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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳逸昆 | zh_TW |
| dc.contributor.advisor | I-Kun Chen | en |
| dc.contributor.author | 蘇哲寬 | zh_TW |
| dc.contributor.author | Jhe-Kuan Su | en |
| dc.date.accessioned | 2025-07-21T16:07:27Z | - |
| dc.date.available | 2025-07-22 | - |
| dc.date.copyright | 2025-07-21 | - |
| dc.date.issued | 2025 | - |
| dc.date.submitted | 2025-07-17 | - |
| dc.identifier.citation | [1] M. Briant. Instantaneous exponential lower bound for solutions to the Boltzmann equation with maxwellian diffusion boundary conditions. Kinet. Relat. Models., 8(2):281–308, 2015.
[2] M. Briant. Instantaneous filling of the vacuum for the full boltzmann equation in convex domains. Archive for Rational Mechanics and Analysis, 218(2):958–1041, 2015. [3] R. E. Caflisch. The boltzmann equation with a soft potential. i. linear, spatially homogeneous. Comm. Math. Phys., 74(1):71–95, 1980. [4] Y. Cao. Rarefied gas dynamics with external fields under specular reflection boundary condition. Commun. Math. Sci., 20(8):2133–2206, 2022. [5] T. Carleman. Sur la theorie de l’equation integrodi erentielle de boltzmann. Acta Math., 60(1), 1933. [6] M. Cessenat. Théorèmes de trace lp pour des espaces de fonctions de la neutronique (french. english summary). C. R. Acad. Sci. Paris Sér. I Math., 299(16):831–834, 1984. [7] H. Chen and C. Kim. Gradient decay in the boltzmann theory of non-isothermal boundary. Arch. Ration. Mech. Anal., 248(2). [8] H. Chen and C. Kim. Regularity of stationary boltzmann equation in convex domains. Arch. Ration. Mech. Anal., 244(3):1099–1222, 2022. [9] I. Chen, P. Chuang, C. Hsia, D. Kawagoe, and J. Su. Geometric effects on w1,p regularity of the stationary linearized boltzmann equation. preprint, arXiv:2311.12387 (accepted by the Indiana University Mathematics Journal). [10] I. Chen, P. Chuang, C. Hsia, D. Kawagoe, and J. Su. On the existence of h1 solutions for stationary linearized boltzmann equation in a small convex domain. preprint, arXiv:2304.08800. [11] I. Chen, C. Hsia, and D. Kawagoe. Regularity for diffuse reflection boundary problem to the stationary linearized boltzmann equation in a convex domain. Ann. Inst. H. Poincaré C Anal., Non Linéaire 36(3):745–782, 2019. [12] I.-K. Chen, P.-H. Chuang, C.-H. Hsia, and J.-K. Su. A revisit of the velocity averaging lemma: on the regularity of stationary boltzmann equation in a bounded convex domain. J. Stat. Phys., 189(17):43, 2022.145 [13] L. Desvillettes and C. Villani. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear fokker-planck equation. Comm. Pure Appl., 54(1):1–42, 2001. [14] L. Desvillettes and C. Villani. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the boltzmann equation. Invent. Math., 159(2):245-316, 2005. [15] R. Esposito, Y. Guo, C. Kim, and R. Marra. Non-isothermal boundary in the boltzmann theory and fourier law. Comm. Math. Phys., 323(1):177–239, 2013. [16] H. Grad. Asymptotic theory of the boltzmann equation. ii. 1963 rarefied gas dynamics (proc. 3rd internat. sympos., palais de l’unesco, paris, 1962). pages 26–59. [17] M. P. Gualdani, S. Mischler, and C. Mouhot. Factorization for non-symmetric operators and exponential H-theorem. 1 edition, 2018. [18] J. P. Guiraud. Problème aux limites intérieur pour l’équation de boltzmann linéaire. J. Mécanique, 9(1):443–490, 1970. [19] J. P. Guiraud. Problème aux limites intérieures pour l’équation de boltzmann. (french. english summary) actes du congrès international des mathématiciens (nice, 1970), tome 3. pages 115–122, 1971. [20] J.-P. Guiraud. Problème aux limites intérieur pour l’équation de boltzmann en régime stationnaire, faiblement non linéaire. J. Mécanique, 11(1):183–231, 1972. [21] Y. Guo, C. Kim, D. Tonon, and A. Trescases. Regularity of the boltzmann equation in convex domains. Invent. Math., 207(1):115–290, 2017. [22] C. Henderson, S. Snelson, and A. Tarfulea. Self-generating lower bounds and continuation for the boltzmann equation. Calc. Var. Partial Differential Equations, 52(6):13, 2020. [23] C. Imbert, C. Mouhot, and L. Silvestre. Gaussian lower bounds for the Boltzmann equation without cutoff. SIAM J. Math. Anal, 52(1):2930–2944, 2020. [24] D. Kawagoe. Regularity of solutions to the stationary transport equation with the incoming boundary data. PhD thesis, Kyoto University, 2018. [25] C. Mouhot. Quantitative lower bounds for the full boltzmann equation. i. periodic boundary conditions. Comm. Partial Differential Equations, 30(5):881–917, 2005. [26] A. Palczewski. Stationary boltzmann’s equation with maxwell’s boundary conditions in a bounded domain. (english summary). Math. Methods Appl. Sci., 15(6):375–393, 1992. [27] A. Pulvirenti and B. Wennberg. A maxwellian lower bound for solutions to the boltzmann equation. Comm. Math. Phys., 183(1):145–160, 1997. [28] J. Su. Quantitative lower bound for solutions to the boltzmann equation in nonconvex domains. preprint, arXiv:2505.03396. [29] C. Yunbai. Regularity of boltzmann equation with external fields in convex domains of diffuse reflection. SIAM J. Math. Anal., 51(4):3195–3275, 2019. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/97882 | - |
