這篇論文將對三維不可壓納維 - 斯托克斯方程的軸對稱解的正則性理論作綜述。 / 在第一章會先重溫一些經典的正則性結果，例如用於 Leray-Hopf 弱解上的Serrin類條件和用於適當弱解上的Caffarelli-Kohn-Nirenberg 類理論。 / 第二章會處理假設為無漩的軸對稱解，由 Uchovskii 和Ladyzhenskaya 提出的整體正則解的唯一存在的結果將包含在其中。 / 從第三章起，解的漩將不再被假設為零，在這熱門的研究領域中，一些值得注意的結果會包含在論文之中。 / In this thesis, a survey on the regularity theory for axisymmetric solutions to the 3D incompressible Navier-Stokes equations is conducted. / In chapter 1, some classical results such as the Serrin-type criteria on the Leray-Hopf weak solutions and the Ca arelli-Kohn-Nirenberg partial regularity theory on the suitable weak solutions to the equations are reviewed. / Chapter 2 deals with the axisymmetric solutions of the equations assuming the unknown velocity has no swirl. The existence result on the unique regular global-in-time solution by Uchovskii and Ladyzhenskaya is included. / In chapter 3, the swirl component of the unknown velocity is no longer assumed to be zero and some remarkable results on this hot research area are presented and discussed in the thesis. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lau, Tsz Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 49-51). / Abstracts also in Chinese. / Chapter 1 --- Classical Results on the 3D Incompressible Navier-Stokes Equations --- p.3 / Chapter 1.1 --- Introduction --- p.3 / Chapter 1.1.1 --- Backgrounds of the Equations --- p.3 / Chapter 1.1.2 --- A Boundary Value Problem on Pressure --- p.4 / Chapter 1.1.3 --- The Helmholtz-Leray Decomposition --- p.5 / Chapter 1.1.4 --- The Energy Equality --- p.6 / Chapter 1.1.5 --- The Leray-Hopf Weak Solution --- p.8 / Chapter 1.1.6 --- Foreword to the Regularity Problem --- p.9 / Chapter 1.2 --- Serrin's Regularity Result --- p.10 / Chapter 1.2.1 --- The Vorticity Form --- p.11 / Chapter 1.2.2 --- Regularity of the Vorticity --- p.12 / Chapter 1.2.3 --- The Biot-Savart Law and Parabolic Regularity --- p.13 / Chapter 1.2.4 --- Later Developments --- p.14 / Chapter 1.3 --- CKN Partial Regularity Theory --- p.14 / Chapter 1.3.1 --- Backgrounds and the Main Result --- p.14 / Chapter 1.3.2 --- A Local Conditions for Regularity of u --- p.16 / Chapter 1.3.3 --- The Blow-up Estimate --- p.19 / Chapter 1.3.4 --- Estimating the Singular Set --- p.22 / Chapter 1.3.5 --- Later Developments --- p.23 / Chapter 2 --- On Axially Symmetric Flows Without Swirl --- p.25 / Chapter 2.1 --- Introduction and the Main Result --- p.25 / Chapter 2.2 --- A Local-in-time Existence Result --- p.26 / Chapter 2.3 --- A Priori Estimate --- p.27 / Chapter 2.4 --- Proving the Global Existence Result --- p.31 / Chapter 3 --- On Axially Symmetric Flows with Non-zero Swirl --- p.32 / Chapter 3.1 --- Serrin's Type Regularity Conditions --- p.32 / Chapter 3.1.1 --- Backgrounds and the Main Result --- p.32 / Chapter 3.1.2 --- A Brief Discussion on the Proof --- p.33 / Chapter 3.1.3 --- Later Developments --- p.36 / Chapter 3.2 --- Lower Bound on the Blow-up Rate --- p.37 / Chapter 3.2.1 --- Backgrounds and the Main Result --- p.37 / Chapter 3.2.2 --- Construction of Suitable Weak Solutions --- p.38 / Chapter 3.2.3 --- Idea of the Proof --- p.40 / Chapter 3.2.4 --- Later Developments --- p.42 / Chapter 3.3 --- An Alternative Proof on Slow Blow-up --- p.43 / Chapter 3.3.1 --- Backgrounds and the Main Result --- p.43 / Chapter 3.3.2 --- Liouville Type Theorems --- p.44 / Chapter 3.3.3 --- The Re-scaling Procedure --- p.46 / Chapter 3.3.4 --- Later Developments --- p.47
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328744 |
| Date | January 2013 |
| Contributors | Lau, Tsz Ho., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource ([3], 51 leaves) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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