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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 夏俊雄(Chun-Hsiung Hsia) | |
| dc.contributor.author | Yung-Hsiang Huang | en |
| dc.contributor.author | 黃詠翔 | zh_TW |
| dc.date.accessioned | 2021-06-17T08:31:07Z | - |
| dc.date.available | 2021-02-22 | |
| dc.date.copyright | 2021-02-22 | |
| dc.date.issued | 2021 | |
| dc.date.submitted | 2021-01-25 | |
| dc.identifier.citation | [1] Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Courier Dover, New York (1964) [2] Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J. Funct. Anal. 262(3), 915–1010 (2012) [3] Alonso, R., Gamba, I.: Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section. J. Stat. Phys. 137(5-6) 1147–1165 (2009) [4] Alonso, R., Carneiro, E., Gamba, I.: Convolution inequalities for the Boltzmann collision operator. Comm. Math. Phys. 298(2) 293–322 (2010) [5] Arkeryd, L. Nouri, A.: A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory. Indiana Univ. Math. 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Phys. 153(1), 93–118 (2013) [12] Chen, I.-K.: Regularity of stationary solutions to the linearized Boltzmann equations. SIAM J. Math. Anal. 50(1) 138–161 (2018) [13] Chen, I.-K., Hsia, C.-H.: Singularity of macroscopic variables near boundary for gases with cutoff hard potential. SIAM J. Math. Anal. 47(6), 4332–4349 (2015) [14] Chen, I.-K., Hsia, C.-H., Kawagoe D.: Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain. Ann. Inst. H. Poincaré Anal. Non Linéaire 36(3) 745–782 (2019) [15] Chen, I.-K., Funagane, H., Takata S., Liu, T.-P.: Singularity of the velocity distribution function in molecular velocity space. Comm. Math. Phys. 341(1), 105–134 (2016) [16] Chen, I K, Liu, T.-P. Takata, S.: Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation. Arch. Ration. Mech. Anal. 212(2), 575–595 (2014) [17] Desvillettes, L., Villani, C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005) [18] Folland, G. Real Analysis. Modern Techniques and Their Applications. New York: John Wiley Sons, Inc. (1999) [19] Golse, F., Poupaud, F.: Stationary solutions of the linearized Boltzmann equation in a half‐space. Math. Methods Appl. Sci. 11(4), 483–502 (1989) [20] Grad H.: Asymptotic theory of the Boltzmann equation, II. In: Laurmann, J.A. (ed.) Rarefied GasDynamics, pp. 26–59. Academic Press, New York (1963) [21] Guo, Y., Strain, R.M.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008) [22] Liu, S., Yang, X.: The initial boundary value problem for the Boltzmann equation with soft potential. Arch. Ration. Mech. Anal. 223(1), 463–541 (2017) [23] Liu, T.-P., Yu, S.-H.: Invariant manifolds for steady Boltzmann flows and applications. Arch. Ration. Mech. Anal. 209(3), 869–997 (2013) [24] Ohwada, T., Sone, Y., Aoki, K.: Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 2042–2049 (1989) [25] Sone, Y.: Kinetic theory analysis of linearized Rayleigh problem. J. Phys. Soc. Japan 19, 1463–1473 (1964) [26] Sone, Y.: Effect of sudden change of wall temperature in a rarefied gas. J. Phys. Soc. Japan 20, 222–229 (1965) [27] Sone, Y.: Molecular Gas Dynamics. Theory, Techniques, and Applications. Birkhäuser, Boston (2007) [28] Sone, Y., Onishi, Y.: Kinetic theory of evaporation and condensation–hydrodynamic equation and slip boundary condition. J. Phys. Soc. Japan 44(6), 1981–1994 (1978) [29] Sone, Y., Onishi, Y.: Kinetic theory of slightly strong evaporation and condensation. Hydrodynamic equation and slip boundary condition for finite Reynolds number. J. Phys. Soc. Japan 47(5), 1676-1685 (1979) [30] Strain, R.M.: Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinet. Relat. Models 5(3), 583–613 (2012) [31] Takata, S., Funagane, H.: Poiseuille and thermal transpiration flows of a highly rarefied gas: Over-concentration in the velocity distribution function. J. Fluid Mech. 669, 242–259 (2011) [32] Takata, S., Funagane, H.: Singular behaviour of a rarified gas on a planar boundary. J. Fluid Mech. 717, 30–47 (2013) [33] Wu, K. C. Personal communication. (2020) [34] Ukai, S., Asano, K.: On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18(2), 477–519 (1982) [35] Ukai, S., Yang T., Yu, S.-H.: Nonlinear boundary layers of the Boltzmann equation. I. Existence. Comm. Math. Phys. 236(3), 373–393 (2003) [36] Ukai, S., Yang, T., Yu, S.-H.: Nonlinear stability of boundary layers of the Boltzmann equation. I. The case M∞ < -1. Comm. Math. Phys. 244(1), 99– 109 (2004) | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/74349 | - |
