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研究生： 張慧慈
Chang, Hui-Tzu
論文名稱： 波茲曼⽅程的時空結構
Space time structure of the Boltzmann equation
指導教授： 吳恭儉
Wu, Kung-Chien
學位類別： 博士
Doctor
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2026
畢業學年度： 114
語文別： 英文
論文頁數： 69
中文關鍵詞： 波茲曼方程 、非截斷 、空間漸近行為 、可壓縮納維－斯托克斯方程式 、一維波茲曼方程 、流體近似 、硬勢 、逐點結構
外文關鍵詞： Boltzmann equation, non cutoff, spatial asymptotic behavior, compressible Navier-Stokes equation, one-dimensional Boltzmann equation, fluid approximation, hard potential, pointwise structure
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    • 本論文主要研究一維 Boltzmann equation 在 Hard potential 和 Maxwellian molecules 情形下，解的逐點估計，我們的研究主要有兩個目標。首先，我們刻畫了解在時間漸近線下的層次結構。其次，我們建立了一維 Boltzmann equation 和 Compressible Navier-Stokes equation 的漸近等價性。 不同於以往在多維度情形中僅需比較線性解，我們需要從 一維 Boltzmann equation 中抽出主要的非線性流體波，並將其與 Nonlinear Navier-Stokes system 進行直接比較。

    此外，我們也完成了 三維 Boltzmann equation without cutoff assumption 在 Hard sphere 的情形下，平衡態附近解的空間行為。 我們對初值施加適當的空間衰減和正則線性假設後，我們推導出解的空間漸進行為。 該行為由初始條件的衰減速率決定。

    This dissertation investigates the quantitative pointwise behavior of solutions to the one-dimensional (1D) Boltzmann equation for hard potentials and Maxwellian molecules. Our study pursues two main objectives. First, we characterize the time-asymptotic hierarchical structure of solutions. Second, we establish the asymptotic equivalence between solutions of the 1D Boltzmann equation and the 1D compressible Navier-Stokes equations. Unlike prior works in the multi-dimensional case, where comparison of linear solutions suffices, our approach extracts the leading nonlinear fluid waves from the 1D Boltzmann equation and directly compares them with solutions of the nonlinear Navier-Stokes system.
    We also investigate the quantitative spatial behavior of solutions to the three-dimensional (3D) Boltzmann equation with hard potentials in the whole space, under the near equilibrium regime and without the angular cutoff assumption. Assuming appropriate spatial decay and Sobolev regularity of the initial data, we derive the spatial asymptotic behavior of the solution, which is determined by the decay rate of the initial condition.

    1 Introduction 1 2 Time-asymptotic expansion and fluid approximation of the 1D Boltzmann equation with hard potential 3 2.1 Themodel 3 2.2 Notations 4 2.3 Main results 5 2.4 Preliminaries 7 2.4.1 Basic properties of L and Γ 8 2.4.2 Review of the Linearized Boltzmann equation 9 2.5 The hierarchy of the solution of the 1D Boltzmann equation 12 2.5.1 Leading waves 12 2.5.2 First order waves 23 2.5.3 Remainder Structure 32 2.6 Spatial asymptotic behavior 36 2.7 Fluid approximation to the Navier-Stokes equation 40 2.7.1 Macro-Micro decomposition and formal derivation 40 2.7.2 Rigorous estimates 41 2.8 Appendix 46 2.8.1 Some convolution estimate 46 2.8.2 Proof of the convolution estimate 47 3 Spatial behavior of the solution to the Boltzmann equation without angular cutoff for γ+2s ≥ 1 50 3.1 The model 50 3.2 Notation 51 3.3 Preliminaries 53 3.4 Proof of the main theorem 56 Bibliography 63

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