Study on transit layers of the Boltzmann equation

玻尔兹曼方程传输层的研究

基本信息

批准号：
16540185
负责人：
KONNO Norio
金额：
$ 1.02万
依托单位：
Yokohama National University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

The purpose of the present study is to clarify some properties of transit layers of the Boltzmann equation. In general, nonlinear partial differential equations, which describe complex phenomena in various fields of mathematical sciences, are investigated on fundamental mathematical structures of solutions, including the existence, uniqueness and asymptotic behavior, with the help of functional analysis, harmonic analysis, operator theory, theory of bifurcation and so on. Applications are made for the Navier-Stokes, Boltzmann and related equations which govern the motion of fluids, on the time-global existence of solutions, multi-scale analysis which establishes the asymptotic relations between these equations, bifurcating solutions, shock wave profiles, and mathematical mechanism of the development of transit layers. The theory of chaos is also investigated, which aims at qualitative and quantitative descriptions of complexity of behaviors of solutions to nonlinear equations. The chaos implies the difficulty of prediction of phenomena governed by the deterministic (non-probabilistic) law of motion, as shown by the famous Lorenz equation. The theory of chaos is now well-established for systems of finite degree. In particular, it is known that the existence of scrambled sets of Li-Yorke type implies the chaos. However, no concrete examples having scrambled sets are known of systems of infinite degree such as nonlinear partial differential equations. The condition for systems of finite degree with infinite dimensional compact perturbations to have the scrambled set is studied. Along this line, we have investigated transit layers of the Boltzmann equation. In addition, relations of between nonlinear differential equations and quantum walks were also discussed from various aspects.

本研究的目的是阐明玻尔兹曼方程传输层的一些性质。一般来说，非线性偏微分方程描述了数学科学各个领域中的复杂现象，借助泛函分析、调和分析、算子理论等研究解的基本数学结构，包括存在性、唯一性和渐近行为。分岔理论等。应用纳维-斯托克斯方程、玻尔兹曼方程和控制流体运动的相关方程、解的时间全局存在性、建立这些方程之间渐近关系的多尺度分析、分叉解、冲击波剖面和传输层发展的数学机制。还研究了混沌理论，旨在定性和定量描述非线性方程解的行为的复杂性。正如著名的洛伦兹方程所示，混沌意味着预测受确定性（非概率性）运动定律支配的现象的难度。对于有限度系统，混沌理论现在已经很成熟。特别是，众所周知，Li-Yorke 型乱集的存在意味着混沌。然而，对于诸如非线性偏微分方程的无限次系统，还没有已知具有扰乱集合的具体例子。研究了具有无限维紧扰动的有限次系统具有置乱集的条件。沿着这条线，我们研究了玻尔兹曼方程的传输层。此外，还从多个方面讨论了非线性微分方程与量子行走之间的关系。