Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems

Frontiers in Mathematics Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît ...

Author: Yuming Qin | Lan Huang

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Frontiers in Mathematics

Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Villani (Institut Henri Poincaré, Paris)

Yuming Qin Lan Huang

Global Well-posedness of Nonlinear Parabolic-Hyperbolic

Coupled Systems

Yuming Qin Department of Applied Mathematics Donghua University Shanghai People’s Republic of China

Lan Huang College of Mathematics and Information Science North China University of Water Sources and Electric Power Zhengzhou People’s Republic of China

ISSN 1660-8046 e-ISSN 1660-8054 ISBN 978-3-0348-0279-6 e-ISBN 978-3-0348-0280-2 DOI 10.1007/978-3-0348-0280-2 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2012931854 Mathematics Subject Classification (2010): 35Q30, 76-XX, 76D05 © Springer Basel AG 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Printed on acid-free paper

Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

To our parents Zhenrong Qin

Xilan Xia

and Shaolin Huang

Chuanfeng Yang

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1 Global Existence of Spherically Symmetric Solutions for Compressible Non-autonomous Navier-Stokes Equations 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.2 Global Existence of Solutions in 𝐻 1 . . . . . . . 1.3 Global Existence of Solutions in 𝐻 2 . . . . . . . 1.4 Global Existence of Solutions in 𝐻 4 . . . . . . . 1.5 Bibliographic Comments . . . . . . . . . . . . . .

Nonlinear . . . . .

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1 4 19 25 32

2 Global Existence and Exponential Stability for a Real Viscous Heat-conducting Flow with Shear Viscosity 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . 2.3 Proof of Theorem 2.1.2 . . . . . . . . . . . . . . . . . . . 2.4 Proof of Theorem 2.1.3 . . . . . . . . . . . . . . . . . . . 2.5 Bibliographic Comments . . . . . . . . . . . . . . . . . .

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33 38 46 53 73

3 Regularity and Exponential Stability Fluid in One Space Dimension 3.1 Introduction . . . . . . . . . . 3.2 Proof of Theorem 3.1.1 . . . . 3.3 Proof of Theorem 3.1.2 . . . . 3.4 Bibliographic Comments . . .

75 78 89 91

of the 𝒑th Power Newtonian . . . .

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4 Global Existence and Exponential Stability for the 𝒑th Power Viscous Reactive Gas 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Global Existence in 𝐻 2 . . . . . . . . . . . . . . . . . . 4.3 Exponential Stability in 𝐻 2 . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . 4.5 Bibliographic Comments . . . . . . . . . . . . . . . . .

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. 93 . 96 . 105 . 112 . 125

vii

viii

5 On a 1D Viscous Reactive and Radiative Gas with First-order Arrhenius Kinetics 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 Global Existence in 𝐻 1 . . . . . . . . . . . . 5.3 Exponential Stability in 𝐻 1 . . . . . . . . . 5.4 Proof of Theorem 5.1.2 . . . . . . . . . . . . 5.5 Proof of Theorem 5.1.3 . . . . . . . . . . . . 5.6 Bibliographic Comments . . . . . . . . . . .

Contents

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127 132 156 157 159 161

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Preface This book aims to present some recent results on a selection of nonlinear parabolichyperbolic coupled systems arising from physics, fluid mechanics and material science such as the compressible Navier-Stokes equations, viscous reactive and radiative gas, etc. Most of the material in this book is based on research carried out by the authors and their collaborators in recent years. Some of it has been previously published only in original papers, and some of the material has never been published until now. There are five chapters in this book. Chapters 1–2 will be concerned with the study of non-isentropic Navier-Stokes equations. In Chapter 1, we shall investigate the global existence of spherically symmetric solutions for nonlinear compressible Navier-Stokes equations of initial boundary value problems with a non-autonomous external force and a heat source in bounded annular domains 𝐺𝑛 = {𝑟 ∈ ℝ𝑛 : 𝑎 ≤ 𝑟 ≤ 𝑏} in ℝ𝑛 (1 ≤ 𝑛 ≤ 3). Thus we stress mainly the affect of non-autonomous terms on the global well-posedness and on the large-time behavior of solutions. Chapter 2 is concerned with the global existence and exponential stability of solutions for a real viscous compressible heat-conducting flow with general constitutive relations between two horizontal plates. Chapters 3–4 are devoted to an investigation of the global existence and exponential stability of solutions to a 𝑝th power viscous reactive gas (Newtonian fluid) without and with a chemical reaction respectively. We should note that these models are quite different from the usual viscous heat-conductive reactive gas in Qin [59]. In detail, the models considered in Chapters 3–4 have more complicated constitutive relationships than those in Qin [59], and so we need to design more delicate and careful estimates to establish the uniform lower and upper bounds of the specific volume. Chapter 5 is devoted to the study of viscous reactive and radiative gas. In Chapter 5, we shall establish the global existence and exponential stability of solutions in 𝐻 𝑖 (𝑖 = 1, 2, 4) for a Stefan-Boltzmann model of a viscous, reactive and radiative gas with first-order Arrhenius kinetics in a bounded interval, which describes classical stellar evolution of a finite mass of a heat-conducting viscous reactive fluid in local equilibrium with thermal radiation: pressure, internal energy and thermal conductivity have Stefan-Boltzmann radiative contributions. We sincerely hope that the reader will learn the main ideas and essence of the basic theories and methods in deriving global well-posedness, asymptotic behavior and regularity of global solutions for the models under consideration in this book. Also we are confident that the reader will be stimulated by some ideas from this book and undertake further study and research after having read the related references and our bibliographic comments. We extend our appreciation to Dr. Thomas Hempfling and Dr. Barbara Hellriegel of Birkhäuser/Springer Basel for their great efforts in publishing this book. We also want to take this opportunity to thank all the people who supported us including our teachers, colleagues and collaborators, etc. Yuming Qin

ix

x

Preface

appreciates his former Ph. D. advisor, Professor Songmu Zheng, for his constant encouragement, great help and support. Also Qin thanks Professors Ta-tsien Li, Boling Guo, Jiaxing Hong, Yiming Long, Weixi Shen, Ling Hsiao, Shuxing Chen, Zhenting Hou, Long-an Ying, Guangjun Yang, Guowang Chen, Jinhua Wang, Junning Zhao, Tiehu Qin, Yongji Tan, Sining Zheng, Zhouping Xin, Tong Yang, Hua Chen, Chaojiang Xu, Jingxue Yin, Liqun Zhang, Weike Wang, Mingxin Wang, Fahuai Yi, Song Jiang, Chengkui Zhong, Xuguang Lu, Yinbin Deng, Daomin Cao, Cheng He, Yi Zhou, Xiangyu Zhou, Quansui Wu, Daoyuan Fang, Ping Zhang, Changjiang Zhu, Changxing Miao, Yongsheng Li, Feimin Huang, Huijiang Zhao, Zheng-an Yao, Changzheng Qu, Yaping Wu, Zhaoli Liu, Huicheng Yin, Xiaozhou Yang, Shu Wang, Yaguang Wang, Zhong Tan, Xingbin Pan, Feng Zhou, Baojun Bian, Shengliang Pan, Wen-an Yong, Boxiang Wang, Lixin Tian, Shangbin Cui, Shijin Ding, Xi-nan Ma, Huaiyu Jian, Yachun Li, Benjin Xuan, Ting Wei, Quansen Jiu, Hailiang Li, Jiabao Su, Kaijun Zhang, Peidong Lei, Yongqian Zhang, Zhaoyang Yin, Wenyi Chen, Zhigui Lin, Xiaochun Liu, Hao Wu, Ting Zhang, Zhenhua Guo, Hongjun Yu, Ning Jiang and Yawei Wei for their constant help. Also Qin would like to thank Professors Herbert Amann, Michel Chipot from Switzerland, Professors J.A. Burns, Taiping Liu, Guiqiang Chen, D. Gilliams, Irena Lasiecka, Joel Spruck, M. Slemrod, Yisong Yang, Zhuangyi Liu, T.H. Otway, Shouhong Wang, Yuxi Zheng, Chun Liu, Changfeng Gui, Shouchuan Hu, Jianguo Liu, Hailiang Liu, Tao Luo from the USA, Professors Roger Temam, Alain Miranville, D. Hilhorst, Vilmos Komornik, Mokhtar Kirane, Patrick Martinez, Fatiha Alabau-Boussouira, Yuejun Peng and Bopeng Rao from France, Professors Hugo Beirao da Veiga, Maurizio Grasselli, Cecilia Cavaterra from Italy, Professors Bert-Wolfgang Schulze, Reinhard Racke, Michael Reissig, Ingo Witt, J¨ urgen Sprekels, H.-D. Alber from Germany, Professors Enrique Zuazua, Peicheng Zhu from Spain, Professors Jaime E. Mu˜ noz Rivera, Tofu Ma, Alexandre L. Madureira, Jinyun Yuan, D. Andrade, M.M. Cavalcanti, Frédéric G. Christian Valentin from Brazil, Professors Tzon Tzer L¨ u, Jyh-Hao Lee, Chun-Kong Law, Ngai-Ching Wong from Chinese Taiwan for their constant help. Yuming Qin also acknowledges the NNSF of China for its support. Currently, this book project is being supported by the NNSF of China with contract no. 11031003 and no. 10871040 and by the Sino-German cooperation grant “Analysis of partial differential equations and applications” with contract no. 446 CHV 113/267/0-1. Last but not least, Yuming Qin wants to express his deepest thanks to his parents (Zhenrong Qin and Xilan Xia), sisters (Yujuan Qin and Yuzhou Qin), brother (Yuxing Qin), wife (Yu Yin) and son (Jia Qin) for their great help, constant concern and advice in his career, and Lan Huang takes this opportunity to express thanks to her husband (Fengxiao Zhai) for his great support in her career. Yuming Qin and Lan Huang

Chapter 1

Global Existence of Spherically Symmetric Solutions for Nonlinear Compressible Non-autonomous Navier-Stokes Equations 1.1 Introduction This chapter concerns the global existence of spherically symmetric solutions for nonlinear compressible non-autonomous Navier-Stokes equations of an initial boundary value problem with an external force and a heat source in bounded annular domains 𝐺𝑛 = {𝑟 ∈ ℝ𝑛 : 𝑎 ≤ 𝑟 ≤ 𝑏} in ℝ𝑛 (1 ≤ 𝑛 ≤ 3). In Eulerian coordinates, equations under consideration are expressed as: (𝑛 − 1) 𝜌𝑣 = 0, 𝑟 [ (𝑛 − 1) (𝑛 − 1) ] 𝐶𝑣 𝜌(∂𝑡 𝑣 + 𝑣∂𝑟 𝑣) − (𝜆 + 2𝜇) ∂𝑟2 𝑣 + ∂𝑟 𝑣 − 𝑣 𝑟 𝑟2 + 𝑅∂𝑟 (𝜌𝑣) = 𝑓 (𝑟, 𝑡), [ (𝑛 − 1) (𝑛 − 1) ] ∂𝑟 𝜃 + 𝑅𝜌𝜃 ∂𝑟 𝑣 + 𝑣 𝐶𝑣 𝜌(∂𝑡 𝜃 + 𝑣∂𝑟 𝜃) − 𝜅∂𝑟2 𝜃 − 𝑘 𝑟 𝑟 ] [ (𝑛 − 1) 2 (𝑛 − 1) 2 𝑣 − 2𝜇(∂𝑟 𝑣)2 − 2𝜇 − 𝜆 ∂𝑟 𝑣 + 𝑣 = 𝑔(𝑟, 𝑡). 𝑟 𝑟2 ∂𝑡 𝜌 + ∂𝑟 (𝜌𝑣) +

(1.1.1)

(1.1.2)

(1.1.3)

We consider equations (1.1.1)–(1.1.3) with initial boundary value conditions 𝜌(𝑟, 0) = 𝜌0 (𝑟),

𝑣(𝑟, 0) = 𝑣0 (𝑟),

𝑣(𝑎, 𝑡) = 𝑣(𝑏, 𝑡) = 0,

𝜃(𝑟, 0) = 𝜃0 (𝑟),

𝑟 ∈ 𝐺𝑛 , 1 ≤ 𝑛 ≤ 3, (1.1.4)

𝜃𝑟 (𝑎, 𝑡) = 𝜃𝑟 (𝑏, 𝑡) = 0, 1 ≤ 𝑛 ≤ 3

(1.1.5)

where constants 𝑅, 𝐶𝑣 , 𝜅, 𝜇 > 0 and 𝛽 = 𝜆 + 2𝜇.

Y. Qin and L. Huang, Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems, Frontiers in Mathematics, DOI 10.1007/978-3-0348-0280-2_1, © Springer Basel AG 2012

1

2

Chapter 1. Global Existence of Spherically Symmetric Solutions

Equations (1.1.1)–(1.1.3) describe the symmetric motion of a viscous polytropic ideal gas in 𝑛-dimension with an external force 𝑓 and a heat source 𝑔 where 𝜌, 𝑣, 𝜃 are the density, velocity, and absolute temperature, 𝜆 and 𝜇 are the constant viscosity coefficients, 𝑅, 𝐶𝑣 , and 𝜅 are the gas constant, specific heat capacity and thermal conductivity, respectively. It is convenient to transform system (1.1.1)–(1.1.3) into Lagrangian coordinates. The Eulerian coordinates (𝑟, 𝑡) are connected to the Lagrangian coordinates (𝜁, 𝑡) by the relation ∫ 𝑡 𝑣˜(𝜁, 𝜏 )𝑑𝜏, (1.1.6) 𝑟(𝜁, 𝑡) = 𝑟0 (𝜁) + 0

where 𝑣˜(𝜁, 𝑡) := 𝑣(𝑟(𝜁, 𝑡), 𝑡), 𝑟 ∈ 𝐺𝑛 ,

𝑟0 (𝜁) := 𝜂 −1 (𝜁),

𝑑𝑛 = 0, (𝑛 = 1),

∫

𝑟(𝜁,𝑡)

𝑑𝑛

𝑏

𝑎

(1.1.1), (1.1.5) and (1.1.6), we obtain ∫

𝑟

𝑑𝑛

𝑠𝑛−1 𝜌0 (𝑠)𝑑𝑠,

𝑑𝑛 = 𝑎, (𝑛 = 2, 3).

Suppose that 𝜌0 (𝑠) > 0, 𝑠 ∈ 𝐺𝑛 . Set 𝐿 =

∂𝑡

∫ 𝜂(𝑟) :=

𝑠𝑛−1 𝜌0 (𝑠)𝑑𝑠 > 0. Using the equation

𝑠𝑛−1 𝜌(𝑠, 𝑡)𝑑𝑠 = 𝛿𝑛1 𝑣(0, 𝑡)𝜌(0, 𝑡),

𝛿𝑖𝑗 = 1, (𝑖 = 𝑗);

𝛿𝑖𝑗 = 0, (𝑖 ∕= 𝑗).

By integration, we derive ∫

𝑟(𝜁,𝑡) 𝑑𝑛

𝑠𝑛−1 𝜌(𝑠, 𝑡)𝑑𝑠 =

∫

𝑟0 (𝜁)

𝑑𝑛

𝑠𝑛−1 𝜌0 (𝑠)𝑑𝑠 + 𝛿𝑛1

= 𝜁 + 𝛿𝑛1

∫ 0

𝑡

∫ 0

𝑡

(𝑣𝜌)(0, 𝜏 )𝑑𝜏

(𝑣𝜌)(0, 𝜏 )𝑑𝜏.

Thus, under the assumption inf{𝜌(𝑠, 𝑡); 𝑠 ∈ 𝐺𝑛 , 𝑡 ≥ 0} > 0, 𝐺𝑛 is transformed to Ω𝑛 , with Ω𝑛 = (0, 𝐿), (𝑛 = 1, 2, 3). Moreover, we have ∂𝜁 𝑟(𝜁, 𝑡) = [𝑟(𝜁, 𝑡)𝑛−1 𝜌(𝑟(𝜁, 𝑡), 𝑡)]−1 .

(1.1.7)

For a function 𝜑(𝑟, 𝑡), we write 𝜑(𝜁, ˜ 𝑡) := 𝜑(𝑟(𝜁, 𝑡), 𝑡). By virtue of (1.1.6) and (1.1.7), we derive ˜ 𝑡) = ∂𝑡 𝜑(𝑟, 𝑡) + 𝑣∂𝑟 𝜑(𝑟, 𝑡), ∂𝑡 𝜑(𝜁, ˜ 𝑡) = ∂𝑟 𝜑(𝑟, 𝑡)∂𝜁 𝑟(𝜁, 𝑡) = ∂𝜁 𝜑(𝜁,

1 ∂𝑟 𝜑(𝑟, 𝑡). 𝑟𝑛−1 𝜌(𝑟, 𝑡)

(1.1.8)

˜ by (𝜌, 𝑣, 𝜃) and (𝜁, 𝑡) by (𝑥, 𝑡). We use 𝑢 := 1 to denote the We still denote (˜ 𝜌, 𝑣˜, 𝜃) 𝜌 specific volume. Therefore, by virtue of (1.1.7)–(1.1.8), equations (1.1.1)–(1.1.5)

1.1. Introduction

3

in the new variables (𝑥, 𝑡) are (1.1.9) 𝑢𝑡 − (𝑟𝑛−1 𝑣)𝑥 = 0, ( ) 𝑛−1 𝜃 𝑣)𝑥 (𝑟 −𝑅 𝑣𝑡 − 𝑟𝑛−1 𝛽 = 𝑓 (𝑟(𝑥, 𝑡), 𝑡), (1.1.10) 𝑢 𝑢 𝑥 ( 2𝑛−2 ) 𝑟 𝜃𝑥 1 𝐶𝑣 𝜃𝑡 − 𝑘 − (𝛽(𝑟𝑛−1 𝑣)𝑥 − 𝑅𝜃)(𝑟𝑛−1 𝑣)𝑥 + 2𝜇(𝑛 − 1)(𝑟𝑛−2 𝑣 2 )𝑥 𝑢 𝑢 𝑥 = 𝑔(𝑟(𝑥, 𝑡), 𝑡),

(1.1.11)

together with 𝑢(𝑥, 0) = 𝑢0 (𝑥),

𝑣(𝑥, 0) = 𝑣0 (𝑥),

𝜃(𝑥, 0) = 𝜃0 (𝑥),

𝑣(0, 𝑡) = 𝑣(𝐿, 𝑡) = 0, 𝜃𝑥 (0, 𝑡) = 𝜃𝑥 (𝐿, 𝑡) = 0,

𝑥 ∈ Ω𝑛 , 1 ≤ 𝑛 ≤ 3, (1.1.12)

𝑡 ≥ 0, 1 ≤ 𝑛 ≤ 3

(1.1.13)

where 𝛽 = 𝜆 + 2𝜇. Then by (1.1.6), we have ∫ 𝑟(𝑥, 𝑡) = 𝑟0 (𝑥) +

0

𝑡

{ 𝑣(𝑥, 𝜏 )𝑑𝜏,

𝑟0 (𝑥) :=

𝑛

(𝑑𝑛 ) + 𝑛

∫

𝑥

0

𝑢0 (𝑦)𝑑𝑦

} 𝑛1

,

i.e., 𝑟𝑡 = 𝑣,

𝑟𝑛−1 𝑟𝑥 = 𝑢,

𝑟∣𝑥=0 = 𝑎,

𝑟∣𝑥=𝐿 = 𝑏.

(1.1.14)

When 𝑛 = 2, 3, for constants 𝜆, 𝜇, we suppose 𝑛𝜆 + 2𝜇 > 0.

(1.1.15)

The aim of this chapter is to investigate the global existence of solutions in 𝐻 𝑖 (𝑖 = 1, 2, 4). The global existence of solutions in 𝐻 4 implies the global existence of classical solutions. We finally obtain the results by the energy method as in [52, 53, 54, 55, 57, 58, 65, 67, 68, 69]. We suppose that 𝑓 (𝑟, 𝑡), 𝑔(𝑟, 𝑡) satisfy: 𝑓 ∈ 𝐿1 (ℝ+ , 𝐿∞ [𝑎, 𝑏]) ∩ 𝐿2 (ℝ+ , 𝐿2 [𝑎, 𝑏]),

(1.1.16)

𝑔 ∈ 𝐿1 (ℝ+ , 𝐿∞ [𝑎, 𝑏]) ∩ 𝐿2 (ℝ+ , 𝐿2 [𝑎, 𝑏]), 𝑔(𝑟, 𝑡) ≥ 0, ∀(𝑟, 𝑡) ∈ [𝑎, 𝑏] × [0, +∞).

(1.1.17)

The notation of this section is as follows: ∥ ⋅ ∥𝐵 denotes the norm of space 𝐵, ∥⋅∥ = ∥⋅∥𝐿2 . 𝐶1 denotes a generic positive constant depending only on the 𝐻 1 norm of initial data (𝑢0 , 𝑣0 , 𝜃0 ), ∥𝑓 ∥𝐿1(ℝ+ ,𝐿∞ [𝑎,𝑏]) , ∥𝑓 ∥𝐿2(ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑔∥𝐿1(ℝ+ ,𝐿∞ [𝑎,𝑏]) , ∥𝑔∥𝐿2(ℝ+ ,𝐿2 [𝑎,𝑏]) , but is independent of 𝑡. The results of this chapter are selected from [72].

4


1.2 Global Existence of Solutions in 𝑯 1 In this section, we shall establish the global existence of solutions in 𝐻 1 . Theorem 1.2.1. Assume that (1.1.16)–(1.1.17) hold. If (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻 1 [0, 𝐿] × 𝐻01 [0, 𝐿] × 𝐻 1 [0, 𝐿], then problem (1.1.9)–(1.1.15) admits a unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐶([0, +∞), 𝐻 1 [0, 𝐿] × 𝐻01 [0, 𝐿] × 𝐻 1 [0, 𝐿]) satisfying 0 < 𝑎 ≤ 𝑟(𝑥, 𝑡) ≤ 𝑏, 0
0, ∀𝑡 ≥ 0. Suppose that 𝑟𝑥 (𝑥, 𝑡) is not always positive over [0, 𝐿] × [0, +∞), i.e., there are 𝑦 ∈ (0, 𝐿], 𝜏 ∈ [0, +∞), such that 𝑟𝑥 (𝑥, 𝑡) > 0 over [0, 𝑦) × [0, 𝜏 ], but 𝑟𝑥 (𝑦, 𝜏 ) = 0. So, by continuity, 𝑟𝑥 (𝑥, 𝑡) ≥ 0 over [0, 𝑦] × [0, 𝜏 ], therefore 𝑟(𝑦, 𝜏 ) ≥ 𝑟(0, 𝜏 ) = 𝑎 > 0. From (1.1.14), 0 = 𝑟𝑥 (𝑦, 𝜏 ) = 𝑟1−𝑛 (𝑦, 𝜏 )𝑢(𝑦, 𝜏 ) > 0, which is a contradiction. This shows that 𝑟𝑥 (𝑥, 𝑡) > 0 over [0, 𝐿] × [0, +∞). Hence, 𝑎 = 𝑟(0, 𝑡) ≤ 𝑟(𝑥, 𝑡) ≤ 𝑟(𝐿, 𝑡) = 𝑏, ∀𝑥 ∈ [0, 𝐿], ∀𝑡 > 0. □ Remark 1.2.1. It follows from Lemma 1.2.1 that assumptions (1.1.16) and (1.1.17) are equivalent to the following conditions: ∀𝑥 ∈ [0, 𝐿], 𝑓 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿1 (ℝ+ , 𝐿∞ [𝑎, 𝑏]) ∩ 𝐿2 (ℝ+ , 𝐿2 [𝑎, 𝑏]), 1

+

∞

2

+

(1.2.5)

2

𝑔(𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]) ∩ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]), 𝑔(𝑟(𝑥, 𝑡), 𝑡) ≥ 0, ∀(𝑥, 𝑡) ∈ [0, 𝐿] × [0, +∞).

(1.2.6)

Thus the generic positive constant 𝐶1 depends only on the 𝐻 1 norm of initial data (𝑢0 , 𝑣0 , 𝜃0 ) and norms of 𝑓 and 𝑔 in the class of functions in (1.2.5)–(1.2.6), but independent of 𝑡.

1.2. Global Existence of Solutions in 𝐻 1

5

Lemma 1.2.2. Under the conditions of Theorem 1.2.1, there is a constant 𝐶1 > 0 such that ) ∫ 𝑡∫ 𝐿( 2 ∫ 𝐿 𝑣𝑥 𝜃2 𝑔 + 𝑥2 + 𝑈 (𝑥, 𝑡)𝑑𝑥 + (1.2.7) (𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 , ∀𝑡 > 0, 𝑢𝜃 𝑢𝜃 𝜃 0 0 0 where { 𝑈 (𝑥, 𝑡) ≜

} 𝑣2 + 𝑅(𝑢 − ln 𝑢 − 1) + 𝐶𝑣 (𝜃 − ln 𝜃 − 1) (𝑥, 𝑡). 2

(1.2.8)

Proof. By equations (1.1.9)–(1.1.11), we can easily obtain 𝛽 𝑛−1 2 𝜅 (𝑟 𝑣)𝑥 + 2 (𝑟𝑛−1 𝜃𝑥 )2 𝑈𝑡 + 𝑢𝜃 { 𝑢𝜃 ( {( )} ) } 𝛽 𝑛−1 𝜃 1 𝑟2𝑛−2 𝜃𝑥 𝑛−1 𝑛−1 = 𝑟 𝑣 𝑣)𝑥 − 𝑅 + 𝑣𝑓 + 𝑅(𝑟 𝑣)𝑥 + 𝜅 1− (𝑟 𝑢 𝑢 𝜃 𝑢 𝑥 𝑥 ( ) 1 𝑔 (1.2.9) − 2(𝑛 − 1)𝜇 1 − (𝑟𝑛−2 𝑣 2 )𝑥 + 𝑔 − . 𝜃 𝜃 By (1.1.14) and (1.1.15), we easily derive 𝛽 ( 𝑛−1 )2 (𝑟𝑛−2 𝑣 2 )𝑥 𝑟 𝑣 𝑥 − 2𝜇(𝑛 − 1) 𝑢𝜃 𝜃 { ( )2 1 𝜆𝑟𝑛−1 𝑣𝑥 = (𝑛 − 1)(2𝜇 + (𝑛 − 1)𝜆) 𝑟−1 𝑢𝑣 + 𝑢𝜃 2𝜇 + (𝑛 − 1)𝜆 } 2𝜇(2𝜇 + 𝑛𝜆) 2𝑛−2 2 𝑟 + 𝑣𝑥 2𝜇 + (𝑛 − 1)𝜆 2𝜇(2𝜇 + 𝑛𝜆) 𝑟2𝑛−2 𝑣𝑥2 . ≥ 2𝜇 + (𝑛 − 1)𝜆 𝑢𝜃

(1.2.10)

Integrating (1.2.9) over [0, 𝐿] × [0, 𝑡], noting that ∫

𝐿

0

( ) 𝑈 (𝑥, 0)𝑑𝑥 ≤ 𝐶1 1 + ∥(𝑢0 , 𝑣0 , 𝜃0 )∥2 ≤ 𝐶1 ,

and considering (1.1.12), (1.2.10), we can obtain ∫ 0

𝐿

𝑈 (𝑥, 𝑡)𝑑𝑥 + ≤ 𝐶1 +

∫ 𝑡∫ 0

∫ 𝑡∫ 0

0

0 𝐿

𝐿

(

𝑣𝑥2 𝜃2 𝑔 + 𝑥2 + 𝑢𝜃 𝑢𝜃 𝜃

(𝑣𝑓 + 𝑔) (𝑥, 𝑠)𝑑𝑥𝑑𝑠

) (𝑥, 𝑠)𝑑𝑥𝑑𝑠 (1.2.11)

6


where ∫ 𝑡 ∫ 0

0

𝐿

∫ 𝑡∫ 𝑣𝑓 𝑑𝑥𝑑𝑠 ≤ 𝐶1 0

∫ ≤ 𝐶1

𝑡

𝐿

0

∣𝑓 ∣𝑣 2 𝑑𝑥𝑑𝑠 + 𝐶1 ∫

sup ∣𝑓 ∣

0 𝑥∈[0,𝐿]

𝐿

∫ 𝑡∫ 0

𝐿

0

𝑣 2 𝑑𝑥𝑑𝑠 + 𝐶1

0

∣𝑓 ∣𝑑𝑥𝑑𝑠

∫ 𝑡∫ 0

0

𝐿

∣𝑓 ∣𝑑𝑥𝑑𝑠.

(1.2.12)

Combining (1.2.11)–(1.2.12), using Gronwall’s inequality and conditions (1.1.16)– (1.1.17), we can obtain (1.2.2). □ Lemma 1.2.3. There are positive constants 𝛼1 and 𝛼2 such that ∫ 𝐿 𝛼1 ≤ 𝜃(𝑥, 𝑡)𝑑𝑥 ≤ 𝛼2 , ∀𝑡 > 0, 0

(1.2.13)

and there is a point 𝑎(𝑡) ∈ [0, 𝐿] satisfying 𝛼2 𝛼1 ≤ 𝜃(𝑎(𝑡), 𝑡) ≤ . 𝐿 𝐿 Proof. From (1.2.7), we have ∫ 𝐿 𝐶𝑣 (𝜃(𝑥, 𝑡) − log 𝜃(𝑥, 𝑡) − 1) (𝑥, 𝑡)𝑑𝑥 ≤ 𝐶1 , 0

(1.2.14)

∀𝑡 > 0.

(1.2.15)

By virtue of the mean value theorem, for any 𝑡 ≥ 0, there is a point 𝑎(𝑡) ∈ [0, 𝐿] such that 𝜃(𝑎(𝑡), 𝑡) − log 𝜃(𝑎(𝑡), 𝑡) − 1 ≤ (𝐶𝑣 𝐿)−1 𝐶1 from which it follows that 𝜁1 ≤ 𝜃(𝑎(𝑡), 𝑡) ≤ 𝜁2 , with 𝜁1 , 𝜁2 being two positive roots of the equation: 𝑦 − ln 𝑦 − 1 = 𝐶𝑣−1 𝐶1 . By Jensen’s inequality, we can obtain ∫ 𝐿 ∫ 𝐿 𝜃(𝑥, 𝑡)𝑑𝑥 − log 𝜃(𝑥, 𝑡)𝑑𝑥 − 1 ≤ 𝐶𝑣−1 𝐶1 , ∀𝑡 > 0. 0

∫

Therefore 0 < 𝜁3 ≤

0

0

𝐿

𝜃(𝑥, 𝑡)𝑑𝑥 ≤ 𝜁4 , where 𝜁3 , 𝜁4 are two positive roots of

equation 𝑦 − ln 𝑦 − 1 = (𝐶𝑣 𝐿)−1 𝐶1 . Taking 𝛼1 = min{𝜁1 , 𝜁3 }, 𝛼2 = max{𝜁2 , 𝜁4 }, we obtain (1.2.13) and (1.2.14). □ Let 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 , 𝜎(𝑥, 𝑡) := 𝛽 𝑢 𝑢 ∫ 𝑥 ∫ 𝑡 −(𝑛−1) 𝜎(𝑥, 𝑠)𝑑𝑠 + 𝑟0 (𝑦)𝑣0 (𝑦)𝑑𝑦 𝜙(𝑥, 𝑡) := 0

+ (𝑛 − 1)

∫ 𝑡∫ 0

0 𝐿

𝑥

𝑟−𝑛 (𝑦, 𝑠)𝑣 2 (𝑦, 𝑠)𝑑𝑦𝑑𝑠.

(1.2.16)

(1.2.17)


7

We deduce after a direct calculation that ∫

𝑡

𝜎𝑥 (𝑥, 𝑠)𝑑𝑠 + 𝑟01−𝑛 (𝑥)𝑣0 (𝑥) − (𝑛 − 1) 0 ∫ 𝑡 𝑟1−𝑛 𝑓 𝑑𝑠, = 𝑟1−𝑛 𝑣(𝑥, 𝑡) −

𝜙𝑥 (𝑥, 𝑡) =

0

∫

𝜙𝑡 (𝑥, 𝑡) = 𝜎(𝑥, 𝑡) + (𝑛 − 1)

𝐿

𝑥

∫ 0

𝑡

𝑟−𝑛 (𝑥, 𝑠)𝑣 2 (𝑥, 𝑠)𝑑𝑠 (1.2.18)

𝑟−𝑛 (𝑦, 𝑡)𝑣 2 (𝑦, 𝑡)𝑑𝑦

𝜃 𝑛 − 1 (𝑟𝑛 )𝑥 (𝑟𝑛−1 𝑣)𝑥 −𝑅 + =𝛽 𝑢 𝑢 𝑛 𝑢

∫

𝐿

𝑥

𝑟−𝑛 𝑣 2 𝑑𝑦.

(1.2.19)

Multiplying (1.2.19) by 𝑢, then from (1.1.9), (1.2.18), we derive (𝑢𝜙)𝑡 − (𝑟𝑛−1 𝑣𝜙)𝑥

∫

2

∫

𝐿

(1.2.20) 𝑡

𝑛−1 𝑛 𝑣 − 𝑅𝜃 + 𝛽(𝑟𝑛−1 𝑣)𝑥 + (𝑟 )𝑥 𝑟−𝑛 𝑣 2 𝑑𝑦 + 𝑟𝑛−1 𝑣 𝑟1−𝑛 𝑓 𝑑𝑠 𝑛 𝑛 0 ] [ ∫ 𝑥 ∫ 𝑡 𝐿 𝑛 − 1 𝑣2 𝑟−𝑛 𝑣 2 𝑑𝑦 + 𝑟𝑛−1 𝑣 𝑟1−𝑛 𝑓 𝑑𝑠. = − − 𝑅𝜃 + 𝛽(𝑟𝑛−1 𝑣)𝑥 + 𝑟𝑛 𝑛 𝑛 𝑥 0 =−

𝑥

Integrating (1.2.20) over [0, 𝐿] × [0, 𝑡], we can get ∫

𝐿

0

) 𝑣2 + 𝑅𝜃 𝑑𝑥𝑑𝑠 (1.2.21) 𝑢𝜙𝑑𝑥 = 𝑢0 (𝑥)𝜙0 (𝑥)𝑑𝑥 − 𝑛 0 0 0 (∫ ) ∫ 𝑡∫ 𝐿 ∫ ∫ 𝑠 𝑛 − 1 𝑛 𝑡 𝐿 −𝑛 2 𝑎 − 𝑟 𝑣 𝑑𝑥𝑑𝑠 + 𝑟𝑛−1 𝑣 𝑟1−𝑛 𝑓 𝑑𝜏 𝑑𝑥𝑑𝑠. 𝑛 0 0 0 0 0 ∫

∫ 𝑡∫

𝐿

𝐿

(

It follows from integration of (1.1.9) that ∫ 0

𝐿

∫ 𝑢𝑑𝑥 =

𝐿 0

𝑢0 𝑑𝑥 := 𝑢∗ ,

∀𝑡 > 0.

(1.2.22)

Applying the mean value theorem, we infer that for any 𝑡 ≥ 0 there exists a point 𝑥0 (𝑡) ∈ [0, 𝐿] such that ∫ 0

𝐿

∫ 𝜙𝑢𝑑𝑥 = 𝜙(𝑥0 (𝑡), 𝑡)

𝐿 0

i.e., 𝜙(𝑥0 (𝑡), 𝑡) =

1 𝑢∗

𝑢𝑑𝑥 = 𝑢∗ 𝜙(𝑥0 (𝑡), 𝑡),

∫ 0

𝐿

𝑢𝜙𝑑𝑥.

(1.2.23)

8


From (1.2.17), (1.2.21) and (1.2.23), we derive ∫ 0

𝑡

∫ 𝜎(𝑥0 (𝑡), 𝑡)𝑑𝑠 = 𝜙(𝑥0 (𝑡), 𝑡) −

𝑥0 (𝑡)

0

𝑟01−𝑛 𝑣0 𝑑𝑦 − (𝑛 − 1)

∫ 𝑡∫ 0

𝐿

𝑥0 (𝑡)

𝑟−𝑛 𝑣 2 𝑑𝑦𝑑𝑠

) ∫ ∫ 𝑣2 (𝑛 − 1)𝑎𝑛 𝑡 𝐿 −𝑛 2 + 𝑅𝜃 𝑑𝑥𝑑𝑠 − 𝑟 𝑣 𝑑𝑥𝑑𝑠 𝑛 𝑛𝑢∗ 0 0 0 0 ∫ 𝑥0 (𝑡) ∫ 𝐿 ∫ 𝑡∫ 𝐿 1 −𝑛 2 𝑟 𝑣 𝑑𝑥𝑑𝑠 + ∗ 𝑢0 𝜙0 𝑑𝑥 − 𝑟01−𝑛 𝑣0 𝑑𝑦 − (𝑛 − 1) 𝑢 0 0 𝑥0 (𝑡) 0 (∫ 𝑠 ) ∫ 𝑡∫ 𝐿 1 + ∗ 𝑟𝑛−1 𝑣 𝑟1−𝑛 𝑓 𝑑𝜏 𝑑𝑥𝑑𝑠, ∀𝑡 ≥ 0. (1.2.24) 𝑢 0 0 0

1 =− ∗ 𝑢

∫ 𝑡∫

𝐿

(

In order to estimate the bounds of 𝑢, we first give a representation of 𝑢. Lemma 1.2.4. We have the following representation: 𝑢(𝑥, 𝑡) =

] [ ∫ 𝐷(𝑥, 𝑡) 𝑅 𝑡 𝜃(𝑥, 𝑠)𝐵(𝑥, 𝑠) 𝑑𝑠 , 1+ 𝐵(𝑥, 𝑡) 𝛽 0 𝐷(𝑥, 𝑠)

𝑥 ∈ [0, 𝐿],

𝑡 > 0,

(1.2.25)

where { [ ∫ 𝐿 ] ∫ 𝑥 ∫ 𝑥 1 1 1−𝑛 1−𝑛 𝑢 𝜙 𝑑𝑥 − 𝑟 𝑣 𝑑𝑦 + 𝑟 𝑣𝑑𝑦 0 0 0 𝛽 𝑢∗ 0 0 𝑥0 (𝑡) (∫ 𝑠 ) } ∫ 𝑥 ∫ 𝑡 ∫ 𝑡∫ 𝐿 1 𝑛−1 1−𝑛 1−𝑛 + ∗ 𝑟 𝑣 𝑟 𝑓 𝑑𝜏 𝑑𝑥𝑑𝑠 − 𝑟 𝑓 𝑑𝑠𝑑𝑦 , 𝑢 0 0 0 𝑥0 (𝑡) 0 (1.2.26) ) { [ ∫ 𝑡 ∫ 𝐿( 2 𝑛 ∫ 𝑡∫ 𝐿 𝑣 1 1 (𝑛 − 1)𝑎 + 𝑅𝜃 𝑑𝑥𝑑𝑠 + 𝑟−𝑛 𝑣 2 𝑑𝑥𝑑𝑠 𝐵(𝑥, 𝑡) = exp 𝛽 𝑢∗ 0 0 𝑛 𝑛𝑢∗ 0 0 ]} ∫ 𝑡∫ 𝐿 𝑟−𝑛 𝑣 2 𝑑𝑦𝑑𝑠 . (1.2.27) + (𝑛 − 1)

𝐷(𝑥, 𝑡) = 𝑢0 (𝑥) exp

0

𝑥

Proof. By (1.1.9) and (1.1.10), we have 𝑟1−𝑛 𝑣𝑡 = 𝛽(ln 𝑢)𝑥𝑡 − 𝑅

( ) 𝜃 + 𝑟1−𝑛 𝑓. 𝑢 𝑥

(1.2.28)

Integrating (1.2.28) over [0, 𝑡] × [𝑥0 (𝑡), 𝑥], we can get ∫ 𝛽 log 𝑢 − 𝑅

0

𝑡

∫ 𝑡 ∫ 𝑥 ∫ 𝑡 𝜃 𝑑𝑠 = 𝛽 log 𝑢0 + 𝜎(𝑥0 (𝑠), 𝑠)𝑑𝑠 + 𝑟1−𝑛 𝑣𝑡 𝑑𝑠𝑑𝑦 𝑢 0 𝑥0 (𝑡) 0 ∫ 𝑥 ∫ 𝑡 − 𝑟1−𝑛 𝑓 𝑑𝑠𝑑𝑥. (1.2.29) 𝑥0 (𝑡)

0


9

From (1.2.24) and (1.2.29), we have ∫ 𝑡 𝜃 𝛽 log 𝑢 − 𝑅 𝑑𝑠 𝑢 0 ) ∫ 𝑡∫ 𝐿( 2 ∫ ∫ 𝑣 (𝑛 − 1)𝑎𝑛 𝑡 𝐿 −𝑛 2 1 + 𝑅𝜃 𝑑𝑥𝑑𝑠 − = 𝛽 log 𝑢0 − ∗ 𝑟 𝑣 𝑑𝑥𝑑𝑠 𝑢 0 0 𝑛 𝑛𝑢∗ 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑥 ∫ 𝑥 ∫ 𝐿 1 + ∗ 𝑢0 𝜙0 𝑑𝑥 − (𝑛 − 1) 𝑟−𝑛 𝑣 2 𝑑𝑦𝑑𝑠 − 𝑟01−𝑛 𝑣0 𝑑𝑦 + 𝑟1−𝑛 𝑣𝑑𝑦 𝑢 0 0 𝑥 0 𝑥0 (∫ 𝑠 ) ∫ 𝑥 ∫ 𝑡 ∫ 𝑡∫ 𝐿 1 − 𝑟1−𝑛 𝑓 𝑑𝑠𝑑𝑦 + ∗ 𝑟𝑛−1 𝑣 𝑟1−𝑛 𝑓 𝑑𝜏 𝑑𝑥𝑑𝑠. 𝑢 0 0 𝑥0 (𝑡) 0 0 With the definition of 𝐵(𝑥, 𝑡) and 𝐷(𝑥, 𝑡), we obtain ( ∫ 𝑡 ) 𝑅 𝐵(𝑥, 𝑡) 1 𝜃(𝑥, 𝑠) = exp 𝑑𝑠 . 𝐷(𝑥, 𝑡) 𝑢(𝑥, .𝑡) 𝛽 0 𝑢(𝑥, 𝑠)

(1.2.30)

𝑅𝜃 , then integrating the result with respect to 𝑡, we derive 𝛽 ) ∫ 𝑅 𝑡 𝜃(𝑥, 𝑡)𝐵(𝑥, 𝑡) 𝜃(𝑥, 𝑡) 𝑑𝑠 = 1 + 𝑑𝑠. (1.2.31) 𝑢(𝑥, 𝑡) 𝛽 0 𝐷(𝑥, 𝑡)

Multiplying (1.2.30) by ( exp

𝑅 𝛽

∫

𝑡

0

Combining (1.2.31) and (1.2.30), we obtain (1.2.25).

□

Lemma 1.2.5. Under the conditions of Theorem 1.2.1, there are positive constants 𝑢 and 𝑢 such that 𝑢 ≤ 𝑢(𝑥, 𝑡) ≤ 𝑢,

∀(𝑥, 𝑡) ∈ [0, 𝐿] × [0, +∞).

Proof. We deduce after a direct calculation that ∫ 𝑡∫ 𝐿 ∫ 𝑥 ∫ 𝑡 𝑟1−𝑛 𝑓 𝑑𝑠𝑑𝑦 ≤ 𝑎1−𝑛 ∣𝑓 ∣𝑑𝑦𝑑𝑠 ≤ 𝐶1 , 𝑥0 (𝑡)

0

∫

𝑥

𝑥0 (𝑡)

𝑟

1−𝑛

∫ 1−𝑛 𝑣𝑑𝑦 ≤ 𝑎

0

(1.2.33)

0

𝐿 0

(1.2.32)

∣𝑣∣𝑑𝑦 ≤ 𝑎1−𝑛 ∥𝑣∥ ≤ 𝐶1 .

(1.2.34)

By Lemma 1.2.1, condition (1.1.16), and (1.2.12), we obtain ) (∫ 𝑠 ) ∫ 𝑡 ∫ 𝐿 (∫ ∞ ∫ 𝑡∫ 𝐿 𝑛−1 1−𝑛 𝑟 𝑣 𝑟 𝑓 𝑑𝜏 𝑑𝑥𝑑𝑠 ≤ 𝐶1 ∣𝑣∣ sup ∣𝑓 ∣𝑑𝜏 𝑑𝑥𝑑𝑠 0

0

≤ 𝐶1

∫ 𝑡∫ 0

∫ = 𝐶1

0

𝐿

0

0

𝐿

𝑣.sgn 𝑣 𝑑𝑥𝑑𝑠 = 𝐶1

∫ 𝑡∫ 0

0

0

𝐿

0

0

𝑥∈[0,𝐿]

(𝑟 sgn 𝑣)𝑠 𝑑𝑥𝑑𝑠

(𝑟 sgn 𝑣 − 𝑟0 sgn 𝑣0 ) 𝑑𝑥 ≤ 𝐶1 .

(1.2.35)

10


From (1.2.33)–(1.2.35), we obtain 0 < 𝐶1−1 ≤ 𝐷(𝑥, 𝑡) ≤ 𝐶1 ,

∀𝑥 ∈ [0, 𝐿],

∀𝑡 > 0.

By the definition of 𝐵(𝑥, 𝑡), (1.2.1) and (1.2.13), we can get { } ∫ 𝑡∫ 𝐿 { 𝑅𝛼 (𝑡 − 𝑠) } 𝐵(𝑥, 𝑠) 𝑅 1 , = exp − ∗ 𝜃(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ exp − 𝐵(𝑥, 𝑡) 𝛽𝑢 𝑠 0 𝛽𝑢∗ Similarly,

𝐵(𝑥, 𝑠) ≥ 𝐶1 𝑒−𝐶1 (𝑡−𝑠) , 𝐵(𝑥, 𝑡)

(1.2.36)

(𝑡 ≥ 𝑠 ≥ 0).

𝑡 ≥ 𝑠 ≥ 0.

(1.2.37)

Therefore, it follows from Lemmas 1.2.1–1.2.3 that 1 ≤ 𝐵(𝑥, 𝑡) ≤ 𝑒𝐶1 𝑡 ,

∀𝑡 > 0.

(1.2.38)

By the Hölder inequality and (1.2.13), we have ∫ 𝐿 1 1 1 2 2 𝜃− 2 (𝑥, 𝑡) ∣ 𝜃𝑥 (𝑥, 𝑡) ∣ 𝑑𝑥 ∣ 𝜃 (𝑥, 𝑡) − 𝜃 (𝑎(𝑡), 𝑡) ∣ ≤ 𝐶1 0

{∫ ≤ 𝐶1

0

{∫ ≤ 𝐶1

𝐿

𝐿

0

} 12 (∫ ) 12 𝐿 𝜃𝑥2 (𝑥, 𝑡)𝑑𝑥 𝜃𝑢𝑑𝑥 𝑢𝜃2 0 } 12 𝜃𝑥2 1 (𝑥, 𝑡)𝑑𝑥 sup 𝑢 2 (𝑥, 𝑡) 𝑢𝜃2 𝑥∈[0,𝐿]

which, together with (1.2.13), implies ∫ 𝐿 2 𝛼1 𝜃𝑥 − 𝐶1−1 sup 𝑢(⋅, 𝑡) 𝑑𝑥 ≤ 𝜃(𝑥, 𝑡) 2 2 𝑥∈[0,𝐿] 0 𝑢𝜃 ∫ 𝐿 2 𝜃𝑥 ≤ 2𝛼2 + 𝐶1 sup 𝑢(⋅, 𝑡) 𝑑𝑥, 2 𝑥∈[0,𝐿] 0 𝑢𝜃

∀(𝑥, 𝑡) ∈ [0, 𝐿] × [0, +∞). (1.2.39)

From (1.2.25), (1.2.36) and (1.2.38), we derive ∫ ∫ 𝑡{ 1 + sup 𝑢(⋅, 𝑠) 𝑢(𝑥, 𝑡) ≤ 𝐶1 + 𝐶1 𝑥∈[0,𝐿]

0

∫ ≤ 𝐶1 + 𝐶1

𝑡

sup 𝑢(⋅, 𝑠)

0 𝑥∈[0,𝐿]

∫ 0

0

𝐿

𝐿

} 𝑡−𝑠 𝜃𝑥2 (𝑥, 𝑠)𝑑𝑥 𝑒− 𝐿 𝑑𝑠 𝑢𝜃2

𝜃𝑥2 (𝑥, 𝑠)𝑑𝑥𝑑𝑠. 𝑢𝜃2

By (1.2.8) and Gronwall’s inequality, we obtain 𝑢(𝑥, 𝑡) ≤ 𝑢¯,

∀(𝑥, 𝑡) ∈ [0, 𝐿] × [0, +∞).


11

From (1.2.7), (1.2.25), (1.2.36), (1.2.38) and (1.2.39), we have ∫ 𝑅𝐷(𝑥, 𝑡) 𝑡 𝜃(𝑥, 𝑠)𝐵(𝑥, 𝑠) 𝑑𝑠 𝑢(𝑥, 𝑡) ≥ 𝛽 0 𝐷(𝑥, 𝑠)𝐵(𝑥, 𝑡) } { ∫ 𝐿 2 ∫ 𝑡 𝜃𝑥 𝛼1 −1 ≥ 𝐶1 − 𝐶1 sup 𝑢(⋅, 𝑠) (𝑥, 𝑠)𝑑𝑥 𝑒−𝐶1 (𝑡−𝑠) 𝑑𝑠 2 2 𝑥∈[0,𝐿] 0 0 𝑢𝜃 ∫ +∞ ∫ 𝐿 2 ) 𝐶1 𝑡 𝜃𝑥 𝛼1 ( 1 − 𝑒−𝐶1 𝑡 − 𝐶1−1 𝑒− 2 ≥ (𝑥, 𝑠)𝑑𝑥𝑑𝑠 2 2 𝑢𝜃 0 0 ∫ 𝑡∫ 𝐿 2 𝜃𝑥 −1 − 𝐶1 (𝑥, 𝑠)𝑑𝑥𝑑𝑠. 2 𝑡 𝑢𝜃 0 2 Note that

∫ 𝑡∫ 𝑡 2

𝐿

0

𝜃𝑥2 (𝑥, 𝑠)𝑑𝑥𝑑𝑠 −→ 0 as 𝑡 −→ +∞; 𝑢𝜃2

therefore there is a sufficiently large time 𝑇0 > 0 such that when 𝑡 > 𝑇0 , 𝑢(𝑥, 𝑡) ≥ 𝛼1 > 0. When 𝑡 ∈ [0, 𝑇0 ], 4 𝑢(𝑥, 𝑡) ≥ Taking 𝑢 = min

{𝛼

1

4

𝐷(𝑥, 𝑡) ≥ 𝐶1−1 𝑒−𝐶1 𝑡 ≥ 𝐶1−1 𝑒−𝐶1 𝑇0 . 𝐵(𝑥, 𝑡)

, 𝐶1−1 𝑒−𝐶1 𝑇0

}

(1.2.40)

> 0, we obtain (1.2.32).

□

Lemma 1.2.6. Under the conditions of Theorem 1.2.1, we have ∫

𝐿

0

(𝜃2 + 𝑣 4 )(𝑥, 𝑡)𝑑𝑥 +

∫ 𝑡∫ 0

0

𝐿

(𝑣 2 𝑣𝑥2 + 𝜃𝑥2 )(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

∀𝑡 > 0.

(1.2.41)

Proof. From equations (1.1.9)–(1.1.11), we obtain (

) 𝑣2 𝐶𝑣 𝜃 + 2 𝑡 ( 2𝑛−2 ) ] 𝑟 𝜃𝑥 1 [ 𝑛−1 𝛽(𝑟 =𝑘 + 𝑣)𝑥 − 𝑅𝜃 (𝑟𝑛−1 𝑣)𝑥 − 2𝜇(𝑛 − 1)(𝑟𝑛−2 𝑣 2 )𝑥 𝑢 𝑢 [𝑥 𝑛−1 ] 𝜃 𝑣)𝑥 (𝑟 𝑛−1 −𝑅 +𝑔+𝑟 𝑣 𝛽 + 𝑣𝑓 𝑢 𝑢 𝑥 [ ( 2𝑛−2 ) ( ] ) 𝑟 𝛽(𝑟𝑛−1 𝑣)𝑥 𝜃 𝜃𝑥 𝑛−1 𝑛−2 2 −𝑅 = 𝑘 𝑣) 𝑣 )) + (𝑟 − 2𝜇(𝑛 − 1)(𝑟 𝑢 𝑢 𝑢 𝑥 + 𝑔 + 𝑣𝑓.

(1.2.42)

12


𝑣2 Multiplying (1.2.42) by 𝐶𝑣 𝜃 + , then integrating the result over [0, 𝐿] × [0, 𝑡], 2 we have )2 ∫ ( 1 𝐿 𝑣2 (𝑥, 𝑡)𝑑𝑥 𝐶𝑣 𝜃 + 2 0 2 ∫ ∫ ∫ 𝑡∫ 𝐿 𝐶𝑣 𝑘 𝑡 𝐿 𝑟2𝑛−2 𝜃𝑥2 ≤ 𝐶1 − 𝑑𝑥𝑑𝑠 + 𝐶1 (𝑟2𝑛−2 𝑣𝑥2 𝑣 2 + 𝑣 4 + 𝜃2 𝑣 2 )𝑑𝑥𝑑𝑠 2 0 0 𝑢 0 0 ) ∫ 𝑡∫ 𝐿( 𝑣2 + 𝐶𝑣 𝜃 + (𝑔 + 𝑣𝑓 )𝑑𝑥𝑑𝑠, (1.2.43) 2 0 0 where ∫ 𝑡∫

( ) ) ∫ 𝐿( ∫ 𝑡 𝑣2 𝑣2 ∥𝑔∥𝐿∞ [0,𝐿] 𝐶𝑣 𝜃 + 𝐶𝑣 𝜃 + 𝑔𝑑𝑥𝑑𝑠 ≤ 𝑑𝑥𝑑𝑠 ≤ 𝐶1 , 2 2 0 0 0 0 (1.2.44) ) ∫ 𝑡∫ 𝐿 ∫ 𝑡 ∫ 𝐿 ( 2 𝑣 (𝜃2 𝑣 2 + 𝑓 2 )𝑑𝑥𝑑𝑠 𝐶𝑣 𝜃 + 𝑣𝑓 𝑑𝑥𝑑𝑠 ≤ 2 0 0 0 0 ∫ 𝐿 ∫ 𝑡 sup ∣𝑓 ∣ (𝑣 4 + 𝑣 2 )𝑑𝑥𝑑𝑠. (1.2.45) +𝐶 𝐿

0 𝑥∈[0,𝐿]

0

Multiplying (1.1.10) by 𝑣 3 , then integrating the result over [0, 𝐿] × [0, 𝑡], we get ( ) ∫ 𝐿∫ 𝑡 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 𝑣 3 𝑣𝑡 𝑑𝑠𝑑𝑥 = 𝑟𝑛−1 𝛽 𝑣 3 𝑑𝑥𝑑𝑠 + 𝑣 3 𝑓 𝑑𝑥𝑑𝑠. 𝑢 𝑢 𝑥 0 0 0 0 0 0 Integrating by parts and using the mean value theorem, we obtain ∫ ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 1 𝐿 4 𝑣 𝑑𝑥 ≤ 𝐶1 − 𝐶1 𝑣 2 𝑣𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1 𝜃2 𝑣 2 𝑑𝑥𝑑𝑠 4 0 0 0 0 0 ∫ 𝐿 ∫ 𝑡 + 𝐶1 sup ∣𝑓 ∣ (𝑣 4 + 𝑣 2 )𝑑𝑥𝑑𝑠 0 𝑥∈[0,𝐿]

≤ 𝐶1 − 𝐶1 ∫ + 𝐶1

∫ 𝑡∫

𝑡

0

𝐿

0

0

𝑣 2 𝑣𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1 ∫

sup ∣𝑓 ∣

0 𝑥∈[0,𝐿]

0

Combining (1.2.43)–(1.2.46), we can get ∫ 𝑡∫ 𝐿 ∫ 𝐿 2 4 (𝜃 + 𝑣 )𝑑𝑥 + (𝑣 2 𝑣𝑥2 + 𝜃𝑥2 )𝑑𝑥𝑑𝑠 0 0 0 )∫ ∫ ( ≤ 𝐶1 + 𝐶1

𝑡

0

2

sup 𝑣 + sup ∣𝑓 ∣

𝑥∈[0,𝐿]

𝑥∈[0,𝐿]

𝐿

0

∫ 𝑡∫ 0

0

𝐿

𝜃2 𝑣 2 𝑑𝑥𝑑𝑠

𝑣 4 𝑑𝑥𝑑𝑠.

(1.2.46)

(1.2.47) 𝐿

2

4

(𝜃 + 𝑣 )𝑑𝑥𝑑𝑠 + 𝐶1

∫ 𝑡∫ 0

0

𝐿

𝑣 4 𝑑𝑥𝑑𝑠.


13

By Gronwall’s inequality and noting that )2 ∫ (∫ ∫ 𝑡

0 𝑥∈[0,𝐿]

∫ 𝑡∫ 0

𝑡

sup 𝑣 2 𝑑𝑠 ≤

0

𝐿

𝑣 4 𝑑𝑥𝑑𝑠 ≤

𝐿

0

∫

0 𝑡

∣𝑣𝑥 ∣𝑑𝑥

sup 𝑣 2

𝑑𝑠 ≤ 𝐶1

(∫

0 𝑥∈[0,𝐿]

0

𝐿

)

∫ 𝑡∫ 0

𝐿

0

𝑣𝑥2 𝑑𝑥 𝑢𝜃

∫

𝐿 0

𝑢𝜃𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

𝑣 2 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 ,

we can obtain (1.2.41).

□

Lemma 1.2.7. Under the conditions of Theorem 1.2.1, we have ∫ 𝑡∫ 𝐿 ∫ 𝐿 2 𝑢𝑥 (𝑥, 𝑡)𝑑𝑥 + (𝑣𝑥2 + 𝑢2𝑥 + 𝜃𝑢2𝑥 )(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 , ∀𝑡 > 0. 0

0

0

(1.2.48)

Proof. Multiplying (1.1.10) by 𝑣, then integrating the result over [0, 𝐿] × [0, 𝑡], we can get ∫ 𝑡∫ 𝐿 ∫ ∫ 1 𝐿 2 1 𝐿 2 (𝑟𝑛−1 𝑣)2𝑥 𝑑𝑥𝑑𝑠 𝑣 𝑑𝑥 = 𝑣0 𝑑𝑥 − 𝛽 2 0 2 0 𝑢 0 0 ∫ 𝑡∫ 𝐿 ( ) ∫ 𝑡∫ 𝐿 𝜃 − 𝑅 (𝑟𝑛−1 𝑣)𝑑𝑥𝑑𝑠 + 𝑣𝑓 𝑑𝑥𝑑𝑠. 𝑢 𝑥 0 0 0 0 By Young’s inequality and the embedding theorem, we have for any 𝜀 > 0, ∫ ∫ ∫ 1 𝐿 2 𝑎2𝑛−2 𝑡 𝐿 2 𝑣 𝑑𝑥 + 𝛽 𝑣𝑥 𝑑𝑥𝑑𝑠 2 0 𝑢 0 0 ) ∫ 𝑡∫ 𝐿( ∫ 𝑡∫ 𝐿 𝜃2 ≤ 𝐶1 + 𝐶1 (𝜀) 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 𝜃2 𝑣 2 + 𝑥2 + 𝜃𝑣 2 𝑑𝑥𝑑𝑠 + 𝐶1 𝜀 𝑢𝜃 0 0 0 0 ∫ 𝑡 ∫ 𝑡∫ 𝐿 ∫ 𝐿 sup ∣𝑣∣2 (𝜃2 + 𝜃)𝑑𝑥𝑑𝑠 + 𝐶1 𝜀 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 (𝜀) 0 𝑥∈[0,𝐿]

≤ 𝐶1 + 𝐶1 𝜀

∫ 𝑡∫ 0

𝐿 0

0

0

0

𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠.

From (1.1.9) and (1.1.10), we derive ( ) (𝑢 ) 𝜃 𝑢𝑥 𝑢𝑡 𝑢𝑡𝑥 𝑥 1−𝑛 −𝛽 2 =𝑟 𝛽 =𝛽 (𝑣𝑡 − 𝑓 ) + 𝑅 . 𝑢 𝑡 𝑢 𝑢 𝑢 𝑥 Multiplying (1.2.50) by ∫

(1.2.49)

(1.2.50)

𝑢𝑥 , then integrating the result over [0, 𝐿] × [0, 𝑡], we have 𝑢

∫ 𝑡 ( )( ) 𝑢𝑥 𝑢𝑥 𝛽 𝑑𝑠𝑑𝑥 𝑢 𝑢 𝑡 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 2 ∫ 𝑡∫ 𝐿 𝜃𝑥 𝑢 𝑥 𝜃𝑢𝑥 𝑢𝑥 𝑑𝑥𝑑𝑠 − 𝑅 = 𝑟1−𝑛 (𝑣𝑡 − 𝑓 ) 𝑑𝑥𝑑𝑠 + 𝑅 𝑑𝑥𝑑𝑠, 𝑢 𝑢 𝑢 𝑢3 0 0 0 0 0 0 𝐿

14


where ∫ 𝑡∫ 𝑅 0

√ 𝜃𝑥 𝜃𝑢𝑥 √ √ 𝑑𝑥𝑑𝑠 𝑢 𝑢 𝑢𝜃 0 0 ∫ 𝑡∫ 𝐿 2 ∫ ∫ 𝜃𝑢𝑥 𝑅 𝑡 𝐿 𝑅 ≤ 𝑑𝑥𝑑𝑠 + 2 0 0 𝑢3 2 0 0 ∫ 𝑡∫ 𝐿 2 ∫ ∫ 𝑅 𝜃𝑢𝑥 𝑅 𝑡 𝐿 ≤ 𝑑𝑥𝑑𝑠 + 2 0 0 𝑢3 4 0 0

𝐿

𝜃𝑥 𝑢 𝑥 𝑑𝑥𝑑𝑠 = 𝑅 𝑢2

0

∫ 𝑡∫

𝐿

(1.2.51) 𝜃𝑥2 𝑑𝑥𝑑𝑠 𝑢𝜃 ) ( 𝜃𝑥2 1 1 + 2 𝑑𝑥𝑑𝑠. 𝑢 𝜃

Using the mean value theorem and (1.2.51), we have ∫ ∫ ∫ 𝑅 𝑡 𝐿 𝜃𝑢2𝑥 𝛽 𝐿 ( 𝑢𝑥 )2 𝑑𝑥 + 𝑑𝑥𝑑𝑠 2 0 𝑢 2 0 0 𝑢3 ) ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 2 ( 𝑢𝑥 𝜃𝑥 1 ≤ 𝐶1 + 𝑟1−𝑛 𝑣𝑡 𝑑𝑥𝑑𝑠 + 𝐶1 1 + 2 𝑑𝑥𝑑𝑠 𝑢 𝑢 𝜃 0 0 0 0 ∫ 𝑡∫ 𝐿 𝑢𝑥 (1.2.52) 𝑟1−𝑛 𝑓 𝑑𝑥𝑑𝑠, − 𝑢 0 0 where ∫ 𝑡 ∫ 𝐿 𝑢𝑥 (1.2.53) 𝑟1−𝑛 𝑣𝑡 𝑑𝑥𝑑𝑠 𝑢 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 ∫ 𝐿 (𝑢 ) 𝑢𝑥 𝑢𝑥 𝑡 𝑡 = 𝑟1−𝑛 𝑣 𝑑𝑥 + (𝑛 − 1) 𝑟−𝑛 𝑣 2 𝑑𝑥𝑑𝑠 − 𝑟1−𝑛 𝑣 𝑑𝑥𝑑𝑠 𝑢 0 𝑢 𝑢 𝑥 0 0 0 0 0 (∫ ( ) ) ∫ ∫ 𝑡 𝐿 𝛽 𝐿 ( 𝑢𝑥 )2 1 𝑑𝑥 + 𝐶1 sup 𝑣 2 𝑢2𝑥 + 2 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 + 4 0 𝑢 𝑢 0 𝑥∈[0,𝐿] 0 ∫ 𝑡∫ 𝐿 2 𝑣 2 𝑑𝑥𝑑𝑠 + 𝑢 0 0 𝑥 ∫ ∫ 𝑡 ∫ 𝐿 ∫ ∫ 𝛽 𝐿 ( 𝑢𝑥 )2 2 𝑡 𝐿 2 2 2 ≤ 𝐶1 + 𝑑𝑥 + 𝐶1 sup 𝑣 𝑢𝑥 𝑑𝑥𝑑𝑠 + 𝑣 𝑑𝑥𝑑𝑠, 4 0 𝑢 𝑢 0 0 𝑥 0 𝑥∈[0,𝐿] 0 ∫ 𝑡∫ − 0

𝐿

0

≤ 𝐶1 + 𝐶1

𝑟

1−𝑛

∫

𝑡

∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 𝑢𝑥 2 𝑓 𝑑𝑥𝑑𝑠 ≤ 𝐶1 ∣𝑓 ∣𝑢𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 ∣𝑓 ∣𝑑𝑥𝑑𝑠 𝑢 0 0 0 0 ∫ 𝐿 sup ∣𝑓 ∣ 𝑢2𝑥 𝑑𝑥𝑑𝑠.

0 𝑥∈[0,𝐿]

0

(1.2.54)

Combining (1.2.51)–(1.2.54), and using Lemmas 1.2.2, 1.2.5 and 1.2.6, we get ∫ 𝐿 ∫ 𝑡∫ 𝐿 𝑅 𝛽 2 𝑢 𝑑𝑥 + 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 (1.2.55) 4(𝑢)2 0 𝑥 2(𝑢)3 0 0 ) ( ) ( ∫ 𝐿 ∫ 𝑡 ∫ ∫ 2 𝑡 𝐿 2 ≤ 𝐶1 + 𝐶1 𝑢2𝑥 𝑑𝑥 𝑑𝑠 + 𝑣 𝑑𝑥𝑑𝑠. sup 𝑣 2 (⋅, 𝑡) + sup ∣𝑓 ∣ 𝑢 0 0 𝑥 𝑥∈[0,𝐿] 𝑥∈[0,𝐿] 0 0


Multiplying (1.2.55) by smallness of 𝜀, we derive ∫

𝐿

0

𝑢2𝑥 𝑑𝑥 +

∫ 𝑡∫ 0

≤ 𝐶1 + 𝐶1

15

𝑢𝛽𝑎2𝑛−2 , then adding the result up to (1.2.49), by the 4𝑢

𝐿

0

∫ 𝑡(

(𝑣𝑥2 + 𝜃𝑢2𝑥 )𝑑𝑥𝑑𝑠

)∫

2

sup 𝑣 (𝑥, 𝑡) + sup ∣𝑓 ∣

𝑥∈[0,𝐿]

0

(1.2.56)

𝑥∈[0,𝐿]

𝐿 0

𝑢2𝑥 𝑑𝑥𝑑𝑠,

∀𝑡 > 0.

Applying Gronwall’s inequality to (1.2.56), we get ∫

𝐿

0

𝑢2𝑥 𝑑𝑥

+

∫ 𝑡∫ 0

𝐿

0

(𝑣𝑥2 + 𝜃𝑢2𝑥 )𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

∀𝑡 > 0.

(1.2.57)

Multiplying (1.2.39) by 𝑢2𝑥 , then integrating the result over [0, 𝑡]×[0, 𝐿], by (1.2.7), (1.2.32) and (1.2.57), we have ) (∫ ) 𝐿 𝜃𝑥2 2 ≤ + 𝐶1 𝑑𝑥 𝑢𝑥 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 . 2 0 0 0 0 0 0 𝑢𝜃 0 (1.2.58) Combining (1.2.57) and (1.2.58), we obtain (1.2.48). □ 𝛼1 2

∫ 𝑡∫

𝐿

𝑢2𝑥 𝑑𝑥𝑑𝑠

∫ 𝑡∫

𝐿

𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠

∫ 𝑡 (∫

𝐿

Lemma 1.2.8. Under the conditions of Theorem 1.2.1, we have ∫ 0

𝐿

𝑣𝑥2 (𝑥, 𝑡)𝑑𝑥

+

∫ 𝑡∫ 0

𝐿

0

∣𝑣(𝑥, 𝑡)∣ ≤ 𝐶1 ,

𝑣𝑡2 (𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

∀𝑡 > 0,

∀(𝑥, 𝑡) ∈ [0, 𝐿] × [0, +∞).

(1.2.59) (1.2.60)

Proof. By the embedding theorem, we have sup 𝜃(𝑥, 𝑡) ≤ 𝐶1 + 𝐶1 ∥𝜃𝑥 (𝑡)∥,

𝑥∈[0,𝐿]

∀𝑡 > 0.

(1.2.61)

Multiplying (1.1.10) by 𝑣𝑡 , integrating the result over [0, 𝐿] × [0, 𝑡], using the estimate (𝑟𝑛−1 𝑣𝑡 )𝑥 = (𝑟𝑛−1 𝑣)𝑥𝑡 − (𝑛 − 1)(𝑟𝑛−2 𝑣 2 )𝑥 ,

∣(𝑟𝑛−2 𝑣 2 )𝑥 ∣ ≤ 𝐶1 (𝑣 2 + 𝑣𝑥2 ),

we get ∫ 𝑡∫ 0

0

𝐿

𝑣𝑡2 𝑑𝑥𝑑𝑠

=

∫ 𝑡∫ 0

0

𝐿

𝑟

𝑛−1

( ) ∫ 𝑡∫ 𝐿 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 𝑣𝑡 𝛽 𝑑𝑥𝑑𝑠 + 𝑣𝑡 𝑓 𝑑𝑥𝑑𝑠, 𝑢 𝑢 𝑥 0 0 (1.2.62)

16


∫ 𝑡∫ 𝐿 𝛽 𝑛−1 𝛽 ∥(𝑟𝑛−1 𝑣)𝑥 ∥2 ≤ 𝐶1 + (𝑛 − 1) (𝑟 𝑣)𝑥 (𝑟𝑛−2 𝑣 2 )𝑥 𝑑𝑥𝑑𝑠 𝑢 𝑢 0 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿( ) 𝜃 + 𝑣𝑡 𝑓 𝑑𝑥𝑑𝑠 − 𝑅 (𝑟𝑛−1 𝑣𝑡 )𝑑𝑥𝑑𝑠 𝑢 𝑥 0 0 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 (∣𝑣∣ + ∣𝑣𝑥 ∣)(𝑣 2 + 𝑣𝑥2 )𝑑𝑥𝑑𝑠 + 𝐶1 𝑓 2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1

∫

𝑡

∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 +

+

∫

1 2

0

𝑡

0

∫ 𝑡∫

∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 + 𝐶1

0

0

𝐿

0

0

(𝜃𝑥2 + 𝜃2 𝑢2𝑥 )𝑑𝑥𝑑𝑠

where ∫ 𝑡∫ 0

𝐿 0

𝜃2 𝑢2𝑥 𝑑𝑥𝑑𝑠

∫ ≤

𝑡

≤

𝑡

0

≤ 𝐶1

𝐼≜

∫ 𝑡∫ 0

≤ 𝐶1

∫

0 𝑡

0

∫ ≤ 𝐶1

0

∫ ≤ 𝐶1

𝐿

sup 𝜃

∫ 𝑡∫ 0

𝐿

𝐿

0

𝑢2𝑥 𝑑𝑥𝑑𝑠

0

(1.2.63)

) 𝑢2𝑥 𝑑𝑥

𝑑𝑠

(∫

𝐿

0

+ 𝐶1

) 𝑢2𝑥 𝑑𝑥 𝑑𝑠

∫ 𝑡∫ 0

𝐿 0

𝜃𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

(∣𝑣∣ + ∣𝑣𝑥 ∣)(𝑣 2 + 𝑣𝑥2 )𝑑𝑥𝑑𝑠

) ( (∥𝑣∥𝐿∞ + ∥𝑣𝑥 ∥𝐿∞ ) ∥𝑣∥2 + ∥𝑣𝑥 ∥2 𝑑𝑠 [

𝑡

( ) ∥𝑣∥2𝐿∞ + ∥𝑣𝑥 ∥2𝐿∞ (𝑠)𝑑𝑠

] ∥𝑣∥𝐿∞ ∥𝑣∥2 + ∥𝑣∥𝐿∞ ∥𝑣𝑥 ∥2 + ∥𝑣𝑥 ∥𝐿∞ ∥𝑣∥2 + ∥𝑣𝑥 ∥𝐿∞ ∥𝑣𝑥 ∥2 (𝑠)𝑑𝑠

(∫

+ 𝐶1 (∫ + 𝐶1

(∫

( ) 𝐶1 + 𝐶1 ∥𝜃𝑥 ∥2

𝑡

0

2

0 𝑥∈[0,𝐿]

∫

0

𝑡 0 𝑡 0

∥𝑣(𝑠)∥2𝐿∞ 𝑑𝑠

) 12 (∫

∥𝑣𝑥 (𝑠)∥2𝐿∞ 𝑑𝑠

2

∥𝑣𝑥 (𝑠)∥ 𝑑𝑠

0

) 12 (∫

≤ 𝐶1 + 𝐶1 sup ∥𝑣𝑥 (𝑠)∥ + 𝐶1 𝑠∈[0,𝑡]

𝑡

𝑡

0

(∫

0

𝑡

) 12

∥𝑣𝑥 (𝑠)∥2 𝑑𝑠

⋅ sup ∥𝑣𝑥 (𝑠)∥

) 12

∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠

𝑠∈[0,𝑡]

⋅ sup ∥𝑣𝑥 (𝑠)∥

) 12

𝑠∈[0,𝑡]

⋅ sup ∥𝑣𝑥 (𝑠)∥. 𝑠∈[0,𝑡]

(1.2.64)

By (1.1.10), we have [ ∥𝑣𝑥𝑥 ∥ ≤ 𝐶1 ∥𝑣𝑡 ∥ + ∥(𝑟𝑛−1 )𝑥𝑥 𝑣∥ + ∥(𝑟𝑛−1 )𝑥 𝑣𝑥 ∥ + ∥𝜃𝑥 ∥ + ∥𝑢𝑥 ∥ + ∥𝑢𝑥 𝑢𝑣∥ ] + ∥𝑢𝑥 𝑣𝑥 ∥ + ∥𝜃𝑢𝑥 ∥ + ∥𝑓 ∥


17

( ≤ 𝐶1 ∥𝑣𝑡 ∥ + ∥𝑣∥𝐿∞ ∥𝑢𝑥 ∥ + ∥𝑣𝑥 ∥ + ∥𝜃𝑥 ∥ + ∥𝑢𝑥 ∥ + ∥𝑣𝑥 ∥𝐿∞ ∥𝑢𝑥∥ ) + ∥𝜃𝑢𝑥 ∥ + ∥𝑓 ∥ ) ( 1 ≤ ∥𝑣𝑥𝑥 ∥ + 𝐶1 ∥𝑣𝑡 ∥ + ∥𝑣∥𝐿∞ + ∥𝑣𝑥 ∥ + ∥𝜃𝑥 ∥ + ∥𝑢𝑥∥ + ∥𝜃𝑢𝑥 ∥ + ∥𝑓 ∥ , 2 i.e., ∥𝑣𝑥𝑥 ∥ ≤ 𝐶1 (∥𝑣𝑡 ∥ + ∥𝑣∥𝐿∞ + ∥𝑣𝑥 ∥ + ∥𝜃𝑥 ∥ + ∥𝑢𝑥 ∥ + ∥𝜃𝑢𝑥 ∥ + ∥𝑓 ∥).

