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This thesis investigates the polynomial collocation method for the numerical solution of Cauchy type integral equations and the use of those equations and the related numerical techniques to solve two practical problem in Acoustics and Aerodynamics. Chapters I and II include the basic background material required for the development of the main body of the thesis. Chapter I discusses a number of practical problems which can be modelled as a singular integral equations. In Chapter II the theory of those equations is given in great detail. In Chapter III the polynomial collocation method for singular integral equations with constant coefficients is presented. A particular set of collocation points, namely the zeros of the first kind Chebyshev polynomials, is shown to give uniform convergence of the numerical approximation for the cases of the index K = 0. 1. The convergence rate for this method is also given. All these results were obtained under slightly stronger assumptions than the minimum required for the existence of an exact solution. Chapter IV contains a generalization of the results in Chapter III to the case of variable coefficients. In Chapter V an example of a practical problem which results in a singular integral equation and which is successfully solved by the collocation method is described in substantial detail. This problem consists of the interaction of a sound wave with an elastic plate freely suspended in a fluid. It can be modelled by a system of two coupled boundary value problems - the Helmholtz equation and the beam equation. The collocation method is then compared with asymptotic results and a quadrature method due to Miller. In Chapter VI an efficient numerical method is developed for solving problems with discontinuous right-hand sides. Numerical comparison with other methods and possible extensions are also discussed.

本論文旨在討論複空間上的特殊拉格朗日子流行的各種建構方法。這些方法主要自R. Harvey, B. Lawson, D. Joyce, R. Bryanl 以及M. Haskins這幾位數學家發展及研究. 本文側重於擁有不同類型對稱位的特殊拉格朗日子流行的結構方法, 其中包指於n維環面群及特殊正交群不變的例子，以及直紋特殊拉格朗日子流行。除此之外,本論文也會討論以上建構方法所給出的具體制子。最後, 本文亦會討論一種可以建構擁有高虧格鏈的特殊拉格朗日子流行的方法。 / This thesis gives a survey on constructions of special Lagrangian submanifolds in C[superscript n]. These construction methods are mainly studied by Harvey and Lawson, D. Joyce, R. Bryant and M. Haskins. We mainly focus on special Lagrangian submanifolds with different kinds of symmetries. These include local constructions of T[superscript n] --and SO(n)--invariant examples, and ruled examples. We also discuss explicit examples arose from those constructions. Besides local constructions, a global gluing construction of special Lagrangian submanifolds with high genus links is also discussed. / Detailed summary in vernacular field only. / Lam, Yi Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 97-99). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Preliminaries --- p.17 / Chapter 2.1 --- Special Lagrangian Geometry --- p.17 / Chapter 2.2 --- Special Legendrian Links --- p.23 / Chapter 2.3 --- Harmonic Maps --- p.25 / Chapter 2.4 --- Moment Maps --- p.28 / Chapter 2.5 --- Evolution Equation --- p.30 / Chapter 2.6 --- Background Materials from Analysis --- p.33 / Chapter 3 --- T[superscript n] invariant Special Lagrangian Submanifolds --- p.36 / Chapter 3.1 --- Basic Example --- p.36 / Chapter 3.2 --- U(1)[superscript n-2] invariant Special Lagrangian Cones in C[superscript n] --- p.37 / Chapter 3.2.1 --- General Construction --- p.37 / Chapter 3.2.2 --- Reduction of O.D.E. System --- p.40 / Chapter 3.2.3 --- The 3--dimensional Case --- p.48 / Chapter 4 --- SO(n)--invariant Special Lagrangian Submanifolds --- p.52 / Chapter 4.1 --- Basic Example --- p.52 / Chapter 4.2 --- Special Lagrangian Submanifolds with Fixed Loci --- p.54 / Chapter 4.2.1 --- Reduction to P.D.E --- p.55 / Chapter 4.2.2 --- The 3--Dimensional Case --- p.63 / Chapter 5 --- Ruled Special Lagrangian Submanifolds --- p.65 / Chapter 5.1 --- Normal Bundle of a Submanifold --- p.65 / Chapter 5.2 --- Twisted Normal Bundles --- p.69 / Chapter 5.3 --- The 3Dimensional Case --- p.72 / Chapter 5.3.1 --- Construction of Special Lagrangian Ruled 3-folds --- p.71 / Chapter 5.3.2 --- Explicit Examples --- p.75 / Chapter 5.4 --- Twisted Special Lagrangian Cones --- p.78 / Chapter 6 --- Other Constructions --- p.81 / Chapter 6.1 --- Analysis on U(1)invariant Special Lagrangian submanifolds --- p.81 / Chapter 6.1.1 --- Non-singular Solutions --- p.83 / Chapter 6.1.2 --- Existence of Singular Solutions --- p.84 / Chapter 6.1.3 --- Properties of Singular Solutions --- p.86 / Chapter 6.2 --- Harmonic Maps and Minimal Immersions --- p.87 / Chapter 6.3 --- Construction of Special Lagrangian 3-folds with High Genus Links --- p.92 / Bibliography --- p.97

Zhou, Qianwen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 80-87). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

by Lin Jun. / "Nov. 28, 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

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

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

ZelenyaK在一九六八年證明了所有二階擬線性拋物方程的有界全域解都會趨向一個穩態解，而其證明中的一個重要部分就是證明所有這類方程都存在一個數土結構，這是高階方程不定會有的。在這篇論文中，我們會證明Zelenyak 定理，以及找出一個四階、六階方程存在變分結構的充分必要條件。 / Zelenyak proved in 1968 that every bounded global solution of a second order quasilinear parabolic equation converges to a stationary solution. An important part in the proof is that every such equation has a variational structure. For higher order parabolic equations, this is not the case. In this thesis, we prove Zelenyak's theorem and find a necessary and sufficient condition for a fourth or sixth order equation to be variational. / Detailed summary in vernacular field only. / Chan, Hon To Hardy. / "October 2012." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leave 66). / Abstracts also in Chinese. / Introduction --- p.1 / Chapter 1 --- Convergence of Global Solutions of Second Order Parabolic Equations --- p.5 / Chapter 1.1 --- Main result --- p.5 / Chapter 1.2 --- Four auxiliary lemmas --- p.6 / Chapter 1.3 --- Proof of main result --- p.15 / Chapter 1.4 --- An extension to fourth order equations --- p.21 / Chapter 1.4.1 --- An example --- p.25 / Chapter 2 --- The Multiplier Problem for the Fourth Order Equa-tion --- p.28 / Chapter 2.1 --- Introduction --- p.28 / Chapter 2.2 --- Main results --- p.31 / Chapter 2.2.1 --- A necessary and sufficient condition for a variational structure --- p.31 / Chapter 2.2.2 --- An algorithm to check the existence of a variational structure --- p.32 / Chapter 2.3 --- Proof of main results --- p.33 / Chapter 2.4 --- Examples --- p.48 / Chapter 3 --- The Multiplier Problem for the Sixth Order Equa-tion --- p.52 / Chapter 3.1 --- Introduction --- p.52 / Chapter 3.2 --- Main results --- p.55 / Chapter 3.2.1 --- A necessary and sufficient condition for a variational structure --- p.55 / Chapter 3.2.2 --- An algorithm to check the existence of a variational structure --- p.56 / Chapter 3.3 --- Proof of main results --- p.59 / Bibliography --- p.66