Boltzmann Equation and Multi-Dimensional Shock Interactions in Gas Dynamics

本项目的目标是研究气体流动中的非线性波动和边界现象。该项目包括两个部分：(1)从运动论的角度研究边界效应，特别是Boltzmann激波层、初始层及其相互作用，以及固体边界的热、曲率和凝结效应；(2)研究具有激波的多维气体流动，特别是固体边界的激波反射对可压缩欧拉方程整体流型的影响。从运动论的角度分析边界效应在物理上是自然的，因为在Boltzmann解中包含微观速度允许对固体边界条件进行物理上真实的建模。它在数学上具有挑战性，因为边界条件，如镜面反射和扩散反射条件的麦克斯韦插值型，使得研究粒子与流体波的丰富相互作用成为可能。该项目将对物理现象进行局部分析。具有激波的多维气体流动是压缩的强烈非线性效应和气体与固体边界整体相互作用的结果。因此，数学分析变得高度非线性和全球化。构造混合型非线性偏微分方程解的自由边界需要新的极值原理。椭圆性原理已经解释了，在远场给定边界条件的情况下，唯一性只对自相似流动是可能的，而对定常流动是不可能的。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，稀薄气体的热梯度流动较强，这是真空泵设计中的一个重要考虑因素。可压缩气体流动中Prandtl猜想和von Neumann悖论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机头和发动机内部的冲击结构之间的差异。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.