一些混合型偏微分方程的定性研究

This program is focus on some properties of solutions of some mixed-type partial differential equations. N-dimensional nonlinear Tricomi type and Keldysh type equations are two typical mixed-type partial differential equations with different degeneracy. Moreover they are closely connected with some foundamental problems in gas dynamics and geometry analysis.Therefore, many attentions from famous mathematician have been attracted to this study, and some important development have been made in this field in recent years. In this program, based on microlocal analysis, singular integrals, Fourier analysis, Littlewood-Paley decomposition, function space, commutators and other mathematical tools, we will analyze the impact of different degeneracy on the regularity of its solution. And according to the structure of the equations, we will choose some proper weighted function spaces to study the existence and uniqueness of solutions for some different kinds of boundary value problems. Moreover, we'll give some positive results on solutions of their initial value problems, including global existence and sigularity propogation.

本项目主要研究了一些混合型偏微分方程解的定性问题。高维非线性的Tricomi 型和Keldysh 型方程是含有不同退化性质的混合型偏微分方程的两类典型代表，并且与几何分析、气体动力学中的一些重要问题有着紧密联系。因此，该类方程的研究成了许多著名数学家关注的焦点，并且在近年来取得了一些重要进展。本项目中，申请人拟采用微局部分析、奇异积分算子、傅里叶分析、Littlewood-Paley 分解和函数空间以及交换子等数学工具对其进行研究，分析不同退化性质对于相应问题解的正则性的影响，根据方程本身的结构选择合适的加权函数空间建立不同边值问题的解的适定性，并分析初值问题解的存在性和奇性传播。

在本项目实施期间，我们主要研究了一些混合型偏微分方程解的定性问题，特别是在Tricomi型方程的闭边值问题的适定性，半线性Keldysh型方程解的存在性，流体力学方程组的超音速激波解理论等方面取得了一些有意义的结果。1.在Hadmade意义下我们分别探讨了了混合型和退化双曲型Tricomi型方程在矩形区域上闭边值问题的解的性态。2.对于可压缩流体的高维锥形超音速激波的稳定性给出了证明。