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
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328575 |
| Date | January 2013 |
| Contributors | Li, Jinkai., 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 (v, ii, 185 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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