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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耳兹曼方程的过渡层的一些性质。一般而言，描述数学科学各个领域中复杂现象的非线性偏微分方程解的基本数学结构，包括解的存在唯一性和渐近性，借助于泛函分析、调和分析、算子理论、分支理论等来研究。应用于控制流体运动的Navier-Stokes、Boltzmann方程、解的时间-整体存在性、建立这些方程之间渐近关系的多尺度分析、分岔解、激波轮廓以及过渡层发展的数学机制。研究了混沌理论，其目的是定性和定量地描述非线性方程组解的行为的复杂性。这种混沌意味着对受确定性(非概率)运动定律支配的现象进行预测的难度，正如著名的洛伦兹方程所表明的那样。混沌理论现在已经很好地适用于有限次系统。特别地，我们知道Li-Yorke型置乱集的存在隐含着混沌。然而，对于无穷次系统，如非线性偏微分方程组，还没有已知具有置乱集的具体例子。研究了具有无限维紧扰动的有限次系统有扰集的条件。沿着这条线，我们研究了玻尔兹曼方程的过渡层。此外，还从多个方面讨论了非线性微分方程与量子游动之间的关系。