在新型飞行器设计、火箭、新型能源乃至天体物理等国民经济和军事工业的核心技术领域中，高维可压缩流体运动的研究起着非常重要的作用。例如，航空发动机设计需要用到可压缩管道流动理论；而飞机和航天器外形的改进和性能的提高则需要可压缩Navier-Stokes方程的相关理论。因此高维可压缩流体运动的研究对航空航天以及国防工业具有重要意义。数学上，高维可压缩流动问题涉及混合型非线性偏微分方程组和退化问题，因此非常有挑战性。由于这些问题无论是在理论上还是应用上都非常重要，所以近几十年以来，吸引了众多学者的关注，取得了丰富的研究成果。但是仍有许多重要问题还未解决。本项目拟研究和高维可压缩流动相关的一些数学问题：1)研究高维管道有旋亚音速流动解的存在唯一性，解的单调性和渐近行为等流动性质；2)研究亚音速流动的音速极限问题。3)研究可压缩Navier-Stokes方程初边值问题强光滑解的整体适定性和大时间行为。

Multi-dimensional compressible fluids have wide applications in engineering such as rocket, wind tunnel, turbine engine etc. Many challenging problems are proposed in mathematics, since the governing euations of multi-dimensional compressible fluids are nonlinear mixed-type partial differential equations. Due to the importance of these problems in both theory and applications, there have been substantial progresses in studying these problems, yet many important issues remain to challenge the field. The theory of subsonic flows in two dimensional and axial symmetric three dimensional nozzles are well developed by using stream function method, which can not be applied to three or higher dimensional problems. The applicant and coworkers proved the existence and uniqueness of subsonic potential flows in nozzle of arbitrary dimension. But the existence and uniqueness of sulutions to subsonic flows with non-zero vorticity in three (or higher) dimensional nozzle remain unknown. The main difficulty is that we must deal with the steady Euler system which is of elliptic-hyperbolic mixed type in this case. As flows in nozzle or around airfoil accelerating, it is important to understand how subsonic flows accelerate from subsonic to sonic and what happen in this procedure. Shiffman claimed without publication that the limits of subsonic flows were subsonic-sonic flows. Unfortunately, his manuscripts was lost. So far, mathematically, we can only take sonic limits in the sense of distribution by using the method of compensate compactness. Therefore, lots of important information is lost such as structure of flow fields and the distribution of sonic sets. Up to now, most results on global wellposedness of smooth solutions with vacuum of compressible Navier-Stokes are about Cauchy problems in whole space. In this project, we will study some problems related to the muitl-dimensional compressible flows. Specifically, we will study, 1) the existence and uniqueness of the smooth solutions to subsonic flows with vorticity in high-dimensional nozzle, and the properties of the subsonic flows in nozzle; 2) the subsonic-sonic limit problems; 3) the global wellposedness and long time behaviors of smooth solutions to initial-boundary value problems of compressible Navier-Stokes equations.