(1.2.65)

Combining (1.2.64) and (1.2.65), using Young’s inequality, we have { ) 12 } (∫ 𝑡 𝐼 ≤ 𝐶1 sup ∥𝑣𝑥 (𝑠)∥ + 𝐶1 1 + ∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 sup ∥𝑣𝑥 (𝑠)∥ 𝑠∈[0,𝑡]

≤ 𝜀 sup ∥𝑣𝑥 (𝑠)∥2 + 𝜀 𝑠∈[0,𝑡]

∫ 0

𝑠∈[0,𝑡]

0

𝑡

∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 + 𝐶1 𝜀−1 ,

∀ 𝜀 > 0.

(1.2.66)

Combining (1.2.64) and (1.2.66), picking 𝜀 small enough, we derive (1.2.59). By the embedding theorem, (1.2.7) and (1.2.59), we have ∣𝑣(𝑥, 𝑡)∣ ≤ 𝐶1 ∥𝑣∥𝐿∞ [0,𝐿] ≤ 𝐶1 ∥𝑣∥𝐻 1 [0,𝐿] ≤ 𝐶1 . Lemma 1.2.9. Under the conditions of Theorem 1.2.1, we have ∫ 𝑡∫ 𝐿 ∫ 𝑡 2 ∥𝑣𝑥 (𝑠)∥2𝐿∞ 𝑑𝑠 + (𝑢2𝑥𝑡 + 𝑣𝑥𝑥 )(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 , ∀𝑡 > 0. 0

0

0

□

(1.2.67)

Proof. By (1.1.9) and (1.1.10), we deduce after a direct calculation that ∫ 𝑡∫ 𝐿 2 (𝑢2𝑥𝑡 + 𝑣𝑥𝑥 )𝑑𝑥𝑑𝑠 0

0

∫ 𝑡∫

𝐿

( 𝑛−1 2 ) (𝑟 𝑣)𝑥𝑥 + 𝑣𝑡2 + 𝑣𝑥2 𝑢2𝑥 + 𝜃𝑥2 + 𝜃2 𝑢2𝑥 + 𝑓 2 𝑑𝑥𝑑𝑠 0 0 ) (∫ ]2 ∫ 𝑡 ∫ 𝐿 [ 𝑛−1 ∫ 𝑡 𝐿 (𝑟 𝑣)𝑥 2 2 ≤ 𝐶1 + 𝐶1 𝑑𝑥𝑑𝑠 + 𝐶1 sup 𝜃 𝑢𝑥 𝑑𝑥 𝑑𝑠 𝑢 0 0 0 𝑥∈[0,𝐿] 0 𝑥 ∫ 𝑡∫ 𝐿 ≤ 𝐶1 + 𝐶1 (𝑣𝑡2 + 𝜃𝑥2 + 𝜃2 𝑢2𝑥 + 𝑓 2 )𝑑𝑥𝑑𝑠

≤ 𝐶1

0

0

≤ 𝐶1 . By the embedding theorem, we have ∫ 𝑡∫ ∫ 𝑡 ∥𝑣𝑥 (𝑠)∥2𝐿∞ 𝑑𝑠 ≤ 𝐶 0

0

0

𝐿

2 𝑣𝑥𝑥 (𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 .

By (1.2.7), (1.2.41), (1.2.48), (1.2.59) and (1.1.16), we obtain (1.2.67).

□

18


Lemma 1.2.10. Under the conditions of Theorem 1.2.1, we have ∫

𝐿

0

𝜃𝑥2 (𝑥, 𝑡)𝑑𝑥 +

∫ 𝑡∫ 0

𝐿

0

2 (𝜃𝑡2 + 𝜃𝑥𝑥 )(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

∀𝑡 > 0,

𝜃(𝑥, 𝑡) ≤ 𝐶1 , ∀(𝑥, 𝑡) ∈ [0, 𝐿] × [0, +∞).

(1.2.68) (1.2.69)

Proof. Multiplying (1.1.11) by 𝜃𝑡 , then integrating the resultant over [0, 𝐿] × [0, 𝑡], we have ∫ 𝑡 ∫ 𝐿 ( 2𝑛−2 ) ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 𝑟 𝜃𝑥 𝛽 𝑛−1 2 (𝑟 𝐶𝑣 𝜃𝑡2 𝑑𝑥𝑑𝑠 = 𝑘 𝜃𝑡 𝑑𝑥𝑑𝑠 + 𝑣)𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 𝑢 0 0 0 0 0 0 𝑢 𝑥 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 + 𝜃𝑡 𝑔𝑑𝑥𝑑𝑠 − 2𝜇(𝑛 − 1) (𝑟𝑛−2 𝑣 2 )𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 0

− where ∫ 𝑡∫

0

∫ 𝑡∫ 0

𝐿

0

0

0

𝜃 𝑅 (𝑟𝑛−1 𝑣)𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 𝑢

(1.2.70)

𝐿

𝛽 𝑛−1 2 (𝑟 𝑣)𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 0 0 𝑢 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 ≤ 𝐶1 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 ∣(𝑟𝑛−1 𝑣)𝑥 ∣2 (𝑣 2 + 𝑣𝑥2 )𝑑𝑥𝑑𝑠 0 0 0 0 )∫ ∫ ∫ ∫ ( ≤ 𝐶1

𝑡

0

𝐿

0

𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1

𝑡

sup ∣𝑣∣2 + sup ∣𝑣𝑥 ∣2

𝑥∈[0,𝐿]

0

𝑥∈[0,𝐿]

0

(1.2.71)

𝐿

∣(𝑟𝑛−1 𝑣)𝑥 ∣2 𝑑𝑥𝑑𝑠,

(

) 𝑟2𝑛−2 𝜃𝑥 𝑘 𝜃𝑡 𝑑𝑥𝑑𝑠 (1.2.72) 𝑢 0 0 𝑥 ∫ 𝑡 ∫ 𝐿 ( 2𝑛−2 ) ∫ 𝑡 ( 2𝑛−2 ) 𝐿 𝑟 𝑟 𝜃𝑥 𝜃𝑥 =𝑘 𝜃𝑡 𝑑𝑠 − 𝑘 𝜃𝑡𝑥 𝑑𝑥𝑑𝑠 𝑢 𝑢 0 0 0 0 ∫ 𝑡 ∫ 𝐿 ( 2𝑛−2 ) ∫ 𝐿 2𝑛−2 𝑟 𝑟 𝜃𝑥 𝑡 𝜃𝑥 𝜃𝑥 𝑑𝑥 + 𝑘 𝜃𝑥 𝑑𝑥𝑑𝑠 = −𝑘 𝑢 𝑢 0 0 0 0 𝑡 ∫ 𝐿 2𝑛−2 𝑟 𝜃𝑥2 𝑑𝑥 ≤ 𝐶1 − 𝑘 𝑢 0 } ∫ 𝑡∫ 𝐿{ (2𝑛 − 2)𝑟2𝑛−3 𝑣𝜃𝑥2 𝑟2𝑛−2 𝜃𝑡𝑥 𝜃𝑥 𝑟2𝑛−2 𝜃𝑥2 𝑢𝑡 + − +𝑘 (𝑥, 𝑠)𝑑𝑥𝑑𝑠, 𝑢 𝑢 𝑢2 0 0 ∫ 𝑡∫

𝐿

− 2𝜇(𝑛 − 1)

∫ 𝑡∫ 0

= −2𝜇(𝑛 − 1)

𝐿

0

∫

0

(𝑟𝑛−2 𝑣 2 )𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠

𝑡

(1.2.73)

∫ 𝑡∫ 𝐿 (𝑟𝑛−2 𝑣 2 )𝜃𝑡 𝑑𝑠 + 2𝜇(𝑛 − 1) 0

0

0

𝐿

(𝑟𝑛−2 𝑣 2 )𝜃𝑡𝑥 𝑑𝑥𝑑𝑠


∫ = 2𝜇(𝑛 − 1)

𝐿

0

𝑡 (𝑟𝑛−2 𝑣 2 )𝜃𝑥 𝑑𝑥

− 2𝜇(𝑛 − 1) ∫ 𝑡∫ −

19

0

∫ 𝑡∫ 0

𝐿

0

[(𝑛 − 2)𝑟𝑛−3 𝑣 3 + 2𝑟𝑛−2 𝑣𝑣𝑡 ]𝜃𝑥 𝑑𝑥𝑑𝑠,

𝜃 (1.2.74) 𝑅 (𝑟𝑛−1 𝑣)𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 𝑢 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 𝐶𝑣 ≤ 𝐶1 𝜃2 𝑣𝑥2 𝑑𝑥𝑑𝑠 + 𝜃2 𝑑𝑥𝑑𝑠 4 0 0 𝑡 0 0 ∫ ∫ ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 𝐶𝑣 𝑡 𝐿 2 2 ¯ 2 𝑣 2 𝑑𝑥𝑑𝑠 + 𝐶1 ≤ 𝐶1 (𝜃 − 𝜃) 𝑣 𝑑𝑥𝑑𝑠 + 𝜃 𝑑𝑥𝑑𝑠 𝑥 𝑥 4 0 0 𝑡 0 0 0 0 ∫ ∫ ∫ 𝑡 ∫ 𝐿 𝐶𝑣 𝑡 𝐿 2 ¯2 ≤ 𝐶1 + 𝐶1 sup (𝜃 − 𝜃) 𝑣𝑥2 𝑑𝑥𝑑𝑠 + 𝜃 𝑑𝑥𝑑𝑠 4 0 0 𝑡 0 𝑥∈[0,𝐿] 0 ∫ ∫ ∫ 𝑡 𝐶𝑣 𝑡 𝐿 2 ≤ 𝐶1 + 𝐶1 ∥𝜃𝑥 (𝑠)∥2 𝑑𝑠 + 𝜃 𝑑𝑥𝑑𝑠 4 0 0 𝑡 0 ∫ ∫ 𝐶𝑣 𝑡 𝐿 2 ≤ 𝐶1 + 𝜃 𝑑𝑥𝑑𝑠, 4 0 0 𝑡

∫ 𝑡 ∫ 0

0

𝐿

𝐿

∫ 𝑡∫ 𝜃𝑡 𝑔𝑑𝑥𝑑𝑠 ≤ 𝐶1 0

0

𝐿

𝑔 2 𝑑𝑥𝑑𝑠 +

𝐶𝑣 4

∫ 𝑡∫ 0

0

𝐿

𝜃𝑡2 𝑑𝑥𝑑𝑠.

(1.2.75)

Combining (1.2.70)–(1.2.75), and using Young’s inequality, (1.2.32), (1.2.36), (1.2.59) and (1.1.17), we can obtain that (1.2.68). (1.2.69) follows from (1.2.68) and (1.2.61). □ Proof of Theorem 1.2.1. Using Lemmas 1.2.1–1.2.10, the embedding theorem and noting that 𝑟𝑥 = 𝑟1−𝑛 𝑢, 𝑟𝑥𝑥 = (1 − 𝑛)𝑟−𝑛 𝑟𝑥 𝑢 + 𝑟1−𝑛 𝑢𝑥 , 𝑟𝑡 = 𝑣, 𝑟𝑡𝑥 = 𝑣𝑥 , 𝑟𝑡𝑥𝑥 = 𝑣𝑥𝑥 , we can obtain (1.2.3).

□

1.3 Global Existence of Solutions in 𝑯 2 For external force 𝑓 (𝑟, 𝑡) and heat source 𝑔(𝑟, 𝑡), besides (1.1.16)–(1.1.17), we suppose that 𝑓 (𝑟, 𝑡), 𝑔(𝑟, 𝑡) are 𝐶 1 functions on [𝑎, 𝑏] × ℝ+ satisfying 𝑓 (𝑟, 𝑡) ∈ 𝐿∞ (ℝ+ , 𝐿2 [𝑎, 𝑏]), 𝑓𝑟 ∈ 𝐿2 (ℝ+ , 𝐿2 [𝑎, 𝑏]), 𝑓𝑡 ∈ 𝐿2 (ℝ+ , 𝐿2 [𝑎, 𝑏]), ∞

+

2

2

+

2

2

+

2

𝑔(𝑟, 𝑡) ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]), 𝑔𝑟 ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]), 𝑔𝑡 ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]).

(1.3.1) (1.3.2)

20


Constant 𝐶2 denotes a generic positive constant depending only on the 𝐻 2 norm of initial data (𝑢0 , 𝑣0 , 𝜃0 ), ∥𝑓 ∥𝐿∞ (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑓𝑟 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑓𝑡 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑔∥𝐿∞ (ℝ+ ,𝐿2 [𝑎,𝑏]) ,

∥𝑔𝑟 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) ,

∥𝑔𝑡 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏])

and constant 𝐶1 , but is independent of 𝑡. Remark 1.3.1. By Remark 1.2.1 and (1.3.1)–(1.3.2), we easily know that assumptions (1.3.1)–(1.3.2) are equivalent to the following conditions: ∀𝑥 ∈ [0, 𝐿], 𝑓 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿∞ (ℝ+ , 𝐿2 [0, 𝐿]), 𝑓𝑟 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿2 (ℝ+ , 𝐿2 [0, 𝐿]), ∞

+

2

𝑔(𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]), 2

+

𝑓𝑡 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿2 (ℝ+ , 𝐿2 [0, 𝐿]), 2

+

(1.3.3)

2

𝑔𝑟 (𝑟(𝑥, 𝑡)) ∈ 𝐿 (ℝ , 𝐿 [0, 𝐿]),

2

𝑔𝑡 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿 (ℝ , 𝐿 [0, 𝐿]).

(1.3.4)

Therefore the generic positive constant 𝐶2 depends only on the norm of initial data (𝑢0 , 𝑣0 , 𝜃0 ) in 𝐻 2 and the norms of 𝑓 and 𝑔 in the class of functions in (1.3.3)–(1.3.4), but is independent of 𝑡. Theorem 1.3.1. Under the conditions (1.1.16), (1.1.17) (1.3.1), (1.3.2), if (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻 2 [0, 𝐿] × 𝐻02 [0, 𝐿] × 𝐻 2 [0, 𝐿], problem (1.1.9)–(1.1.15) admits a unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐶([0, +∞), 𝐻 2 [0, 𝐿] × 𝐻02 [0, 𝐿] × 𝐻 2 [0, 𝐿]) satisfying that for any 𝑡 > 0, ¯ 22 ∥𝑟(𝑡) − 𝑟¯∥2𝐻 3 + ∥𝑟𝑡 (𝑡)∥2𝐻 2 + ∥𝑢(𝑡) − 𝑢 ¯∥2𝐻 2 + ∥𝑣(𝑡) − 𝑣¯∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 2 2 2 2 ¯ + ∥𝑢(𝑡) − 𝑢¯∥𝑊 1,∞ + ∥𝑣(𝑡) − 𝑣¯∥𝑊 1,∞ + ∥𝜃(𝑡) − 𝜃∥𝑊 1,∞ + ∥𝑢𝑡 (𝑡)∥𝐻 1 ∫ 𝑡( + ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑢 − 𝑢 ¯∥2𝐻 2 + ∥𝑢𝑡 ∥2𝐻 2 + ∥𝑣 − 𝑣¯∥2𝐻 3 0 ) 2 2 ¯ + ∥𝜃 − 𝜃∥𝐻 3 + ∥𝑣𝑡 ∥𝐻 1 + ∥𝜃𝑡 ∥2𝐻 1 + ∥𝑟 − 𝑟¯∥2𝐻 3 + ∥𝑟𝑡 ∥2𝐻 3 (𝑠)𝑑𝑠 ≤ 𝐶2 . (1.3.5) The proof of Theorem 1.3.1 can be divided into the following several lemmas. Lemma 1.3.1. Under the conditions of Theorem 1.3.1, we have that for any 𝑡 > 0, ∫ 𝑡 ∥𝑣𝑡 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2𝐿∞ + ∥𝑣𝑡𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 , (1.3.6) 0 ∫ 𝑡 ∥𝜃𝑡𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 . (1.3.7) ∥𝜃𝑡 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2𝐿∞ + 0


21

Proof. Differentiating (1.1.10) with respect to 𝑡, we have ( ( ) ) 𝑑𝑓 𝜃 𝜃 (𝑟𝑛−1 𝑣)𝑥 (𝑟𝑛−1 𝑣)𝑥 𝑛−2 𝑛−1 𝑟𝑡 𝛽 +𝑟 + . −𝑅 −𝑅 𝑣𝑡𝑡 = (𝑛 − 1)𝑟 𝛽 𝑢 𝑢 𝑥 𝑢 𝑢 𝑡𝑥 𝑑𝑡 (1.3.8) Multiplying (1.3.8) by 𝑣𝑡 , then integrating the result over [0, 𝑡] × [0, 𝐿], by 𝑑𝑓 (1.1.9) and Theorem 1.2.1, and noting that = 𝑓𝑟 𝑣 + 𝑓𝑡 , we can derive 𝑑𝑡 ∫ 𝐿 ∫ 𝑡∫ 𝐿 1 𝑣𝑡𝑡 𝑣𝑡 𝑑𝑥𝑑𝑠 ≥ 𝑣 2 𝑑𝑥 − 𝐶2 , (1.3.9) 2 0 𝑡 0 0 ( ) ∫ 𝑡∫ 𝐿 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 𝑣𝑡 𝑟𝑛−1 𝛽 𝑑𝑥𝑑𝑠 𝑢 𝑢 𝑡𝑥 0 0 ( ) ∫ 𝑡∫ 𝐿 (𝑟𝑛−1 𝑣)𝑥 𝑢𝑡 𝜃𝑢𝑡 𝜃𝑡 (𝑟𝑛−1 𝑣)𝑡𝑥 𝑛−1 − = 𝑣𝑡 𝑟 −𝑅 +𝑅 2 𝑑𝑥𝑑𝑠 𝛽 𝑢 𝑢2 𝑢 𝑢 𝑥 0 0 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 2 2 𝑣𝑡𝑥 𝑑𝑥𝑑𝑠 + 𝐶2 (𝜀) 𝑣𝑡𝑥 𝑑𝑥𝑑𝑠 + 𝐶2 (𝑟𝑛−1 𝑣)4𝑥 𝑑𝑥𝑑𝑠 ≤ −𝐶2−1 +

∫ 𝑡∫ 0

≤ 𝐶2 −

[

0

𝐶2−1 ∫

+ 𝐶2

0 𝐿

𝑡

0

0

∫ 𝑡∫ 0

𝐿

0

sup 𝜃

2

2 𝑣𝑡𝑥 𝑑𝑥𝑑𝑠

(∫

0 𝑥∈[0,𝐿]

≤ 𝐶2 − 𝐶2−1

0

] 𝜃𝑡2 + 𝜃2 (𝑟𝑛−1 𝑣)2𝑥 𝑑𝑥𝑑𝑠

∫ 𝑡∫ 0

0

𝐿

0

𝐿

∫ + 𝐶2 )

𝑣𝑥2 𝑑𝑥

𝑡

sup

0

0 𝑥∈[0,𝐿]

𝑣𝑥2

(∫ 0

𝐿

𝑑𝑠

2 𝑣𝑡𝑥 𝑑𝑥𝑑𝑠,

(1.3.10)

𝑛−2

𝑡

≤ 𝐶2 + 𝐶2 ∫ ≤ 𝐶2 + 𝐶2

) 𝑣𝑥2 𝑑𝑥

𝑑𝑠

( ) 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 (𝑛 − 1)𝑟 𝑣𝑣𝑡 𝛽 𝑑𝑥𝑑𝑠 𝑢 𝑢 𝑥 0 0 ∫ 𝑡∫ 𝐿 ( 2 ) 2 𝑣𝑡 + 𝑣𝑥2 𝑢2𝑥 + 𝑣𝑥𝑥 + 𝜃𝑥2 𝑢2𝑥 + 𝜃𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶2 0 0 ) (∫ ∫ ∫

∫ 𝑡∫

𝐿

0

sup 𝑣𝑥2

0 𝑥∈[0,𝐿] 𝑡 0

𝐿

0

𝑡

𝑢2𝑥 𝑑𝑥 𝑑𝑠 + 𝐶2

( ) ∥𝑣𝑥 ∥2 + ∥𝑣𝑥𝑥 ∥2 (𝑠)𝑑𝑠 + 𝐶2

∫ 0

sup 𝜃𝑥2

0 𝑥∈[0,𝐿] 𝑡

(∫ 0

𝐿

) 𝑢2𝑥 𝑑𝑥 𝑑𝑠

( ) ∥𝜃𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 (𝑠)𝑑𝑠

(1.3.11) ≤ 𝐶2 , ∫ ∫ ∫ ∫ ∫ ∫ 𝑡 𝐿 𝑑𝑓 𝑡 𝐿 𝑡 𝐿 𝑣𝑡 𝑑𝑥𝑑𝑠 ≤ 𝐶1 𝑣𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 (𝑓𝑟2 + 𝑓𝑡2 )𝑑𝑥𝑑𝑠 ≤ 𝐶2 . 0 0 𝑑𝑡 0 0 0 0 (1.3.12)

22


Combining (1.3.8)–(1.3.12), we obtain ∫ 𝑡 2 ∥𝑣𝑡𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 ∥𝑣𝑡 (𝑡)∥ +

(1.3.13)

0

which, by noting that from (1.1.10) ∥𝑣𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝑣𝑡 (𝑡)∥ + ∥𝑢𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃(𝑡)∥𝐻 1 + ∥𝑓 ∥) ≤ 𝐶2 , gives

∥𝑣𝑥𝑥 (𝑡)∥ ≤ 𝐶2 .

(1.3.14)

Hence (1.3.6) follows from (1.3.13) by using the embedding theorem. Similarly, differentiating (1.1.11) with respect to 𝑡, we can get ( 2𝑛−2 ) ( ) 𝑟 𝜃 𝜃𝑥 (𝑟𝑛−1 𝑣)𝑥 𝐶𝑣 𝜃𝑡𝑡 = 𝑘 −𝑅 + 𝛽 (𝑟𝑛−1 𝑣)𝑥 𝑢 𝑢 𝑢 𝑡𝑥 𝑡 ( ) 𝜃 (𝑟𝑛−1 𝑣)𝑥 𝑑𝑔 𝑛−1 −𝑅 + 𝛽 𝑣)𝑡𝑥 − 2𝜇(𝑛 − 1)(𝑟𝑛−2 𝑣 2 )𝑡𝑥 + . (1.3.15) (𝑟 𝑢 𝑢 𝑑𝑡 Multiplying (1.3.15) by 𝜃𝑡 , then integrating the result over [0, 𝑡]×[0, 𝐿], and noting that 𝑑𝑔 = 𝑔𝑟 𝑣 + 𝑔𝑡 , 𝑑𝑡 we have ∫ ∫ 𝑡∫ 𝐿 𝐶𝑣 𝐿 2 𝐶𝑣 𝜃𝑡 𝜃𝑡𝑡 𝑑𝑥𝑑𝑠 ≥ 𝜃 𝑑𝑥 − 𝐶2 , (1.3.16) 2 0 𝑡 0 0 ( 2𝑛−2 ) ∫ 𝑡∫ 𝐿 𝑟 𝜃𝑥 𝑘𝜃𝑡 𝑑𝑥𝑑𝑠 𝑢 0 0 𝑡𝑥 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 ∫ 𝑡∫ 𝐿 −1 2 2 ≤ −𝐶2 𝜃𝑡𝑥 𝑑𝑥𝑑𝑠 + 𝐶2 (𝜀) 𝜃𝑡𝑥 𝑑𝑥𝑑𝑠 + 𝐶2 𝜃𝑥2 𝑢2𝑡 𝑑𝑥𝑑𝑠 ≤ −𝐶2−1 ≤ 𝐶2 −

0

0

0

0

∫ 𝑡∫

𝐶2−1

𝐿

2 𝜃𝑡𝑥 𝑑𝑥𝑑𝑠 + 𝐶2

∫ 𝑡∫ 0

0

𝐿

∫

0

𝑡

0

sup 𝑣𝑥2

0 𝑥∈[0,𝐿]

∫ 0

𝐿

0

𝜃𝑥2 𝑑𝑥𝑑𝑠

2 𝜃𝑡𝑥 𝑑𝑥𝑑𝑠,

(1.3.17)

( ) 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 (𝑟𝑛−1 𝑣)𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 𝛽 𝑢 𝑢 0 0 𝑡 ∫ 𝑡∫ 𝐿 2 (𝑣𝑡𝑥 + 𝑣𝑥4 + 𝜃𝑡2 + 𝜃2 𝑣𝑥2 + 𝑣𝑥2 𝜃𝑡2 )𝑑𝑥𝑑𝑠 ≤ 𝐶2 0 0 (∫ ) ∫ ∫ 𝑡 𝐿 2 2 2 ≤ 𝐶2 + 𝐶2 sup (𝑣𝑥 + 𝜃 ) 𝑣𝑥 𝑑𝑥 𝑑𝑠 +

∫ 𝑡∫

0

𝐿

0 𝑥∈[0,𝐿]

0

𝑡

sup

0 𝑥∈[0,𝐿]

𝑣𝑥2

(∫ 0

𝐿

) 𝜃𝑡2 𝑑𝑥

𝑑𝑠


∫ ≤ 𝐶2 + 𝐶2

𝑡

sup 𝑣𝑥2

(∫

0 𝑥∈[0,𝐿]

𝐿

0

23

) 𝜃𝑡2 𝑑𝑥 𝑑𝑠.

(1.3.18)

From (1.1.14) and Lemmas 1.2.1, 1.2.5 and 1.2.7–1.2.10, we derive (𝑟𝑛−1 𝑣)𝑥 𝜃 𝛽 −𝑅 ≤ 𝐶2 , 𝑢 𝑢 𝐿∞

(1.3.19)

(1.3.20) ∥(𝑟𝑛−1 𝑣)𝑡𝑥 ∥ ≤ 𝐶2 (∥𝑣𝑥 ∥ + ∥𝑣𝑡𝑥 ∥), ∫ 𝐿 ∫ 𝐿 ( ) 1 𝛽(𝑟𝑛−1 𝑥)𝑥 − 𝑅𝜃 (𝑟𝑛−1 𝑣)𝑥 𝜃𝑡 𝑑𝑥 ≤ 𝐶1 𝑢 0 0 ≤ 𝐶2 ∥𝑣𝑥 ∥

(1.3.21)

which, with the Poincaré inequality, gives ∫ ∥𝜃𝑡 ∥ ≤

𝐿 0

𝜃𝑡 𝑑𝑥 + 𝐶∥𝜃𝑡𝑥 ∥ ≤ 𝐶2 (∥𝑣𝑥 ∥ + ∥𝜃𝑡𝑥 ∥).

(1.3.22)

Thus from (1.3.19)–(1.3.22) we derive for 𝜀 > 0 small enough, ∫ 𝑡∫ 0

𝐿

0

( ) 𝜃 (𝑟𝑛−1 𝑣)𝑥 −𝑅 𝛽 (𝑟𝑛−1 𝑣)𝑡𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 𝑢 𝑢

≤ 𝐶2 ∥(𝑟𝑛−1 𝑣)𝑡𝑥 ∥∥𝜃𝑡 ∥ ≤ 𝐶2 (∥𝑣𝑥 ∥ + ∥𝑣𝑡𝑥 ∥)(∥𝑣𝑥 ∥ + ∥𝜃𝑡𝑥 ∥) ≤ 𝜀∥𝜃𝑡𝑥 ∥2 + 𝐶2 (∥𝑣𝑥 ∥2 + ∥𝑣𝑡𝑥 ∥2 ),

∫ 𝑡∫ 0

𝐿

0

≤ 𝐶2

(1.3.23)

−2𝜇(𝑛 − 1)(𝑟𝑛−2 𝑣 2 )𝑡𝑥 𝜃𝑡 𝑑𝑥𝑑𝑠 ∫ 𝑡∫ 0

𝐿

2 (𝜃𝑡2 + 𝑣𝑥2 𝑣𝑡2 + 𝑣 2 𝑣𝑡𝑥 )𝑑𝑥𝑑𝑠 ) ( ∫ ∫

0

≤ 𝐶2 + 𝐶2

𝑡

sup

0 𝑥∈[0,𝐿]

𝐿

𝑣𝑡2

0

𝑣𝑥2 𝑑𝑥

𝑑𝑠 + 𝐶2

∫ 𝑡∫ 0

𝐿

0

2 𝑣𝑡𝑥 𝑑𝑥𝑑𝑠

≤ 𝐶2 , ∫ 𝑡∫ 0

𝐿

0

(1.3.24)

𝑑𝑔 𝜃𝑡 𝑑𝑥𝑑𝑠 ≤ 𝐶1 𝑑𝑡

∫ 𝑡∫ 0

0

𝐿

(𝑔𝑟2 + 𝑔𝑡2 + 𝜃𝑡2 )𝑑𝑥𝑑𝑠 ≤ 𝐶2 .

Combining (1.3.15)–(1.3.25), we derive for 𝜀 > 0 small enough, (∫ ∫ ∫ ∫ ∫ 𝐿

0

𝜃𝑡2 𝑑𝑥

+

𝑡

0

𝐿

0

2 𝜃𝑡𝑥 𝑑𝑥𝑑𝑠

≤ 𝐶2 + 𝐶2

𝑡

sup

0 𝑥∈[0,𝐿]

𝑣𝑥2

𝐿

0

(1.3.25)

) 𝜃𝑡2 𝑑𝑥

𝑑𝑠.

24


By Gronwall’s inequality, we obtain ∫ 𝑡 2 ∥𝜃𝑡𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 , ∥𝜃𝑡 (𝑡)∥ + 0

(1.3.26)

which, by noting that from (1.1.11) ∥𝜃𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑡 (𝑡)∥ + ∥𝜃(𝑡)∥𝐻 1 + ∥𝑢(𝑡)∥𝐻 1 + ∥𝑣(𝑡)∥𝐻 1 + ∥𝑔(𝑡)∥) ≤ 𝐶2 , and by the embedding theorem, implies (1.3.7). Lemma 1.3.2. Under the conditions of Theorem 1.3.1, we have ∫ 𝑡 2 ∥𝑢𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 , ∀𝑡 > 0. ∥𝑢𝑥𝑥(𝑡)∥ + 0

□

(1.3.27)

Proof. Differentiating (1.1.10) with respect to 𝑥, we have 𝛽

𝑑 ( 𝑢𝑥𝑥 ) 𝑅𝜃𝑢𝑥𝑥 (𝑟𝑛−1 𝑣)𝑥𝑥 𝑢𝑥 1−𝑛 1−2𝑛 + = 𝑟 𝑣 + (1 − 𝑛)𝑟 𝑢𝑣 + 2𝛽 𝑡𝑥 𝑡 𝑑𝑡 𝑢 𝑢2 𝑢2 𝑛−1 2 2 𝜃𝑥 𝑢 𝑥 𝑣)𝑥 𝑢𝑥 (𝑟 𝜃𝑥𝑥 𝜃𝑢 − 2𝑅 2 + 2𝑅 3𝑥 − 𝑟1−𝑛 𝑓𝑟 𝑢 − (1 − 𝑛)𝑟1−2𝑛 𝑢𝑓 − 2𝛽 +𝑅 𝑢3 𝑢 𝑢 𝑢 := 𝑀 (1.3.28)

where ∥𝑀 ∥ ≤ 𝐶2 (∥𝑣𝑡𝑥 ∥ + ∥𝑣𝑡 ∥ + ∥𝑢𝑥∥ + ∥𝑣𝑥 ∥𝐻 1 + ∥𝜃∥𝐻 2 + ∥𝑓𝑟 ∥ + ∥𝑓 ∥). By Theorem 1.2.1, condition (1.3.1) and Lemma 1.3.1, we get ∫ 𝑡 ∥𝑀 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 , ∀𝑡 > 0. 0

(1.3.29)

𝑢𝑥𝑥 , then integrating the result over [0, 𝑡] × [0, 𝐿], using 𝑢 Young’s inequality and (1.3.29), we can obtain (1.3.27). □ Multiplying (1.3.28) by

Lemma 1.3.3. Under conditions of Theorem 1.3.1, we have ∫ 𝑡 ( ) ∥𝑣𝑥𝑥𝑥∥2 + ∥𝜃𝑥𝑥𝑥∥2 (𝑠)𝑑𝑠 ≤ 𝐶2 , ∀𝑡 > 0. 0

(1.3.30)

Proof. Differentiating (1.1.10) with respect to 𝑥, we can get ∥𝑣𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝑣𝑡𝑥 ∥ + ∥𝑣𝑥 ∥𝐻 1 + ∥𝑢𝑥 ∥𝐻 1 + ∥𝜃∥𝐻 2 + ∥𝑓𝑟 𝑢∥) . By Theorem 1.2.1, Lemmas 1.3.1–1.3.2, we have ∫ 𝑡 ∥𝑣𝑥𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 . 0

(1.3.31)


25

Similarly, differentiating (1.1.11) with respect to 𝑥, we can obtain (𝑟2𝑛−2 𝜃𝑥 )𝑥 𝑢𝑥 (𝑟2𝑛−2 𝜃𝑥 )𝑥𝑥 (𝑟2𝑛−2 𝜃𝑥 )𝑢𝑥𝑥 (𝑟2𝑛−2 𝜃𝑥 )𝑢2𝑥 + 2𝑘 + 𝑘 − 2𝑘 𝑢 𝑢2 𝑢2 𝑢3 𝑛−1 2 𝑛−1 2 𝑛−1 𝑛−1 (𝑟 [(𝑟 𝜃𝑥 (𝑟 𝑣)𝑥 ]𝑥 𝑣)𝑥 𝑢𝑥 𝑣)𝑥 + 𝜃(𝑟 𝑣)𝑥𝑥 +𝛽 −𝛽 +𝑅 𝑢 𝑢2 𝑢 𝜃(𝑟𝑛−1 𝑣)𝑥 𝑢𝑥 −𝑅 + 2𝜇(𝑛 − 1)(𝑟𝑛−2 𝑣 2 )𝑥𝑥 𝑢2 = 𝑔𝑟 𝑢𝑟1−𝑛 ,

𝐶𝑣 𝜃𝑡𝑥 − 𝑘

∥𝜃𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶2 (∥𝜃𝑡𝑥 ∥ + ∥𝜃𝑥 ∥𝐻 1 + ∥𝑢𝑥∥𝐻 1 + ∥𝑣𝑥 ∥𝐻 1 + ∥𝑔𝑟 𝑢∥) . By Theorem 1.2.1, condition (1.3.1), (1.3.2) and Lemmas 1.3.1, 1.3.2, we derive (1.3.30). □ Proof of Theorem 1.3.1. By equations (1.1.9)–(1.1.11), Theorem 1.2.1, Lemmas 1.3.1–1.3.3 and the embedding theorem, we have ∥𝑢𝑡𝑥 (𝑡)∥ ≤ 𝐶2 ∥𝑣(𝑡)∥𝐻 2 ≤ 𝐶2 , ∥𝑢𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶2 ∥𝑣(𝑡)∥𝐻 3 , ∫ 𝑡 ∫ 𝑡 ∥𝑢𝑡𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶2 ∥𝑣(𝑠)∥2𝐻 3 𝑑𝑠 ≤ 𝐶2 , 0

0

∥𝑟(𝑡)∥𝐻 3 ≤ 𝐶2 ∥𝑢(𝑡)∥𝐻 2 ≤ 𝐶2 , ∥𝑢(𝑡)∥𝑊 1,∞ ≤ 𝐶2 ∥𝑢(𝑡)∥𝐻 2 ≤ 𝐶2 , ∥𝜃(𝑡)∥𝑊 1,∞ ≤ 𝐶2 ∥𝜃(𝑡)∥𝐻 2 ≤ 𝐶2 .

∥𝑟𝑡 (𝑡)∥𝐻 2 ≤ 𝐶2 ∥𝑣(𝑡)∥𝐻 2 ≤ 𝐶2 , ∥𝑣(𝑡)∥𝑊 1,∞ ≤ 𝐶2 ∥𝑣(𝑡)∥𝐻 2 ≤ 𝐶2 ,

Combining Lemmas 1.3.1–1.3.3 with the above estimates, we obtain (1.3.5).

□

1.4 Global Existence of Solutions in 𝑯 4 For external force 𝑓 (𝑟, 𝑡) and heat source 𝑔(𝑟, 𝑡), besides (1.1.16)–(1.1.17) and (1.3.1)–(1.3.2), we suppose that 𝑓 (𝑟, 𝑡), 𝑔(𝑟, 𝑡) are 𝐶 3 functions on [𝑎, 𝑏] × ℝ+ satisfying 𝑓𝑟𝑟 , 𝑓𝑟𝑡 , 𝑓𝑡𝑡 , 𝑓𝑟𝑟𝑟 ∈ 𝐿2 (ℝ+ , 𝐿2 [𝑎, 𝑏]), 𝑓𝑟 , 𝑓𝑡 , 𝑓𝑟𝑟 ∈ 𝐿∞ (ℝ+ , 𝐿2 [𝑎, 𝑏]), 2

+

2

𝑔𝑟𝑟 , 𝑔𝑟𝑡 , 𝑔𝑡𝑡 , 𝑔𝑟𝑟𝑟 ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]),

∞

+

2

𝑔𝑟 , 𝑔𝑡 , 𝑔𝑟𝑟 ∈ 𝐿 (ℝ , 𝐿 [𝑎, 𝑏]).

(1.4.1) (1.4.2)

Constant 𝐶4 denotes a generic positive constant depending only on the 𝐻 4 norm of initial data (𝑢0 , 𝑣0 , 𝜃0 ), ∥𝑓𝑟𝑟 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑓𝑟𝑡 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑓𝑡𝑡 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑓𝑟𝑟𝑟 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑔𝑟𝑟 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑔𝑟𝑡 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑔𝑡𝑡 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , ∥𝑔𝑟𝑟𝑟 ∥𝐿2 (ℝ+ ,𝐿2 [𝑎,𝑏]) , and constants 𝐶1 , 𝐶2 , but is independent of 𝑡.

26


Remark 1.4.1. By Remarks 1.2.1 and 1.3.1, we easily know that assumptions (1.4.1)–(1.4.2) are equivalent to the following conditions: ∀𝑥 ∈ [0, 𝐿], 𝑓𝑟𝑟 (𝑟(𝑥, 𝑡), 𝑡), 𝑓𝑟𝑡 (𝑟(𝑥, 𝑡), 𝑡), 𝑓𝑡𝑡 (𝑟(𝑥, 𝑡), 𝑡), 𝑓𝑟𝑟𝑟 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿2 (ℝ+ , 𝐿2 [0, 𝐿]), ∞

+

(1.4.3)

2

𝑓𝑟 (𝑟(𝑥, 𝑡), 𝑡), 𝑓𝑡 (𝑟(𝑥, 𝑡), 𝑡), 𝑓𝑟𝑟 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿 (ℝ , 𝐿 [0, 𝐿]), 𝑔𝑟𝑟 (𝑟(𝑥, 𝑡), 𝑡), 𝑔𝑟𝑡 (𝑟(𝑥, 𝑡), 𝑡), 𝑔𝑡𝑡 (𝑟(𝑥, 𝑡), 𝑡),

(1.4.4)

𝑔𝑟𝑟𝑟 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿2 (ℝ+ , 𝐿2 [0, 𝐿]), ∞

+

(1.4.5)

2

𝑔𝑟 (𝑟(𝑥, 𝑡), 𝑡), 𝑔𝑡 (𝑟(𝑥, 𝑡), 𝑡), 𝑔𝑟𝑟 (𝑟(𝑥, 𝑡), 𝑡) ∈ 𝐿 (ℝ , 𝐿 [0, 𝐿]).

(1.4.6)

Theorem 1.4.1. Under conditions (1.1.16), (1.1.17), (1.3.1), (1.3.2), (1.4.1) and (1.4.2), if (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻 4 [0, 𝐿] × 𝐻04 [0, 𝐿] × 𝐻 4 [0, 𝐿], problem (1.1.9)–(1.1.15) admits a unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐶([0, +∞), 𝐻 4 [0, 𝐿] × 𝐻04 [0, 𝐿] × 𝐻 4 [0, 𝐿]) satisfying, for any 𝑡 > 0, ∥𝑢(𝑡) − 𝑢 ¯∥2𝐻 4 + ∥𝑢𝑡 (𝑡)∥2𝐻 3 + ∥𝑢𝑡𝑡 (𝑡)∥2𝐻 1 + ∥𝑢𝑥𝑥 (𝑡)∥2𝑊 1,∞ + ∥𝑣(𝑡)∥2𝐻 4 ¯ 2 4 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑣𝑡 (𝑡)∥2 2 + ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 1,∞ + ∥𝜃(𝑡) − 𝜃∥ 𝐻

𝑊

𝐻

𝐻2

+ ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2𝑊 1,∞ ≤ 𝐶4 , (1.4.7) ∫ 𝑡{ ∥𝑢 − 𝑢¯∥2𝐻 4 + ∥𝑢𝑡 ∥2𝐻 4 + ∥𝑢𝑡𝑡 ∥2𝐻 2 + ∥𝑢𝑡𝑡𝑡 ∥2 + ∥𝑢𝑥𝑥 ∥2𝑊 1,∞ + ∥𝑣∥2𝐻 5 0

¯ 2 5 + ∥𝜃𝑡 ∥2 3 + ∥𝜃𝑡𝑡 ∥2 1 + ∥𝑣𝑡 ∥2𝐻 3 + ∥𝑣𝑡𝑡 ∥2𝐻 1 + ∥𝑣𝑥𝑥 ∥2𝑊 2,∞ + ∥𝜃 − 𝜃∥ 𝐻 𝐻 𝐻 } 2 + ∥𝜃𝑥𝑥 ∥𝑊 2,∞ (𝜏 )𝑑𝜏 ≤ 𝐶4 . (1.4.8)

The proof of Theorem 1.4.1 can be done by a series of lemmas as follows. Lemma 1.4.1. Under conditions of Theorem 1.4.1, it holds that ∫ 𝑡 2 ∥𝑣𝑡𝑡 (𝑡)∥ + ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 (∥𝑣𝑡𝑥𝑥 ∥2 + 𝜀∥𝜃𝑡𝑡𝑥 ∥2 )(𝜏 )𝑑𝜏, ∀𝑡 > 0, ≤ 𝐶4 + 𝐶4 0 ∫ 𝑡 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥𝑑𝜏 ∥𝜃𝑡𝑡 (𝑡)∥2 + 0 ∫ 𝑡 ( ) ∥𝜃𝑡𝑥𝑥 ∥2 + 𝜀∥𝑣𝑡𝑡𝑥 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶4 + 𝐶4 0 ∫ 𝑡 (∥𝑣𝑥 ∥2 + ∥𝑣𝑡𝑥 ∥2 + ∥𝜃∥2 + ∥𝜃𝑡 ∥2 )∥𝜃𝑡𝑡 ∥2 (𝜏 )𝑑𝜏, ∀𝑡 > 0. + 𝐶4 0

(1.4.9)

(1.4.10)

Proof. Differentiating (1.1.10), (1.1.11) with respect to 𝑡, we can get ∥𝑣𝑡𝑡 (𝑡)∥ ≤ 𝐶4 (∥𝜃𝑥 ∥ + ∥𝑢𝑥 ∥ + ∥𝑣𝑡𝑥𝑥 ∥ + ∥𝜃𝑡 ∥𝐻 1 + ∥𝑓𝑟 ∥ + ∥𝑓𝑡 ∥),

(1.4.11)

∥𝜃𝑡𝑡 (𝑡)∥ ≤ 𝐶4 (∥𝜃𝑡 ∥𝐻 2 + ∥𝑣𝑥 ∥ + ∥𝑣𝑡𝑥 ∥ + ∥𝜃𝑥 ∥𝐻 1 + ∥𝑔𝑟 ∥ + ∥𝑔𝑡 ∥).

(1.4.12)


27

Differentiating (1.1.10) twice with respect to 𝑡, multiplying the resultant by 𝑣𝑡𝑡 , then integrating the result over [0, 𝐿], using Young’s inequality, we obtain 𝑑 ∥𝑣𝑡𝑡 (𝑡)∥2 + 𝐶1 ∥𝑣𝑡𝑡𝑥 (𝑡)∥2 𝑑𝑡 ( ≤ 𝐶4

(1.4.13) 2 2 ) 𝑑 𝑓 ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑡𝑡 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 + 𝑑𝑡2 .

Integrating (1.4.13) with respect to 𝑡, and noting that 𝑑2 𝑓 = 𝑓𝑟𝑟 𝑣 2 + 2𝑓𝑟𝑡 𝑣 + 𝑓𝑟 𝑣𝑡 + 𝑓𝑡𝑡 , 𝑑𝑡2 then by (1.4.11), Theorems 1.2.1 and 1.3.1, and conditions (1.3.25), (1.3.1), we can get (1.4.9). By the same method, differentiating (1.1.11) twice with respect to 𝑡, multiplying the resultant by 𝜃𝑡𝑡 , then integrating the resultant over [0, 𝐿], using Young’s inequality, we have { 𝑑 ∥𝜃𝑡𝑡 (𝑡)∥2 ≤ 𝐶4 −∥𝜃𝑡𝑡𝑥 ∥2 + 𝜀∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑣𝑥 ∥2 𝑑𝑡 2 2 } 𝑑 𝑔 ( ) 2 2 2 2 2 2 + ∥𝜃∥ + 𝑑𝑡2 + 𝐶4 ∥𝑣𝑥 ∥ + ∥𝑣𝑡𝑥 ∥ + ∥𝜃∥ + ∥𝜃𝑡 ∥ ∥𝜃𝑡𝑡 ∥ . (1.4.14) Integrating (1.4.14) with respect to 𝑡, using (1.4.12), conditions (1.4.2), (1.3.2), Theorems 1.2.1, 1.3.1 and noting that 𝑑2 𝑔 = 𝑔𝑟𝑟 𝑣 2 + 2𝑔𝑟𝑡 𝑣 + 𝑔𝑟 𝑣𝑡 + 𝑔𝑡𝑡 , 𝑑𝑡2 we derive (1.4.10).

□

Lemma 1.4.2. Under conditions of Theorem 1.4.1, we have that for any 𝑡 > 0, and for any 𝜀 > 0, ∫ 𝑡 ∫ 𝑡 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 𝜀∥𝜃𝑡𝑥𝑥 ∥2 (𝜏 )𝑑𝜏, (1.4.15) ∥𝑣𝑡𝑥 (𝑡)∥2 + 0 0 ∫ 𝑡 2 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ∥𝜃𝑡𝑥 (𝑡)∥ + 0 ∫ 𝑡 [ ] ≤ 𝐶4 + 𝐶4 𝜀∥𝑣𝑡𝑥𝑥 ∥2 + (∥𝑣𝑥 ∥2 + ∥𝜃∥2𝐻 1 + ∥𝑣𝑥𝑥 ∥2 )∥𝜃𝑡𝑥 ∥2 (𝜏 )𝑑𝜏. (1.4.16) 0

Proof. Differentiating (1.1.10) with respect to 𝑡 and 𝑥, multiplying by 𝑣𝑡𝑥 , then integrating the result with respect to 𝑥 by parts, using Young’s inequality, we have

28


for any 𝜀 > 0, ( 𝑑 2 2 ∥𝑣𝑡𝑥 (𝑡)∥ + 𝐶1 ∥𝑣𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶4 ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥 ∥2𝐻 1 + 𝜀∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑥 ∥2 𝑑𝑡 2 2 ) 𝑑 𝑓 + ∥𝑣𝑥 ∥2𝐻 2 + 𝜀∥𝜃𝑡𝑥𝑥 ∥2 + (1.4.17) 𝑑𝑡𝑑𝑥 . 𝑑2 𝑓 = 𝑓𝑟𝑟 𝑢𝑟1−𝑛 𝑣 + 𝑓𝑟𝑡 𝑟1−𝑛 𝑢 + 𝑑𝑡𝑑𝑥 𝑓𝑟 𝑣𝑥 , then by Theorems 1.2.1–1.3.1 and conditions (1.4.1), (1.3.1), we have for 𝜀 > 0 small enough,

Integrating (1.4.17) with respect to 𝑡, noting that

∥𝑣𝑡𝑥 (𝑡)∥2 +

∫ 0

𝑡

∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 𝜀

∫ 0

𝑡

∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏.

By the same method, differentiating (1.1.11) with respect to 𝑡 and 𝑥, multiplying the result by 𝜃𝑡𝑥 , then integrating with respect to 𝑥, using Young’s inequality, we obtain 2 2 ) ( 𝑑 𝑔 𝑑 2 2 2 2 2 ∥𝜃𝑡𝑥 (𝑡)∥ ≤ 𝐶4 −∥𝜃𝑡𝑥𝑥 ∥ + 𝜀∥𝑣𝑡𝑥𝑥 ∥ + ∥𝑣𝑡𝑥 ∥ + ∥𝜃𝑡 ∥ + 𝑑𝑡𝑑𝑥 𝑑𝑡 ( ) + 𝐶4 ∥𝑣𝑥 ∥2 + ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥 ∥2𝐻 1 ∥𝜃𝑡𝑥 ∥2 . (1.4.18) Integrating (1.4.18) with respect to 𝑡, then by Theorems 1.2.1–1.3.1 and conditions (1.4.2), (1.3.2), we derive ∥𝜃𝑡𝑥 (𝑡)∥2 +

∫

𝑡

∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ∫ 𝑡 [ ] 𝜀∥𝑣𝑡𝑥𝑥 ∥2 + (∥𝑣𝑥 ∥2 + ∥𝜃𝑥 ∥2𝐻 1 + ∥𝑣𝑥𝑥 ∥2 )∥𝜃𝑡𝑥 ∥2 (𝜏 )𝑑𝜏. ≤ 𝐶4 + 𝐶4 0

0

□

Lemma 1.4.3. Under the conditions of Theorem 1.4.1, we have that for any 𝑡 > 0, ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 ∫ 𝑡 ( ) ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶4 . + 0

(1.4.19)

Proof. Adding (1.4.15) to (1.4.16), we have ∫ 𝑡 ( ) ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 (𝜏 )𝑑𝜏 ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + 0 ∫ 𝑡 ( ) ∥𝑣𝑥 ∥2 + ∥𝜃𝑥 ∥2𝐻 1 + ∥𝑣𝑥𝑥 ∥2 ∥𝜃𝑡𝑥 ∥2 (𝜏 )𝑑𝜏. ≤ 𝐶4 + 𝐶4 0

(1.4.20)


29

Using Gronwall’s inequality, and Theorems 1.2.1–1.3.1, we can derive ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 +

∫

𝑡

0

( ) ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶4 .

(1.4.21)

Multiplying (1.4.9), (1.4.10) by 𝜀 respectively, then adding the results up to (1.4.13), we obtain ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 ∫ 𝑡 ( ) ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 (𝜏 )𝑑𝜏 + 0 ∫ 𝑡 ( ) ∥𝑣𝑥 ∥2 + ∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝜃𝑡 ∥2 ∥𝜃𝑡𝑡 ∥2 (𝜏 )𝑑𝜏. ≤ 𝐶4 + 𝐶4 (𝜀) 0

(1.4.22)

Applying Gronwall’s inequality to (1.4.22), and Theorems 1.2.1, 1.3.2, we get (1.4.19). □ Lemma 1.4.4. Under conditions of Theorem 1.4.1, it holds that for any 𝑡 > 0, ∥𝑢𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑢𝑥𝑥(𝑡)∥2𝑊 1,∞ ∫ 𝑡 ( ) + ∥𝑢𝑥𝑥𝑥∥2𝐻 1 + ∥𝑢𝑥𝑥∥2𝑊 1,∞ (𝜏 )𝑑𝜏 ≤ 𝐶4 , 0 ∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1

+ ∥𝑣𝑥𝑥 (𝑡)∥2𝑊 1,∞ + ∥𝜃𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥 (𝑡)∥2𝑊 1,∞ 2

2

(1.4.23) (1.4.24)

2

+ ∥𝑢𝑡𝑥𝑥𝑥(𝑡)∥ + ∥𝑣𝑡𝑥𝑥 (𝑡)∥ + ∥𝜃𝑡𝑥𝑥 (𝑡)∥ ∫ 𝑡 + (∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝑢𝑡𝑥𝑥𝑥∥2𝐻 1 )(𝜏 )𝑑𝜏 ≤ 𝐶4 , ∫ 0

𝑡

0

(∥𝑣𝑥𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥∥2𝐻 1 )(𝜏 )𝑑𝜏 ≤ 𝐶4 .

(1.4.25)

Proof. Differentiating (1.1.10) with respect to 𝑥, and using equation (1.1.9), we get 𝛽

𝜃𝑢𝑥𝑥𝑥 𝑑 ( 𝑢𝑥𝑥𝑥 ) +𝑅 2 𝑑𝑡 𝑢 𝑢

{ 𝑛−1 𝑣)𝑥𝑥𝑥 𝑢𝑥 (𝑟 (𝑟𝑛−1 𝑣)𝑥𝑥 𝑢𝑥𝑥 = (𝑟 )𝑥𝑥 𝑣𝑡 + 2(𝑟 )𝑥 𝑣𝑡𝑥 + 𝑟 𝑣𝑡𝑥𝑥 + 𝛽 3 + 3 𝑢2 𝑢2 } { 𝑛−1 3 𝑛−1 2 𝑛−1 2 𝜃𝑥𝑥𝑥 𝜃𝑥𝑥𝑢𝑥 𝑣)𝑥 𝑢𝑥 𝑣)𝑥𝑥 𝑢𝑥 𝑣)𝑥 𝑢𝑥𝑥 𝑢𝑥 (𝑟 (𝑟 (𝑟 −3 2 +6 −6 −6 +𝑅 4 3 3 𝑢 𝑢 𝑢 𝑢 𝑢 } 𝜃𝑥 𝑢𝑥𝑥 𝜃𝑥 𝑢2𝑥 𝜃𝑢𝑥𝑥𝑢𝑥 𝜃𝑥 𝑢2𝑥 𝜃𝑢𝑥 𝑢𝑥𝑥 𝑢3𝑥𝜃 −3 2 +4 3 +2 +2 3 +4 − 6 4 − (𝑓𝑟𝑟 𝑢2 + 𝑓𝑟 𝑢𝑥 ) 𝑢 𝑢 𝑢3 𝑢 𝑢3 𝑢 1−𝑛

: = 𝐸(𝑥, 𝑡)

1−𝑛

1−𝑛

(1.4.26)

30


with ∥𝐸(𝑡)∥ ≤ 𝐶4 (∥𝑣𝑡 ∥𝐻 2 + ∥𝑢𝑥 ∥𝐻 1 + ∥𝑣𝑥 ∥𝐻 2 + ∥𝜃𝑥 ∥𝐻 2 + ∥𝑓𝑟𝑟 ∥ + ∥𝑓𝑟 ∥) . By Theorem 1.3.1, Lemma 1.4.3 and condition (1.4.1), we have ∫ 0

𝑡

∥𝐸(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀𝑡 > 0.

(1.4.27)

𝑢𝑥𝑥𝑥 , then integrating the resultant over [0, 𝐿] × [0, 𝑡], we 𝑢 ∫ 𝑡 2 ∥𝑢𝑥𝑥𝑥 (𝑡)∥ + ∥𝑢𝑥𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 . (1.4.28)

Multiplying (1.4.26) by obtain

0

Differentiating (1.1.10) with respect to 𝑥, by Theorem 1.3.1 and Lemma 1.4.3, we can deduce that ∥𝑣𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝑣𝑡𝑥 ∥ + ∥𝜃∥𝐻 2 + ∥𝑣∥𝐻 2 + ∥𝑢∥𝐻 2 + ∥𝑓𝑟 𝑢∥) ≤ 𝐶4 .

(1.4.29)

Differentiating (1.1.10) with respect to 𝑥 twice, we have ∥𝑣𝑥𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝑣𝑡𝑥𝑥 ∥ + ∥𝜃∥𝐻 3 + ∥𝑣∥𝐻 3 + ∥𝑢∥𝐻 3 + ∥𝑓𝑟𝑟 ∥ + ∥𝑓𝑟 ∥) .

(1.4.30)

By the same method, differentiating (1.1.11) with respect to 𝑥 once and twice respectively, we get ∥𝜃𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝜃𝑡𝑥 ∥ + ∥𝑣∥𝐻 2 + ∥𝜃∥𝐻 2 + ∥𝑢∥𝐻 2 + ∥𝑔𝑟 ∥) ≤ 𝐶4 , ∥𝜃𝑥𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝜃𝑡𝑥𝑥 ∥ + ∥𝑣∥𝐻 3 + ∥𝜃∥𝐻 3 + ∥𝑢∥𝐻 3 + ∥𝑔𝑟𝑟 ∥ + ∥𝑔𝑟 ∥).

(1.4.31) (1.4.32)

By Theorem 1.3.1 and (1.4.20), we have ∥𝑣𝑥𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥𝑥(𝑡)∥2 +

∫ 0

𝑡

( ) ∥𝑣𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥∥2𝐻 1 (𝜏 )𝑑𝜏 ≤ 𝐶4 .

(1.4.33)

Differentiating (1.1.10), (1.1.11) with respect to 𝑡 respectively, using Theorem 1.3.1 and Lemma 1.4.3, we can obtain ∥𝑣𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝑣𝑡𝑡 ∥ + ∥𝜃𝑡𝑥 ∥ + ∥𝑢𝑡𝑥 ∥ + ∥𝑓𝑟 ∥ + ∥𝑓𝑡 ∥ + ∥𝑣𝑥 ∥𝐻 1 ) ≤ 𝐶4 ,

(1.4.34)

∥𝜃𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝜃𝑡𝑡 ∥ + ∥𝜃𝑡 ∥ + ∥𝑣𝑡𝑥 ∥ + ∥𝑢𝑡𝑥 ∥ + ∥𝑔𝑟 ∥ + ∥𝑔𝑡 ∥) ≤ 𝐶4 .

(1.4.35)

By (1.4.30), (1.4.32)–(1.4.35), Theorem 1.3.1 and conditions (1.4.1), (1.4.2), we have ∫ 𝑡 (∥𝑣𝑥𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥𝑥∥2 )(𝜏 )𝑑𝜏 ≤ 𝐶4 . (1.4.36) ∥𝑣𝑥𝑥𝑥𝑥(𝑡)∥2 + ∥𝜃𝑥𝑥𝑥𝑥(𝑡)∥2 + 0


31

Differentiating (1.4.26) with respect to 𝑥, we can get 𝛽

𝜃𝑢𝑥𝑥𝑥𝑥 𝑑 ( 𝑢𝑥𝑥𝑥𝑥 ) +𝑅 𝑑𝑡 𝑢 𝑢2 𝑛−1 (𝑟 𝑢𝑥𝑥𝑥(𝑟𝑛−1 𝑣)𝑥𝑥 𝑢𝑥𝑥𝑥𝑢2𝑥 𝜃𝑢𝑥𝑥𝑥𝑢𝑥 𝜃𝑥 𝑢𝑥𝑥𝑥 𝑣)𝑥𝑥𝑥𝑥 𝑢𝑥 = + − 2 +2 − + 𝐸𝑥 2 2 3 3 𝑢 𝑢 𝑢 𝑢 𝑢2 = 𝐸1 (𝑥, 𝑡) (1.4.37)

with

3 ) ( 𝑑 𝑓 ∥𝐸1 (𝑡)∥ ≤ 𝐶4 ∥𝑢∥𝐻 3 + ∥𝑣∥𝐻 4 + ∥𝜃∥𝐻 4 + ∥𝑣𝑡 ∥𝐻 3 + 𝑑𝑥3 .

(1.4.38)

Differentiating (1.1.10) with respect to 𝑡 and 𝑥, we can derive ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥ (1.4.39) ≤ 𝐶4 (∥𝑣𝑡𝑡𝑥 ∥ + ∥𝜃𝑡𝑥𝑥 ∥ + ∥𝑓𝑟 ∥ + ∥𝑓𝑟𝑡 ∥ + ∥𝑓𝑟𝑟 ∥ + ∥𝑣𝑡 ∥𝐻 2 + ∥𝑣∥𝐻 3 + ∥𝜃∥𝐻 2 ) . 𝑢𝑥𝑥𝑥𝑥 , then integrating the resultant over [0, 𝐿] × [0, 𝑡], 𝑢 using Theorem 1.3.1, Lemma 1.4.3, (1.4.28), (1.4.36), (1.4.39) and (1.4.1)–(1.4.2), we obtain ∫ 𝑡 2 ∥𝑢𝑥𝑥𝑥𝑥(𝑡)∥ + ∥𝑢𝑥𝑥𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 , ∀𝑡 > 0. (1.4.40) Multiplying (1.4.37) by

0

Differentiating (1.1.10) and (1.1.11) three times with respect to 𝑥 respectively, we get 3 ) ( 𝑑 𝑓 ∥𝑣𝑥𝑥𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶4 ∥𝑣𝑡 ∥𝐻 3 + ∥𝑣∥𝐻 4 + ∥𝜃∥𝐻 4 + ∥𝑢∥𝐻 4 + (1.4.41) 𝑑𝑥3 , 3 ) ( 𝑑 𝑔 ∥𝜃𝑥𝑥𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶4 ∥𝜃𝑡 ∥𝐻 3 + ∥𝑣∥𝐻 4 + ∥𝜃∥𝐻 4 + ∥𝑢∥𝐻 3 + (1.4.42) 𝑑𝑥3 . Differentiating (1.1.11) with respect to 𝑡 and 𝑥, we have (1.4.43) ∥𝜃𝑡𝑥𝑥𝑥(𝑡)∥ 2 ) ( 𝑑 𝑔 ≤ 𝐶4 ∥𝜃𝑡𝑡𝑥 ∥ + ∥𝜃∥𝐻 3 + ∥𝜃𝑡 ∥𝐻 2 + ∥𝑣𝑡 ∥𝐻 2 + ∥𝑣∥𝐻 2 + ∥𝑢𝑡𝑥 ∥ + 𝑑𝑡𝑑𝑥 . By (1.1.9), we obtain ∥𝑢𝑡𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶4 ∥𝑣(𝑡)∥𝐻 4 ,

∥𝑢𝑡𝑥𝑥𝑥(𝑡)∥𝐻 1 ≤ 𝐶4 ∥𝑣(𝑡)∥𝐻 5 .

(1.4.44)

Combining (1.4.28) and (1.4.40), by the embedding theorem, we get (1.4.23). Combining (1.4.41) and (1.4.42), by (1.4.36), (1.4.39), (1.4.40), (1.4.43) and conditions (1.4.1), (1.4.2), we obtain (1.4.25). By (1.4.33)–(1.4.36), (1.4.11), (1.4.12), (1.4.39), (1.4.43), (1.4.44) and the embedding theorem, we obtain (1.4.24). □

32


Remark 1.4.2. The results in this chapter indicate that suitable regularities of the non-autonomous terms can ensure the global existence of solutions in 𝐻 𝑖 (𝑖 = 1, 2, 4). Remark 1.4.3. Based on the uniform estimates in this chapter, we can established the asymptotic behavior of global solutions in 𝐻 𝑖 (𝑖 = 1, 2, 4), and even exponential stability of global solutions if we impose some proper exponential decay on nonautonomous terms 𝑓 and 𝑔. Proof of Theorem 1.4.1. By Lemmas 1.4.1–1.4.4, and noting the following estimates: ∥𝑢𝑡𝑡𝑥 (𝑡)∥ ≤ 𝐶4 ∥𝑣𝑡𝑡𝑥 (𝑡)∥,

∥𝑢𝑡𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶4 ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥,

we can obtain Theorem 1.4.1.

∥𝑢𝑡𝑡𝑡 (𝑡)∥ ≤ 𝐶4 ∥𝑣𝑡𝑡𝑥 (𝑡)∥, □

1.5 Bibliographic Comments For the case 𝑓 = 𝑔 ≡ 0, Fujita-Yashima and Benabidallah [80, 81] established the global existence of solutions to problem (1.1.9)–(1.1.15), Jiang [29] proved the large-time behavior of global solutions in 𝐻 1 , Zheng and Qin [84] obtained the global existence of universal attractors in 𝐻 1 and 𝐻 2 , Qin et al. [65] established the exponential stability of global solutions in 𝐻 4 . We also refer the reader to the related results in Cho, Choe and Kim [6], Qin and Mu˜ noz Rivera [71], Xu and Yang [78], Yanagi [79], Zheng [82], Zheng and Qin [83] and Zimmer [86].

Chapter 2

Global Existence and Exponential Stability for a Real Viscous Heat-conducting Flow with Shear Viscosity 2.1 Introduction In this chapter we shall study the global existence and exponential stability of weak solutions for a real viscous compressible heat-conducting flow between two horizontal plates. The system describing this type of flow is derived from the following general 3𝐷 Navier-Stokes equations: 𝜌𝑡 + div(𝜌⃗𝑢) = 0,

(2.1.1)

(2.1.2) (𝜌⃗𝑢)𝑡 + div (𝜌⃗𝑢 ⊗ ⃗𝑢) + ∇𝑝 = div (𝜆′ (div ⃗𝑢)) 𝐼⃗ + 𝜇(∇⃗𝑢 + (∇⃗𝑢)𝑇 ), ( ′ ) 𝑇 𝜁𝑡 + div (⃗𝑢(𝜁 + 𝑝)) = div 𝜆 (div ⃗𝑢)⃗𝑢 + 𝜇⃗𝑢(∇⃗𝑢 + (∇⃗𝑢) ) + 𝜅∇𝜃 (2.1.3) where 𝑥 ∈ ℝ3 is the spatial variable and 𝑡 > 0 is the time, 𝜌 ∈ ℝ, ⃗𝑢 ∈ ℝ3 and 𝜃 ∈ ℝ+ denote the )density, velocity and temperature, respectively, the total energy ( is 𝜁 = 𝜌 𝑒 + 12 ∣⃗𝑢∣2 , with 𝑒 the internal energy and 12 ∣⃗𝑢∣2 the kinetic energy, the equations of state 𝑝 = 𝑝(𝜌, 𝜃), 𝑒 = 𝑒(𝜌, 𝜃) relate this pressure 𝑝 and the internal energy 𝑒 with the density and temperature of the flow, (∇⃗𝑢)𝑇 is the transpose of the matrix ∇⃗𝑢, 𝜆′ = 𝜆′ (𝜌, 𝜃) and 𝜇 = 𝜇(𝜌, 𝜃) are the viscosity coefficients of the flow and 𝜅 = 𝜅(𝜌, 𝜃) is the heat conductivity. The fluid in question is a Newtonian fluid, i.e., the stress tensor −𝑝𝐼⃗ + 𝜆′ (div ⃗𝑣 )𝐼⃗ + 𝜇(∇⃗𝑢 + (∇⃗𝑢)𝑇 ) is a linear function of the deformation tensor 12 (∇⃗𝑢 + (∇⃗𝑢)𝑇 ). The viscosity and heat conduction terms describe the dissipative processes in viscous flows. Consider a 3𝐷 flow (2.1.1)–(2.1.3) with spatial variable ⃗𝑥 = (𝑥, 𝑥2 , 𝑥3 ), which is moving in the 𝑥 direction and uniform in the transverse direction (𝑥2 , 𝑥3 ), 𝜌 = 𝜌(𝑥, 𝑡),

𝜃 = 𝜃(𝑥, 𝑡),

⃗𝑢 = (𝑢, 𝑤)(𝑥, ⃗ 𝑡),

𝑤 ⃗ = (𝑢2 , 𝑢3 ),


(2.1.4)

33

34

Chapter 2. Global Existence and Exponential Stability

where 𝑢 ∈ ℝ is the longitudinal velocity and 𝑤 ⃗ ∈ ℝ2 is the transverse velocity. With this structure (2.1.4), equations (2.1.1)–(2.1.3) can be written as the following in one space dimension with 𝜆 = 𝜆′ + 2𝜇 > 0: 𝜌𝑡 + (𝜌𝑢)𝑥 = 0,

(2.1.5)

2

(𝜌𝑢)𝑡 + (𝜌𝑢 + 𝑝)𝑥 = (𝜆𝑢𝑥 )𝑥 ,

(2.1.6)

(𝜌𝑤) ⃗ 𝑡 + (𝜌𝑢𝑤) ⃗ 𝑥 = (𝜇𝑤 ⃗ 𝑥 )𝑥 , ⃗ ⋅𝑤 ⃗ 𝑥 + 𝜅𝜃𝑥 )𝑥 , 𝜁𝑡 + (𝑢(𝜁 + 𝑝))𝑥 = (𝜆𝑢𝑢𝑥 + 𝜇𝑤

(2.1.7) (2.1.8)

where as in (2.1.1)–(2.1.3), 𝑥 ∈ ℝ is the spatial variable and 𝑡 > 0 is the time ⃗ ∈ ℝ2 , 𝜃 ∈ ℝ, 𝑝 denote the density, longitudinal velocity, variable, 𝜌 ∈ ℝ+ , 𝑢 ∈ ℝ, 𝑤 transverse velocity, temperature, pressure respectively and the total energy of the plane viscous flow is ) ( 1 ⃗ 2) , (2.1.9) 𝜁 = 𝜌 𝑒 + (𝑢2 + ∣𝑤∣ 2 with internal energy 𝑒. The pressure 𝑝 and internal energy 𝑒 are expressed by the density and the temperature of the flow according to the state equations 𝑝 = 𝑝(𝜌, 𝜃),

𝑒 = 𝑒(𝜌, 𝜃)

(2.1.10)

where 𝜆 = 𝜆(𝜌, 𝜃) and 𝜇 = 𝜇(𝜌, 𝜃) are the viscosity coefficients of the flow and 𝜅 = 𝜅(𝜌, 𝜃) is the heat conductivity; 𝜇 = 𝜇(𝜌, 𝜃) is particularly called the shear viscosity. Consider the initial boundary value problem (2.1.5)–(2.1.8) in a bounded spatial domain Ω = (0, 1) with the following initial condition and boundary conditions: ⃗ 0 , 𝜃0 )(𝑥), 𝑥 ∈ Ω, (𝜌, 𝑢, 𝑤, ⃗ 𝜃)∣𝑡=0 = (𝜌0 , 𝑢0 , 𝑤 (𝑢, 𝑤) ⃗ ∂Ω = 0, 𝜃𝑥 ∂Ω = 0, 𝑜𝑟 𝜃∂Ω = 𝑇0 = const . > 0,

(2.1.11) (2.1.12)

where the initial data are compatible with each other. We assume that 𝜃0 (𝑥) > 0, 𝜌0 (𝑥) > 0, for any 𝑥 ∈ (0, 1), ∫ 1 𝜌0 (𝑥)𝑑𝑥 = 1.

