Chu Shun Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 63-65). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Preliminaries --- p.8 / Chapter 2 --- Singular Integrals --- p.15 / Chapter 2.1 --- Marcinkiewicz Integral --- p.15 / Chapter 2.2 --- Decomposition in cubes of open sets in Rn --- p.17 / Chapter 2.3 --- Interpolation Theorem for Lp --- p.18 / Chapter 2.4 --- Singular Integrals on homogeneous of degree 0 --- p.25 / Chapter 3 --- Solutions to the Euler Equations --- p.36 / Chapter 3.1 --- Existence and Uniqueness of smooth solutions for Euler Equations --- p.36 / Chapter 3.2 --- Rate of Convergence and Decay in Time --- p.43 / Chapter 3.2.1 --- Rate of Convergence --- p.43 / Chapter 3.2.2 --- Lp Decay for Solutions of the Navier-Stokes Equations --- p.46 / Chapter 3.3 --- Weak Solution to the Euler Equations --- p.48 / Chapter 3.3.1 --- Weak Solution to the Velocity Formulation --- p.49 / Chapter 3.3.2 --- Weak Solution to the Vorticity Formulation --- p.52 / Bibliography --- p.63

by Ho Wing Kin. / Thesis submitted in: July 1997. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-92). / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- The Euler-Lagrange Equation for Capillary Surfaces --- p.6 / Chapter 1.2 --- The Capillary Tube --- p.12 / Chapter 2 --- Comparison Principles --- p.16 / Chapter 2.1 --- The General Comparison Principle --- p.16 / Chapter 2.2 --- Applications --- p.24 / Chapter 3 --- Existence Criteria --- p.30 / Chapter 3.1 --- Necessary Conditions --- p.31 / Chapter 3.2 --- Sufficient Conditions --- p.35 / Chapter 4 --- Uniqueness --- p.53 / Chapter 4.1 --- General Bounded Domains Case --- p.53 / Chapter 4.2 --- Infinite Strip Case --- p.56 / Chapter 5 --- Gradient Estimate of Surfaces of Constant Mean Curvature --- p.66 / Chapter 5.1 --- The Gradient Estimate --- p.67 / Chapter 5.2 --- Behavior as R→ 1 --- p.70 / Chapter 5.3 --- Existence of the Comparison Surfaces --- p.76 / Chapter 5.4 --- Ro is Best Possible --- p.82 / Appendix Mean Curvature --- p.85

