臺灣博碩士論文加值系統

波茲曼方程式在研究熱傳導與稀薄氣體的領域中有著重要的地位。本文探討
穩態波茲曼方程式在擴散反射邊界條件下解的存在性，並討論當邊界溫度不為定
值的情形。我們給出一個直接的方法去估計線性化波茲曼算子的核空間，並藉由
此證明了L 2 空間的解存在性與估計。我們也推廣了傳統的特徵線方法，討論了與
邊界多次碰撞的情形，並證明了L ∞ 空間的解存在性與估計。最後，我們證明了
當邊界的溫度與均衡溫度差距小的情形下，穩態波茲曼方程式在邊界非常溫的擴
散反射邊界條件下解的存在性。

In this thesis, we consider the steady Boltzmann equation with diffuse reflection
boundary condition. We study the case of hard potential and the non-isothermal
boundary. We prove the existence and the uniqueness of solution and their estimate
in both L 2 and L ∞ space.
In L 2 the Theorem, we provide a direct way to estimate the kernel of the
linearized Boltzmann operator. In the L ∞ Theorem, we introduce the stochastic
cycles and prove the estimate that is valid for both steady and dynamic cases. And
we provide a iteration scheme for the non-isothermal boundary temperature to prove
the existence result and the L ∞ estimate when the wall temperature do not oscillate
too much.

口 試 委 員 會 審 定 書 i
誌 謝 ii
中 文 摘 要 iii
Abstract 英 文 摘 要 iv
1 Introduction 1
2 Preliminaries 7
2.1 Notations and Background . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Some Basic Properties and Lemmas . . . . . . . . . . . . . . . . . . . 8
3 L 2 Estimate on Stationary Linearized Boltzmann Equation 10
3.1 Construction of Solution for Damping Transport Equation . . . . . . 10
3.2 Construction of Solution for the Equation with Cut-off Linearized
Boltzmann Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Key Estimate for L 2 Estimate . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Proof of the L 2 Estimate Theorem . . . . . . . . . . . . . . . . . . . 20
4 L ∞ Estimate on Stationary Linearized Boltzmann Equation 22
4.1 Estimate on Integration on Stochastic Cycles . . . . . . . . . . . . . . 23
4.2 Iteration Scheme for L ∞ Case . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Key Estimate for L ∞ Case . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Proof of the L ∞ Estimate Theorem . . . . . . . . . . . . . . . . . . . 33
5 Existence and L ∞ Estimate for Non-isothermal Boundary 38
5.1 Iteration for Non-isothermal Boundary . . . . . . . . . . . . . . . . . 38
5.2 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 38
6 Conclusion 42
Reference 43

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