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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/93811完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 阮文先 | zh_TW |
| dc.contributor.advisor | Van Tien Nguyen | en |
| dc.contributor.author | 吳家豪 | zh_TW |
| dc.contributor.author | Jia-Hao Wu | en |
| dc.date.accessioned | 2024-08-08T16:20:40Z | - |
| dc.date.available | 2024-08-09 | - |
| dc.date.copyright | 2024-08-08 | - |
| dc.date.issued | 2024 | - |
| dc.date.submitted | 2024-07-31 | - |
| dc.identifier.citation | [1] G. Acosta, R. G. Durán, and J. D. Rossi. An adaptive time step procedure for a parabolic problem with blow-up. Computing, 68(4):343–373, 2002.
[2] M. Berger and R. V. Kohn. A rescaling algorithm for the numerical calculation of blowing-up solutions. Comm. Pure Appl. Math., 41(6):841–863, 1988. [3] C. J. Budd, W. Huang, and R. D. Russell. Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput., 17(2):305–327, 1996. [4] A. Cangiani, E. H. Georgoulis, I. Kyza, and S. Metcalfe. Adaptivity and blow-up detection for nonlinear evolution problems. SIAM J. Sci. Comput., 38(6):A3833–A3856, 2016. [5] I. Dimov, I. Faragó, and L. Vulkov, editors. Numerical analysis and its applications, volume 10187 of Lecture Notes in Computer Science. Springer, Cham, 2017. Revised selected papers of the 6th International Conference (NAA 2016) held in Lozenetz, June 15–22, 2016. [6] C. L. Fefferman. Existence and smoothness of the Navier-Stokes equation. In The millennium prize problems, pages 57–67. Clay Math. Inst., Cambridge, MA, 2006. [7] S. Filippas and W. X. Liu. On the blowup of multidimensional semilinear heat equations. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 10(3):313–344, 1993. [8] Y. Giga and R. V. Kohn. Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 38(3):297–319, 1985. [9] P. Groisman. Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions. Computing, 76(3-4):325–352, 2006. [10] M. A. Herrero and J. J. L. Velázquez. Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 10(2):131–189, 1993. [11] T. Y. Hou. Potentially singular behavior of the 3D Navier-Stokes equations. Found.Comput. Math., 23(6):2251–2299, 2023. [12] T. Y. Hou and C. Li. Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl. Comm. Pure Appl. Math., 61(5):661–697, 2008. [13] J. C. Robinson, J. L. Rodrigo, and W. Sadowski. The three-dimensional Navier-Stokes equations, volume 157 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. Classical theory. [14] J. J. L. Velázquez. Higher-dimensional blow up for semilinear parabolic equations. Comm. Partial Differential Equations, 17(9-10):1567–1596, 1992. [15] J. J. L. Velázquez, V. A. Galaktionov, and M. A. Herrero. The space structure near a blow-up point for semilinear heat equations: a formal approach. Zh. Vychisl. Mat. i Mat. Fiz., 31(3):399–411, 1991. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/93811 | - |
| dc.description.abstract | 不可壓縮 Navier-Stokes 方程是否能夠從平滑的初始數據展現出有限時間奇異行為，是非線性偏微分方程中的一個具有挑戰性的問題之一。本文提出了一些數值證據，針對 Thomas Y. Hou 使用的平滑初始數據的軸對稱 Navier-Stokes方程進行數值模擬，得到其可能在原點發展出潛在的奇異行為。我們的方法遵循了Berger 和 Kohn 的結果，這套方法是基於方程的尺度不變性。與 Thomas Y. Hou 的情況不同，我們固定黏性項進行數值模擬，並且最後呈現不同的結果。由於這種基於方程自身性質的方法，它可以在迭代過程中保持結構。 | zh_TW |
| dc.description.abstract | Whether the 3D incompressible Navier-Stokes equation can exhibit finite-time singularity from smooth initial data is one of the challenging problems in nonlinear PDEs. In this paper, we present numerical evidence that the axisymmetric Navier-Stokes equations, with Thomas Y. Hou’s smooth initial data, can develop potential singular behavior at the origin. Our method follows the approach of Berger and Kohn's study, which is based on the scaling invariance of the equation. Different from Thomas Y. Hou’s situation, we fix the viscosity and then obtain some different results. Due to the method’s reliance on the self-similarity properties of the equation, it can preserve the structure throughout the iteration process. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2024-08-08T16:20:40Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2024-08-08T16:20:40Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | Verification Letter from the Oral Examination Committee i
Acknowledgements iii 摘要v Abstract vii Contents ix Chapter 1 Introduction 1 Chapter 2 Preliminaries 7 Chapter 3 A Refinement Method 11 3.1 Refinement Process in One Dimension . . . . . . . . . . . . . . . . 12 3.2 Extend the coarser region . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Application 1: the Energy-Subcritical Semilinear Heat Equation . . . 23 3.3.1 Convergence of rescaled solutions . . . . . . . . . . . . . . . . . . 24 3.3.2 One Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.3 Two Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Application 2: A random path . . . . . . . . . . . . . . . . . . . . . 41 Chapter 4 Numerical Results of 3D Navier-Stokes equations 45 Chapter 5 Conclusion 53 References 55 | - |
| dc.language.iso | en | - |
| dc.subject | 尺度不變性 | zh_TW |
| dc.subject | 網格細化方法 | zh_TW |
| dc.subject | 潛在爆破 | zh_TW |
| dc.subject | 軸對稱Navier-Stokes方程 | zh_TW |
| dc.subject | axisymmetry Navier-Stokes equations | en |
| dc.subject | potentially blow-up | en |
| dc.subject | scaling invariance | en |
| dc.subject | refinemenft method | en |
| dc.title | 一種用於三維不可壓縮 Navier-Stokes 方程潛在爆破解的網格細化方法 | zh_TW |
| dc.title | A Refinement Method for Potential Blowup Solutions to the 3D incompressible Navier-Stokes equations | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 112-2 | - |
| dc.description.degree | 碩士 | - |
| dc.contributor.oralexamcommittee | 陳俊全;沈俊嚴 | zh_TW |
| dc.contributor.oralexamcommittee | Chiun Chuan Chen;Chun Yen Shen | en |
| dc.subject.keyword | 網格細化方法,尺度不變性,軸對稱Navier-Stokes方程,潛在爆破, | zh_TW |
| dc.subject.keyword | refinemenft method,scaling invariance,axisymmetry Navier-Stokes equations,potentially blow-up, | en |
| dc.relation.page | 56 | - |
| dc.identifier.doi | 10.6342/NTU202402879 | - |
| dc.rights.note | 同意授權(全球公開) | - |
| dc.date.accepted | 2024-08-02 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 應用數學科學研究所 | - |
| 顯示於系所單位： | 應用數學科學研究所 | |
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