EDP Sciences
1711636508
20240328
fre
laboutique.edpsciences.fr-R001755
03
01
R001755
00
ED
E107
00
01
01
1D Radiative Fluid and Liquid Crystal Equations
08
154
03
22
2058780
17
01
P15
EDP Sciences & Science Press
01
01
P15
EDP Sciences & Science Press
11
00
20221011
02
01
002141
03
9782759829040
15
9782759829040
03
01
D1
EDP Sciences
45
03
laboutique.edpsciences.fr-002141
03
01
002141
03
9782759829040
15
9782759829040
10
EA
E107
00
01
R001755
ED
E107
1
10
01
02
Current Natural Sciences
01
01
1D Radiative Fluid and Liquid Crystal Equations
1
A01
01
A2052
Yuming Qin
Qin, Yuming
Yuming
Qin
<p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-family:MilibusRg-Regular;mso-bidi-font-family:MilibusRg-Regular;mso-ansi-language:EN-US">Dr. Yuming QIN </span><span lang="EN-US" style="font-family:MilibusLt-Regular;mso-bidi-font-family:MilibusLt-Regular;mso-ansi-language:EN-US">is full professor, head of Mathematics Department and director of Institute of Nonlinear Sciences of Donghua University. His research interests are global (local) wellposedness of solutions and infinite-dimensional dynamical systems for nonlinear evolutionary equations including fluid equations such as Navier–Stokes equations, MHD, etc, and thermo(visco)elasticequations.<o:p></o:p></span></p>
01
eng
08
152
03
01
24
Izibook:Subject
Mathématiques
20
Mathematics;equations;fluid equations;liquid crystal equations
10
MAT007020
29
3052
01
510
05
06
03
00
This book presents recent results on nonlinear evolutionary fluid equations, in particular the global well-posedness and asymptotic behavior of solutions to 1D radiative fluid equations, as well as liquid crystal equations. Most of the material in this book was prepared by the author over the past few years.This book has two main features. Firstly, there are more known results on higher dimensional radiative fluid systems but only on the local existence and explosion of solutions; while the existing findings on the one-dimensional case present some shortcomings, this book introduces corrections and improvements of these shortcomings. Secondly, the current findings on the high-dimensional compressible liquid crystal fluid equations are few and include only globally existing solutions but not the asymptotic behavior of the solutions; the author developed not only the global existence and regularity of the solutions, but also the asymptotic behavior of the solutions for the one-dimensional case in the chapter 3 of this book. Therefore, this work provides the reader with complete elements related to the one-dimensional compressible liquid crystal fluid system.
02
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<p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:9.0pt;font-family:MilibusLt-Regular;mso-bidi-font-family:MilibusLt-Regular;mso-ansi-language:EN-US">This book presents recent results on nonlinear evolutionary fluid equations,in particular the global well-posedness and asymptotic behavior of solutions to1D radiative fluid equations, as well as liquid crystal equations. <o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:9.0pt;font-family:MilibusLt-Regular;mso-bidi-font-family:MilibusLt-Regular;mso-ansi-language:EN-US">It provides complete elements related to the one-dimensional compressibleliquid crystal fluid system.<o:p></o:p></span></p><p></p><p class="MsoNormal"><span lang="EN-US"> </span></p>
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<p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Foreword </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">VII<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">CHAPTER 1<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Preliminary </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1 Some Basic Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.1 The Sobolev Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.2 The Interpolation Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . .. . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">5<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.3 The Poincar</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTTec369687",sans-serif;mso-bidi-font-family:AdvTTec369687;mso-ansi-language:EN-US">é </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Inequality </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">6<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.4 The Classical Bellman</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0+20",sans-serif;mso-bidi-font-family:AdvTT691e30a0+20;mso-ansi-language:EN-US">–</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Gronwall Inequality </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">7<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.5 The Generalized Bellman</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0+20",sans-serif;mso-bidi-font-family:AdvTT691e30a0+20;mso-ansi-language:EN-US">–</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Gronwall Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">8<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.6 The Uniform Bellman</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0+20",sans-serif;mso-bidi-font-family:AdvTT691e30a0+20;mso-ansi-language:EN-US">–</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Gronwall Inequality</span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">9<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.7 The Young Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">12<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.8 The H</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTTec369687",sans-serif;mso-bidi-font-family:AdvTTec369687;mso-ansi-language:EN-US">ö</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">lder Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">13<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1.1.9 The Minkowski Inequalities </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . .. . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">14<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">CHAPTER 2<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Asymptotic Behavior of Solutions for the One-DimensionalInfrarelativistic<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Model of a Compressible Viscous Gas with Radiation </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">17<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.1 Main Results</span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">17<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.2 Global Existence and Uniform-in-Time Estimates in </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">22<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.3 Asymptotic Behavior of Solutions in </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">48<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.4 Global Existence and Uniform-in-Time Estimates in </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">53<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.5 Asymptotic Behavior of Solutions in </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">60<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.6 Global Existence and Uniform-in-Time Estimates in </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">62<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.7 Asymptotic Behavior of Solutions in </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">81<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2.8 Bibliographic Comments </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">85<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">CHAPTER 3<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Global Existence and Regularity of a One-Dimensional Liquid Crystal System</span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">89<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">3.1 Main Results</span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">89<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">3.2 Global Existence in </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1 </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">10</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">91</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US"><o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">3.3 Proof of Theorem 3.1.2 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">100<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">3.4 Proof of Theorem 3.1.3 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">103<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">3.5 Bibliographic Comments </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">109<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">CHAPTER 4<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">Large-time Behavior of Solutions to a One-Dimensional Liquid Crystal System</span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">111<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4.1 Introduction </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">111<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4.2 Uniform Estimates in </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">0</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i </span><span lang="EN-US" style="font-size:7.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">þ</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1 </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">ð</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">¼</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvP4C4E51",sans-serif;mso-bidi-font-family:AdvP4C4E51;mso-ansi-language:EN-US">; </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">Þ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">and </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4 </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">40</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. . </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">113</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US"><o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4.3 Large-time Behavior in </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">0</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i </span><span lang="EN-US" style="font-size:7.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">þ</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1 </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">ð</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">i </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">¼</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">1</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvP4C4E51",sans-serif;mso-bidi-font-family:AdvP4C4E51;mso-ansi-language:EN-US">; </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">2</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">Þ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">and </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4 </span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_</span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">40</span><span lang="EN-US" style="font-size:10.0pt;font-family:AdvP4C4E74;mso-bidi-font-family:AdvP4C4E74;mso-ansi-language:EN-US">_ </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT7846fc6b.I",serif;mso-bidi-font-family:"AdvTT7846fc6b\.I";mso-ansi-language:EN-US">H</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">4 </span><span lang="EN-US" style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">. </span><span lang="EN-US" style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US">122</span><span lang="EN-US" style="font-size:7.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0;mso-ansi-language:EN-US"><o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">4.4 Bibliographic Comments </span><span style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . </span><span style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">134<o:p></o:p></span></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">Bibliography </span><span style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . </span><span style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">135<o:p></o:p></span></p><p></p><p class="MsoNormal" style="margin-bottom:0cm;line-height:normal;mso-layout-grid-align:none;text-autospace:none"><span style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">Index </span><span style="font-size:9.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . </span><span style="font-size:10.0pt;font-family:"AdvTT691e30a0",serif;mso-bidi-font-family:AdvTT691e30a0">143<o:p></o:p></span></p>
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