(2.1.13)

Now we introduce the Lagrangian variable, ∫ 𝑥 𝜌(𝜉, 𝑡)𝑑𝜉. 𝑦 = 𝑦(𝑥, 𝑡) =

(2.1.14)

Then we have from (2.1.5), (2.1.13) and (2.1.14), ∫ 1 ∫ 1 0 ≤ 𝑦 ≤ 1, 𝜌(𝑥, 𝑡)𝑑𝑥 = 𝜌0 (𝑥)𝑑𝑥 = 1.

(2.1.15)

0

0

0

0

2.1. Introduction

35

Therefore we can translate the problem (2.1.1)–(2.1.3) in Euler coordinates into the following initial boundary value problem in Lagrangian coordinates (𝑦, 𝑡), 𝑦 ∈ Ω = (0, 1), a moving of coordinates along the particle path, 𝑣𝑡 − 𝑢𝑦 = 0, ( 𝑢 ) 𝑦 𝑢 𝑡 + 𝑝𝑦 = 𝜆 , 𝑣 𝑦 ( ) 𝑤 ⃗𝑦 , 𝑤 ⃗𝑡 = 𝜇 𝑣 𝑦 ( ) 𝜆𝑢𝑢𝑦 + 𝜇𝑤 ⃗𝑤 ⃗ 𝑦 + 𝜅𝜃𝑦 , 𝐸𝑡 + (𝑢𝑝)𝑦 = 𝑣 𝑦 with the initial boundary conditions (𝑣, 𝑢, 𝑤, ⃗ 𝜃)𝑡=0 = (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 )(𝑦), (𝑢, 𝑤) ⃗ = 0,

𝜃𝑦 = 0,

(2.1.16) (2.1.17) (2.1.18) (2.1.19)

𝑦 ∈ Ω,

(2.1.20)

on ∂Ω × [0, +∞),

(2.1.21)

or (𝑢, 𝑤) ⃗ = 0, where 𝑣 =

𝜃𝑦 = 𝑇0 > 0,

on ∂Ω × [0, +∞),

(2.1.22)

1 is specific volume, 𝑒 = 𝑒(𝑣, 𝜃), 𝑝 = 𝑝(𝑣, 𝜃), and 𝜌 1 𝐸 = 𝑒 + (𝑢2 + ∣𝑤∣ ⃗ 2 ). 2

(2.1.23)

The second law of thermodynamics states the relation between 𝑝 and 𝑒, 𝑒𝑣 (𝑣, 𝜃) + 𝑝(𝑣, 𝜃) = 𝜃𝑝𝜃 (𝑣, 𝜃).

(2.1.24)

Now we assume that 𝑒, 𝑝 and 𝜅 are 𝐶 3 functions on 0 < 𝑢 < +∞ and 0 ≤ 𝜃 < +∞. Let 𝑞 and 𝑟 be two positive constants (exponents of growth) satisfying one of the following relations: 0 ≤ 𝑟 ≤ 1/3,

1/3 < 𝑞,

(2.1.25)

1/3 < 𝑟 < 4/7, (2𝑟 + 1)/5 < 𝑞,

(2.1.26)

4/7 ≤ 𝑟 ≤ 1, 1 < 𝑟 ≤ 13/3,

(5𝑟 + 1)/9 < 𝑞, (9𝑟 + 1)/15 < 𝑞,

(2.1.27) (2.1.28)

13/3 < 𝑟,

(11𝑟 + 3)/19 < 𝑞.

(2.1.29)

Moreover, we further assume that there are positive constants 𝜈, 𝑝0 , 𝑝1 , 𝑘0 and for any 𝑣 > 0, there are positive constants 𝑁 (𝑣), 𝑝2 (𝑣), 𝑝3 (𝑣) and 𝑘1 (𝑣) such that for any 𝑣 ≥ 𝑣 and 𝜃 ≥ 0 the following conditions hold: 0 ≤ 𝑒(𝑣, 0), 𝑝0 𝜃

𝑟+1

𝜈(1 + 𝜃𝑟 ) ≤ 𝑒𝜃 (𝑣, 𝜃) ≤ 𝑁 (𝑣)(1 + 𝜃𝑟 ),

< 𝑣𝑝(𝑣, 𝜃) ≤ 𝑝1 (1 + 𝜃

𝑟+1

),

(2.1.30) (2.1.31)

36


− 𝑝2 (𝑣)[𝑙 + (1 − 𝑙)𝜃 + 𝜃𝑟+1 ] ≤ 𝑝𝑣 (𝑣, 𝜃) ≤ −𝑝3 (𝑣)[𝑙 + (1 − 𝑙)𝜃 + 𝜃𝑟+1 ], 𝑙 = 0 𝑜𝑟 1,

(2.1.32)

𝑟

∣𝑝𝜃 (𝑣, 𝜃)∣ ≤ 𝑝3 (𝑣)(1 + 𝜃 ),

(2.1.33)

𝑘0 (1 + 𝜃𝑞 ) ≤ 𝑘(𝑣, 𝜃) ≤ 𝑘1 (𝑣)(1 + 𝜃𝑞 ), ∣𝑘𝑣 (𝑣, 𝜃)∣ + ∣𝑘𝑣𝑣 (𝑣, 𝜃)∣ ≤ 𝑘1 (𝑣)(1 + 𝜃𝑞 ).

(2.1.34) (2.1.35)

For the viscosity 𝜆(𝑣, 𝜃), we assume that 𝜆(𝑣, 𝜃) = 𝜆0 > 0

(2.1.36)

is a constant. The notation in the chapter is standard. We put ∥ ⋅ ∥ = ∥ ⋅ ∥𝐿2 and denote by 𝐶 𝑘 (𝐼, 𝐵), 𝑘 ∈ ℕ0 ≡ {0} ∪ ℕ, the space of 𝑘-times continuously differentiable functions from 𝐼 ⊆ ℝ into a Banach space 𝐵, and likewise by 𝐿𝑝¯(𝐼, 𝐵), 1 ≤ 𝑝¯ ≤ +∞, the corresponding Lebesgue spaces. Subscripts 𝑡, 𝑥 and 𝑦 denote the (partial) derivatives with respect to 𝑡, 𝑥 and 𝑦, respectively. We use 𝐶𝑖 (𝑖 = 1, 2, 3, 4) to denote the universal positive constants depending only on the 𝐻 𝑖 norm of initial data, min 𝑦∈[0,1] 𝑣0 (𝑦), min 𝑦∈[0,1] 𝑤 ⃗ 0 (𝑦) and min 𝑦∈[0,1] 𝜃0 (𝑦). Without danger of confusion, we will use the same symbol to denote the state functions as well as their values along the thermodynamic process, e.g., 𝑝(𝑢, 𝜃) and 𝑝(𝑢(𝑥, 𝑡), 𝜃(𝑥, 𝑡)) and 𝑝(𝑥, 𝑡). Define the following three spaces: { 1 = (𝑣, 𝑢, 𝑤, ⃗ 𝜃) ∈ (𝐻 1 [0, 1])5 : 𝑣(𝑥) > 0, 𝜃(𝑥) > 0, ∀𝑥 ∈ [0, 1], 𝐻+

} 𝑢(0) = 𝑢(1) = 0, 𝑤(0) ⃗ = ⃗0, 𝜃(0) = 𝜃(1) = 𝑇0 for (2.1.22) ,

{ 𝑖 𝐻+ = (𝑣, 𝑢, 𝑤, ⃗ 𝜃) ∈ (𝐻 𝑖 [0, 1])5 : 𝑣(𝑥) > 0, 𝜃(𝑥) > 0, ∀𝑥 ∈ [0, 1], 𝑢(0) = 𝑢(1) = 0, 𝑤(0) ⃗ = ⃗0, 𝜃(0) } = 𝜃(1) = 𝑇0 for (2.1.22), ′ ′ 𝜃 (0) = 𝜃 (1) = 0 for (2.1.21) , 𝑖 = 2, 4. Our results read as follows, which are selected from [60, 64]. Theorem 2.1.1. Assume that 𝑒, 𝑝 and 𝜅 are 𝐶 2 functions on 0 < 𝑣 < +∞ and 0 ≤ 𝜃 < +∞, and assumptions (2.1.23)–(2.1.36) hold. Then for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 1 1 , there exists a unique global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem 𝐻+ (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) verifying that, 0 < 𝐶1−1 ≤ 𝜃(𝑦, 𝑡) ≤ 𝐶1 ,

0 < 𝐶1−1 ≤ 𝑢(𝑦, 𝑡) ≤ 𝐶1 ,

∀(𝑦, 𝑡) ∈ [0, 1] × [0, +∞) (2.1.37)

2.1. Introduction

37

and for any 𝑡 > 0, 2 ¯ 2 ⃗ ∥𝑣(𝑡) − 𝑣¯∥2𝐻 1 + ∥𝑢(𝑡)∥2𝐻 1 + ∥𝑤(𝑡)∥ 𝐻 1 + ∥𝜃(𝑡) − 𝜃∥𝐻 1 ∫ 𝑡( ¯ 2 2 + ∥𝑢𝑡 ∥2 ∥𝑣 − 𝑣¯∥2𝐻 1 + ∥𝑢∥2𝐻 2 + ∥𝑤∥ + ⃗ 2𝐻 2 + ∥𝜃 − 𝜃∥ 𝐻 0 ) + ∥𝑤 ⃗ 𝑡 ∥2 + ∥𝜃𝑡 ∥2 (𝜏 ) d𝜏 ≤ 𝐶1 .

(2.1.38)

Moreover, there exist constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) > 0 such that and for any 𝑡 > 0, ( ) 2 ¯ 2 ⃗ 𝑒𝛾𝑡 ∥𝑣(𝑡) − 𝑣¯∥2𝐻 1 + ∥𝑢(𝑡)∥2𝐻 1 + ∥𝑤(𝑡)∥ 𝐻 1 + ∥𝜃(𝑡) − 𝜃∥𝐻 1 ∫ 𝑡 ( ¯ 22 + 𝑒𝛾𝜏 ∥𝑣 − 𝑣¯∥2𝐻 1 + ∥𝑢∥2𝐻 2 + ∥𝑤∥ ⃗ 2𝐻 2 + ∥𝜃 − 𝜃∥ 𝐻 0 ) + ∥𝑢𝑡 ∥2 + ∥𝑤 ⃗ 𝑡 ∥2 + ∥𝜃𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶1 ∫ ∫1 ¯ = 1 (𝑒(𝑣0 , 𝜃0 ) + where 𝑣¯ = 0 𝑣0 (𝑥) 𝑑𝑥, 𝜃¯ = 𝑇0 for (2.1.22), 𝑒(¯ 𝑣 , 𝜃) 0 for (2.1.21).

(2.1.39) 𝑣02 2 )(𝑥) 𝑑𝑥

Theorem 2.1.2. Assume that 𝑒, 𝑝 and 𝜅 are 𝐶 3 functions on 0 < 𝑣 < +∞, 0 ≤ 𝜃 < +∞, and assumptions (2.1.23)–(2.1.36) hold. Then for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 2 2 𝐻+ , there exists a unique global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) verifying that for any 𝑡 > 0, 2 2 ¯ 2 ⃗ ∥𝑣(𝑡) − 𝑣¯∥2𝐻 2 + ∥𝑢(𝑡)∥2𝐻 2 + ∥𝑤(𝑡)∥ 𝐻 2 + ∥𝜃(𝑡) − 𝜃∥𝐻 2 + ∥𝑢𝑡 (𝑡)∥ ∫ 𝑡( ∥𝑣 − 𝑣¯∥2𝐻 2 + ∥𝑢∥2𝐻 3 + ∥𝑤∥ + ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ⃗ 2𝐻 3 0 ) 2 2 ¯ 2 3 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑤 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 , + ∥𝜃 − 𝜃∥ ⃗ ∥ + ∥𝜃 ∥ 𝑡𝑦 𝑡𝑦 𝐻

(2.1.40)

and there exist constants 𝐶2 > 0 and 𝛾2 = 𝛾2 (𝐶2 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2 ], the following estimates hold for any 𝑡 > 0: ( 2 2 ¯ 2 ⃗ 𝑒𝛾𝑡 ∥𝑣(𝑡) − 𝑣¯∥2𝐻 2 + ∥𝑢(𝑡)∥2𝐻 2 + ∥𝑤(𝑡)∥ 𝐻 2 + ∥𝜃(𝑡) − 𝜃∥𝐻 2 + ∥𝑢𝑡 (𝑡)∥ ) ∫ 𝑡 ( + ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + 𝑒𝛾𝑡 ∥𝑣 − 𝑣¯∥2𝐻 2 + ∥𝑢∥2𝐻 3 + ∥𝑤∥ ⃗ 2𝐻 3 0 ) 2 2 ¯ + ∥𝜃 − 𝜃∥𝐻 3 + ∥𝑢𝑡𝑦 ∥ + ∥𝑤 ⃗ 𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶2 . (2.1.41) Theorem 2.1.3. Assume that 𝑒, 𝑝 are 𝐶 5 functions on 0 < 𝑣 < +∞ and 0 ≤ 𝜃 < +∞, and assumptions (2.1.23)–(2.1.36) hold. Then for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 4 4 𝐻+ , there exists a unique global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem

38


(2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) verifying that for any 𝑡 > 0, ∥𝑣(𝑡) − 𝑣¯∥2𝐻 4 + ∥𝑣(𝑡) − 𝑣¯∥2𝑊 3,∞ + ∥𝑣𝑡 (𝑡)∥2𝐻 3 + ∥𝑣𝑡𝑡 (𝑡)∥2𝐻 1 + ∥𝑢(𝑡)∥2𝐻 4 ¯ 24 + ∥𝑢(𝑡)∥2 3,∞ + ∥𝑢𝑡 (𝑡)∥2 2 + ∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝜃(𝑡) − 𝜃∥ 𝑊

𝐻

𝐻

2 ¯ 2 3,∞ + ∥𝜃𝑡 (𝑡)∥2 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑤(𝑡)∥ + ∥𝜃(𝑡) − 𝜃∥ ⃗ 𝑊 𝐻 𝐻4 2 + ∥𝑤(𝑡)∥ ⃗ ⃗ 𝑡 (𝑡)∥2𝐻 2 + ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 ≤ 𝐶4 , 𝑊 3,∞ + ∥𝑤 ∫ 𝑡( ∥𝑣 − 𝑣¯∥2𝐻 4 + ∥𝑣 − 𝑣¯∥2𝑊 3,∞ + ∥𝑣𝑡 ∥2𝐻 4 + ∥𝑣𝑡𝑡 ∥2𝐻 2 + ∥𝑣𝑡𝑡𝑡 ∥2 0

¯ 25 + ∥𝑢∥2𝐻 5 + ∥𝑢∥2𝑊 4,∞ + ∥𝑢𝑡 ∥2𝐻 3 + ∥𝑢𝑡𝑡 ∥2𝐻 1 + ∥𝜃 − 𝜃∥ 𝐻 2 2 2 2 ¯ + ∥𝜃 − 𝜃∥𝑊 4,∞ + ∥𝜃𝑡 ∥𝐻 3 + ∥𝜃𝑡𝑡 ∥𝐻 1 + ∥𝑤∥ ⃗ 𝐻 5 + ∥𝑤∥ ⃗ 2𝑊 4,∞ ) + ∥𝑤 ⃗ 𝑡 ∥2𝐻 3 + ∥𝑤 ⃗ 𝑡𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 ,

(2.1.42)

(2.1.43)

and there exist constants 𝐶4 > 0 and 𝛾4 = 𝛾4 (𝐶4 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0: ∥𝑣(𝑡) − 𝑣¯∥2𝐻 4 + ∥𝑣(𝑡) − 𝑣¯∥2𝑊 3,∞ + ∥𝑣𝑡 (𝑡)∥2𝐻 3 + ∥𝑣𝑡𝑡 (𝑡)∥2𝐻 1 + ∥𝑢(𝑡)∥2𝐻 4 ¯ 24 + ∥𝑢(𝑡)∥2 3,∞ + ∥𝑢𝑡 (𝑡)∥2 2 + ∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝜃(𝑡) − 𝜃∥ 𝑊

𝐻

𝐻

2 ¯ 2 3,∞ + ∥𝜃𝑡 (𝑡)∥2 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑤(𝑡)∥ + ∥𝜃(𝑡) − 𝜃∥ ⃗ 𝑊 𝐻 𝐻4 2 + ∥𝑤(𝑡)∥ ⃗ ⃗ 𝑡 (𝑡)∥2𝐻 2 + ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 ≤ 𝐶4 𝑒−𝛾𝑡 , 𝑊 3,∞ + ∥𝑤 ∫ 𝑡 ( 𝑒𝛾𝜏 ∥𝑣 − 𝑣¯∥2𝐻 4 + ∥𝑣 − 𝑣¯∥2𝑊 3,∞ + ∥𝑣𝑡 ∥2𝐻 4 + ∥𝑣𝑡𝑡 ∥2𝐻 2 + ∥𝑣𝑡𝑡𝑡 ∥2 0

¯ 25 + ∥𝑢∥2𝐻 5 + ∥𝑢∥2𝑊 4,∞ + ∥𝑢𝑡 ∥2𝐻 3 + ∥𝑢𝑡𝑡 ∥2𝐻 1 + ∥𝜃 − 𝜃∥ 𝐻 2 2 2 2 ¯ + ∥𝜃 − 𝜃∥𝑊 4,∞ + ∥𝜃𝑡 ∥𝐻 3 + ∥𝜃𝑡𝑡 ∥𝐻 1 + ∥𝑤∥ ⃗ 𝐻 5 + ∥𝑤∥ ⃗ 2𝑊 4,∞ ) + ∥𝑤 ⃗ 𝑡 ∥2𝐻 3 + ∥𝑤 ⃗ 𝑡𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(2.1.44)

(2.1.45)

2.2 Proof of Theorem 2.1.1 In this section, we shall complete the proof of Theorem 2.1.1 and take that the assumptions in Theorem 2.1.1 to be valid. We begin with the following lemma. Lemma 2.2.1. Assume that 𝑒, 𝑝 and 𝜅 are 𝐶 2 functions on 0 < 𝑣 < +∞ and 0 ≤ 𝜃 < +∞, and assumptions (2.1.23)–(2.1.36) hold. Then for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 1 1 , there exists a unique global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem 𝐻+ (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) verifying that, 2 ¯ 2 ∥𝑣(𝑡) − 𝑣¯∥2𝐻 1 + ∥𝑢(𝑡)∥2𝐻 1 + ∥𝑤(𝑡)∥ ⃗ 𝐻 1 + ∥𝜃(𝑡) − 𝜃∥𝐻 1 ∫ 𝑡( ) ¯ 2 2 (𝜏 ) d𝜏 ≤ 𝐶1 , ∥𝑣 − 𝑣¯∥2𝐻 1 + ∥𝑢∥2𝐻 2 + ∥𝑤∥ + ⃗ 2𝐻 2 + ∥𝜃 − 𝜃∥ 𝐻 0

(2.2.1) ∀𝑡 > 0

2.2. Proof of Theorem 2.1.1

39

and for any (𝑦, 𝑡) ∈ [0, 1] × [0, +∞), 0 < 𝐶1−1 ≤ 𝜃(𝑦, 𝑡) ≤ 𝐶1 ,

0 < 𝐶1−1 ≤ 𝑣(𝑦, 𝑡) ≤ 𝐶1 .

(2.2.2)

Proof. For the case 𝑤 ⃗ = 0, Qin [55] proved the result to problems (2.1.16), (2.1.17), (2.1.19)–(2.1.21) or (2.1.16), (2.1.17), (2.1.19)–(2.1.20), (2.1.22) under the assumptions (2.1.23)–(2.1.36). The proof is the same as that in Qin [50, 51, 52, 54] and Chapter 3 of [59] for the case of 𝑤 ⃗ = 0. The proof is complete. □ 1 In what follows we shall prove the exponential stability in 𝐻+ . Set

Ψ(𝑣, 𝜃) = 𝑒(𝑣, 𝜃) − 𝜃𝜂(𝜃, 𝑣)

(2.2.3)

where 𝜂(𝜃, 𝑣) is defined by the relations 𝑒𝜃 = 𝜃𝜂𝜃 ,

𝜂𝑣 = 𝑝𝜃 .

(2.2.4)

Now we introduce the density of the gas, 𝜌 = 1/𝑣, then 𝜂 = 𝜂(1/𝜌, 𝜃) satisfies ∂𝜂 −𝑝𝜃 ∂𝜂 𝑒𝜃 = 2 , = . ∂𝜌 𝜌 ∂𝜃 𝜃

(2.2.5)

We consider the transform 𝐴 : (𝜌, 𝜃) ∈ 𝐷𝜌,𝜃 = {(𝜌, 𝜃) : 𝜌 > 0, 𝜃 > 0} → (𝑣, 𝜂) ∈ 𝐴𝐷𝜌,𝜃 .

(2.2.6)

Owing to the Jacobian, ∣∂(𝑣, 𝜂)/∂(𝜌, 𝜃)∣ = −𝑒𝜃 /𝜌2 𝜃 < 0 on 𝐴𝐷𝜌,𝜃 . Thus the functions 𝑒, 𝑝 can also be regarded as the smooth functions of (𝑣, 𝜂). We denote them by 𝑒 = 𝑒(𝑣, 𝜂) :≡ 𝑒(𝑣, 𝜃(𝑣, 𝜂)) = 𝑒(1/𝜌, 𝜃), 𝑝 = 𝑝(𝑣, 𝜂) :≡ 𝑝(𝑣, 𝜃(𝑣, 𝜂)) = 𝑝(1/𝜌, 𝜃).

(2.2.7)

Then it follows from (2.2.3)–(2.2.7) that 𝑒𝑣 = −𝑝,

𝑒𝜂 = 𝜃, 2

𝑝𝑣 = −(𝜌 𝑝𝜌 + 𝜃𝑝2𝜃 /𝑒𝜃 ), 𝑝𝜂 = 𝜃𝑝𝜃 /𝑒𝜃 , 𝜃𝑣 = −𝜃𝑝𝜃 /𝑒𝜃 ,

𝜃𝜂 = 𝜃/𝑒𝜃 .

(2.2.8)

We define the energy form 𝑉 (𝑣, 𝑢, 𝑤, ⃗ 𝜂) = where 𝑣¯ = 1/𝜌¯,

𝑢2 ∣𝑤∣ ⃗2 ∂𝑒 ∂𝑒 + + 𝑒(𝑣, 𝜂) − 𝑒(¯ 𝑣 , 𝜂¯) − (¯ 𝑣 , 𝜂¯)(𝑣 − 𝑣¯) − (¯ 𝑣 , 𝜂¯)(𝜂 − 𝜂¯), 2 2 ∂𝑣 ∂𝜂 (2.2.9) ¯ 𝜂¯ = 𝜂(1/𝜌¯, 𝜃),

𝜃¯ = 𝑇0 , for (2.1.22),

¯ = 𝑒(¯ 𝑣 , 𝜃)

∫ 0

(2.2.10) 1

(𝑒(𝑣0 , 𝜃0 ) +

𝑣02 )(𝑥) 𝑑𝑥 for (2.1.21). 2

□

40


1 Lemma 2.2.2. The unique global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) satisfies the following estimates:

[ ] 𝑢2 ∣𝑤∣ ⃗2 + + 𝐶1−1 (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 ≤ 𝑉 (𝑣, 𝑢, 𝑤, ⃗ 𝜂) (2.2.11) 2 2 [ ] ∣𝑤∣ ⃗2 𝑢2 + + 𝐶1 (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 . ≤ 2 2 Proof. By the mean value problem theorem, there exists a point (˜ 𝑣 , 𝜂˜) between (𝑣, 𝜂) and (¯ 𝑣 , 𝜂¯) such that [ ∂𝑒 1 ∂2𝑒 ∂𝑒 (¯ 𝑣 , 𝜂¯)(𝑣 − 𝑣¯) + (¯ 𝑣 , 𝜂¯)(𝜂 − 𝜂¯) + (˜ 𝑣 , 𝜂˜)(𝑣 − 𝑣¯)2 𝑒(𝑣, 𝜂) = 𝑒(¯ 𝑣 , 𝜂¯) + ∂𝑣 ∂𝜂 2 ∂𝑣 2 ] ∂2𝑒 ∂2𝑒 2 (˜ 𝑣 , 𝜂˜)(𝑣 − 𝑣¯)(𝜂 − 𝜂¯) + 2 (˜ +2 𝑣 , 𝜂˜)(𝜂 − 𝜂¯) . (2.2.12) ∂𝑣∂𝜂 ∂𝜂 By (2.2.9) and (2.2.12), we get

[ ∣𝑤∣ ⃗2 1 ∂2𝑒 𝑢2 + + (˜ 𝑣 , 𝜂˜)(𝑣 − 𝑣¯)2 𝑉 (𝑣, 𝑢, 𝑤, ⃗ 𝜂) = 2 2 2 ∂𝑣 2 ] ∂2𝑒 ∂2𝑒 2 (˜ 𝑣 , 𝜂˜)(𝑣 − 𝑣¯)(𝜂 − 𝜂¯) + 2 (˜ +2 𝑣 , 𝜂˜)(𝜂 − 𝜂¯) ∂𝑣∂𝜂 ∂𝜂

where

𝑣˜ = 𝜆0 𝑣¯ + (1 − 𝜆0 )𝑣, 𝜂˜ = 𝜆0 𝜂¯ + (1 − 𝜆0 )𝜂,

By (2.2.2), we get

˜ ≤ 𝐶1 , 0 < 𝐶1−1 ≤ ˜ 𝑣 (1/𝜌˜, 𝜃)

which implies

(2.2.13)

0 ≤ 𝜆0 ≤ 1.

˜ ≤ 𝐶1 , 0 < 𝐶1−1 ≤ ˜ 𝜂 (1/𝜌˜, 𝜃)

2 2 2 ∂ 𝑒 ∂ 𝑒 ∂ 𝑒 (˜ 𝑣 , 𝜂˜) + 2 (˜ 𝑣 , 𝜂˜) + 𝑣 , 𝜂˜) ≤ 𝐶1 . ∂𝑣 2 (˜ ∂𝑣∂𝜂 ∂𝜂

(2.2.14)

Thus (2.2.14) and the Cauchy inequality give [ ] ∣𝑤∣ ⃗2 𝑢2 + + 𝐶1 (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 . 2 2 On the other hand, we infer from (2.2.8) that 𝑉 (𝑢, 𝑣, 𝑤, ⃗ 𝜂) ≤

(2.2.15)

𝑒𝑣𝑣 = −𝑝𝑣 = 𝜌2 𝑝𝜌 + 𝜃𝑝2𝜃 /𝑒𝜃 , 𝑒𝑣𝜂 = −𝑝𝜂 = 𝜃𝑣 = −𝜃𝑝𝜃 /𝑒𝜃 , 𝑒𝜂𝜂 = 𝜃𝜂 = 𝜃/𝑒𝜃 , which yields that the Hessian of 𝑒(𝑣, 𝜂) is positive definite for any 𝑣 > 0 and 𝜃 > 0. Thus we deduce from (2.2.12) that [ ] ∣𝑤∣ ⃗2 𝑢2 + + 𝜆min (˜ 𝑣 , 𝜂˜) (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 2 2 [ ] ∣𝑤∣ ⃗2 𝑢2 + + 𝐶1−1 (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 , ≥ 2 2

𝑉 (𝑣, 𝑢, 𝑤, ⃗ 𝜂) ≥

(2.2.16)


41

where 𝜆min (˜ 𝑣 , 𝜂˜) (≥ 𝐶1−1 ) is the smaller characteristic root of the Hessian of 𝑒(˜ 𝑣 , 𝜂˜). Thus by the combination of (2.2.15) and (2.2.16), we deduce that [ ] 𝑢2 ∣𝑤∣ ⃗2 + + 𝐶1−1 (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 ≤ 𝑉 (𝑣, 𝑢, 𝑤, ⃗ 𝜂) 2 2 [ ] ∣𝑤∣ ⃗2 𝑢2 + + 𝐶1 (𝑣 − 𝑣¯)2 + (𝜂 − 𝜂¯)2 . ≤ 2 2 The proof is complete.

□

Lemma 2.2.3. There exist constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) > 0 such that for 1 any fixed 𝛾 ∈ (0, 𝛾1 ], the global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) satisfies the estimate ( ) 2 ¯ 2 ⃗ + ∥𝜃(𝑡) − 𝜃∥ 𝑒𝛾𝑡 ∥𝑢(𝑡)∥2 + ∥𝑣(𝑡) − 𝑣¯∥2𝐻 1 + ∥𝑤(𝑡)∥ ∫ 𝑡 ( ) 𝑒𝛾𝑡 ∥𝑢𝑦 ∥2 + ∥𝑣𝑦 ∥2 + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝜃𝑦 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶1 , ∀ 𝑡 > 0. + 0

(2.2.17)

Proof. By (2.1.19) and (2.1.23), we get [

] 1 𝑒 + (𝑢2 + ∣𝑤∣ ⃗ 2 ) = (−𝑢𝑝 + 𝜆𝜌𝑢𝑢𝑦 + 𝜇𝑤 ⃗ ⋅𝑤 ⃗ 𝑦 + 𝜅𝜌𝜃𝑦 )𝑦 . 2 𝑡

That is, 𝑒𝜂 𝜂𝑡 + 𝑒𝑣 𝑣𝑡 + 𝑢𝑢𝑡 + 𝑤 ⃗ ⋅𝑤 ⃗ 𝑡 = (−𝑢𝑝 + 𝜆𝜌𝑢𝑢𝑦 + 𝜇𝑤 ⃗ ⋅𝑤 ⃗ 𝑦 + 𝜅𝜌𝜃𝑦 )𝑦 .

(2.2.18)

Thus it follows from (2.1.17)–(2.1.18) that 𝑒𝜂 𝜂𝑡 + 𝑒𝑣 𝑣𝑡 = −𝑝𝑢𝑦 + 𝜆𝜌𝑢2𝑦 + (𝜅𝜌𝜃𝑦 )𝑦 .

(2.2.19)

By (2.1.16) and (2.2.8), we get 𝜂𝑡 =

𝜆𝜌𝑢2𝑦 + 𝜃

(

𝜅𝜌𝜃𝑦 𝜃

(

) 𝑦

+ 𝜅𝜌

𝜃𝑦 𝜃

)2 .

(2.2.20)

¯ 𝑒(¯ 𝑣 , 𝜂¯) = 0, by (2.2.8), (2.2.9) and Since 𝑢 ¯, 𝜃, 𝑣 , 𝜂¯) are constants, 𝑢¯𝑡 = 0, 𝜃¯𝑡 = 0, 𝑒𝑡 (¯ (2.2.20), we get ( 𝑉𝑡 = ( =

∣𝑤∣ ⃗2 𝑢2 + +𝑒 2 2 ∣𝑤∣ ⃗2 𝑢2 + +𝑒 2 2

) )𝑡 𝑡

−

∂𝑒 ∂𝑒 (¯ 𝑣 , 𝜂¯)𝑣𝑡 − (¯ 𝑣 , 𝜂¯)𝜂𝑡 ∂𝑣 ∂𝜂

¯ 𝑡 + 𝑝¯𝑣𝑡 , − 𝜃𝜂

(2.2.21)

42


which, together with (2.2.20), gives ( ) [ ] 𝜅𝜌𝜃𝑦2 𝜃¯ 𝜃¯ 2 𝑉𝑡 + 𝜆𝜌𝑢𝑦 + = 𝜆𝜌𝑢𝑢𝑦 + 𝜇𝜌𝑤 ⃗ ⋅𝑤 ⃗ 𝑦 + (1 − )𝜅𝜌𝜃𝑦 − (𝑝 − 𝑝¯)𝑢 . 𝜃 𝜃 𝜃 𝑦 (2.2.22) Differentiating 𝜌𝜃 with respect to 𝑦, we have ∂ (𝜌𝜃 ) = −2𝜌𝜌𝑦 𝑢𝑦 − 𝜌2 𝑢𝑦𝑦 ∂𝑦 which implies

(

𝜌𝑦 𝜌

) 𝑡

= −𝜌𝑦 𝑢𝑦 − 𝜌𝑢𝑦𝑦 .

(2.2.23)

Multiplying (2.1.17) by 𝜆𝜌𝑦 /𝜌, we have 𝜆2 𝜌2𝑦 𝑢𝑦 𝜆𝑝𝜌 𝜌2𝑦 𝜆𝜌𝑦 𝑢𝑡 𝜆2 𝜌𝜌𝑦 𝑢𝑦𝑦 𝜆𝑝𝜃 𝜃𝑦 𝜌𝑦 = + − − . 𝜌 𝜌 𝜌 𝜌 𝜌

(2.2.24)

We can also get 𝜆(−𝜌𝑦 𝑢𝑦 − 𝜌𝑢𝑦𝑦 )𝑢 = −𝜆(𝜌𝑢𝑢𝑦 )𝑦 + 𝜆𝜌𝑢2𝑦 .

(2.2.25)

𝜆2 𝜌𝑦 2 𝜆𝜌𝑦 𝑢 ( ) + with respect to 𝑡, we derive from (2.2.23), 2 𝜌 𝜌 [ 2 ] 𝜆 𝜌𝑦 2 𝜆𝜌𝑦 𝑢 𝜌𝑦 𝜆𝜌𝑦 𝑢𝑡 ( ) + + 𝜆(−𝜌𝑦 𝑢𝑦 − 𝜌𝑢𝑦𝑦 )𝑢. = 𝜆2 ( )(−𝜌𝑦 𝑢𝑦 − 𝜌𝑢𝑦𝑦 ) + 2 𝜌 𝜌 𝑡 𝜌 𝜌 (2.2.26) By (2.2.25) and (2.2.26), we derive [ 2 ] 𝜆𝑝𝜌 𝜌2𝑦 𝜆 𝜌𝑦 2 𝜆𝜌𝑦 𝑢 𝜆𝑝𝜃 𝜃𝑦 𝜌𝑦 ( ) + + − 𝜆𝜌𝑢2𝑦 = −𝜆(𝜌𝑢𝑢𝑦 )𝑦 . + (2.2.27) 2 𝜌 𝜌 𝑡 𝜌 𝜌 Differentiating

Taking the inner product of 𝑤 ⃗ in ℝ2 on both sides of (2.1.18), we get 1 ∂ ∣𝑤∣ ⃗ 2 + 𝜇𝜌∣𝑤 ⃗ 𝑦 ∣2 = (𝜇𝜌𝑤 ⃗ ⋅𝑤 ⃗ 𝑦 )𝑦 . 2 ∂𝑡

(2.2.28)

Multiplying (2.2.22), (2.2.27), (2.2.28) by 𝑒𝛾𝑡 , 𝛽𝑒𝛾𝑡 , 𝑒𝛾𝑡 respectively, and then adding the results up, we deduce that ) ( [ ( ) ] ¯ 𝜆𝑝𝜌 𝜌2𝑦 𝜅𝜌𝜃𝑦2 𝜆𝑝𝜃 𝜃𝑦 𝜌𝑦 ∂ 𝛾𝑡 𝜃 2 2 2 𝐺(𝑡) + 𝑒 𝜆𝜌𝑢𝑦 + +𝛽 + − 𝜆𝜌𝑢𝑦 + 𝜇𝜌∣𝑤 ⃗𝑦∣ ∂𝑡 𝜃 𝜃 𝜌 𝜌 ( ) 𝜃¯ 𝛾𝑡 ¯ = 𝛾𝐺(𝑡) + 𝑒 [(1 − 𝛽)𝜆𝜌𝑢𝑢𝑦 + 2𝜇𝜌𝑤 ⃗ ⋅𝑤 ⃗𝑦 + 𝜅 1 − 𝜌, 𝜃))𝑢] 𝜌𝜃𝑦 − (𝑝 − 𝑝(¯ 𝑦, 𝜃 (2.2.29)


where

43

[ ( ( ( ) ) ] ) 2 1 𝜆𝜌𝑦 𝑢 𝜆2 𝜌𝑦 𝐺(𝑡) = 𝑒 𝑉 + , 𝑢, 𝑤, ⃗ 𝜂 +𝛽 . 𝜌 2 𝜌 𝜌 𝛾𝑡

Integrating (2.2.29) over [0, 1] × [0, 𝑡], we get ∫ 0

1

𝐺(𝑡) 𝑑𝑦

[ ( ] ) ) ( 𝜅𝜌𝜃𝑦2 𝜆𝑝𝜌 𝜌2𝑦 𝜆𝑝𝜃 𝜃𝑦 𝜌𝑦 𝜃¯ 2 2 2 + + − 𝜆𝜌𝑢𝑦 + 𝜇𝜌∣𝑤 𝑒 ⃗ 𝑦 ∣ 𝑑𝑦𝑑𝜏 𝜆𝜌𝑢𝑦 + +𝛽 𝜃 𝜃 𝜌 𝜌 0 0 ( ] [ ) ∫ 𝑡∫ 1 𝜃¯ 𝛾𝑡 ¯ +𝛾 𝑒 ⃗ ⋅𝑤 ⃗𝑦 + 𝜅 1 − 𝜌, 𝜃))𝑢 𝑑𝜏 (1 − 𝛽)𝜆𝜌𝑢𝑢𝑦 + 2𝜇𝜌𝑤 𝜌𝜃𝑦 − (𝑝 − 𝑝(¯ 𝜃 0 0 𝑦 ∫ 𝑡∫ 1 ∫ 1 𝐺(0) 𝑑𝑦 + 𝛾 𝐺(𝜏 ) 𝑑𝑦d𝜏. (2.2.30) = ∫ 𝑡∫

0

1

𝛾𝜏

0

0

Using Cauchy’s and Poincaré’s inequalities, we deduce that for small 𝛽 > 0 and for any 𝛾 > 0, ( ) 2 𝑒𝛾𝑡 ∥𝜌(𝑡) − 𝜌¯∥2 + ∥𝑢(𝑡)∥2 + ∥𝑤(𝑡)∥ (2.2.31) ⃗ + ∥𝜂(𝑡) − 𝜂¯∥2 + ∥𝜌𝑦 (𝑡)∥2 ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑢𝑦 ∥2 + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝜃𝑦 ∥2 + ∥𝑣𝑦 ∥2 (𝜏 ) 𝑑𝜏 0 ∫ 𝑡 ( ) ≤ 𝐶1 + 𝐶1 𝛾 𝑒𝛾𝜏 ∥𝜌 − 𝜌¯∥2 + ∥𝑢∥2 + ∥𝑤∥ ⃗ 2 + ∥𝜂 − 𝜂¯∥2 + ∥𝜌𝑦 ∥2 (𝜏 ) 𝑑𝜏. 0

By Lemma 2.2.1, boundary conditions (2.1.21)–(2.1.22) together with the Poincaré inequality, we get ∥𝑣(𝑡) − 𝑣¯∥ ≤ 𝐶1 ∥𝑣𝑦 (𝑡)∥,

∥𝑤(𝑡)∥ ⃗ ≤ 𝐶1 ∥𝑤 ⃗ 𝑦 (𝑡)∥, ¯ ∥𝜃(𝑡) − 𝜃∥ ≤ 𝐶1 ∥𝜃𝑦 (𝑡)∥.

∥𝑢(𝑡)∥ ≤ 𝐶1 ∥𝑢𝑦 (𝑡)∥,

(2.2.32)

Using the mean value theorem, we infer that ˜ − 𝑣¯) + 𝜂𝜃 (˜ ˜ − 𝜃). ¯ 𝜂(𝑣, 𝜃) − 𝜂¯ = 𝜂𝑣 (˜ 𝑣 , 𝜃)(𝑣 𝑣 , 𝜃)(𝜃 Hence, ¯ 2 ) ≤ ∥𝜂 − 𝜂¯∥2 ≤ 𝐶1 (∥𝑣 − 𝑣¯∥2 + ∥𝜃 − 𝜃∥ ¯ 2 ), 𝐶1−1 (∥𝑣 − 𝑣¯∥2 + ∥𝜃 − 𝜃∥ 𝐶1−1 ∥𝑣

2

2

2

− 𝑣¯∥ ≤ ∥𝜌 − 𝜌¯∥ ≤ 𝐶1 ∥𝑣 − 𝑣¯∥ .

(2.2.33) (2.2.34)

44


By (2.2.31)–(2.2.34), we get ( ) 2 ⃗ + ∥𝜂(𝑡) − 𝜂¯∥2 + ∥𝜌𝑦 (𝑡)∥2 𝑒𝛾𝑡 ∥𝜌(𝑡) − 𝜌¯∥2 + ∥𝑢(𝑡)∥2 + ∥𝑤(𝑡)∥ ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑢2𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝜃𝑦 ∥2 + ∥𝑣𝑦 ∥2 (𝜏 ) 𝑑𝜏 0 ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢2𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝜃𝑦 ∥2 + ∥𝑣𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶1 + 𝐶1 𝛾 0

which gives that there exists a constant 𝛾1′ = 𝛾1′ (𝐶1 ) > 0, such that for any fixed 𝛾 ∈ (0, 𝛾 ′ ], ( ) 2 ¯ 2 ⃗ + ∥𝜃(𝑡) − 𝜃∥ 𝑒𝛾𝑡 ∥𝑢(𝑡)∥2 + ∥𝑣(𝑡) − 𝑣¯∥2𝐻 1 + ∥𝑤(𝑡)∥ ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢𝑦 ∥2 + ∥𝑣𝑦 ∥2 + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝜃𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶1 . + 0

The proof is complete.

□

Lemma 2.2.4. There exist constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) ≤ 𝛾1′ such that for 1 any fixed 𝛾 ∈ (0, 𝛾1 ], the global solution (𝑢(𝑡), 𝑣(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) satisfies the following estimates: ( ) ∫ 𝑡 ( ⃗ 𝑦 (𝑡)∥2 + ∥𝜃𝑦 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑢𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑦𝑦 ∥2 𝑒𝛾𝑡 ∥𝑢𝑦 (𝑡)∥2 + ∥𝑤 0 ) + ∥𝜃𝑦𝑦 ∥2 + ∥𝑢𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑤 ⃗ 𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶1 , ∀ 𝑡 > 0. (2.2.35) Proof. By Lemma 2.2.1’s boundary conditions together with the Poincaré inequality, we get ⃗ 𝑦𝑦 (𝑡)∥, ∥𝑢𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑦𝑦 (𝑡)∥, ∥𝜃𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝜃𝑦𝑦 (𝑡)∥. (2.2.36) ∥𝑤 ⃗ 𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑤 Multiplying (2.1.17) by −𝑢𝑦𝑦 and integrating the resulting equality over [0, 1], we have ) ∫ 1( ∫ 1 2 𝑢𝑦𝑦 𝜆𝑢𝑦 𝑣𝑦 1 𝑑 ∥𝑢𝑦 (𝑡)∥2 + 𝜆 𝑑𝑦 = + 𝑝 𝜃 + 𝑝 𝑣 (2.2.37) 𝜃 𝑦 𝑣 𝑦 𝑢𝑦𝑦 𝑑𝑦. 2 𝑑𝑡 𝑣 𝑣2 0 0 Using the interpolation inequality, we have that for any 𝜀 > 0, ∫ 1 𝜆𝑢𝑦 𝑣𝑦 𝑢𝑦𝑦 𝑑𝑦 ≤ 𝐶1 ∥𝑣𝑦 ∥∥𝑢𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦 ∥ 𝑣2 0 1 3 ≤ 𝐶1 ∥𝑣𝑦 ∥∥𝑢𝑦 ∥ 2 ∥𝑢𝑦𝑦 ∥ 2 ≤ 𝐶1 (∥𝑣𝑦 ∥2 + ∥𝑢𝑦 ∥2 ) + 𝜀∥𝑢𝑦𝑦 ∥2 .

(2.2.38)

By Young’s inequality and Cauchy’s inequality, we have 𝑑 ∥𝑢𝑦 (𝑡)∥2 + 𝐶1−1 ∥𝑢𝑦𝑦 (𝑡)∥2 ≤ 𝐶1 (∥𝜃𝑦 (𝑡)∥2 + ∥𝑣𝑦 (𝑡)∥2 + ∥𝑢𝑦 (𝑡)∥2 ). 𝑑𝑡

(2.2.39)


45

Multiplying (2.2.39) by 𝑒𝛾𝑡 and integrating the resulting equality over [0, 𝑡], implies that there exists a 𝛾1 = 𝛾1 (𝐶1 ) ≤ 𝛾1′ , such that for any fixed 𝛾 ∈ (0, 𝛾1 ], ∫ 𝑡 𝑒𝛾𝑡 ∥𝑢𝑦 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑢𝑦𝑦 (𝜏 )∥2 𝑑𝜏 (2.2.40) 0 ∫ 𝑡 ∫ 𝑡 ( ) 𝛾𝜏 2 2 2 𝑒 𝑒𝛾𝜏 ∥𝑢𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶1 . ≤ 𝐶1 ∥𝜃𝑦 ∥ + ∥𝑣𝑦 ∥ + ∥𝑢𝑦 ∥ (𝜏 ) 𝑑𝜏 + 𝛾 0

0

By (2.1.17), ∥𝑢𝑡 (𝑡)∥ ≤ 𝐶1 (∥𝑢𝑦𝑦 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥), which, along with (2.2.40), gives ∫ 𝑡 𝑒𝛾𝜏 ∥𝑢𝑡 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶1 , 0

∀𝑡 > 0.

Similarly to (2.2.40)–(2.2.41), we have for 𝛾 ∈ (0, 𝛾1 ] that ∫ 𝑡 𝛾𝑡 2 𝑒 ∥𝑤 ⃗ 𝑦 (𝑡)∥ + 𝑒𝛾𝜏 (∥𝑤 ⃗ 𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡 (𝜏 )∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶1 , ∀𝑡 > 0. 0

(2.2.41)

(2.2.42)

Multiplying (2.1.19) by −𝑒𝜃 𝜃𝑦𝑦 and integrating the resulting equality over [0,1], we get ∫ 1 2 𝜅𝜃𝑦𝑦 1 𝑑 ∥𝜃𝑦 (𝑡)∥2 + 𝑑𝑦 2 𝑑𝑡 0 𝑒𝜃 𝑣 ) ∫ 1( 𝜆𝑢2𝑦 + 𝜇𝑤⃗𝑦 2 𝑝𝑢𝑦 𝜃𝑦𝑦 𝜅𝜃𝑦 𝑣𝑦 𝑒𝑣 𝑣𝑡 = 𝜃𝑦𝑦 + 𝜃𝑦𝑦 − + 𝜃𝑦𝑦 𝑑𝑦. (2.2.43) 𝑒𝜃 𝑒𝜃 𝑣 𝑒𝜃 𝑒𝜃 𝑣 2 0 Using the interpolation inequality, we have that for any 𝜀 > 0, ∫ 1 𝜅𝜃𝑦 𝑣𝑦 𝜃𝑦𝑦 𝑑𝑦 ≤ 𝐶1 ∥𝑣𝑦 (𝑡)∥∥𝜃𝑦 (𝑡)∥𝐿∞ ∥𝜃𝑦𝑦 (𝑡)∥ 𝑒𝜃 𝑣 2 0 1 3 ≤ 𝐶1 ∥𝑣𝑦 (𝑡)∥∥𝜃𝑦 (𝑡)∥ 2 ∥𝜃𝑦𝑦 (𝑡)∥ 2 ≤ 𝐶1 (∥𝑣𝑦 (𝑡)∥2 + ∥𝜃𝑦 (𝑡)∥2 ) + 𝜀∥𝜃𝑦𝑦 (𝑡)∥2 , ∫

1 0

𝜆𝑢2𝑦 𝜃𝑦𝑦 𝑑𝑦 ≤ 𝐶1 ∥𝑢𝑦 (𝑡)∥∥𝑢𝑦 (𝑡)∥𝐿∞ ∥𝜃𝑦𝑦 (𝑡)∥ 𝑒𝜃 𝑣 1 1 ≤ 𝐶1 ∥𝑢𝑦 (𝑡)∥∥𝑢𝑦 (𝑡)∥ 2 ∥𝑢𝑦𝑦 (𝑡)∥ 2 ∥𝜃𝑦𝑦 (𝑡)∥ ≤ 𝐶1 (∥𝑢𝑦 (𝑡)∥2 + ∥𝑢𝑦𝑦 (𝑡)∥2 ) + 𝜀∥𝜃𝑦𝑦 (𝑡)∥2 ,

∫ 0

1

(2.2.44)

(2.2.45)

𝜇∣𝑤 ⃗ 𝑦 ∣2 𝜃𝑦𝑦 𝑑𝑦 ≤ 𝐶1 ∥𝑤 ⃗ 𝑦 (𝑡)∥∥𝑤 ⃗ 𝑦 (𝑡)∥𝐿∞ ∥𝜃𝑦𝑦 (𝑡)∥ 𝑒𝜃 𝑣 1 1 ≤ 𝐶1 ∥𝑤 ⃗ 𝑦 (𝑡)∥∥𝑤 ⃗ 𝑦 (𝑡)∥ 2 ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥ 2 ∥𝜃𝑦𝑦 (𝑡)∥ ⃗ 𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥2 ) + 𝜀∥𝜃𝑦𝑦 (𝑡)∥2 . ≤ 𝐶1 (∥𝑤

(2.2.46)

46


Inserting (2.2.44)–(2.2.46) into (2.2.43), and using Lemma 2.2.1, we deduce that 𝑑 ∥𝜃𝑦 (𝑡)∥2 + 𝐶1−1 ∥𝜃𝑦𝑦 (𝑡)∥2 ≤ 𝐶1 (∥𝑢𝑦 (𝑡)∥2 + ∥𝜃𝑦 (𝑡)∥2 + ∥𝑣𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑦 (𝑡)∥2 ). 𝑑𝑡 (2.2.47) Multiplying (2.2.47) by 𝑒𝛾𝑡 , and integrating the resulting equation over [0, 𝑡], we have that there exists a constant 𝛾1 = 𝛾1 (𝐶1 ) ≤ 𝛾1′ , such that for any fixed 𝛾 ∈ (0, 𝛾1 ], ∫ 𝑡 𝑒𝛾𝜏 ∥𝜃𝑦𝑦 (𝜏 )∥2 𝑑𝜏 (2.2.48) 𝑒𝛾𝑡 ∥𝜃𝑦 (𝑡)∥2 + 0 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑢𝑦 ∥2 + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝑣𝑦 ∥2 )(𝜏 )𝑑𝜏 + 𝛾 𝑒𝛾𝜏 ∥𝜃𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶1 ≤ 𝐶1 0

0

which with ∥𝜃𝑡 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑦𝑦 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑦 (𝑡)∥) yields

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝜃𝑡 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶1 .

(2.2.49)


□

2.3 Proof of Theorem 2.1.2 In this section we shall complete the proof of Theorem 2.1.2 and take that the assumptions in Theorem 2.1.2 to be valid. We begin with the following lemma. ⃗ 0 , 𝜃0 ) ∈ Lemma 2.3.1. Under the assumptions of Theorem 2.1.2, for any (𝑣0 , 𝑢0 , 𝑤 2 𝐻+ , the following estimates hold for any 𝑡 > 0, ∥𝑢𝑡 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 ∫ 𝑡 ( ) ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝑤 + ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 (𝜏 )𝑑𝜏 ≤ 𝐶2 ,

(2.3.1)

2 ¯ 2 ∥𝑢(𝑡)∥2𝐻 2 + ∥𝑣(𝑡) − 𝑣¯∥2𝐻 2 + ∥𝑤(𝑡)∥ ⃗ 𝐻 2 + ∥𝜃(𝑡) − 𝜃∥𝐻 2 ≤ 𝐶2 .

(2.3.2)

0

Proof. Differentiating (2.1.17) with respect to 𝑡, multiplying the result by 𝑢𝑡 and integrating over [0,1], we infer that 1 𝑑 ∥𝑢𝑡 (𝑡)∥2 = − 2 𝑑𝑡

∫ 0

1

𝜆

𝑢2𝑡𝑦 𝑑𝑦 + 𝑣

∫ 0

1

𝜆

𝑢2𝑦 𝑢𝑡𝑦 𝑑𝑦 + 𝑣2

∫ 0

1

∫ 𝑝𝜃 𝜃𝑡 𝑢𝑡𝑦 𝑑𝑦 +

1 0

𝑝𝑣 𝑣𝑡 𝑢𝑡𝑦 𝑑𝑦. (2.3.3)


47

Using Cauchy’s inequality, Young’s inequality and the embedding theorem, we conclude that for any 𝜀 > 0, 1 𝑑 ∥𝑢𝑡 (𝑡)∥2 + 𝐶1−1 ∥𝑢𝑡𝑦 (𝑡)∥2 ≤ 𝜀∥𝑢𝑡𝑦 (𝑡)∥2 + 𝐶1 (∥𝑢𝑦 (𝑡)∥4𝐿4 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑢𝑦 (𝑡)∥2 ) 2 𝑑𝑡 ≤ 𝜀∥𝑢𝑡𝑦 (𝑡)∥2 + 𝐶2 (∥𝑢𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑢𝑦 (𝑡)∥2 ). (2.3.4) By equations (2.1.16)–(2.1.19), we get ∥𝑢𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑡 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥), ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥ ≤ 𝐶2 ∥𝑤 ⃗ 𝑡 (𝑡)∥, (2.3.5) ∥𝜃𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑡 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑦 (𝑡)∥)

(2.3.6)

or ∥𝑢𝑡 ∥ ≤ 𝐶2 (∥𝑢𝑦𝑦 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥), ∥𝑤 ⃗ 𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥, (2.3.7) ⃗ 𝑦 (𝑡)∥). ∥𝜃𝑡 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑦𝑦 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥ + ∥𝑤

(2.3.8)

2 Since (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃 0 ) ∈ 𝐻+ , we can infer from (2.3.7)–(2.3.8) that

∥𝑢𝑡 (𝑦, 0)∥ ≤ 𝐶2 , ∥𝑤 ⃗ 𝑡 (𝑦, 0)∥ ≤ 𝐶2 , ∥𝜃𝑡 (𝑦, 0)∥ ≤ 𝐶2 .

(2.3.9)

Integrating (2.3.4) on [0, 𝑡], using Lemma 2.2.1 and (2.3.7)–(2.3.9), we get ∫ 𝑡 ∥𝑢𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0, (2.3.10) ∥𝑢𝑡 (𝑡)∥2 + 0

which, together with (2.3.5), implies ∥𝑢𝑦𝑦 (𝑡)∥ ≤ 𝐶2 , ∀𝑡 > 0.

(2.3.11)

Differentiating (2.1.18) with respect to 𝑡, multiplying the resulting equation by 𝑤 ⃗ 𝑡 and integrating the resulting equality over [0,1], we infer that ∫ 1 ∫ 1 1 𝑑 ⃗ 𝑡𝑦 𝑣𝑡 ∣𝑤 ⃗ 𝑡𝑦 ∣2 𝑤 ⃗𝑦 ⋅ 𝑤 2 ∥𝑤 ⃗ 𝑡 (𝑡)∥ = − 𝑑𝑦 + 𝜇 𝜇 𝑑𝑦, 2 2 𝑑𝑡 𝑣 𝑣 0 0 which, by using Cauchy’s inequality, Young’s inequality and the embedding theorem, gives that for 𝜀 > 0 small enough, 1 𝑑 ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + 𝐶1−1 ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 ≤ 𝜀∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + 𝐶1 ∥𝑣𝑡 𝑤 ⃗ 𝑦 ∥2 2 𝑑𝑡 ≤ 𝜀∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + 𝐶1 ∥𝑣𝑡 (𝑡)∥2𝐿∞ ∥𝑤 ⃗ 𝑦 (𝑡)∥2 .

(2.3.12)

Thus integrating (2.3.12) over [0, 𝑡] and using Lemma 2.2.1, we get ∫ 𝑡 ∫ 𝑡 ∥𝑤 ⃗ 𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 ∥𝑤 ⃗ 𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶1 ∥𝑤 ⃗ 𝑡 (𝑦, 0)∥2 ≤ 𝐶2 , (2.3.13) ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + 0

0

48


which, along with (2.3.5), gives ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥ ≤ 𝐶2 , ∀𝑡 > 0.

(2.3.14)

By (2.1.17)–(2.1.19) and (2.1.23), we get 𝑒𝜃 𝜃𝑡 + 𝑒𝑣 𝑣𝑡 = −𝑝𝑢𝑦 +

𝜆𝑢2𝑦 𝜇∣𝑤 ⃗ 𝑦 ∣2 + + 𝑣 𝑣

(

𝜅𝜃𝑦 𝑣

) 𝑦

.

(2.3.15)

Differentiating (2.3.15) with respect to 𝑡, multiplying the result 𝜃𝑡 , and finally integrating the resultant over [0,1], we get 1 𝑑 √ ∥ 𝑒 𝜃 𝜃𝑡 ∥ 2 + 2 𝑑𝑡

∫

1

0

∫ 1( 𝑢2𝑦 𝑣𝑡 ∣𝑤 ⃗ 𝑦 ∣2 𝑣𝑡 ⃗ 𝑦𝑡 𝑤 ⃗𝑦 ⋅ 𝑤 𝑢𝑦 𝑢𝑦𝑡 − 𝜆 2 + 2𝜇 −𝜇 2𝜆 𝑣 𝑣 𝑣 𝑣2 0

2 𝜅𝜃𝑡𝑦 𝑑𝑦 = 𝑣

− 𝑢𝑦𝑡 𝑝 − 𝑢𝑦 𝑝𝜃 𝜃𝑡 − 𝑢𝑦 𝑝𝑣 𝑣𝑡 − 𝑒𝑣 𝑣𝑡𝑡 − 𝑒𝑣𝑣 𝑣𝑡2 ) 1 3 𝜅𝜃𝑡𝑦 𝜃𝑦 𝑣𝑡 − 𝑒𝜃𝜃 𝜃𝑡2 − 𝑒𝜃𝑣 𝑣𝑡 𝜃𝑡 + 𝜃𝑡 𝑑𝑦. (2.3.16) 2 2 𝑣2

Integrating (2.3.16) over [0,t], using the Cauchy inequality, the Young inequality and the embedding theorem, we conclude that for any 𝜀 > 0, ∫ 𝑡 1 𝑑 √ ∥ 𝑒𝜃 𝜃𝑡 ∥2 + 𝐶1−1 ∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 2 𝑑𝑡 0 ∫ 𝑡( ∫ 𝑡 ∥𝜃𝑦 ∥2 + ∥𝜃𝑦𝑦 ∥2 + ∥𝜃𝑡 ∥2 ∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶1 ≤𝜀 0 0 ) (2.3.17) + ∥𝑢𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 + ∥𝜃𝑡 (𝜏 )∥3𝐿3 𝑑𝜏. By the Nirenberg interpolation inequality, we can get for any 𝜀 > 0, ∫ 𝑡 ∫ 𝑡( ) 5 1 3 ∥𝜃𝑡 ∥ 2 ∥𝜃𝑡𝑦 ∥ 2 + ∥𝜃𝑡 ∥3 (𝜏 )𝑑𝜏 ∥𝜃𝑡 ∥𝐿3 (𝜏 )𝑑𝜏 ≤ 𝐶1 0

0

≤ 𝐶1

∫ 𝑡( ∫ 𝑡 ) 5 2 3 sup ∥𝜃𝑡 (𝜏 )∥ ∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ∥𝜃𝑡 ∥ + ∥𝜃𝑡 ∥ (𝜏 )𝑑𝜏 + 𝜀 4 3

0≤𝜏 ≤𝑡

0

4

≤ 𝐶1 sup ∥𝜃𝑡 (𝜏 )∥ 3 + 𝜀 0≤𝜏 ≤𝑡

1 sup ∥𝜃𝑡 (𝜏 )∥2 + 𝜀 ≤ 2 0≤𝜏 ≤𝑡

0

∫ ∫ 0

𝑡

0 𝑡

∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏

∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶2 .

(2.3.18)

Hence, inserting (2.3.18) into (2.3.17), and taking 𝜀 > 0 small enough, we get ∥𝜃𝑡 (𝑡)∥2 +

∫ 0

𝑡

∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0.

(2.3.19)


49

By (2.3.6) and (2.3.19), we conclude that ∥𝜃𝑦𝑦 (𝑡)∥ ≤ 𝐶2 , ∀𝑡 > 0. The proof is complete.

□

Lemma 2.3.2. Under the assumptions of Theorem 2.1.2, for any (𝑢0 , 𝑣0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 2 𝐻+ , the following estimate holds for any 𝑡 > 0: ∥𝑣(𝑡)∥2𝐻 2 +

∫ 0

𝑡

∥𝑣(𝜏 )∥2𝐻 2 𝑑𝜏 ≤ 𝐶2 .

Proof. Differentiating (2.1.17) with respect to 𝑦, by equation (2.1.16), 𝑣𝑡𝑦𝑦 = 𝑢𝑦𝑦𝑦 , we get 𝑢𝑦 𝑣𝑦2 ∂ ( 𝑣𝑦𝑦 ) 𝑢𝑦𝑦 𝑣𝑦 − 𝑝𝑣 𝑣𝑦𝑦 = 𝑢𝑡𝑦 + 𝑝𝑣𝑣 𝑣𝑦2 + 2𝑝𝑣𝜃 𝑣𝑦 𝜃𝑦 + 𝑝𝜃 𝜃𝑦𝑦 + 𝑝𝜃𝜃 𝜃𝑦2 + 2𝜆 2 − 2𝜆 3 ∂𝑡 𝑣 𝑣 𝑣 (2.3.20) where 𝑣𝑦𝑦𝑡 𝑣𝑦𝑦 𝑣𝑡 ∂ ( 𝑣𝑦𝑦 ) =𝜆 −𝜆 2 . 𝜆 ∂𝑡 𝑣 𝑣 𝑣 𝑣𝑦𝑦 Multiplying (2.3.20) by , we get 𝑣 𝑣 2 𝐶 −1 𝑣 2 ( 𝜆 𝑑 𝑦𝑦 𝑣𝑦𝑦 2 𝑦𝑦 + 𝐶1−1 ≤ 1 + 𝐶2 ∥𝑢𝑦𝑦 𝑣𝑦 ∥2 + ∥𝑢𝑦 𝑣𝑦2 ∥2 + ∥𝜃𝑦 𝑣𝑦 ∥2 2 𝑑𝑡 𝑣 𝑣 4 𝑣 ) + ∥𝑣𝑦 ∥4𝐿4 + ∥𝜃𝑦 ∥4𝐿4 + ∥𝜃𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 . 𝜆

(2.3.21) Integrating (2.3.21) over [0, 𝑡] gives ∫ 𝑡 ∫ 𝑡( 2 2 ∥𝑣𝑦𝑦 (𝑡)∥ + ∥𝑣𝑦𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶2 ∥𝑢𝑦𝑦 ∥2 + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥4𝐿4 + ∥𝜃𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥2 0 0 ) +∥𝑣𝑦 ∥4𝐿4 + ∥𝜃𝑦 ∥4𝐿4 + ∥𝜃𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 (𝜏 )𝑑𝜏. Using the Young inequality, the Poincaré inequality, the Nirenberg interpolation inequality, Lemmas 2.2.1 and 2.3.1, we infer that ∫ 𝑡 ∫ 𝑡 ( ) 2 2 ∥𝑣𝑦𝑦 (𝑡)∥ + ∥𝑣𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦𝑦 ∥2 + ∥𝜃𝑦𝑦 ∥2 (𝜏 )𝑑𝜏 ∥𝑣𝑦𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶2 0

0

≤ 𝐶2 . The proof is complete.

□

50


Lemma 2.3.3. Under the assumptions of Theorem 2.1.2, for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 2 , the following estimate holds for any 𝑡 > 0: 𝐻+ ∫ 𝑡 ( ) ¯ 2 3 𝑑𝜏 ≤ 𝐶2 . ∥𝑢(𝜏 )∥2𝐻 3 + ∥𝑤(𝜏 ⃗ )∥2𝐻 3 + ∥𝜃(𝜏 ) − 𝜃∥ (2.3.22) 𝐻 0

Proof. Differentiating (2.1.17) with respect to 𝑦, we infer that 𝜆𝑢𝑦𝑦𝑦 = 𝑢𝑡𝑦 + 𝑝𝜃 𝜃𝑦𝑦 + 𝑝𝑣 𝑣𝑦𝑦 + 2𝑝𝜃𝑣 𝜃𝑦 𝑣𝑦 + 𝑝𝜃𝜃 𝜃𝑦2 + 𝑝𝑣𝑣 𝑣𝑦2 𝑣 2𝜆𝑢𝑦 𝑣𝑦2 2𝜆𝑢𝑦𝑦 𝑣𝑦 𝜆𝑢𝑦 𝑣𝑦𝑦 + + − . 𝑣2 𝑣2 𝑣3

(2.3.23)

Multiplying (2.3.23) by 𝑢𝑦𝑦𝑦 , integrating the resultant over [0,1], using Cauchy’s inequality and Young’s inequality, we get for any 𝜀 > 0, ( ∥𝑢𝑦𝑦𝑦 (𝑡)∥2 ≤ 𝐶2 ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑦𝑦 (𝑡)∥2 + ∥𝑣𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑦 𝑣𝑦 ∥2 + ∥𝜃𝑦 (𝑡)∥4𝐿4 ) + ∥𝑣𝑦 (𝑡)∥4𝐿4 + ∥𝑢𝑦𝑦 𝑣𝑦 ∥2 + ∥𝑣𝑦𝑦 𝑢𝑦 ∥2 + ∥𝑢𝑦 𝑣𝑦2 ∥2 + 𝜀∥𝑢𝑦𝑦𝑦 (𝑡)∥2 . (2.3.24) Integrating (2.3.24) over [0, 𝑡], by Lemma 2.2.1 and Lemma 2.3.1, we get ∫ 𝑡 ∫ 𝑡 ( ) 2 ∥𝑢𝑦𝑦𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶2 ∥𝑢𝑡𝑦 ∥2 + ∥𝜃𝑦𝑦 ∥2 + ∥𝑣𝑦𝑦 ∥2 + ∥𝑢𝑦𝑦 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶2 . 0

0

Differentiating (2.1.18) with respect to 𝑦, we infer that 2𝜇𝑤 ⃗ 𝑦 𝑣𝑦2 𝜇𝑤 ⃗ 𝑦𝑦𝑦 2𝜇𝑤 ⃗ 𝑦𝑦 𝑣𝑦 𝜇𝑤 ⃗ 𝑦 𝑣𝑦𝑦 =𝑤 ⃗ 𝑡𝑦 + + − . 𝑣 𝑣2 𝑣2 𝑣3

(2.3.25)

(2.3.26)

Multiplying (2.3.26) by 𝑤 ⃗ 𝑦𝑦𝑦 , integrating the resulting equality over [0, 1], using Cauchy’s inequality and Young’s inequality, we get that for any small 𝜀 > 0, ( ) ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝑣𝑦𝑦 (𝑡)∥2 + ∥𝑤 (2.3.27) ⃗ 𝑦𝑦 (𝑡)∥2 . ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2 ≤ 𝐶2 ∥𝑤 Integrating (2.3.27) over [0, 𝑡], by Lemma 2.2.1 and Lemma 2.3.1, we arrive at ∫ 𝑡 ∫ 𝑡 ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 (∥𝑤 ⃗ 𝑡𝑦 ∥2 + ∥𝑣𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 . (2.3.28) 0

0

Differentiating (2.1.19) with respect to 𝑦, we obtain 𝜅𝜃𝑦𝑦𝑦 = 𝑒𝜃 𝜃𝑡𝑦 + 𝑒𝑣 𝑣𝑡𝑦 + 𝑒𝜃𝑣 𝜃𝑡 𝑣𝑦 + 𝑒𝜃𝑣 𝜃𝑦 𝑣𝑡 + 𝑒𝜃𝜃 𝜃𝑦 𝜃𝑡 + 𝑒𝑣𝑣 𝑣𝑦 𝑣𝑡 𝑣 𝜆𝑣𝑦 𝑢2𝑦 ⃗𝑦 ⃗ 𝑦 ∣2 2𝜆𝑢𝑦𝑦 𝑢𝑦 2𝜇𝑤 ⃗ 𝑦𝑦 ⋅ 𝑤 𝜇𝑣𝑦 ∣𝑤 2𝜅𝜃𝑦𝑦 𝑣𝑦 − + − + + 2 2 2 2 𝑣 𝑣 𝑣 𝑣 𝑣2 2 2𝜅𝜃𝑦 𝑣𝑦 𝜅𝜃𝑦 𝑣𝑦𝑦 + − + 𝑢𝑦𝑦 𝑝 + 𝑝𝜃 𝜃𝑦 𝑢𝑦 + 𝑝𝑣 𝑢𝑦 𝑣𝑦 . (2.3.29) 𝑣2 𝑣3


51

Multiplying (2.3.29) by 𝜃𝑦𝑦𝑦 , integrating the resultant over [0,1] and using Cauchy’s inequality, we deduce that for any small 𝜀 > 0, ( ∥𝜃𝑦𝑦𝑦 (𝑡)∥2 ≤ 𝐶2 ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 + ∥𝑣𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑦 𝑣𝑦 ∥2 + ∥𝜃𝑦 (𝑡)∥4𝐿4 ) + ∥𝑣𝑦 (𝑡)∥4𝐿4 + ∥𝑢𝑦𝑦 𝑣𝑦 ∥2 + ∥𝑣𝑦𝑦 𝑢𝑦 ∥2 + ∥𝑢𝑦 𝑣𝑦2 ∥2 + 𝜀∥𝜃𝑦𝑦𝑦 (𝑡)∥2 , i.e.,

( ∥𝜃𝑦𝑦𝑦 (𝑡)∥2 ≤ 𝐶2 ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 + ∥𝑣𝑦𝑦 (𝑡)∥2 + ∥𝑣𝑦𝑦 (𝑡)∥2 ) + ∥𝑢𝑦 (𝑡)∥2 + ∥𝑣𝑦 (𝑡)∥2 + ∥𝜃𝑦 (𝑡)∥2 .

(2.3.30)

Integrating (2.3.30) over [0, 𝑡], and using Lemma 2.2.1 and Lemma 2.3.1, we get ∫ 𝑡( ∫ 𝑡 ∥𝑢𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦 ∥2 + ∥𝑣𝑦𝑦 ∥2 + ∥𝑢𝑦𝑦 ∥2 + ∥𝜃𝑦 ∥2 ∥𝜃𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 0 0 ) + ∥𝑢𝑦 ∥2 + ∥𝑣𝑦 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶2 . The proof is complete.

