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In this thesis, we consider the non-uniqueness of weak solutions to the compressible isentropic Euler equations on a bounded domain. We establish a criterion for the existence of infinitely many solutions with the same initial data. Using this criterion, we prove that for general smooth initial density and velocity, there exists infinitely many weak solutions. Furthermore, given smooth initial density bounded away from zero, we can construct a initial velocity field such taht there exists infinitely many weak solutions with this initial data, and these solutions satisfy the entropy inequality for a positive finite time. The results also generalizes to the free boundary value problems. / 本論文考慮可壓等熵歐拉方程在有界區域上其弱解的非唯一性。我們建立了關於存在無窮多個有同一初始狀態的弱解的判別準則。利用此準則，我們證明了對於一般的光滑的初始密度和速度場，存在無窮多個弱解:進一步的，對於光滑有正下界的初始密度場，我們構造了一小初始速度場，證明存在滿足這樣初始資料的無窮多個弱解，並且，在一個有限時間內，這些弱解滿足熵不等式。這些結果可以推廣到自由邊界情形。 / Detailed summary in vernacular field only. / Luo, Tianwen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 55-58). / Abstracts also in Chinese. / Introduction --- p.6 / Chapter 1 --- Preliminaries --- p.18 / Chapter 1.1 --- Notational Conventions and Function Spaces --- p.18 / Chapter 1.2 --- Elementary Inequalities --- p.19 / Chapter 1.3 --- Fundamental Lemmas --- p.21 / Chapter 2 --- Formulation and Main Results --- p.22 / Chapter 2.1 --- Mathematical Formulation --- p.23 / Chapter 2.2 --- Main Theorems and the Criterion --- p.24 / Chapter 2.3 --- The Space of Subsolutions --- p.27 / Chapter 3 --- Proof of the Main Theorems --- p.30 / Chapter 3.1 --- Proof of the Subsolution Lemma --- p.31 / Chapter 3.2 --- The Perturbation Property --- p.34 / Chapter 3.3 --- Construction of Suitable Initial Data --- p.39 / Chapter 3.4 --- Convex Integration --- p.43 / Chapter 4 --- Free Boundary Problem --- p.47 / Chapter 4.1 --- Formulation and Results --- p.48 / Chapter 4.2 --- Remarks on the Proofs --- p.51 / Chapter 5 --- Discussions and Future Work --- p.54 / Bibliography --- p.55

Li Yeping. / "December 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 138-147) / 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.

