Kinetics Theory and Multidimensional Gas Flow

摘要：DMS-0406089，T-P Liu，斯坦福大学动力学理论和多维气体流动本项目的目标是发展新的数学方法来研究多维气体流动与冲击和动力学理论。对于多维气体流动的研究，最具挑战性的是需要解决的混合型和自由边界的非线性方程组的数学问题。近年来平面波研究的巨大进展以及大量的数值实验使人们能够确定基本的关键问题。一个这样的问题是时间相关的，自相似制定的普朗特-迈耶反射了斜坡。固体边界对椭圆域有着至关重要的稳定作用。这些基本的、易处理的问题的解决将为非线性偏微分方程中的其他问题提供必要的分析工具。 随着现代设备变得越来越小，并且随着人们远离地球表面，气体的动力学方面变得更加明显。玻尔兹曼方程的粒子和流体方面的双重属性是拟议的研究的主要问题。对于流体方面的研究，守恒定律的思想被推广到玻尔兹曼方程。粒子方面将研究的基础上的玻尔兹曼绿色函数的建设和估计，其中有最近的进展的基础上的组合的频谱分析和建设的动力学类型的波。 重要的，长期存在的开放问题，如热蠕变现象正在理解与玻尔兹曼方程的精确分析。新的数学方法正在开发中，并应证明对动力学理论中的其他问题也是有用的。

Abstract: DMS-0406089, T-P Liu, Stanford UniversityKinetics Theory and Multidimensional Gas FlowThe goal of this project is to develop new mathematical methods for the study of multi-dimensional gas flows with shocks and the kinetic theory. For the study of multi-dimensional gas flows, the mathematical problem of nonlinear equations of mixed types and free boundary that one needs to address are most challenging. Recent tremendous progress on the study of planary waves as well as the extensive numerical experiments allows one to identify the basic key problems. One such problem is the time-dependent, self-similar formulation of the Prandtl-Meyer reflection off a ramp. The solid boundary has the crucial stabilizing effect on the domain of ellipticity. The resolution of these basic, tractable problems would provide the necessary analytical tools for other problems in nonlinear partial differential equations. As modern devices are getting smaller and as one travels further from the earth surface, the kinetic aspects of the gases become more pronounced. The dual property of the particle and fluid aspects of the Boltzmann equation is the main issue of the proposed research. For the study of the fluid aspect, the ideas from conservation laws are being generalized to bear on the Boltzmann equation. The particle aspect will be studied based on the construction and estimates of the Boltzmann Green function, for which there is the recent progress based on the combination of spectral analysis and the construction of the kinetic-types waves. Important, long-standing open problems such as the thermal creep phenomena are being understood with the exact analysis of the Boltzmann equation. New mathematical methods are being developed and should prove useful also for other problems in the kinetic theory.