| dc.description.abstract | 在各種氣體模型中，氣體動力學適合用來模擬其靠近邊界的現象。在研究由邊界所造成的影響時，關於稀薄氣體在兩片平面夾層間的靜態問題是最簡化卻必要的。在本論文中，我們研究具截斷軟位勢的線性化波茲曼方程之邊界奇異性。我們對於其巨觀變量的微分建立了漸進展開式。對於硬球模式與硬位勢，類似的展開式已經發表在陳逸昆2013年的文章以及陳逸昆和夏俊雄2015年的文章中。本研究將把他們的結果推廣到軟位勢模型 -3/2 < γ < 0。但因為已知的解所坐落的函數空間對於速度變數的二次加權可積性之權重的特性，在軟位勢與硬位勢是有所不同，所以我們不能直接採用陳夏兩位的論證方式。為了要克服這個癥結點，我們針對該加權二次可積空間採用了一個與之前不同版本的平滑化估計，該估計是來自於高爾斯與珀波德1989年的文章。利用此一估計及陳夏2015年文章的想法，我們成功地建立了對於軟位勢模型之巨觀變量的邊界奇異性。 | zh_TW |
| dc.description.abstract | Among various models of gases, the kinetic theory is suitable for modelling boundary phenomena. In the studies of the effects of the boundary, problems related to the steady behavior of a rarefied gas bounded by a pair of planar walls are simplest yet indispensable. In this thesis, the boundary singularity for stationary solutions of the linearized Boltzmann equation with cutoff soft potential in a slab is studied. An asymptotic formula for the gradient of the moments is established, which reveals the logarithmic singularity near the planar boundary. Similar results for cutoff hard-sphere and hard potential were proved in [Chen, I.-K.: J. Stat. Phys. 153(1), 93--118] and [Chen, I.-K., Hsia, C.-H.: SIAM J. Math. Anal. 47(6) 4332--4349 (2015)]. We extend their results to the case of soft potential -3/2< γ <0. Since the solution space from the known existence theory is equipped with a weighted L2 integrability for the velocity variables that behaves differently from the solution space for hard potential case, we cannot apply their arguments directly. To overcome this crux, we employed a different version of smoothing property for weighted L2 space in [Golse, F., Poupaud, F.: Math. Methods Appl. Sci. 11(4), 483--502 (1989)] to carry out the idea of Chen and Hsia. We then successfully extend the boundary singularity result to the soft potential case -3/2 < γ < 0. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T08:31:07Z (GMT). No. of bitstreams: 1 U0001-2201202107200000.pdf: 1101319 bytes, checksum: b5a0756634e9b24c1d3d1f4c6ef1f4f0 (MD5) Previous issue date: 2021 | en |
| dc.description.tableofcontents | 口試委員審定書i 致謝ii 中文摘要iii Abstract iv 1 Introduction 1 2 Preliminaries: Properties of the Integral Operator K 7 2.1 Pointwise estimates of the kernels and the proofs of Lemma 1.1 and Lemma 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Proof of the pointwise estimates . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 The case γ < -1 . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 The case γ = -1 . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Asymptotic Formula Associated with the Logarithmic Singularity 20 4 Upper Bound Estimates via Lipschitz-Type Continuity of the Integral Operator 25 5 Conclusion 34 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A Derivation of the Kernel of K(f) 36 Bibliography 38 | |
| dc.language.iso | en | |
| dc.subject | 氣體動力學 | zh_TW |
| dc.subject | 波茲曼方程 | zh_TW |
| dc.subject | 截斷軟位勢 | zh_TW |
| dc.subject | 平面邊界 | zh_TW |
| dc.subject | 對數奇異性 | zh_TW |
| dc.subject | Boltzmann equation | en |
| dc.subject | Logarithmic singularity | en |
| dc.subject | Planar boundary | en |
| dc.subject | Cutoff soft potential | en |
| dc.subject | Kinetic theory | en |
| dc.title | 線性化波茲曼方程之巨觀變量的邊界奇異性 | zh_TW |
| dc.title | Boundary Singularity of Macroscopic Variables for Linearized Boltzmann Equation with Cutoff Soft Potential | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 109-1 | |
| dc.description.degree | 博士 | |
| dc.contributor.author-orcid | 0000-0002-1172-4368 | |
| dc.contributor.oralexamcommittee | 陳逸昆(I-Kun Chen),吳恭儉(Kung-Chien Wu),郭鴻文(Hung-Wen Kuo),江金城(Jin-Cheng Jiang) | |
| dc.subject.keyword | 氣體動力學,波茲曼方程,截斷軟位勢,平面邊界,對數奇異性, | zh_TW |
| dc.subject.keyword | Kinetic theory,Boltzmann equation,Cutoff soft potential,Planar boundary,Logarithmic singularity, | en |
| dc.relation.page | 42 | |
| dc.identifier.doi | 10.6342/NTU202100122 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2021-01-26 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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