□

Lemma 2.3.4. There exist constants 𝐶2 > 0 and 𝛾2′ = 𝛾2′ (𝐶2 ) > 0 such that 2 for any fixed 𝛾 ∈ (0, 𝛾2′ ], the global solution (𝑣(𝑡), 𝑢(𝑡), 𝑤(𝑡), ⃗ 𝜃(𝑡)) ∈ 𝐻+ to problem (2.1.16)–(2.1.21) or (2.1.16)–(2.1.20), (2.1.22) satisfies that the following estimates: ( ) 2 ¯ 2 (2.3.31) ⃗ 𝑒𝛾𝑡 ∥𝑣(𝑡) − 𝑣¯∥2𝐻 2 + ∥𝑢(𝑡)∥2𝐻 2 + ∥𝑤(𝑡)∥ 𝐻 2 + ∥𝜃(𝑡) − 𝜃∥𝐻 2 ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑣 − 𝑣¯∥2𝐻 2 + ∥𝑢∥2𝐻 3 + ∥𝑤∥ ⃗ 2𝐻 3 + ∥𝜃∥2𝐻 3 (𝜏 )𝑑𝜏 ≤ 𝐶2 , ∀ 𝑡 > 0, 0

( ) 𝑒 ∥𝑢𝑡 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑢𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶2 , ∀ 𝑡 > 0. 𝛾𝑡

(2.3.32)

0

Proof. Multiplying (2.3.4) by 𝑒𝛾𝑡 , and integrating the resulting equality over [0, 𝑡], we have ∫ 𝑡 𝑒𝛾𝑡 ∥𝑢𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑢𝑡𝑦 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑢𝑦𝑦 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑢𝑦 ∥2 )(𝜏 )𝑑𝜏 + 𝛾 𝑒𝛾𝜏 ∥𝑢𝑡 (𝜏 )∥2 𝑑𝜏. (2.3.33) ≤ 𝐶2 0

0

By Lemma 2.2.4 and (2.3.33), we can infer that there exists a constant 𝛾2′ = 𝛾2′ (𝐶1 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2′ ], ∫ 𝑡 𝑒𝛾𝜏 ∥𝑢𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 . (2.3.34) 𝑒𝛾𝑡 ∥𝑢𝑡 (𝑡)∥2 + 0

52


Multiplying (2.3.24) by 𝑒𝛾𝑡 , and integrating the resulting equality over [0, 𝑡], we get ∫ 𝑡 𝑒𝛾𝜏 ∥𝑢𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 . (2.3.35) 0

𝛾𝑡

Multiplying (2.3.12) by 𝑒 , and integrating the result over [0, 𝑡], we have ∫ 𝑡 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝑡 ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑦 (𝜏 )∥2 𝑑𝜏 + 𝛾 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡 (𝜏 )∥2 𝑑𝜏. 0

0

0

(2.3.36) By Lemma 2.2.4 and (2.3.36), we can derive that there exists a constant 𝛾2′ = 𝛾2′ (𝐶1 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2′ ], ∫ 𝑡 𝑒𝛾𝑡 ∥𝑤 ⃗ 𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 . (2.3.37) 0

Multiplying (2.3.28) by 𝑒𝛾𝑡 , and integrating the resultant over [0, 𝑡], we get ∫ 𝑡 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑦𝑦𝑦 𝜏 )∥2 (𝑑𝜏 ≤ 𝐶2 . (2.3.38) 0

𝛾𝑡

Multiplying (2.3.17) by 𝑒 , and integrating the resultant over [0, 𝑡], we have ∫ 𝑡 𝛾𝑡 2 𝑒 ∥𝜃𝑡 (𝑡)∥ + 𝑒𝛾𝜏 ∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 ( 𝑒𝛾𝜏 ∥𝜃𝑦 ∥2 + ∥𝜃𝑦𝑦 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑢𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 ≤ 𝐶2 0 ) (2.3.39) + ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 + ∥𝜃𝑡 ∥3𝐿3 (𝜏 ) 𝑑𝜏. By Lemma 2.2.4 and (2.3.37), we get that there exists a constant 𝛾2′ = 𝛾2′ (𝐶1 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2′ ], ∫ 𝑡 𝑒𝛾𝑡 ∥𝜃𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝜃𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 . (2.3.40) 0

𝛾𝑡

Multiplying (2.3.30) by 𝑒 , and integrating the resultant over [0, 𝑡], we get ∫ 𝑡 𝑒𝛾𝜏 ∥𝜃𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0. (2.3.41) 0

Multiplying (2.3.21) by 𝑒𝛾𝑡 , and integrating over [0, 𝑡], we have ∫ 𝑡 𝑒𝛾𝑡 ∥𝑣𝑦𝑦 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑣𝑦𝑦 (𝜏 )∥2 𝑑𝜏 (2.3.42) 0 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦𝑦𝑦 ∥2 + ∥𝜃𝑦𝑦 ∥2 )(𝜏 )𝑑𝜏 + 𝛾 𝑒𝛾𝜏 ∥𝑣𝑡 (𝜏 )∥2 𝑑𝜏. ≤ 𝐶2 0

0


53

By Lemma 2.2.3 and (2.3.42), we can deduce that there exists a constant 𝛾2′ = 𝛾2′ (𝐶1 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2′ ], ∫ 𝑡 𝛾𝑡 2 𝑒 ∥𝑣𝑦𝑦 (𝑡)∥ + 𝑒𝛾𝜏 ∥𝑣𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 . 0


□

2.4 Proof of Theorem 2.1.3 Lemma 2.4.1. Under the assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 4 𝐻+ and for any 𝜀 ∈ (0, 1) small enough, we have for any 𝑡 > 0, ∥𝑢𝑡𝑦 (𝑦, 0)∥ + ∥𝑤 ⃗ 𝑡𝑦 (𝑦, 0)∥ + ∥𝜃𝑡𝑦 (𝑦, 0)∥ ≤ 𝐶3 , ∥𝑢𝑡𝑡 (𝑦, 0)∥ + ∥𝑤 ⃗ 𝑡𝑡 (𝑦, 0)∥ + ∥𝜃𝑡𝑡 (𝑦, 0)∥ + ∥𝑢𝑡𝑦𝑦 (𝑦, 0)∥ + ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑦, 0)∥ + ∥𝜃𝑡𝑦𝑦 (𝑦, 0)∥ ≤ 𝐶4 , ∫ 𝑡 ∫ 𝑡 2 2 ∥𝑢𝑡𝑡𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 + 𝐶4 ∥𝜃𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏, ∥𝑢𝑡𝑡 (𝑡)∥ + ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 + 2

∫

0 ∫ 𝑡 0

∥𝑤 ⃗ 𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4

∫

0

0

𝑡

∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏,

(2.4.1) (2.4.2) (2.4.3) (2.4.4)

𝑡

∥𝜃𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 (2.4.5) ∫ 𝑡 ∫ 𝑡 ( ) ⃗ 𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏. ≤ 𝐶4 𝜀−3 + 𝐶2 𝜀−1 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶1 𝜀 ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤

∥𝜃𝑡𝑡 (𝑡)∥ +

0

0

0

Proof. By (2.1.17), we can derive ∥𝑢𝑡 ∥ ≤ 𝐶1 (∥𝑢𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥ + ∥𝑣𝑦 ∥ + ∥𝜃𝑦 ∥) ≤ 𝐶2 (∥𝑢𝑦 ∥𝐻 1 + ∥𝑣𝑦 ∥ + ∥𝜃𝑦 ∥).

(2.4.6)

We differentiate (2.1.17) with respect to 𝑦, and use Theorems 2.1.1–2.1.2 to derive ( ∥𝑢𝑡𝑦 ∥ ≤ 𝐶1 ∥𝑣𝑦 ∥2𝐿4 + ∥𝑣𝑦 ∥𝐿∞ ∥𝜃𝑦 ∥ + ∥𝑣𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥2𝐿4 + ∥𝜃𝑦 ∥2𝐿4 ) (2.4.7) + ∥𝑢𝑦𝑦𝑦 ∥ + ∥𝜃𝑦𝑦 ∥ + ∥𝑣𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦 ∥ + ∥𝑣𝑦𝑦 ∥∥𝑢𝑦 ∥𝐿∞ ∥ . Using the Gagliardo-Nirenberg inequality and the Young inequality, we conclude 1

1

3

1

∥𝑢𝑦 ∥𝐿∞ ≤ 𝐶(∥𝑢𝑦 ∥ 2 ∥𝑢𝑦𝑦 ∥ 2 + ∥𝑢𝑦𝑦 ∥) ≤ 𝐶(∥𝑢𝑦 ∥ + ∥𝑢𝑦𝑦 ∥), 3

1

∥𝑢𝑦 ∥2𝐿4 ≤ 𝐶(∥𝑢𝑦 ∥ 4 ∥𝑢𝑦𝑦 ∥ 4 + ∥𝑢𝑦 ∥)2 ≤ 𝐶(∥𝑢𝑦 ∥ 2 ∥𝑢𝑦𝑦 ∥ 2 + ∥𝑢𝑦 ∥2 ) ≤ 𝐶(∥𝑢𝑦 ∥2 + ∥𝑢𝑦𝑦 ∥2 ).

54


Finally, using (2.4.7), we can obtain ∥𝑢𝑡𝑦 ∥ ≤ 𝐶2 (∥𝑣𝑦 ∥𝐻 1 + ∥𝑢𝑦 ∥𝐻 2 + ∥𝜃𝑦 ∥𝐻 1 )

(2.4.8)

or ∥𝑢𝑦𝑦𝑦 ∥ ≤ 𝐶2 (∥𝑢𝑡𝑦 ∥ + ∥𝑣𝑦 ∥𝐻 1 + ∥𝑢𝑦 ∥𝐻 1 + ∥𝜃𝑦 ∥𝐻 1 ).

(2.4.9)

We differentiate (2.1.17) with respect to 𝑦 twice to derive ( ∥𝑢𝑡𝑦𝑦 ∥ ≤ 𝐶1 ∥𝑣𝑦 ∥3𝐿6 + ∥𝑣𝑦 ∥2𝐿4 ∥𝜃𝑦 ∥𝐿∞ + ∥𝑣𝑦 ∥𝐿∞ ∥∥𝑣𝑦𝑦 ∥ + ∥𝜃𝑦 ∥𝐿∞ ∥∥𝑣𝑦𝑦 ∥ + ∥𝑣𝑦 ∥𝐿∞ ∥𝜃𝑦2 ∥ + ∥𝑣𝑦 ∥𝐿∞ ∥∥𝜃𝑦𝑦 ∥ + ∥𝜃𝑦 ∥3𝐿6 + ∥𝑣𝑦𝑦𝑦 ∥ + ∥𝜃𝑦𝑦𝑦 ∥ + ∥𝜃𝑦 ∥𝐿∞ ∥𝜃𝑦𝑦 ∥ + ∥𝜃𝑦𝑦 ∥ + ∥𝑣𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦𝑦 ∥ + ∥𝑢𝑦𝑦 ∥𝐿∞ ∥𝑣𝑦𝑦 ∥ + ∥𝑢𝑦𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥2𝐿4 + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦𝑦 ∥∥𝑣𝑦 ∥ ) (2.4.10) + ∥𝑢𝑦𝑦𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥3𝐿6 . Using the Gagliardo-Nirenberg and the Young inequality, we conclude ∥𝑢𝑡𝑦𝑦 ∥ ≤ 𝐶2 (∥𝑢𝑦 ∥𝐻 3 + ∥𝑣𝑦 ∥𝐻 2 + ∥𝜃𝑦 ∥𝐻 2 )

(2.4.11)

or ∥𝑢𝑦𝑦𝑦𝑦 ∥ ≤ 𝐶2 (∥𝑢𝑡𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐻 2 + ∥𝑣𝑦 ∥𝐻 2 + ∥𝜃𝑦 ∥𝐻 2 ) .

(2.4.12)

By (2.1.18), we can derive ∥𝑤 ⃗ 𝑡 ∥ ≤ 𝐶1 (∥𝑤 ⃗ 𝑦𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥) ≤ 𝐶2 (∥𝑤 ⃗ 𝑦 ∥𝐻 1 + ∥𝑣𝑦 ∥).

(2.4.13)

We differentiate (2.1.18) with respect to 𝑦, and use Theorems 2.1.1–2.1.2 to get ) ( ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑤 ⃗ 𝑦𝑦 ∥∥𝑣𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑣𝑦𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑣𝑦2 ∥ . ⃗ 𝑦𝑦𝑦 ∥ + ∥𝑤 Using the Gagliardo-Nirenberg inequality and the Young inequality, we conclude

or

∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑤 ⃗ 𝑦 ∥𝐻 2 + ∥𝑣𝑦 ∥𝐻 1 )

(2.4.14)

∥𝑤 ⃗ 𝑦𝑦𝑦 ∥ ≤ 𝐶2 (∥𝑤 ⃗ 𝑦 ∥𝐻 1 + ∥𝑣𝑦 ∥𝐻 1 + ∥𝑤 ⃗ 𝑡𝑦 ∥).

(2.4.15)

We differentiate (2.1.18) with respect to 𝑦 twice to derive ( ⃗ 𝑦𝑦𝑦𝑦 ∥ + ∥𝑣𝑦 ∥𝐿∞ ∥𝑤 ⃗ 𝑦𝑦𝑦 ∥ + ∥𝑣𝑦𝑦 ∥𝐿∞ ∥𝑤 ⃗ 𝑦𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑤 ⃗ 𝑦𝑦𝑦 ∥ ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑤 ) ⃗ 𝑦 ∥𝐿∞ ∥𝑣𝑦3 ∥ + ∥𝑤 ⃗ 𝑦 𝑣𝑦 ∥∥𝑣𝑦𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑦𝑦 ∥𝐿∞ ∥𝑣𝑦2 ∥ + ∥𝑤 ≤ 𝐶2 (∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 3 + ∥𝑣𝑦 (𝑡)∥𝐻 2 )

(2.4.16)


or

⃗ 𝑡𝑦𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 2 + ∥𝑣𝑦 (𝑡)∥𝐻 2 ). ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑤

55

(2.4.17)

By (2.1.19), we can derive ( ) ∥𝜃𝑡 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑦 (𝑡)∥ + ∥𝑢2𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑦2 (𝑡)∥ + ∥𝜃𝑦𝑦 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥𝐿∞ ∥𝑣𝑦 (𝑡)∥ ⃗ 𝑦 (𝑡)∥𝐻 1 + ∥𝜃𝑦 (𝑡)∥𝐻 1 ). ≤ 𝐶2 (∥𝑢𝑦 (𝑡)∥𝐻 1 + ∥𝑤

(2.4.18)

Differentiate (2.1.19) with respect to 𝑦, and use Theorems 2.1.1–2.1.2 to obtain ∥𝜃𝑡𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑦 (𝑡)∥𝐻 1 + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 1 + ∥𝜃𝑦 (𝑡)∥𝐻 2 + ∥𝑣𝑦 (𝑡)∥𝐻 1 )

(2.4.19)

or ∥𝜃𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑡𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥𝐻 1 + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 1 + ∥𝜃𝑦 (𝑡)∥𝐻 1 + ∥𝑣𝑦 (𝑡)∥𝐻 1 ) . (2.4.20) Differentiate (2.1.18) with respect to 𝑦 twice to derive ∥𝜃𝑡𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑦 (𝑡)∥𝐻 2 + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 2 + ∥𝜃𝑦 (𝑡)∥𝐻 3 + ∥𝑣𝑦 (𝑡)∥𝐻 2 )

(2.4.21)

or ∥𝜃𝑦𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑡𝑦𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥𝐻 2 + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 2 + ∥𝜃𝑦 (𝑡)∥𝐻 2 + ∥𝑣𝑦 (𝑡)∥𝐻 2 ) . (2.4.22) Differentiate (2.1.17) with respect to 𝑡 to obtain ( ∥𝑢𝑡𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝜃𝑡𝑦 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥𝐿∞ ∥𝜃𝑦 (𝑡)∥ + ∥𝜃𝑦 (𝑡)∥𝐿∞ ∥𝑢𝑦 (𝑡)∥ + ∥𝑣𝑡𝑦 (𝑡)∥ + ∥𝑢𝑡𝑦𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥𝐿∞ ∥𝜃𝑡 (𝑡)∥ + ∥𝑢𝑡𝑦 (𝑡)∥∥𝑣𝑦 (𝑡)∥𝐿∞ ) + ∥𝑣𝑦 (𝑡)∥𝐿∞ ∥𝑢𝑦𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥𝐿∞ ∥𝑣𝑦2 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥𝐿∞ ∥𝑣𝑦 (𝑡)∥ ( ) (2.4.23) ≤ 𝐶2 ∥𝜃𝑦 (𝑡)∥𝐻 2 + ∥𝑣𝑦 (𝑡)∥𝐻 2 + ∥𝑢𝑦 (𝑡)∥𝐻 3 . We differentiate (2.1.18) with respect to 𝑡 to deduce ( ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑡)∥ + ∥𝑢𝑦 (𝑡)∥𝐿∞ ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥∥𝑣𝑦 (𝑡)∥𝐿∞ ) + ∥𝑢𝑦𝑦 (𝑡)∥∥𝑤 ⃗ 𝑦 (𝑡)∥𝐿∞ + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐿∞ ∥𝑣𝑦 (𝑡)𝑢𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 3 + ∥𝑢𝑦 (𝑡)∥𝐻 2 + ∥𝑣𝑦 (𝑡)∥𝐻 2 ) .

(2.4.24)

We differentiate (2.1.19) with respect to 𝑡 to infer ( ∥𝜃𝑡𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝜃𝑡 ∥2𝐿4 + ∥𝑢𝑦 ∥2𝐿4 + ∥𝜃𝑡 ∥∥𝑢𝑦 ∥𝐿∞ + ∥𝑢𝑡𝑦 ∥ + ∥𝑢𝑡𝑦 ∥∥𝑢𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑡𝑦 ∥∥𝑤 ⃗ 𝑦 ∥𝐿∞ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑤 ⃗ 𝑡 ∥2𝐿4 + ∥𝜃𝑡𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝜃𝑦𝑦 ∥ ) + ∥𝜃𝑦𝑡 ∥∥𝑣𝑦 ∥𝐿∞ + ∥𝜃𝑦 ∥𝐿∞ ∥𝑣𝑦𝑡 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦 𝜃𝑦 ∥

56


( ≤ 𝐶2 ∥𝑢𝑦 ∥𝐻 1 + ∥𝑣𝑦 ∥𝐻 1 + ∥𝜃𝑦 ∥𝐻 2 + ∥𝜃𝑡𝑦𝑦 ∥ ) + ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑡𝑦 ∥ + ∥𝑤 ⃗ 𝑡𝑦 ∥ , ( ) ≤ 𝐶2 ∥𝑢𝑦 (𝑡)∥𝐻 2 + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 2 + ∥𝜃𝑦 (𝑡)∥𝐻 3 + ∥𝑣𝑦 (𝑡)∥𝐻 2 .

(2.4.25) (2.4.26)

Thus estimates (2.4.1)–(2.4.2) follow from (2.4.6)–(2.4.25). Differentiating (2.1.17) with respect to 𝑡 twice, multiplying the resulting equation by 𝑢𝑡𝑡 in 𝐿2 (0, 1), performing an integration by parts, using Theorems 2.1.1–2.1.2, we arrive at ∫ 1 ( ∣𝑢𝑡𝑡𝑦 ∣2 1 𝑑 ∥𝑢𝑡𝑡 (𝑡)∥2 ≤ −𝜆 𝑑𝑦 + 𝐶2 ∥𝑢𝑦 ∥2𝐿4 + ∥𝜃𝑡𝑡 ∥ + ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑡 ∥2𝐿4 2 𝑑𝑡 𝑣 0 ) + ∥𝜃𝑡 ∥𝐿∞ ∥𝑢𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑢𝑡𝑦 ∥ + ∥𝑢𝑦 ∥3𝐿6 ∥𝑢𝑡𝑡𝑦 ∥ ( ) ≤ −𝐶1−1 ∥𝑢𝑡𝑡𝑦 ∥2 + 𝐶2 ∥𝑢𝑦 ∥2𝐻 2 + ∥𝜃𝑡 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑣𝑦 ∥2 + ∥𝜃𝑡𝑡 ∥2 . (2.4.27) Thus, by Theorems 2.1.1–2.1.2, we deduce ∫ 𝑡 ∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝑢𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡( ) ∥𝑢𝑦 ∥2𝐻 2 + ∥𝜃𝑡 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑣𝑦 ∥2 + ∥𝜃𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ ∥𝑢𝑡𝑡 (𝑦, 0)∥2 + 𝐶 ∫ ≤ 𝐶4 + 𝐶2

0

𝑡

0

∥𝜃𝑡𝑡 (𝜏 )∥2 𝑑𝜏.

By (2.4.25), we conclude ∫ 𝑡 ∫ 𝑡 ∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝑢𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 0

0

which, along with Theorems 2.1.1–2.1.2, gives estimate (2.4.3). Differentiating (2.1.18) with respect to 𝑡 twice, multiplying the resulting equation by 𝑤 ⃗ 𝑡𝑡 in 𝐿2 (0, 1), performing an integration by parts, and using Theorems 2.1.1–2.1.2, we derive 1 𝑑 ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 (2.4.28) 2 𝑑𝑡 ∫ 1 ( ) ∣𝑤 ⃗ 𝑡𝑡𝑦 ∣2 ⃗ 𝑡𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥ + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥ 𝑑𝑦 + 𝐶2 ∥𝑤 ≤ −𝜇 ⃗ 𝑦 ∥𝐿∞ ∥𝑢𝑡𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥2𝐿4 ∥𝑤 𝑣 0 ( ) ⃗ 𝑡𝑡𝑦 ∥2 + 𝐶2 ∥𝑤 ⃗ 𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 . ≤ −(2𝐶)−1 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝑣𝑦 ∥2 + ∥𝑤 1 ∥𝑤 By (2.4.28), we can obtain ∫ 𝑡 ∫ 𝑡 2 2 ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥ + ∥𝑤 ⃗ 𝑡𝑡𝑦 (𝜏 )∥ 𝑑𝑦 ≤ 𝐶4 + 𝐶2 ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 0

0

which, along with Theorems 2.1.1–2.1.2, gives estimate (2.4.4).


57

Differentiating (2.1.19) with respect to 𝑡 twice, multiplying the resulting equation by 𝜃𝑡𝑡 in 𝐿2 (0, 1), performing an integration by parts, and using Theorems 2.1.1–2.1.2, we infer that ∫ 1 𝑑 1 2 𝑒𝜃 𝜃𝑡𝑡 𝑑𝑦 2 𝑑𝑡 0 ) ∫ ∫ 1 ∫ 1( 𝜅𝜃𝑦 3 1 2 𝜃𝑡𝑡𝑦 𝑑𝑦 − (𝑒𝜃𝑡𝑡 𝜃𝑡 + 𝑒𝑢𝑡𝑡 𝑣𝑡 ) 𝜃𝑡𝑡 𝑑𝑦 − 𝑒𝜃 𝜃𝑡𝑡 𝑑𝑦 =− 𝑣 𝑡𝑡 2 0 0 0 ( )] ) ∫ 1( ∫ 1[ 𝜆𝑢𝑦 𝜆𝑢𝑦 −2 𝑒𝑣𝑡 − −𝑝 + 𝑢𝑡𝑦 𝜃𝑡𝑡 𝑑𝑦 − 𝑒𝑣 + 𝑝 − 𝑢𝑡𝑡𝑦 𝜃𝑡𝑡 𝑑𝑦 𝑣 𝑡 𝑣 0 0 ) ) ∫ 1( ∫ 1( 𝑤 ⃗𝑦 ⋅ 𝑤 ⃗ 𝜆𝑢𝑦 𝑢𝑦 𝜃𝑡𝑡 𝑑𝑦 + 𝜇 𝜃𝑡𝑡 𝑑𝑦 + −𝑝 + 𝑣 𝑡𝑡 𝑣 0 0 𝑡𝑡 = 𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + 𝐴6 + 𝐴7 .

(2.4.29)

By virtue of Theorems 2.1.1–2.1.2, (2.4.1)–(2.4.2) and using the embedding theorem, we deduce that for any 𝜀 ∈ (0, 1), ∫ 1( ) 𝜃𝑦 𝐴1 = −𝜅 𝜃𝑡𝑡𝑦 𝑑𝑦 (2.4.30) 𝑣 𝑡𝑡 0 ) ∫ 1( ∫ 1 2 𝜃𝑦 𝑢2𝑦 𝜃𝑡𝑡𝑦 2𝜃𝑡𝑦 𝑢𝑦 𝜃𝑦 𝑢𝑡𝑦 𝑑𝑦 + 𝜅 + − 2 = −𝜅 𝜃𝑡𝑡𝑦 𝑑𝑦 𝑣 𝑣2 𝑣2 𝑣3 0 0 ) ( ≤ −𝐶1−1 ∥𝜃𝑡𝑡𝑦 ∥2 + 𝐶2 ∥𝜃𝑡𝑡𝑦 ∥ ∥𝜃𝑡𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥ + ∥𝜃𝑦 ∥𝐿∞ ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥2𝐿4 ( ) ≤ −(2𝐶1 )−1 ∥𝜃𝑡𝑡𝑦 ∥2 + 𝐶2 ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 1 + ∥𝜃𝑡𝑦𝑦 ∥2 , ∫ 1 (𝑒𝜃𝑡𝑡 𝜃𝑡 + 𝑒𝑢𝑡𝑡 𝑣𝑡 ) 𝜃𝑡𝑡 𝑑𝑦 (2.4.31) 𝐴2 = − 0

∫ ≤ 𝐶1

0

1

] [ (∣𝑣𝑡 ∣ + ∣𝜃𝑡 ∣)2 + ∣𝑣𝑡𝑡 ∣ + ∣𝜃𝑡𝑡 ∣ (∣𝜃𝑡 ∣ + ∣𝑣𝑡 ∣)∣𝜃𝑡𝑡 ∣ 𝑑𝑦

≤ 𝐶1 ∥𝜃𝑡𝑡 ∥𝐿∞ (∥𝜃𝑡 ∥ + ∥𝑢𝑦 ∥)[(∥𝑢𝑦 ∥𝐿∞ + ∥𝜃𝑡 ∥𝐿∞ )(∥𝜃𝑡 ∥ + ∥𝑢𝑦 ∥) + ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑡𝑡 ∥] ≤ 𝜀∥𝜃𝑡𝑡𝑦 ∥2 + 𝐶2 𝜀−1 (∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 1 + ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝜃𝑡 ∥2 ), ∫ ∫ 1 3 1 2 𝑒𝜃 𝜃𝑡𝑡 𝑑𝑦 ≤ 𝐶1 (∣𝜃𝑡 ∣ + ∣𝑢𝑦 ∣)∣𝜃𝑡𝑡 ∣2 𝑑𝑦 ≤ 𝜀∥𝜃𝑡𝑡𝑦 ∥2 + 𝐶2 𝜀−1 ∥𝜃𝑡𝑡 ∥2 , 𝐴3 = − 2 0 0 (2.4.32) ] ∫ 1[ ) ( 𝜆𝑢𝑦 𝐴4 = −2 (2.4.33) 𝑒𝑣𝑡 − −𝑝 + 𝑢𝑡𝑦 𝜃𝑡𝑡 𝑑𝑦 𝑣 𝑡 0 ∫ 1 ) ( ≤ 𝐶1 ∣𝜃𝑡 ∣ + ∣𝑢𝑦 ∣ + ∣𝑢𝑡𝑦 ∣ + ∣𝑢2𝑦 ∣ ∣𝑢𝑡𝑦 ∣∣𝜃𝑡𝑡 ∣ 𝑑𝑦 0 ( ) ≤ 𝐶2 ∥𝑢𝑡𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥ + ∥𝜃𝑡 ∥ + ∥𝑢𝑡𝑦 ∥ + ∥𝑢𝑦 ∥2𝐿4 ∥𝜃𝑡𝑡 ∥ 1

1

≤ 𝐶2 ∥𝑢𝑡𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2 (∥𝑢𝑦 ∥𝐻 1 + ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑡 ∥) ∥𝜃𝑡𝑡 ∥

58


which implies ∫ 0

𝑡

∫ 𝐴4 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 sup ∥𝜃𝑡𝑡 (𝜏 )∥ 0≤𝜏 ≤𝑡

0≤𝜏 ≤𝑡

× ( ≤𝜀

𝑡

0

𝑡

1

2

∥𝑢𝑡𝑦 (𝜏 )∥ 𝑑𝜏

(∥𝑢𝑦 ∥2𝐻 1

2

2

) 14 (∫

𝑡

0

2

∥𝑢𝑡𝑦𝑦 (𝜏 )∥ 𝑑𝜏

∫

1

0

) 14

) 12

+ ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑡 ∥ )(𝜏 ) 𝑑𝜏 ) ∫ 𝑡 sup ∥𝜃𝑡𝑡 (𝜏 )∥2 + ∥𝑢𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶2 𝜀−3 , 0

0≤𝜏 ≤𝑡

𝐴5 = −

0

1

∥𝑢𝑡𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2 (∥𝑢𝑦 ∥𝐻 1 + ∥𝑢𝑡𝑦 ∥ + ∥𝜃𝑡 ∥) 𝑑𝜏

(∫

≤ 𝐶2 sup ∥𝜃𝑡𝑡 (𝜏 )∥ (∫

𝑡

0

(2.4.34)

( ) 𝜆𝑢𝑦 𝑒𝑣 + 𝑝 − 𝑢𝑡𝑡𝑦 𝜃𝑡𝑡 𝑑𝑦 ≤ 𝜀∥𝑢𝑡𝑡𝑦 ∥2 + 𝐶2 𝜀−1 ∥𝜃𝑡𝑡 ∥2 , 𝑣 (2.4.35)

( ) 𝜆𝑢𝑦 𝐴6 = 𝑢𝑦 𝜃𝑡𝑡 𝑑𝑦 −𝑝 + 𝑣 𝑡𝑡 0 ∫ 1 ( ) (∣𝑣𝑡 ∣ + ∣𝜃𝑡 ∣)2 + ∣𝜃𝑡𝑡 ∣ + ∣𝑢𝑡𝑡𝑦 ∣ + ∣𝑢𝑡𝑦 𝑢𝑦 ∣ + ∣𝑢𝑦 ∣3 ∣𝑢𝑦 ∣∣𝜃𝑡𝑡 ∣ 𝑑𝑦 ≤ 𝐶1 0 ( ≤ 𝐶1 ∥𝑢𝑦 ∥𝐿∞ ∥𝜃𝑡𝑡 ∥ (∥𝑣𝑡 ∥𝐿∞ + ∥𝜃𝑡 ∥𝐿∞ )(∥𝑣𝑡 ∥ + ∥𝜃𝑡 ∥) + ∥𝑢𝑡𝑡𝑦 ∥ + ∥𝑢𝑦 ∥ ) + ∥𝜃𝑡𝑡 ∥ + ∥𝑢𝑡𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ + ∥𝑢𝑦 ∥3𝐿6 ( ) ≤ 𝜀∥𝑢𝑡𝑡𝑦 ∥2 + 𝐶2 𝜀−1 ∥𝑢𝑡𝑡 ∥2 + ∥𝑢𝑦 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑡 ∥2 , (2.4.36) ∫

1

(

) 𝑤 ⃗𝑦 ⋅ 𝑤 ⃗ 𝐴7 = 𝜇 𝜃𝑡𝑡 𝑑𝑦 𝑣 0 𝑡𝑡 ( ≤ 𝐶2 ∥𝜃𝑡𝑡 ∥ ∥𝑤 ⃗ 𝑡𝑦 ∥2𝐿4 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑤 ⃗ 𝑡𝑦 ∥∥𝑢𝑦 ∥𝐿∞ ) + ∥𝑤 ⃗ 𝑦 ∥2𝐿∞ ∥𝑢𝑦𝑦 ∥ + ∥𝑢𝑦 ∥2𝐿∞ ∥𝑤 ⃗ 𝑦 ∥2𝐿4 ( ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 ≤ 𝐶2 𝜀−1 ∥𝜃𝑡𝑡 ∥2 + 𝜀 ∥𝑤 ) + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑦 ∥2𝐻 1 + ∥𝑢𝑦 ∥2𝐻 1 . ∫

1

(2.4.37)

Integrating with respect to 𝑡, using Theorems 2.1.1–2.1.2 and (2.4.35)–(2.4.37), we can derive ∫ 𝑡 √ ∥ 𝑒𝜃 𝜃𝑡𝑡 ∥2 ≤ 𝐶4 + (𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + 𝐴6 + 𝐴7 )(𝜏 ) 𝑑𝜏. 0


Thus

59

∫ 𝑡 ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 0 { } ∫ 𝑡 ≤ 𝐶1 𝜀 sup ∥𝜃𝑡𝑡 (𝜏 )∥2 + (∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏 0≤𝜏 ≤𝑡

∫

0

𝑡

+ 𝐶4 𝜀−3 + 𝐶2 𝜀−1 (∥𝜃𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 )(𝜏 ) 𝑑𝜏 0 ∫ 𝑡 ( ) ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 (𝜏 ) 𝑑𝜏 + 𝐶1 𝜀 0

(2.4.38)

which, along with Theorems 2.1.1–2.1.2 and (2.4.4), (2.4.25), (2.4.38), gives estimate (2.4.5). □ 4 Lemma 2.4.2. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃0 ) ∈ 𝐻+ and for any 𝜀 ∈ (0, 1), we have for any 𝑡 > 0, ∫ 𝑡 ∫ 𝑡 2 2 −6 2 ∥𝑢𝑡𝑦 (𝑡)∥ + ∥𝑢𝑡𝑦𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶3 𝜀 + 𝐶1 𝜀 (∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏, 0

∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 +

∫

∫

0

𝑡

0 𝑡

∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶3 𝜀−6 + 𝐶1 𝜀2

∫ 0

(2.4.39) 𝑡

∥𝑤 ⃗ 𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏,

∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 −6 (∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 )(𝜏 ) 𝑑𝜏. ≤ 𝐶3 𝜀 + 𝐶2 𝜀2 0

(2.4.40)

(2.4.41)

Proof. Differentiating (2.1.17) with respect to 𝑦 and 𝑡, multiplying the resulting equation by 𝑢𝑡𝑦 and integrating by parts in 𝐿2 (0, 1), we arrive at 𝑦=1 ∫ 1 ( ( ) ) 1 𝑑 𝜆𝑢𝑦 𝜆𝑢𝑦 ∥𝑢𝑡𝑦 (𝑡)∥2 = −𝑝 + 𝑢𝑡𝑦 − 𝑢𝑡𝑦𝑦 𝑑𝑦 −𝑝 + 2 𝑑𝑡 𝑣 𝑡𝑦 𝑣 𝑡𝑦 (2.4.42) 0 𝑦=0 = 𝐵1 + 𝐵 2 . We employ Theorems 2.1.1–2.1.2 and Lemma 2.4.1, the interpolation inequality and Poincaré’s inequality to get ( 𝐵1 ≤ 𝐶1 ∥𝑢𝑦𝑦 ∥𝐿∞ + (∥𝑢𝑦 ∥𝐿∞ + ∥𝜃𝑡 ∥𝐿∞ )(∥𝑣𝑦 ∥𝐿∞ + ∥𝜃𝑦 ∥𝐿∞ ) + ∥𝜃𝑡𝑦 ∥𝐿∞ + ∥𝑢𝑡𝑦𝑦 ∥𝐿∞ + ∥𝑢𝑡𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥𝐿∞ + ∥𝑢𝑦𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥𝐿∞ ) + ∥𝑢𝑦 ∥2𝐿∞ ∥𝑣𝑦 ∥𝐿∞ ∥𝑢𝑡𝑦 ∥𝐿∞ 1

1

≤ 𝐶2 (𝐵11 + 𝐵12 )∥𝑢𝑡𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2

(2.4.43)

60


where 𝐵11 = ∥𝑣𝑦 ∥𝐻 2 + ∥𝜃𝑡 ∥ + ∥𝜃𝑡𝑦 ∥, 1

1

1

1

1

1

𝐵12 = ∥𝑢𝑡𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2 + ∥𝑢𝑡𝑦𝑦𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2 + ∥𝜃𝑡𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 . Applying Young’s inequality several times, we have that for any 𝜀 ∈ (0, 1), ( ) 1 1 2 𝜀2 𝐶2 𝐵11 ∥𝑢𝑡𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2 ≤ ∥𝑢𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀− 3 ∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝜃𝑡 ∥2 + ∥𝜃𝑡𝑦 ∥2 , 2 (2.4.44) 2 ( ) 1 1 𝜀 𝐶2 𝐵11 ∥𝑢𝑡𝑦 ∥ 2 ∥𝑢𝑡𝑦𝑦 ∥ 2 ≤ ∥𝑢𝑡𝑦𝑦 ∥2 + 𝜀2 ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦𝑦 ∥2 2 ( ) (2.4.45) + 𝐶2 𝜀−6 ∥𝜃𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 . Thus we infer from (2.4.43)–(2.4.45) and Theorems 2.1.1–2.1.2 and Lemma 2.4.1, 𝐵1 ≤ 𝜀2 (∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦𝑦 ∥2 ) + 𝐶2 𝜀−6 (∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝜃𝑡 ∥2 + ∥𝜃𝑡𝑦 ∥2 ) (2.4.46) which, together with Lemmas 2.2.1–2.2.3, further leads to ∫ 𝑡( ∫ 𝑡 ) ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 + 𝐶2 𝜀−6 , ∀ 𝑡 > 0, 𝐵1 (𝜏 ) 𝑑𝜏 ≤ 𝜀2 0

0

(2.4.47) [ 𝑑𝑦 + 𝐶1 (∥𝑢𝑦 ∥ + ∥𝜃𝑡 ∥)(∥𝑣𝑦 ∥𝐿∞ + ∥𝜃𝑦 ∥𝐿∞ ) + ∥𝑢𝑦𝑦 ∥ + ∥𝜃𝑡𝑦 ∥ 𝐵2 ≤ −𝜆 𝑣 0 ] + ∥𝑣𝑦 ∥𝐿∞ ∥𝑢𝑡𝑦 ∥ + ∥𝑢𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦 ∥ + ∥𝑢𝑦 ∥2𝐿∞ ∥𝑣𝑦 ∥ ∥𝑢𝑡𝑦𝑦 ∥ ( ) −1 ≤ (−2𝐶1 ) ∥𝑢𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀2 ∥𝑢𝑡𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 1 + ∥𝑣𝑦 ∥2𝐻 1 + ∥𝜃𝑡 ∥2𝐻 1 . (2.4.48) ∫

1

𝑢2𝑡𝑦𝑦

By (2.4.42), (2.4.47)–(2.4.48) and Theorems 2.1.1–2.1.2 and Lemma 2.4.1, for any 𝜀 ∈ (0, 1) small enough, we have ∫ 𝑡 ∫ 𝑡 2 2 −6 2 ∥𝑢𝑡𝑦 (𝑡)∥ + ∥𝑢𝑡𝑦𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶2 𝜀 + 𝐶1 𝜀 (∥𝑢𝑡𝑦𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏. 0

0

(2.4.49) On the other hand, differentiating (2.1.17) with respect to 𝑦 and 𝑡, and using Theorems 2.1.1–2.1.2 and Lemma 2.4.1, we derive ∥𝑢𝑡𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑡𝑡𝑦 ∥ + ∥𝑢𝑡𝑦𝑦 ∥ + ∥𝜃𝑡 ∥𝐻 2 + ∥𝜃𝑦 ∥𝐻 1 + ∥𝑢𝑦 ∥𝐻 2 + ∥𝑢𝑦 ∥𝐻 1 ) . (2.4.50) Thus inserting (2.4.50) into (2.4.49) implies estimate (2.4.39). Differentiating (2.1.18) with respect to 𝑦 and 𝑡, multiplying the resulting equation by 𝑤 ⃗ 𝑡𝑦 in 𝐿2 (0, 1), and integrating by parts, we can obtain ( ) ∫ 1( ) 𝑦=1 𝑤 ⃗𝑦 𝑤 ⃗𝑦 1 𝑑 2 ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥ = 𝜇 𝑤 ⃗ 𝑡𝑦 𝑦=0 − 𝜇 𝑤 ⃗ 𝑡𝑦𝑦 𝑑𝑦 2 𝑑𝑡 𝑣 𝑡𝑦 𝑣 𝑡𝑦 (2.4.51) 0 = 𝐷1 + 𝐷2


where

61

( ⃗ 𝑡𝑦𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑡𝑦 ∥𝐿∞ ∥𝑤 ⃗ 𝑡𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑦𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥𝐿∞ 𝐷1 ≤ 𝐶1 ∥𝑤 ) + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦 ∥𝐿∞ + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑣𝑦 𝑢𝑦 ∥𝐿∞ ( ⃗ 𝑡𝑦 ∥2 + ∥𝑤 ≤ 𝐶1 𝜀2 (∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 ∥2 ) + 𝐶2 𝜀−6 ∥𝑤 ⃗ 𝑦 ∥2𝐻 2 ) (2.4.52) + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝑣𝑦 ∥2𝐻 1 , ∫ 1 ( ∣𝑤 ⃗ 𝑡𝑦𝑦 ∣2 𝑑𝑦 + 𝐶1 ∥𝑤 𝐷2 ≤ −𝜇 ⃗ 𝑡𝑦𝑦 ∥ ∥𝑤 ⃗ 𝑡𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥ + ∥𝑤 ⃗ 𝑦𝑦 ∥∥𝑢𝑦 ∥𝐿∞ 𝑣 0 ) + ∥𝑤 ⃗ 𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦 ∥ + ∥𝑤 ⃗ 𝑡𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥∥𝑢𝑦 ∥𝐿∞ ( −1 ⃗ 𝑡𝑦 ∥2 + ∥𝑣𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 1 ⃗ 𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀2 ∥𝑤 ≤ −(2𝐶1 ) ∥𝑤 ) (2.4.53) + ∥𝑤 ⃗ 𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 .

By (2.4.51)–(2.4.53) and Theorems 2.1.1–2.1.2 and Lemma 2.4.1, we have ∫ 𝑡 ∫ 𝑡 ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 𝜀−6 + 𝐶1 𝜀2 ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏. (2.4.54) 0

0

Differentiating (2.1.18) with respect to 𝑦 and 𝑡, and using Lemmas 2.2.1–2.2.4, we derive ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶2 (∥𝑤 ⃗ 𝑡𝑡𝑦 ∥ + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥ + ∥𝑤 ⃗ 𝑦 ∥𝐻 2 + ∥𝑢𝑦 ∥𝐻 2 + ∥𝑣𝑦 ∥𝐻 2 + ∥𝑤 ⃗ 𝑡𝑦 ∥). (2.4.55) Thus inserting (2.4.55) into (2.4.54) implies estimate (2.4.40). Analogously, we get from (2.1.19), ∫ 1 𝑑 1 2 𝑒𝜃 𝜃𝑡𝑦 𝑑𝑦 = 𝐸1 + 𝐸2 + 𝐸3 + 𝐸4 + 𝐸5 2 𝑑𝑡 0 where

(

) 𝑦=1 𝜃𝑦 𝜃𝑡𝑦 𝑦=0 , 𝑣 𝑡𝑦 ∫ 1( ) 𝜃𝑦 = −𝜅 𝜃𝑡𝑦𝑦 𝑑𝑦, 𝑣 𝑡𝑦 0 ) ] ∫ 1 [( 𝜆𝑢𝑦 =− 𝜃𝑡𝑦 𝑑𝑦, 𝑒𝑣 − 𝑝 + 𝑢𝑦 𝑣 0 𝑡𝑦 ) ∫ 1( 1 =− 𝑒𝜃𝑡𝑦 𝜃𝑡 + 𝑒𝜃𝑦 𝜃𝑡𝑡 + 𝑒𝜃𝑡 𝜃𝑡𝑦 𝜃𝑡𝑦 𝑑𝑦, 2 0 ) ∫ 1( 𝑤 ⃗𝑦 ⋅ 𝑤 ⃗ =𝜇 𝜃𝑡𝑦 𝑑𝑦. 𝑣 0 𝑡𝑦

𝐸1 = 𝜅 𝐸2 𝐸3 𝐸4 𝐸5

(2.4.56)

62


It follows that

( 𝐸1 ≤ 𝐶2 ∥𝜃𝑡𝑦 ∥𝐿∞ ∥𝜃𝑡𝑦𝑦 ∥𝐿∞ + ∥𝜃𝑡𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥𝐿∞ + ∥𝜃𝑦𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥𝐿∞ ) + ∥𝜃𝑦 ∥𝐿∞ ∥𝑢𝑦𝑦 ∥𝐿∞ + ∥𝜃𝑦 ∥𝐿∞ ∥𝑢𝑦 ∥𝐿∞ ∥𝑣𝑦 ∥𝐿∞ ( 1 1 1 1 1 1 ≤ 𝐶2 ∥𝜃𝑡𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦𝑦 ∥ 2 + ∥𝜃𝑡𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 ) + ∥𝜃𝑦 ∥𝐻 2 + ∥𝑢𝑦 ∥𝐻 2 + ∥𝑣𝑦 ∥𝐻 1 ,

) 1 1 1 1 𝜀2 ( ∥𝜃𝑡𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦𝑦 ∥ 2 ≤ ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦𝑦 ∥2 + 𝐶2 𝜀−6 ∥𝜃𝑡𝑦 ∥2 , 3 )2 ( 1 1 𝜀2 ∥𝜃𝑡𝑦 ∥ 2 ∥𝜃𝑡𝑦𝑦 ∥ 2 ≤ ∥𝜃𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀−2 ∥𝜃𝑡𝑦 ∥2 , 3 1 1 2 2 ∥𝜃𝑡𝑦 ∥ ∥𝜃𝑡𝑦𝑦 ∥ (∥𝜃𝑦 ∥𝐻 2 + ∥𝑢𝑦 ∥𝐻 2 + ∥𝑣𝑦 ∥𝐻 1 ) ( ) 𝜀2 ≤ ∥𝜃𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀−2 ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 2 + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝑣𝑦 ∥2𝐻 1 . 3 Thus ( 𝐸1 ≤ 𝜀2 (∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦𝑦 ∥2 ) + 𝐶2 𝜀−6 ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 2 ) + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝑣𝑦 ∥2𝐻 1 , (2.4.57) ∫ 1 2 ( 𝜃𝑡𝑦𝑦 𝑑𝑦 + 𝐶2 ∥𝜃𝑡𝑦𝑦 ∥ ∥𝜃𝑡𝑦 ∥∥𝑣𝑦 ∥𝐿∞ + ∥𝜃𝑦𝑦 ∥∥𝑢𝑦 ∥𝐿∞ 𝐸2 ≤ −𝜅 𝑣 0 ) + ∥𝑢𝑦𝑦 ∥∥𝜃𝑦 ∥𝐿∞ + ∥𝑢𝑦 𝑣𝑦 ∥∥𝜃𝑦 ∥𝐿∞ ( ) (2.4.58) ≤ −(2𝐶1 )−1 ∥𝜃𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀2 ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 1 + ∥𝑢𝑦 ∥2𝐻 1 . Similarly,

( ) 𝐸3 ≤ 𝜀2 ∥𝑢𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀−2 ∥𝑢𝑦 ∥2𝐻 2 + ∥𝑢𝑡𝑦 ∥2 + ∥𝜃𝑡 ∥2𝐻 1 , (2.4.59) ( 𝐸4 ≤ 𝜀2 ∥𝜃𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀−2 ∥𝑢𝑦 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 + ∥𝜃𝑡 ∥2𝐻 1 ) (2.4.60) + ∥𝜃𝑦 ∥2𝐻 2 + ∥𝑢𝑡𝑦 ∥2 , ( ) ⃗ 𝑦 ∥2𝐻 2 + ∥𝑤 𝐸5 ≤ 𝜀2 ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀2 ∥𝜃𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀−2 ∥𝑤 ⃗ 𝑡𝑦 ∥2 + ∥𝑢𝑦 ∥2𝐻 1 . (2.4.61)

By (2.4.56)–(2.4.61) and Theorems 2.1.1–2.1.2 and Lemma 2.4.1, we have ∫ 𝑡 ∫ 𝑡 ∥𝜃𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶1 𝜀2 (∥𝜃𝑡𝑦𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏 0 0 ∫ 𝑡 (∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 2 + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝑣𝑦 ∥2𝐻 1 )(𝜏 ) 𝑑𝜏 + 𝐶2 𝜀−6 0 ∫ 𝑡 (∥𝑤 ⃗ 𝑦 ∥2𝐻 2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 )(𝜏 ) 𝑑𝜏. (2.4.62) + 𝐶2 𝜀−2 0


63

Differentiating (2.1.19) with respect to 𝑦 and 𝑡, and using Theorems 2.1.1–2.1.2 and Lemma 2.4.1, we derive ( ∥𝜃𝑡𝑦𝑦𝑦 ∥ ≤ 𝐶2 ∥𝑢𝑡𝑦 ∥𝐻 1 + ∥𝜃𝑡 ∥𝐻 1 + ∥𝜃𝑦 ∥𝐻 2 + ∥𝜃𝑡𝑡 ∥𝐻 1 + ∥𝑤 ⃗ 𝑡𝑦 ∥𝐻 1 ) + ∥𝑤 ⃗ 𝑦 ∥𝐻 1 + ∥𝑣𝑦 ∥𝐻 1 . (2.4.63) By (2.4.62)–(2.4.63) and Theorems 2.1.1–2.1.2 and Lemma 2.4.1, we derive estimate (2.4.41). □ 4 ⃗ 0 , 𝜃0 ) ∈ 𝐻+ Lemma 2.4.3. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 and for any 𝜀 ∈ (0, 1), we have for any 𝑡 > 0,

∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 (2.4.64) ∫ 𝑡( ) ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 + ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , 0

∥𝑢𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑢𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝜃𝑦𝑦 (𝑡)∥2𝑤1,∞ + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝑣𝑡𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑢𝑡𝑦𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦𝑦 (𝑡)∥2 ∫ 𝑡( ∥𝑢𝑡𝑡 ∥2 + ∥𝑤 + ⃗ 𝑡𝑡 ∥2 + ∥𝜃𝑦𝑦 ∥2𝑤1,∞ + ∥𝜃𝑡𝑡 ∥2 + ∥𝑢𝑦𝑦 ∥2𝑊 1,∞ + ∥𝑤 ⃗ 𝑦𝑦 ∥2𝑊 1,∞ 0

+ ∥𝑢𝑡𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑡𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2𝑊 1,∞ + ∥𝜃𝑡𝑦 ∥2𝑤1,∞ ) + ∥𝑤 ⃗ 𝑡𝑦 ∥2𝑊 1,∞ + ∥𝑣𝑡𝑦𝑦𝑦 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∥𝑣𝑦𝑦𝑦 (𝑡)∥2𝐻 1 ∫ 0

𝑡

+

∥𝑣𝑦𝑦 (𝑡)∥2𝑊 1,∞

∫ +

𝑡 0

(

(2.4.65)

) ∥𝑣𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑣𝑦𝑦 ∥2𝑊 1,∞ (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , (2.4.66)

( ) ∥𝑢𝑦𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(2.4.67)

Proof. Adding up (2.4.39), (2.4.40) and (2.4.41), picking 𝜀 ∈ (0, 1) small enough, we arrive at ∫ 𝑡 ( ) ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 + ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 0 ∫ 𝑡 ( ) ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑤 (2.4.68) ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 (𝜏 ) 𝑑𝜏. ≤ 𝐶3 𝜀−6 + 𝐶1 𝜀2 0

Now multiplying (2.4.3) and (2.4.4) by 𝜀 respectively, multiplying (2.4.5) by 𝜀3/2 , then adding the resultant to (2.4.68), and choosing 𝜀 ∈ (0, 1) small enough, we obtain (2.4.64). Differentiating (2.1.17) with respect to 𝑦, and using (2.1.16), we derive ∂ ( 𝑣𝑦𝑦 ) − 𝑝𝑣 𝑣𝑦𝑦 = 𝑢𝑡𝑦 + 𝐸(𝑦, 𝑡) (2.4.69) 𝜆 ∂𝑡 𝑣

64


where 𝐸(𝑦, 𝑡) = 2𝜆

𝑢𝑦 𝑣𝑦2 𝑢𝑦𝑦 𝑢𝑦 − 2𝜆 + 𝑝𝑣𝑣 𝑣𝑦2 + 2𝑝𝑣𝜃 𝑣𝑦 𝜃𝑦 + 𝑝𝜃𝜃 𝜃𝑦2 + 𝑝𝜃 𝜃𝑦𝑦 . 𝑣2 𝑣3

Differentiating (2.4.69) with respect to 𝑦, we have 𝜆

∂ ( 𝑣𝑦𝑦𝑦 ) − 𝑝𝑣 𝑣𝑦𝑦𝑦 = 𝐸1 (𝑦, 𝑡) ∂𝑡 𝑣

(2.4.70)

where (𝑣 𝑣 ) 𝑦𝑦 𝑦 , 𝑣2 𝑡 + ∥𝑣𝑦 ∥𝐻 1 + ∥𝜃𝑦 ∥𝐻 2 ) .

𝐸1 (𝑦, 𝑡) = 𝐸𝑦 (𝑦, 𝑡) + 𝑝𝑣𝑦 𝑣𝑦𝑦 + 𝑢𝑡𝑦𝑦 + 𝜆 ∥𝐸1 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑡𝑦𝑦 ∥ + ∥𝑢𝑦 ∥𝐻 2 Thus

∫

𝑡

0

Multiplying (2.4.70) by

𝑣𝑦𝑦𝑦 𝑣

∥𝐸1 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 .

(2.4.71)

in 𝐿2 (0, 1), we can obtain

𝑣 2 𝑑 𝑦𝑦𝑦 𝑣𝑦𝑦𝑦 2 + 𝐶1−1 ≤ 𝐶1 ∥𝐸1 (𝑡)∥2 . 𝑑𝑡 𝑣 𝑣

(2.4.72)

We infer from (2.4.71)–(2.4.72) that ∥𝑣𝑦𝑦𝑦 (𝑡)∥2 +

∫ 0

𝑡

∥𝑣𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀𝑡 > 0

(2.4.73)

which, together with (2.4.9), (2.4.15), (2.4.20) and (2.4.64), gives ∥𝑢𝑦𝑦𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥ + ∥𝜃𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶4 .

(2.4.74)

By (2.4.12), (2.4.17), (2.4.22) and (2.4.64), we have ∫ 0

𝑡

(∥𝑢𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦 ∥2𝐻 1 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(2.4.75)

Using the embedding theorem and (2.4.73)–(2.4.75), we have ∥𝑢𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑢𝑦𝑦 (𝑡)∥2𝐿∞ + ∥𝜃𝑦𝑦 (𝑡)∥2𝐿∞ + ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥2𝐿∞ ∫ 𝑡 ( ) ∥𝑢𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑤 + ⃗ 𝑦𝑦𝑦 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (2.4.76) 0


65

Differentiating (2.1.17), (2.1.19) with respect to 𝑡 respectively, and using Lemmas 2.4.1–2.4.2, we have ∥𝑢𝑡𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑡𝑡 (𝑡)∥ + 𝐶2 (∥𝑢𝑦 (𝑡)∥𝐻 1 + ∥𝑣𝑦 (𝑡)∥𝐻 1 + ∥𝑢𝑡𝑦 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥) ≤ 𝐶4 , (2.4.77) ⃗ 𝑡𝑡 (𝑡)∥ + 𝐶2 (∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 1 + ∥𝑢𝑦 (𝑡)∥𝐻 1 + ∥𝑣𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥) ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑤 ≤ 𝐶4 , (2.4.78) ∥𝜃𝑡𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝜃𝑡𝑡 (𝑡)∥ + 𝐶2 (∥𝜃𝑦 (𝑡)∥𝐻 1 + ∥𝑢𝑦 (𝑡)∥𝐻 1 + ∥𝑢𝑡𝑦 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥𝐻 1 + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥ + ∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 1 ) ≤ 𝐶4 .

(2.4.79)

By (2.4.12), (2.4.17), (2.4.22), (2.4.75)–(2.4.79) and Lemmas 2.4.1–2.4.2, for any 𝑡 > 0, we get ⃗ 𝑦𝑦𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑦𝑦𝑦𝑦 (𝑡)∥2 (2.4.80) ∥𝑢𝑦𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑤 ∫ 𝑡( ) + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝑢𝑦𝑦𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦 ∥2 + ∥𝜃𝑦𝑦𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 0

which, together with (2.4.76) and the interpolation inequality, gives ∥𝑢𝑦𝑦𝑦 (𝑡)∥2𝐿∞ + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2𝐿∞ + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2𝐿∞ ) ∫ 𝑡( + ⃗ 𝑦𝑦𝑦 ∥2𝐿∞ + ∥𝜃𝑦𝑦𝑦 ∥2𝐿∞ (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . ∥𝑢𝑦𝑦𝑦 ∥2𝐿∞ + ∥𝑤

(2.4.81)

0

Differentiating (2.4.70) with respect to 𝑦, we get ∂ ( 𝑣𝑦𝑦𝑦𝑦 ) 𝜆 − 𝑝𝑣 𝑣𝑦𝑦𝑦𝑦 = 𝐸2 (𝑡) ∂𝑡 𝑣 where

𝐸2 (𝑡) = 𝐸1𝑦 (𝑡) + 𝑝𝑣𝑦 𝑣𝑦𝑦𝑦 + 𝜆

(𝑣

𝑦𝑦𝑦 𝑣𝑦 𝑣2

(2.4.82) ) 𝑡

.

By Lemmas 2.4.1–2.4.2, (2.4.77)–(2.4.80) and the embedding theorem, the interpolation inequality, we deduce ∥𝐸1𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑡𝑦𝑦𝑦 ∥ + 𝐶4 (∥𝑢𝑦 ∥𝐻 3 + ∥𝑣𝑦 ∥𝐻 2 + ∥𝜃𝑦 ∥𝐻 3 ) , ∥𝐸2 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑡𝑦𝑦𝑦 ∥ + 𝐶4 (∥𝑢𝑦 ∥𝐻 3 + ∥𝑣𝑦 ∥𝐻 2 + ∥𝜃𝑦 ∥𝐻 3 ) . By (2.4.23), (2.4.24), (2.4.26) and (2.4.80), we have ∫ 𝑡 ( ) ∥𝑢𝑡𝑡 ∥2 + ∥𝑤 ⃗ 𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , 0

∀𝑡 > 0

which, together with (2.4.50), (2.4.55), (2.4.63) and (2.4.64), gives ∫ 𝑡 ( ) ∥𝑢𝑡𝑦𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀𝑡 > 0. 0

(2.4.83)

(2.4.84)

(2.4.85)

66


Thus

∫ 0


𝑡

∥𝐸2 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

𝑣𝑦𝑦𝑦𝑦 𝑣

∀ 𝑡 > 0.

(2.4.86)

in 𝐿2 (0, 1), we can obtain

𝑣 𝑑 𝑦𝑦𝑦𝑦 2 𝑣𝑦𝑦𝑦𝑦 2 + 𝐶1−1 ≤ 𝐶1 ∥𝐸2 (𝑡)∥2 . 𝑑𝑡 𝑣 𝑣 Thus

∫ 𝑡 𝑣 𝑦𝑦𝑦𝑦 2 𝑣𝑦𝑦𝑦𝑦 2 (𝑡) + (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , 𝑣 𝑣 0

∀ 𝑡 > 0.

(2.4.87)

(2.4.88)

Differentiating (2.1.17), (2.1.18), (2.1.19) with respect to 𝑦 three times respectively, using Lemmas 2.4.1–2.4.2 and Poincaré’s inequality, we infer ∥𝑢𝑦𝑦𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑡𝑦𝑦𝑦 (𝑡)∥ + 𝐶2 (∥𝑢𝑦 (𝑡)∥𝐻 3 + ∥𝑣𝑦 (𝑡)∥𝐻 3 + ∥𝜃𝑦 (𝑡)∥𝐻 3 ) , (2.4.89) ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 (𝑡)∥ + 𝐶2 (∥𝑤 ⃗ 𝑦 (𝑡)∥𝐻 3 + ∥𝑣𝑦 (𝑡)∥𝐻 3 ) , ( ⃗ 𝑦 (𝑡)∥𝐻 3 + ∥𝜃𝑦 (𝑡)∥𝐻 3 ∥𝜃𝑦𝑦𝑦𝑦𝑦 (𝑡)∥ ≤ 𝐶1 ∥𝜃𝑡𝑦𝑦𝑦 (𝑡)∥ + 𝐶2 ∥𝑢𝑦 (𝑡)∥𝐻 3 + ∥𝑤 ) + ∥𝜃𝑡𝑦𝑦 (𝑡)∥ + ∥𝑣𝑦 (𝑡)∥𝐻 3 .

(2.4.90) (2.4.91)

Using (2.1.16), (2.4.76), (2.4.85), (2.4.88)–(2.4.91) and Lemmas 2.4.1–2.4.2, the interpolation inequality, we have ∫ 𝑡 ( ) ∥𝑢𝑦𝑦𝑦𝑦𝑦 ∥2 + ∥𝑣𝑡𝑦𝑦𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦𝑦 ∥2 + ∥𝜃𝑦𝑦𝑦𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀𝑡 > 0, 0

∫ 0

(2.4.92) 𝑡

( ) ∥𝑢𝑦𝑦 ∥2𝑊 2,∞ + ∥𝜃𝑦𝑦 ∥2𝑊 2,∞ + ∥𝑤 ⃗ 𝑦𝑦 ∥2𝑊 2,∞ (𝜏 ) 𝑑𝜏 ≤ 𝐶4 ,

∀𝑡 > 0.

(2.4.93)

Finally, using (2.1.16), (2.4.76)–(2.4.81), (2.4.84)–(2.4.85), (2.4.92)–(2.4.93) and Sobolev’s interpolation inequality, we can derive the desired estimates (2.4.65)– (2.4.67). □ 4 ⃗ 0 , 𝜃0 ) ∈ 𝐻+ Lemma 2.4.4. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 and for any 𝜀 ∈ (0, 1), we have for any 𝑡 > 0,

∥𝑣(𝑡) − 𝑣¯∥2𝐻 4 + ∥𝑣𝑡 (𝑡)∥2𝐻 3 + ∥𝑣𝑡𝑡 (𝑡)∥2𝐻 1 + ∥𝑢(𝑡)∥2𝐻 4 + ∥𝑢𝑡 (𝑡)∥2𝐻 2 + ∥𝑢𝑡𝑡 (𝑡)∥2 2 ¯ 2 4 + ∥𝜃𝑡 (𝑡)∥2 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑤(𝑡)∥ ⃗ ⃗ 𝑡 (𝑡)∥2𝐻 2 + ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 4 + ∥𝑤 𝐻 𝐻 ∫ 𝑡( ∥𝑣 − 𝑣¯∥2𝐻 4 + ∥𝑢∥2𝐻 5 + ∥𝑢𝑡 ∥2𝐻 3 + ∥𝑢𝑡𝑡 ∥2𝐻 1 + ∥𝑤∥ + ⃗ 2𝐻 5 + ∥𝑤 ⃗ 𝑡 ∥2𝐻 3 + ∥𝑤 ⃗ 𝑡𝑡 ∥2 0 ) ¯ 2 5 + ∥𝜃𝑡 ∥2 3 + ∥𝜃𝑡𝑡 ∥2 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , + ∥𝜃 − 𝜃∥ (2.4.94) 𝐻 𝐻 𝐻 ∫ 𝑡 ( ) ∥𝑣𝑡 ∥2𝐻 4 + ∥𝑣𝑡𝑡 ∥2𝐻 2 + ∥𝑣𝑡𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (2.4.95) 0


67

Proof. Exploiting (2.1.16) and Lemmas 2.4.1–2.4.3, we easily obtain Lemma 2.4.4. The proof is complete. □ 4 ⃗ 0 , 𝜃0 ) ∈ 𝐻+ Lemma 2.4.5. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 (1) (1) , there exists a positive constant 𝛾4 = 𝛾4 (𝐶4 ) ≤ 𝛾2 (𝐶2 ) such that for any fixed (1) 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0, and 𝜀 ∈ (0, 1) small enough, ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝑡 ∥𝑢𝑡𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 𝑒𝛾𝜏 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏, (2.4.96) 0 0 ∫ 𝑡 ∫ 𝑡 ⃗ 𝑡𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏, (2.4.97) 𝑒𝛾𝑡 ∥𝑤 0

𝛾𝑡

2

∫

𝑡

0

2

−3

−1

∫

𝑡

𝑒 ∥𝜃𝑡𝑡𝑦 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 𝜀 + 𝐶2 𝜀 𝑒𝛾𝜏 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 0 0 ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏. + 𝐶1 𝜀

𝑒 ∥𝜃𝑡𝑡 (𝑡)∥ +

𝛾𝜏

0

(2.4.98)

Proof. Multiplying (2.4.27) by 𝑒𝛾𝑡 , we have ( ) 𝛾 1 𝑑 ( 𝛾𝑡 𝑒 ∥𝑢𝑡𝑡 (𝑡)∥2 ≤ 𝑒𝛾𝑡 ∥𝑢𝑡𝑡 (𝑡)∥2 − 𝐶1−1 𝑒𝛾𝑡 ∥𝑢𝑡𝑡𝑦 (𝑡)∥2 + 𝐶2 𝑒𝛾𝑡 ∥𝑢𝑦 (𝑡)∥2𝐻 2 2 𝑑𝑡 2

) + ∥𝜃𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 + ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝑣𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 . (2.4.99)

Using (2.1.12) and Poincaré’s inequality, we can derive ∥𝑢𝑡𝑡 (𝑡)∥ ≤ 𝐶1 ∥𝑢𝑡𝑡𝑦 (𝑡)∥. Exploiting (2.4.25) and Lemmas 2.2.3–2.2.4, we arrive at ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝑡 ∥𝑢𝑡𝑡 (𝑡)∥2 ≤ 𝐶4 − (𝐶1−1 − 𝐶1 𝛾) 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶2 𝑒𝛾𝜏 ∥𝜃𝑡𝑡 (𝜏 )∥2 𝑑𝜏 0 0 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶4 𝑒𝛾𝜏 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 − (𝐶1−1 − 𝐶1 𝛾) 0

0

0

0

] [ which gives (2.4.96) if we take 𝛾 > 0 so small that 0 < 𝛾 ≤ min 4𝐶1 2 , 𝛾2 (𝐶2 ) . 1 Multiplying (2.4.28) by 𝑒𝛾𝑡 , using (2.4.24), Theorems 2.1.1–2.1.3 and Poincaré’s inequality, we have ∫ 𝑡 ∫ 𝑡 ⃗ 𝑡𝑡 (𝑡)∥2 ≤ 𝐶4 − (𝐶1−1 − 𝐶1 𝛾) 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶4 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 𝑒𝛾𝑡 ∥𝑤 [ ] which gives (2.4.97) if we take 𝛾 > 0 so small that 0 < 𝛾 ≤ min 4𝐶1 2 , 𝛾2 (𝐶2 ) . 1

68


Multiplying (2.4.29) by 𝑒𝛾𝜏 , using (2.4.30)–(2.4.37), (2.4.25), Theorems 2.1.1– 2.1.3 and Poincaré’s inequality, for any 𝜀 ∈ (0, 1) small enough, we have ∫ 1 𝛾𝑡 √ 𝛾 𝑡 𝛾𝜏 √ 2 𝑒 ∥ 𝑒𝜃 𝜃𝑡𝑡 ∥ ≤ 𝐶4 + 𝑒 ∥ 𝑒𝜃 𝜃𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 2 2 0 ∫ 𝑡 𝑒𝛾𝜏 (𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + 𝐴6 + 𝐴7 )(𝜏 ) 𝑑𝜏 + 0

∫ ∫ 𝑡 𝛾 𝑡 𝛾𝜏 √ ≤ 𝐶4 𝜀−3 + 𝑒 ∥ 𝑒𝜃 𝜃𝑡𝑡 (𝜏 )∥2 𝑑𝜏 − (2𝐶1 )−1 𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏 2 0 0 ∫ 𝑡 [ 𝛾𝜏 ] 𝑒 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 + 2𝜀𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑦(𝜏 )∥2 𝑑𝜏 + 𝐶2 0

+ 𝐶2 𝜀 (∫ × ∫ +𝜀

−1

𝑡

0 𝑡

0 𝑡

0

∫

𝛾𝜏

(∫

2

𝑒 ∥𝜃𝑡𝑡 (𝜏 )∥ 𝑑𝜏 + sup ∥𝜃𝑡𝑡 (𝜏 )∥ 0≤𝜏 ≤𝑡

𝑒𝛾𝜏 ∥𝑢𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏

)1/4 [∫

0

𝑡

0

𝑡

𝛾𝜏

2

)1/4

𝑒 ∥𝑢𝑡𝑦 (𝜏 )∥ 𝑑𝜏

𝑒𝛾𝜏 (∥𝑢𝑦 ∥2𝐻 1 + ∥𝜃𝑦 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 )(𝜏 ) 𝑑𝜏

]1/2

𝑒𝛾𝜏 (∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏

)∫ 𝑡 ∫ 𝑡 1 𝛾𝜏 2 −1 ≤ 𝐶4 𝜀 − − 2𝜀 − 𝐶1 𝛾 𝑒 ∥𝜃𝑡𝑡𝑦 (𝜏 )∥ 𝑑𝜏 + 𝐶2 𝜀 𝑒𝛾𝜏 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 2𝐶1 0 0 ∫ 𝑡 ( ) +𝜀 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 + 𝜀 sup ∥𝜃𝑡𝑡 (𝜏 )∥2 −3

(

0≤𝜏 ≤𝑡

0

which gives (2.4.98) if we take 𝛾 > 0 small enough.

□

4 ⃗ 0 , 𝜃 0 ) ∈ 𝐻+ , Lemma 2.4.6. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 (2) (1) (2) there exists a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0, and 𝜀 ∈ (0, 1) small enough, 𝛾𝑡

2

∫

𝑡

𝑒𝛾𝜏 ∥𝑢𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ∫ 𝑡 ( ) −6 2 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏, ≤ 𝐶2 𝜀 + 𝐶1 𝜀

𝑒 ∥𝑢𝑡𝑦 (𝑡)∥ +

0

0

𝑒𝛾𝑡 ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 +

∫

(2.4.100)

𝑡

𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑡𝑦 (𝜏 )∥2 𝑑𝜏, ≤ 𝐶2 𝜀−6 + 𝐶1 𝜀2 0

0

(2.4.101)


𝑒𝛾𝑡 ∥𝜃𝑡𝑦 (𝑡)∥2 +

∫

69

𝑡

𝑒𝛾𝜏 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ∫ 𝑡 ( ) ≤ 𝐶2 𝜀−6 + 𝐶1 𝜀2 𝑒𝛾𝜏 ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏, 0 ( ) 𝑒𝛾𝑡 ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 + 0 ∫ 𝑡 −6 2 𝑒𝛾𝜏 (∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 )(𝜏 ) 𝑑𝜏. ≤ 𝐶2 𝜀 + 𝐶1 𝜀 0

0

(2.4.102)

(2.4.103)

Proof. Adding up (2.4.100), (2.4.101) and (2.4.102), picking 𝜀 ∈ (0, 1) small enough, we obtain (2.4.103). Multiplying (2.4.42) by 𝑒𝛾𝑡 , using (2.4.47)–(2.4.48) and Lemmas 2.2.3–2.2.4, for any 𝜀 ∈ (0, 1) small enough, we have ) 1 𝑑 ( 𝛾𝑡 𝑒 ∥𝑢𝑡𝑦 ∥2 2 𝑑𝑡 [ 1 ≤ 𝛾𝑒𝛾𝑡 ∥𝑢𝑡𝑦 ∥2 + 𝑒𝛾𝑡 𝜀2 (∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦𝑦 ∥2 ) 2 ( )] + 𝐶2 𝜀−6 ∥𝑢𝑦 ∥2𝐻 2 + ∥𝜃𝑡 ∥2 + ∥𝜃𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 − (2𝐶1 )−1 𝑒𝛾𝑡 ∥𝑢𝑡𝑦𝑦 ∥2 ( ) (2.4.104) + 𝐶2 𝑒𝛾𝑡 ∥𝑢𝑦 ∥2𝐻 1 + ∥𝜃𝑡 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑣𝑦 ∥2𝐻 1 . Integrating (2.4.104) with respect to 𝑡 and using Poincaré’s inequality, we have 𝑒𝛾𝑡 ∥𝑢𝑡𝑦 (𝑡)∥2 ≤ 𝐶2 𝜀−6 − [(2𝐶1 )−1 − 2𝜀2 − 𝐶1 𝛾] + 𝐶1 𝜀

2

∫ 0

𝑡

𝛾𝜏

2

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑢𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏

2

𝑒 (∥𝑢𝑡𝑡𝑦 ∥ + ∥𝜃𝑡𝑦𝑦 ∥ )(𝜏 ) 𝑑𝜏

which gives (2.4.100) if we take 𝛾 > 0 and 𝜀 ∈ (0, 1) so small that 0 < 𝜀 < [ ] [ (1) ] (2) min 1, 8𝐶1 1 and 0 < 𝛾 < min 𝛾4 , 8𝐶11 2 ≡ 𝛾4 . Similarly, multiplying (2.4.51) by 𝑒𝛾𝑡 , using (2.4.52)–(2.4.53), (2.4.55) and Theorems 2.1.1–2.1.3, for any 𝜀 ∈ (0, 1) small enough , we obtain 𝛾𝑡

2

−1

2

∫

𝑡

− [(2𝐶1 ) − 𝐶1 𝜀 ] 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶1 𝜀2

𝑒 ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥ ≤ 𝐶2 𝜀

−6

0

which gives (2.4.101) if we take 𝜀 ∈ (0, 1) small enough.

70


Multiplying (2.4.56) by 𝑒𝛾𝑡 , using (2.4.57)–(2.4.61), (2.4.63) and Theorems 2.1.1–2.1.3, we derive ) 1 √ 1 𝑑 ( 𝛾𝑡 √ 𝑒 ∥ 𝑒𝜃 𝜃𝑡𝑦 ∥2 ≤ 𝛾𝑒𝛾𝑡 ∥ 𝑒𝜃 𝜃𝑡𝑦 ∥2 + 𝜀2 𝑒𝛾𝑡 (∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦𝑦 ∥2 ) 2 𝑑𝑡 2 ( ) + 𝐶2 𝜀−6 𝑒𝛾𝑡 ∥𝜃𝑡𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 2 + ∥𝑢𝑦 ∥2𝐻 2 + ∥𝑣𝑦 ∥2𝐻 1 − (2𝐶1 )−1 𝑒𝛾𝑡 ∥𝜃𝑡𝑦𝑦 ∥2 ( + 𝐶2 𝜀−2 𝑒𝛾𝑡 ∥𝑢𝑦 ∥2𝐻 2 + ∥𝜃𝑡 ∥2𝐻 1 + ∥𝑢𝑡𝑦 ∥2 + ∥𝜃𝑦 ∥2𝐻 2 + ∥𝑣𝑦 ∥2 ) + ∥𝑤 ⃗ 𝑦 ∥2𝐻 2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 + 𝜀2 𝑒𝛾𝑡 ∥𝜃𝑡𝑦𝑦 ∥2 + 𝐶2 𝜀2 𝑒𝛾𝑡 ∥𝑣𝑡𝑦 ∥2 . (2.4.105) Integrating (2.4.105) with respect to 𝑡, we can deduce ∫

𝑡

( ) 𝑒𝛾𝜏 ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 0 ∫ ) 𝑡 𝛾𝜏 ( −6 −1 2 𝑒 ∥𝜃𝑡𝑦𝑦 (𝜏 )∥2 𝑑𝜏 + 𝐶2 𝜀 − (2𝐶1 ) − 𝜀

𝑒𝛾𝑡 ∥𝜃𝑡𝑦 (𝑡)∥2 ≤ 𝜀2

0

which gives (2.4.102) for 𝜀 ∈ (0, 1) small enough.

