Kinetics Theory and Multidimensional Gas Flow

文摘：DMS-0406089，刘天平，斯坦福大学动力学理论与多维气体流动本项目的目标是发展新的数学方法来研究含激波的多维气体流动和动力学理论。对于多维气体流动的研究，需要解决的混合类型和自由边界的非线性方程的数学问题是最具挑战性的。最近在平面波研究方面取得的巨大进展以及广泛的数值实验使人们能够确定基本的关键问题。其中一个问题是依赖于时间的、自相似的Prandtl-Meyer反射公式。固体边界对椭圆度区域具有重要的稳定化作用。这些基本的、容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，随着人们离开地球表面的距离越来越远，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的双重性质是所提出的研究的主要问题。对于流体方面的研究，来自守恒定律的思想正在被推广到与玻尔兹曼方程有关的方面。粒子方面的研究将基于玻尔兹曼格林函数的构造和估计，该函数的最新进展是基于谱分析和动力学类型波的构造相结合的。通过对玻尔兹曼方程的精确分析，人们正在理解诸如热蠕变现象等重要的、长期存在的未决问题。新的数学方法正在开发中，应该对运动论中的其他问题也有用。

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.