频率空间的分解，现代函数空间和色散型非线性方程

Nonlinear dispersive equations (NDE), including nonlinear Schrodinger equation, nonlinear wave equation, KdV equation are one of the most important part of modern PDE. In the past twenty yers nonlinear dispersive equations have made a great progress by using the harmonic analysis techniques, which play a centeral role in the study of NDE. To study the global well posedness of NDE, one needs to carry out the following four step works. First, one should choose the suitable function spaces. Secondly, one needs to make the linear dispersive estimates and the nonlinear mapping estimates in the corresponding function spaces. Thirdly, we should have some a priori estimates according to the energy structure of a NDE. Finally, we need to find the essential connections between the a priori estimates and the resolution function spaces, up to get the global well posedness. The developments in the past twenty years indicate that the first two steps are due to harmonic analysis theory, the last two steps are also connected with the frequency decomposition techniques in harmonic analysis. In this project, we will study a class of derivative nonlinear dispersive equations by using the harmonic analysis techniques, including the global well posedness, scattering theory.

色散波方程，包括非线性Schrodinger、非线性波动、KdV等方程，是当代PDE的核心内容之一，近二十年发展的调和分析方法是色散波方程的核心研究方法。研究非线性色散波方程，我们首先需要研究解的适定性理论，解的适定性研究（粗略地说）有四方面的工作需要做：第一需要我们选取适当的函数空间；第二需要在选定的函数空间做出线性波方程的色散估计和非线性项的估计；第三需要根据方程的自身结构，得到一些先验估计；第四需要对在解的适定空间和先验估计寻找内在联系，至最后解决问题。近二十年的发展表明，前两方面是纯调和分析的内容，后两方面也和调和分析的频率空间分解也有很大的联系。本项目将对（导数型）非线性色散方程，用调和分析的方法展开研究，特别是本项目将对一类函数空间- - - - 与频率空间分解生成的函数空间，结合色散方程做出深入研究，力争解决导数型色散方程的1-2个前沿公开问题。

本项目主要研究了下面的问题: (1) Navier-Stokes方程在临界Besov空间的不适定问题, 著名数学家Bourgain J. 等人得到Navier-Stokes 方程在临界Besov空间B^{-1}_{\infty.q}当q>2时不适定, 我们得到Navier-Stokes 方程在所有的临界Besov空间B^{-1}_{\infty.q}都不适定, 这一结果当q<2是出乎意料的. 这一结果最终回答了Navier-Stokes方程在临界Besov空间的适定性这一长期公开问题. (2) 负责人发展了频率一致分解方法来研究非线性偏微分方程, 针对(四阶, 导数型)非线性Schrodinger方程, 得到整体适定性和散射结果, 这些结论可以推出在临界Sobolev空间H^s中的一类大初值的整体适定性结果. (3) 研究了alpha-模空间的性质, 得到它们和Besov空间的相互嵌入, scaling 的充分必要条件. 也得到alpha-模空间的代数结构的充分条件.