亚音速和亚音速-音速可压Euler流

This project concerns compressible subsonic and subsonic-sonic flows governed by the steady Euler equations, possessing physical background and theoretic meaning. The studied flows include steady compressible subsonic-sonic potential flows in nozzles, subsonic Euler flows and subsonic-sonic Euler flows. From the mathematical point of view, studies of these steady compressible subsonic and subsonic-sonic flows require us to solve boundary value problems of quasilinear elliptic partial differential equations or quasilinear elliptic-hyperbolic partial differential equations of mixed type. The aim is to establish reasonable mathematical formulations with physical conditions for these equations from specific problems, to give the reasonable definition and space of solutions, and to study their qualitative theory, such as well-posedness of problems, regularity, singularity, behavior of fluids through the sonic interfaces. The study of these equations needs not only the classical theory of partial differential equation, but also choosing proper research framework according to different equations with constantly innovating and developing research tools and methods. The project and the research results in this project can supply the important reference and guidance for practical problems, and also play a unique role in our understanding of more complex flow patterns in nature, as well as enrich and develop the mathematical theory of partial differential equations.

本项目旨在研究具有鲜明实际背景和重要理论价值的亚音速和亚音速-音速定常可压Euler流，主要包含亚音速-音速定常可压位势管道流，亚音速定常可压Euler流以及亚音速-音速定常可压Euler流。这些亚音速和亚音速-音速定常可压流问题导致了求解拟线性椭圆型方程或拟线性椭圆-双曲混合型方程组的边值问题。我们将针对具体的问题，建立符合物理条件的数学描述，提出合理的定解问题和解的定义，确定合适的求解空间，研究问题的适定性以及解的正则性和奇性，刻画流体在音速界面附近的性态。研究这些方程，不仅需要经典的偏微分方程理论知识，而且需要根据不同的方程选择合适的研究框架，以及研究工具和研究方法的不断拓展和创新。本项目的开展及其研究成果一方面为实际生活生产和航天航空等方面提供重要的参考信息和理论依据，另一方面对理解其它更加复杂的流体的运动形式有着重要的理论意义，同时也发展和完善了偏微分方程的数学理论体系。

该项目主要研究来源于流体动力学领域的具有鲜明实际背景和重要理论价值的亚音速和亚音速-音速定常可压Euler流问题和来源于物理学、几何学、力学、生物学以及材料科学和工程技术等领域的非线性偏微分方程，包括拟线性退化椭圆方程、非线性椭圆-双曲混合型偏微分方程、非线性退化和奇异扩散方程和方程组、半线性退化扩散-对流方程和方程组。项目主要研究成果有：拟线性退化椭圆方程的自由边界问题和一般二维弯曲管道内连续亚音速-音速流的适定性，来源于物理学、几何学和工程技术领域的非线性扩散-对流方程和方程组的适定性和渐近理论，来源于生物数学领域的半线性扩散-对流方程和方程组的适定性和控制理论。在Acta Math. Sin.、Electronic J. Differential Equations、J. Nonlinear Sci. Appl.等杂志发表论文9篇，于2019年获得吉林省科学技术奖二等奖，获奖项目名称为“物理和几何中的非线性偏微分方程的定性理论”，其中部分研究成果是该项目资助的一些论文。