Study on transit layers of the Boltzmann equation

本研究的目的是澄清玻尔兹曼方程的传输层的一些性质。一般来说，非线性偏微分方程描述了数学科学各个领域的复杂现象，借助泛函分析、调和分析、算子理论、分岔理论等，研究了解的基本数学结构，包括解的存在性、唯一性和渐近性。研究了控制流体运动的Navier-Stokes、Boltzmann及相关方程的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、分岔解、激波剖面和过渡层发展的数学机制。本文还研究了混沌理论，该理论旨在定性和定量地描述非线性方程解的行为复杂性。正如著名的洛伦兹方程所显示的那样，混沌意味着预测由确定性（非概率）运动定律支配的现象的困难。有限次系统的混沌理论现在已经建立起来了。特别是，已知Li-Yorke型乱集的存在意味着混沌。然而，对于非线性偏微分方程这样的无限次系统，目前还没有具体的例子来证明乱集的存在。研究了具有无限维紧摄动的有限次系统存在置乱集的条件。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的关系。

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.