本論文討論了具有非平凡旋度的穩態亞音速流體的適定性問題。 / 首先，我們研究了通過無限長週期管道的二維亞音速流禮。當管道某一週期位置伯努利函數擾動很小，且質量數介於與適當的範圍時，有且僅有唯一的亞音速流禮。特別地，對於伯努利函數為常值的情形，我們還通過結構緊性的方法證明了亞音速-音速流體的存在性。此時，質量數可以達到一臨界值。謝春景和辛周平在處理二維司壓歐拉方程時曾引入了一個重要的處理方法一一流函數表達式。然而，對於週期流體的問題，伯努利函數和流函數的相互關係是無法事先確定的。為此，我們建立了一個關於流函數的非線性映射。該映射的不動點給出了相應歐拉方程的解。 / 其次，對於二維亞音速流體通過對稱障礙物的問題，當來流的伯努利函數關於y方向對稱，且擾動很小時，我們給出了流体的存在性和唯一性的証明。这里，我們利用歐拉方程的流函數方法，得到了對應于流函數的二階方程的解。能量方法以及動量場與來流動量場之差的L2可積性給出了流函數的漸進行為。這一漸進行為結合障礙物外無駐點的事實說明了流函數表示與原先歐拉方程是相容的。 / 最后，我們研究了當給定管道壁上法向动量時，三維穩態流體通過方體管道的問題。如果入口處伯努利函數的擾動和旋度的法向分量為零，則當邊界的法向動量不超過一臨界值時，無旋的亞音速流體存在。對於一般情形，若伯努利函數的擾動和旋度的法向分量很小時，我們利用將速度均分解均無旋部分和旋度部分的方法給出了流體存在性的證明。這裡，我們通過求解一加權的散旋系統得到了旋度部份的解:而無旋部份則由一擬線性橢圓方程的解給出。 / In this thesis, the wellposedness theory of steady subsonic flows with nontrivial vorticities is studied in various aspects. / First, we study 2-D subsonic flows through infinitely long periodic nozzles. It is showed that when mass flux lies in a suitable regime and the variation of Bernoulli's function at some given section is sufficiently small, there exists a unique global subsonic flow in the periodic nozzle. In particular, if Bernoulli's function is a constant, the existence of subsonic flow is also obtained when mass flux takes the critical number by a compensated compactness framework. One of the main tools to handle 2-D compressible Euler equations is the stream function formulation first established by Xie and Xin. The main difficulty in adapting this formulation in periodic nozzles is that the relation between Bernoulli's function and stream function cannot be fixed. We resolve this difficulty via setting up a nonlinear map from stream function at the given section to itself. The fixed point of this map induces a solution of corresponding Euler equations. / Second, the existence and uniqueness of 2-D subsonic flows past a symmetric body are established under the assumption that Bernoulli's function is given symmetrically in the upstream with small variation. By the stream function formulation for 2-D compressible Euler equations, one obtains the solution of the Euler equations via solving a quasilinear second order equation for stream function. This is achieved with the help of the theory of elliptic equations of two variables. Asymptotic behavior for the stream function is obtained via energy method and L²-integral of the difference between the momentum and its asymptotic behavior in the upstream. The asymptotic behavior, together with the property that stagnation points are absent outside the body, yields that the stream function formulation is consistent with the original Euler system. / Finally, we study the existence of 3-D steady subsonic flows in rectangular nozzles when prescribing the normal component of the momentum on the boundary. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the exit vanish, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. In general, if the normal component of the vorticity and the variation of Bernoulli's function are both sufficiently small, we prove the existence of Euler flows by decomposing the velocity into the vortical part and the potential part. A div-curl system with given weighted function is used to obtain the vortical part and the potential part is induced by the solution to a quasilinear elliptic equation. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chen, Chao. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 111-120). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.12 / Chapter 3 --- 2-D subsonic flows through in finitely long periodic nozzles --- p.23 / Chapter 3.1 --- Introduction and main result --- p.23 / Chapter 3.2 --- Stream function formulation of potential flows --- p.27 / Chapter 3.2.1 --- Bernoulli's law and stream function formulation --- p.27 / Chapter 3.2.2 --- Potential flows and proof of Theorem 3.1.1 --- p.30 / Chapter 3.3 --- Analysis of the well-posedness of Euler flows --- p.32 / Chapter 3.3.1 --- Existence, uniqueness, and periodicity of truncated flows --- p.34 / Chapter 3.3.2 --- Existence and uniqueness of Euler flflows --- p.41 / Chapter 4 --- 2-D subsonic flows past a symmetric body --- p.47 / Chapter 4.1 --- Motivation and mathematical formulation --- p.47 / Chapter 4.2 --- Truncated problem --- p.53 / Chapter 4.3 --- Asymptotic behavior at upstream and downstream --- p.59 / Chapter 4.4 --- Existence and uniqueness of Euler flflows --- p.61 / Chapter 5 --- 3-D subsonic Euler flows through nitely long nozzles --- p.67 / Chapter 5.1 --- Mathematical formulation and main results --- p.67 / Chapter 5.2 --- Some preliminaries --- p.71 / Chapter 5.3 --- 3-D potential flows --- p.76 / Chapter 5.3.1 --- Apriori estimates for truncated potential flows --- p.77 / Chapter 5.3.2 --- Existence and uniqueness of potential flows --- p.91 / Chapter 5.4 --- General 3-D steady Euler systems --- p.94 / Chapter 6 --- Further discussions and future work --- p.109 / Bibliography --- p.111