□

4 Lemma 2.4.7. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃 0 ) ∈ 𝐻+ , (2) there exists a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0, and 𝜀 ∈ (0, 1),

⃗ 𝑡𝑡 (𝑡)∥2 + ∥𝑢𝑡𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦 (𝑡)∥2 ) 𝑒𝛾𝑡 (∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑤 ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑢𝑡𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑡𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 0

≤ 𝐶4 , 𝑒𝛾𝑡 (∥𝑣𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑣𝑦𝑦 (𝑡)∥2𝑊 1,∞ ) +

(2.4.106) ∫ 0

𝑡

𝑒𝛾𝜏 (∥𝑣𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑣𝑦𝑦 ∥2𝑊 1,∞ )(𝜏 ) 𝑑𝜏

≤ 𝐶4 ,

(2.4.107)

𝑒𝛾𝑡 (∥𝑢𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑢𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝜃𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝑣𝑡𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑢𝑡𝑦𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦𝑦 (𝑡)∥2 ) ∫ 𝑡 ( + 𝑒𝛾𝜏 ∥𝑢𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑤 ⃗ 𝑡𝑡 ∥2 + ∥𝑢𝑦𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑢𝑡𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦 ∥2𝐻 1 0

+ ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑡𝑦𝑦 ∥2𝐻 1 + ∥𝑢𝑦𝑦 ∥2𝑊 2,∞ + ∥𝑢𝑡𝑦 ∥2𝑊 2,∞

) + ∥𝑤 ⃗ 𝑦𝑦 ∥2𝑊 2,∞ + ∥𝑤 ⃗ 𝑡𝑦 ∥2𝑊 2,∞ + ∥𝜃𝑦𝑦 ∥2𝑊 2,∞ + ∥𝜃𝑡𝑦 ∥2𝑊 2,∞ + ∥𝑣𝑡𝑦𝑦𝑦 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(2.4.108)


71 3

Proof. (2.4.96) × 𝜀 + (2.4.97) × 𝜀 + (2.4.98) × 𝜀 2 + (2.4.103), and taking 𝜀 ∈ (0, 1) small enough, we derive (2.4.106). Multiplying (2.4.72) by 𝑒𝛾𝑡 , we can calculate ( 𝑣 2 2 ) ( ) 𝑑 𝑦𝑦𝑦 𝛾𝑡 𝑣𝑦𝑦𝑦𝑦 (2.4.109) 𝑒 + 𝐶1−1 − 𝛾 𝑒𝛾𝑡 ≤ 𝐶1 𝑒𝛾𝑡 ∥𝐸1 (𝑡)∥2 . 𝑑𝑡 𝑣 𝑣 [ (2) ] Choose 𝛾 > 0 so small that 0 < 𝛾 < 𝛾4 ≡ min 2𝐶1 1 , 𝛾4 . Integrating (2.4.109) with respect to 𝑡, we have ∫ 𝑡 ∫ 𝑡 2 𝑣 𝑣 1 𝑦𝑦𝑦 2 𝑦𝑦𝑦 (𝑡) + (𝜏 ) 𝑑𝜏 ≤ 𝐶3 +𝐶1 𝑒𝛾𝑡 𝑒𝛾𝜏 𝑒𝛾𝜏 ∥𝐸1 (𝜏 )∥2 𝑑𝜏, ∀ 𝑡 > 0. 𝑣 2𝐶1 0 𝑣 0 Using (2.4.106), we have 𝑒𝛾𝑡 ∥𝑣𝑦𝑦𝑦 (𝑡)∥2 +

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑣𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀ 𝑡 > 0.

(2.4.110)

By (2.4.9), (2.4.15), (2.4.20), (2.4.106) and Theorems 2.1.1–2.1.3, we have 𝑒𝛾𝑡 (∥𝑢𝑦𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2 ) ≤ 𝐶4 .

(2.4.111)

By (2.4.12), (2.4.17), (2.4.22), (2.4.106) and Theorems 2.1.1–2.1.3, we get ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (2.4.112) 0

Using the embedding theorem, the interpolation inequality and (2.4.109)– (2.4.112), we conclude ( ⃗ 𝑦𝑦𝑦 (𝑡)∥2 𝑒𝛾𝑡 ∥𝑢𝑦𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2 + ∥𝑤 ) + ∥𝑢𝑦𝑦 (𝑡)∥2𝐿∞ + ∥𝜃𝑦𝑦 (𝑡)∥2𝐿∞ + ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥2𝐿∞ ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (2.4.113) + 0

Using (2.4.77)–(2.4.79), (2.4.106) and Theorems 2.1.1–2.1.3, we derive ( ) ⃗ 𝑡𝑦𝑦 (𝑡)∥2 ) ≤ 𝑒𝛾𝑡 ∥𝑢𝑡𝑡 ∥2 + ∥𝑤 ⃗ 𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 𝑒𝛾𝑡 (∥𝑢𝑡𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦𝑦 (𝑡)∥2 + ∥𝑤 ( ) + 𝑒𝛾𝑡 ∥𝑢𝑦 ∥2𝐻 1 + ∥𝑣𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦 ∥2𝐻 1 + ∥𝜃𝑡 ∥2 + ∥𝜃𝑡𝑦 ∥2 + ∥𝑢𝑡𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦 ∥2 ≤ 𝐶4 . (2.4.114) By (2.4.12), (2.4.17), (2.4.22), (2.4.111)–(2.4.114), we conclude ( 𝑒𝛾𝑡 ∥𝑢𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦 (𝑡)∥2𝐻 1 + ∥𝑢𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝑤 ⃗ 𝑦𝑦 (𝑡)∥2𝑊 1,∞ ) + ∥𝜃𝑦𝑦 (𝑡)∥2𝑊 1,∞ + ∥𝑢𝑡𝑦𝑦 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 (𝑡)∥2 + ∥𝜃𝑡𝑦𝑦 (𝑡)∥2 ∫ 𝑡 ( + 𝑒𝛾𝜏 ∥𝑢𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝜃𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑦𝑦𝑦 ∥2𝐻 1 + ∥𝑢𝑦𝑦 ∥2𝑊 1,∞ + ∥𝑤 ⃗ 𝑦𝑦 ∥2𝑊 1,∞ 0 ) + ∥𝜃𝑦𝑦 ∥2𝑊 1,∞ + ∥𝑢𝑡𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀𝑡 > 0. (2.4.115)

72


Using (2.4.23)–(2.4.24), (2.4.26), (2.4.115) and Theorem 2.1.3, we can deduce ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑢𝑡𝑡 ∥2 + ∥𝑤 ⃗ 𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀𝑡 > 0. (2.4.116) 0

Analogously, by (2.4.49), (2.4.55), (2.4.63), (2.4.106), (2.4.115) and Theorem 2.1.3, we derive ∫ 𝑡 𝑒𝛾𝜏 (∥𝑢𝑡𝑦𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑡𝑦𝑦𝑦 ∥2 + ∥𝜃𝑡𝑦𝑦𝑦 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀ 𝑡 > 0. (2.4.117) 0

Multiplying (2.4.87) by 𝑒𝛾𝑡 , using (2.4.83), (2.4.106), (2.4.110)–(2.4.117) and Theorem 2.1.3, for any fixed 𝛾 ∈ (0, 𝛾4 ), we have ∫ 𝑡 ∫ 𝑡 𝑣 𝑣 1 𝑦𝑦𝑦𝑦 2 𝑦𝑦𝑦𝑦 2 𝑒𝛾𝜏 𝑒𝛾𝜏 ∥𝐸2 (𝜏 )∥2 𝑑𝜏 𝑒𝛾𝑡 + (𝜏 ) 𝑑𝜏 ≤ 𝐶4 + 𝐶1 𝑣 2𝐶1 0 𝑣 0 ≤ 𝐶4 , ∀𝑡 > 0. Thus 𝑒𝛾𝑡 ∥𝑣𝑦𝑦𝑦𝑦 (𝑡)∥2 +

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑣𝑦𝑦𝑦𝑦 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀𝑡 > 0

(2.4.118)

which, combined with (2.4.110), (2.4.118) and the embedding theorem, gives (2.4.107). On the other hand, by (2.4.89)–(2.4.91), (2.4.115), (2.4.117) and Theorem 2.1.3, we have ∫ 𝑡 ( 𝑒𝛾𝜏 ∥𝑢𝑦𝑦𝑦𝑦𝑦 ∥2 + ∥𝜃𝑦𝑦𝑦𝑦𝑦 ∥2 + ∥𝑤 ⃗ 𝑦𝑦𝑦𝑦𝑦 ∥2 + ∥𝑢𝑦𝑦 ∥2𝑊 2,∞ 0 ) + ∥𝑤 ⃗ 𝑦𝑦 ∥2𝑊 2,∞ + ∥𝜃𝑦𝑦 ∥2𝑊 2,∞ (𝜏 ) 𝑑𝜏 ≤ 𝐶4 (2.4.119) which, combined with (2.4.115)–(2.4.119), gives (2.4.108).

□

4 Lemma 2.4.8. Under assumptions of Theorem 2.1.3, for any (𝑣0 , 𝑢0 , 𝑤 ⃗ 0 , 𝜃 0 ) ∈ 𝐻+ , for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0, ( ¯ 2 4 + ∥𝑣𝑡 ∥2 3 𝑒𝛾𝑡 ∥𝑣(𝑡) − 𝑣¯∥2𝐻 4 + ∥𝑢(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 𝐻

+ ∥𝑣𝑡𝑡 (𝑡)∥2𝐻 1 + ∥𝑢𝑡 (𝑡)∥2𝐻 2 + ∥𝑢𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 ) + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑤 ⃗ 𝑡 (𝑡)∥2𝐻 3 + ∥𝑤 ⃗ 𝑡 (𝑡)∥2𝐻 2 + ∥𝑤 ⃗ 𝑡𝑡 (𝑡)∥2 ≤ 𝐶4 , ∫ 0

𝑡

(2.4.120)

( ¯ 2 5 + ∥𝑤∥ 𝑒𝛾𝜏 ∥𝑣(𝑡) − 𝑣¯∥2𝐻 4 + ∥𝑢∥2𝐻 5 + ∥𝜃(𝑡) − 𝜃∥ ⃗ 2𝐻 5 + ∥𝑢𝑡 ∥2𝐻 3 𝐻

+ ∥𝑢𝑡𝑡 ∥2𝐻 1 + ∥𝑤 ⃗ 𝑡 ∥2𝐻 3 + ∥𝑤 ⃗ 𝑡𝑡 ∥2𝐻 1 + ∥𝜃𝑡 ∥2𝐻 3 + ∥𝜃𝑡𝑡 ∥2𝐻 1 ) + ∥𝑣𝑡 ∥2𝐻 4 + ∥𝑣𝑡𝑡 ∥2𝐻 2 + ∥𝑣𝑡𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(2.4.121)

2.5. Bibliographic Comments

73

Proof. Using (2.1.16), Theorems 2.1.1–2.1.2 and Lemmas 2.4.1–2.4.7, we can prove the conclusion. □ Proof of Theorem 2.1.3. Exploiting Lemmas 2.4.1–2.4.8, we complete the proof of Theorem 2.1.3. □

2.5 Bibliographic Comments In this section, we shall recall some related known results in this direction. For the case 𝑤 ⃗ = 0, there are many results on the global existence and asymptotic behavior of solutions to problem (2.1.16)–(2.1.17), (2.1.19)–(2.1.22) with different constitutive assumptions; we refer to Jiang [26, 27, 28, 29], Kawashima and Nishida [31], Kawohl [32], Kazhikhov [34], Kazhikhov and Shelukhin [36], Okada and Kawashima [47], Qin [49, 50, 52, 55, 54], Qin and Mu˜ noz Rivera [70], and Wang [76]. Among these cases we would like to mention two classes of models: an ideal gas and a real viscous gas. For the former case, i.e., for the case of 𝑤 ⃗ =0 and an ideal gas whose constitutive relations take the following form, 1 𝜃 ⃗ 2 ), 𝑝 = 𝑅 , 𝑒 = 𝐶𝑉 𝜃, 𝐸 = 𝐶𝑉 𝜃 + (𝑢2 + ∣𝑤∣ 2 𝑣

(2.5.122)

with suitable positive constants 𝐶𝑉 , 𝑅, the global existence and asymptotic behavior of smooth (generalized) solutions to the system (2.1.16), (2.1.17), (2.1.19) were established by many authors; we refer to Jiang [28, 29], Kawashima and Nishida [31], Kazhikhov [34], Kazhikhov and Shelukhin [36], Okada and Kawashima [47], Qin [49, 50, 51, 52, 55, 54, 56, 57], Qin, Wu and Liu [73] on the initial boundary value problems and the Cauchy problem. In detail, Qin [49, 50] established the existence and asymptotic behavior solutions in 𝐻 1 to (2.1.16), (2.1.17), (2.1.19)– (2.1.21) for a viscous ideal gas (2.5.122) in bounded domain in ℝ, for which Zheng and Qin [83] obtained the existence of maximal attractors (see also for a viscous ideal gas (2.5.122) in bounded annular domains 𝐺𝑛 = {𝑥 ∈ ℝ𝑛 ∣0 < 𝑎 < ∣𝑥∣ < 𝑏} (𝑛 = 2, 3) in ℝ𝑛 for a viscous spherically symmetric ideal gas). For the latter case, i.e., for the case of 𝑤 ⃗ = 0 and a real gas with the same assumptions as those in (2.1.25)–(2.1.36), Qin [55] (see also, Qin [51, 52, 54] with some stronger growth assumptions) established the existence and exponential stability of a 𝐶0 -semigroup generated by the solutions to (2.1.16), (2.1.17), (2.1.19)–(2.1.21) in the subspace of 𝐻 𝑖 × 𝐻 𝑖 × 𝐻 𝑖 (𝑖 = 1, 2) for a viscous ideal gas (2.5.122) in a bounded domain in ℝ. For the case of 𝑤 ⃗ ∕= 0, an ideal flow (2.5.122) which is the special case of 𝑞 = 𝑟 = 0 of the problem (2.1.16)–(2.1.21), Qin [70] proved the exponential stability and existence of attractors; Wang [76] investigated the global existence, uniqueness, regularity in 𝐻 1 . In this chapter, under more general assumptions (2.1.25)–(2.1.36) on the constitutive relations than those in [73], we establish the global existence uniqueness and asymptotic behavior of solutions in 𝐻 1 and 𝐻 2 .

74


The novelties of this chapter consist of the following aspects: (1) the more general constitutive relations and growth assumptions (2.1.25)–(2.1.36) are studied, the related results in 𝐻 1 in this chapter have improved and extended those in [73]; (2) the global existence and exponential stability of solutions in 𝐻 1 and 𝐻 2 are established for the model under consideration; (3) the results in 𝐻 2 and 𝐻 4 are obtained first for the model under consideration.

Chapter 3

Regularity and Exponential Stability of the 𝒑th Power Newtonian Fluid in One Space Dimension 3.1 Introduction In this chapter, we are interested in the regularity and exponential stability of solutions in 𝐻 𝑖 (𝑖 = 2, 4) for a 𝑝th power Newtonian fluid undergoing one-dimensional longitudinal motions. We assume that the pressure 𝒫, in terms of the absolute temperature 𝜃 and the specific volume 𝑢, is given by 𝒫=

𝜃 𝑢𝑝

(3.1.1)

with the pressure exponent 𝑝 ≥ 1. The balance laws of mass, momentum, and energy in Lagrangian form are as follows: 𝑢𝑡 = 𝑣𝑥 , ( 𝑣𝑥 ) , 𝑣𝑡 = −𝒫 + 𝜇 𝑢 𝑥 ( ) ) ( 𝑣𝑥 𝜃𝑥 𝑐𝑣 𝜃𝑡 = −𝒫 + 𝜇 𝑣𝑥 + 𝜅 . 𝑢 𝑢 𝑥

(3.1.2) (3.1.3) (3.1.4)

Here, 𝑢, 𝑣, 𝜃 are specific volume, velocity, and absolute temperature, respectively. The positive constants 𝑐𝑣 , 𝜇, 𝜅 represent specific heat, viscosity and conductivity, respectively. Since the magnitude of the specific heat 𝑐𝑣 plays no role in the mathematical analysis of the system, in what follows we will assume the scaling 𝑐𝑣 = 1.


75

76

Chapter 3. Regularity and Exponential Stability

We consider a typical initial boundary value problem for (3.1.2)–(3.1.4) in the reference domain {(𝑥, 𝑡) : 0 < 𝑥 < 1, 𝑡 ≥ 0} under the initial conditions 𝑢(𝑥, 0) = 𝑢0 (𝑥), 𝑣(𝑥, 0) = 𝑣0 (𝑥), 𝜃(𝑥, 0) = 𝜃0 (𝑥),

𝑥 ∈ [0, 1]

(3.1.5)

and boundary conditions 𝑣(0, 𝑡) = 𝑣(1, 𝑡) = 0, 𝜃𝑥 (0, 𝑡) = 𝜃𝑥 (1, 𝑡) = 0.

(3.1.6)

Obviously, when 𝑝 = 1, (3.1.1) reduces to the case of a polytropic ideal gas (see (3.4.1)). This chapter mainly continues to the case of 𝑝 > 1, which was selected from [61]. The notation in this chapter is standard. We put ∥ ⋅ ∥ = ∥ ⋅ ∥𝐿2 [0,1] . Subscripts 𝑡 and 𝑥 denote the (partial) derivatives with respect to 𝑡 and 𝑥, respectively. We use 𝐶𝑖 (𝑖 = 1, 2, 4) to denote a generic positive constant depending on the 𝐻 𝑖 [0, 1] norm of initial data (𝑢0 , 𝑣0 , 𝜃0 ), min 𝑢0 (𝑥) and min 𝜃0 (𝑥), but independent of 𝑥∈[0,1]

𝑥∈[0,1]

time variable 𝑡. ∫1 For convenience and without loss of generality, we may assume 0 𝑢0 (𝑥) 𝑑𝑥 = 1. Then from conservation of mass and boundary condition (3.1.6), we have ∫ 1 𝑢(𝑥, 𝑡) 𝑑𝑥 = 1. (3.1.7) 0

We define two spaces as { 2 𝐻+ = (𝑢, 𝑣, 𝜃) ∈ 𝐻 2 [0, 1] × 𝐻 2 [0, 1] × 𝐻 2 [0, 1] : 𝑢(𝑥) > 0, 𝜃(𝑥) > 0, } ∀𝑥 ∈ [0, 1], 𝑣(0) = 𝑣(1) = 0, 𝜃′ (0) = 𝜃′ (1) = 0 and

{ 4 = (𝑢, 𝑣, 𝜃) ∈ 𝐻 4 [0, 1] × 𝐻 4 [0, 1] × 𝐻 4 [0, 1] : 𝑢(𝑥) > 0, 𝜃(𝑥) > 0, 𝐻+ } ∀𝑥 ∈ [0, 1], 𝑣(0) = 𝑣(1) = 0, 𝜃′ (0) = 𝜃′ (1) = 0 . Now our main results in this chapter read as follows.

2 and the compatibility conditions Theorem 3.1.1. Suppose that (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ 2 hold. Then there exists a unique generalized global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐻+ to the problem (3.1.2)–(3.1.6) verifying that for any (𝑥, 𝑡) ∈ [0, 1] × [0, +∞),

0 < 𝐶1−1 ≤ 𝑢(𝑥, 𝑡),

𝜃(𝑥, 𝑡) ≤ 𝐶1

(3.1.8)

and for any 𝑡 > 0, ∥𝑢(𝑡) − 1∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥2𝐻 2 + ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 (3.1.9) ∫ 𝑡( ) ∥𝑢 − 1∥2𝐻 2 + ∥𝑣∥2𝐻 3 + ∥𝜃 − 𝜃∥2𝐻 3 + ∥𝑣𝑡 ∥2𝐻 1 + ∥𝜃𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 . + 0

3.1. Introduction

77

Moreover, there exists a constant 𝛾2 = 𝛾2 (𝐶2 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2 ], the following inequality holds for any 𝑡 > 0, ( ) (3.1.10) 𝑒𝛾𝑡 ∥𝑢(𝑡) − 1∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥2𝐻 2 + ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑢 − 1∥2𝐻 2 + ∥𝑣∥2𝐻 3 + ∥𝜃 − 𝜃∥2𝐻 3 + ∥𝑣𝑡 ∥2𝐻 1 + ∥𝜃𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 0

where 𝜃 =

) ∫1( 1 2 0 𝜃0 + 2 𝑣0 𝑑𝑥.

4 Theorem 3.1.2. Suppose that (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ and the compatibility conditions 4 hold. Then there exists a unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐻+ to the problem (3.1.2)–(3.1.6) such that for any 𝑡 > 0,

∥𝑢(𝑡) − 1∥2𝐻 4 + ∥𝑣(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃∥2𝐻 4 + ∥𝑣𝑡 (𝑡)∥2𝐻 2 + ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 ∫ 𝑡( ∥𝑢 − 1∥2𝐻 4 + ∥𝑣∥2𝐻 5 + ∥𝜃 − 𝜃∥2𝐻 5 + ∥𝑣𝑡 ∥2𝐻 3 + 0 ) + ∥𝜃𝑡 ∥2𝐻 3 + ∥𝑣𝑡𝑡 ∥2𝐻 1 + ∥𝜃𝑡𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(3.1.11)

Moreover, there exists a constant 𝛾4 = 𝛾4 (𝐶4 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following inequality holds for any 𝑡 > 0, ( 𝑒𝛾𝑡 ∥𝑢(𝑡) − 1∥2𝐻 4 + ∥𝑣(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃∥2𝐻 4 + ∥𝑣𝑡 (𝑡)∥2𝐻 2 ) + ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑢 − 1∥2𝐻 4 + ∥𝑣∥2𝐻 5 + ∥𝜃 − 𝜃∥2𝐻 5 + ∥𝑣𝑡 ∥2𝐻 3 + ∥𝜃𝑡 ∥2𝐻 3 + 0

+ ∥𝑣𝑡𝑡 ∥2𝐻 1 + ∥𝜃𝑡𝑡 ∥2𝐻 1 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(3.1.12)

Corollary 3.1.1. ( ) Under assumptions of Theorem 3.1.2, (3.1.12) implies that 𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡) is the classical solution verifying that for any fixed 𝛾 ∈ (0, 𝛾4 ] and for any 𝑡 > 0, ¯ 2 3+1/2 3 ≤ 𝐶4 𝑒−𝛾𝑡 . ∥(𝑢(𝑡) − 1, 𝑣(𝑡), 𝜃(𝑡) − 𝜃)∥ (𝐶 ) Remark 3.1.1. Obviously, it is easy to see that the similar results hold for the boundary conditions 𝑣(0, 𝑡) = 𝑣(1, 𝑡) = 0, 𝜃(0, 𝑡) = 𝜃(1, 𝑡) = const . > 0. Remark 3.1.2. Theorems 3.1.1–3.1.2 have improved the results in [37].

78


3.2 Proof of Theorem 3.1.1 In this section, we shall complete the proof of Theorem 3.1.1 and assume that the assumptions in Theorem 3.1.1 are valid. We begin with a technical lemma selected from [46]. Lemma 3.2.1. Let 𝜆(𝑡) (≥ 0) and 𝜔(𝑡) be continuous functions satisfying that there exist positive constants 𝐶𝑖 (𝑖 = 1, 2, 3, 4) such that 𝐶1 𝑒𝐶2 (𝑡−𝜏 ) ≤ exp

{∫

𝑡

𝜏

}

≤ 𝐶3 𝑒𝐶4 (𝑡−𝜏 ) ,

𝜔(𝑠)𝑑𝑠

We denote Λ(𝑡) by

∫ Λ(𝑡) =

𝑡+1 𝑡

0 ≤ 𝜏 ≤ 𝑡.

(3.2.1)

𝜆(𝜏 )𝑑𝜏.

Then { ∫ 𝑡 } exp − 𝜔(𝑠)𝑑𝑠 𝜆(𝜏 )𝑑𝜏 (3.2.2) 𝑡→+∞ 0 𝜏 { ∫ 𝑡 } ∫ 𝑡 ≤ lim sup exp − 𝜔(𝑠)𝑑𝑠 𝜆(𝜏 )𝑑𝜏 ≤ 𝐶 lim sup Λ(𝑡)

𝐶 −1 lim inf Λ(𝑡) ≤ lim inf 𝑡→+∞

𝑡→+∞

∫

𝑡

0

𝑡→+∞

𝜏

holds. Proof. Only the estimate from above in (3.2.2) will be shown, the estimate from below is derived similarly. By use of (3.2.1), ∫ 0

𝑡

{ ∫ 𝑡 } exp − 𝜔(𝑠)𝑑𝑠 𝜆(𝜏 )𝑑𝜏 𝜏

}∫ { ∫ 𝑡 𝜔(𝑠)𝑑𝑠 ≤ exp − 0

+

max{0,[𝑡−𝑇 ]−1} ∫ 𝑡−𝑗 ∑

𝑇 +1

0

𝑡−𝑗−1

𝑗=0

{∫ exp

{ ∫ exp −

0 𝜏

0

𝜏

} 𝜔(𝑠)𝑑𝑠 𝜆(𝜏 )𝑑𝜏 }

𝜔(𝑠)𝑑𝑠 𝜆(𝜏 )𝑑𝜏

{ ∫ 𝑡 } ) max{0,[𝑡−𝑇 ( ∑ ]−1} ≤ 𝐶(𝑇 ) − 𝜔(𝑠)𝑑𝑠 + 𝐶1−1 sup Λ(𝑡) 𝑒−𝐶2 𝑗 0

𝑡≥𝑇

𝑗=0

holds. Here [⋅] is a Gaussian symbol. Therefore the desired estimate easily follows. □ Lemma 3.2.2. If (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻 1 [0, 1] × 𝐻01 [0, 1] × 𝐻 1 [0, 1], then there exists a unique generalized global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐻 1 [0, 1] × 𝐻01 [0, 1] × 𝐻 1 [0, 1]


79

to the problem (3.1.2)–(3.1.6) satisfying 0 < 𝐶1−1 ≤ 𝑢(𝑥, 𝑡) ≤ 𝐶1 ,

∀(𝑥, 𝑡) ∈ [0, 1] × [0, +∞), 2

(3.2.3)

2

𝑢𝑡 , 𝑣𝑡 , 𝜃𝑡 , 𝜃𝑥 , 𝑣𝑥 , 𝑢𝑥 , 𝑣𝑥𝑥 , 𝜃𝑥𝑥 ∈ 𝐿 ([0, +∞), 𝐿 [0, 1]),

(3.2.4)

0 < 𝜃(𝑥, 𝑡) ≤ 𝐶1 ,

(3.2.5)

∥𝑢(𝑡)∥2𝐻 1

∀(𝑥, 𝑡) ∈ [0, 1] × [0, +∞), ∫ 𝑡 + ∥𝑣(𝑡)∥2𝐻 1 + ∥𝜃(𝑡)∥2𝐻 1 + (∥𝑢𝑥 ∥2 + ∥𝑣∥2𝐻 2 0

+ ∥𝜃𝑥 ∥2𝐻 1 + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶1 ,

∀𝑡 > 0,

(3.2.6)

and there exist positive constants 𝜆, 𝑡0 , Λ, independent of 𝑡, such that as 𝑡 ≥ 𝑡0 , ∥𝑢𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 ≤ Λ𝑒−𝜆𝑡 , max (∣𝑢(𝑥, 𝑡) − 1∣ + ∣𝑣(𝑥, 𝑡)∣ + ∣𝜃(𝑥, 𝑡) − 𝜃∣) ≤ Λ𝑒

−𝜆𝑡

𝑥∈[0,1]

.

(3.2.7) (3.2.8)

Proof. The existence of a generalized global solution in 𝐻 1 [0, 1] and estimates (3.2.4)–(3.2.8) were obtained by Lewicka and Watson [37]. For convenience of the reader, we give a simple proof. First, we note that the entropy 𝜂 of a 𝑝th power Newtonian fluid is a concave function 𝜂(𝑢, 𝜃) = log 𝜃 + ℎ(𝑢), where

{ ℎ(𝑢) =

(3.2.9)

log 𝑢, 𝑝 = 1, 1 1−𝑝 ), 𝑝 > 1, 𝑝−1 (1 − 𝑢

which satisfies the following standard entropy identity: 𝜂𝑡 = 𝜇

(𝑞 ) 𝜃2 𝑣𝑥2 + 𝜅 𝑥2 − . 𝑢𝜃 𝑢𝜃 𝜃 𝑥

Set 𝑒 = 𝑐𝑣 𝜃, 𝜎 = −

(3.2.10)

𝜃𝑥 𝜃 𝑣𝑥 + 𝜇 , 𝑞 = −𝜅 . 𝑢𝑝 𝑢 𝑢

By combining (3.1.2)–(3.1.4) and (3.2.10), we obtain the identity, ) ( ( ( 2 ( ) ) ) 𝜃𝑥2 𝜃¯ 𝑣𝑥 1 2 ¯ ¯ +𝜅 2 . −𝜃 𝜇 𝑒 + 𝑣 − 𝜃𝜂 = 𝜎𝑣 − 1 − 𝑞 2 𝜃 𝑢𝜃 𝑢𝜃 𝑡 𝑥

(3.2.11)

Integrating (3.2.9) with respect to 𝑥 and then using Jensen’s inequality and (3.1.7), ∫1 we infer from the fact 0 𝑢𝑑𝑥 = 1, ∫ 0

1

(∫ 𝜂𝑑𝑥 ≤ log

0

1

) (∫ 𝜃𝑑𝑥 + ℎ

0

1

) (∫ 𝑢𝑑𝑥 = log

0

1

) 𝜃𝑑𝑥 .

(3.2.12)

80


Integrating (3.2.11) over [0, 1] × [0, 𝑡], and noting the boundary conditions, we arrive at ) ) ∫ 𝑡∫ 1( 2 ∫ 1 ∫ 1( 𝜃2 1 𝑣 ¯ 𝜃 + 𝑣 2 𝑑𝑥 + 𝜃¯ 𝜇 𝑥 + 𝜅 𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝜃𝜂𝑑𝑥 2 𝑢𝜃 𝑢𝜃 0 0 0 0 which, along with (3.2.12), gives ∫ 𝜆≤ ∫

1

0

Set

𝑣 2 𝑑𝑥 + 𝜃¯

∫ 𝑡∫ 0

1

0 1 0

𝜃(𝑥, 𝑡)𝑑𝑥 ≤ Λ, ( 2 ) 𝜃2 𝑣 𝜇 𝑥 + 𝜅 𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 . 𝑢𝜃 𝑢𝜃

(3.2.13) (3.2.14)

𝜃𝑚 = max 𝜃(𝑥, 𝑡), 𝑢𝑚 = max 𝑢(𝑥, 𝑡), 𝑣𝑚 = max 𝑣(𝑥, 𝑡). 𝑥∈[0,1]

𝑥∈[0,1]

𝑥∈[0,1]

By a straightforward calculation, we have

[ ]2 )2 ] (∫ 1 ∫ 1 1 ∣𝜃𝑥 ∣ 1 ∣𝜃𝑥 ∣ 𝜃(𝑥, 𝑡) ≤ 𝜃 (𝑦, 𝑡) + 𝑑𝑥 ≤ 2 𝜃(𝑦, 𝑡) + 𝑑𝑥 1/2 2 0 𝜃1/2 4 0 𝜃 [ ) (∫ 1 2 )] (∫ 1 1 𝜃𝑥 ≤ 2 𝜃(𝑦, 𝑡) + 𝑢𝜃𝑑𝑥 𝑑𝑥 . (3.2.15) 2 4 0 0 𝑢𝜃 [

1/2

Integrating (3.2.15) with respect to 𝑦 over [0, 1], by (3.2.13), we obtain ) ( ∫ 1 2 𝜃𝑥 𝜃𝑚 (𝑡) ≤ 𝐶1 1 + 𝑢𝑚 (𝑡) 𝑑𝑥 . 2 0 𝑢𝜃

(3.2.16)

In a similar manner, we have 𝜃(𝑥, 𝑡) ≥ 𝐶1−1 − 𝐶1

∫

1 0

𝜃𝑥2 𝑑𝑥. 𝜃2

(3.2.17)

To prove (3.2.3), we divide the proof into three steps. Step 1. Integrating (3.1.3) over [𝑥, 1] × [𝜏, 𝑡], we get ∫ 𝑥 ∫ 𝑡 ∫ 𝑡 (𝑣(𝑟, 𝑡) − 𝑣(𝑟, 𝜏 )) 𝑑𝑟 = 𝜎(𝑥, 𝑠)𝑑𝑠 − 𝜎(1, 𝑠)𝑑𝑠 1

𝜏

=− Setting 𝑀 (𝑥, 𝜏, 𝑡) :=

∫𝑥 1

∫ 𝜏

𝜏

𝑡

∫

𝜎(1, 𝑠)𝑑𝑠 −

𝜏

𝑡

𝑠=𝑡 𝜃 (𝑥, 𝑠)𝑑𝑠 + 𝜇 ln 𝑢(𝑥, 𝑠) . 𝑝 𝑢 𝑠=𝜏

[𝑣(𝑟, 𝑡) − 𝑣(𝑟, 𝜏 )]𝑑𝑟, and the impulse ∫ 𝐼(𝜏, 𝑡) :=

𝜏

𝑡

𝜎(1, 𝑠)𝑑𝑠,


81

we rewrite the above equation in the form ∫ 𝑡 𝜃 (𝑥, 𝑠)𝑑𝑠 = 𝜇 log 𝑢(𝑥, 𝑡) − 𝜇 log 𝑢(𝑥, 𝜏 ) − 𝐼(𝜏, 𝑡) − 𝑀 (𝑥, 𝜏, 𝑡). 𝑝 𝜏 𝑢

(3.2.18)

Multiplying (3.2.18) by 𝑝/𝜇 and taking exponentials, we have readily that }] { } { [ 𝑑 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝 = 𝜃 exp − 𝑀 − 𝐼 . 𝑢 ⋅ exp − 𝑀 − 𝐼 𝑑𝑡 𝜇 𝜇 𝜇 𝜇 𝜇 Hence, } { 𝑝 𝑝 𝑢𝑝 (𝑥, 𝑡) ⋅ exp − 𝑀 (𝑥, 𝜏, 𝑡) − 𝐼(𝜏, 𝑡) 𝜇 𝜇 { } ∫ 𝑡 𝑝 𝑝 𝑝 = 𝑢𝑝 (𝑥, 𝜏 ) + 𝜃(𝑥, 𝑠) exp − 𝑀 (𝑥, 𝜏, 𝑠) − 𝐼(𝜏, 𝑠) 𝑑𝑠. 𝜇 𝜇 𝜏 𝜇

(3.2.19)

By (3.2.14), we have ∣𝑀 (𝑥, 𝜏, 𝑡)∣2 ≤ 2

∫

1

0

[

] 𝑣 2 (𝑥, 𝑡) + 𝑣 2 (𝑥, 𝜏 ) 𝑑𝑥 ≤ 𝐶1 .

(3.2.20)

Introducing (3.2.20) in (3.2.19), we arrive at ] [ ∫ 𝑡 𝑝 𝑝 𝜃(𝑥, 𝑠)𝑒− 𝜇 𝐼(𝜏,𝑠) 𝑑𝑠 ≤ 𝑢𝑝 (𝑥, 𝑡)𝑒− 𝜇 𝐼(𝜏,𝑡) (3.2.21) 𝐶1 𝑢𝑝 (𝑥, 𝜏 ) + 𝜏 ] [ ∫ 𝑡 𝑝 −𝜇 𝐼(𝜏,𝑠) 𝑝 ≤ 𝐶1 𝑢 (𝑥, 𝜏 ) + 𝜃(𝑥, 𝑠)𝑒 𝑑𝑠 . 𝜏

Step 2. In this step, we shall prove the uniform upper bound on 𝑢. First, from (3.1.7) it follows that ∫

1 0

𝑝

𝑢 (𝑥, 𝑡)𝑑𝑥 ≤

𝑢𝑝−1 𝑚 (𝑡)

∫ 0

1

𝑝−1 𝑢(𝑥, 𝑡)𝑑𝑥 = 𝑢𝑚 .

On the other hand, by Jensen’s inequality, we have (∫ 1 )𝑝 ∫ 1 𝑝 𝑢 (𝑥, 𝑡)𝑑𝑥 ≥ 𝑢(𝑥, 𝑡)𝑑𝑥 = 1. 0

0

(3.2.22)

(3.2.23)

From the right-hand inequality in (3.2.21) with 𝜏 = 0, (3.2.16) shows that, 𝑝

𝑢𝑝𝑚 (𝑡) ⋅ 𝑒− 𝜇 𝐼(0,𝑡) [ (∫ ∫ 𝑡 ∫ 𝑡 𝑝 𝑝 ≤ 𝐶1 1 + 𝑒− 𝜇 𝐼(0,𝑠) 𝑑𝑠 + 𝑢𝑚 (𝑠)𝑒− 𝜇 𝐼(0,𝑠) 0

0

0

1

) ] 𝜃𝑥2 𝑑𝑥 (𝑠)𝑑𝑠 . (3.2.24) 𝑢𝜃2

82


Since by (3.2.23) 𝑢𝑝𝑚 ≥ 1, it is clear that 𝑢𝑚 (𝑡) ≤ 𝑢𝑝𝑚 (𝑡), and thus from (3.2.24), by means of the Gronwall inequality, we obtain ( ) {∫ 𝑡 ∫ 1 2 } ∫ 𝑡 𝑝 𝑝 𝜃𝑥 −𝜇 𝐼(0,𝑡) −𝜇 𝐼(0,𝑠) 𝑝 𝑢𝑚 (𝑡) ⋅ 𝑒 ≤ 𝐶1 1 + 𝑒 𝑑𝑠 ⋅ exp 𝑑𝑥𝑑𝑠 . 2 0 0 0 𝑢𝜃 From (3.2.14) it follows that 𝑢𝑝𝑚 (𝑡)

≤ 𝐶1 𝑒

𝑝 𝜇 𝐼(0,𝑡)

) ( ∫ 𝑡 𝑝 −𝜇 𝐼(0,𝑠) 𝑒 𝑑𝑠 . 1+ 0

(3.2.25)

On the other hand, setting 𝜏 = 0 in the left-hand inequality (3.2.21), then integrating over the spatial interval [0, 1], and utilizing the estimates (3.2.22), (3.2.23) and (3.2.13), we have the following bound, ) ( ∫ 𝑡 𝑝 𝑝 𝐼(0,𝑡) −𝜇 𝐼(0,𝑠) −1 𝜇 𝑝−1 𝑒 𝑑𝑠 . (3.2.26) 𝑢𝑚 ≥ 𝐶1 𝑒 1+ 0

Now, (3.2.25) and (3.2.26) give 𝑝−1 𝑢𝑝𝑚 (𝑡) ≤ 𝐶1 𝑢𝑚 (𝑡),

from which we conclude the existence of a constant 𝐶1 > 0 such that 𝑢(𝑥, 𝑡) ≤ 𝐶1 .

(3.2.27)

Step 3. Our next concern will be the lower bound on 𝑢. Integrating (3.2.21) in 𝑥 over [0, 1] and recalling (3.2.23), (3.2.27) and (3.2.13), we see that ) ( ∫ 𝑡 𝑝 𝑝 𝑒− 𝜇 𝐼(𝜏,𝑠) 𝑑𝑠 ≤ 𝐶1 . (3.2.28) 𝐶1−1 ≤ 𝑒 𝜇 𝐼(𝜏,𝑡) 1 + 𝜏

Setting 𝜏 = 0 in the left-hand inequality in (3.2.21) while utilizing (3.2.17) and (3.2.28), we have [ ] ∫ 𝑡 𝑝 𝑝 𝑝 𝐼(0,𝑡) −𝜇 𝐼(0,𝑠) −1 𝜇 𝑝 𝑢 (𝑥, 𝑡) ≥ 𝐶1 𝑒 ⋅ 𝑢0 (𝑥) + 𝑒 𝜃(𝑥, 𝑠)𝑑𝑠 0 [ ∫ 𝑡 𝑝 𝑝 ≥ 𝐶1−1 𝑒 𝜇 𝐼(0,𝑡) ⋅ 𝑢𝑝0 (𝑥) + 𝑒− 𝜇 𝐼(0,𝑠) 𝑑𝑠 0

∫

−𝐶1 ≥ 𝐶1−1 − 𝐶1

∫

𝑡

0

𝑒

𝑝 𝜇 𝐼(𝑠,𝑡)

(∫ 0

1

𝑡

0

𝑝

𝑒− 𝜇 𝐼(0,𝑠)

(∫

)

𝜃𝑥2 𝑑𝑥 𝜃2

1

0

(𝑠)𝑑𝑠.

Now, by Gronwall’s inequality applied to (3.2.28), we obtain −1

𝐶1−1 𝑒𝐶1

(𝑡−𝜏 )

𝑝

≤ 𝑒− 𝜇 𝐼(𝜏,𝑡) ≤ 𝐶1 𝑒𝐶1 (𝑡−𝜏 ) .

) ] 𝜃𝑥2 𝑑𝑥 (𝑠)𝑑𝑠 𝜃2 (3.2.29)


83

∫ 1 𝜃2 Thus, by virtue of Lemma 3.2.1 with 𝜔(𝑡) = −𝜎(1, 𝑡) and 𝜆(𝑡) = 0 𝜃𝑥2 (𝑥, 𝑡)𝑑𝑥, we conclude that (∫ 1 2 ) ∫ 𝑡 ∫ 𝑡+1 ∫ 1 2 𝑝 𝜃𝑥 𝜃𝑥 𝐼(𝑠,𝑡) 𝜇 𝑒 𝑑𝑥 (𝑠)𝑑𝑠 ≤ 𝐶1 lim 𝑑𝑥𝑑𝑠. (3.2.30) lim 2 2 𝑡→+∞ 0 𝑡→+∞ 𝜃 0 𝑡 0 𝜃 ∫ 1 𝜃2 On the other hand, (3.2.14) and (3.2.27) imply that the function 0 𝜃𝑥2 𝑑𝑥 is integrable in [0, +∞), so the right-hand side of (3.2.30) equals zero. Now since the right-hand side of the first inequality in (3.2.29) is a continuous and positive function of 𝑡, in view of (3.2.30), it implies that 𝑢(𝑥, 𝑡) ≥ 𝐶1−1 which, together with (3.2.27), gives (3.2.3). The balance of momentum (3.1.3) can be rewritten in the form: ( ) ( 𝑢 ) 𝜃 𝑥 𝜇 −𝑣 = . 𝑢 𝑢𝑝 𝑥 𝑡 Multiplying by 𝜇𝑢𝑥 /𝑢 − 𝑣 and integrating over [0, 1] × [0, 𝑡], using the interpolation inequality and the Young inequality, we derive ∫ 1 ∫ 𝑡∫ 1 𝑢2𝑥 𝑑𝑥 + (𝑢2𝑥 + 𝜃𝑥2 + 𝜃2 𝑢2𝑥 )𝑑𝑥𝑑𝑠 ≤ 𝐶1 . (3.2.31) 0

0

0

In a similar manner, multiplying (3.1.3) by 𝑣, integrating over [0, 1] × [0, 𝑡] and using integrations by parts, using (3.2.3) and (3.2.31), we have ∫ 𝑡∫ 1 𝑣𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 . (3.2.32) 0

0

By (3.1.3), integrating by parts and using Young’s inequality gives ∫ 1 ∫ 𝑡∫ 1 2 𝑣𝑥2 (𝑥, 𝑡)𝑑𝑥 + (𝑣𝑥𝑥 + 𝜃2 𝑣𝑥2 + 𝑢2𝑥 𝑣𝑥2 + 𝑣𝑥4 )(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 .

(3.2.33)

Similarly, from (3.1.4), we infer that ∫ 1 ∫ 𝑡∫ 1 2 𝜃𝑥2 (𝑥, 𝑡)𝑑𝑥 + (𝜃𝑥𝑥 + 𝜃𝑥2 𝑢2𝑥 )(𝑥, 𝑠)𝑑𝑥𝑑𝑠 ≤ 𝐶1 .

(3.2.34)

0

0

0

0

0

0

By (3.2.31)–(3.2.34) and (3.2.3), we obtain (3.2.4) and (3.2.6). By (3.1.2)–(3.1.4) and (3.2.3), we observe that ∫ 1( ∫ 1 )2 𝑑 ( 2 ) 𝑢𝑥 𝜇 − 𝑣 𝑑𝑥 ≤ 𝐶1 𝜃𝑥 + 𝑢2𝑥 + 𝑣 2 + 𝜃2 𝑢2𝑥 𝑑𝑥, 𝑑𝑡 𝑢 0 0 ∫ 1 ∫ 1 𝑑 ( 2 ) 𝑣𝑥2 𝑑𝑥 ≤ 𝐶1 𝜃𝑥 + 𝜃2 𝑢2𝑥 + 𝑢2𝑥 𝑣𝑥2 𝑑𝑥, 𝑑𝑡 0 0 ∫ 1 ∫ 1 𝑑 ( 2 2 ) 𝜃𝑥2 𝑑𝑥 ≤ 𝐶1 𝜃 𝑣𝑥 + 𝑣𝑥4 + 𝜃𝑥2 𝑢2𝑥 𝑑𝑥 𝑑𝑡 0 0

84


which, along with (3.2.31)–(3.2.34) and (3.2.4), gives ∫ 1( ( 𝑢 )2 ) 𝑥 2 2 lim −𝑣 𝑣𝑥 + 𝜃𝑥 + 𝜇 𝑑𝑥 = 0. 𝑡→+∞ 0 𝑢 In particular, since the boundary condition gives )1/2 (∫ 1 𝑣𝑥2 𝑑𝑥 , ∣𝑣(𝑥, 𝑡)∣ ≤ 0

(3.2.35)

(3.2.36)

which, together with (3.2.35), yields lim

max ∣𝑣(𝑥, 𝑡)∣ = 0.

𝑡→+∞ 𝑥∈[0,1]

Thus, in view of (3.2.35), we conclude ∫ 1 ( 2 ) lim 𝑣𝑥 + 𝜃𝑥2 + 𝑢2𝑥 (𝑥, 𝑡)𝑑𝑥 = 0. 𝑡→+∞

0

(3.2.37)

(3.2.38)

Now, for the Neumann boundary conditions, we have ∫ ∫ 1 1 1 2 ¯ ≤ 𝜃(𝑥, 𝑡) − + ∣𝜃(𝑥, 𝑡) − 𝜃∣ 𝜃𝑑𝑥 𝑣 (𝑥, 𝑡)𝑑𝑥. 2 0 0 Employing (3.2.36) and (3.2.38), we see that in both cases ∫ 1 ¯ 2 ≤ 𝐶1 ∣𝜃(𝑥, 𝑡) − 𝜃∣ (𝜃𝑥2 + 𝑣𝑥2 )𝑑𝑥, 0

(3.2.39)

and thus (3.2.35) implies lim

¯ = 0. max ∣𝜃(𝑥, 𝑡) − 𝜃∣

(3.2.40)

(∫ 𝑢𝑑𝑥 ≤

(3.2.41)

𝑡→+∞ 𝑥∈[0,1]

Finally, (3.1.7) yields ∫ ∣𝑢(𝑥, 𝑡) − 1∣ ≤ 𝑢(𝑥, 𝑡) −

0

1

0

1

)1/2 𝑢2𝑥 𝑑𝑥 ,

and so recalling (3.2.37)–(3.2.38) and (3.2.40), we deduce that ( ) ¯ + ∣𝑢(𝑥, 𝑡) − 1∣ = 0. lim max ∣𝑣(𝑥, 𝑡)∣ + ∣𝜃(𝑥, 𝑡) − 𝜃∣ 𝑡→+∞ 𝑥∈[0,1]

Set

∫ 𝑉 (𝑡) :=

1 0

∫ 1( )2 ( 2 ) 𝑢𝑥 2 − 𝑣 𝑑𝑥, 𝑣𝑥 + 𝜃𝑥 𝑑𝑥, 𝐷(𝑥, 𝑡) := 𝜇 𝑢 0 ) ∫ 1( 1 2 ¯ 𝜃 + 𝑣 − 𝜃𝜂 + 𝛾 𝑑𝑥 𝐴(𝑡) := 2 0

(3.2.42)


85

¯ 𝜃 − 1). Integrating the availability identity (3.2.11) over [0, 1], we where 𝛾 = 𝜃(ln have 𝑑 𝐴(𝑡) + 𝜆𝑉 (𝑡) ≤ 0. (3.2.43) 𝑑𝑡 Observing the boundedness of 𝜃, due to (3.2.40), it follows from the Taylor expansion of the function ln, that ¯ 2 ≤ (𝜃 − 𝜃¯ ln 𝜃) + 𝛾 ≤ Λ(𝜃 − 𝜃) ¯ 2. 𝜆(𝜃 − 𝜃)

(3.2.44)

Analogously, using (3.1.7), the boundedness of 𝑢, and the concavity of ℎ, ∫ 𝜆

1

0

(𝑢 − 1)2 𝑑𝑥 ≤ −

∫

1

0

∫ ℎ(𝑢)𝑑𝑥 ≤ Λ

1

0

(𝑢 − 1)2 𝑑𝑥.

(3.2.45)

Adding (3.2.44) and (3.2.45) yields, ∫ 𝜆

0

1

( ) ¯ 2 + (𝑢 − 1)2 + 𝑣 2 𝑑𝑥 ≤ 𝐴 ≤ Λ (𝜃 − 𝜃)

∫

1

0

( ) ¯ 2 + (𝑢 − 1)2 + 𝑣 2 𝑑𝑥. (𝜃 − 𝜃)

Hence, by (3.2.39), (3.2.41), and (3.2.36), we obtain ∫ 𝐴(𝑡) ≤ Λ

1

0

(𝜃𝑥2 + 𝑣 2 + 𝑢2𝑥 )𝑑𝑥

] [ ∫ 1 (𝜃𝑥2 + 𝑣 2 )𝑑𝑥 ≤ Λ(𝐷(𝑡) + 𝑉 (𝑡)). ≤ Λ 𝐷(𝑡) +

(3.2.46)

0

In addition, from (3.2.36), (3.2.40) and Young’s inequality, we see that 𝑑 𝐷(𝑡) + 𝜆𝐷(𝑡) ≤ Λ𝑉 (𝑡). 𝑑𝑡

(3.2.47)

From (3.1.3)–(3.1.4), we infer 𝑑 𝑉 (𝑡) + 𝜆 𝑑𝑡

∫

1 0

2 (𝑣𝑥𝑥

+

2 𝜃𝑥𝑥 )𝑑𝑥

[ ≤ Λ 𝑉 (𝑡) +

∫ 0

1

(𝑣𝑥4

+

𝑢2𝑥

+

𝜃𝑥2 𝑢2𝑥

]

+

𝑢2𝑥 𝑣𝑥2 )𝑑𝑥

.

(3.2.48) Noting the boundedness of the interpolation inequalities imply the integral on the right-hand side of (3.2.48) is estimated by ∫1

2 2 0 (𝑢𝑥 + 𝑣𝑥 )𝑑𝑥,

∫ 𝜆 Thus, by (3.2.36),

0

1

( ∫ 2 2 (𝑣𝑥𝑥 + 𝜃𝑥𝑥 )𝑑𝑥 + Λ 𝑉 (𝑡) +

1

0

𝑑 𝑉 (𝑡) ≤ Λ(𝐷(𝑡) + 𝑉 (𝑡)). 𝑑𝑡

) 𝑢2𝑥 𝑑𝑥 .

(3.2.49)

86


Finally, multiplying (3.2.47) by a small constant 𝜀 > 0 and then adding the result to (3.2.43), we deduce 𝑑 (𝐴 + 𝜀𝐷) + 𝜆(𝐷 + 𝑉 ) ≤ 0. 𝑑𝑡 For sufficiently small 𝜀, by (3.2.46) and (3.2.49) and the above inequality, we may conclude 𝑑 (𝐴 + 𝜀𝐷 + 𝜀𝑉 ) + 𝜆(𝐴 + 𝜀𝐷 + 𝜀𝑉 ) ≤ 0. 𝑑𝑡 Thus, (𝐴 + 𝜀𝐷 + 𝜀𝑉 )(𝑡) ≤ Λ𝑒−𝜆𝑡 . Recalling (3.2.36), (3.2.40) and (3.2.41), we deduce (3.2.7)–(3.2.8). And (3.2.5) follows from (3.2.7)–(3.2.8). The proof is complete. □ Lemma 3.2.3. Under the assumptions of Theorem 3.1.1, the following estimate holds: (3.2.50) 0 < 𝐶1−1 ≤ 𝜃(𝑥, 𝑡), ∀(𝑥, 𝑡) ∈ [0, 1] × [0, +∞). Proof. To prove (3.2.50), we recall (3.2.8). Then we conclude that there exists a positive constant 𝑇 , such that for any 𝑡 > 𝑇 , 𝜃(𝑥, 𝑡) ≥ 𝐶1−1 > 0.

(3.2.51) 1 𝜃

For the lower bound of 𝜃(𝑥, 𝑡) on the interval [0, 𝑇 ], we introduce 𝜔 = and find that 𝜔 satisfies (𝜔 ) 𝜔2 𝜔 𝜔 2 𝑣𝑥2 𝑥 𝜔𝑡 = 𝑝 𝑣𝑥 − 𝜇 − 2𝜅 𝑥 + 𝜅 . (3.2.52) 𝑢 𝑢 𝑢𝜔 𝑢 𝑥 Multiplying (3.2.52) by 𝜔 2𝑝−1 and integrating the resultant over (0, 1), we obtain ∫ 1 2𝑝 ∫ 1 2𝑝+1 2 ∫ 1 𝑑 1 2𝑝 𝜔 𝑣𝑥 𝜔 𝑑𝑥 ≤ 𝜔 𝑑𝑥 + 𝜇 𝑣 𝑑𝑥 𝑝 𝑥 2𝑝 𝑑𝑡 0 𝑢 0 0 𝑢 ∫ ∫ 1 𝜇 1 𝜔 2𝑝+1 𝑣𝑥2 𝑑𝑥 + 𝐶1 𝜔 2𝑝−1 𝑑𝑥. (3.2.53) ≤ 2 0 𝑢 0 Let 𝑦(𝑡) = ∥𝜔(⋅, 𝑡)∥𝐿2𝑝 . Then we derive from (3.2.53) that

Hence,

𝑑 𝑦(𝑡) ≤ 𝐶1 . 𝑑𝑡

(3.2.54)

∥𝜔(⋅, 𝑡)∥𝐿2𝑝 ≤ 𝐶1 (1 + 𝑡).

(3.2.55)

Passing to limit, as 𝑝 → +∞, we have ∥𝜔(⋅, 𝑡)∥𝐿∞ ≤ 𝐶1 (1 + 𝑡), i.e,

𝜃(𝑥, 𝑡) ≥ 𝐶1 (1 + 𝑡)−1 ≥ 𝐶1 (1 + 𝑇 )−1 ,

which, along with (3.2.51), gives (3.2.50).

(3.2.56) ∀𝑡 ∈ [0, 𝑇 ] □


87

In what follows we shall use the idea of [55] to prove exponential stability in 𝐻 1 . Set Ψ(𝑢, 𝜃) = 𝑒(𝑢, 𝜃) − 𝜃𝜂(𝜃, 𝑢) (3.2.57) where 𝜂(𝑢, 𝜃) verifies 1 = 𝑐𝑣 𝑒𝜃 = 𝜃𝜂𝜃 ,

𝜂𝑢 = 𝒫𝜃 .

(3.2.58)

Now we introduce the density of Newtonian fluid, 𝜌 = 1/𝑢, then 𝜂 = 𝜂(1/𝜌, 𝜃) satisfies { −𝒫𝜃 𝑒𝜃 1 ∂𝜂 ∂𝜂 𝜌2 , 𝑝 = 1 = = = . (3.2.59) , 1 ∂𝜌 ∂𝜃 𝜃 𝜃 , 𝑝>1 𝒫𝜃

We consider the transform 𝐴 : (𝜌, 𝜃) ∈ 𝐷𝜌,𝜃 = {(𝜌, 𝜃) : 𝜌 > 0, 𝜃 > 0} → (𝑢, 𝜂) ∈ 𝐴𝐷𝜌,𝜃 .

(3.2.60)

Owing to the Jacobian ∣∂(𝑢, 𝜂)/∂(𝜌, 𝜃)∣ = −𝑒𝜃 /𝜌2 𝜃 < 0 on 𝐴𝐷𝜌,𝜃 , there is a unique inverse function 𝜃 = 𝜃(𝑢, 𝜂) as the smooth function of (𝑢, 𝜂) ∈ 𝐴𝐷𝜌,𝜃 . Thus the functions 𝑒, 𝒫 can also be regarded as the smooth functions of (𝑢, 𝜂). We denote them by 𝑒 = 𝑒(𝑢, 𝜂) :≡ 𝑒(𝑢, 𝜃(𝑢, 𝜂)) = 𝑒(1/𝜌, 𝜃), 𝒫 = 𝒫(𝑢, 𝜂) :≡ 𝒫(𝑢, 𝜃(𝑢, 𝜂)) = 𝒫(1/𝜌, 𝜃).

(3.2.61)

Then it follows from (3.2.57)–(3.2.61) that 𝑒𝑢 = −𝒫,

𝑒𝜂 = 𝜃,

2

𝒫𝑢 = −(𝜌 𝒫𝜌 + 𝜃𝒫𝜃2 /𝑒𝜃 ), 𝒫𝜂 = 𝜃𝒫𝜃 /𝑒𝜃 , 𝜃𝑢 = −𝜃𝒫𝜃 /𝑒𝜃 ,

𝜃𝜂 = 𝜃/𝑒𝜃 .

(3.2.62)

We define the following energy form, 𝑉 (𝑢, 𝑣, 𝜂) =

𝑣2 ∂𝑒 ∂𝑒 + 𝑒(𝑢, 𝜂) − 𝑒(¯ 𝑢, 𝜂¯) − (¯ 𝑢, 𝜂¯)(𝑢 − 𝑢¯) − (¯ 𝑢, 𝜂¯)(𝜂 − 𝜂¯). (3.2.63) 2 ∂𝑢 ∂𝜂

Lemma 3.2.4. The unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ (𝐻 1 [0, 1])3 to problem (3.1.2)–(3.1.6) satisfies the following estimates: 𝑣2 + 𝐶1−1 [(𝑢 − 1)2 + (𝜂 − 𝜂¯)2 ] ≤ 𝑉 (𝑢, 𝑣, 𝜂) 2 𝑣2 + 𝐶1 [(𝑢 − 𝑢¯)2 + (𝜂 − 𝜂¯)2 ]. ≤ 2 Proof. The proof is similar to that of Lemma 2.2.2 (see also Qin [55]).

(3.2.64) □

88


Lemma 3.2.5. There exist constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) > 0 such any fixed 𝛾 ∈ (0, 𝛾1 ], the global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐻 1 to problem (3.1.6) satisfies the following estimate: ( ) ¯ 21 𝑒𝛾𝑡 ∥𝑣(𝑡)∥2𝐻 1 + ∥𝑢(𝑡) − 1∥2𝐻 1 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 ∫ 𝑡 ( ) + 𝑒𝛾𝑡 ∥𝑢𝑥 ∥2 + ∥𝑣𝑥 ∥2𝐻 1 + ∥𝜃𝑥 ∥2𝐻 1 + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶1 , ∀ 0

that for (3.1.2)– (3.2.65) 𝑡 > 0.

Proof. The proof is similar to that of Lemmas 2.2.3–2.2.4 (see also Qin [55]).

□

Lemma 3.2.6. Under the assumptions of Theorem 3.1.1, the following estimates hold: ∫ 𝑡 2 2 (∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0, (3.2.66) ∥𝑣𝑡 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + 0 ∫ 𝑡 2 ∥𝑢𝑥𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0, (3.2.67) ∥𝑢𝑥𝑥(𝑡)∥ + 0 ∫ 𝑡 2 2 (∥𝑣𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0. (3.2.68) ∥𝑣𝑥𝑥 (𝑡)∥ + ∥𝜃𝑥𝑥(𝑡)∥ + 0

Proof. The proofs of (3.2.66) and (3.2.68) are similar to that of (2.3.1). We only need to prove (3.2.67). Differentiating (3.1.3) with respect to 𝑥, using (3.1.2) (𝑢𝑡𝑥𝑥 = 𝑣𝑥𝑥𝑥 ), we see that ∂ ( 𝑢𝑥𝑥 ) − 𝒫𝑢 𝑢𝑥𝑥 = 𝑣𝑡𝑥 + 𝐸(𝑥, 𝑡) 𝜇 (3.2.69) ∂𝑡 𝑢 where 𝐸(𝑥, 𝑡) = 𝒫𝑢𝑢 𝑢2𝑥 + 2𝒫𝜃𝑢 𝜃𝑥 𝑢𝑥 + 𝒫𝜃𝜃 𝜃𝑥2 + 𝒫𝜃 𝜃𝑥𝑥 − 2𝜇𝑣𝑥 𝑢2𝑥 /𝑢3 + 2𝜇𝑢𝑥 𝑣𝑥𝑥 /𝑢2 ( ) 𝑣𝑥𝑥 𝑢𝑥 𝜃 𝜃𝑥 𝑢 𝑥 1 𝑣𝑥 𝑢2𝑥 = 𝑝(𝑝 + 1) 𝑝+2 𝑢2𝑥 − 2𝑝 𝑝+1 + 𝑝+1 𝜃𝑥𝑥 − 2𝜇 − . 𝑢 𝑢 𝑢 𝑢2 𝑢3 Multiplying (3.2.69) by 𝑢𝑥𝑥 /𝑢, and by Young’s inequality, by Lemmas 3.2.2–3.2.3, we can deduce that 𝑢 2 𝑑 𝑥𝑥 𝑢𝑥𝑥 2 + 𝐶1−1 𝑑𝑡 𝑢 𝑢 ( 1 𝑢𝑥𝑥 2 ≤ + 𝐶1 ∥𝜃𝑥 ∥2𝐿∞ ∥𝑢𝑥 ∥2 + ∥𝑢𝑥 ∥4𝐿4 + ∥𝑣𝑡𝑥 ∥2 4𝐶1 𝑢 ) + ∥𝑢𝑥 ∥2𝐿∞ ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥∥2 + ∥𝑣𝑥 ∥2𝐿∞ ∥𝑢𝑥 ∥4𝐿4 ( ) 1 𝑢𝑥𝑥 2 ≤ (3.2.70) + 𝐶2 ∥𝜃𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 + ∥𝑢𝑥 ∥2 + ∥𝑣𝑡𝑥 ∥2 2𝐶1 𝑢 which, combined with Lemma 3.2.2, gives (3.2.67).

□


89

Lemma 3.2.7. There exist constants 𝐶2 > 0 and 𝛾2 = 𝛾2 (𝐶2 ) > 0 such that for 2 to problem (3.1.2)– any fixed 𝛾 ∈ (0, 𝛾2 ], the global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡)) ∈ 𝐻+ (3.1.6) satisfies that the following estimates: ( ) ¯ 22 𝑒𝛾𝑡 ∥𝑢(𝑡) − 1∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 ∫ 𝑡 ( ) ¯ 2 3 (𝜏 )𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0, (3.2.71) + 𝑒𝛾𝜏 ∥𝑢 − 1∥2𝐻 2 + ∥𝑣∥2𝐻 3 + ∥𝜃 − 𝜃∥ 𝐻 0 ∫ 𝑡 ( ) ( ) 𝛾𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0. 𝑒 ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + 0

(3.2.72)

Proof. The proof is similar to that of Lemma 2.3.4.

□

Proof of Theorem 3.1.1. By Lemmas 3.2.1–3.2.7, we complete the proof of Theorem 3.1.1. □

3.3 Proof of Theorem 3.1.2 In this section, we shall complete the proof of Theorem 3.1.2 and take the assumptions in Theorem 3.1.2 to be valid. We begin with the following lemma. 4 , we Lemma 3.3.1. Under assumptions of Theorem 3.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ have

∥𝑣𝑡𝑥 (𝑥, 0)∥ + ∥𝜃𝑡𝑥 (𝑥, 0)∥ ≤ 𝐶3 ,

(3.3.1)

∥𝑣𝑡𝑡 (𝑥, 0)∥ + ∥𝜃𝑡𝑡 (𝑥, 0)∥ + ∥𝑣𝑡𝑥𝑥 (𝑥, 0)∥ + ∥𝜃𝑡𝑥𝑥 (𝑥, 0)∥ ≤ 𝐶4 , ∫ 𝑡 ∫ 𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, ∀𝑡 > 0, 0 0 ∫ 𝑡 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ∥𝜃𝑡𝑡 (𝑡)∥2 + 0 ∫ 𝑡 ∫ 𝑡 ( ) ∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 (𝜏 ) 𝑑𝜏, ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 + 𝐶1 𝜀 ≤ 𝐶4 + 𝐶2 𝜀−1

(3.3.2)

0

0

(3.3.3) (3.3.4) ∀𝑡 > 0

for 𝜀 > 0 small enough. Proof. The proof is similar to that of Lemma 2.4.1.

□

4 , the Lemma 3.3.2. Under assumptions of Theorem 3.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ following estimates hold for any 𝑡 > 0 and for 𝜀 ∈ (0, 1) small enough, ∫ 𝑡 ∫ 𝑡 ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶1 𝜀2 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, (3.3.5) 0 0 ∫ 𝑡 ∫ 𝑡 2 2 2 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏. (3.3.6) ∥𝜃𝑡𝑥 (𝑡)∥ + 0

0


□

90


4 Lemma 3.3.3. Under assumptions of Theorem 3.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ , we have for any 𝑡 > 0,

∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 ∫ 𝑡( ) ∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , +

0 ∥𝑢𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∫ 𝑡(

+

∫ 0

𝑡

∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 ) ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑡𝑥𝑥 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 ,

0

(∥𝑢𝑥𝑥𝑥∥2𝐻 1 + ∥𝑣𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥 ∥2𝐻 1 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(3.3.7)

(3.3.8) (3.3.9)


□

4 Lemma 3.3.4. Under assumptions of Theorem 3.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ , (1) (1) there exists a positive constant 𝛾4 = 𝛾4 (𝐶4 ) ≤ 𝛾2 (𝐶2 ) such that for any fixed (1) 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0 and 𝜀 ∈ (0, 1) small enough, ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝑒𝛾𝜏 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, 𝑒𝛾𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + 0

𝑒𝛾𝑡 ∥𝜃𝑡𝑡 (𝑡)∥2 +

∫

0

𝑡

0

∫

(3.3.10) 𝑡

𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 𝜀−3 + 𝐶2 𝜀−1 𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏. +𝜀 0

(3.3.11)


□ 4 𝐻+ ,

Lemma 3.3.5. Under assumptions of Theorem 3.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 ) ∈ (2) (1) (2) there exists a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0 and 𝜀 ∈ (0, 1) small enough, ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀2 𝑒𝛾𝜏 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, 𝑒𝛾𝑡 ∥𝑣𝑡𝑥 (𝑡)∥2 + 0

𝑒𝛾𝑡 ∥𝜃𝑡𝑥 (𝑡)∥2 +

∫

0

𝑡

∫

𝑡

𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝜀2 𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, 0 0 ∫ 𝑡 𝛾𝑡 2 2 𝛾𝜏 2 𝑒 (∥𝑣𝑡𝑥𝑥 ∥ + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 𝑒 (∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥ ) + 0 ∫ 𝑡 2 𝛾𝜏 𝑒 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏. ≤ 𝐶4 + 𝐶2 𝜀 0


(3.3.12) (3.3.13)

(3.3.14) □


91

4 Lemma 3.3.6. Under assumptions of Theorem 3.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 ) ∈ 𝐻+ , (2) there is a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0,

𝑒𝛾𝑡 (∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 ) ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , 0 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑢𝑥𝑥𝑥(𝜏 )∥2𝐻 1 𝑑𝜏 ≤ 𝐶4 , 𝑒𝛾𝑡 ∥𝑢𝑥𝑥𝑥(𝑡)∥2𝐻 1 + 0 ( ) 𝑒𝛾𝑡 ∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 ∫ 𝑡 ( + 𝑒𝛾𝜏 ∥𝑣𝑥𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 0 ) + ∥𝜃𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . Proof. The proof is similar to that of Lemmas 2.4.7–2.4.8.

(3.3.15) (3.3.16)

(3.3.17) □

Proof of Theorem 3.1.2. By Lemmas 3.3.1–3.3.6, Theorem 3.1.1 and Sobolev’s embedding theorem, we complete the proof of Theorem 3.1.2. □

3.4 Bibliographic Comments Now let’s first recall some previous works in this direction. For the case of an ideal gas, i.e., 𝜃 (3.4.1) 𝑒 = 𝐶𝑣 𝜃, 𝒫 = 𝑅 𝑢 with suitable positive constants 𝐶𝑣 and 𝑅, the global existence and asymptotic behavior of smooth (generalized) solutions in 𝐻 𝑖 (𝑖 = 1, 2) to the system (3.1.2)– (3.1.4) have been investigated by many authors (see, e.g., Antontsev, Kazhikhov and Monakhov [1], Chen, Hoff and Trivisa [4], Hoff [22], Hsiao and Luo [24], Jiang [26, 27], Kawashima and Nishida [31], Kawohl [32], Matsumura and Nishida [40, 41, 42, 43], Nagasawa [45], Okada and Kawashima [47], Qin [59, 50, 49, 51, 52], Qin and Mu˜ noz Rivera [66], Qin, Wu and Liu [73]) on the initial boundary value problems. For the Cauchy problem with (3.4.1), we refer to the works Itaya [25], Kawashima and Nishida [31], Kazhikhov [35], Kazhikhov and Shelukhin [36], Matsumura and Nishida [40, 41], Okada and Kawashima [47], Qin [55], Zheng and Shen [85]. For a nonlinear one-dimensional heat-conductive viscous real gas with some constitutive equations and special forms of functions 𝜇, 𝜅 and 𝒫, the classical solutions (weak solutions in 𝐻 1 ) exist globally in time and converge exponentially to a steady state in 𝐻 1 with some strong assumptions for (3.1.2)–(3.1.6) (see, Jiang

92


[26, 27], Kawohl [32]). Later on, Qin [49, 50, 51, 52, 54] established the same results as above on the global existence and exponential stability with some weaker assumptions. For multidimensional initial boundary value problems and Cauchy problems, the global existence and asymptotic behavior of smooth solutions have been investigated for general domains only in case of “small initial data” (see, e.g, Antontsev, Kazhikhov and Monakhov [1], Deckelnick [9], Fujita-Yashima and Benabidallah [80, 81], Hoff [23], Itaya [25], Kawashima and Nishida [31], Kazhikhov [35], Kazhikhov and Shelukhin [36], Matsumura and Nishida [40, 41, 42, 43], Okada and Kawashima [47], Qin [59, 55], Zheng and Shen [85] and references cited therein). However, in our case, the form of the pressure 𝒫 in (3.1.1) can be regarded as a generalization of the constitutive equation for an ideal gas, as well as a modification of the relation for a barotropic gas, where 𝒫 = 𝑢−𝑝 . From this point of view, (3.1.1) is an interpolation between these models. In this direction, based on the result obtained in [37] with 𝒫 = 𝑢𝜃𝑝 , we have established in this chapter the regularity and exponential stability of global solutions in 𝐻 𝑖 (𝑖 = 2, 4), which are two new ingredients of this chapter. As a result, by the embedding theorem, the global solution obtained in 𝐻 4 is actually a classical one in 𝐶 3+1/2 when it is subjected to the corresponding compatibility conditions. Thus the exponential stability of classical solutions is obtained, which is a new result for this model. The aim of this chapter is to prove the global existence and exponential of solutions in 𝐻 𝑖 (𝑖 = 2, 4) for equations (3.1.2)– (3.1.4) for boundary conditions (3.1.6) with pressure (3.1.1).

