有关跨音速激波及接触间断的若干偏微分方程问题

This project devotes to studying the existence, stability and uniqueness of several phenomena in gas dynamics involving transonic shocks and transonic contact discontinuities by partial differential equations. The typical problems are: (1) For the three-dimensional steady complete compressible Euler equations, study the stability of spherical symmetric transonic shock solutions under perturbations of the upcoming supersonic flows and downstream pressure, as well as the stability of the transonic contact discontinuities under perturbations of the supersonic flows at one side, for the case that the subsonic flow at the other side is quiescent gas; (2) Study well-posedness of transonic shocks in steady compressible perfect flows with frictions or other effects; (3) For the two-dimensional steady complete compressible Euler equations, study the stability of transonic shocks and transonic contact discontinuities under the perturbations of the supersonic flows in the class of functions of bounded variations (hence it may contain discontinuities). These problems are proposed to attack many typical difficulties in the present researches of this field, such as characteristic or non-characteristic free boundaries, and nonlinear elliptic-hyperbolic coupled-composite systems in non-smooth domains. The research of this project is useful to solve related problems in applications, and is also valuable for the development of the mathematical theory of partial differential equations. Based on previous works, we will utilize methods from complex analysis and harmonic analysis to develop the necessary new theory of first-order elliptic systems and hyperbolic systems, to solve the above typical problems or other related problems.

本项目利用偏微分方程研究气体动力学中若干跨音速激波和跨音速接触间断现象的存在性、稳定性及唯一性。典型问题是：（1）对三维定常可压缩欧拉方程组，研究其球对称跨音速激波特解在超音速来流和下游压强扰动下的稳定性；对一侧亚音速流是静止气体的情形，研究跨音速接触间断对于另一侧超音速流光滑小扰动的局部稳定性；（2）含摩擦等效应的定常可压缩理想流体中的跨音速激波的适定性；（3）对两维定常欧拉方程组，研究当超音速流在有界变差函数类里扰动时（从而含有间断）跨音速激波及跨音速接触间断的稳定性。这些问题是围绕着如特征及非特征自由边界，非光滑区域上非线性双曲-椭圆耦合-复合型方程组等本领域当前研究的关键难点展开的,对其研究不但有助于解决相关实际应用问题，对偏微分方程的理论发展也有重要意义。在前期工作的基础上,我们拟结合复分析,调和分析等方法，发展所需一阶椭圆组及双曲组理论，解决上述典型问题或相关问题。

跨音速激波和接触间断是气体高速运动中出现的两种基本流动形态，在超音速进气道、冲压发动机、超音速喷管设计中起着重要作用。若以定常可压缩欧拉方程组为控制方程，其理论研究涉及双曲—椭圆复合混合型偏微分方程组、特征或非特征自由边界、强波干涉、区域几何效应等典型数学难点。本项目发展了若干偏微分方程方法，研究了欧拉方程组的涉及跨音速激波和接触间断的若干典型边值问题的适定性。所得结果包括：（1）对三维定常非等熵可压缩欧拉方程组，证明了球对称跨音速激波在来流和下游压强高维小扰动下的稳定性。我们结合微分几何方法，发展了一套分解三维定常欧拉方程组的方法，不但证明了区域几何扩张对于定常跨音速激波的稳定性效应，而且可以用来研究摩擦、热交换等其它工程需中考虑的因素对定常跨音速激波稳定性的影响。此外，我们在研究中自然地导出了非经典的二阶线性非局部椭圆型方程及其Venttsel型边值问题、球面上的椭圆型方程等，我们综合运用和发展调和分析技术得到了这类线性问题的适定性，这在偏微分方程理论中也饶有趣味。（2）对二维定常非等熵可压缩欧拉方程组，证明了一侧为静止气体,一侧为超音速流的跨音速接触间断当超音速流场含有强激波或强疏散波时，在上游超音速来流不光滑小扰动下，整体弱熵解的存在性。我们构造了新的Glimm 泛函，仔细处理了不同类型基本波的干涉问题。（3）对相关三维定常亚音速流的稳定性和唯一性，以及平面定常等熵无旋流中的脱体激波问题，做了一些探索性研究。这些结果有助于人们深入理解相关物理现象，对工程实践和数值计算有一定借鉴价值，也为相关数学理论进一步发展打下了良好的基础。

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