非线性高维双曲方程和双曲方程组是偏微分方程中的重要研究领域，关于它们解性质的研究不但富有数学理论意义，而且与可压缩流体动力学、广义相对论等学科密切相关。非线性高维双曲方程和双曲方程组的典型代表分别是非线性波动方程和可压缩Euler方程组。关于双曲方程（组）经典解爆破和爆破机制以及爆破后解发展性质的研究始终是高维双曲方程理论中的重大课题。本项目将围绕如下问题进行：（1）系统研究并力争解决2维Euler方程组或3维Euler方程组环流或饶流的长时间性态。（2）研究一般拟线性波动方程（即系数依赖于解本身及解导数）小初值或带有Neumann边值问题解的爆破或整体存在性。（3）研究半线性广义退化Tricomi双曲方程低正则解的适定性及长时间性态，并对可压缩Euler方程组当初始值含有间断以及阻尼项随时间有较慢的衰减时，证明整体间断解（含有高维激波或中心疏散波或高维接触间断）的稳定性。

Nonlinear hyperbolic equations and systems are important research areas for nonlinear partial differential equations. It is always an important topic to study the blowup or blowup mechanism or the development of solutions after the blowup time of classical solutions to nonlinear hyperbolic equations (systems). The typical representations of nonlinear hyperbolic equations and systems are nonlinear wave equations and compressible Euler systems respectively. This project focuses on the following problems: (1) Systematically study and solve the long time behaviors for the compressible Euler flows around bounded or unbounded bodies.(2) Study the blowup or global existence of solutions to quasilinear wave equations with small initial data or small initial-boundary values of Neumann types. (3) Study the well-posedness of low regularity solutions and long time behaviors of solutions to general Tricomi equations. Meanwhile, study the global discontinuous solutions (including shock waves, rarefaction waves or contact discontinuities) for the compressible Euler systems with time-depending damping terms.