向量值函数空间上退化微分方程的适定性

向量值函数空间上微分方程的适定性研究具有重要的理论意义，在物理学、经济学、人口统计学以及工程技术等领域有很多直接的应用。本项目拟研究几类退化微分方程在Lebesgue- Bochner空间、周期Besov空间、周期Triebel-Lizorkin空间、 Holder连续函数空间上的适定性。利用向量值函数空间上的算子值傅里叶乘子定理工具，将微分方程的适定性转化为相应函数空间上的傅里叶乘子问题，从而给出这些微分方程具有适定性的充分条件，必要条件，甚至充分必要条件。进一步，我们将所得抽象结果应用到一些具体的退化微分方程。另外，我们也研究这些微分方程的适定性与所在巴拿赫空间几何结构的联系。

The research for the well-posedness of differential equations in vector-valued function spaces has important theoretical significance. There are many direct applications in physics, economics, demography, and engineering technology and so on. In this project, we study the well-posedness of some degenerate differential equations in Lebesgue-Bochner spaces, periodic Besov spaces, periodic Triebel- Lizorkin spaces and Holder continuous function spaces. Using operator-valued Fourier multiplier theorems in vector-valued function spaces, we will transform the well-posedness of differential equations to an operator-valued Fourier multiplier problem in the corresponding vector-valued function spaces. Thus we will give sufficient conditions, necessary conditions and necessary and sufficient conditions for these differential equations to have the well-posedness. Furthermore, we apply our abstract results to some concrete degenerate differential equations. In addition, we also study the relations between the well-posedness of these problems and the geometry of the underlying Banach space.