Chapter 4

Global Existence and Exponential Stability for the 𝒑th Power Viscous Reactive Gas 4.1 Introduction In this chapter, we prove the global existence and exponential stability of solutions in 𝐻 𝑖 (𝑖 = 2, 4) for the compressible Navier-Stokes equations, which arise in the study of a thermal explosion and describe the dynamic combustion for a reactive Newtonian fluid, confined between two infinite parallel plates. We assume that the pressure 𝒫, in terms of the absolute temperature 𝜃 and the specific volume 𝑢, is given by 𝜃 (4.1.1) 𝒫= 𝑝 𝑢 with the pressure exponent 𝑝 ≥ 1. The balance laws of mass, momentum, and energy, coupled with the description of the chemical reaction for one-dimensional (in Lagrangian form) case are 𝑢𝑡 = 𝑣𝑥 , ( 𝑣𝑥 ) =: 𝜎𝑥 , 𝑣𝑡 = −𝒫 + 𝜇 𝑢 𝑥 ( ) ( 𝑣𝑥 ) 𝜃𝑥 𝑐𝑣 𝜃𝑡 = −𝒫 + 𝜇 𝑣𝑥 + 𝜅 + 𝛿𝑓 (𝑢, 𝜃, 𝑧), 𝑢 𝑢 𝑥 ( 𝑧 ) 𝑥 𝑧𝑡 = 𝑑 2 − 𝑓 (𝑢, 𝜃, 𝑧). 𝑢 𝑥

(4.1.2) (4.1.3) (4.1.4) (4.1.5)

Here, 𝑢, 𝑣, 𝜃, 𝑧, 𝑓 are the specific volume, velocity, the absolute temperature, the mass fraction of the unburned fuel, the function of chemical reaction, respectively. All the above quantities are assumed to vary spatially only in the direction perpendicular to the plates. The positive constants 𝑐𝑣 , 𝜇, 𝜅, 𝑑, 𝛿 represent specific heat, viscosity, conductivity, the species diffusion coefficient, and the reactive


93

94


rate, respectively. Since the magnitude of the specific heat 𝑐𝑣 plays no role in the mathematical analysis of the system, in what follows we will assume the scaling 𝑐𝑣 = 1. has

The function 𝑓 describes the intensity of the chemical reaction, typically one 𝑓 (𝑢, 𝜃, 𝑧) = 𝜀𝑢1−𝑚 𝑧 𝑚 exp

𝜃−1 𝜀𝜃

(4.1.6)

which is called the Arrhenius rate law for chemical reaction. Here, 𝜀−1 , 𝑚 are positive constants and denote the activation energy and the overall sum of the individual reaction orders for the fuel and oxidizer, respectively. The physically interesting case is that 𝑚 ≥ 1. We consider a typical initial boundary value problem for (4.1.2)–(4.1.6) in the reference domain {(𝑥, 𝑡) : 0 < 𝑥 < 1, 𝑡 ≥ 0} under the initial conditions and boundary conditions 𝑢(𝑥, 0) = 𝑢0 (𝑥), 𝑣(𝑥, 0) = 𝑣0 (𝑥), 𝜃(𝑥, 0) = 𝜃0 (𝑥), 𝑧(𝑥, 0) = 𝑧0 (𝑥),

(4.1.7)

𝑣(0, 𝑡) = 𝑣(1, 𝑡) = 0, 𝜃𝑥 (0, 𝑡) = 𝜃𝑥 (1, 𝑡) = 0, 𝑧𝑥 (0, 𝑡) = 𝑧𝑥 (1, 𝑡) = 0.

(4.1.8)

The physical meaning of the boundary conditions (4.1.8) are clear, namely, both ends of the rods are clamped, thermally insulated and impermeable. The notation in this chapter is standard. We put ∥⋅∥ = ∥⋅∥𝐿2 [0,1] . Subscripts 𝑡 and 𝑥 denote the (partial) derivatives with respect to 𝑡 and 𝑥, respectively. We use 𝐶𝑖 (𝑖 = 1, 2, 4) to denote the generic positive constant depending on the 𝐻 𝑖 [0, 1] norm of initial data (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ), min 𝑢0 (𝑥), min 𝜃0 (𝑥) and min 𝑧0 (𝑥), but 𝑥∈[0,1]

𝑥∈[0,1]

𝑥∈[0,1]

independent of variable 𝑡. 𝐶 stands for the absolute positive constant independent of 𝐶𝑖 (𝑖 = 1, 2, 4) and initial data. ∫1 For convenience and without loss of generality, we may assume 0 𝑢0 (𝑥) 𝑑𝑥 = 1. Then from conservation of mass and boundary condition (4.1.8), we have ∫ 0

1

∫ 𝑢(𝑥, 𝑡) 𝑑𝑥 =

0

1

𝑢0 (𝑥) 𝑑𝑥 = 1.

(4.1.9)

In Chapter 3, we discussed the case 𝛿 = 0 (without chemical reaction). In this chapter, we shall investigate the case 𝛿 > 0 and the contribution of a chemical reaction to the existence and exponential stability of global solutions. Hence we have improved the previous results in [37]. Due to the involvement of a chemical reaction, the present situation is more complicated than that in Chapter 3, and more delicate and careful analyses are needed. Now we are in a position to state our main results which are chosen from [62].

4.1. Introduction

95

Define { 1 𝐻+ = (𝑢, 𝑣, 𝜃, 𝑧) ∈ (𝐻 1 [0, 1])4 : 𝑢(𝑥) > 0, 𝜃(𝑥) > 0, 𝑧(𝑥) ≥ 0, ∀𝑥 ∈ [0, 1], } 𝑣(0) = 𝑣(1) = 0 , { 𝑖 𝐻+ = (𝑢, 𝑣, 𝜃, 𝑧) ∈ (𝐻 𝑖 [0, 1])4 : 𝑢(𝑥) > 0, 𝜃(𝑥) > 0, 𝑧(𝑥) ≥ 0, ∀𝑥 ∈ [0, 1], } 𝑣(0) = 𝑣(1) = 𝜃′ (0) = 𝜃′ (1) = 𝑧 ′ (0) = 𝑧 ′ (1) = 0 , 𝑖 = 2, 4. 2 Theorem 4.1.1. Suppose that (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ and the compatibility conditions 2 hold. Then there exists a unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑧(𝑡)) ∈ 𝐻+ to the problem (4.1.2)–(4.1.8) verifying that for any (𝑥, 𝑡) ∈ [0, 1] × [0, +∞),

0 < 𝐶1−1 ≤ 𝑢(𝑥, 𝑡) ≤ 𝐶1 ,

0 < 𝐶1−1 ≤ 𝜃(𝑥, 𝑡) ≤ 𝐶1 ,

0 ≤ 𝑧(𝑥, 𝑡) ≤ max 𝑧0 (𝑥) 𝑥∈[0,1]

and for any 𝑡 > 0,

(4.1.10)

∥𝑢(𝑡) − 1∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥2𝐻 2 + ∥𝑧(𝑡)∥2𝐻 2 + ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑧𝑡 (𝑡)∥2 ∫ 𝑡 + (∥𝑢 − 1∥2𝐻 2 + ∥𝑣∥2𝐻 3 + ∥𝜃 − 𝜃∥2𝐻 3 + ∥𝑧𝑥 ∥2𝐻 2 + ∥𝑣𝑡 ∥2𝐻 1 0

+ ∥𝜃𝑡 ∥2𝐻 1 + ∥𝑧𝑡 ∥2𝐻 1 )(𝜏 )𝑑𝜏 ≤ 𝐶2 .

(4.1.11)

Moreover, there are constants 𝐶2 > 0 and 𝛾2 = 𝛾2 (𝐶2 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2 ], the following estimate holds for any 𝑡 > 0, ( ¯ 22 𝑒𝛾𝑡 ∥𝑢(𝑡) − 1∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 ) 2 2 2 + ∥𝑧(𝑡)∥𝐻 2 + ∥𝑣𝑡 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝑧𝑡 (𝑡)∥2 ∫ 𝑡 ( + 𝑒𝛾𝜏 ∥𝑢 − 1∥2𝐻 2 + ∥𝑣∥2𝐻 3 0 ) ¯ 2 3 + ∥𝑧∥2 3 + ∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝑧𝑡𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 + ∥𝜃 − 𝜃∥ (4.1.12) 𝐻 𝐻 where 𝜃 =

) ∫1( 1 2 0 𝜃0 + 2 𝑣0 + 𝛿𝑧0 𝑑𝑥 .

4 Theorem 4.1.2. Suppose that (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ and the compatibility conditions 4 hold. Then there exists a unique global solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑧(𝑡)) ∈ 𝐻+ to the problem (4.1.2)–(4.1.8) such that for any 𝑡 > 0,

∥𝑢(𝑡) − 1∥2𝐻 4 + ∥𝑣(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃(𝑡)∥2𝐻 4 + ∥𝑧(𝑡)∥2𝐻 4 + ∥𝑣𝑡 (𝑡)∥2𝐻 2 + ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑧𝑡 (𝑡)∥2𝐻 2 + ∥𝑧𝑡𝑡 (𝑡)∥2 ∫ 𝑡( ∥𝑢 − 1∥2𝐻 4 + ∥𝑣∥2𝐻 5 + ∥𝜃 − 𝜃∥2𝐻 5 + ∥𝑧∥2𝐻 5 + ∥𝑣𝑡 ∥2𝐻 3 + ∥𝜃𝑡 ∥2𝐻 3 + 0 ) + ∥𝑧𝑡 ∥2𝐻 3 + ∥𝑣𝑡𝑡 ∥2𝐻 1 + ∥𝜃𝑡𝑡 ∥2𝐻 1 + ∥𝑧𝑡𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (4.1.13)

96


Moreover, there are constants 𝐶4 > 0 and 𝛾4 = 𝛾4 (𝐶4 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimate holds for any 𝑡 > 0, ( ¯ 2 4 + ∥𝑧(𝑡)∥2 4 + ∥𝑣𝑡 (𝑡)∥2 2 𝑒𝛾𝑡 ∥𝑢(𝑡) − 1∥2𝐻 4 + ∥𝑣(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 𝐻 𝐻 ) + ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑧𝑡 (𝑡)∥2𝐻 2 + ∥𝑧𝑡𝑡 (𝑡)∥2 ∫ 𝑡 ( ¯ 2 5 + ∥𝑧𝑥 ∥2 4 + ∥𝑣𝑡 ∥2 3 + ∥𝜃𝑡 ∥2 3 𝑒𝛾𝜏 ∥𝑢 − 1∥2𝐻 4 + ∥𝑣∥2𝐻 5 + ∥𝜃 − 𝜃∥ + 𝐻 𝐻 𝐻 𝐻 0 ) + ∥𝑧𝑡 ∥2𝐻 3 + ∥𝑣𝑡𝑡 ∥2𝐻 1 + ∥𝜃𝑡𝑡 ∥2𝐻 1 + ∥𝑧𝑡𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (4.1.14) Remark 4.1.1. Obviously, it is easy to see that similar results hold for the boundary conditions 𝑣(0, 𝑡) = 𝑣(1, 𝑡) = 0, 𝜃(0, 𝑡) = 𝜃(1, 𝑡) = const . > 0, 𝑧𝑥 (0, 𝑡) = 𝑧𝑥 (1, 𝑡) = 0. Remark 4.1.2. Theorems 4.1.1–4.1.2 have improved the results in [38].

4.2 Global Existence in 𝑯 2 In this section we establish the global existence in 𝐻 2 by a series of lemmas. The 1 next lemma concerns the estimate in 𝐻+ . Lemma 4.2.1. Under the conditions in Theorem 4.1.1, the following estimates hold: 0 < 𝐶1−1 ≤ 𝑢(𝑥, 𝑡) ≤ 𝐶1 , 0 < 𝐶1−1 ≤ 𝜃(𝑥, 𝑡) ≤ 𝐶1 , ∀(𝑥, 𝑡) ∈ [0, 1] × [0, +∞), 0 ≤ 𝑧(𝑥, 𝑡) ≤ max 𝑧0 (𝑥), (𝑥, 𝑡) ∈ [0, 1] × [0, +∞), 𝑥∈[0,1]

∥𝑢(𝑡)

− 1∥2𝐻 1 ∫ 𝑡

+

0

+ ∥𝑣(𝑡)∥2𝐻 1 + ∥𝜃(𝑡) − 𝜃∥2𝐻 1 + ∥𝑧(𝑡)∥2𝐻 1 + ∥𝑓 (𝑡)∥2

(4.2.1) (4.2.2) (4.2.3)

(∥𝑢 − 1∥2𝐻 1 + ∥𝑣∥2𝐻 2 + ∥𝜃 − 𝜃∥2𝐻 2 + ∥𝑓 ∥𝐿1 + ∥𝑧𝑥∥2𝐻 1 )(𝜏 ) 𝑑𝜏 ≤ 𝐶1 .

Moreover, as 𝑡 → +∞, we have ∥𝑢𝑥(𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 + ∥𝑧𝑥 (𝑡)∥2 → 0, ) ( max ∣𝑢(𝑥, 𝑡) − 1∣ + ∣𝑣(𝑥, 𝑡)∣ + ∣𝜃(𝑥, 𝑡) − 𝜃∣ + ∣𝑧(𝑥, 𝑡)∣ → 0.

𝑥∈[0,1]

(4.2.4) (4.2.5)

Proof. Set 𝑧− (𝑥, 𝑡) = min{𝑧(𝑥, 𝑡), 0} and 𝑧+ (𝑥, 𝑡) = max{𝑧(𝑥, 𝑡), max𝑦∈[0,1] 𝑧0 (𝑦)}. In order to prove (4.2.2), we multiply (4.1.5) by 𝑧+ and integrate in space, obtaining ∫ 1 ∫ 1 ∫ (𝑧+ )2𝑥 1 𝑑 1 2 𝑧+ 𝑑𝑥 = −𝑑 𝑑𝑥 − 𝑓 𝑧+ 𝑑𝑥 ≤ 0. 2 𝑑𝑡 0 𝑢2 0 0

4.2. Global Existence in 𝐻 2

97

∫1 2 ∫1 2 In view of the initial condition 0 𝑧+ (𝑥, 0) = 0, we conclude that 0 𝑧+ 𝑑𝑥 = 0. So we obtain (4.2.2). We note that the entropy 𝜂 of a 𝑝th power viscous reactive gas is a concave function 𝜂(𝑢, 𝜃) = log 𝜃 + ℎ(𝑢), (4.2.6) where

{ ℎ(𝑢) =

log 𝑢, 𝑝 = 1, 1 1−𝑝 (1 − 𝑢 ), 𝑝 > 1, 𝑝−1

which satisfies the following standard entropy identity: 𝜂𝑡 = 𝜇 Thus, 𝑑 𝑑𝑡

∫ 0

1

𝜃2 𝑣𝑥2 𝑓 (𝑞) + 𝜅 𝑥2 + 𝛿 − . 𝑢𝜃 𝑢𝜃 𝜃 𝜃 𝑥

(4.2.7)

) ( ) ∫ 1( 2 𝜃𝑥2 𝑓 1 2 𝑣𝑥 ¯ ¯ +𝜅 2 +𝛿 𝜃 + 𝑣 + 𝛿𝑧 − 𝜃𝜂 𝑑𝑥 = −𝜃 𝜇 𝑑𝑥. 2 𝑢𝜃 𝑢𝜃 𝜃 0

(4.2.8)

Integrating (4.2.8) over [0, 𝑡], we get ) ) ∫ 1( 2 ∫ 1( 𝜃2 𝑓 1 𝑣 ¯ 𝑑𝑥 + 𝜃¯ 𝜃 + 𝑣 2 + 𝛿𝑧 − 𝜃𝜂 𝜇 𝑥 + 𝜅 𝑥2 + 𝛿 𝑑𝑥 ≤ 𝐶1 . 2 𝑢𝜃 𝑢𝜃 𝜃 0 0

(4.2.9)

Integrating (4.2.6) in space and then using Jensen’s inequality, we receive (∫ 1 ) (∫ 1 ) (∫ 1 ) ∫ 1 𝜂𝑑𝑥 ≤ log 𝜃𝑑𝑥 + ℎ 𝑢𝑑𝑥 = log 𝜃𝑑𝑥 . 0

0

0

0

Thus, in view of (4.2.9), we see that ) (∫ 1 ) ) ∫ 1( 2 ∫ 1( 𝜃2 𝑓 1 𝑣 𝜃𝑑𝑥 . 𝜃 + 𝑣 2 + 𝛿𝑧 𝑑𝑥 + 𝜃¯ 𝜇 𝑥 + 𝜅 𝑥2 + 𝛿 𝑑𝑥 ≤ 𝐶1 + 𝜃¯ log 2 𝑢𝜃 𝑢𝜃 𝜃 0 0 0 (4.2.10) In particular,

∫ 0

1

𝜃𝑑𝑥 ≤ 𝐶1 + 𝜃¯ log

which yields 0
1. 𝐶 1 0 Since

∫ 𝑧(𝑥, 𝑡) −

0

by (4.2.4), we receive

1

(∫ 𝑧(𝑥, 𝑡)𝑑𝑥 ≤

0

lim

1

)1/2 𝑧𝑥2 𝑑𝑥 ,

max 𝑧(𝑥, 𝑡) = 0.

(4.2.32)

𝑡→+∞ 𝑥∈[0,1]

Finally, we note that the quantity and thus ∫ 1 ∫ 1 + 𝜃 − 𝜃¯ ≤ 𝜃(𝑥, 𝑡) − 𝜃𝑑𝑥 2 0

∫1 0

1 0

(4.2.31)

(𝜃 + 𝑣 2 /2 + 𝛿𝑧)𝑑𝑥 is a constant in time,

𝑣 2 (𝑥, 𝑡)𝑑𝑥 + 𝛿

∫ 0

1

𝑧(𝑥, 𝑡)𝑑𝑥,

(4.2.33)

which, together with (4.2.33), (4.2.29) and (4.2.32), gives lim

¯ = 0. max ∣𝜃(𝑥, 𝑡) − 𝜃∣

𝑡→+∞ 𝑥∈[0,1]

(4.2.34)

Combining (4.2.29), (4.2.32) and (4.2.34), we obtain (4.2.5). The proof is complete. □

102


Lemma 4.2.2. Under the assumptions of Theorem 4.1.1, the following estimate holds for any 𝑡 > 0: ∫ 𝑡 (∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑧𝑡 ∥2 + ∥𝑓 ∥2 + ∥𝑓𝑥 ∥2 + ∥𝑓𝑡 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶1 . (4.2.35) 0

Proof. Since

( 0 < exp

𝜃−1 𝜀𝜃

it is easy to see that

( exp

)

𝜃−1 𝜀𝜃

≤ 𝐶,

∀𝜃 > 0,

)2 ≤ 𝐶.

Combining this with (4.1.6) and (4.2.1)–(4.2.3), we have ∫ 𝑡∫ 1 ∫ 𝑡∫ 1 ∫ 𝑡 ∥𝑓 (𝜏 )∥2 𝑑𝜏 = 𝑓 2 (𝑥, 𝜏 ) 𝑑𝑥𝑑𝜏 ≤ 𝐶 𝑓 (𝑥, 𝜏 ) 𝑑𝑥𝑑𝜏 ≤ 𝐶1 . 0

0

0

0

0

(4.2.36)

We easily infer from (4.1.3)–(4.1.6) and Lemma 4.2.1 that ∥𝑣𝑡 (𝑡)∥ ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝑢𝑥 (𝑡)∥ + ∥𝑣𝑥𝑥 (𝑡)∥) (4.2.37) ≤ 𝐶2 (∥𝑢𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥), ∥𝜃𝑡 (𝑡)∥ ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝑣𝑥 (𝑡)∥ + ∥𝑢𝑥 (𝑡)∥∥𝜃𝑥 (𝑡)∥𝐿∞ + ∥𝜃𝑥𝑥 (𝑡)∥ + ∥𝑓 (𝑡)∥) ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑓 (𝑡)∥), ∥𝑧𝑡 (𝑡)∥ ≤ 𝐶1 (∥𝑧𝑥𝑥(𝑡)∥ + ∥𝑢𝑥 (𝑡)∥∥𝑧𝑥 (𝑡)∥𝐿∞ + ∥𝑓 (𝑡)∥)

(4.2.38)

≤ 𝐶1 (∥𝑧𝑥 (𝑡)∥𝐻 1 + ∥𝑓 (𝑡)∥), ∥𝑓𝑥 (𝑡)∥ ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑥 (𝑡)∥ + ∥𝑧𝑥 (𝑡)∥),

(4.2.39) (4.2.40)

∥𝑓𝑡 (𝑡)∥ ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝑧𝑡 (𝑡)∥).

(4.2.41)

Utilizing (4.2.3) and (4.2.36)–(4.2.41), we obtain (4.2.35).

□

Lemma 4.2.3. Under the assumptions of Theorem 4.1.1, the following estimates hold for any 𝑡 > 0: ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑧𝑡 (𝑡)∥2 + ∥𝑓𝑥 (𝑡)∥2 + ∥𝑓𝑡 (𝑡)∥2 ∫ 𝑡 + (∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝑧𝑡𝑥 ∥2 + ∥𝑓𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 , 0 ∫ 𝑡 ∥𝑢𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∥𝑢𝑥𝑥(𝑡)∥2 + 0

∥𝑣𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝑧𝑥𝑥 (𝑡)∥2 ∫ 𝑡 + (∥𝑣𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥∥2 + ∥𝑧𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 . 0

(4.2.42) (4.2.43)

(4.2.44)


103

Proof. Since the proof is similar to that of Lemma 3.2.5 with a little modification, we only sketch the proof here. Only the estimates on 𝜃 are different due to the involvement of 𝑓 . Differentiating (4.1.3) with respect to 𝑡, multiplying the resulting equation by 𝑣𝑡 in 𝐿2 (0, 1), performing an integration by parts, we obtain ) ∫ 1( 1 𝑑 𝑣𝑥 𝜃 ∥𝑣𝑡 (𝑡)∥2 = − 𝑣𝑡𝑥 𝑑𝑥 − 𝑝 +𝜇 2 𝑑𝑡 𝑢 𝑢 𝑡 0 ∫ 1 2 𝑣𝑡𝑥 𝑑𝑥 + 𝐶1 (∥𝑣𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥2𝐿4 )∥𝑣𝑡𝑥 (𝑡)∥ ≤ −𝜇 𝑢 0 ≤ −𝐶1−1 ∥𝑣𝑡𝑥 (𝑡)∥2 + 𝐶1 (∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 )

which, together with Lemmas 4.2.1–4.2.2, yields ∫ 𝑡 ∥𝑣𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∥𝑣𝑡 (𝑡)∥2 + 0

∀ 𝑡 > 0.

(4.2.45)

(4.2.46)

On the other hand, using (4.1.3) and (4.2.46), Lemmas 4.2.1–4.2.2, Sobolev’s embedding theorem and Young’s inequality, we have ∥𝑣𝑥𝑥 (𝑡)∥ ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥∥𝑣𝑥 (𝑡)∥𝐿∞ + ∥𝜃𝑥 (𝑡)∥ + ∥𝑣𝑡 ∥ + ∥𝑢𝑥 (𝑡)∥) 1 ≤ ∥𝑣𝑥𝑥 (𝑡)∥ + 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥ + ∥𝜃𝑥 (𝑡)∥ + ∥𝑣𝑡 (𝑡)∥) 2 which leads to ∥𝑣𝑥𝑥 (𝑡)∥ ≤ 𝐶2 , Similarly,

∥𝑣𝑥 (𝑡)∥𝐿∞ ≤ 𝐶2 ,

∀ 𝑡 > 0.

(4.2.47)

∥𝜃𝑥𝑥 (𝑡)∥ ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥ + ∥𝑓 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥), ∥𝑧𝑥𝑥 (𝑡)∥ ≤ 𝐶1 (∥𝑓 (𝑡)∥ + ∥𝑧𝑥 (𝑡)∥ + ∥𝑧𝑡 (𝑡)∥).

(4.2.48) (4.2.49)

Similarly, differentiating (4.1.4) with respect to 𝑡, multiplying the resultant by 𝜃𝑡 and integrating by parts, we deduce 1 𝑑 ∥𝜃𝑡 (𝑡)∥2 + 𝐶1−1 ∥𝜃𝑡𝑥 (𝑡)∥2 2 𝑑𝑡 (

≤ (2𝐶1 )−1 ∥𝜃𝑡𝑥 (𝑡)∥2 + 𝐶1 ∥𝜃𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2 ) + ∥𝜃𝑡 (𝑡)∥2 ∥𝑣𝑥 (𝑡)∥2𝐿∞ + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑓𝑡 (𝑡)∥2

which, combined with (4.2.46) and Lemmas 4.2.1–4.2.2, implies ∫ 𝑡 ∥𝜃𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∀ 𝑡 > 0. 0

(4.2.50)

(4.2.51)

By (4.2.48) and (4.2.51), we easily get ∥𝜃𝑥𝑥 (𝑡)∥ ≤ 𝐶2 ,

∥𝜃𝑥 (𝑡)∥𝐿∞ ≤ 𝐶2 ,

∀ 𝑡 > 0.

(4.2.52)

104


Similarly to (4.2.46) and (4.2.51), by equation (4.1.5), we have 1 𝑑 (4.2.53) ∥𝑧𝑡 (𝑡)∥2 + 𝐶1−1 ∥𝑧𝑡𝑥 (𝑡)∥2 2 𝑑𝑡 −1 2 2 2 2 2 ≤ (2𝐶1 ) ∥𝑧𝑡𝑥 (𝑡)∥ + 𝐶1 (∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝑧𝑥 (𝑡)∥ + ∥𝑧𝑡 (𝑡)∥ + ∥𝑓𝑡 (𝑡)∥ ) which, combined with (4.2.47), (4.2.49) and Lemmas 4.2.1–4.2.2, implies ∫ 𝑡 2 ∥𝑧𝑡 (𝑡)∥ + ∥𝑧𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , ∀ 𝑡 > 0, (4.2.54) 0

∥𝑧𝑥𝑥 (𝑡)∥ + ∥𝑧𝑥 (𝑡)∥𝐿∞ ≤ 𝐶2 ,

∀ 𝑡 > 0.

(4.2.55)

Differentiating (4.1.3) with respect to 𝑥, using (4.1.2) (𝑢𝑡𝑥𝑥 = 𝑣𝑥𝑥𝑥 ), we see that 𝜇

∂ ( 𝑢𝑥𝑥 ) − 𝒫𝑢 𝑢𝑥𝑥 = 𝑣𝑡𝑥 + 𝐸(𝑥, 𝑡) ∂𝑡 𝑢

(4.2.56)

where 𝑣𝑥 𝑢2𝑥 𝑢𝑥 𝑣𝑥𝑥 + 2𝜇 2 𝑢(3 𝑢 ) 𝑣 𝑢 𝑢 𝜃 𝜃 1 𝑣𝑥 𝑢2𝑥 𝑥 𝑥 𝑥𝑥 𝑥 2 = 𝑝(𝑝 + 1) 𝑝+2 𝑢𝑥 − 2𝑝 𝑝+1 + 𝑝+1 𝜃𝑥𝑥 − 2𝜇 − 3 . 𝑢 𝑢 𝑢 𝑢2 𝑢

𝐸(𝑥, 𝑡) = 𝒫𝑢𝑢 𝑢2𝑥 + 2𝒫𝜃𝑢 𝜃𝑥 𝑢𝑥 + 𝒫𝜃𝜃 𝜃𝑥2 + 𝒫𝜃 𝜃𝑥𝑥 − 2𝜇

Multiplying (4.2.56) by 𝑢𝑥𝑥 /𝑢, and by Young’s inequality, by (4.2.46)–(4.2.47), (4.2.51)–(4.2.52), (4.2.54)–(4.2.55) and Lemmas 4.2.1–4.2.2, we can deduce that 𝑢 𝑑 𝑥𝑥 2 𝑢𝑥𝑥 2 (𝑡) + 𝐶1−1 (𝑡) 𝑑𝑡 𝑢 𝑢 ( 1 𝑢𝑥𝑥 2 (𝑡) + 𝐶1 ∥𝜃𝑥 (𝑡)∥2𝐿∞ ∥𝑢𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥4𝐿4 + ∥𝑣𝑡𝑥 (𝑡)∥2 ≤ 4𝐶1 𝑢 ) + ∥𝑢𝑥 (𝑡)∥2𝐿∞ ∥𝑣𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2𝐿∞ ∥𝑢𝑥 (𝑡)∥4𝐿4 1 𝑢𝑥𝑥 2 ≤ + 𝐶2 (∥𝜃𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 ) (4.2.57) 2𝐶1 𝑢 which, combined with Lemma 4.2.1 and (4.2.46), gives (4.2.43). Differentiating (4.1.3), (4.1.4) and (4.1.5) with respect to 𝑥, respectively, and using (4.2.47), (4.2.52), (4.2.55) and Lemmas 4.2.1–4.2.2 to get ∥𝑣𝑡𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝑢𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥𝐻 1 )

(4.2.58)

∥𝜃𝑡𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑢𝑥 (𝑡)∥𝐻 1 + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑓𝑥 (𝑡)∥), ∥𝑧𝑡𝑥 (𝑡)∥ ≤ 𝐶1 (∥𝑓𝑥 (𝑡)∥ + ∥𝑧𝑥𝑥𝑥(𝑡)∥ + ∥𝑢𝑥 (𝑡)∥𝐿∞ ∥𝑧𝑥𝑥 (𝑡)∥

(4.2.59)

+ ∥𝑧𝑥 (𝑡)∥𝐿∞ ∥𝑢𝑥𝑥(𝑡)∥ + ∥𝑢𝑥 (𝑡)∥2𝐿∞ ∥𝑧𝑥 (𝑡)∥) ≤ 𝐶2 (∥𝑧𝑥 (𝑡)∥𝐻 2 + ∥𝑓𝑥 (𝑡)∥ + ∥𝑢𝑥 (𝑡)∥𝐻 1 )

(4.2.60)

4.3. Exponential Stability in 𝐻 2

105

or (4.2.61) ∥𝑣𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥(𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑣𝑡𝑥 (𝑡)∥), ∥𝜃𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶2 (∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥 (𝑡)∥𝐻 1 + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑓𝑥 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥), (4.2.62) ∥𝑧𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶2 (∥𝑧𝑥 (𝑡)∥𝐻 1 + ∥𝑓𝑥 (𝑡)∥ + ∥𝑢𝑥 (𝑡)∥𝐻 1 + ∥𝑧𝑡𝑥 (𝑡)∥). We derive from (4.1.6), (4.2.47), (4.2.52) and (4.2.55) that ∥𝑓𝑡𝑥 (𝑡)∥ ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑧𝑡 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥ + ∥𝑧𝑡𝑥 (𝑡)∥).

(4.2.63) (4.2.64)

By (4.2.40)–(4.2.41), (4.2.46)–(4.2.47), (4.2.51)–(4.2.52), (4.2.54)–(4.2.55), (4.2.64) and Lemmas 4.2.1–4.2.2, we obtain (4.2.42), and from (4.2.42)–(4.2.43) and (4.2.61)–(4.2.63), we conclude (4.2.44). The proof is complete. □

4.3 Exponential Stability in 𝑯 2 2 In this section, we shall establish the exponential stability of solutions in 𝐻+ . 1 Let 𝜌 = 𝑢 , and we easily get that the specific entropy (4.2.6) satisfies { −𝒫𝜃 /𝜌2 , 𝑝 = 1, ∂𝜂 𝑒𝜃 1 ∂𝜂 = = = (4.3.1) , ∂𝜌 ∂𝜃 𝜃 𝜃 1/𝒫𝜃 , 𝑝 > 1,

with 𝒫 = 𝜌𝑝 𝜃. We consider the transform 𝒜 : (𝜌, 𝜃) ∈ 𝒟𝜌,𝜃 = {(𝜌, 𝜃) : 𝜌 > 0, 𝜃 > 0} → (𝑢, 𝜂) ∈ 𝒜𝒟𝜌,𝜃 , where 𝑢 = 1/𝜌 and 𝜂 = 𝜂(1/𝜌, 𝜃). Owing to the Jacobian 𝑒𝜃 1 ∂(𝑢, 𝜂) = − 2 = − 2 < 0, ∂(𝜌, 𝜃) 𝜌 𝜃 𝜌 𝜃

(4.3.2)

there is a unique inverse function (𝑢, 𝜂) ∈ 𝒜𝒟𝜌,𝜃 . Thus the function 𝑒, 𝑝 can be also regarded as the smooth functions of (𝑢, 𝜂). We write 𝑒 = 𝑒(𝑢, 𝜂) :≡ 𝑒(𝑢, 𝜃(𝑢, 𝜂)) = 𝑒(1/𝜌, 𝜃), 𝒫 = 𝒫(𝑢, 𝜂) :≡ 𝒫(𝑢, 𝜃(𝑢, 𝜂)) = 𝒫(1/𝜌, 𝜃). Let ∂𝑒 ∂𝑒 𝑣2 +𝛿𝑧 +𝑒(𝑢, 𝜂)−𝑒(¯ 𝑢, 𝜂¯)− (¯ 𝑢, 𝜂¯)(𝑢− 𝑢 ¯)− (¯ 𝑢, 𝜂¯)(𝜂− 𝜂¯), (4.3.3) 2 ∂𝑢 ∂𝜂 ∫1 where 𝑒(𝑢, 𝜂) = 𝑒(𝑢, 𝜃) = 𝜃, 𝑢 ¯ = 0 𝑢0 𝑑𝑥 = 1 and 𝜃¯ > 0 is determined by ) ∫ 1( 1 2 ¯ 𝑣 + 𝑒(𝑢0 , 𝜃0 ) + 𝛿𝑧0 𝑑𝑥 𝑒(¯ 𝑢, 𝜃) = 𝑒(¯ 𝑢, 𝜂¯) = (4.3.4) 2 0 0 ℰ(𝑢, 𝑣, 𝜂, 𝑧) =

and

¯ = 𝜂(1, 𝜃). ¯ 𝜂¯ = 𝜂(¯ 𝑢, 𝜃)

(4.3.5)

106


Lemma 4.3.1. Under assumptions of Theorem 4.1.1, there holds ( ) 𝑣2 + 𝛿𝑧 + 𝐶1−1 ∣𝑢 − 1∣2 + ∣𝜂 − 𝜂¯∣2 2 ( ) 𝑣2 + 𝛿𝑧 + 𝐶1 ∣𝑢 − 1∣2 + ∣𝜂 − 𝜂¯∣2 . ≤ ℰ(𝑢, 𝑣, 𝜂, 𝑧) ≤ 2 Proof. The proof is similar to that of Lemma 3.2.3.

(4.3.6) □

Lemma 4.3.2. Under assumptions of Theorem 4.1.1, there are positive constants 𝐶1 > 0 and 𝛾1′ = 𝛾1′ (𝐶1 ) < 𝛾0 /2 such that for any fixed 𝛾 ∈ (0, 𝛾1′ ], there holds for any 𝑡 > 0, ( 𝑒𝛾𝑡 ∥𝑣(𝑡)∥2 + ∥𝑧(𝑡)∥𝐿1 [0,1] + ∥𝑧(𝑡)∥2 + ∥𝑢(𝑡) − 1∥2 ) + ∥𝜂(𝑡) − 𝜂¯∥2 + ∥𝜌𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2 ∫ 𝑡 ( 𝑒𝛾𝑡 ∥𝑧∥2𝐻 1 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥2 + 0

∫ +

1

0

) 𝑓 (𝑢, 𝜃, 𝑧) 𝑑𝑥 + ∥𝑢𝑥 ∥2 + ∥𝜌𝑥 ∥2 (𝑠) 𝑑𝑠 ≤ 𝐶1 .

(4.3.7)

Proof. By (4.1.1)–(4.1.6) and (4.2.6), we easily infer that the following equations hold: ( ) ( ) 𝑣2 𝛿𝑑 𝑘𝜃𝑥 𝜃+ + 𝛿𝑧 = 𝜎𝑣 + + 2 𝑧𝑥 , (4.3.8) 2 𝑢 𝑢 𝑡 𝑥 2

𝜂𝑡 = (𝑘𝜌𝜃𝑥 /𝜃)𝑥 + 𝑘𝜌 (𝜃𝑥 /𝜃) +

𝛿𝑓 𝜇𝜌𝑣𝑥2 + . 𝜃 𝜃

From (4.1.1)–(4.1.6), (4.3.8)–(4.3.9), we can infer [ ] 𝛿𝑓 ℰ𝑡 + 𝜃¯ 𝑘𝜌(𝜃𝑥 /𝜃)2 + 𝜇𝜌𝑣𝑥2 /𝜃 + 𝜃 ] ] [ ( 2 ) [ ¯ ¯ = 𝜇𝜌𝑣𝑣𝑥 − 𝑣(𝒫 − 𝒫) 𝑥 + (1 − 𝜃/𝜃)𝑘𝜌𝜃 𝑥 𝑥 + 𝛿 𝑑𝜌 𝑧𝑥 𝑥

(4.3.9)

(4.3.10)

¯ and with 𝒫¯ = 𝒫(¯ 𝑢, 𝜃) (

𝜌𝑥 𝑣 (𝜌𝑥 /𝜌)2 𝜇2 +𝜇 2 𝜌

) +𝜇 𝑡

𝒫𝜌 𝜌2𝑥 𝒫 𝜃 𝜌𝑥 𝜃𝑥 = −𝜇 − 𝜇(𝜌𝑣𝑣𝑥 )𝑥 + 𝜇𝜌𝑣𝑥2 . 𝜌 𝜌

(4.3.11)

For any 𝛽, 𝛾 > 0, let )] [ ( 𝐺(𝑡) = 𝑒𝛾𝑡 ℰ + 𝛽 𝜇2 (𝜌𝑥 /𝜌)2 /2 + 𝜇𝜌𝑥 𝑣/𝜌 .

(4.3.12)


107

Multiplying (4.3.10) and (4.3.11) by 𝑒𝛾𝑡 , 𝛽𝑒𝛾𝑡 respectively, and then adding the resulting equations, we conclude [¯ ] ) 𝜃( 𝑘𝜌𝜃𝑥2 /𝜃 + 𝜇𝜌𝑣𝑥2 + 𝛿𝑓 (𝑢, 𝜃, 𝑧) 𝐺′ (𝑡) + 𝑒𝛾𝑡 𝜃 [ ] 𝛾𝑡 + 𝛽𝑒 𝜇𝒫𝜌 𝜌2𝑥 /𝜌 + 𝜇𝒫𝜃 𝜌𝑥 𝜃𝑥 /𝜌 − 𝜇𝜌𝑣𝑥2 [ ( )] = 𝛾𝑒𝛾𝑡 ℰ + 𝛽 𝜇2 (𝜌𝑥 /𝜌)2 /2 + 𝜇𝜌𝑥 𝑣/𝜌 ( ] [ ) 𝜃¯ ¯ . (4.3.13) + 𝑒𝛾𝑡 𝜇(1 − 𝛽)𝜌𝑣𝑣𝑥 + 𝑘 1 − 𝜌𝜃𝑥 + 𝛿𝑑𝜌2 𝑧𝑥 − (𝒫 − 𝒫)𝑣 𝜃 𝑥 By the mean value theorem and Lemmas 4.2.1–4.2.3, we get ¯ ≤ 𝐶1 (∥𝑢(𝑡) − 𝑢 ¯ ∥𝜂(𝑡) − 𝜂¯∥ = ∥𝜂(𝑢, 𝜃) − 𝜂(¯ 𝑢, 𝜃)∥ ¯∥ + ∥𝜃(𝑡) − 𝜃∥) ¯ ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝜃(𝑡) − 𝜃∥). (4.3.14) Noting that ¯ = 𝑒(¯ 𝑢, 𝜃)

∫ 0

1

(

1 2 𝑣 + 𝜃0 + 𝛿𝑧0 2 0

)

∫ 𝑑𝑥 =

0

1

(

1 2 𝑣 + 𝜃 + 𝛿𝑧 2

) 𝑑𝑥,

(4.3.15)

we infer, by the mean value theorem, ¯ ¯ ≤ 𝐶1 (∥𝑢(𝑡) − 𝑢 ¯∥ + ∥𝑒(𝑢, 𝜃) − 𝑒(¯ 𝑢, 𝜃)∥) ∥𝜃(𝑡) − 𝜃∥ ¯∥ + ∥𝑒(𝑢, 𝜃) − 𝑒(𝑢, 𝜃)∥ + ∥𝑣(𝑡)∥ + ∥𝑧(𝑡)∥𝐿1[0,1] ) ≤ 𝐶1 (∥𝑢(𝑡) − 𝑢 ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝑒𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥ + ∥𝑧(𝑡)∥) ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥ + ∥𝑧(𝑡)∥)

(4.3.16)

which, along with (4.3.14) and the Poincaré inequality, gives ∥𝜂 − 𝜂¯∥ ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥ + ∥𝑧(𝑡)∥) (4.3.17) ∫1 where 𝑒(𝑢, 𝜃) = 0 𝑒(𝑢, 𝜃) 𝑑𝑥. The following estimate is crucial to our next argument, which implies that ∥𝑧∥ decays exponentially. Multiplying (4.1.5) by 𝑧, integrating the resulting equation in 𝐿2 (0, 1), and using Lemmas 4.2.1–4.2.3, we infer ∫ 1 ∫ 1 1 𝑑 𝑑 2 𝑧 𝑑𝑥 + 𝑓 (𝑢, 𝜃, 𝑧)𝑧 𝑑𝑥 = 0 ∥𝑧(𝑡)∥2 + 2 𝑥 2 𝑑𝑡 0 𝑢 0 which implies i.e.,

𝑑 ∥𝑧(𝑡)∥2 + 𝛾0 ∥𝑧𝑥 (𝑡)∥2 + 𝛾0 ∥𝑧(𝑡)∥2 ≤ 0 𝑑𝑡 𝑒𝛾0 𝑡 ∥𝑧(𝑡)∥2 +

∫

𝑡 0

𝑒𝛾0 𝑠 ∥𝑧𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 ,

with a positive constant 𝛾0 = 𝛾0 (𝐶1 ) > 0.

∀𝑡 > 0

(4.3.18)

108


Integrating (4.3.13) with respect to 𝑥 and 𝑡, using (4.3.17) and (4.3.18), we conclude ] ∫ 1[¯ ∫ 𝑡 ∫ 1 ) 𝜃( 𝛾𝑠 2 2 𝑘𝜌𝜃𝑥 /𝜃 + 𝜇𝜌𝑣𝑥 + 𝛿𝑓 (𝑢, 𝜃, 𝑧) (𝑥, 𝑠) 𝑑𝑥𝑑𝑠 𝐺(𝑡)𝑑𝑥 + 𝑒 𝜃 0 0 0 ∫ 𝑡∫ 1 [ ] 𝑒𝛾𝑠 𝜇𝒫𝜌 𝜌2𝑥 /𝜌 + 𝜇𝒫𝜃 𝜌𝑥 𝜃𝑥 /𝜌 − 𝜇𝜌𝑣𝑥2 (𝑥, 𝑠) 𝑑𝑥𝑑𝑠 +𝛽 0

0

∫ 𝑡∫ 1 [ ( )] 𝐺(0) 𝑑𝑥 + 𝛾 𝑒𝛾𝑠 ℰ + 𝛽 𝜇2 (𝜌𝑥 /𝜌)2 /2 + 𝜇𝜌𝑥 𝑣/𝜌 (𝑥, 𝑠) 𝑑𝑥𝑑𝑠 0 0 0 ∫ 𝑡 ( ) 𝛾𝑠 ∥𝑣∥2 + ∥𝑧∥𝐿1(0,1) + ∥𝑢 − 𝑢¯∥2 + ∥𝜂 − 𝜂¯∥2 + ∥𝜌𝑥 ∥2 (𝑠) 𝑑𝑠 𝑒 ≤ 𝐶1 + 𝐶1 𝛾 0 ∫ 𝑡 ( ) 𝑒𝛾𝑠 ∥𝑣𝑥 ∥2 + ∥𝑧∥𝐿1(0,1) + ∥𝑢𝑥∥2 + ∥𝜃𝑥 ∥2 + ∥𝑧∥2 + ∥𝜌𝑥 ∥2 (𝑠) 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝛾 0 ∫ 𝑡 ( ) 𝑒𝛾𝑠 ∥𝑣𝑥 ∥2 + ∥𝑧∥ + ∥𝑢𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑧∥2 + ∥𝜌𝑥 ∥2 (𝑠) 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝛾 ∫

=

1

0

≤ 𝐶1 (𝛾) + 𝐶1 𝛾

∫ 0

𝑡

( ) 𝑒𝛾𝑠 ∥𝑣𝑥 ∥2 + ∥𝑢𝑥 ∥2 + ∥𝜌𝑥 ∥2 + ∥𝜃𝑥 ∥2 (𝑠) 𝑑𝑠

(4.3.19)

if we choose 0 < 𝛾 < 𝛾0 /2. From Lemma 4.3.1 and Lemmas 4.2.1–4.2.3, we easily infer ∫ 0

1

{ 𝐺(𝑡) 𝑑𝑥 ≥ 𝑒𝛾𝑡 𝐶1−1 (∥𝑣(𝑡)∥2 + ∥𝑧(𝑡)∥𝐿1 (0,1) + ∥𝑢(𝑡) − 1∥2 + ∥𝜂(𝑡) − 𝜂¯∥2 ) ∫ +𝛽

1

( 2 ) } 𝜇 (𝜌𝑥 /𝜌)2 + 𝜇𝑣𝜌𝑥 /𝜌 𝑑𝑥

0 { −1 ≥ 𝑒 𝐶1 (∥𝑣(𝑡)∥2 + ∥𝑧(𝑡)∥𝐿1 (0,1) + ∥𝑢(𝑡) − 1∥2 + ∥𝜂(𝑡) − 𝜂¯∥2 ) } + 𝛽𝐶1−1 ∥𝜌𝑥 (𝑡)∥2 − 𝐶1 𝛽∥𝑣(𝑡)∥2 ( ≥ 𝐶1−1 𝑒𝛾𝑡 ∥𝑣(𝑡)∥2 + ∥𝑧(𝑡)∥𝐿1 (0,1) + ∥𝑢(𝑡) − 1∥2 + ∥𝜂(𝑡) ) − 𝜂¯∥2 + 𝛽∥𝜌𝑥 (𝑡)∥2 (4.3.20) 𝛾𝑡

if 𝛽 is small enough. The Young inequality yields 1 𝜌2 𝒫 𝜃 𝜌𝑥 𝜃𝑥 ≥ − 𝒫𝜌 𝑥 − 𝐶1 𝜃𝑥2 𝜌 2 𝜌

(4.3.21)

with 𝒫𝜌 = 𝑝𝜌𝑝−1 𝜃 > 0. Thus it follows from (4.3.16), (4.3.19)–(4.3.21) that for 𝛽 = 𝛾 1/2 , there are constants 𝐶1 > 0 and 0 < 𝛾1′ = 𝛾1′ (𝐶1 ) < 𝛾0 /2 such that


109

for 𝛾 ∈ (0, 𝛾1′ ], ( 𝑒𝛾𝑡 ∥𝑣(𝑡)∥2 + ∥𝑧(𝑡)∥𝐿1(0,1) + ∥𝑢(𝑡) − 1∥2 + ∥𝜂(𝑡) − 𝜂¯∥2 ) 1 1 + 𝛾 2 ∥𝜌𝑥 (𝑡)∥2 + 𝛾 2 ∥𝑢𝑥(𝑡)∥2 ∫ 𝑡 ∫ 1 1 1 −1 𝛾𝑠 2 2 + 𝑒 (∥𝑣𝑥 ∥ + ∥𝜃𝑥 ∥ + 𝐶1 𝑓 (𝑢, 𝜃, 𝑧) 𝑑𝑥 + 𝛾 2 ∥𝜌𝑥 ∥2 + 𝛾 2 ∥𝑢𝑥 ∥2 )(𝑠) 𝑑𝑠 0 0 ∫ 𝑡 𝑒𝛾𝑠 (∥𝑣𝑥 ∥2 + ∥𝑢𝑥 ∥2 + ∥𝜌𝑥 ∥2 + ∥𝜃𝑥 ∥2 )(𝑠) 𝑑𝑠 ≤ 𝐶1 (𝛾) + 𝐶1 𝛾 0

which, by choosing 0 < 𝛾 < 𝛾1′ small enough and using (4.3.18), yields (4.3.7).

□

Lemma 4.3.3. Under assumptions of Theorem 4.1.1, there are positive constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) ≤ 𝛾1′ such that for any fixed 𝛾 ∈ (0, 𝛾1 ], the following estimate holds, (4.3.22) 𝑒𝛾𝑡 (∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 + ∥𝑧𝑥 (𝑡)∥2 ) ∫ 𝑡 ( ) + 𝑒𝛾𝑠 ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 + ∥𝑧𝑥𝑥 ∥2 + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑧𝑡 ∥2 (𝑠) 𝑑𝑠 ≤ 𝐶1 . 0

Proof. Multiplying (4.1.3) by 𝑒𝛾𝑡 𝑣𝑥𝑥 , respectively, using Lemmas 4.2.1–4.2.3, we infer from ∫ 1 2 1 𝑑 𝛾𝑡 𝑣𝑥𝑥 (𝑒 ∥𝑣𝑥 ∥2 ) + 𝜇𝑒𝛾𝑡 𝑑𝑥 2 𝑑𝑡 𝑢 0 ∫ 1( 𝛾 𝜇𝑣𝑥 𝑢𝑥 ) 𝛾𝑡 𝒫𝑥 + 𝑑𝑥 + 𝑒𝛾𝑡 ∥𝑣𝑥 ∥2 =𝑒 2 𝑢 2 0 𝛾 𝛾𝑡 ≤ 𝑒 ∥𝑣𝑥 ∥2 + 𝐶1 𝑒𝛾𝑡 (∥𝑢𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥𝐿∞ ∥𝑢𝑥 ∥) 2 1 1 𝛾 𝛾𝑡 ≤ 𝑒 ∥𝑣𝑥 ∥2 + +𝐶1 𝑒𝛾𝑡 (∥𝑢𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥ 2 ∥𝑣𝑥𝑥 ∥ 2 ∥𝑢𝑥 ∥) 2 ≤ 𝐶1 𝛾𝑒𝛾𝑡∥𝑣𝑥𝑥 ∥2 + 𝐶1 𝑒𝛾𝑡 (∥𝑢𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥2 ), i.e., for 𝛾 > 0 small enough, 1 𝑑 𝛾𝑡 (𝑒 ∥𝑣𝑥 ∥2 ) + (2𝐶1 )−1 𝑒𝛾𝑡 ∥𝑣𝑥𝑥 ∥2 2 𝑑𝑡 ≤ 𝐶1 𝛾 −1 𝑒𝛾𝑡 (∥𝑢𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥2 ).

(4.3.23)

Multiplying (4.1.4) by 𝑒𝛾𝑡 𝜃𝑥𝑥 , using Lemmas 4.2.1–4.2.3, we infer from ∫ 1 2 𝑘𝜃𝑥𝑥 1 𝑑 𝛾𝑡 (𝑒 ∥𝜃𝑥 (𝑡)∥2 ) + 𝑒𝛾𝑡 𝑑𝑥 2 𝑑𝑡 𝑢 0 ) ( ∫ 1 𝛾 𝑓 (𝑢, 𝜃, 𝑧) 𝑑𝑥 ∥𝜃𝑥𝑥∥ ≤ 𝑒𝛾𝑡 ∥𝜃𝑥 ∥2 + 𝐶1 𝑒𝛾𝑡 ∥𝜃𝑥 ∥𝐿∞ ∥𝑢𝑥∥ + ∥𝑣𝑥 ∥ + ∥𝑣𝑥 ∥𝐿∞ + 2 0

110


( 1 3 𝛾 𝛾𝑡 2 𝛾𝑡 ≤ 𝑒 ∥𝜃𝑥 ∥ + 𝐶1 𝑒 ∥𝜃𝑥 ∥ 2 ∥𝜃𝑥𝑥∥ 2 ∥𝑢𝑥 ∥ 2 ) ∫ 1 1 1 + ∥𝑣𝑥 ∥∥𝜃𝑥𝑥 ∥ + ∥𝑣𝑥 ∥ 2 ∥𝑣𝑥𝑥 ∥ 2 ∥𝜃𝑥𝑥 ∥ + 𝑓 (𝑢, 𝜃, 𝑧) 𝑑𝑥∥𝜃𝑥𝑥 ∥ 0

≤

[ ] 𝛾 𝛾𝑡 𝛾 𝑒 ∥𝜃𝑥𝑥 ∥2 + 𝛾𝑒𝛾𝑡 ∥𝑣𝑥𝑥 ∥2 + 𝑒𝛾𝑡 ∥𝜃𝑥𝑥 ∥2 + 𝐶1 (𝛾)𝑒𝛾𝑡 ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥2 + ∥𝑧∥2 , 2 2

i.e., for 𝛾 > 0 small enough, 1 𝑑 𝛾𝑡 1 𝛾𝑡 (𝑒 ∥𝜃𝑥 ∥2 ) + 𝑒 ∥𝜃𝑥𝑥 ∥2 ≤ 𝛾𝑒𝛾𝑡 (∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 ) + 𝐶1 𝑒𝛾𝑡 ∥𝑧∥2 . (4.3.24) 2 𝑑𝑡 2𝐶1 Similarly, multiplying (4.1.5) by 𝑒𝛾𝑡 𝑧𝑥𝑥 , using Lemmas 4.2.1–4.2.3, we conclude for 𝛾 > 0 small enough, 1 𝑑 𝛾𝑡 (𝑒 ∥𝑧𝑥 ∥2 ) + (2𝐶1 )−1 𝑒𝛾𝑡 ∥𝑧𝑥𝑥∥2 ≤ 𝐶1 𝛾 −1 𝑒𝛾𝑡 (∥𝑢𝑥 ∥2 + ∥𝑧∥2). 2 𝑑𝑡

(4.3.25)

Adding (4.3.23), (4.3.24) and (4.3.25), we arrive at ] 1 𝛾𝑡 1 𝑑 [ 𝛾𝑡 𝑒 (∥𝑣𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑧𝑥 ∥2 ) + 𝑒 (∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 + ∥𝑧𝑥𝑥 ∥2 ) 2 𝑑𝑡 2𝐶1 ≤ 𝛾𝑒𝛾𝑡(∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 ) + 𝐶1 𝛾 −1 𝑒𝛾𝑡 (∥𝑢𝑥 ∥2 + ∥𝑣𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑧∥2) (4.3.26) for 𝛾 > 0 small enough. This, along with Lemma 4.3.2, gives (4.3.22) by using □ (4.3.18) and (4.1.2)–(4.1.5) for 𝛾 ∈ (0, 𝛾1′ ] small enough. Lemma 4.3.4. Under assumptions of Theorem 4.1.1, there exists a positive constant 𝛾2′ = 𝛾2′ (𝐶2 ) ≤ 𝛾1 such that for any fixed 𝛾 ∈ (0, 𝛾2′ ], the following estimate holds, ( ) 𝑒𝛾𝑡 ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑧𝑡 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝑧𝑥𝑥 (𝑡)∥2 ∫ 𝑡 + 𝑒𝛾𝜏 (∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝑧𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0. (4.3.27) 0

Proof. Multiplying (4.2.45) by 𝑒𝛾𝑡 and integrating the resulting equation over [0, 𝑡], using Young’s inequality, we easily conclude 𝑒𝛾𝑡 ∥𝑣𝑡 (𝑡)∥2 +

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑣𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 +

∫ 0

𝑡

( ) 𝑒𝛾𝜏 ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑣𝑥 ∥2𝐻 1 (𝜏 ) 𝑑𝜏

which, combined with Lemmas 4.3.2–4.3.3 and Lemmas 4.2.1–4.2.3, implies that there exists a constant 𝛾2′ = 𝛾2′ (𝐶2 ) ≤ 𝛾1 such that for any fixed 𝛾 ∈ (0, 𝛾2′ ] 𝑒𝛾𝑡 (∥𝑣𝑡 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 ) +

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑣𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 ,

∀ 𝑡 > 0.

(4.3.28)


111

In the same manner, multiplying (4.2.50) and (4.2.53) by 𝑒𝛾𝑡 , respectively, integrating the resulting equations over [0, 𝑡] and using Lemmas 4.3.1–4.3.3 and (4.2.48)–(4.2.49), we infer that 𝑒𝛾𝑡 (∥𝜃𝑡 (𝑡)∥2 +∥𝑧𝑡 (𝑡)∥2 +∥𝜃𝑥𝑥 (𝑡)∥2 +∥𝑧𝑥𝑥 (𝑡)∥2 )+

∫ 0

𝑡

𝑒𝛾𝜏 (∥𝜃𝑡𝑥 ∥2 +∥𝑧𝑡𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2

which, together with (4.3.28), yields (4.3.27). The proof is complete.

□

Lemma 4.3.5. Under assumptions of Theorem 4.1.1, there exists a positive constant 𝛾2 = 𝛾2 (𝐶2 ) ≤ 𝛾2′ such that for any fixed 𝛾 ∈ (0, 𝛾2 ], the following estimate holds, 𝛾𝑡

2

𝑒 ∥𝑢𝑥𝑥(𝑡)∥ +

∫

𝑡

𝑒𝛾𝜏 (∥𝑢𝑥𝑥 ∥2 + ∥𝑣𝑥𝑥𝑥∥2 + ∥𝜃𝑥𝑥𝑥∥2 + ∥𝑧𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 ,

0

∀𝑡 > 0. (4.3.29)

Proof. Multiplying (4.2.57) by 𝑒𝛾𝑡 and choosing 𝛾 so small that 𝛾 ≤ 𝛾2 = min[𝛾2′ , 1/(4𝐶1 )], and using Lemmas 4.3.1–4.3.4 and (4.2.61)–(4.2.63), we conclude that 𝛾𝑡

2

∫

𝑡

𝑒 ∥𝑢𝑥𝑥(𝑡)∥ + 𝑒𝛾𝜏 ∥𝑢𝑥𝑥(𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝜃𝑥𝑥 ∥2 + ∥𝑢𝑥 ∥2 + ∥𝑣𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 + 𝐶1 0

≤ 𝐶2 .

(4.3.30)

Differentiating (4.1.3), (4.1.4) and (4.1.5) with respect to 𝑥, respectively, using Lemmas 4.3.1–4.3.4, and (4.3.30), we easily deduce ∫ 0

𝑡

𝑒𝛾𝜏 (∥𝑣𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥 ∥2 + ∥𝑧𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2

which, along with (4.3.30), gives (4.3.29). The proof is complete.

□

Proof of Theorem 4.1.1. By Lemmas 4.2.1–4.2.3, Lemmas 4.3.1–4.3.5 and Sobolev’s embedding theorem, we complete the proof of Theorem 4.1.1. □

112


4.4 Proof of Theorem 4.1.2 In this section, we shall give the proof of Theorem 4.1.2 by a series of a priori estimates. 4 Lemma 4.4.1. Under assumptions of Theorem 4.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ , we have

∥𝑣𝑡𝑥 (𝑥, 0)∥ + ∥𝜃𝑡𝑥 (𝑥, 0)∥ + ∥𝑧𝑡𝑥 (𝑥, 0)∥ ≤ 𝐶3 , ∥𝑣𝑡𝑡 (𝑥, 0)∥ + ∥𝜃𝑡𝑡 (𝑥, 0)∥ + ∥𝑧𝑡𝑡 (𝑥, 0)∥ + ∥𝑣𝑡𝑥𝑥 (𝑥, 0)∥ + ∥𝜃𝑡𝑥𝑥 (𝑥, 0)∥ + ∥𝑧𝑡𝑥𝑥(𝑥, 0)∥ ≤ 𝐶4 , ∫ 𝑡 ∫ 𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, ∀𝑡 > 0, 0 0 ∫ 𝑡 2 2 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥ 𝑑𝜏 ∥𝜃𝑡𝑡 (𝑡)∥ + 0 ∫ 𝑡 −1 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀 0 ∫ 𝑡 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, ∀𝑡 > 0, + 𝐶1 𝜀 0 ∫ 𝑡 ∫ 𝑡 2 2 ∥𝑧𝑡𝑡𝑥 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 + 𝐶4 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, ∥𝑧𝑡𝑡 (𝑡)∥ + 0

0

(4.4.1) (4.4.2) (4.4.3)

(4.4.4) ∀𝑡 > 0 (4.4.5)

for 𝜀 small enough. Proof. By virtue of (4.2.58)–(4.2.60) and (4.2.40), we conclude (4.4.1). Differentiating (4.1.3), (4.1.4) and (4.1.5) with respect to 𝑥 twice, respectively, and using Lemmas 4.2.1–4.2.3 to get ∥𝑣𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 3 + ∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝑢𝑥𝑥𝑥(𝑡)∥ + ∥𝑢𝑥 (𝑡)∥𝐿∞ ∥𝑣𝑥𝑥𝑥(𝑡)∥ + ∥𝑣𝑥𝑥 (𝑡)∥𝐿∞ ∥𝑢𝑥𝑥(𝑡)∥) ≤ 𝐶2 (∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 3 ), ∥𝜃𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥(𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝑓𝑥𝑥(𝑡)∥)

(4.4.6)

≤ 𝐶2 (𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝑧𝑥 (𝑡)∥𝐻 1 ), ∥𝑧𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝑧𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑓𝑥𝑥 (𝑡)∥)

(4.4.7)

≤ 𝐶2 (∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥(𝑡)∥𝐻 2 + ∥𝑧𝑥 (𝑡)∥𝐻 3 )

(4.4.8)

or ∥𝑣𝑥𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥),

(4.4.9)

∥𝜃𝑥𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶2 (∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑢𝑥(𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝑧𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥),

(4.4.10)

∥𝑧𝑥𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶2 (∥𝑧𝑥 (𝑡)∥𝐻 2 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑧𝑡𝑥𝑥 (𝑡)∥).

(4.4.11)


113

Differentiating (4.1.3) with respect to 𝑡, using Theorem 4.1.1, (4.2.37)–(4.2.39), (4.4.6)–(4.4.8) and (4.2.58)–(4.2.60), we have ( ∥𝑣𝑡𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥ ) (4.4.12) + ∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝑣𝑡𝑥𝑥 (𝑡)∥ ( ) ≤ 𝐶2 ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 3 + ∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑧𝑥 (𝑡)∥ + ∥𝑓 (𝑡)∥ . (4.4.13) Analogously, we get ( ∥𝜃𝑡𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 2 ) + ∥𝑓𝑡 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥ (4.4.14) ) ( ≤ 𝐶2 ∥𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝑧𝑥 (𝑡)∥𝐻 1 + ∥𝑓 (𝑡)∥ , (4.4.15)

(

∥𝑧𝑡𝑡 (𝑡)∥ ≤ 𝐶2 ∥𝑧𝑡𝑥𝑥 (𝑡)∥ + ∥𝑧𝑥𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥𝐻 1 ) + ∥𝑢𝑥(𝑡)∥ + ∥𝑓𝑡 (𝑡)∥ (4.4.16) ) ( ≤ 𝐶2 ∥𝑧𝑥 (𝑡)∥𝐻 3 + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑓 (𝑡)∥ . (4.4.17) Thus estimate (4.4.2) follows from (4.4.6)–(4.4.8), (4.4.13), (4.4.15) and (4.4.17). Differentiating (4.1.3) with respect to 𝑡 twice, multiplying the resulting equation by 𝑣𝑡𝑡 in 𝐿2 (0, 1), performing an integration by parts, using Theorem 4.1.1, (4.4.1)–(4.4.2), we obtain ) ∫ 1( 𝜃 𝑣𝑥 1 𝑑 2 − 𝑝 +𝜇 𝑣𝑡𝑡𝑥 𝑑𝑥 (4.4.18) ∥𝑣𝑡𝑡 (𝑡)∥ = − 2 𝑑𝑡 𝑢 𝑢 𝑡𝑡 0 ∫ 1 2 ) ( 𝑣𝑡𝑡𝑥 𝑑𝑥 + 𝐶2 ∥𝜃𝑡𝑡 ∥ + ∥𝑣𝑡𝑥 𝑣𝑥 ∥ + ∥𝑣𝑥3 ∥ + ∥𝜃𝑡 𝑣𝑥 ∥ + ∥𝑣𝑡𝑥 ∥ + ∥𝑣𝑥2 ∥ ∥𝑣𝑡𝑡𝑥 ∥ ≤ −𝜇 𝑢 0 ( ) −1 ≤ −𝐶1 ∥𝑣𝑡𝑡𝑥 (𝑡)∥2 + 𝐶2 ∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 . Thus, by Theorem 4.1.1, 2

∥𝑣𝑡𝑡 (𝑡)∥ +

∫ 0

𝑡

2

∥𝑣𝑡𝑡𝑥 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 + 𝐶4

∫ 0

𝑡

∥𝜃𝑡𝑡 (𝜏 )∥2 𝑑𝜏

which, along with (4.4.14) and Lemmas 4.2.1–4.2.3, gives estimate (4.4.3).

114


Similarly, differentiating (4.1.4) with respect to 𝑡 twice, multiplying the resulting equation by 𝜃𝑡𝑡 in 𝐿2 (0, 1) and integrating by parts, we arrive at ( ) ) ∫ 1( 𝑣𝑥 𝜃𝑥 𝜃 𝜃𝑡𝑡𝑥 𝑑𝑥 + 𝜅 − 𝑝 +𝜇 𝑣𝑡𝑡𝑥 𝜃𝑡𝑡 𝑑𝑥 𝑢 𝑡𝑡 𝑢 𝑢 0 0 ) ∫ 1( ∫ 1 𝑣𝑥 𝜃 𝑓𝑡𝑡 𝜃𝑡𝑡 𝑑𝑥 + 𝑣𝑥 𝜃𝑡𝑡 𝑑𝑥 + − 𝑝 +𝜇 𝑢 𝑢 𝑡𝑡 0 0 ) ∫ 1( 𝑣𝑥 𝜃 𝑣𝑡𝑥 𝜃𝑡𝑡 𝑑𝑥 + − 𝑝 +𝜇 𝑢 𝑢 𝑡 0

1 𝑑 ∥𝜃𝑡𝑡 (𝑡)∥2 = − 2 𝑑𝑡

∫

1

= 𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 .

(4.4.19)

By virtue of Lemmas 4.2.1–4.2.3 and (4.4.1)–(4.4.2), and using the embedding theorem, we deduce that for any 𝜀 ∈ (0, 1), 𝐴1 ≤ −𝐶1−1 ∥𝜃𝑡𝑡𝑥 (𝑡)∥2 + 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝜃𝑡𝑥 (𝑡)∥ + ∥𝜃𝑥 (𝑡)∥𝐿∞ ∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥2𝐿∞ ∥𝜃𝑥 (𝑡)∥)∥𝜃𝑡𝑡𝑥 (𝑡)∥ ≤ −(2𝐶1 )−1 ∥𝜃𝑡𝑡𝑥 (𝑡)∥2 + 𝐶2 (∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2𝐻 1 ),

(4.4.20)

𝐴2 ≤ 𝜀∥𝑣𝑡𝑡𝑥 (𝑡)∥2 + 𝐶2 𝜀−1 ∥𝜃𝑡𝑡 (𝑡)∥2 , ∫ 1 [(∣𝑣𝑥 ∣ + ∣𝜃𝑡 ∣ + ∣𝑧𝑡 ∣)2 + ∣𝑣𝑡𝑥 ∣ + ∣𝜃𝑡𝑡 ∣ + ∣𝑧𝑡𝑡 ∣]∣𝜃𝑡𝑡 ∣ 𝑑𝑥 𝐴3 ≤ 0 [ ≤ 𝐶1 ∥𝜃𝑡𝑡 (𝑡)∥𝐿∞ (∥𝑣𝑥 (𝑡)∥𝐿∞ + ∥𝜃𝑡 (𝑡)∥𝐿∞ + ∥𝑧𝑡 (𝑡)∥𝐿∞ )(∥𝑣𝑥 (𝑡)∥ ] + ∥𝜃𝑡 (𝑡)∥ + ∥𝑧𝑡 (𝑡)∥) + 𝐶1 (∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝜃𝑡𝑡 (𝑡)∥ + ∥𝑧𝑡𝑡 (𝑡)∥)∥𝜃𝑡𝑡 (𝑡)∥

(4.4.21)

≤ 𝐶1 (∥𝜃𝑡𝑡 ∥ + ∥𝜃𝑡𝑡𝑥 ∥)(∥𝑣𝑥 ∥𝐻 1 + ∥𝜃𝑡 ∥𝐻 1 + ∥𝑧𝑡 ∥𝐻 1 ) + 𝐶1 (∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝜃𝑡𝑡 (𝑡)∥ + ∥𝑧𝑡𝑡 (𝑡)∥)∥𝜃𝑡𝑡 (𝑡)∥ ≤ 𝜀(∥𝜃𝑡𝑡𝑥 (𝑡)∥2 + ∥𝑧𝑡𝑥𝑥 (𝑡)∥2 ) + 𝐶2 𝜀−1 (∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2𝐻 1 + ∥𝑧𝑡 (𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 ), (4.4.22) { 𝐴4 ≤ 𝐶2 ∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝜃𝑡𝑡 (𝑡)∥ ∥𝑣𝑥 (𝑡)∥𝐿∞ (∥𝜃𝑡 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥) } + ∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝜃𝑡𝑡 (𝑡)∥ + ∥𝑣𝑡𝑡𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐻 1 ≤ 𝐶2 ∥𝜃𝑡𝑡 (𝑡)∥(∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑡 (𝑡)∥ + ∥𝜃𝑡𝑡 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝑣𝑡𝑡𝑥 (𝑡)∥) ≤ 𝜀∥𝑣𝑡𝑡𝑥 (𝑡)∥2 + 𝐶2 𝜀−1 (∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 ), (4.4.23) ∫ 1 𝐴5 ≤ 𝐶2 (∣𝜃𝑡 ∣ + ∣𝑣𝑥 ∣ + ∣𝑣𝑡𝑥 ∣ + ∣𝑣𝑥2 ∣)∣𝜃𝑡𝑡 ∣∣𝑣𝑡𝑥 ∣ 𝑑𝑥 0

≤ 𝐶2 ∥𝑣𝑡𝑥 (𝑡)∥1/2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥1/2 (∥𝑣𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥)∥𝜃𝑡𝑡 (𝑡)∥


115

which implies ∫ 0

𝑡

(∫ 𝐴5 𝑑𝜏 ≤ 𝐶2 sup ∥𝜃𝑡𝑡 (𝜏 )∥ 0≤𝜏 ≤𝑡

{∫

0

𝑡

∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏

)1/4 (∫ } 1/2

𝑡

𝑡 0

∥𝑣𝑡𝑥 (𝜏 )∥2 𝑑𝜏

× (∥𝑣𝑥 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑣𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 0 ( ) ∫ 𝑡 2 2 ≤ 𝜀 sup ∥𝜃𝑡𝑡 (𝜏 )∥ + ∥𝑣𝑡𝑥𝑥 (𝜏 )∥ 𝑑𝜏 + 𝐶2 𝜀−3 . 0≤𝜏 ≤𝑡

0

)1/4

(4.4.24)

Thus we infer from (4.4.19)–(4.4.24) that for 𝜀 ∈ (0, 1) small enough, ∫ 𝑡 ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 0 ∫ 𝑡 ∥𝜃𝑡𝑡 (𝜏 )∥2 𝑑𝜏 (4.4.25) ≤ 𝐶4 𝜀−3 + 𝐶2 𝜀−1 0 { } ∫ 𝑡 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 . + 𝐶1 𝜀 sup ∥𝜃𝑡𝑡 (𝜏 )∥2 + 0≤𝜏 ≤𝑡

0

Thus taking supremum in 𝑡 on the left-hand side of (4.4.25), picking 𝜀 ∈ (0, 1) small enough, and using (4.4.14), we can derive estimate (4.4.4). Differentiating (4.1.5) with respect to 𝑡 twice, multiplying the resulting equation by 𝑧𝑡𝑡 in 𝐿2 (0, 1) and integrating by parts, we obtain ) ∫ 1( ∫ 1 𝑑 1 𝑑 2 ∥𝑧𝑡𝑡 ∥ = − 𝑧𝑥 𝑧𝑡𝑡𝑥 𝑑𝑥 − 𝑓𝑡𝑡 𝑧𝑡𝑡 𝑑𝑥 2 𝑑𝑡 𝑢2 0 0 𝑡𝑡 ∫ 1 2 [( 𝑧 ∥𝑧𝑡𝑥 (𝑡)∥ 𝑑 𝑡𝑡𝑥 𝑑𝑥 + 𝐶 ≤− 2 𝑢2 0 ) ] + ∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝑧𝑥 (𝑡)∥ ∥𝑧𝑡𝑡𝑥 (𝑡)∥ + ∥𝑓𝑡𝑡 (𝑡)∥∥𝑧𝑡𝑡 (𝑡)∥ ≤ −(2𝐶1 )−1 ∥𝑧𝑡𝑡𝑥 (𝑡)∥2 ( ) + 𝐶2 ∥𝑧𝑡𝑡 (𝑡)∥2 + ∥𝑓𝑡𝑡 (𝑡)∥2 + ∥𝑧𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑧𝑥 (𝑡)∥2 ( ≤ −(2𝐶1 )−1 ∥𝑧𝑡𝑡𝑥 (𝑡)∥2 + 𝐶2 ∥𝑧𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 ) (4.4.26) + ∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2𝐻 1 + ∥𝑧𝑡 (𝑡)∥2𝐻 1 + ∥𝑧𝑥 (𝑡)∥2 . Thus, by Lemmas 4.2.1–4.2.3 and (4.4.2), ∫ 𝑡 ∫ 𝑡 ∥𝑧𝑡𝑡 (𝑡)∥2 + ∥𝑧𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 (∥𝑧𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 )(𝜏 ) 𝑑𝜏 0

0

(4.4.27)

which, together with (4.4.14) and (4.4.16), gives (4.4.5). The proof is complete. □

116


4 Lemma 4.4.2. Under assumptions of Theorem 4.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ , the following estimates hold for any 𝑡 > 0, ∫ 𝑡 ∫ 𝑡 2 2 2 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 + 𝐶1 𝜀 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, (4.4.28) ∥𝑣𝑡𝑥 (𝑡)∥ + 0 0 ∫ 𝑡 ∫ 𝑡 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀2 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, (4.4.29) ∥𝜃𝑡𝑥 (𝑡)∥2 + 0 0 ∫ 𝑡 ∥𝑧𝑡𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 (4.4.30) ∥𝑧𝑡𝑥 (𝑡)∥2 + 0

for 𝜀 ∈ (0, 1) small enough. Proof. Differentiating (4.1.3) with respect to 𝑥 and 𝑡, multiplying the resulting equation by 𝑣𝑡𝑥 in 𝐿2 (0, 1), and integrating by parts, we have 1 𝑑 ∥𝑣𝑡𝑥 (𝑡)∥2 = 𝐵0 (𝑡) + 𝐵1 (𝑡) 2 𝑑𝑡 with

( ) 𝑥=1 𝜃 𝑣𝑥 𝐵0 (𝑡) = − 𝑝 + 𝜇 𝑣𝑡𝑥 , 𝑢 𝑢 𝑡𝑥 𝑥=0

∫ 𝐵1 (𝑡) = −

0

1

(4.4.31)

( ) 𝑣𝑥 𝜃 𝑣𝑡𝑥𝑥 𝑑𝑥. − 𝑝 +𝜇 𝑢 𝑢 𝑡𝑥

We employ Lemmas 4.2.1–4.2.3 and Lemma 4.4.1, the interpolation inequality and Poincaré’s inequality to get [ 𝐵0 (𝑡) ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥𝐿∞ + ∥𝜃𝑡 (𝑡)∥𝐿∞ )(∥𝑢𝑥 (𝑡)∥𝐿∞ + ∥𝜃𝑥 (𝑡)∥𝐿∞ ) + ∥𝑣𝑥𝑥 (𝑡)∥𝐿∞ + ∥𝜃𝑡𝑥 (𝑡)∥𝐿∞ + ∥𝑣𝑡𝑥𝑥 (𝑡)∥𝐿∞ + ∥𝑢𝑥 (𝑡)∥𝐿∞ ∥𝑣𝑡𝑥 (𝑡)∥𝐿∞ ] + ∥𝑣𝑥 (𝑡)∥𝐿∞ ∥𝑣𝑥𝑥 (𝑡)∥𝐿∞ + ∥𝑣𝑥2 (𝑡)∥𝐿∞ ∥𝑣𝑡𝑥 (𝑡)∥𝐿∞ ≤ 𝐶2 (𝐵01 + 𝐵02 )∥𝑣𝑡𝑥 (𝑡)∥1/2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥1/2 where

(4.4.32)

𝐵01 = ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑡 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥

and 𝐵02 = ∥𝜃𝑡𝑥 (𝑡)∥1/2 ∥𝜃𝑡𝑥𝑥 (𝑡)∥1/2 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥1/2 ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥1/2 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥1/2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥1/2 . Applying Young’s inequality several times, we have that for 𝜀 ∈ (0, 1), 𝐶2 𝐵01 ∥𝑣𝑡𝑥 (𝑡)∥1/2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥1/2 ( ) 𝜀2 ≤ ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + 𝐶2 𝜀−1 ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2𝐻 2 + ∥𝜃𝑡 (𝑡)∥2𝐻 1 2

(4.4.33)


117

and 𝐶2 𝐵02 ∥𝑣𝑡𝑥 (𝑡)∥1/2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥1/2 ≤

( 𝜀2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + 𝜀2 ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥2 ) 2 ) + 𝐶2 𝜀−6 (∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 .

