题进行研究。具体研究内容主要包括两个方面。一方面，我们将研究质量超临界NLS方程解的整体适定性与散射理论。具体来说，对一般的低于基态解的“动能”与“能量”门槛的非径向初值，我们将研究3维聚焦型能量临界NLS方程H^1解的整体适定与散射性质；我们还将对“散射猜想”进行研究，即，尝试对H^s（0<s<1）临界散焦型NLS方程证明其H^s解的整体适定性与散射。另一方面，本课题还将对关于不同空间变量具有不同维数（0维或1维）奇异点集（线性或非线性）的多参数、双线性乘子的L^p估计进行研究，并且尝试证明双线性Hilbert变换当p小于或等于2/3时的L^p有界性。这些都是NLS方程与现代调和分析研究中的基本问题，具有重要理论意义和研究价值。

在本项目研究中，我们对各种类型的分数阶与高阶静态半线性椭圆型方程非负解建立了分类结果，并在此基础上进一步研究分数阶与高阶聚焦型非线性薛定谔方程的整体适定性与散射。此外，我们还对对由多线性拟微分算子构成的极大函数建立了Sharp 的L^p 估计。