权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Boundary Problems in the Boltzmann theory

玻尔兹曼理论中的边界问题

基本信息

批准号：
1501031
负责人：
Chanwoo Kim
金额：
$ 17.8万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-06-01 至 2019-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501031&HistoricalAwards=false
关键词：
Boundary Problems Boltzmann theory

项目摘要

The kinetic theory and its models such as the Boltzmann equation and Vlasov equations have played an important role in the understanding of problems in gas dynamics, plasma physics, and fluid equations. In many physical situations, the particles in various models interact with boundaries, and this interaction creates several interesting phenomena such as the formation of singularities. In general, boundary effects may not stay only near the boundary but also impact on the whole interior dynamics. Therefore the boundary effects are important and have rich application in many cases. However, boundary problems in kinetic models are mathematically challenging due to their singular nature. This project aims to develop new mathematical tools to handle these problems.This research project studies several important boundary problems arising in kinetic models and fluid equations. The first topic regards optimal regularity of Boltzmann solutions in bounded domains with several physical boundary conditions for both dynamical and stationary problems. The second topic concerns the boundary-field interaction in kinetic models such as Vlasov-Poisson-Boltzmann systems. The third topic is to understand the relation between steady Boltzmann solutions and the incompressible Navier-Stokes-Fourier systems in the presence of a boundary when the mean free path is sufficiently small. The last topic concerns fluid-material interaction such as surfactant dynamics on viscous surface waves. Results of the project are expected to improve modeling capabilities for a wide range of problems of interest to other scientists and engineers.

动力学理论及其模型，如玻尔兹曼方程和弗拉索夫方程，在理解气体动力学、等离子体物理和流体方程中的问题方面发挥了重要作用。在许多物理情况下，各种模型中的粒子与边界相互作用，这种相互作用产生了一些有趣的现象，例如奇点的形成。一般来说，边界效应可能不仅停留在边界附近，而且会影响整个内部动力学。因此，边界效应在许多情况下是重要的，并且有着丰富的应用. 然而，动力学模型中的边界问题由于其奇异性而在数学上具有挑战性。本研究计画旨在发展新的数学工具来处理这些问题，本研究计画针对动力学模型与流体方程式中的几个重要边界问题进行研究。第一个主题是关于动力学和定常问题的有界区域中具有几个物理边界条件的Boltzmann解的最优正则性。第二个主题涉及的边界场相互作用的动力学模型，如Vlasov-Poisson-Boltzmann系统。第三个主题是了解定常玻尔兹曼解和不可压缩的Navier-Stokes-Fourier系统在边界的存在下，当平均自由程是足够小的关系。最后一个主题涉及流体-材料相互作用，如粘性表面波上的表面活性剂动力学。该项目的结果预计将提高其他科学家和工程师感兴趣的广泛问题的建模能力。