本論文致力於棒狀液晶體動力學系統的幾個方面的研究，其中包括:空間週期問題及初邊值問題弱解的整體存在性，初值問題及初邊值問題強解的局部存在唯一性及相應的爆破準則。 / 首先，借助於一類新的逼近系統，我們證明了非等溫液晶體系統的二維空間週期問題存在整體弱解。與經典的Ginzburg Landau逼近系統不同的是，我們所採用的系統不是奇異型逼近。由於無法從基本能量等式中獲得方向場的二階導數的估計，我們採用局部能量不等式，於局部時間內，重新獲得了這些估計。與等溫系統不同，我們所獲得的弱解保持整體能量不變。由此可以看出，系統所損失的內能和動能全部轉化為熱能。此外，我們所獲得的弱解至多含有限個奇異時間點。而且，在每個奇異時間點上，由於能量集中現象的發生，系統的溫度必然會在某些區域內突然升高。 / 其次，通過Ginzburg- Landau逼近，我們證明了三維Ericksen-Leslie系統的初值問題存在唯一的局部強解，並建立了相應的爆破準則。我們共建立了如下四種爆破準則: (i) Serrin型準則; (ii) Beale-Kato-Majda (縮寫為BKM) 型準則; (iii) 混合型準則，即對速度場和方向場之一提Serrin型條件，而對另一場提BMK型條件;(iv) 一個新型準則，即用Ginzburg-Landau逼近系統的強解的Serrin型範數來刻畫Ericksen- Leslie 系統的強解的最大存在區間。其中，借助於一新的對數型Sobolev不等式，我們對速度場所建立的BKM型條件是經典條件的一個BMO型弱化。此外，我們還證明了，于強解的存在區間內， Ginzburg-Landau系統強收斂至Ericksen- Leslie 系統。 / 然後，我們將我們的關於強解的局部存在唯一性的結果推廣至有界域的情形。我們採用的依舊是Ginzburg-Landau逼近。為克服由於缺乏方向場的法向三階導教估計而帶來的困難，我們採用了一新的Sobolev型嵌入不等式。該嵌入不等式將Sobolev函數嵌入至一適當的混合範教LP 空間。此外，我們亦證明了局部強解的相應爆破準則 。 / 最後，通過利用前述提及的技街，我們證明了二維有界域上Ericksen-Leslie 系統的初邊值問題存在唯一的局部強解並存在一整體弱解。特別地，通過利用局部能量不等式，我們證明了強解的存在區間以及局部時間內的估計只依賴於初值的基本能量及其L²積分連續性。借助於此，通過對強解取極限，我們獲得了Ericksen-Leslie 系統的弱解。 / This thesis is devoted to some studies on the hydrodynamical model of nematic liquid crystals, including: the global existence of weak solutions to the spacial periodic and the initial-boundary value problems, the local well-posedness and blow-up criteria of strong solutions to the Cauchy and the initial-boundary value problems. / First, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system in T², based on a new approximate system. Different from the classic Ginzburg-Landau approximation, this new system is not a singular type one. Local energy inequalities are employed to recover the estimates on the second order spacial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. Different from the isothermal case, the weak solutions we obtained conserve the total energy, and thus the kinetic and potential energies transfer to nothing but the heat energy. Also our weak solutions have at most finite many singular times at which the energy concentration occurs, and as a result, the temperature must increase suddenly at each singular time on some part of T². / Next, we prove the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in R³, based on the Ginzburg-Landau approximation. Four kinds of blow-up criteria are established, including: (i) the Serrin type; (ii) the Beale-Kato-Majda (BKM for short) type; (iii) a mixed type, i.e., a Serrin type condition for one field and a BKM type condition on the other one; (iv) a new one, which characterizes the maximal existence time of a strong solution to the Ericksen-Leslie system in terms of the Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Besides, thanks to a new logarithmic Sobolev type inequality, our BKM type condition for the velocity is a BMO type improvement version. We also show that the strong solutions of the Ginzburg-Landau approximate system converge to a strong solution of the Ericksen-Leslie system up to the maximal existence time of this solution. / Then, we generalize our results on the local well-posedness of strong solutions to the Ericksen-Leslie system for the whole space to bounded domains of R³, still by the Ginzburg-Landau approximation method. A new Sobolev embedding inequality into mixed-norm L{U+1D3E} space is exploited to overcome the difficulty caused by the lack of the uniform estimates on the third order normal derivative of the director field to the Ginzburg-Landau approximate system. We also establish a blow-up criterion of the local strong solutions to the Ericksen-Leslie system. / Finally, using the technics exploited in the previous results, we obtain the local existence of strong solutions and the global existence of weak solutions to the Ericksen-Leslie system in bounded domains of R². In particular, by employing the local energy inequality, we prove that the lower bound of the existence time and the local in time estimates of a strong solution depend only on the basic energy and the L² integral continuity of the initial data. Thanks to these properties, by taking the limit of a sequence of strong solutions, we obtain a weak solution to the Ericksen-Leslie system. / 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. / Li, Jinkai. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 179-185). / Abstracts also in Chinese. / Introduction --- p.1 / Chapter 1 --- Global existence of weak solutions to the nematic liquid crystals in T² --- p.14 / Chapter 1.1 --- Introduction and main results --- p.14 / Chapter 1.2 --- Preliminaries --- p.21 / Chapter 1.3 --- Faedo-Galerkin scheme --- p.24 / Chapter 1.4 --- The limit n → ∞ --- p.27 / Chapter 1.5 --- The limit M → ∞ --- p.33 / Chapter 1.6 --- The limit N → ∞ and the local existence --- p.47 / Chapter 1.7 --- The global existence --- p.51 / Chapter 2 --- Local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in R³ --- p.60 / Chapter 2.1 --- Introduction and main results --- p.60 / Chapter 2.2 --- Local existence --- p.67 / Chapter 2.3 --- Blow-up criteria --- p.83 / Chapter 2.4 --- Convergence of Ginzburg-Landau to Ericksen-Leslie --- p.99 / Chapter 3 --- Local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in bounded domains of R³ --- p.112 / Chapter 3.1 --- Introduction and main results --- p.112 / Chapter 3.2 --- Local well-posedness --- p.119 / Chapter 3.3 --- Blow-up criteria --- p.141 / Chapter 4 --- Global existence of weak solutions to the Ericksen-Leslie system in bounded domains of R² --- p.156 / Chapter 4.1 --- Introduction and main results --- p.156 / Chapter 4.2 --- Strong solutions and blow-up criteria --- p.161 / Chapter 4.3 --- Global weak solutions --- p.172 / Chapter 5 --- Discuss on future works --- p.177 / Bibliography --- p.178