(4.4.34)

Thus we infer from (4.4.32)–(4.4.34) and Lemmas 4.2.1–4.2.3, Lemma 4.4.1 that 𝐵0 ≤ 𝜀2 (∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 ) + 𝐶2 𝜀−6 (∥𝑣𝑥 (𝑡)∥2𝐻 2 + ∥𝜃𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 )

(4.4.35)

which, together with Theorem 4.1.1, further leads to ∫ 0

𝑡

𝐵0 𝑑𝜏 ≤ 𝜀2

∫ 0

𝑡

(∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 + 𝐶2 𝜀−6 ,

∀ 𝑡 > 0.

(4.4.36) Analogously, by Lemma 4.4.1 and Theorem 4.1.1 and the embedding theorem, we get that for any 𝜀 ∈ (0, 1), ∫ 𝐵1 ≤ − 𝜇

1

{ 2 𝑣𝑡𝑥𝑥 𝑑𝑥 + 𝐶1 (∥𝑣𝑥 ∥ + ∥𝜃𝑡 ∥)(∥𝑢𝑥 ∥𝐿∞ + ∥𝜃𝑥 ∥𝐿∞ ) 𝑢

(4.4.37) } + ∥𝑣𝑥𝑥 ∥ + ∥𝜃𝑡𝑥 ∥ + ∥𝑢𝑥∥𝐿∞ ∥𝑣𝑡𝑥 ∥ + ∥𝑣𝑥 ∥𝐿∞ ∥𝑣𝑥𝑥 ∥ + ∥𝑣𝑥 ∥2𝐿∞ ∥𝑢𝑥 ∥ ∥𝑣𝑡𝑥𝑥 ∥ 0

≤ − (2𝐶1 )−1 ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + 𝐶2 (∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2 ) which, combined with (4.4.31), (4.4.36) and Theorem 4.1.1, gives that for 𝜀 ∈ (0, 1) small enough, 2

∥𝑣𝑡𝑥 (𝑡)∥ +

∫ 0

𝑡

2

∥𝑣𝑡𝑥𝑥 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶3 𝜀

−6

+ 𝐶1 𝜀

2

∫ 0

𝑡

(∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏.

(4.4.38) On the other hand, differentiating (4.1.3) with respect to 𝑥 and 𝑡, and using Theorem 4.1.1 and Lemma 4.4.1, we derive ∥𝑣𝑡𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶1 ∥𝑣𝑡𝑡𝑥 (𝑡)∥ + 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥(𝑡)∥𝐻 1 + ∥𝜃𝑡 (𝑡)∥𝐻 2 ). (4.4.39) Thus inserting (4.4.39) into (4.4.38) implies (4.4.28). Similarly, we derive from (4.1.4) that 1 𝑑 ∥𝜃𝑡𝑥 (𝑡)∥2 = 𝐷1 (𝑡) + 𝐷2 (𝑡) + 𝐷3 (𝑡) 2 𝑑𝑡

(4.4.40)

118


with

∫ 𝐷1 (𝑡) = −

1

(𝜅

)

𝜃𝑡𝑥𝑥 𝑑𝑥, ) ∫ 1 (( 𝜃 𝑣𝑥 ) − 𝑝 +𝜇 𝑣𝑥 𝐷2 (𝑡) = 𝜃𝑡𝑥 𝑑𝑥, 𝑢 𝑢 0 𝑡𝑥 ∫ 1 𝑓𝑡𝑥 𝜃𝑡𝑥 𝑑𝑥. 𝐷3 (𝑡) = 0

𝑢

𝜃𝑥

𝑡𝑥

0

By virtue of Lemmas 4.2.1–4.2.3, Lemma 4.4.1 and (4.4.28), and using the embedding theorem, we deduce that for any 𝜀 ∈ (0, 1), ( ) 𝐷1 ≤ − (2𝐶1 )−1 ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 + 𝐶2 ∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑡 (𝑡)∥2𝐻 1 , (4.4.41) 𝐷2 ≤ 𝜀2 ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 ( ) + 𝐶2 𝜀−2 ∥𝑣𝑥 (𝑡)∥2𝐻 2 + ∥𝜃𝑡 (𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2𝐻 1 , 2

(4.4.42)

2

𝐷3 ≤ 𝐶2 (∥𝑓𝑡𝑥 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥ ) which, along with (4.4.40)–(4.4.42), implies (4.4.29). Differentiating (4.1.5) with respect to 𝑥 and 𝑡, multiplying the resulting equation by 𝑧𝑡𝑥 , and integrating by parts, we arrive at ) ∫ 1( ∫ 1 1 𝑑 𝑑 2 ∥𝑧𝑡𝑥 (𝑡)∥ = − 𝑧𝑥 𝑧𝑡𝑥𝑥 𝑑𝑥 − 𝑓𝑡𝑥 𝑧𝑡𝑥 𝑑𝑥 (4.4.43) 2 𝑑𝑡 𝑢2 0 0 𝑡𝑥 ∫ 1 2 ( ) 𝑧𝑡𝑥𝑥 1 ∥𝑧 ∥𝑧𝑡𝑥𝑥(𝑡)∥ + ∥𝑧𝑡𝑥 (𝑡)∥∥𝑓𝑡𝑥 (𝑡)∥ 𝑑𝑥 + 𝐶 (𝑡)∥ + ∥𝑧 (𝑡)∥ + ∥𝑣 (𝑡)∥ ≤ −𝑑 1 𝑡𝑥 𝑥𝑥 𝑥 𝐻 2 0 𝑢 ( ) ≤ −(2𝐶1 )−1 ∥𝑧𝑡𝑥𝑥 (𝑡)∥2 + 𝐶2 ∥𝑣𝑥 (𝑡)∥2𝐻 1 + ∥𝑧𝑥𝑥 (𝑡)∥2 + ∥𝑓𝑡𝑥 (𝑡)∥2𝐻 1 + ∥𝑧𝑡𝑥 (𝑡)∥2 which, together with Lemmas 4.2.1–4.2.3 and Lemma 4.4.1, gives (4.4.30). The proof is complete. □ 4 Lemma 4.4.3. Under assumptions of Theorem 4.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ , we have for any 𝑡 > 0, ∫ 𝑡( ∥𝑣𝑡𝑡𝑥 ∥2 ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑧𝑡𝑡 (𝑡)∥2 + ∥𝑧𝑡𝑥 (𝑡)∥2 + 0 ) + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑡𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , (4.4.44)

∥𝑢𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑣𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 ∫ 𝑡( 2 2 ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑧𝑡𝑡 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥ + ∥𝑧𝑡𝑥𝑥(𝑡)∥ + 0 ) 2 2 + ∥𝜃𝑡𝑥𝑥 ∥𝐻 1 + ∥𝑧𝑡𝑥𝑥 ∥𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∫ 𝑡( ) ∥𝑢𝑥𝑥𝑥∥2𝐻 1 + ∥𝑣𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥𝑥∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . 0

(4.4.45) (4.4.46)


119

Proof. Adding up (4.4.28), (4.4.29) and (4.4.30), choosing 𝜀 small enough, we conclude ∫ 𝑡 2 2 2 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ∥𝑣𝑡𝑥 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥ + ∥𝑧𝑡𝑥 (𝑡)∥ + 0 ∫ 𝑡 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏. (4.4.47) ≤ 𝐶4 + 𝐶2 𝜀2 0

Now multiplying (4.4.3), (4.4.4) and (4.4.5) by 𝜀, 𝜀2 and 𝜀, respectively; then adding the resultant to (4.4.47), and picking 𝜀 small enough, we obtain (4.4.44). Differentiating (4.2.56) with respect to 𝑥, we get 𝜇

∂ ( 𝑢𝑥𝑥𝑥 ) − 𝒫𝑢 𝑢𝑥𝑥𝑥 = 𝐸1 (𝑥, 𝑡) ∂𝑡 𝑢

where 𝐸1 (𝑥, 𝑡) = 𝑣𝑡𝑥𝑥 + 𝐸𝑥 (𝑥, 𝑡) + 𝒫𝑢𝑥 𝑢𝑥𝑥 + 𝜇

(4.4.48)

(𝑢 𝑢 ) 𝑥𝑥 𝑥 . 𝑢2 𝑡

Obviously, we can infer from Lemmas 4.2.1–4.2.3 and Lemmas 4.4.1–4.4.2 that ∥𝐸1 (𝑡)∥ ≤ 𝐶2 (∥𝑢𝑥 (𝑡)∥𝐻 1 + ∥𝑣𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥) leading to

∫

𝑡

0


𝑢𝑥𝑥𝑥 𝑢

∥𝐸1 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀𝑡 > 0.

(4.4.49)

(4.4.50)

in 𝐿2 (0, 1), we obtain

𝑢 𝑑 𝑥𝑥𝑥 2 𝑢𝑥𝑥𝑥 2 + 𝐶1−1 ≤ 𝐶1 ∥𝐸1 (𝑡)∥2 𝑑𝑡 𝑢 𝑢

(4.4.51)

which, combined with (4.4.50), gives 2

∥𝑢𝑥𝑥𝑥 (𝑡)∥ +

∫ 0

𝑡

∥𝑢𝑥𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀𝑡 > 0.

(4.4.52)

By (4.2.61)–(4.2.63), (4.4.9)–(4.4.11), (4.4.44), (4.4.52), Lemmas 4.2.1–4.2.3 and Lemmas 4.4.1–4.4.2, and using the embedding theorem, we obtain for any 𝑡 > 0, ∥𝑣𝑥𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥𝑥 (𝑡)∥2 + ∥𝑧𝑥𝑥𝑥(𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2𝐿∞ + ∥𝜃𝑥𝑥 (𝑡)∥2𝐿∞ + ∥𝑧𝑥𝑥(𝑡)∥2𝐿∞ ∫ 𝑡( ∥𝑣𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥∥2𝐻 1 + ∥𝑣𝑥𝑥 ∥2𝑊 1,∞ + 0 ) + ∥𝜃𝑥𝑥 ∥2𝑊 1,∞ + ∥𝑧𝑥𝑥∥2𝑊 1,∞ (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (4.4.53)

120


Differentiating (4.1.3)–(4.1.5) with respect to 𝑡, using (4.4.44), (4.4.52)–(4.4.53), Lemmas 4.4.1–4.4.2 and Lemmas 4.2.1–4.2.2, we conclude ∥𝑣𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶1 ∥𝑣𝑡𝑡 (𝑡)∥ + 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥ + ∥𝜃𝑡𝑥 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥) ≤ 𝐶4 ,

(4.4.54)

∥𝜃𝑡𝑥𝑥 (𝑡)∥ ≤ 𝐶1 ∥𝜃𝑡𝑡 (𝑡)∥ + 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥 (𝑡)∥ + ∥𝜃𝑡 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥𝐻 2 + ∥𝑓𝑡 (𝑡)∥ + ∥𝑣𝑡𝑥 (𝑡)∥) ≤ 𝐶4 , (4.4.55) ∥𝑧𝑡𝑥𝑥(𝑡)∥ ≤ 𝐶1 ∥𝑧𝑡𝑡 (𝑡)∥ + 𝐶2 (∥𝑧𝑥𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐻 1 + ∥𝜃𝑥 (𝑡)∥𝐻 1 + ∥𝑢𝑥(𝑡)∥ + ∥𝑓𝑡 (𝑡)∥) ≤ 𝐶4

(4.4.56)

which, combined with (4.4.9)–(4.4.11), implies ∫ 𝑡 2 2 2 ∥𝑣𝑥𝑥𝑥𝑥(𝑡)∥ + ∥𝜃𝑥𝑥𝑥𝑥(𝑡)∥ + ∥𝑧𝑥𝑥𝑥𝑥(𝑡)∥ + (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 0

+ ∥𝑣𝑥𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥𝑥∥2 + ∥𝑧𝑥𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 ,

∀ 𝑡 > 0.

(4.4.57)

Therefore it follows from (4.4.53), (4.4.57) and the embedding theorem that ∫ 𝑡 ∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐿∞ + ∥𝜃𝑥𝑥𝑥(𝑡)∥2𝐿∞ + ∥𝑧𝑥𝑥𝑥(𝑡)∥2𝐿∞ + (∥𝑣𝑥𝑥𝑥 ∥2𝐿∞ +

∥𝜃𝑥𝑥𝑥 ∥2𝐿∞

+

∥𝑧𝑥𝑥𝑥∥2𝐿∞ )(𝜏 ) 𝑑𝜏

0

≤ 𝐶4 ,

∀ 𝑡 > 0.

(4.4.58)

Now differentiating (4.4.48) with respect to 𝑥, we find ∂ ( 𝑢𝑥𝑥𝑥𝑥 ) − 𝒫𝑢 𝑢𝑥𝑥𝑥𝑥 = 𝐸2 (𝑥, 𝑡) 𝜇 ∂𝑡 𝑢 where

𝐸2 (𝑥, 𝑡) = 𝐸1𝑥 (𝑥, 𝑡) + 𝒫𝑢𝑥 𝑢𝑥𝑥𝑥 + 𝜇

(𝑢

(4.4.59)

𝑥𝑥𝑥 𝑢𝑥 𝑢2

) 𝑡

.

Using the embedding theorem, Lemmas 4.4.1–4.4.2, (4.4.53) and (4.4.54)–(4.4.58), we derive that ∥𝐸𝑥𝑥 (𝑡)∥ ≤ 𝐶4 (∥𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 3 ), ( 𝑢 𝑢 ) ) ( 𝑥𝑥 𝑥 (𝑡) ∥𝐸1𝑥 (𝑡)∥ ≤ 𝐶1 ∥𝐸𝑥𝑥 (𝑡)∥ + ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥ + ∥(𝑝𝑢𝑥 𝑢𝑥𝑥 )𝑥 (𝑡)∥ + 𝑢2 𝑡𝑥 ≤ 𝐶1 ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥ + 𝐶4 (∥𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 3 ), whence ∥𝐸2 (𝑡)∥ ≤ 𝐶1 ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥ + 𝐶4 (∥𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝑣𝑥 (𝑡)∥𝐻 3 ). We infer from (4.4.12)–(4.4.17) that ∫ 𝑡 (∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑧𝑡𝑡 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 , 0

∀𝑡 > 0

(4.4.60)

(4.4.61)


121

which, together with (4.4.39) and (4.4.44), gives ∫ 𝑡 ∥𝑣𝑡𝑥𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 , ∀ 𝑡 > 0.

(4.4.62)

0

Thus it follows from (4.4.53), (4.4.57), (4.4.60), (4.4.62), Lemmas 4.2.1–4.2.2 and Lemmas 4.4.1–4.4.2 that ∫ 𝑡 ∥𝐸2 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 , ∀ 𝑡 > 0. (4.4.63) 0

Multiplying (4.4.59) by 𝑢𝑥𝑥𝑥𝑥 in 𝐿2 (0, 1), we get 𝑢 𝑢 𝑑 𝑥𝑥𝑥𝑥 2 𝑢𝑥𝑥𝑥𝑥 2 + 𝐶1−1 ≤ 𝐶1 ∥𝐸2 ∥2 , 𝑑𝑡 𝑢 𝑢

(4.4.64)

whence by (4.4.63), ∥𝑢𝑥𝑥𝑥𝑥(𝑡)∥2 +

∫

𝑡 0

∥𝑢𝑥𝑥𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀ 𝑡 > 0.

(4.4.65)

Differentiating (4.1.4) and (4.1.5) with respect to 𝑥 and 𝑡, respectively, and using Lemmas 4.2.1–4.2.3 and Lemmas 4.4.1–4.4.2, we derive

Thus,

∥𝜃𝑡𝑥𝑥𝑥 (𝑡)∥ ≤ 𝐶1 ∥𝜃𝑡𝑡𝑥 (𝑡)∥ + 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 2 + ∥𝜃𝑥 (𝑡)∥𝐻 3 + ∥𝑓𝑡𝑥 (𝑡)∥),

(4.4.66)

∥𝑧𝑡𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶1 ∥𝑧𝑡𝑡𝑥 (𝑡)∥ + 𝐶2 (∥𝑣𝑥 (𝑡)∥𝐻 3 + ∥𝑢𝑥 (𝑡)∥𝐻 3 + ∥𝑧𝑥 (𝑡)∥𝐻 2 + ∥𝑓𝑡𝑥 (𝑡)∥).

(4.4.67)

∫ 0

𝑡

(∥𝜃𝑡𝑥𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 ,

∀ 𝑡 > 0.

(4.4.68)

Differentiating (4.1.3) with respect to 𝑥 three times, using Lemmas 4.4.1– 4.4.2 and Lemmas 4.2.1–4.2.3, applying Poincaré’s inequality, we deduce ∥𝑣𝑥𝑥𝑥𝑥𝑥(𝑡)∥ ≤ 𝐶1 ∥𝑣𝑡𝑥𝑥𝑥 (𝑡)∥ + 𝐶2 (∥𝑢𝑥 (𝑡)∥𝐻 3 + ∥𝑣𝑥 (𝑡)∥𝐻 3 + ∥𝜃𝑥 (𝑡)∥𝐻 3 ). (4.4.69) Thus we derive from (4.4.52)–(4.4.53), (4.4.57)–(4.4.58), (4.4.62) and (4.4.65) that ∫ 𝑡 ∥𝑣𝑥𝑥𝑥𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 , ∀ 𝑡 > 0. (4.4.70) 0

Similarly, we can deduce from (4.1.4)–(4.1.5), (4.4.65), (4.4.68) and (4.4.57) that ∫ 𝑡 (∥𝜃𝑥𝑥𝑥𝑥𝑥∥2 + ∥𝑧𝑥𝑥𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀ 𝑡 > 0. (4.4.71) 0

Hence, (4.4.45)–(4.4.46) follow from (4.4.52)–(4.4.71).The proof is complete.

□

122


4 Lemma 4.4.4. Under assumptions of Theorem 4.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ , (1) (1) there exists a positive constant 𝛾4 = 𝛾4 (𝐶4 ) ≤ 𝛾2 (𝐶2 ) such that for any fixed (1) 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0 and 𝜀 ∈ (0, 1) small enough: ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, (4.4.72) 0 0 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 𝜀−3 + 𝐶2 𝜀−1 𝑒𝛾𝜏 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏 𝑒𝛾𝑡 ∥𝜃𝑡𝑡 (𝑡)∥2 + 0 0 ∫ 𝑡 + 𝜀 𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 )(𝜏 )𝑑𝜏, (4.4.73) 0 ∫ 𝑡 ∫ 𝑡 𝛾𝑡 2 𝛾𝜏 2 𝑒 ∥𝑧𝑡𝑡𝑥 (𝜏 )∥ 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝑒𝛾𝜏 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏. 𝑒 ∥𝑧𝑡𝑡 (𝑡)∥ + 0

0

(4.4.74)

Proof. Multiplying (4.4.18) by 𝑒𝛾𝑡 and using (4.4.14) and Theorem 4.1.1, we have ∫ 𝑡 ∫ 𝑡 1 𝛾𝑡 −1 2 𝛾𝜏 2 𝑒 ∥𝑣𝑡𝑡 (𝑡)∥ ≤ 𝐶4 − (𝐶1 − 𝐶1 𝛾) 𝑒 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥ 𝑑𝜏 + 𝐶2 𝑒𝛾𝜏 ∥𝜃𝑡𝑡 (𝜏 )∥2 𝑑𝜏 2 0 0 ∫ 𝑡 ∫ 𝑡 −1 𝛾𝜏 2 𝑒 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥ 𝑑𝜏 + 𝐶4 𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 − (𝐶1 − 𝐶1 𝛾) 0

0

] [ which gives (4.4.72) if we take 𝛾 > 0 so small that 0 < 𝛾 ≤ min 4𝐶1 2 , 𝛾2 (𝐶2 ) . 1 Similarly, multiplying (4.4.19) by 𝑒𝛾𝑡 and using (4.4.14), (4.4.16) and Theorem 4.1.1, we have 1 𝛾𝑡 √ 𝑒 ∥ 𝑒𝜃 𝜃𝑡𝑡 ∥2 2 ∫ ∫ 𝑡 𝛾 𝑡 𝛾𝜏 √ ≤ 𝐶4 + 𝑒 ∥ 𝑒𝜃 𝜃𝑡𝑡 ∥2 (𝜏 )𝑑𝜏 + 𝑒𝛾𝜏 (𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + 𝐴6 + 𝐴7 ) 2 0 0 ∫ 𝑡 ≤ 𝐶4 𝜀−3 − (𝐶1−1 − 2𝜀) 𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑥 ∥2 (𝜏 )𝑑𝜏 0 ∫ 𝑡 ∫ 𝑡 −1 𝛾𝜏 𝑒 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏 + 𝜀 𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 ∥2 (𝜏 )𝑑𝜏 + 𝐶2 𝜀 0

+ 𝐶2 𝑒𝛾𝑡/2 sup ∥𝜃𝑡𝑡 (𝜏 )∥ [∫ ×

0

0≤𝜏 ≤𝑡

𝑡

(∫ 0

0

𝑡

𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 ∥2 (𝜏 )𝑑𝜏

𝑒𝛾𝜏 (∥𝑣𝑡𝑥 ∥2 + ∥𝑣𝑥 ∥2 + ∥𝜃𝑡 ∥2 )(𝜏 )𝑑𝜏

≤ 𝐶4 𝜀−3 − (𝐶1−1 − 2𝜀) 2

∫

+ ∥𝑧𝑡𝑥𝑥∥ )(𝜏 )𝑑𝜏 + 𝜀

𝑡

0

∫

0

)1/4

]1/2 (∫

𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 + 𝐶2 𝜀−1

𝑡

∫

𝑡

0 𝑡

0

𝑒𝛾𝜏 ∥𝑣𝑡𝑥 ∥2 (𝜏 )𝑑𝜏

)1/4

𝑒𝛾𝜏 (∥𝜃𝑡𝑥𝑥 ∥2

𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 )(𝜏 )𝑑𝜏 + 𝜀𝑒𝛾𝑡 sup ∥𝜃𝑡𝑡 (𝜏 )∥2 0≤𝜏 ≤𝑡


123

which, by taking supremum on the right-hand side and choosing 𝜀 ∈ (0, 1) small enough, implies (4.4.73). Multiplying (4.4.26) by 𝑒𝛾𝑡 and using (4.4.14), (4.4.16) and Theorem 4.1.1, we have ∫ 𝑡 ∫ 𝑡 1 𝛾𝑡 𝑒 ∥𝑧𝑡𝑡 (𝑡)∥2 + 𝐶1−1 𝑒𝛾𝜏 ∥𝑧𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝑒𝛾𝜏 (∥𝑧𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 )(𝜏 )𝑑𝜏 2 0 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑧𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏 ≤ 𝐶4 + 𝐶2 0

which implies (4.4.74). The proof is complete.

□

4 , Lemma 4.4.5. Under assumptions of Theorem 4.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ (2) (1) (2) there exists a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0 and 𝜀 ∈ (0, 1) small enough: ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀2 𝑒𝛾𝜏 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏, 𝑒𝛾𝑡 ∥𝑣𝑡𝑥 (𝑡)∥2 + 0


𝑒𝛾𝑡 ∥𝑧𝑡𝑥 (𝑡)∥2 +

∫

0

𝑡

0

∫

𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝜀2

∫ 0

𝑡

(4.4.75)

𝑒𝛾𝜏 (∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏, (4.4.76)

𝑡

𝑒𝛾𝜏 ∥𝑧𝑡𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 , (4.4.77) ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏 𝑒𝛾𝑡 (∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑧𝑡𝑥 (𝑡)∥2 ) + 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 )(𝜏 )𝑑𝜏. (4.4.78) ≤ 𝐶4 + 𝐶2 𝜀2 0

0

Proof. Multiplying (4.4.31) by 𝑒𝛾𝑡 and using (4.4.35), (4.4.37) and Theorem 4.1.1, we infer that for any 𝜀 ∈ (0, 1) small enough, ∫ 𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 𝑒𝛾𝑡 ∥𝑣𝑡𝑥 (𝑡)∥2 ≤ 𝐶4 − [(2𝐶1 )−1 − 𝜀2 ] 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 + 𝜀2 0

which, together with (4.4.39), gives (4.4.75) if we take 𝛾 > 0 and 𝜀 ∈ (0, 1) so (1) (2) small enough that 0 < 𝜀 < min[1, 1/(8𝐶1 )] and 0 < 𝛾 ≤ min[𝛾4 , 1/(8𝐶12 )] ≡ 𝛾4 . In the same manner, we easily derive (4.4.76) from (4.4.40)–(4.4.42). And we deduce (4.4.77) from (4.4.43) and Theorem 4.1.1. Adding (4.4.75), (4.4.76) to (4.4.77) and picking 𝜀 ∈ (0, 1) small enough give (4.4.78). The proof is complete. □

124


4 Lemma 4.4.6. Under assumptions of Theorem 4.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑧0 ) ∈ 𝐻+ , (2) there is a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0:

𝑒𝛾𝑡 (∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑧𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑧𝑡𝑥(𝑡)∥2 ) (4.4.79) ∫ 𝑡 + 𝑒𝛾𝜏 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝑧𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥 ∥2 )(𝜏 )𝑑𝜏 ≤ 𝐶4 , 0 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑢𝑥𝑥𝑥(𝜏 )∥2𝐻 1 𝑑𝜏 ≤ 𝐶4 , (4.4.80) 𝑒𝛾𝑡 ∥𝑢𝑥𝑥𝑥(𝑡)∥2𝐻 1 + 0

𝑒𝛾𝑡 (∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 + ∥𝑧𝑡𝑥𝑥 (𝑡)∥2 ) ∫ 𝑡 ( + 𝑒𝛾𝜏 ∥𝑣𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 0 ) + ∥𝜃𝑡𝑥𝑥∥2𝐻 1 + ∥𝑧𝑡𝑥𝑥∥2𝐻 1 + ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑧𝑡𝑡 ∥2 (𝜏 )𝑑𝜏 ≤ 𝐶4 . (4.4.81) Proof. Multiplying (4.4.72), (4.4.73) and (4.4.74) by 𝜀, 𝜀3/2 and 𝜀, respectively; then adding the resultant to (4.4.78), and picking 𝜀 small enough, we can obtain the desired estimate (4.4.79). Multiplying (4.4.51) by 𝑒𝛾𝑡 , using (4.4.49), (4.4.79) and Theorem 4.1.1, and [ (2) ] picking 𝛾 > 0 so small that 0 < 𝛾 ≤ 𝛾4 ≡ min 2𝐶1 1 , 𝛾4 , we conclude that for any 𝑡 > 0, ∫ 𝑡 2 𝑢 𝑥𝑥𝑥 2 −1 (𝑡) + (2𝐶1 ) (𝜏 ) 𝑑𝜏 𝑒𝛾𝜏 𝑢 𝑢 0 ∫ 𝑡 𝑒𝛾𝜏 ∥𝐸1 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ≤ 𝐶3 + 𝐶1

𝑒

𝛾𝑡 𝑢𝑥𝑥𝑥

0

whence 𝑒𝛾𝑡 ∥𝑢𝑥𝑥𝑥(𝑡)∥2 +

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑢𝑥𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀ 𝑡 > 0.

(4.4.82)

Similarly to (4.4.53), (4.4.57)–(4.4.58), (4.4.61)–(4.4.62), (4.4.66)–(4.4.67), using (4.4.79) and (4.4.82) and Theorem 4.1.1, we deduce that for any fixed 𝛾 ∈ (0, 𝛾4 ], ( 𝑒𝛾𝑡 ∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥(𝑡)∥2𝐻 1 ) + ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 + ∥𝑧𝑡𝑥𝑥 (𝑡)∥2 ∫ 𝑡 ( + 𝑒𝛾𝜏 ∥𝑣𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝑧𝑥𝑥𝑥 ∥2𝐻 1 0 ) + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 (4.4.83)


and ∫ 𝑡 0

125

( ) 𝑒𝛾𝜏 ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑧𝑡𝑡 ∥2 + ∥𝑣𝑡𝑥𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥𝑥 ∥2 + ∥𝑧𝑡𝑥𝑥𝑥∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(4.4.84) Similarly to (4.4.82), multiplying (4.4.64) by 𝑒𝛾𝑡 , using (4.4.60), (4.4.79), (4.4.82)– (4.4.84) and Theorem 4.1.1, we conclude that for any fixed 𝛾 ∈ (0, 𝛾4 ], ∫ 𝑡 𝑢 𝑢 𝑥𝑥𝑥𝑥 2 𝑥𝑥𝑥𝑥 2 (𝑡) + (2𝐶1 )−1 (𝜏 ) 𝑑𝜏 𝑒𝛾𝑡 𝑒𝛾𝜏 𝑢 𝑢 0 ∫ 𝑡 ≤ 𝐶3 + 𝐶1 𝑒𝛾𝜏 ∥𝐸2 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 0

whence 𝑒𝛾𝑡 ∥𝑢𝑥𝑥𝑥𝑥(𝑡)∥2 +

∫ 0

𝑡

𝑒𝛾𝜏 ∥𝑢𝑥𝑥𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 ,

∀ 𝑡 > 0.

Similarly to (4.4.68)–(4.4.70), we deduce that for any fixed 𝛾 ∈ (0, 𝛾4 ], ∫ 𝑡 ( ) 𝑒𝛾𝜏 ∥𝑣𝑥𝑥𝑥𝑥𝑥∥2 + ∥𝜃𝑥𝑥𝑥𝑥𝑥∥2 + ∥𝑧𝑥𝑥𝑥𝑥𝑥∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , ∀ 𝑡 > 0. 0

(4.4.85)

(4.4.86)

Finally, we derive the desired estimates (4.4.80)–(4.4.81) by (4.4.79) and (4.4.82)– (4.4.86). The proof is complete. □ Proof of Theorem 4.1.2. By Lemmas 4.4.1–4.4.6, Theorem 4.1.1 and Sobolev’s embedding theorem, we complete the proof of Theorem 4.1.2. □ Remark 4.4.1. It is easy to find that some proofs of some lemmas in this chapter are applicable to the corresponding lemmas in Chapter 3.

4.5 Bibliographic Comments In this section, we shall recall some related results. When the material is a polytropic ideal linear viscous gas with constant viscosity and heat conductivity, e.g., 𝑒 = 𝐶𝑣 𝜃,

𝒫=𝑅

𝜃 𝑢

(4.5.1)

with suitable positive constants 𝐶𝑣 and 𝑅, the global existence and asymptotic behavior of smooth (generalized) solutions to the system (4.1.2)–(4.1.4) with 𝛿 = 0 have been investigated by many authors (see, e.g., Kazhikhov [33], Kazhikhov and Shelukhin [36], Nagasawa [45, 46], Okada and Kawashima [47], Qin [59, 52, 55], see also Chapter 3) on the initial boundary value problems and the Cauchy problem. Zheng and Qin [83] established the existence of maximal attractors for the problem (4.1.2)–(4.1.4), (4.1.7) and (4.1.2)–(4.1.8) with 𝛿 = 0.

126


The global existence and asymptotic behavior for the system (4.1.2)–(4.1.4) with 𝛿 = 0 (without chemical reactions) have been established only for some constitutive equations and special forms of functions 𝜇, 𝜅 and 𝒫 before the results of Chapter 3 were published (see [62]). Kazhikhov and Shelukhin [36] proved the global existence and uniqueness of a solution to the initial boundary value problem of a system consisting of a viscous heat-conductive ideal gas bounded by two infinite parallel plates with thermally insulated boundaries, but does not allow heat to be generated by chemical reactions. However, Bebernes and Bressan [2] investigated the case that permits generation by a chemical reaction, so the system contains the chemical species equation (4.1.5). The system (4.1.2)–(4.1.6) was essentially proposed by Kassoy and Poland in [30] for a polytropic ideal gas (𝑝 = 1), and was studied in Guo and Zhu [21] for global existence, regularity and asymptotic behavior of solutions, in particular, the well-posedness and large time behavior for discontinuous initial data was investigated in Chen, Hoff and Trivisa [5]. In this direction, we would like to mention the works of Qin [57], Qin, Ma, Cavalcanti and Andrade [65], Qin and Mu˜ noz Rivera [66], Qin, Wu and Liu [73], Qin [57, 65, 66, 73] who produced some works on global well-posedness in 𝐻 𝑖 (𝑖 = 1, 2, 4) and existence of global attractors for heat-conducting real gas of the compressible Navier-Stokes equations with a constant viscosity coefficient. As mentioned above, the global existence and the asymptotic behavior in (𝐻 2 [0, 1])4 of the generalized (global) solutions have never been investigated for equations (4.1.2)–(4.1.4) of the 𝑝th power viscous reactive gas with boundary conditions (3.1.7), nor have the existence of classical solutions. Therefore, we continue the work by Lewicka and Mucha [38] (𝑝 ≥ 1) and study the global existence and exponential stability of the solutions in 𝐻 𝑖 (𝑖 = 2, 4) in this chapter and have improved the works in [38].

Chapter 5

On a 1D Viscous Reactive and Radiative Gas with First-order Arrhenius Kinetics 5.1 Introduction In this chapter, we establish the global existence and exponential stability of solutions in 𝐻 𝑖 (𝑖 = 1, 2, 4) for a Stefan-Boltzmann model of a viscous, reactive and radiative gas with first-order Arrhenius kinetics in a bounded interval. In so doing we describe the classical stellar evolution [11] of a finite mass of a heat-conducting viscous reactive fluid in local equilibrium with thermal radiation: pressure, internal energy and thermal conductivity have Stefan-Boltzmann radiative contributions. In order to mimic chemical exchanges inside the fluid, we may consider a simple reacting process with a first-order kinetics, commonly used in combustion theory [12]. The results of this chapter are chosen from [63]. The system under consideration in Lagrangian coordinates reads as follows: 𝑢𝑡 = 𝑣𝑥 ,

(5.1.1)

𝑣𝑡 = 𝜎𝑥 , 𝑒𝑡 − 𝜎𝑣𝑥 + 𝑄𝑥 − 𝜆𝜙(𝜃, 𝑍) = 0, ( ) 𝑑 𝑍 + 𝜙(𝜃, 𝑍) = 0 𝑍𝑡 − 𝑥 𝑢2 𝑥

(5.1.2) (5.1.3) (5.1.4)

where 𝜇 is a constant viscous coefficient, 𝜆 ≥ 0 and 𝑑 ≥ 0 are two “chemical” constants, and 𝑥 ∈ [0, 1] denotes the mass variable, 𝑢(𝑥, 𝑡) the specific volume, 𝑣(𝑥, 𝑡) the velocity, 𝜃(𝑥, 𝑡) and 𝑍(𝑥, 𝑡) denote the temperature and the fraction of reactant, respectively. We consider system (5.1.1)–(5.1.4) subject to the boundary conditions 𝑣(0, 𝑡) = 𝑣(1, 𝑡) = 0, 𝑄(0, 𝑡) = 𝑄(1, 𝑡) = 0, 𝑍𝑥 (0, 𝑡) = 𝑍𝑥 (1, 𝑡) = 0


(5.1.5)

127

128

Chapter 5. On a 1D Viscous Reactive and Radiative Gas

and the initial condition (𝑢, 𝑣, 𝜃, 𝑍)(𝑥, 0) = (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 )(𝑥).

(5.1.6)

For the constitutive relations, we consider (see, e.g., [12]) the Stefan-Boltzmann model, i.e., the pressure 𝑝(𝑢, 𝜃), internal energy 𝑒(𝑢, 𝜃), the stress 𝜎(𝑢, 𝑣𝑥 , 𝜃) and the thermo-radiative flux 𝑄(𝑢, 𝜃) take the following forms respectively, 𝑝(𝑢, 𝜃) = 𝑅

𝜃 𝑎 𝑣𝑥 + 𝜃4 , 𝑒(𝑢, 𝜃) = 𝐶𝑣 𝜃 + 𝑎𝑢𝜃4 , 𝜎(𝑢, 𝑣𝑥 , 𝜃) = −𝑝(𝑢, 𝜃) + 𝜇 , 𝑢 3 𝑢 (5.1.7)

𝑄(𝑢, 𝜃) = −

𝑘(𝑢, 𝜃)𝜃𝑥 , 𝑢

𝑘(𝑢, 𝜃) = 𝜅1 + 𝜅2 𝑢𝜃𝑞

(5.1.8)

where 𝑅, 𝑎, 𝐶𝑣 , 𝜇, 𝜅1 , 𝜅2 and 𝑞 are positive constants (the values 𝑘2 = 0, 𝑎 = 0 correspond to the ideal gas). Finally, the function 𝜙 in the equations (5.1.3)–(5.1.4) mimics the simplest first-order Arrhenius kinetics (see, e.g., [12]): 𝜙(𝜃, 𝑍) = 𝐴𝑍𝜃𝛽 𝑒−𝐸/(𝐵𝜃)

(5.1.9)

where 𝐴, 𝛽, 𝐵 and 𝐸 are positive constants. For a one-dimensional homogeneous real gas, 𝑒, 𝜎, 𝜂 and 𝑄 are given by the constitutive relations (see, e.g., [7, 8, 27, 32]): 𝑒 = 𝑒(𝑢, 𝜃), 𝜎 = 𝜎(𝑢, 𝜃, 𝑣𝑥 ), 𝜂 = 𝜂(𝑢, 𝜃), 𝑄 = 𝑄(𝑢, 𝜃, 𝜃𝑥 )

(5.1.10)

which, in order to be consistent with the Clausius-Duhem inequality (see below (5.1.17)), must satisfy 𝜎(𝑢, 𝜃, 0) = Ψ𝑢 (𝑢, 𝜃),

𝜂(𝑢, 𝜃) = −Ψ𝜃 (𝑢, 𝜃),

(𝜎(𝑢, 𝜃, 𝑤) − 𝜎(𝑢, 𝜃, 0))𝑤 ≥ 0,

𝑄(𝑢, 𝜃, 𝑔)𝑔 ≤ 0

(5.1.11) (5.1.12)

where Ψ = 𝑒 − 𝜃𝜂 is the Helmholtz free energy function, and 𝑒, 𝜎, 𝜂 are internal energy, stress and specific entropy, respectively. For the case of an ideal gas without radiative effect (i.e., 𝜙(𝜃, 𝑍) = 0, 𝑍 = 0), 𝑒 = 𝐶𝑣 𝜃, 𝜎 = −𝑅

𝑣𝑥 𝜃 𝜃 + 𝜇 , 𝑄 = −𝜆 𝑢 𝑢 𝑢

(5.1.13)

with suitable positive constants 𝐶𝑣 , 𝑅, 𝜇 and 𝜆, the global existence and asymptotic behavior of smooth (generalized) solutions to the system (5.1.1)–(5.1.3) have been investigated by many authors (see, e.g., [33, 36, 45, 46, 47, 49, 50, 59, 73]) on the initial boundary value problems and the Cauchy problem. However, under very high temperature and densities, equation (5.1.13) becomes inadequate. Thus, a more realistic model would be a linearly viscous gas (or Newtonian fluid), 𝜎(𝑢, 𝜃, 𝑣𝑥 ) = −𝑝(𝑢, 𝜃) +

𝜇(𝑢, 𝜃) 𝑣𝑥 𝑢

(5.1.14)

5.1. Introduction

129

satisfying Fourier’s law of heat flux, 𝑄(𝑢, 𝜃, 𝜃𝑥 ) = −

𝑘(𝑢, 𝜃) 𝜃𝑥 𝑢

(5.1.15)

where the internal energy 𝑒 and the pressure 𝑝 are coupled by the standard thermodynamical relations 𝑒𝑢 (𝑢, 𝜃) = −𝑝(𝑢, 𝜃) + 𝜃𝑝𝜃 (𝑢, 𝜃) to comply with the Clausius-Duhem inequality, ( ) 𝑄 ≥ 0. 𝜂𝑡 + 𝜃 𝑥

(5.1.16)

(5.1.17)

We define three spaces as follows: { ( )4 1 𝐻+ = (𝑢, 𝑣, 𝜃, 𝑍) ∈ 𝐻 1 [0, 1] : 𝑢(𝑥) > 0, 𝜃(𝑥) > 0, 0 ≤ 𝑍(𝑥) ≤ 1, 𝑥 ∈ [0, 1], } 𝑣(0) = 𝑣(1) = 0 , { ( )4 𝑖 𝐻+ = (𝑢, 𝑣, 𝜃, 𝑍) ∈ 𝐻 𝑖 [0, 1] : 𝑢(𝑥) > 0, 𝜃(𝑥) > 0, 0 ≤ 𝑍(𝑥) ≤ 1, 𝑥 ∈ [0, 1], } 𝑣(0) = 𝑣(1) = 0, 𝜃′ (0) = 𝜃′ (1) = 0, 𝑍 ′ (0) = 𝑍 ′ (1) = 0 , 𝑖 = 2, 4. Let 𝐸 = 𝐸1 ∪ 𝐸2 be a set in the (𝛽, 𝑞)-plane in ℝ2 , where } { 12 < 𝑞, 0 < 𝛽 ≤ 8 , 𝐸1 = (𝛽, 𝑞) ∈ ℝ2 : 5 } { 5 < 𝑞 ≤ 3, 𝛽 < 2𝑞 + 6, 𝛽 > 8 𝐸2 = (𝛽, 𝑞) ∈ ℝ2 : 2 } ∪{ (𝛽, 𝑞) ∈ ℝ2 : 3 ≤ 𝑞, 𝛽 < 𝑞 + 9, 𝛽 > 8 . The notation in this chapter is standard. We put ∥ ⋅ ∥ = ∥ ⋅ ∥𝐿2 [0,1] and denote by 𝐶 𝑘 (𝐽, 𝐵), 𝑘 ∈ ℕ0 , the space of 𝑘-times continuously differentiable functions from 𝐽 ⊆ ℝ into a Banach space 𝐵, and likewise by 𝐿𝑝 (𝐽, 𝐵), (1 ≤ 𝑝 ≤ +∞) the corresponding Lebesgue spaces. Subscripts 𝑡 and 𝑥 denote the (partial) derivatives with respect to 𝑡 and 𝑥, respectively. We use 𝐶𝑖 (𝑖 = 1, 2, 4) to denote the generic positive constant depending on the 𝐻 𝑖 [0, 1] norm of initial datum (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ), min 𝑢0 (𝑥), min 𝜃0 (𝑥) and min 𝑍0 (𝑥), but independent of variable 𝑡. Constant 𝑥∈[0,1]

𝑥∈[0,1]

𝑥∈[0,1]

𝐶 stands for the absolute positive constant independent of initial data. Without danger of confusion, we shall use the same symbol to denote the state functions as well as their values along a dynamic process, e.g., 𝑝(𝑢, 𝜃), 𝑝(𝑢(𝑥, 𝑡), 𝜃(𝑥, 𝑡)) and 𝑝(𝑥, 𝑡).

130


Our main results read as follows. 1 Theorem 5.1.1. Let (𝛽, 𝑞) ∈ 𝐸. Assume that the initial data (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ and the compatibility conditions are satisfied, and there exists a constant 𝜀0 > 0 ∫1 such that 𝑢¯0 = 0 𝑢0 𝑑𝑥 ≤ 𝜀0 , then problem (5.1.1)–(5.1.6) admits a unique global 1 solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑍(𝑡)) ∈ 𝐻+ verifying

0 < 𝐶1−1 ≤ 𝑢(𝑥, 𝑡) ≤ 𝐶1 ,

0 < 𝐶1−1 ≤ 𝜃(𝑥, 𝑡) ≤ 𝐶1 on [0, 1] × [0, +∞), (5.1.18)

and

∫ 𝑡( ¯ 2 1 + ∥𝑍(𝑡)∥2 1 + ∥𝑢(𝑡) − 𝑢¯∥2𝐻 1 + ∥𝑣(𝑡)∥2𝐻 1 + ∥𝜃(𝑡) − 𝜃∥ ∥𝑢 − 𝑢 ¯∥2𝐻 1 + ∥𝑣∥2𝐻 2 𝐻 𝐻 0 ) ¯ 2 2 + ∥𝑍∥2 2 + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑍𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶1 , ∀ 𝑡 > 0 + ∥𝜃 − 𝜃∥ 𝐻 𝐻

(5.1.19) ∫1 ¯ = where 𝑢 ¯ = 0 𝑢(𝑥) 𝑑𝑥 = 0 𝑢0 (𝑥) 𝑑𝑥, constant 𝜃¯ > 0 is determined by 𝑒(¯ 𝑢, 𝜃) ) ∫ 1 (1 2 0 2 𝑣0 + 𝑒(𝑢0 , 𝜃0 ) + 𝜆𝑍0 𝑑𝑥. Moreover, there are constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾1 ], the following estimate holds for any 𝑡 > 0, ( ) ¯ 2 1 + ∥𝑍(𝑡)∥2 1 ¯∥2𝐻 1 + ∥𝑣(𝑡)∥2𝐻 1 + ∥𝜃(𝑡) − 𝜃∥ 𝑒𝛾𝑡 ∥𝑢(𝑡) − 𝑢 𝐻 𝐻 ∫ 𝑡 ( ¯ 2 2 + ∥𝑍∥2 2 𝑒𝛾𝜏 ∥𝑢 − 𝑢 ¯∥2𝐻 1 + ∥𝑣∥2𝐻 2 + ∥𝜃 − 𝜃∥ + 𝐻 𝐻 0 ) + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑍𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶1 . (5.1.20) ∫1

2 Theorem 5.1.2. Let (𝛽, 𝑞) ∈ 𝐸. Assume that the initial data (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ and the compatibility conditions are satisfied, and there exists a constant 𝜀0 > ∫1 0 such that as 𝑢 ¯0 = 0 𝑢0 𝑑𝑥 ≤ 𝜀0 , then there exists a unique global solution 2 (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑍(𝑡)) ∈ 𝐻+ to problem (5.1.1)–(5.1.6) verifying that for any 𝑡 > 0,

¯ 2 2 + ∥𝑍(𝑡)∥2 2 + ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 ∥𝑢(𝑡) − 𝑢 ¯∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 𝐻 ∫ 𝑡( ¯ 2 3 + ∥𝑍∥2 3 + ∥𝑍𝑡 (𝑡)∥2 + ∥𝑢 − 𝑢 ¯∥2𝐻 2 + ∥𝑣∥2𝐻 3 + ∥𝜃 − 𝜃∥ 𝐻 𝐻 0 ) + ∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝑍𝑡𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 . (5.1.21) Moreover, there are constants 𝐶2 > 0 and 𝛾2 = 𝛾2 (𝐶2 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾2 ], the following estimate holds for any 𝑡 > 0, ( ¯ 2 2 + ∥𝑍(𝑡)∥2 2 + ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 ¯∥2𝐻 2 + ∥𝑣(𝑡)∥2𝐻 2 + ∥𝜃(𝑡) − 𝜃∥ 𝑒𝛾𝑡 ∥𝑢(𝑡) − 𝑢 𝐻 𝐻 ) ∫ 𝑡 ( ¯ 2 3 + ∥𝑍∥2 3 ¯∥2𝐻 2 + ∥𝑣∥2𝐻 3 + ∥𝜃 − 𝜃∥ + ∥𝑍𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑢 − 𝑢 𝐻 𝐻 0 ) + ∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝑍𝑡𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶2 . (5.1.22)

5.1. Introduction

131

4 Theorem 5.1.3. Let (𝛽, 𝑞) ∈ 𝐸. Assume that the initial data (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ and the compatibility conditions are satisfied, and there exists a constant 𝜀0 > ∫1 0 such that as 𝑢 ¯0 = 0 𝑢0 𝑑𝑥 ≤ 𝜀0 , then there exists a unique global solution 4 to problem (5.1.1)–(5.1.6) verifying that for any 𝑡 > 0, (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑍(𝑡)) ∈ 𝐻+

¯ 2 4 + ∥𝑍(𝑡)∥2 4 + ∥𝑣𝑡 (𝑡)∥2 2 + ∥𝑣𝑡𝑡 (𝑡)∥2 ∥𝑢(𝑡) − 𝑢¯∥2𝐻 4 + ∥𝑣(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃∥ 𝐻 𝐻 𝐻 ∫ 𝑡( ∥𝑢 − 𝑢 ¯∥2𝐻 4 + ∥𝑣∥2𝐻 5 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑍𝑡 (𝑡)∥2𝐻 2 + ∥𝑍𝑡𝑡 (𝑡)∥2 + 0

¯ 2 5 + ∥𝑍∥2 5 + ∥𝑣𝑡 ∥2 3 + ∥𝜃𝑡 ∥2 3 + ∥𝑍𝑡 ∥2 3 + ∥𝜃 − 𝜃∥ 𝐻 𝐻 𝐻 𝐻 𝐻 ) 2 2 2 + ∥𝑣𝑡𝑡 ∥𝐻 1 + ∥𝜃𝑡𝑡 ∥𝐻 1 + ∥𝑍𝑡𝑡 ∥𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(5.1.23)

Moreover, there are constants 𝐶4 > 0 and 𝛾4 = 𝛾4 (𝐶4 ) > 0 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimate holds for any 𝑡 > 0, ( ¯ 2 4 + ∥𝑍(𝑡)∥2 4 + ∥𝑣𝑡 (𝑡)∥2 2 + ∥𝑣𝑡𝑡 (𝑡)∥2 ¯∥2𝐻 4 + ∥𝑣(𝑡)∥2𝐻 4 + ∥𝜃(𝑡) − 𝜃∥ 𝑒𝛾𝑡 ∥𝑢(𝑡) − 𝑢 𝐻 𝐻 𝐻 ) ∫ 𝑡 ( ¯∥2𝐻 4 + ∥𝜃𝑡 (𝑡)∥2𝐻 2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑍𝑡 (𝑡)∥2𝐻 2 + ∥𝑍𝑡𝑡 (𝑡)∥2 + 𝑒𝛾𝜏 ∥𝑢 − 𝑢 0

¯ 2 5 + ∥𝑍∥2 5 + ∥𝑣𝑡 ∥2 3 + ∥𝜃𝑡 ∥2 3 + ∥𝑍𝑡 ∥2 3 + ∥𝑣∥2𝐻 5 + ∥𝜃 − 𝜃∥ 𝐻 𝐻 𝐻 𝐻 𝐻 ) + ∥𝑣𝑡𝑡 ∥2𝐻 1 + ∥𝜃𝑡𝑡 ∥2𝐻 1 + ∥𝑍𝑡𝑡 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 .

(5.1.24)

Corollary 5.1.1. Under assumptions of Theorem 5.1.3, the global solution 4 (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑍(𝑡)) ∈ 𝐻+

obtained in Theorem 5.1.3 is in fact a classical solution verifying for any 𝛾 ∈ (0, 𝛾4 ], ¯ 𝑍(𝑡))∥(𝐶 3+1/2 )4 ≤ 𝐶4 𝑒−𝛾𝑡 . ∥(𝑢(𝑡) − 𝑢 ¯, 𝑣(𝑡), 𝜃(𝑡) − 𝜃, Remark 5.1.1. Theorems 5.1.1–5.1.3 still hold for the boundary conditions 𝑣(0, 𝑡) = 𝑣(1, 𝑡) = 0, 𝜃(0, 𝑡) = 𝜃(1, 𝑡) = 𝑇0 = const . > 0, 𝑍𝑥 (0, 𝑡) = 𝑍𝑥 (1, 𝑡) = 0 where 𝜃¯ should be replaced by 𝑇0 .

132


5.2 Global Existence in 𝑯 1 In this section we study the global existence of problem (5.1.1)–(5.1.6) in 𝐻 1 [0, 1] by establishing a series of a priori estimates. Lemma 5.2.1. Under the assumptions in Theorem 5.1.1, the following estimates hold: ∫ 1 ∫ 𝑡∫ 1 ∫ 1 𝑍(𝑥, 𝑡) 𝑑𝑥 + 𝜙(𝜃, 𝑍)(𝑥, 𝑠) 𝑑𝑥𝑑𝑠 = 𝑍0 (𝑥) 𝑑𝑥, (5.2.1) 0 0 0 0 ] [ [ ] ∫ 1 ∫ 1 1 2 1 2 𝑣 + 𝑒 + 𝜆𝑍 (𝑥, 𝑡) 𝑑𝑥 = 𝑣 + 𝑒0 + 𝜆𝑍0 (𝑥) 𝑑𝑥 = 𝐸0 , (5.2.2) 2 2 0 0 0 𝜃(𝑥, 𝑡) > 0, 0 ≤ 𝑍(𝑥, 𝑡) ≤ 1, ∀ (𝑥, 𝑡) ∈ [0, 1] × [0, +∞), ∫ ∫ 𝑡∫ 1 𝑑 2 1 1 2 𝑍 (𝑥, 𝑡) 𝑑𝑥 + 𝑍 𝑑𝑥𝑑𝑠 2 𝑥 2 0 0 0 𝑢 ∫ ∫ 𝑡∫ 1 1 1 2 𝑍𝜙(𝜃, 𝑍) 𝑑𝑥𝑑𝑠 = 𝑍 (𝑥) 𝑑𝑥, + 2 0 0 0 0 ∫ 𝑡 Ψ(𝑠) 𝑑𝑠 ≤ 𝐶1 Φ(𝑡) + 0

(5.2.3)

(5.2.4) (5.2.5)

where ∫ Φ(𝑡) =

1

[𝑅(𝑢 − log 𝑢 − 1) + 𝐶𝑣 (𝜃 − log 𝜃 − 1)] (𝑥, 𝑡) 𝑑𝑥, ) ∫ 1( 2 𝑣𝑥 𝑘𝜃2 𝜙(𝜃, 𝑍) + 𝑥2 + 𝜆 Ψ(𝑡) = (𝑥, 𝑡) 𝑑𝑥. 𝑢𝜃 𝑢𝜃 𝜃 0 0

Proof. Relation (5.2.1) is obtained by integrating (5.1.4) on [0, 1] × (0, 𝑡), by using boundary condition (5.1.5). Multiplying by 𝑣 in (5.1.2), we find the conservation law ) ( ) ( 1 2 𝜆𝑑 𝑣 + 𝑒 + 𝜆𝑍 = 𝜎𝑣 − 𝑄 + 2 𝑍𝑥 . (5.2.6) 2 𝑢 𝑡 𝑥 Now integrating (5.2.6) on [0, 1], and using (5.1.1)–(5.1.6), we obtain (5.2.2). To get the positivity of 𝜃 in (5.2.3), we apply the maximum principle to (5.1.3), rewritten as 𝜇 𝑒𝜃 𝜃𝑡 + 𝜃𝑝𝜃 𝑣𝑥 − 𝑣𝑥2 = 𝑢

(


) 𝑥

+ 𝜆𝜙(𝜃, 𝑍),

(5.2.7)

together with (5.1.5), and we use the positivity of 𝜃0 . To get the positivity of 𝑍, we apply the same principle to the equation (5.1.4), together with (5.1.5), and we use the positivity of 𝑍0 .


133

To get (5.2.5), we multiply (5.2.7) by 𝜃−1 , ( ) 𝑘(𝑢, 𝜃)𝜃𝑥 𝜃𝑡 𝜇 2 𝑘(𝑢, 𝜃) 2 𝜙(𝜃, 𝑍) 𝑒 𝜃 + 𝑝𝜃 𝑢 𝑡 = 𝜃 + +𝜆 𝑣 + . 𝜃 𝑢𝜃 𝑥 𝑢𝜃2 𝑥 𝑢𝜃 𝜃 𝑥

(5.2.8)

A standard thermodynamical computation of the entropy 𝑆 takes the form 4 𝑆(𝑢, 𝜃) = 𝑅 log 𝑢 + 𝐶𝑣 log 𝜃 + 𝑎𝑢𝜃3 + 𝑆0 . 3

(5.2.9)

By using the thermodynamical formulation 𝑆𝜃 = 𝑒𝜃 /𝜃, and 𝑆𝑢 = 𝑝𝜃 , and by integrating (5.2.8) on [0, 1] × [0, 𝑡], we get ) ∫ 1∫ 𝑡( ∫ 1 ∫ 1 𝜇 2 𝑘(𝑢, 𝜃) 2 𝜙(𝜃, 𝑍) 𝑣𝑥 + 𝜃 + 𝜆 𝑆(𝑥, 𝑡)𝑑𝑥 − 𝑆0 (𝑥)𝑑𝑥 𝑑𝑠𝑑𝑥 = 𝑢𝜃 𝑢𝜃2 𝑥 𝜃 0 0 0 0 which, with (5.2.9), yields the identity ) ∫ 𝑡∫ 1( 𝜇 2 𝑘(𝑢, 𝜃) 2 𝜙(𝜃, 𝑍) 𝑣 + 𝜃 +𝜆 𝑑𝑥𝑑𝑠 𝑢𝜃 𝑥 𝑢𝜃2 𝑥 𝜃 0 0 ∫ 1 (𝑅(𝑢 − log 𝑢 − 1) + 𝐶𝑣 (𝜃 − log 𝜃 − 1)) 𝑑𝑥 + 0 ) ∫ 1( 4 𝑅(𝑢 − 1) + 𝐶𝑣 (𝜃 − 1) + 𝑎𝑢𝜃3 𝑑𝑥. = 3 0 By using estimate (5.1.1) and (5.2.2), we bound the first two terms of the right-hand side. For the last one, we use the Cauchy-Schwartz inequality ∫ 0

1

𝑢𝜃3 𝑑𝑥 ≤

(∫

1

0

)1/4 (∫ 𝑢𝑑𝑥

0

1

)3/4 𝑢𝜃4 𝑑𝑥

and we obtain (5.2.5), by using once more (5.2.2). Relation (5.2.4) is obtained by multiplying the equation (5.1.4) by 𝑍, integrating the result on [0, 1] × [0, 𝑡], and using (5.1.5) and (5.2.1). □ The next lemma is a generalization of the Bellman-Gronwall inequality which was established in Qin [52]. Lemma 5.2.2. Assume that 𝑓 (𝑡), 𝑔(𝑡) and 𝑦(𝑡) are nonnegative integrable functions in [𝜏, 𝑇 ] (𝜏 < 𝑇 ) verifying the integral inequality ∫ 𝑡 𝑦(𝑡) ≤ 𝑔(𝑡) + 𝑓 (𝑠)𝑦(𝑠)𝑑𝑠, 𝑡 ∈ [𝜏, 𝑇 ]. (5.2.10) 𝜏

Then we have 𝑦(𝑡) ≤ 𝑔(𝑡) +

∫ 𝜏

𝑡

(∫ exp

𝑠

𝑡

) 𝑓 (𝜃)𝑑𝜃 𝑓 (𝑠)𝑔(𝑠)𝑑𝑠, 𝑡 ∈ [𝜏, 𝑇 ].

(5.2.11)

134


In addition, if 𝑔(𝑡) is a nondecreasing function in [𝜏, 𝑇 ], then we conclude [ (∫ 𝑡 ) ] ∫ 𝑡 𝑦(𝑡) ≤ 𝑔(𝑡) 1 + exp 𝑓 (𝜃)𝑑𝜃 𝑓 (𝑠)𝑑𝑠 , 𝑡 ∈ [𝜏, 𝑇 ], 𝜏 𝑠 (∫ 𝑡 )] [ ∫ 𝑡 ≤ 𝑔(𝑡) 1 + 𝑓 (𝑠)𝑑𝑠 exp 𝑓 (𝜃)𝑑𝜃 , 𝑡 ∈ [𝜏, 𝑇 ]. (5.2.12) 𝜏

If further 𝑇 = +∞ and

∫ +∞ 𝜏

𝜏

𝑓 (𝑠)𝑑𝑠 < +∞, then we conclude

𝑦(𝑡) ≤ 𝐶𝑔(𝑡) { } ∫ +∞ ∫ +∞ where 𝐶 = 1 + 𝜏 𝑓 (𝑠)𝑑𝑠 exp 𝜏 𝑓 (𝜃)𝑑𝜃 is a positive constant. Proof. Let

∫ 𝐺(𝑡) =

𝑡

0

𝑓 (𝑠)𝑦(𝑠)𝑑𝑠.

(5.2.13)

(5.2.14)

Differentiating (5.2.14) with respect to 𝑡, using (5.2.10), we obtain ( ) 𝑑 𝐺(𝑡) = 𝑓 (𝑡)𝑦(𝑡) ≤ 𝑓 (𝑡) 𝑔(𝑡) + 𝐺(𝑡) . 𝑑𝑡

(5.2.15)

Thus,

) ∫𝑡 ∫𝑡 𝑑 ( 𝐺(𝑡)𝑒− 0 𝑓 (𝑠)𝑑𝑠 ≤ 𝑓 (𝑡)𝑔(𝑡)𝑒− 0 𝑓 (𝑠)𝑑𝑠 . 𝑑𝑡 Integrating (5.2.16) over (𝜏, 𝑡) with respect to 𝑡, we have (∫ 𝑡 ) ∫ 𝑡 𝐺(𝑡) ≤ exp 𝑓 (𝜃)𝑑𝜃 𝑓 (𝑠)𝑔(𝑠)𝑑𝑠 𝜏

(5.2.16)

𝑠

which, along with (5.2.10), gives (5.2.11). (5.2.12) and (5.2.13) are obvious. The proof is complete. □ Lemma 5.2.3. For any 𝑡 ≥ 0, there exists one point 𝑥1 = 𝑥1 (𝑡) ∈ [0, 1] such that specific volume 𝑢(𝑥, 𝑡) in problem (5.1.1)–(5.1.6) possesses the following expression: for any 𝛿 ≥ 0, ] [ ∫ 𝑡 𝑢(𝑥, 𝑠)[𝑝(𝑥, 𝑠) − 𝛿] −1 𝑑𝑠 (5.2.17) 𝑢(𝑥, 𝑡) = 𝐷(𝑥, 𝑡)𝐵(𝑡) 1 + 𝜇 𝐷(𝑥, 𝑠)𝐵(𝑠) 0 where

[ 𝐷(𝑥, 𝑡) = 𝑢0 (𝑥) exp 𝜇

−1

(∫

𝑥

𝑥1 (𝑡)

∫ 𝑣(𝑦, 𝑡)𝑑𝑦 −

0

𝑥

𝑣0 (𝑦) 𝑑𝑦

)] ∫ 𝑥 ∫ 1 1 + 𝑢0 (𝑥) 𝑣0 (𝑦) 𝑑𝑦𝑑𝑥 , 𝑢 ¯0 (𝑥) 0 0 [ ] ∫ 𝑡∫ 1 1 2 𝐵(𝑡) = exp − (𝑣 + 𝑢𝑝)(𝑥, 𝑠) 𝑑𝑦𝑑𝑠 + 𝛿𝑡/𝜇 𝜇¯ 𝑢0 0 0 ∫1 with 𝑢 ¯0 = 0 𝑢0 (𝑥)𝑑𝑥.

(5.2.18) (5.2.19)


135

Proof. For 𝛿 = 0, (5.2.17) was proved in [1, 3, 33, 36, 45, 46, 47] for a polytropic ideal gas and in [27, 32, 49, 51, 52, 54, 55, 57, 59] for viscous heat-conductive real gas (see also, Lemma 1.2.4). For 𝛿 > 0, we shall borrow some ideas from [20, 27, 32, 49, 51, 52, 54, 55, 57, 59]. In fact, for any 𝛿 ≥ 0, we can rewrite (5.1.1) as 1 1 𝑢𝑡 = 𝑣𝑥 = (𝜎 + 𝛿)𝑢 + 𝑢(𝑝 − 𝛿), 𝜇 𝜇 i.e., 1 1 𝑢𝑡 − (𝜎 + 𝛿)𝑢 = 𝑢(𝑝 − 𝛿). (5.2.20) 𝜇 𝜇 ∫𝑡 Multiplying (5.2.20) by exp{− 𝜇1 0 (𝜎 + 𝛿)𝑑𝑠}, we infer )( ( ) ) ( ∫ 𝑡 ∫ ∫ 1 𝑡 1 𝑠 1 (𝜎 + 𝛿)𝑑𝑠 𝑢0 + 𝑢(𝑝 − 𝛿)exp − (𝜎 + 𝛿)𝑑𝜏 𝑑𝑠 . 𝑢(𝑥,𝑡) = exp 𝜇 0 𝜇 0 𝜇 0 (5.2.21) Let ∫ 𝑡 ∫ 𝑥 ℎ(𝑥, 𝑡) = 𝜎(𝑥, 𝑠)𝑑𝑠 + 𝑣0 (𝑦)𝑑𝑦. (5.2.22) 0

Then we have

0

ℎ𝑥 = 𝑣, ℎ𝑡 = 𝜎 = which, along with (5.1.1) and (5.1.5), implies

𝜇ℎ𝑥𝑥 −𝑝 𝑢

(5.2.23)

(𝑢ℎ)𝑡 = ℎ𝑣𝑥 + 𝜇ℎ𝑥𝑥 − 𝑝𝑢, ℎ𝑥 ∣𝑥=0,1 = 0.

(5.2.24) (5.2.25)

Integrating (5.2.24) with respect to 𝑡 and using (5.2.22)–(5.2.25), we get ∫ 1 ∫ 𝑡∫ 1 ∫ 1 𝑢ℎ𝑑𝑥 = 𝑢0 ℎ0 𝑑𝑥 − (𝑣 2 + 𝑢𝑝)𝑑𝑥𝑑𝑠 0

0

0

0

which, by using the mean value theorem, implies that there exists a point 𝑥1 (𝑡) ∈ [0, 1] for any 𝑡 ≥ 0 such that ∫ 1 ∫ 1 ∫ 𝑡∫ 1 𝑢𝑑𝑥 = 𝑢0 ℎ0 𝑑𝑥 − (𝑣 2 + 𝑢𝑝)𝑑𝑥𝑑𝑠, ℎ(𝑥1 (𝑡), 𝑡) 0

0

0

0

i.e.,

] [∫ 1 ∫ 𝑡∫ 1 1 2 𝑢0 ℎ0 𝑑𝑥 − (𝑣 + 𝑢𝑝)𝑑𝑥𝑑𝑠 . ℎ(𝑥1 (𝑡, 𝑡)) = 𝑢 ¯0 0 0 0 Thus by (5.2.23)–(5.2.26), we deduce ∫ 𝑥1 (𝑡) ∫ 𝑡 𝜎(𝑥1 (𝑡), 𝑠)𝑑𝑠 = ℎ(𝑥1 (𝑡), 𝑡) − 𝑣0 (𝑦)𝑑𝑦 0

=

1 𝑢 ¯0

[∫ 0

0

1

𝑢0 ℎ0 𝑑𝑥 −

∫ 𝑡∫ 0

0

1

(5.2.26)

(5.2.27)

] ∫ (𝑣 2 + 𝑢𝑝)𝑑𝑥𝑑𝑠 −

0

𝑥1 (𝑡)

𝑣0 (𝑦)𝑑𝑦.

136


Integrating (5.1.2) over [𝑥1 (𝑡), 𝑥] × [0, 𝑡] (for any fixed 𝑡 ≥ 0), we infer ∫ 𝑡 ∫ 𝑡 ∫ 𝑥 𝜎(𝑥, 𝑠)𝑑𝑠 = 𝜎(𝑥1 (𝑡), 𝑠)𝑑𝑠 + (𝑣 − 𝑣0 )𝑑𝑦. 0

0

𝑥1 (𝑡)

(5.2.28)

Then we infer from (5.2.23), (5.2.27) and (5.2.28) that } {∫ 1 ∫ 𝑡 ∫ 𝑡∫ 1 1 2 𝜎(𝑥, 𝑠)𝑑𝑠 = 𝑢0 ℎ0 𝑑𝑥 − (𝑣 + 𝑢𝑝)𝑑𝑥𝑑𝑠 𝑢 ¯0 0 0 0 0 ∫ 𝑥1 (𝑡) ∫ 𝑥 − 𝑣0 (𝑦)𝑑𝑦 + (𝑣 − 𝑣0 )𝑑𝑦 0

1 = 𝑢 ¯0 +

{∫ ∫

𝑥

1 0

𝑥1 (𝑡)

𝑢0 ℎ0 𝑑𝑥 −

𝑥1 (𝑡) 1

∫ 𝑡∫ 0 ∫

𝑣(𝑦, 𝑡)𝑑𝑦 −

0

𝑥

0

2

}

(𝑣 + 𝑢𝑝)𝑑𝑥𝑑𝑠

𝑣0 (𝑦)𝑑𝑦

which, together with (5.2.21), gives (5.2.17). The proof is complete.

