与频率分解有关的现代函数空间的理论及应用

Using the theory of function space to solve the Cauchy problem of nonlinear evolution equations is the typical application of the method of harmonic analysis. This program will consider the Cauchy problem of (derivative) Schrödinger equation in some function space under the framework of α-decomposition of the frequency space. This program shall continue to improve the study on the structure and property of this type of function space. This program shall study the global well-posedness and scattering problem of (derivative) Schrödinger equation in the framework of these function spaces. This program tries to provide a fresh perspective to interpret the difficulties of some problems from partial differential equation, and it tries to contribute to this field. Meanwhile, the study of the program also develops the classical function theory.

运用函数空间的理论去解决非线性发展方程的初值问题是调和分析方法的典型应用. 该项目拟在一些与频率空间的α-分解有关的函数空间框架中考察(导数)Schrödinger方程的初值问题. 该项目将在我们已有的研究基础上继续完善空间框架结构、性质的研究. 该项目将开展(导数)Schrödinger方程在这样的空间框架下整体适定性及散射问题研究. 该项目拟对方程学的一些难点问题从新的视角给出诠释，以期对问题的解决作出一点贡献. 同时，该项目所开展的研究也是对经典函数论课题的发展.

这个研究项目是我们早期工作的一个继续。我们运用之前得到的alpha-模空间的性质与技术处理非线性（导数）薛定谔方程的初值问题。我们得到了非线性（导数）薛定谔方程在alpha-模空间框架下的临界正则性指标。具体地说，对携带非线性项|u|^{2k}u的非线性薛定谔方程，只要s>=(d*alpha)/2-(alpha)/k并且初值很小，我们证明了它在alpha-模空间M^{s,alpha}_{2,1}中是整体适定的；对携带非线性项nabla(|u|^{2k}u)的非线性导数薛定谔方程，只要s>=(d*alpha)/2-(alpha)/k+1/(2k)并且初值很小，我们证明了它在alpha-模空间M^{s,alpha}_{2,1}中是整体适定的。我们还指出这些指标是临界的。我们还对一般导数薛定谔方程开展了类似研究。