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A seletividade e eficiência do processo de flotação não estão fundamentadas tão somente nas diferenças das propriedades físico-químicas das superfícies dos minerais, mas também na hidrodinâmica do sistema de concentração. Via de regra, a eficácia da flotação diminui quando o tamanho de partícula ultrapassa 150 micrômetros. Esta pesquisa visou estudar a influência de variáveis hidrodinâmicas sobre mecanismos de flotação de partículas grossas em células mecânicas e suas implicações cinéticas. Os ensaios utilizaram células com agitação mecânica e cada situação experimental esteve associada a um conjunto de variáveis hidrodinâmicas: números de hidrodinâmicos adimensionais, características geométricas do impelidor, velocidade mínima de suspensão da polpa, energia dissipada média e energia de preservação do agregado partícula-bolha. Desta forma, a influência dos parâmetros hidrodinâmicos na flotação de apatita e esferas de vidro (diâmetro médio248 micrômetros) foi determinada na presença de oleato de sódio e acetato de eteramina em pH 10. Em condições hidrodinâmicas pouco turbulenta, a recuperação de partículas grossas decresce em função da falta de suspensão de partículas, por outro lado para condições hidrodinâmicas mais severas a recuperação foi quase nula devido à destruição do agregado partícula/bolha. A cinética de flotação da apatita foi estudada em escala de laboratório sob severas condições de turbulência, os resultados indicaram que partículas grossas de apatita têm flotabilidade inferior a finas, corroborando que a flotação dessas partículas grossas não se ajustou ao modelo de primeira ordem. Estes resultados evidenciaram que grossas demandam um ambiente hidrodinâmico apropriado para flotar eficientemente. / Coarse particles badly float mostly because the bubbleparticle aggregates are not strong enough to prevent the particle detachment from the bubble surface caused by the particle weight and turbulence eddies during the rise of the aggregates in the pulp phase to the froth phase and then to the froth launder. The aim this work was to determine effects and degrees of significance of hydrodynamic variables on coarse particle flotation and their kinetic implications. The tests were carried out in flotation cells with mechanical stirring system and each experimental condition was related to group of hydrodynamic variables: hydrodynamic dimensionless numbers, impeller geometries, minimum impeller rotational speed, energy dissipation rate and detachment energy. The influence of hydrodynamic parameters on flotation of the apatite and glass spheres (diameter average248 micrometers) was determined in the presence of sodium oleate and etheramine acetate at pH 10. For more quiescent hydrodynamic conditions, recovery of coarse particles decreased due to a lack in particle suspension, on the other hand, in the most severe hydrodynamic conditions the recovery was almost nil because the bubbleparticle aggregates detachment. Flotation kinetics of apatite was studied in laboratory scale under strong hydrodynamic conditions, the results indicated that coarse particles of apatite float at a lower rate than finer particles and the flotation kinetics of coarse particles did not fit to first order model. Those results evidenced that coarse particles demand a suitable hydrodynamic environment to float efficiently.

From September 2009 to July 2011 I collected vertically stratified zooplankton samples and recorded estuarine water parameters on a monthly basis in the Coos Bay estuary, Oregon during flood and ebb tides. I identified five taxa of juvenile bivalves in the plankton:

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In this study, the hydrodynamic performance of anguilliform swimming motion is computed using Morison’s equation. This method was shown to predict the servo motor torques well. The anguilliform swimming motion is sinusoidal with increasing amplitude from head to tail. A “wakeless” swimming motion proposed by Vorus and Taravella (2011) with zero net circulation is considered. This method is compared to the existing slender body theory and is validated with reference to the experimental results of NEELBOT-1.1 (Potts, 2015). The results for the study indicates that self-propulsion speed of the motion is independent of the oscillating tail amplitude at a constant advance ratio. At a constant wave speed, the self-propulsion speed attains a local maximum at an advance ratio of 0.5. Where the nominal length is equal to half the wavelength.

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