| dc.description.abstract | 在本文中，我們將探討在有界域上各種邊界條件的線性與非線性波茲曼方程式的解。在第二章中，我們探討在硬勢假設下，在具有高斯正曲率的足夠小有界域上，在入射邊界條件下，我們證明穩態 H^1 解的存在性。在第三章中，我們進一步探討在特殊情況下，在硬球假設下，在入射邊界非線性下 W^{1,p} 穩態解的存在性。在第四章中我們研究在硬球假設下，鏡面和擴散反射邊界條件下的行為。我們證明無凸域假設下在任意短時間下高斯分布下界的存在性，並提供在非截斷假設下較弱的下界。 | zh_TW |
| dc.description.abstract | We investigate solutions of the Boltzmann equations, linearized and nonlinear ones, with various boundary conditions on bounded domains. In Chapter 2, we consider the incoming boundary problem for the linearized Boltzmann equation with the hard potential cutoff assumption. With the assumption of the C2 bounded domain with a positive Gaussian curvature boundary, we provide the H1 existence of stationary solutions provided that the diameter of the domain Ω is small. In Chapter 3, we consider the incoming boundary problem for the Boltzmann equation with the hard potential cutoff assumption. With some additional assumptions to the space domain, we establish a W 1,p stationary solution provided that the domain and the boundary data are small enough. In Chapter 4, we investigate the time evolutionary behavior of mild solutions on a bounded connected domain without convex assumption. For specular or diffusive reflection boundary condition, we prove that a Maxwellian lower bound in cut-off case and a weaker lower bound for non-cutoff case can generate instantly. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2025-07-21T16:07:27Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2025-07-21T16:07:27Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | Acknowledgements i
摘要 iii Abstract v Contents vii List of Figures ix Chapter 1 Introduction 1 1.1 Introduction 1 Chapter 2 The H1 Existence and regularity over stationary solutions on a small domain 3 2.1 Preliminary and basic estimate 7 2.2 Existence and regularity result 9 2.3 Sufficient condition on boundary data 21 Chapter 3 The W 1,p Existence and regularity over stationary solutions on small domain 27 3.1 Prelimilary estimate 32 3.1.1 Estimates for the linear integral kernel 32 3.1.2 Estimates for the nonlinear cross section 34 3.1.3 Some geometrical estimates on bounded convex domains 35 3.2 The W 1,p existence and regularity for solutions to linearized equation 41 3.3 The W 1,p existence and regularity for solutions to non-linear equations on a small domain 46 Chapter 4 Lower bound on time evolutionary solution on non-convex domain 59 4.1 Geometry property near the boundary 63 4.2 Operater estimate and spreading property 87 4.3 Maxwellian bound for the cut-off case for non-fully specular reflection condition 113 4.4 Maxwellian bound in the cut-off case for fully specular reflection condition 125 4.5 Lower bound for the non-cutoff case 136 References 145 Appendix — Characteristic line 147 | - |
| dc.language.iso | en | - |
| dc.subject | 波茲曼方程 | zh_TW |
| dc.subject | 偏微分方程 | zh_TW |
| dc.subject | 波茲曼方程 | zh_TW |
| dc.subject | 偏微分方程 | zh_TW |
| dc.subject | Boltzmann equation | en |
| dc.subject | PDE | en |
| dc.subject | Boltzmann equation | en |
| dc.subject | PDE | en |
| dc.title | 波茲曼方程式的解的存在性與性質 | zh_TW |
| dc.title | On the solutions to Boltzmann equations on bounded domains | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 113-2 | - |
| dc.description.degree | 博士 | - |
| dc.contributor.oralexamcommittee | 夏俊雄;Jerry Lloyd Bona;劉太平;Kazuo Aoki;江金城;陳俊全;吳恭儉 | zh_TW |
| dc.contributor.oralexamcommittee | Chun-Hsiung Hsia;Jerry Lloyd Bona;Tai-Ping Liu;Kazuo Aoki;Jin-Cheng Jiang;Chiun-Chuan Chen;Kung-Chien Wu | en |
| dc.subject.keyword | 偏微分方程,波茲曼方程, | zh_TW |
| dc.subject.keyword | PDE,Boltzmann equation, | en |
| dc.relation.page | 148 | - |
| dc.identifier.doi | 10.6342/NTU202501891 | - |
| dc.rights.note | 同意授權(全球公開) | - |
| dc.date.accepted | 2025-07-18 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 數學系 | - |
| dc.date.embargo-lift | 2025-07-22 | - |
| 顯示於系所單位： | 數學系 | |
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