本文研究了不可壓Navier-Stokes 方程和其他相關係統的解的正則性，包括适当弱解和Leray-Hopf 弱解。 / 首先，我們考慮了不可壓穩態Navier-Stokes 方程和磁流体方程的適當弱解的部分正則性，包括內部和邊界情況。與五維情況不同，最關鍵的爆破方法在這種情況下不適用，我們用了靴帶法。但是對於邊界情況，在壓力的迭代中出現了困難項。為了克服它們，我們充分利用了改進的局部能量不等式並且找到了一個新的迭代量。這在本文定理的證明中起到了關鍵的作用。此外，這給我們提供了一種處理適當弱解的邊界正則性的新方法。 / 接著，我們得到了一些不依賴於磁場的四維磁流體方程的內部正則性準則。考慮到四維是臨界情況，我們仍然採用靴帶法。為了保證解的正則性準則不依賴於磁場，我們要做更精細的估計。 / 再次，对于三維轴对称磁流体方程，我們讨论了兩种LerayHopf弱解的正則性准則。利用标准的能量方法和爆破准則，我們得到了一些充分条件。这些条件保证了解的光滑性。 / 最後，對於帶有部分粘性的三維軸對稱磁流體方程，我們得到了Leray-Hopf 弱解的整體正則性。在部分粘性缺失的情況下，相關方向失了光滑性效應。在這種情況下，我們充分利用了磁流體的方程的特殊結構去彌補這個困難。 / In this thesis, we study the regularity of solutions to the incompressible Navier- Stokes equations and other related systems, including both suitable weak solutions and Leray-Hopf weak solutions. / Firstly, we consider the partial regualrity of suitable weak solutions to the 6D steady-state Navier-Stokes and MHD equations, including both interior and boundary case. Be different from the five dimensional case, the key blow-up arguments don't hold in this case, we use the bootstrap arguments instead. However, for the boundary case, hard terms appear in the interation of pressure. To overcome them, we make full use of the revised local energy inequality and find a new iteration quantity which plays a crucial role in the proof. Moreover, this provides a new method to deal with the boundary regularity of suitable weak solutions. / Secondly, we obtain some interior regularity criterias for the four dimensional MHD equations indepedent of magnetic field. Considering that four dimension is the critical case, we still use the bootstrap arguments. To guarantee that the criterias is independent of magnetic, we should do more subtle estimates. / Thirdly, we discuss two kinds of regularity criterias of Leray-Hopf weak solutions to 3D axisymmetric MHD equations. By use of standard energy method and blow-up arguments, we derive some sufficient conditions which guarantee the smoothness of solutions. / Finally, we get the global regualrity of Leray-Hopf weak solutions of 3D axisymmetric MHD equations with partial viscosities. In the absence of partial viscosities, there is no smoothing effect on that directions. Under this circumstance, we take full advantage of the special structure of MHD equations to make up this shortcoming. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Liu, Jitao. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 132-142). / Abstracts also in Chinese. / Introduction --- p.4 / Chapter 1 --- Interior and Boundary Regularity Criterias for 6D Steady-State incompressible Navier-Stokes and MHD equations --- p.17 / Chapter 1.1 --- Introduction --- p.18 / Chapter 1.2 --- Notations --- p.18 / Chapter 1.3 --- Interior Regualrity --- p.20 / Chapter 1.3.1 --- Main results --- p.20 / Chapter 1.3.2 --- Proof of Theorem 2.1.2 --- p.23 / Chapter 1.3.3 --- Proof of Theorem 2.1.3 --- p.27 / Chapter 1.3.4 --- Proof of Theorem 2.5.4 --- p.29 / Chapter 1.4 --- Boundary Regularity --- p.36 / Chapter 1.4.1 --- Main Results --- p.36 / Chapter 1.4.2 --- Some technical lemmas --- p.39 / Chapter 1.4.3 --- Proof of Theorem 1.4.2 --- p.42 / Chapter 1.4.4 --- Proof of Proposition 1.4.8 --- p.45 / Chapter 2 --- Interior Regularity Criterias for the incompressible MHD Equations in four diemnsion --- p.54 / Chapter 2.1 --- Introduction --- p.55 / Chapter 2.2 --- Notations and some technical lemmas --- p.57 / Chapter 2.3 --- Proof of Theorem 2.1.2: Estimates of the velocity --- p.60 / Chapter 2.4 --- Proof of Theorem 2.1.3: Estimates of the gradient of the velocity --- p.65 / Chapter 2.5 --- Proof of Proposition 2.1.4: "-regularity --- p.67 / Chapter 3 --- Regularity Criterias of the 3D axisymmetric MHD Equations --- p.73 / Chapter 3.1 --- Introduction --- p.74 / Chapter 3.2 --- Preliminaries --- p.76 / Chapter 3.2.1 --- Notations --- p.76 / Chapter 3.2.2 --- Some useful estimates --- p.77 / Chapter 3.3 --- Proof of Theorem 3.1.1 --- p.82 / Chapter 3.4 --- Proof of Theorem 3.1.2 and 3.1.3 --- p.86 / Chapter 3.4.1 --- Proof of Theorem 3.1.2 --- p.86 / Chapter 3.4.2 --- Proof of Theorem 3.1.3 --- p.91 / Chapter 3.5 --- Proof of Theorem 3.1.4 --- p.94 / Chapter 4 --- Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic dicusion --- p.103 / Chapter 4.1 --- Introduction --- p.104 / Chapter 4.2 --- Notations and some technical lemmas --- p.104 / Chapter 4.2.1 --- Notations --- p.105 / Chapter 4.2.2 --- Some estimates about axisymmetric structure --- p.106 / Chapter 4.2.3 --- Some estimates about partial viscosities --- p.109 / Chapter 4.3 --- A Priori Estimates --- p.110 / Chapter 4.3.1 --- L² and H¹ Estimates --- p.110 / Chapter 4.3.2 --- H² Estimates --- p.117 / Chapter 4.4 --- Proof of Theorem 4.1.1 --- p.126 / Chapter 5 --- Discussions on the Future Research --- p.130 / Chapter 5.1 --- Incompressible Navier-Stokes equations --- p.130 / Chapter 5.2 --- Incompressible MHD equations --- p.131 / Bibliography --- p.131
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328604 |
| Date | January 2013 |
| Contributors | Liu, Jitao., 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, 142 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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