高维可压缩流体力学中的一些问题

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.

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。

本项目主要研究高维可压缩流体力学相关的数学问题，包括定常可压缩Euler方程和非定常可压缩/不可压缩Navier-Stokes方程的解的局部和整体适定性，光滑解的爆破现象，亚音速-音速极限，粘性消失极限等问题；同时本项目还研究了高维可压缩流体力学中的数值计算方法。本项目证明了二维可压缩Navier-Stokes方程组任意含真空大初值的整体强解存在性；在补偿列紧框架下证明了任意高维的定常非等熵Euler方程的亚音速-音速极限，该项工作是国际上第一个在任意空间维数定常非等熵 Euler 方程亚音速-音速极限方面取得的结果；证明了等熵可压Navier-Stokes方程的Navier-slip初边值问题解到Euler方程的解的消失粘性极限，并且得到了消失粘性极限的收敛速率；研究并提出了具有高维特征的黎曼解法器，该解法器非常健壮，适用于多介质问题的模拟；证明了三维球对称可压缩Navier-Stokes方程组弱解的存在性和唯一性；非等熵可压缩Navier-Stokes流体光滑解的延拓准则，彻底解决并推广了Nash在1958年提出的问题；本项目建立了可压Euler方程收敛到不可压方程的紧性框架；证明了有界域上密度大扰动的三维非齐次不可压Navier-Stokes方程整体光滑解的存在性；本项目共发表SCI论文14篇。