关于一类退化双曲型方程解的适定性研究及相关应用

Tricomi equation is a typical representation of degenerate partial differential equations, which arises from the continuous transonic problem in compressible dynamical fluids. There are the important physical backgrounds on Tricomi equation, moreover, there exists some essencial relations with other basic mathematical problems: For the potential equation in compressible Euler gases,if the solution produces breakdown,the vacuum will be formed, meanwhile the potential equation becomes the nonlinear Tricomi-type. As well known, it is a basic open question for the formation and development of vacuum in multidimensional compressible gases. On the other hand, under various assumptions of metrics, the Einstein’s equation in general relativity theory will become many kinds of semilinear wave equations with time-dependent damping, for examples, Klein-Gorden equation, Einstein’s equation in FLRW metrics, the cosmic explosive models in de Sitter line element and so on. These equations are closely related to the semilinear Tricomi equations by suitable transformations of variables and unknown functions. Based on our previous works on the semilinear Tricomi equations, we hope that we can take the joint researches on the well-posedness of problems mentioned above by making full use of harmonic analysis.

Tricomi方程是退化方程的典型代表，它来源于空气动力学中的连续跨音速流问题。关于它的研究不但有重要的物理背景，而且近年来还发现与其它重要数学问题存在密切联系：对于可压缩气体的位势流方程，若其解产生断裂(breakdown)，则意味着真空形成，此时方程将退化成（非线性）Tricomi方程，而关于高维可压缩气体如何产生和发展真空一直是大家关注的公开问题；对于广义相对论中的爱因斯坦方程，根据度量的不同条件，可以演化出许多带有时间阻尼项的半线性波动方程，如Klein-Gorden方程，FLRW度量下的Einstein方程，de Sitter线素（line element）度量下的宇宙爆炸模型等，而这些方程通过合适的自变量或未知函数的变换都与半线性Tricomi方程存在密切的本质联系。利用我们以前建立的半线性Tricomi方程解的适定性，同时运用深入的调和分析工具，合作研究上述问题的适定性。

本项目的研究内容是与调和分析方法在偏微分方程中的应用相关的一些问题. 对于可压缩气体的位势流方程，若其解产生断裂(breakdown)，则意味着真空形成，此时方程将退化成（非线性）Tricomi方程，而Tricomi方程是非退化方程的典型代表，运用调和分析工具来研究半线性Tricomi方程是本项目关注的重点。在本项目中我们还研究了非交换Calderon-Lozanovskii空间的复插值空间和Yosida-Hewitt分解定理等，同时我们还研究了带Dini核的多线性Calderon-Zygmund算子的迭代交换子的有界性、分数次Hardy算子的交换子在Lipschitz空间上的端点估计、双线性分数次极大算子的交换子的紧性.