高维激波致稳机制及守恒律方程测度值解的数学研究

This project is devoted to mathematical studies of two important problems in aerodynamics. The first problem is, based on previous research, to investigate macroscopic effects like heat transfer and mass addition encountered in designation of inlets of ramjets, analyze their roles in stabilization of transonic normal shocks in steady flows passing three-dimensional ducts, and explore the mathematical mechanisms that guarantee stable transonic shocks. The second problem is to establish a rigorous mathematical foundation for the method of infinite-thin shock layers, which is used in physics to solve the problem of hypersonic flow passing an obstacle, by developing the theory of measure-valued solutions of boundary value problems of conservation laws. The former is related to typical difficulties in mathematics such as systems of equations of elliptic-hyperbolic composite-mixed type, free-boundary, non-smooth domain, and Venttsel problems of nonlocal elliptic equations. For the latter, we need to propose a reasonable definition of the limit problem and its measure-valued solution, and study in which sense the sequence of solutions converges to the measure-valued solution, for a sequence of boundary value problems of the steady compressible Euler system describing flows passing the obstacle, for them the Mach numbers of the upcoming flows increase to infinity. To overcome these difficulties, we designed a series of simplified problems, and shall utilize methods from the regularity theory of elliptic equations and transport equations, harmonic analysis, differential geometry, optimal transport and Wasserstein space etc. The project will promote development of mathematical theory of multi-dimensional systems of conservation laws.

本项目将从数学理论角度研究两类重要的空气动力学问题。一是在前期研究基础上，考察热交换、添质等冲压发动机进气道设计中常见的宏观效应，分析它们对三维管道内定常流中跨音速正激波稳定性的作用，探索保证跨音速激波稳定的数学机制；二是对物理中求解高超音速绕流的无限薄激波层方法，通过发展守恒律方程边值问题测度值解的理论，从数学上予以严格阐述。前者涉及双曲—椭圆复合混合型偏微分方程组、自由边界、非光滑区域、非局部椭圆型方程Venttsel边值问题等典型数学难点，后者需要考虑在来流马赫数趋于无穷大时，定常可压缩欧拉方程组绕流边值问题的极限问题及其测度值解的合理定义，以及解序列在何种意义下收敛。针对典型难点，我们设计了一系列过渡问题以逐步推进研究，所用方法涉及二阶椭圆型方程及输运方程的解的正则性、调和分析、微分几何、最优输运、Wasserstein空间等。本项目研究将有助于促进高维守恒律方程组数学理论的发展。

本项目研究了数学气体动力学中有关可压缩欧拉方程组的三类问题。.一是与亚燃冲压发动机进气道设计相关的管道内定常跨音激波及一类亚音流的稳定机制。证明了具有摩擦阻尼或给定单位质量气体的热交换量时，跨音激波对上游超音流和下游压强的小扰动是稳定的；给定单位体积的热交换量不具有致稳效应；对含添质的管道内稳态亚音流，提出并证明了适定的定解问题。从数学角度讲，建立了综合运用非线性迭代、调和分析、常微分方程、实解析函数、二阶椭圆型方程正则性理论等求解一类拟线性双曲—椭圆复合型方程组的边值问题的一般方法，特别是克服了由于背景解是非平凡大解带来的含多种非局部项的椭圆型方程定解问题的新难点。这也为后续研究燃烧等效应对跨音激波稳定性的影响打下了数学方法上的基础。.二是高超音绕流问题与可压缩欧拉方程组的Radon测度解。证明了在适当无量纲化后，高超音极限流就是零压欧拉流；推广了国内外学者对delta波的定义，严格给出了具有一般状态方程的高维稳态可压缩欧拉方程组边值问题的Radon测度解的概念，将高超音绕流的牛顿无限薄激波层理论数学化，对于楔形体、带攻角直圆锥、轴对称弯曲锥体等构造出了质量含有集中在物面的Dirac测度的Radon测度解，严格推导出了著名的牛顿—Busemann定律，并将之推广到了具有一般外形的物体。对于绕有限长物体的高超音极限流，研究了其与零压射流或静止气体的作用，首次发现了delta波在物体后缘有限距离处的终止现象，在其下游已无法构造测度解，暗示欧拉方程组将完全失效。这些结果对于从纯粹力学角度严格分析高超音飞行器的外形设计提供了数学工具。.三是黎曼问题测度解的构造及应用。对一维等熵可压缩欧拉方程组，完整地构造出了不同于经典黎曼解的新的含有delta波的解。对这类新的解给出了物理解释：其中的delta波可看作是能吸收或放出气体的自由活塞。对零压欧拉方程组，严格证明了这种等价性。这表明Radon测度解理论将为解决一类复杂的流固耦合和多相流问题提供一种新的思路。