Kinetics Theory and Multidimensional Gas Flow

摘要：DMS-0406089，刘t - p，斯坦福大学，动力学理论与多维气体流动。该项目的目标是为研究具有激波的多维气体流动和动力学理论建立新的数学方法。对于多维气体流动的研究，需要解决的非线性混合型方程和自由边界的数学问题是最具挑战性的。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认识到基本的关键问题。一个这样的问题是时间依赖的，自相似的公式的普朗特-迈耶反射斜坡。固体边界对椭圆域具有重要的稳定作用。这些基本的、可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，随着人们离地球表面越来越远，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方面的对偶性质是提出的研究的主要问题。对于流体方面的研究，从守恒定律的思想被推广到玻尔兹曼方程。粒子方面的研究将基于玻尔兹曼格林函数的构造和估计，这方面的研究最近取得了基于谱分析和动力学型波的构造相结合的进展。重要的，长期开放的问题，如热蠕变现象正在理解与精确分析玻尔兹曼方程。新的数学方法正在发展中，对动力学理论中的其他问题也将证明是有用的。

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.