This thesis concerns two topics: the kinematics of Pacific cross-equatorial flow ??? the location, timing and magnitude of the flow; and their dynamics???what are the driving forces controlling the flow? Despite extensive observations in the central and eastern Pacific, observations of these flows remain contradictory. We use output from an Ocean General Circulation Model (OGCM) viewed from a Lagrangian framework on density layers. This addresses the problem of high variability due to features such as Tropical Instability Waves. The annual mean flow is found to be southward nearly everywhere, east of 140??W. Flow becomes stronger in the second half of the year due to a bolus transport of very light surface water, introduced by Tropical Instability Waves. A Tropical Cell pattern occurs along the equator that does not require diapycnal downwelling. From 160??E to 160??W the annual mean flow is northward, occurring mostly in the mixed layer, appearing to originate partly from the Equatorial Undercurrent surfacing in the east. The northward flow is strongest in March and becomes southward in September. The wind stress and nonlinear terms are shown to be the key driving features, with a prescribed biharmonic Smagorinsky horizontal friction scheme having negligible impact. From 160??E to 160??W, the flow is partly accounted for by an Ekman forcing, with the curl of the nonlinear term providing a crucial additional torque, more than doubling the magnitude in some instances. From 160??W to 120??W the wind stress curl provides a weak southward flow of about 1 Sv, which increases by the nonlinear addition to around 5 Sv. The curl of the steady component of the nonlinear term, derived from annual mean currents, is similar in structure to the total nonlinear term, but higher in magnitude. The structure of the variable term, which was mostly of opposite sign to the steady term, suggests damping occurs in place of friction. While our study is limited to an examination of the model's characteristics, our results provide important clues to the observed flow patterns not resolved by present-day measurements. This study also highlights the importance of time-space variability and both horizontal and vertical density structure in controlling the flow and its feedback on the system.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (M.S.)--University of Hawaii at Manoa, 2004. / Includes bibliographical references (leaves 48-51).

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 212-219). Available also in a digital version from Dissertation Abstracts.

The Lagrangian formalism utilized by Ilinskii, Hamilton and Zabolotskaya [J. Acoust. Soc. Am. 121, 786-795 (2007)] to derive equations for the radial and translational motion of interacting bubbles is extended here to obtain a model for the dynamics of interacting bubbles and elastic particles. The bubbles and particles are assumed to be spherical but are otherwise free to pulsate and translate. The model is accurate to fifth order in terms of a nondimensional expansion parameter R/d, where R is a characteristic radius and d is a characteristic distance between neighboring bubbles or particles. The bubbles and particles may be of nonuniform size, the particles elastic or rigid, and external acoustic sources are included to an order consistent with the accuracy of the model. Although the liquid is assumed initially to be incompressible, corrections accounting for finite liquid compressibility are developed to first order in the acoustic Mach number for a cluster of bubbles and particles, and to second order in the acoustic Mach number for a single bubble. For a bubble-particle pair consideration is also given to truncation of the model at fifth order in R/d via automated derivation of the model equations to arbitrary order. Numerical simulation results are presented to demonstrate the effects of key parameters such as particle density and size, liquid compressibility, particle elasticity and model order on the dynamics of single bubbles, pairs of bubbles, bubble-particle pairs and clusters of bubbles and particles under both free response conditions and sinusoidal or shock wave excitation. / text

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This thesis is mainly concerned with problems in the areas of the Calculus of Variations and Partial Differential Equations (PDEs). The properties of the functional to minimize with respect to the given topology play an important role in the existence of minimizers of integral problems. We will introduce the important concepts of quasiconvexity and polyconvexity. Inspired by finite element methods from Numerical Analysis, we introduce a perturbed problem which has some surprising uniqueness properties.