Lorentz流形中理想流体动力学方程及相关模型的研究

Lorentz流形中理想流体动力学方程与广义相对论、宇宙学、流体力学等学科具有紧密的联系，是双曲偏微分方程研究领域的热门课题之一。本项目主要从流体具有的“线性退化”结构以及Lorentz流形几何的弯曲性质两个方面研究可压缩理想流体及相关模型小解的整体存在性以及大解的奇性形成机制。具体研究如下几个方面的问题：（1）具有“线性退化”特征的高维相对论理想流体方程小解的整体存在性；（2）具有“线性退化”特征的相对论欧拉方程大解的奇性形成及传播机制；（3）一般的Lorentz（如Schwarzschild、Schwarzschild-de Sitter）流形中理想流体动力学方程小初值解的长时间行为；（4）自引力相对论欧拉（Euler-Einstein）系统小扰动解的长时间行为。这些问题的解决对于双曲偏微分方程以及流体力学的发展都具有重要的理论意义及应用价值。

The dynamical equations of the perfect fluids in Lorentz manifolds are colsely related to general relativity, cosmology and fluid dynamics, and is one of the hot research topics in the field of hyperbolic partial differential equations. This project mainly study the global existence of the smooth solution and the blowup phenomenon of the compressible perfect fluids depending on the ``lineary degenerate'' character of the fluids and the geometric property of the Lorentz manifolds. Concretely speaking, we study: (1)The global small solutions to the relativistic Euler equations for a linearly degenerate equation of state in high dimensions; (2)The formation and propagation mechanism of the large solutions to relativistic Euler equations for a linearly degenerate equation of state; (3)The long time behavior of the small solutions to perfect fluids in general Lorentz manifolds, such as Schwarzschild and Schwarzschild-de Sitter spacetimes; (4)The long time behavior of the small solution to Euler-Einstein system. The solvability of these problems has important theoretical and applicable significances for the development of hyperbolic partial differential equations and fluid dynamics.