关于高维非线性双曲偏微分方程低正则解的局部和整体适定性

The nonlinear multidimensional hyperbolic partial differential equations and systems are closely related to the compressible fluid dynamics,general relativity theory and so on,meanwhile,they have the basic mathematically theoretical meanings.The related typical models are the nonlinear wave equations and the compressible Euler equations.In addition,for the Einstein equations in the general relativity theory,under the various metrics,it can be reduced into the quasilinear or semilinear wave equations (systems),for examples,the Klein-Gorden equation,the semilinear generalized Tricomi equation,the universal explosion model of the de Sitter line element.Generally speaking,with the development of the time,the classical solutions to these nonlinear equations can blow up in finite time.Therefore,it is important to study the local or global well-posedness of the related low regularity solutions.In this subject,we will apply the harmonic analysis,the theory of function spaces,microlocal analysis and the theory of partial differential equations to study the local or global well-posedness of optimal low regularity solutions for the compressible Euler equations of polytropic gases or Chaplygin gases,for the quasilinear wave equations satisfying the null conditions and the semilinear Tricomi equations.One of our main purposes is to establish the related Strichartz estimates for the corresponding non-flat metrics.

高维非线性双曲方程和方程组与可压缩流体动力学、广义相对论等学科密切相关,同时富有数学理论意义.其典型代表是非线性波动方程(组)和可压缩Euler方程组.另外对于广义相对论中的爱因斯坦方程,根据不同度量条件,可以演化出拟线性或半线性波动方程(组),如Klein-Gorden方程,半线性广义Tricomi方程,de Sitter线素度量下的宇宙爆炸模型等.一般说来,随着时间的发展,这些非线性双曲方程经典解将在有限时间内爆破,因此研究低正则解的局部或整体适定性是非常必要的.我们将综合运用调和分析、函数空间理论、微局部分析和偏微分方程理论等,研究可压缩Euler方程组(包括多方气体和Chaplygin气体)、满足零条件结构的拟线性波动方程和半线性广义Tricomi方程等最佳低正则解的局部或整体适定性,核心目的之一是建立相关非平坦度量下的Strichartz估计.

本项目研究的背景是可压缩Euler方程组满足零条件结构的拟线性波动方程最佳低正则解的局部或整体适定性。在项目执行期间，主要研究了两方面的内容。第一，在两维可压缩Euler方程组方面得到了相关边值问题解的存在性和正则性；第二，本项目在流形上的调和分析方面得到了与方程相关的结果，得到的主要结论与本项目相关的微局部分析方法方面有密切的联系，对应高阶微分方程的正则解问题。目的是在仿射对称空间上利用微局部分析法来考虑Helgason猜想。在项目执行期间，项目组发表学术论文2篇。