Boltzmann Equation and Multi-Dimensional Shock Interactions in Gas Dynamics

本计画的目的是研究气体流动中的非线性波与边界现象。 该项目有两个部分：（1）从运动论的观点研究边界效应，特别是玻尔兹曼激波层、初始层及其相互作用，以及固体边界的热效应、曲率效应和凝结效应;（2）研究多维气体激波流动，特别是激波从固体边界反射对可压缩欧拉方程的整体流动模式的影响。 从动力学理论的角度分析边界效应在物理上是自然的，因为在玻尔兹曼解中包含微观速度允许物理上真实的固体边界条件建模。 这在数学上是具有挑战性的，因为边界条件，例如镜面反射和扩散反射条件的麦克斯韦型插值，使得研究粒子和流体波的丰富相互作用成为可能。 本项目将对物理现象进行局部分析。带激波的多维气体流动是压缩的强非线性效应和气体与固体边界的全局相互作用的结果。 数学分析因此变得高度非线性和全球化。 构造混合型非线性偏微分方程的自由边界解需要新的极值原理。 椭圆性原理已经帮助解释了，在远场给定边界条件下，唯一性仅对自相似流是可能的，而对定常流是不可能的。研究气体流动的边界效应对工程实践具有重要意义。 为了定量和定性地理解物理过程，需要进行数学分析。 一个经典的例子是真空泵基于热梯度流动现象，而不是机械装置，来产生压差。 初步的分析研究表明，热梯度流是更强的稀薄气体，在真空泵的设计中的一个重要考虑。 可压缩气流中普朗特猜想和冯·诺依曼佯谬的研究为超音速飞行的激波结构提供了基本的认识。 例如，数学分析量化了飞机头部和发动机内部的激波结构之间的差异。 该项目旨在促进对这些重要过程的数学模型的理解。

The goal of this project is to study nonlinear waves and boundary phenomena in gas flow. The project has two parts: (1) The study of boundary effects from the point of view of kinetic theory, in particular, the Boltzmann shock layer, initial layer, and their interactions, and the thermal, curvature, and condensation effects of the solid boundary; (2) Study of multidimensional gas flow with shocks, in particular, the effects of shock reflections from a solid boundary on the overall flow patterns for the compressible Euler equations. To analyze boundary effects from the point of view of kinetic theory is physically natural, as the inclusion of the microscopic velocity in the Boltzmann solutions allows for physically realistic modeling of the solid boundary conditions. It is mathematically challenging, as the boundary condition, such as the Maxwell type of the interpolation of specular and diffusion reflection conditions, makes possible the study of the rich interactions of particles and fluid waves. This project will pursue local analysis of the physical phenomena. Multi-dimensional gas flow with shocks is the consequence of the strongly nonlinear effect of compression and the global interaction of the gas with the solid boundary. Mathematical analysis thereby becomes highly nonlinear and global. New extremal principles are needed for the construction of solutions with free boundary for nonlinear partial differential equations of mixed types. Already, an Ellipticity Principle has helped to explain that, with given boundary condition at far field, the uniqueness is possible only for self-similar flow, and not for stationary flows. The study of the boundary effects on gas flow is of great importance for engineering practices. Mathematical analysis is needed for quantitative and qualitative understanding of the physical process. A classical example is the vacuum pump based on the phenomenon of thermal gradient flow, rather than on mechanical devices, to generate pressure differences. Preliminary analytical studies show that the thermal gradient flow is stronger for more rarefied gases, an important consideration in the design of vacuum pumps. The study of Prandtl's conjecture and the von Neumann paradox in compressible gas flows provides basic understanding for the shock structure of supersonic flight. For instance, mathematical analysis quantifies the difference between the shock structure at the nose and that inside the engine of an aircraft. This project aims to advance understanding of the mathematical models of these important processes.