□

Lemma 5.2.4. Under the assumptions in Theorem 5.1.1, the following estimate holds: 0 < 𝐶1−1 ≤ 𝑢(𝑥, 𝑡) ≤ 𝐶1 , ∀ (𝑥, 𝑡) ∈ [0, 1] × [0, +∞). (5.2.29) Proof. The main idea of the proof is similar to that of Lemma 1.2.5. But the different key point here is to estimate the higher order term 𝑢𝜃4 which is a contribution of radiative effects from the pressure 𝑝 and internal energy 𝑒 (see also (5.1.7)). Thus we need a detailed and careful analysis. Noting the convexity of function − log 𝑦 and using (5.2.5), we have ∫ 1 ∫ 1 ∫ 1 𝜃 𝑑𝑥 − log 𝜃 𝑑𝑥 − 1 ≤ (𝜃 − log 𝜃 − 1) 𝑑𝑥 ≤ 𝐶1 /𝐶𝑣 0

0

0

which implies that there exists a point 𝑏(𝑡) ∈ [0, 1] and two positive constants 𝑟1 , 𝑟2 such that ∫ 1 0 < 𝑟1 ≤ 𝜃(𝑥, 𝑡) 𝑑𝑥 = 𝜃(𝑏(𝑡), 𝑡) =: 𝜃1 (𝑡) ≤ 𝑟2 (5.2.30) 0

with 𝑟1 , 𝑟2 being two positive roots of equation 𝑦 − log 𝑦 − 1 = 𝐶1 /𝐶𝑣 . Thus we infer from (5.2.2) that ∫ 1 ∫ 1 ∫ max(𝑅, 𝑎/3) 1 max[𝑅, 𝑎/3] 4 𝐸0 𝑢𝑝𝑑𝑥 ≤ max[𝑅, 𝑎/3] (𝜃 + 𝑢𝜃 )𝑑𝑥 ≤ 𝑒𝑑𝑥 ≤ min(𝐶 , 𝑎) min(𝐶𝑣 , 𝑎) 𝑣 0 0 0 and

∫ 0

1

𝑣 2 𝑑𝑥 ≤ 2𝐸0


137

which, along with (5.2.30), implies 1 0 < 𝑎1 ≤ 𝜇¯ 𝑢0

∫ 0

1

(𝑢𝑝 + 𝑣 2 )(𝑥, 𝑠) 𝑑𝑥 ≤ 𝑎2

(5.2.31)

with

[ ( ) ] 𝑢0 . 𝑢0 , 𝑎2 = 𝑅𝑟2 + 2 + (max(𝑅, 𝑎/3)𝐸0 )/ min(𝐶𝑣 , 𝑎) 𝐸0 /𝜇¯ 𝑎1 = 𝑅𝑟1 /𝜇¯

Using Lemmas 5.2.1–5.2.2, we derive for any [0, 1] × [0, +∞), 0 < 𝐶1−1 ≤ 𝐷(𝑥, 𝑡) ≤ 𝐶1 .

(5.2.32)

On the other hand, for 0 < 𝑚1 ≤ 2, we infer from Lemma 5.2.1, ∫ 𝑥 ∣𝜃𝑚1 (𝑥, 𝑡) − 𝜃𝑚1 (𝑏(𝑡), 𝑡)∣ ≤ 𝐶1 𝜃𝑚1 −1 𝜃𝑥 𝑑𝑦 𝑏(𝑡)

)1/2 )1/2 (∫ 1 𝑘(𝑢, 𝜃)𝜃𝑥2 𝑢𝜃2𝑚1 𝑑𝑥 𝑑𝑥 ≤ 𝐶1 𝑢𝜃2 0 0 𝑘(𝑢, 𝜃) (∫ 1 ) 1/2 𝑢(1 + 𝜃4 ) ≤ 𝐶1 𝑉 1/2 (𝑡) 𝑑𝑥 𝜅1 0 (∫

1

≤ 𝐶1 𝑉 1/2 (𝑡) where 𝑉 (𝑡) =

∫1

2 𝑘𝜃𝑥 0 𝑢𝜃 2

(5.2.33)

𝑑𝑥. Thus, for 0 < 𝑚1 ≤ 2,

1 2𝑚1 𝑟 − 𝐶1 𝑉 (𝑡) ≤ 𝜃2𝑚1 (𝑥, 𝑡) ≤ 2𝑟22𝑚1 + 𝐶1 𝑉 (𝑡). 2 1

(5.2.34)

Obviously, we derive from Lemmas 5.2.1, 5.2.3 and (5.2.31) for 𝛿 ≥ 0 and 0 ≤ 𝑠 ≤ 𝑡, 𝑒−(𝑎2 −𝛿/𝜇)(𝑡−𝑠) ≤ 𝐵(𝑡)𝐵 −1 (𝑠) ≤ 𝑒−(𝑎1 −𝛿/𝜇)(𝑡−𝑠) .

(5.2.35)

Thus for 𝛿 = 0 in Lemma 5.2.3, we use (5.2.32)–(5.2.35) to derive that there exists a large time 𝑡0 > 0 such that, as 𝑡 ≥ 𝑡0 > 0, ∫ 𝑡 −1 𝑢(𝑥, 𝑡) ≥ 𝐶1 𝜃𝑒−𝑎2 (𝑡−𝑠) 𝑑𝑠 0 ) ∫ 𝑡( 1 2 −1 𝑟 − 𝐶1 𝑉 (𝑠) 𝑒−𝑎2 (𝑡−𝑠) 𝑑𝑠 ≥ 𝐶1 2 1 0 ∫ 𝑡 𝑟12 𝑟12 ≥ (1 − 𝑒−𝑎2 𝑡 ) − 𝐶1 𝑉 (𝑠)𝑒−𝑎2 (𝑡−𝑠) 𝑑𝑠 ≥ > 0, (5.2.36) 2𝑎2 𝐶1 4𝑎2 𝐶1 0 where we have used the estimate as 𝑡 → +∞, ∫ 𝑡 ∫ ∫ ∞ 𝑉 (𝑠)𝑒−𝑎2 (𝑡−𝑠) 𝑑𝑠 ≤ 𝑒−𝑎2 𝑡/2 𝑉 (𝑠)𝑑𝑠 + 0

0

𝑡

𝑡/2

𝑉 (𝑠)𝑑𝑠 → 0.

138


On the other hand, we can also derive from Lemma 5.2.3 and (5.2.35) (for 𝑠 = 0) that for 𝛿 = 0 and for any 𝑡 ∈ [0, 𝑡0 ], 𝑢(𝑥, 𝑡) ≥

𝐷(𝑥, 𝑡) ≥ 𝐶1−1 exp(−𝑎2 𝑡) ≥ 𝐶1−1 exp(−𝑎2 𝑡0 ) > 0 𝐵(𝑡)

which, together with (5.2.36), gives 𝑢(𝑥, 𝑡) ≥ 𝐶1−1 > 0, ∀(𝑥, 𝑡) ∈ [0, 1] × [0, +∞).

(5.2.37)

Now we begin to show the positive lower bound of 𝑢 in (5.2.29). To prove it, we need the smallness of the total initial mass 𝑢 ¯0 , which is only used to prove the positive lower bound of 𝑢. 1 , then as 𝑢 ¯0 ≤ 𝜀0 , we get Now choosing 0 < 𝜀0 ≤ 3𝑅𝑟 𝑎𝑟 4 2

𝑎𝜃4 𝑅𝑟1 𝑅𝑟1 𝑎𝑟4 ≥ > 2 ≥ 1. 𝑢¯0 𝜀0 3 3

(5.2.38)

Now from (5.2.38), we can pick 𝛿 > 0 such that 𝑅𝑟1 𝑎𝜃4 𝑅𝑟1 𝑎𝑟4 ≥ >𝛿> 2 ≥ 1 𝑢 ¯0 𝜀0 3 3 which gives ] [ 1 𝛿 1 𝑅𝑟1 = [𝜇𝑎1 − 𝛿] = − 𝛿 > 0, 𝜇 𝜇 𝜇 𝑢 ¯0 3 −4 3 −4 𝛿1 ≡ 𝜃1 𝛿 − 1 ≥ 𝑟2 𝛿 − 1 ≡ 𝛿0 > 0. 𝑎 𝑎 𝜆 ≡ 𝑎1 −

(5.2.39) (5.2.40)

Noting that by the Young inequality (𝑏1 𝑏2 ≤ 𝜀𝑏21 + 𝑏22 /(2𝜀), ∀𝜀, 𝑏1 , 𝑏2 > 0) and inequality 𝑏4 − 1 − 𝜀 ≤ (𝑏2 − 1)2 (1 + 𝜀−1 ) with 𝑏 ∈ ℝ and any 𝜀 > 0, we derive from (5.2.39)–(5.2.40) that ) (𝑎 𝜃4 − 𝛿 (5.2.41) 𝑢(𝑝 − 𝛿) = 𝑅𝜃 + 𝑢 3 and [ ] (𝑎 ) 𝑎 3 𝑢 𝜃4 − 𝛿 = 𝜃14 𝑢 𝜃˜4 − 𝜃1−4 𝛿 3 3 𝑎 𝑎 4 ≤ 𝜃1 𝑢 max[𝜃˜4 − 1 − 𝛿1 , 0] 3 𝑎 4 ≤ 𝜃1 𝑢 max[𝜃˜4 − 1 − 𝛿0 , 0] 3 𝑎 ≤ 𝜃14 𝑢(𝜃˜2 − 1)2 (1 + 𝛿0−1 ) 3 (5.2.42) ≤ 𝐶1 𝑢(𝜃˜2 − 1)2


139

where ˜ 𝑡) = 𝜃(𝑥, 𝑡)/𝜃1 , 𝜃1 ≡ 𝜃(𝑏(𝑡), 𝑡) = 𝜃(𝑥,

∫

1 0

𝜃(𝑥, 𝑡) 𝑑𝑥 ∈ [𝑟1 , 𝑟2 ].

˜ Noting that 𝜃(𝑏(𝑡), 𝑡) = 1 and by the Poincaré inequality, we deduce (𝜃˜2 − 1)2 ≤

(∫

1

0

)2 (∫ ˜ ˜ 2∣𝜃𝜃𝑥 ∣𝑑𝑥 ≤ 𝐶1

0

1

)2 ∣𝜃𝜃𝑥 ∣𝑑𝑥

) ) (∫ 1 𝑘(𝑢, 𝜃)𝜃𝑥2 𝑢𝜃4 𝑑𝑥 ≤ 𝐶1 𝑑𝑥 𝑢𝜃2 0 0 𝑘(𝑢, 𝜃) ∫ 1 ≤ 𝐶1 𝑉 (𝑡) 𝑢𝜃4 𝑑𝑥 ≤ 𝐶1 𝑉 (𝑡) (∫

1

0

which, together with (5.2.41)–(5.2.42), gives 𝑢(𝑝 − 𝛿) ≤ 𝑅𝜃 + 𝐶1 𝑉 (𝑡)𝑢.

(5.2.43)

Thus it follows from Lemma 5.2.3, (5.2.32)–(5.2.34), (5.2.42) and (5.2.43) that ∫ 𝑡 ( ) [𝜃(𝑥, 𝑠) + 𝑢(𝑥, 𝑠)𝑉 (𝑠)] exp − 𝜆(𝑡 − 𝑠) 𝑑𝑠 𝑢(𝑥, 𝑡) ≤ 𝐶1 + 𝐶1 0 ∫ 𝑡 ( ) [1 + 𝑉 (𝑠) + 𝑢(𝑥, 𝑠)𝑉 (𝑠)] exp − 𝜆(𝑡 − 𝑠) 𝑑𝑠 ≤ 𝐶1 + 𝐶1 0 ∫ 𝑡 ( ) 𝑢(𝑥, 𝑠)𝑉 (𝑠) exp − 𝜆(𝑡 − 𝑠) 𝑑𝑠 ≤ 𝐶1 + 𝐶1 0

that is, 𝐹 (𝑡) ≤ 𝐶1 𝑒

𝜆𝑡

∫ + 𝐶1

0

𝑡

𝑉 (𝑠)𝐹 (𝑠) 𝑑𝑠

(5.2.44)

with 𝐹 (𝑡) = 𝑒𝜆𝑡 max𝑥∈[0,1] 𝑢(𝑥, 𝑡) ≡ 𝑒𝜆𝑡 𝑀𝑢 (𝑡). Therefore, by Lemma 5.2.2, we deduce ( ∫ 𝑡 ) 𝜆𝑡 𝐹 (𝑡) ≤ 𝐶1 𝑒 exp 𝐶1 𝑉 (𝑠)𝑑𝑠 ≤ 𝐶1 𝑒𝜆𝑡 , 0

i.e.,

𝑀𝑢 (𝑡) ≤ 𝐶1

which, along with (5.2.37), completes the proof of (5.2.29).

□

Lemma 5.2.5. Under the assumptions in Theorem 5.1.1, the following estimate holds: ∫ 𝑡∫ 1 (𝑢2𝑥 + 𝜃𝑢2𝑥 )(𝑥, 𝑠) 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞0 , ∀𝑡 > 0 (5.2.45) ∥𝑢𝑥 (𝑡)∥2 + 0

0

with 𝐴 = sup ∥𝜃(𝑠)∥𝐿∞ and 𝑞0 = max(4 − 𝑞, 0). 0≤𝑠≤𝑡

140


Proof. Equation (5.1.2) can be rewritten as ( 𝑢𝑥 ) 𝑣−𝜇 = −𝑝𝑥 . 𝑢 𝑡

(5.2.46)

Multiplying (5.2.46) by 𝑣 − 𝜇 𝑢𝑢𝑥 and then integrating the resulting equation over 𝑄𝑡 ≡ [0, 1] × [0, 𝑡], we have ∫ 𝑡∫ 1 2 𝜃𝑢𝑥 1 𝑢𝑥 2 𝑑𝑥𝑑𝑠 𝑣 − 𝜇 + 𝑅𝜇 2 𝑢 𝑢3 0 0 2 ∫ 𝑡 ∫ 1 [ ( ) ( ] 𝑢𝑥 ) 𝑢0𝑥 1 4 3 1 𝜃𝑢𝑥 𝑣 + = 𝑣0 − 𝜇 𝑅 2 − 𝑅 + 𝑎𝜃 𝜃𝑥 𝑣 − 𝜇 𝑑𝑥𝑑𝑠 2 𝑢0 𝑢 𝑢 3 𝑢 0 0 implying ∫ 𝑡∫ 1 𝑢𝑥 2 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 (5.2.47) 𝑣 − 𝜇 + 𝑢 0 0 ∫ 𝑡∫ 1 ] [ ≤ 𝐶1 + 𝐶1 ∣𝜃𝑢𝑥 𝑣∣ + ∣𝜃𝑥 𝑣∣ + ∣𝜃3 𝜃𝑥 𝑣∣ + ∣𝜃𝑥 𝑢𝑥 ∣ + ∣𝜃3 𝜃𝑥 𝑢𝑥 ∣ 𝑑𝑥𝑑𝑠. 0

0

Noting the following facts ∫ 𝑡 ∫ 𝑡 (∫ 2 ∥𝑣(𝑠)∥𝐿∞ 𝑑𝑠 ≤ 0

∫ ∫

1

0

1

0

0

1

0

) (∫ 1 ) ∫ 𝑡∫ 1 2 𝑣𝑥2 𝑣𝑥 𝑑𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 , 𝜃 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 𝜃 0 0 0 𝜃

𝜃4 𝑑𝑥 ≤ 𝐶1 ,

𝑢𝜃𝑟 𝑑𝑥 ≤ 𝐶1 𝑘(𝑢, 𝜃)

∫

1 0

(1 + 𝜃)𝑟−𝑞 𝑑𝑥 ≤ 𝐶1 + 𝐶1 𝐴max(𝑟−𝑞−4,0) ,

𝑟 ≥ 0,

and using Lemmas 5.2.1–5.2.4, we easily derive that for any 𝜀 > 0, ∫ 𝑡∫ 1 ∫ 𝑡∫ 1 ∫ 𝑡 ∫ 1 2 2 ∣𝜃𝑢𝑥 𝑣∣𝑑𝑥𝑑𝑠 ≤ 𝜀 𝜃𝑢𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 (𝜀) ∥𝑣(𝑠)∥𝐿∞ 𝜃𝑑𝑥𝑑𝑠 0

0

0

0

≤ 𝐶1 (𝜀) + 𝜀 ∫ 𝑡∫ 0

0

∫ 𝑡∫ 0

1

0

1

∫ 𝑡∫ 0

1

0 )1/2

0

0

𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠,

(5.2.48)

)1/2 𝑢𝜃2 𝑣 2 𝑑𝑥𝑑𝑠 ∣𝜃𝑥 𝑣∣𝑑𝑥𝑑𝑠 ≤ 𝑉 (𝑠)𝑑𝑠 0 0 0 𝑘(𝑢, 𝜃) ) }1/2 {∫ 𝑡 (∫ 1 2 2 ≤ 𝐶1 ∥𝑣(𝑠)∥𝐿∞ 𝜃 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 , (∫

3

𝑡

0

(∫

∣𝜃 𝜃𝑥 𝑣∣𝑑𝑥𝑑𝑠 ≤

𝑡 0

𝑉 (𝑠)𝑑𝑠

(∫ 𝑡 ∫

0

)1/2 (∫ 𝑡 ∫

≤ 𝐶1 + 𝐶1 𝐴𝑞0 /2 ,

1

0

0

1

𝑢𝜃8 𝑣 2 𝑑𝑥𝑑𝑠 𝑘(𝑢, 𝜃)

(5.2.49)

)1/2 (5.2.50)


∫ 𝑡∫ 0

1

0

141

(∫ ∣𝜃𝑥 𝑢𝑥 ∣ 𝑑𝑥𝑑𝑠 ≤

𝑡 0

0

1

𝑢𝜃 ≤ 𝐶1 sup 0≤𝑠≤𝑡 𝑘(𝑢, 𝜃) 𝐿∞ 0 0 ∫ 𝑡∫ 1 ≤𝜀 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 𝐴max(1−𝑞,0) + 𝐶1 , 0

∫ 𝑡∫

𝑉 (𝑠)𝑑𝑠

)1/2 𝑢𝜃2 𝑢2𝑥 𝑑𝑥𝑑𝑠 0 0 𝑘(𝑢, 𝜃) 1/2 (∫ 𝑡 ∫ 1 )1/2 2 𝜃𝑢𝑥 𝑑𝑥𝑑𝑠

)1/2 (∫ 𝑡 ∫

(5.2.51)

0

2 (𝜃 − 𝜃12 )𝜃𝜃𝑥 𝑢𝑥 𝑑𝑥𝑑𝑠 0 ∫ 1 ∫ 𝑡∫ 1 ∣𝜃𝜃𝑥 ∣ 𝑑𝑥 ∣𝜃𝜃𝑥 𝑢𝑥 ∣ 𝑑𝑥𝑑𝑠 ≤ 𝐶1 0 0 0 {(∫ )1/2 (∫ 1 )1/2 } ∫ 𝑡 1 𝑢𝜃4 𝑢𝜃4 𝑢2𝑥 𝑑𝑥 𝑑𝑥 ≤ 𝐶1 sup 𝑉 (𝑠)𝑑𝑠 0≤𝑠≤𝑡 0 𝑘(𝑢, 𝜃) 0 𝑘(𝑢, 𝜃) 0 ) )1/2 ( (∫ 1 𝑢𝜃4 1/2 4 𝑢𝜃 𝑑𝑥 ≤ 𝐶1 + 𝐶1 sup 𝑘(𝑢, 𝜃) ∞ ∥𝑢𝑥 ∥ 0≤𝑠≤𝑡 0 𝐿 1

≤ 𝜀 sup ∥𝑢𝑥 (𝑠)∥2 + 𝐶1 (𝜀)𝐴𝑞0 + 𝐶1 ,

(5.2.52)

0≤𝑠≤𝑡

∫ 𝑡∫ 0

1

0

≤

∣𝜃3 𝜃𝑥 𝑢𝑥 ∣𝑑𝑥𝑑𝑠 ∫ 𝑡∫ 0

1

0

∣(𝜃2 − 𝜃12 )𝜃𝜃𝑥 𝑢𝑥 ∣𝑑𝑥𝑑𝑠 +

≤ 𝜀 sup ∥𝑢𝑥 (𝑠)∥2 + 𝐶1 + 𝐶1 𝐴𝑞0 0≤𝑠≤𝑡

(∫

+ 𝐶1

0

𝑡

𝑉 (𝑠)𝑑𝑠

)1/2 (∫ 𝑡 ∫ 0

2

1

0 𝑞0

∫ 𝑡∫ 0

1

0

𝜃12 ∣𝜃𝜃𝑥 𝑢𝑥 ∣𝑑𝑥𝑑𝑠

𝑢𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 𝑘(𝑢, 𝜃)

)1/2

≤ 𝜀 sup ∥𝑢𝑥 (𝑠)∥ + 𝐶1 + 𝐶1 𝐴 0≤𝑠≤𝑡

(∫ 𝑡 ∫ 1 )1/2 𝑢𝜃3 1/2 2 + 𝐶1 sup 𝜃𝑢𝑥 𝑑𝑥𝑑𝑠 0≤𝑠≤𝑡 𝑘(𝑢, 𝜃) 𝐿∞ 0 0 ∫ 𝑡∫ 1 ≤ 𝜀 sup ∥𝑢𝑥 (𝑠)∥2 + 𝜀 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴𝑞0 . 0≤𝑠≤𝑡

0

(5.2.53)

0

Inserting (5.2.48)–(5.2.52) into (5.2.47), we have ∥𝑢𝑥(𝑡)∥2 +

∫ 𝑡∫ 0

0

1

𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞0 + 3𝜀

∫ 𝑡∫ 0

0

1

𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 + 𝜀 sup ∥𝑢𝑥(𝑠)∥2 0≤𝑠≤𝑡

142


which, by taking the supremum on the left-hand side and choosing 𝜀 > 0 small enough, gives ∫ 𝑡∫ 1 𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞0 . (5.2.54) ∥𝑢𝑥 (𝑡)∥2 + 0

0

Moreover, noting that, from (5.2.34) for 𝑚1 = 1/2, 1 ≤ 𝐶1 𝜃 + 𝐶1 𝑉 (𝑡),

(5.2.55)

we derive from (5.2.54)–(5.2.55) that ∫ 𝑡∫ 0

0

1

𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1

∫ 𝑡∫ 0

0

1

𝜃𝑢2𝑥 𝑑𝑥𝑑𝑠 + 𝐶1

∫

𝑡

0

≤ 𝐶1 + 𝐶1 𝐴𝑞0

𝑉 (𝑠)∥𝑢𝑥 (𝑠)∥2 𝑑𝑠

which, together with (5.2.54), gives (5.2.44).

□

Now repeating the derivation of (5.2.33)–(5.2.34), we conclude ∫ 0

1

𝑢𝜃2𝑚 𝑑𝑥 ≤ 𝐶1 𝑘(𝑢, 𝜃) ≤ 𝐶1

∫ 0

∫

0

1

1

𝜃2𝑚 𝑑𝑥 ≤ 𝐶1 1 + 𝜃𝑞

∫

1 0

(1 + 𝜃4 )𝑑𝑥 ≤ 𝐶1 ,

(1 + 𝜃)2𝑚−𝑞 𝑑𝑥

0 ≤ 𝑚 ≤ (𝑞 + 4)/2.

We readily obtain the next corollary. Corollary 5.2.1. Under the assumptions in Theorem 5.1.1, the following estimate holds: 𝐶1−1 − 𝐶1 𝑉 (𝑡) ≤ 𝜃2𝑚 (𝑥, 𝑡) ≤ 𝐶1 + 𝐶1 𝑉 (𝑡),

0 ≤ 𝑚 ≤ (𝑞 + 4)/2.

(5.2.56)

Lemma 5.2.6. Under the assumptions in Theorem 5.1.1, the following estimates hold: ∫ 𝑡∫ 1 (1 + 𝜃)2𝑚 𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞0 , ∀𝑡 > 0, (5.2.57) 0

0

0

0

∫ 𝑡∫

1

(1 + 𝜃)2𝑚 𝑣 2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 ,

∀𝑡 > 0

(5.2.58)

with 0 ≤ 𝑚 ≤ (𝑞 + 4)/2. Proof. It follows from Corollary 5.2.1 and Lemma 5.2.5 that ∫ 𝑡 ∫ 𝑡 ∫ 𝑡∫ 1 (1 + 𝜃)2𝑚 𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 ∥𝑢𝑥 (𝑠)∥2 𝑑𝑠 + 𝑉 (𝑠)∥𝑢𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞0 . 0

0

0

0

The proof of (5.2.58) is similar to that of (5.2.57).

□


143

Lemma 5.2.7. Under the assumptions in Theorem 5.1.1, the following estimates hold for any 𝑡 > 0: ∫ 𝑡 ∥𝑣𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞0 , (5.2.59) 0 ∫ 𝑡 ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞2 , (5.2.60) ∥𝑣𝑥 (𝑡)∥2 + 0 ∫ 𝑡 ∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞3 (5.2.61) 0

where 𝑞1 = max(8 − 𝑞, 0), 𝑞2 = max(3𝑞0 , 𝑞1 ), 𝑞3 = max[𝑞1 , (𝑞2 + 3𝑞0 )/2](≥ 𝑞2 ). Proof. Multiplying (5.1.2) by 𝑣, 𝑣𝑥𝑥 and 𝑣𝑡 , respectively, and then integrating the resulting equations over 𝑄𝑡 , using Lemmas 5.2.1–5.2.6, we get ∫ 𝑡 ∥𝑣𝑥 (𝑠)∥2 𝑑𝑠 ∥𝑣(𝑡)∥2 + 0

∫ 𝑡 ∫ ≤ 𝐶1 + 𝐶1

] (1 + 𝜃3 )𝜃𝑥 𝑣 + 𝜃𝑢𝑥 𝑣 𝑑𝑥𝑑𝑠 0 0 ) ∫ 𝑡 ∫ 𝑡∫ 1( (1 + 𝜃3 )2 𝜃2 2 2 2 ≤ 𝐶1 + 𝐶1 𝑉 (𝑠) 𝑑𝑠 + 𝐶1 𝑣 𝜃𝑢𝑥 + 𝜃𝑣 + 𝑑𝑥𝑑𝑠 1 + 𝜃𝑞 0 0 0 ] [∫ 1 ∫ 𝑡 ∫ 1 𝑞0 2 𝑞0 4 ∥𝑣(𝑠)∥𝐿∞ 𝜃𝑑𝑥 + (1 + ∥𝜃∥𝐿∞ ) (1 + 𝜃) 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴 + 𝐶1 1

[

0

0

≤ 𝐶1 + 𝐶1 𝐴𝑞0 , ∫ 𝑡 2 ∥𝑣𝑥 (𝑡)∥ + ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠

0

(5.2.62)

0

∫ 𝑡 ∫ 1 [ ] (1 + 𝜃3 )𝜃𝑥 𝑣𝑥𝑥 + 𝜃𝑢𝑥 𝑣𝑥𝑥 + 𝑣𝑥 𝑢𝑥 𝑣𝑥𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 0 0 ∫ ∫ 𝑡∫ 1 [ ] 1 𝑡 (1 + 𝜃3 )2 𝜃𝑥2 + 𝑣𝑥2 𝑢2𝑥 + 𝜃2 𝑢2𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 + 𝐶1 4 0 0 0 ∫ ∫ 𝑡 1 𝑡 ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 + 𝐶1 𝐴𝑞0 + 𝐶1 (1 + 𝐴𝑞0 ) ∥𝑣𝑥 (𝑠)∥2𝐿∞ 𝑑𝑠 ≤ 𝐶1 + 4 0 0 ∫ 𝑡 𝑉 (𝑠) 𝑑𝑠 + 𝐶1 (1 + 𝐴)𝑞1 0 ∫ 1 𝑡 ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 4 0 (∫ 𝑡 )1/2 (∫ 𝑡 )1/2 + 𝐶1 (1 + 𝐴𝑞0 ) ∥𝑣𝑥 (𝑠)∥2 𝑑𝑠 ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 + 𝐶1 (1 + 𝐴)𝑞1 0

≤ 𝐶1 + 𝐶1 𝐴3𝑞0 + 𝐶1 𝐴𝑞1 +

1 2

∫ 0

𝑡

0

∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠,

144


i.e., ∥𝑣𝑥 (𝑡)∥2 +

∫

𝑡

0

∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 (1 + 𝐴)𝑞2

(5.2.63)

with 𝑞2 = max(3𝑞0 , 𝑞1 ), and 2

∥𝑣𝑥 (𝑡)∥ +

∫ 0

𝑡

∥𝑣𝑡 (𝑠)∥2 𝑑𝑠

∫ 𝑡 ∫ ≤ 𝐶1 + 𝐶1 0

0

1

[ ] (1 + 𝜃)3 𝜃𝑥 𝑣𝑡 + 𝜃𝑢𝑥 𝑣𝑡 + 𝑣𝑥3 𝑑𝑥𝑑𝑠

∫ ∫ 𝑡∫ 1 [ ] 1 𝑡 (1 + 𝜃)6 𝜃𝑥2 + 𝜃2 𝑢2𝑥 + ∣𝑣𝑥 ∣3 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + ∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 + 𝐶1 2 0 0 0 ∫ 𝑡 ∫ 𝑡 1 ∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 + 𝐶1 (1 + 𝐴)𝑞1 𝑉 (𝑠) 𝑑𝑠 ≤ 𝐶1 + 2 0 0 ∫ 𝑡 ∥𝑣𝑥 (𝑠)∥5/2 ∥𝑣𝑥𝑥 (𝑠)∥1/2 𝑑𝑠 + 𝐶1 𝐴𝑞0 + 𝐶1 0 ∫ 1 𝑡 𝑞0 ∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴 + 𝐶1 𝐴𝑞1 + 2 0 (∫ 𝑡 )3/4 (∫ 𝑡 )1/4 + 𝐶1 sup ∥𝑣𝑥 (𝑠)∥ ∥𝑣𝑥 (𝑠)∥2 𝑑𝑠 ∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠 0≤𝑠≤𝑡 𝑞1

≤ 𝐶1 + 𝐶1 𝐴

0

0

(𝑞2 +3𝑞0 )/2

+ 𝐶1 𝐴

1 + 𝜀 sup ∥𝑣𝑥 (𝑠)∥ + 2 0≤𝑠≤𝑡 2

∫

𝑡

0

∥𝑣𝑡 (𝑠)∥2 𝑑𝑠.

Thus for sufficiently small 𝜀 > 0, 2

∫

∥𝑣𝑥 (𝑡)∥ +

𝑡 0

∥𝑣𝑡 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞3

which, along with (5.2.62)–(5.2.63), yields (5.2.59)–(5.2.61).

□

Corollary 5.2.2. Under the assumptions in Theorem 5.1.1, the following estimate holds: ∫ 𝑡∫ 1 (1 + 𝜃)2𝑚 𝑣𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 (1 + 𝐴)𝑞2 , ∀𝑡 > 0, 0 ≤ 𝑚 ≤ (𝑞 + 4)/2. (5.2.64) 0

0

Proof. We easily derive from (5.2.56), (5.2.59) and (5.2.60) that ∫ 𝑡∫ 0

0

1

(1 + 𝜃)2𝑚 𝑣𝑥2 𝑑𝑥𝑑𝑠 ≤ 𝐶1

∫ 𝑡∫ 0

0

1

𝑣𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1

∫ 𝑡∫ 0

0

1

𝑉 (𝑠)𝑣𝑥2 𝑑𝑥𝑑𝑠

≤ 𝐶1 + 𝐶1 𝐴𝑞0 + 𝐶1 𝐴𝑞2 ≤ 𝐶1 + 𝐶1 𝐴𝑞2 .

(5.2.65) □


145

Lemma 5.2.8. Under the assumptions in Theorem 5.1.1, the following estimate holds: ∫ 𝑡∫ 1 ∥(𝜃 + 𝜃4 )(𝑡)∥2 + (1 + 𝜃)𝑞+3 𝜃𝑥2 (𝑥, 𝑠) 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝑞4 , ∀𝑡 > 0 (5.2.66) 0

where

{ 𝑞4 =

0

max[𝑞5 , 2𝑞0 , 𝑞0 + 1, (𝑞0 + 𝑞2 )/2] if max[𝑞5 , 2𝑞0 , 𝑞0 + 1, (𝑞0 + 𝑞2 )/2, 4] if

0 < 𝛽 ≤ 8, 𝛽>8

and 𝑞5 = max(7 − 2𝑞, 0). Proof. Multiplying (5.1.3) by 𝑒 and integrating the resulting equation over 𝑄𝑡 , we have ∫ 𝑡∫ 1 4 2 ∥𝜃 + 𝜃 ∥ + (1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠 0

≤ 𝐶1 + 𝐶1

0

∫ 𝑡∫ 0

0

∫ 𝑡∫

1

(𝑝𝑒)𝑥 𝑣 + 𝑒𝑣𝑥2 + 𝑘(𝑢, 𝜃)𝑢−1 𝜃𝑥 𝜃4 𝑢𝑥 + 𝜆𝜙𝑒 𝑑𝑥𝑑𝑠

1

[

(1 + 𝜃)7 ∣𝜃𝑥 𝑣∣ + (1 + 𝜃)5 ∣𝑢𝑥 𝑣∣ + (1 + 𝜃)𝑞+4 ∣𝜃𝑥 𝑢𝑥 ∣ ] + (1 + 𝜃) ∣𝑢𝑥 𝑣∣ + 𝑒𝑣𝑥2 + ∣𝜙𝑒∣ 𝑑𝑥𝑑𝑠. (5.2.67)

≤ 𝐶1 + 𝐶1

0

0 8

By Lemmas 5.2.1–5.2.7, we infer ∫ 𝑡∫ 1 (1 + 𝜃)7 ∣𝜃𝑥 𝑣∣𝑑𝑥𝑑𝑠 0 0 ∫ ∫ ∫ 𝑡∫ 1 𝑡 1 𝑞+3 2 (1 + 𝜃) 𝜃𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝜃)11−𝑞 𝑣 2 𝑑𝑥𝑑𝑠 ≤𝜀 ≤𝜀 ∫ 𝑡∫ 0

0

0

0

0

∫ 𝑡∫

1

0

0

(1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴𝑞5 ,

(5.2.68)

1

(1 + 𝜃)8 ∣𝑢𝑥 𝑣∣𝑑𝑥𝑑𝑠 ∫ 𝑡∫ 1 ∫ 𝑡∫ 1 (1 + 𝜃)8 𝑢2𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝜃)8 𝑣 2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 0

0

≤ 𝐶1 𝐴𝑞0

0

∫ 𝑡∫ 0

0 2𝑞0

1

0

0

(1 + 𝜃)𝑞+4 𝑢2𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 𝐴𝑞0

∫ 𝑡∫ 0

0

1

(1 + 𝜃)𝑞+4 𝑣 2 𝑑𝑥𝑑𝑠

≤ 𝐶1 + 𝐶1 𝐴 , ∫ 𝑡∫ 1 (1 + 𝜃)𝑞+4 ∣𝜃𝑥 𝑢𝑥 ∣ 𝑑𝑥𝑑𝑠 0 0 ∫ ∫ ∫ 𝑡∫ 1 𝑡 1 ≤𝜀 (1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝜃)𝑞+5 𝑢2𝑥 𝑑𝑥𝑑𝑠 0

0

≤ 𝐶1 + 𝐶1 𝐴𝑞0 +1 + 𝜀

∫ 𝑡∫ 0

1 0

0

(5.2.69)

0

(1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠,

(5.2.70)

146


∫ 𝑡∫ 0

1

0

𝑒𝑣𝑥2 𝑑𝑥𝑑𝑠

∫

≤

𝑡

0

(∫

∥𝑣𝑥 (𝑠)∥2𝐿∞ (∫

≤ 𝐶1

𝑡

1

0

∥𝑣𝑥 (𝑠)∥2 𝑑𝑠

0

) 𝑒𝑑𝑥 𝑑𝑠

)1/2 (∫ 0

𝑡

∥𝑣𝑥𝑥 (𝑠)∥2 𝑑𝑠

)1/2

≤ 𝐶1 + 𝐶1 𝐴(𝑞0 +𝑞2 )/2 , ∫ 𝑡∫ 1 ∣𝜙𝑒∣ 𝑑𝑥𝑑𝑠 0

0

∫ 𝑡∫

1

(5.2.71)

∫ 𝑡∫

1

∫ 𝑡∫

1

𝜃4 𝜙 𝑑𝑥𝑑𝑠 (∫ 1 ) ∫ 𝑡∫ 1 ∫ 𝑡 𝛽 (1 + 𝑉 (𝑠))𝜙 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝑉 (𝑠) 𝜃 𝑑𝑥 𝑑𝑠 ≤ 𝐶1 + 𝐶1 ≤ 𝐶1

0

4

(1 + 𝜃 )𝜙 𝑑𝑥𝑑𝑠 ≤ 𝐶1

0

0

∫ ≤ 𝐶1 + 𝐶1

0

𝑡

(∫

𝑉 (𝑠)

0

1

0

) 𝜃8 𝑑𝑥 𝑑𝑠

0

0

𝜙 𝑑𝑥𝑑𝑠 + 𝐶1

0

0

0

0

for 0 < 𝛽 ≤ 8

(5.2.72)

or ∫ 𝑡∫ 0

1

∫ 𝑡∫

∣𝜙𝑒∣ 𝑑𝑥𝑑𝑠 ≤ sup ∥𝑒(𝑠)∥𝐿∞ 0≤𝑠≤𝑡

0

≤ 𝐶1 + 𝐶1 𝐴4

0

1

0

𝜙 𝑑𝑥𝑑𝑠

for 𝛽 > 8.

(5.2.73)

Thus for 0 < 𝛽 ≤ 8, we infer from (5.2.67)–(5.2.72) that ∫ 0

1

𝜃8 𝑑𝑥 + ∫

≤ 2𝜀

∫ 𝑡∫

0 𝑡∫ 1

0

0

1 0

(1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠

(1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴𝑞5 + 𝐶1 𝐴2𝑞0

+ 𝐶1 𝐴𝑞0 +1 + 𝐶1 𝐴(𝑞0 +𝑞2 )/2 + 𝐶1

∫ 0

𝑡

∫ 𝑉 (𝑠)

1 0

𝜃8 𝑑𝑥𝑑𝑠

which, by taking 𝜀 > 0 sufficiently small and using the Gronwall inequality, gives (5.2.66). While for 𝛽 > 8, taking 𝜀 > 0 small enough, we infer from (5.2.67)– (5.2.71) and (5.2.73) that ∫ 0

1

8

𝜃 𝑑𝑥 +

∫ 𝑡∫ 0

0

1

(1 + 𝜃)𝑞+3 𝜃𝑥2 𝑑𝑥𝑑𝑠

≤ 𝐶1 + 𝐶1 𝐴𝑞5 + 𝐶1 𝐴2𝑞0 + 𝐶1 𝐴𝑞0 +1 + 𝐶1 𝐴(𝑞0 +𝑞2 )/2 + 𝐶1 𝐴4 which also implies the desired estimate (5.2.66).

□


147

Lemma 5.2.9. Under the assumptions in Theorem 5.1.1, the following estimate holds: ∫ 𝑡∫ 1 ∫ 1 (1 + 𝜃)2𝑞 𝜃𝑥2 𝑑𝑥 + (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 ≤ 𝐶1 (1 + 𝐴)𝑞14 , ∀𝑡 > 0 (5.2.74) 0

0

0

where

{ 𝑞14 = max 𝑞0 + 𝑞6 , 𝑞8 , 1 + 𝑞2 , 𝑞9 + 𝑞2 , 1 + 3(𝑞0 + 𝑞2 )/4,

} (𝑞4 + 𝑞10 + 𝑞2 )/2, (𝑞4 + 𝑞10 )/2 + (3𝑞0 + 𝑞2 )/4, 𝑞12 , 𝑞13 ,

𝑞6 = max(1 − 2𝑞, 0), 𝑞7 = max((𝑞 − 10)/2, 0), 𝑞8 = 𝑞7 + 2𝑞2 , 𝑞9 = max(1 − 𝑞, 0), ′ 𝑞10 = max(𝑞 − 1, 0), 𝑞11 = max(𝑞 − 3, 0), 𝑞11 = max(3 − 𝑞, 0), { ′ , 𝑞0 + (𝑞11 + 𝑞4 )/2 + 𝛽/2, 𝑞12 = max 2𝑞0 + 𝑞11 + 𝑞4 + 𝑞11 } 𝑞0 + (𝑞11 + 𝑞4 )/2 + (3𝑞2 + 𝑞0 )/4 ,

𝑞13 = max[𝑞11 + 𝛽, (𝛽 + 𝑞2 )/2]. Proof. Let ∫ 𝐻(𝑥, 𝑡) = 𝐻(𝑢, 𝜃) =

𝜃

0

𝜅1 𝜃 𝜅2 𝜃𝑞+1 𝑘(𝑢, 𝜉) 𝑑𝜉 = + . 𝑢 𝑢 𝑞+1

Then it is easy to verify that 𝑘(𝑢, 𝜃)𝜃𝑡 , 𝐻𝑡 = 𝐻𝑢 𝑣𝑥 + 𝑢 [ ( ) ] 𝑘𝜃𝑥 𝑘 𝐻𝑥𝑡 = + 𝐻𝑢 𝑣𝑥𝑥 + 𝐻𝑢𝑢 𝑣𝑥 𝑢𝑥 + 𝑢 𝑥 𝜃𝑡 , 𝑢 𝑡 𝑢 𝑢 ∣𝐻𝑢 ∣ + ∣𝐻𝑢𝑢 ∣ ≤ 𝐶1 (1 + 𝜃).

(5.2.75)

We rewrite (5.1.3) as 𝑣2 𝑒𝜃 𝜃𝑡 + 𝜃𝑝𝜃 𝑣𝑥 − 𝜇 𝑥 = 𝑢

(


) 𝑥

+ 𝜆𝜙(𝜃, 𝑍).

(5.2.76)

Multiplying (5.2.76) by 𝐻𝑡 and integrating the resulting equation over 𝑄𝑡 , we obtain ) ∫ 𝑡∫ 1( ∫ 𝑡∫ 1 𝑘(𝑢, 𝜃) 𝑣2 𝜃𝑥 𝐻𝑡𝑥 𝑑𝑥𝑑𝑠 𝑒𝜃 𝜃𝑡 + 𝜃𝑝𝜃 𝑣𝑥 − 𝜇 𝑥 𝐻𝑡 𝑑𝑥𝑑𝑠 + 𝑢 𝑢 0 0 0 0 ∫ 𝑡∫ 1 𝜆𝜙(𝜃, 𝑍)𝐻𝑡 𝑑𝑥𝑑𝑠. (5.2.77) = 0

0

148


Now we estimate each term in (5.2.77) by using Lemmas 5.2.1–5.2.8 and Corollaries 5.2.1–5.2.2. We have first ∫ 𝑡∫ 1 ∫ 𝑡∫ 1 𝑒𝜃 𝜃𝑡2 𝑘(𝑢, 𝜃) 𝑑𝑥𝑑𝑠 ≥ 𝐶0 (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠. (5.2.78) 𝑢 0 0 0 0 Now, ∫ 𝑡 ∫ 0

∫ 𝑡∫ 1 𝑒𝜃 𝜃𝑡 𝐻𝑢 𝑣𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶0 (1 + 𝜃)4 ∣𝑣𝑥 𝜃𝑡 ∣ 𝑑𝑥𝑑𝑠

1

0

𝐶0 ≤ 8 𝐶0 ≤ 8 𝐶0 ≤ 8

∫ 𝑡∫ 0

1

0

∫ 𝑡∫ 0

0

0

0

∫ 𝑡∫

1

1

0

(1 +

𝜃)𝑞+3 𝜃𝑡2

(1 +

𝜃)𝑞+3 𝜃𝑡2

0

𝑑𝑥𝑑𝑠 + 𝐶1

∫ 𝑡∫ 0

1

0

(1 + 𝜃)5−𝑞 𝑣𝑥2 𝑑𝑥𝑑𝑠 𝑞6

𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴)

∫ 𝑡∫ 0

0

1

(1 + 𝜃)𝑞+4 𝑣𝑥2 𝑑𝑥𝑑𝑠

(1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴)𝑞6 +𝑞0 .

(5.2.79)

Next, ) ∫ 𝑡 ∫ 1 ( 𝑣 2 𝑘𝜃𝑡 𝑑𝑥𝑑𝑠 𝜃𝑝𝜃 𝑣𝑥 − 𝜇 𝑥 𝑢 𝑢 0 0 ∫ 𝑡∫ 1 [ ] ≤ 𝐶1 (1 + 𝜃)𝑞+4 ∣𝑣𝑥 𝜃𝑡 ∣ + (1 + 𝜃)𝑞 ∣𝜃𝑡 ∣𝑣𝑥2 𝑑𝑥𝑑𝑠 0

𝐶0 ≤ 8 𝐶0 ≤ 8

0

0

1

1

(1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1

0

0

0

0 𝑡∫ 1

∫ 𝑡∫

+ 𝐶1 But ∫ 𝑡∫

0

∫ 𝑡∫

∫ 0

1

∫ 𝑡∫ 0

1[

0

] (1 + 𝜃)𝑞+5 𝑣𝑥2 + (1 + 𝜃)𝑞−3 𝑣𝑥4 𝑑𝑥𝑑𝑠

(1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴)(1+𝑞2 )

0

(1 + 𝜃)𝑞−3 𝑣𝑥4 𝑑𝑥𝑑𝑠.

(5.2.80)

(1 + 𝜃)𝑞−3 𝑣𝑥4 𝑑𝑥𝑑𝑠 𝑞7

≤ 𝐶1 (1 + 𝐴)

≤ 𝐶1 (1 + 𝐴)𝑞7 ≤ 𝐶1 (1 + 𝐴)𝑞7

∫ 𝑡∫ 0

1 0

(1 + 𝜃)(𝑞+4)/2 𝑣𝑥4 𝑑𝑥𝑑𝑠

(∫ 𝑡 ∫ (∫

0

0

𝑡

0

1

𝑣𝑥4 𝑑𝑥𝑑𝑠 +

∫ 0

𝑡

∫

1

𝑉 2 (𝑠)

∥𝑣𝑥 (𝑠)∥3 ∥𝑣𝑥𝑥 (𝑠)∥ 𝑑𝑠 +

∫ 0

1

0 𝑡

𝑣𝑥4 𝑑𝑥𝑑𝑠 1

)

𝑉 2 (𝑠)∥𝑣𝑥 (𝑠)∥3 ∥𝑣𝑥𝑥 (𝑠)∥ 𝑑𝑠

)


149

{ 𝑞7

sup ∥𝑣𝑥 (𝑠)∥

≤ 𝐶1 (1 + 𝐴)

2

(∫

0≤𝑠≤𝑡

+ sup ∥𝑣𝑥 (𝑠)∥

3

(∫

0≤𝑠≤𝑡

𝑡

0

𝑡

0

𝑉 (𝑠) 𝑑𝑠

2


) 12 (∫

𝑡

0

) 12 (∫ 0

2

∥𝑣𝑥𝑥 (𝑠)∥ 𝑑𝑠

𝑡

2


) 12

) 12 }

≤ 𝐶1 + 𝐶1 𝐴𝑞7 +(3𝑞2 +𝑞0 )/2 + 𝐶1 𝐴2𝑞2 +𝑞7 ≤ 𝐶1 + 𝐶1 𝐴2𝑞2 +𝑞7 ≤ 𝐶1 + 𝐶1 𝐴𝑞8 which, along with (5.2.80), gives ( ) 𝑣𝑥2 𝑘𝜃𝑡 𝑑𝑥𝑑𝑠 𝜃𝑝𝜃 𝑣𝑥 − 𝜇 𝑢 𝑢 0 0 ∫ ∫ 𝐶0 𝑡 1 (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴𝑞8 + 𝐶1 𝐴1+𝑞2 . ≤ 8 0 0

∫ 𝑡∫

1

(5.2.81)

It follows from Lemma 5.2.7 and Corollary 5.2.2 that ∫ 𝑡 ∫ 1 𝑣2 (𝜃𝑝𝜃 𝑣𝑥 − 𝜇 𝑥 )𝐻𝑢 𝑣𝑥 𝑑𝑥𝑑𝑠 𝑢 0 0 ∫ 𝑡∫ 1 [ ] ≤ 𝐶1 (1 + 𝜃)5 𝑣𝑥2 + (1 + 𝜃)𝑣𝑥3 𝑑𝑥𝑑𝑠 0

0

𝑞9

≤ 𝐶1 (1 + 𝐴)

∫ 𝑡∫ 0

𝑞9 +𝑞2

≤ 𝐶1 (1 + 𝐴)

0

1

(1 + 𝜃)𝑞+4 𝑣𝑥2 𝑑𝑥𝑑𝑠 + 𝐶1

+ 𝐶1 (1 + 𝐴)

≤ 𝐶1 (1 + 𝐴)𝑞9 +𝑞2 + 𝐶1 (1 + 𝐴) 𝑞9 +𝑞2

≤ 𝐶1 (1 + 𝐴)

∫ 𝑡∫ 0

∫

0

𝑡

1

0

∫ 𝑡∫ 0

1

0

(1 + 𝜃)∣𝑣𝑥3 ∣𝑑𝑥𝑑𝑠

∣𝑣𝑥 ∣3 𝑑𝑥𝑑𝑠 5

1

∥𝑣𝑥 (𝑠)∥ 2 ∥𝑣𝑥𝑥 (𝑠)∥ 2 𝑑𝑠 (∫

+ 𝐶1 (1 + 𝐴) sup ∥𝑣𝑥 (𝑠)∥ 0≤𝑠≤𝑡

0

𝑡

2


) 34 (∫ 0

≤ 𝐶1 + 𝐶1 𝐴𝑞9 +𝑞2 + 𝐶1 𝐴1+3(𝑞0 +𝑞2 )/4 .

𝑡

2


) 14

(5.2.82)

Let us consider now the various contributions in the second integral of (5.2.77). By (5.2.64) and Lemmas 5.2.1–5.2.8, we have ∫ 𝑡∫ 0

0

1

𝑘𝜃𝑥 𝑢

(

𝑘𝜃𝑥 𝑢

) 𝑡

1 𝑑𝑥𝑑𝑠 = 2

∫ 0

≥ 𝐶1−1

1

(

∫ 0

𝑘𝜃𝑥 𝑢

1

)2

𝑡 𝑑𝑥

0

(1 + 𝜃)2𝑞 𝜃𝑥2 𝑑𝑥 − 𝐶1 ,

(5.2.83)

150


∫ 𝑡∫

∫ 𝑡∫ 1 𝑘𝜃𝑥 𝐻𝑢 𝑣𝑥𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 (1 + 𝜃)𝑞+1 ∣𝜃𝑥 𝑣𝑥𝑥 ∣ 𝑑𝑥𝑑𝑠 𝑢 0 0 0 0 (∫ 𝑡 ∫ 1 ) 12 (∫ 𝑡 ∫ 1 ) 12 𝑞+3 2 𝑞−1 2 ≤ 𝐶1 (1 + 𝜃) 𝜃𝑥 𝑑𝑥𝑑𝑠 (1 + 𝜃) 𝑣𝑥𝑥 𝑑𝑥𝑑𝑠 1

0

0

0

0

≤ 𝐶1 (1 + 𝐴)(𝑞4 +𝑞10 +𝑞2 )/2

(5.2.84)

and ∫ 𝑡 ∫

∫ 𝑡∫ 1 𝑘𝜃𝑥 𝐻𝑢𝑢 𝑣𝑥 𝑢𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 (1 + 𝜃)𝑞+1 ∣𝜃𝑥 𝑢𝑥 𝑣𝑥 ∣ 𝑑𝑥𝑑𝑠 𝑢 0 0 0 0 ) 12 (∫ 𝑡 ∫ 1 ) 12 (∫ 𝑡 ∫ 1 𝑞+3 2 𝑞−1 2 2 (1 + 𝜃) 𝜃𝑥 𝑑𝑥𝑑𝑠 (1 + 𝜃) 𝑣𝑥 𝑢𝑥 𝑑𝑥𝑑𝑠 ≤ 𝐶1 1

0

0

(𝑞4 +𝑞10 )/2

(∫

≤ 𝐶1 (1 + 𝐴)

0

𝑡

0

∥𝑣𝑥 (𝑠)∥2𝐿∞ ∥𝑢𝑥(𝑠)∥2

0 (𝑞4 +𝑞10 )/2+(3𝑞0 +𝑞2 )/4

≤ 𝐶1 (1 + 𝐴)

𝑑𝑠

) 12

.

(5.2.85)

Noting the following facts ∫ 𝑡∫ 0

1

∫

𝑣𝑥4 𝑑𝑥𝑑𝑠

0

≤ 𝐶1

𝑡

0

≤ 𝐶1 sup ∥𝑣𝑥 (𝑠)∥ 0≤𝑠≤𝑡

2

∥𝑣𝑥 (𝑠)∥3 ∥𝑣𝑥𝑥 (𝑠)∥ 𝑑𝑠

(∫ 0

𝑡

2


≤ 𝐶1 (1 + 𝐴)(3𝑞2 +𝑞0 )/2 , ∫ 𝑡∫ 1 ∫ 𝑡∫ 2 𝛽 𝜙 𝑑𝑥𝑑𝑠 ≤ 𝐶1 𝐴 0

0

0

1

0

) 12 (∫ 0

𝑡

2


) 12

𝑍𝜙 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 𝐶1 𝐴𝛽

and using equation (5.1.3), we obtain ) ∫ 𝑡 ( ∫ 𝑡 𝑘(𝑢,𝜃)𝜃𝑥 2 ( ) 𝑑𝑠 ≤ 𝐶1 ∥𝑒𝜃 𝜃𝑡 ∥2 + ∥𝜃𝑝𝜃 𝑣𝑥 ∥2 + ∥𝑣𝑥2 ∥2 + ∥𝜙∥2 (𝑠)𝑑𝑠 𝑢 0 0 𝑥 ∫ 𝑡∫ 1 [ ] (1 + 𝜃)6 𝜃𝑡2 + (1 + 𝜃)8 𝑣𝑥2 + 𝑣𝑥4 + 𝑍𝜃𝛽 𝜙 𝑑𝑥𝑑𝑠 ≤ 𝐶1 0

0

′ 𝑞11

≤ 𝐶1 (1 + 𝐴)

∫ 𝑡∫ 0

1 0

+ 𝐶1 sup ∥𝑣𝑥 (𝑠)∥ 0≤𝑠≤𝑡

′

≤ 𝐶1 (1 + 𝐴)𝑞11

∫ 𝑡∫ 0

2

1 0

(1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴)𝑞0 (∫

𝑡 0

2


)1/2 (∫

𝑡 0

∫ 𝑡∫ 0

0 2

1

(1 + 𝜃)𝑞+4 𝑣𝑥2 𝑑𝑥𝑑𝑠


)1/2

+ 𝐶1 (1 + 𝐴)𝛽

(1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴(𝑞0 +3𝑞2 )/2 + 𝐶1 𝐴𝛽 . (5.2.86)


151

Thus, by the Sobolev inequality, Lemmas 5.2.5 and 5.2.8, we conclude ∫ 𝑡∫ 1 ∫ 𝑡 ∫ 1 𝑘(𝑢, 𝜃)𝜃 𝑘(𝑢, 𝜃) 𝑥 ≤ 𝐶 𝑢 𝜃 𝑑𝑥𝑑𝑠 (1 + 𝜃)𝑞 ∣𝜃𝑥 𝑢𝑥 𝜃𝑡 ∣ 𝑑𝑥𝑑𝑠 ( ) 𝑢 𝑥 𝑡 1 𝑢 𝑢 0 0 0 0 ( )2 ∫ ∫ ∫ 𝑡∫ 1 (1 + 𝜃)𝑞−3 𝑘𝜃𝑥 𝐶0 𝑡 1 𝑞+3 2 ≤ (1 + 𝜃) 𝜃𝑡 𝑑𝑥𝑑𝑠 + 𝐶1 𝑢2𝑥 𝑑𝑥𝑑𝑠 8 0 0 𝑘 2 (𝑢, 𝜃) 𝑢 0 0 ∫ ∫ ∫ 𝑡 𝑘𝜃𝑥 2 𝐶0 𝑡 1 2 ≤ (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 𝑢 ∞ ∥𝑢𝑥 ∥ 𝑑𝑠 8 0 0 0 𝐿 ( ) ∫ ∫ ∫ 𝑡 𝑘𝜃𝑥 𝑘𝜃𝑥 𝐶0 𝑡 1 𝑞+3 2 𝑞0 𝑑𝑠 ≤ (1 + 𝜃) 𝜃𝑡 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴) 𝑢 𝑢 8 0 0 0 𝑥 ∫ 𝑡∫ 1 𝐶0 (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 ≤ 8 0 0 ) 12 (∫ 2 ) 12 (∫ 𝑡 ( ) 𝑡 𝑘𝜃𝑥 2 𝑘𝜃 𝑥 𝑞0 𝑑𝑠 + 𝐶1 (1 + 𝐴) 𝑢 𝑑𝑠 𝑢 0 0 𝑥 ∫ ∫ 𝐶0 𝑡 1 ≤ (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 8 0 0 ) 12 ) 12 (∫ 𝑡 ( (∫ 𝑡 ∫ 1 ) 𝑘𝜃𝑥 2 𝑞11 /2+𝑞0 𝑞+3 2 (1 + 𝜃) 𝜃𝑥 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴) 𝑢 𝑑𝑠 0 0 0 𝑥 ∫ ∫ ′ 𝐶0 𝑡 1 ≤ (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴2𝑞0 +𝑞11 +𝑞4 +𝑞11 4 0 0 + 𝐶1 𝐴𝑞0 +𝑞11 /2+𝑞4 /2+𝛽/2 + 𝐶1 𝐴(3𝑞2 +𝑞0 )/4+𝑞0 +𝑞11 /2+𝑞4 /2 ∫ ∫ 𝐶0 𝑡 1 ≤ (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴𝑞12 . 4 0 0

(5.2.87)

The last contribution is ∫ 𝑡∫ 1 ∫ 𝑡 ∫ 1 𝑘 𝜆𝜙(𝐻𝑢 𝑣𝑥 + 𝜃𝑡 ) 𝑑𝑥𝑑𝑠 ≤ 𝐶1 [(1 + 𝜃)∣𝑣𝑥 𝜙∣ + (1 + 𝜃)𝑞 ∣𝜃𝑡 𝜙∣] 𝑑𝑥𝑑𝑠 𝑢 0 0 0 0 ∫ ∫ ∫ 𝑡∫ 1 𝐶0 𝑡 1 𝑞+3 2 (1 + 𝜃) 𝜃𝑡 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝜃)𝑞−3 𝜙2 𝑑𝑥𝑑𝑠 ≤ 8 0 0 0 0 )1/2 (∫ 𝑡 ∫ 1 )1/2 (∫ 𝑡 ∫ 1 2 2 2 + 𝐶1 𝜙 𝑑𝑥𝑑𝑠 (1 + 𝜃) 𝑣𝑥 𝑑𝑥𝑑𝑠 0

𝐶0 ≤ 8 ≤

𝐶0 8

∫ 𝑡∫ 0

0

0

0

∫ 𝑡∫

1

1

0

0

0

(1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 + 𝐶1 𝐴𝑞11 +𝛽 + 𝐶1 𝐴(𝛽+𝑞2 )/2 (1 + 𝜃)𝑞+3 𝜃𝑡2 𝑑𝑥𝑑𝑠 + 𝐶1 (1 + 𝐴)𝑞13 .

Therefore estimate (5.2.74) follows from (5.2.77)–(5.2.88).

(5.2.88) □

152


Lemma 5.2.10. Under the assumptions in Theorem 5.1.1, the following estimates hold: (5.2.89) ∥𝜃(𝑡)∥𝐿∞ ≤ 𝐶1 , ∀𝑡 > 0, ∥𝑢𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 + ∥𝑍𝑥 (𝑡)∥2 ∫ 𝑡( + ∥𝑢𝑥 ∥2 + ∥𝑣𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑍𝑥 ∥2 + ∥𝑣𝑡 ∥2 0 ) + ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝜃𝑥𝑥 ∥2 + ∥𝑍𝑡 ∥2 + ∥𝑍𝑥𝑥∥2 (𝑠)𝑑𝑠 ≤ 𝐶1 ,

(5.2.90)

∀𝑡 > 0.

Proof. By Lemmas 5.2.1–5.2.9 and the Young inequality, we infer ∫ 1 𝑞+5 𝑞+5 − 𝜃1 ≤ 𝐶1 ∣𝜃𝑞+4 𝜃𝑥 ∣ 𝑑𝑥 𝜃 0

(∫ ≤ 𝐶1

0

1

)1/2 (∫ 𝜃2𝑞 𝜃𝑥2 𝑑𝑥

0

1

)1/2 𝜃8 𝑑𝑥

≤ 𝐶1 (1 + 𝐴)(𝑞14 +𝑞4 )/2 which gives

1 𝑞+5 𝐴 + 𝐶1 (5.2.91) 2 where we have used that (𝛽, 𝑞) ∈ 𝐸 implies 𝑞14 + 𝑞4 < 2𝑞 + 10. Therefore, we derive from (5.2.91) and the Young inequality that 𝐴𝑞+5 ≤ 𝐶1 + 𝐶1 𝐴(𝑞14 +𝑞4 )/2 ≤

𝐴 ≤ 𝐶1 ,

∥𝜃(𝑡)∥𝐿∞ ≤ 𝐶1

which, together with Lemmas 5.2.5–5.2.9, gives ∫ 𝑡 ( ∥𝑢𝑥 ∥2 + ∥𝑣𝑥 ∥2 ∥𝑢𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 + 0 ) + ∥𝜃𝑥 ∥2 + +∥𝑣𝑥𝑥 ∥2 + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 (𝑠)𝑑𝑠 ≤ 𝐶1 ,

∀𝑡 > 0.

(5.2.92)

Multiplying (5.1.4) by 𝑍𝑥𝑥 , 𝑍𝑡 , respectively, and then integrating the resulting equations over [0, 1] × [0, 𝑡], using (5.2.92), Lemmas 5.2.1 and 5.2.4, we obtain ∫ 𝑡 2 ∥𝑍𝑥𝑥(𝑠)∥2 𝑑𝑠 ∥𝑍𝑥 (𝑡)∥ + 0

∫ ∫ 𝑡∫ 1 1 𝑡 ∥𝑍𝑥𝑥(𝑠)∥2 𝑑𝑠 + 𝐶1 (𝑢2𝑥 𝑍𝑥2 + 𝜙2 )(𝑥, 𝑠) 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 4 0 0 0 ∫ ∫ 𝑡 1 𝑡 ∥𝑍𝑥𝑥(𝑠)∥2 𝑑𝑠 + 𝐶1 ∥𝑍𝑥 (𝑠)∥2𝐿∞ 𝑑𝑥𝑑𝑠 ≤ 𝐶1 + 4 0 0 ∫ 1 𝑡 ∥𝑍𝑥𝑥(𝑠)∥2 𝑑𝑠, ≤ 𝐶1 + 2 0


153

i.e., ∥𝑍𝑥 (𝑡)∥2 + and ∥𝑍𝑥 (𝑡)∥2 +

∫

𝑡 0

∫

𝑡

0

∥𝑍𝑡 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1 + 𝐶1

∥𝑍𝑥𝑥 (𝑠)∥2 𝑑𝑠 ≤ 𝐶1

∫ 𝑡∫ 0

∫

𝑡

1

0

[

(5.2.93)

] 𝑣𝑥2 + 𝑍𝑥4 + 𝜙2 𝑑𝑥𝑑𝑠

∥𝑍𝑥 (𝑠)∥2𝐿∞ ∥𝑍𝑥 (𝑠)∥2 𝑑𝑠 0 ∫ 𝑡 ∫ 𝑡 2 ∥𝑍𝑥 (𝑠)∥ 𝑑𝑠 + 𝐶1 ∥𝑍𝑥𝑥(𝑠)∥2 𝑑𝑠 ≤ 𝐶1 ≤ 𝐶1 + 𝐶1 ≤ 𝐶1 + 𝐶1

0

0

which, along with (5.2.92)–(5.2.93), implies (5.2.90).

□

Remark 5.2.1. We also use the following estimate to derive 𝐴 ≤ 𝐶1 . ∫ 1 𝑞+3 − 𝜃1𝑞+3 ≤ 𝐶1 ∣𝜃𝑞+2 𝜃𝑥 ∣ 𝑑𝑥 𝜃 0

(∫ ≤ 𝐶1

1

0

)1/2 (∫ 𝜃2𝑞 𝜃𝑥2 𝑑𝑥

0

1

)1/2 𝜃4 𝑑𝑥

𝑞14 /2

≤ 𝐶1 (1 + 𝐴) which implies

𝐴𝑞+3 ≤ 𝐶1 + 𝐶1 𝐴𝑞14 /2 ≤ i.e.,

1 𝑞+3 𝐴 + 𝐶1 , 2

𝐴 ≤ 𝐶1 ,

provided that 𝑞14 < 2𝑞 + 6. However, after a lengthy calculation we easily know that the range of (𝛽, 𝑞) derived by 𝑞14 < 2𝑞 + 6 is smaller than that derived by 𝑞14 + 𝑞4 < 2𝑞 + 10. More precisely, { } (𝛽, 𝑞) ∈ 𝐷 ≡ (𝛽, 𝑞) ∈ ℝ2 : 𝛽 > 0, 𝑞 > 0, 𝑞14 < 2𝑞 + 6 = 𝐷1 ∪ (𝐷2 ∩ 𝐷3 ∩ 𝐷4 ) ⊂ 𝐸 ≡ (𝐸1 ∪ 𝐸2 ) = {(𝛽, 𝑞) ∈ ℝ2 : 𝛽 > 0, 𝑞 > 0, 𝑞14 + 𝑞4 < 2𝑞 + 10} where

} { 5 𝐷1 = (𝛽,𝑞) ∈ ℝ2 : < 𝑞,0 < 𝛽 ≤ 8 , 2 }∪ { { } 5 (𝛽,𝑞) ∈ ℝ2 : 3 < 𝑞 ≤ 4,8 < 𝛽 < 5𝑞 + 3 𝐷2 = (𝛽,𝑞) ∈ ℝ2 : < 𝑞 ≤ 3,8 < 𝛽 < 6𝑞 2 } ∪{ (𝛽,𝑞) ∈ ℝ2 : 4 < 𝑞,8 < 𝛽 < 3𝑞 + 11 ,

154


}∪ { { } 2 5 2 (𝛽,𝑞) ∈ ℝ : 3 ≤ 𝑞,8 < 𝛽 < 𝑞 + 9 , 𝐷3 = (𝛽,𝑞) ∈ ℝ : < 𝑞 ≤ 3,8 < 𝛽 < 2𝑞 + 6 2 }∪ { { } 2 5 2 (𝛽,𝑞) ∈ ℝ : 8 ≤ 𝑞,8 < 𝛽 < 4𝑞 + 2 . 𝐷4 = (𝛽,𝑞) ∈ ℝ : < 𝑞 ≤ 8,8 < 𝛽 < 5𝑞 + 4 2 Lemma 5.2.11. Under the assumptions in Theorem 5.1.1, the following estimates hold for any 𝑡 > 0: 𝑑 ∥𝑢𝑥(𝑡)∥2 ≤ 𝐶1 (∥𝑢𝑥 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 ), 𝑑𝑡 𝑑 ∥𝑣𝑥 (𝑡)∥2 ≤ 𝐶1 (∥𝜃𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2 + ∥𝑣𝑥 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 ), 𝑑𝑡 ∫ 1 𝑑 2 2 ∥𝜃𝑥 (𝑡)∥ + (1 + 𝜃)𝑞−3 𝜃𝑥𝑥 𝑑𝑥 𝑑𝑡 0 ≤ 𝐶1 (∥𝑣𝑥 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 + ∥𝜙(𝑡)∥2 ), 𝑑 ∥𝑍𝑥 (𝑡)∥2 ≤ 𝐶1 (∥𝑍𝑥 (𝑡)∥2 + ∥𝑍𝑥𝑥 (𝑡)∥2 + ∥𝜙(𝑡)∥2 ). 𝑑𝑡

(5.2.94) (5.2.95)

(5.2.96) (5.2.97)

Proof. Differentiating (5.1.1) with respect to 𝑥 and multiplying the resulting equation by 𝑢𝑥 , we get estimate (5.2.94). Similarly, multiplying (5.1.2) and (5.1.4) by 𝑣𝑥𝑥 , 𝑍𝑥𝑥 , respectively, and then integrating the resulting equation on [0, 1], we obtain estimates (5.2.95) and (5.2.97). Multiplying (5.2.76) by 𝑒−1 𝜃 𝜃𝑥𝑥 and integrating the resulting equation on [0,1], we deduce ) ∫ 1 2 ∫ 1( 𝜃𝑝𝜃 𝑣𝑥 𝑘𝜃𝑥𝑥 𝑑 𝑣2 𝑘𝑥 𝜃𝑥 𝑘𝜃𝑥 𝑢𝑥 𝜃𝑥𝑥 𝜙 ∥𝜃𝑥 (𝑡)∥2 + 2 𝑑𝑥 = −𝜇 𝑥 − + + 𝜃𝑥𝑥 𝑑𝑥 𝑑𝑡 𝑒𝜃 𝑢𝑒𝜃 𝑢𝑒𝜃 𝑢2 𝑒𝜃 𝑒𝜃 0 𝑢𝑒𝜃 0 ( ) ≤ 𝜀∥𝜃𝑥𝑥∥2 + 𝐶1 ∥𝑣𝑥 ∥2 + ∥𝑣𝑥 ∥4𝐿4 + ∥𝜃𝑥 ∥4𝐿4 + ∥𝑢𝑥 𝜃𝑥 ∥2 + ∥𝜙∥2 ( ) ≤ 𝜀∥𝜃𝑥𝑥∥2 + 𝐶1 ∥𝑣𝑥 ∥2 + ∥𝑣𝑥 ∥3 ∥𝑣𝑥𝑥 ∥ + ∥𝜃𝑥 ∥3 ∥𝜃𝑥𝑥 ∥ + ∥𝜃𝑥∥4 + ∥𝜃𝑥 ∥2𝐿∞ + ∥𝜙∥2 ∫ 1 2 𝑘𝜃𝑥𝑥 ≤ 2𝜀 𝑑𝑥 + 𝐶1 (∥𝑣𝑥 ∥2 + ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝜙∥2 ). 0 𝑢𝑒𝜃 Taking 𝜀 > 0 sufficiently small in the above inequality, we obtain (5.2.96).

□

Lemma 5.2.12. Under the assumptions in Theorem 5.1.1, the following estimates hold as 𝑡 → +∞: ∥𝑢(𝑡) − 𝑢 ¯∥𝐻 1 → 0, ∥𝑣(𝑡)∥ → 0, ∥𝑣(𝑡)∥𝐿∞ → 0, ¯ 𝐻 1 → 0, ∥𝜃(𝑡) − 𝜃∥ ¯ 𝐿∞ → 0, ∥𝜃𝑥 (𝑡)∥ → 0, ∥𝜃(𝑡) − 𝜃∥

(5.2.98) (5.2.99)

∥𝑍(𝑡)∥𝐻 1 → 0,

(5.2.100)

∀(𝑥, 𝑡) ∈ [0, 1] × [0, +∞).

(5.2.101)

0
0 such that for any 𝑡 ≥ 𝑡0 , 1 𝜃(𝑥, 𝑡) ≥ 𝜃¯ > 0. (5.2.104) 2 On the other hand, if we put Θ := 1𝜃 , then (5.2.76) becomes [ ] ( ) ( )2 𝑘 2𝑘Θ2𝑥 Θ2 𝑢𝑝2𝜃 𝑢𝑝𝜃 2 Θ𝑥 − +𝜇 𝑒𝜃 Θ𝑡 = + + 𝜆Θ 𝜙 𝑣𝑥 − 𝑢 4𝜇 𝑢Θ 𝑢 2𝜇Θ 𝑥 which, together with (5.2.29) and (5.2.89), implies that there exists a positive constant 𝐶1 such that ( ) 1 𝑘 Θ𝑡 ≤ Θ𝑥 + 𝐶1 . 𝑒𝜃 𝑢 𝑥 1 ˜ − Θ(𝑥, 𝑡) and a parabolic operator ℒ : = Defining Θ(𝑥, 𝑡) := 𝐶1 𝑡 + max [0,1] 𝜃0 (𝑥) (𝑘 ∂ ) ∂ 1 ∂ − ∂𝑡 + 𝑒𝜃 ∂𝑥 𝑢 ∂𝑥 , we have a system ⎧ ˜ on 𝑄𝑡0 ,  ⎨ ℒΘ ≤ 0, ˜ Θ∣𝑡=0 ≥ 0, on [0, 1],  ⎩ ˜ Θ𝑥 ∣𝑥=0,1 = 0, on [0, 𝑡0 ]. The standard comparison argument implies min

¯𝑡 (𝑥,𝑡)∈𝑄 0

˜ Θ(𝑥, 𝑡) ≥ 0

¯ 𝑡0 , which gives for any (𝑥, 𝑡) ∈ 𝑄 ( 𝜃(𝑥, 𝑡) ≥ 𝐶1 𝑡 + max

1 𝑥∈[0,1] 𝜃0 (𝑥)

Thus,

( 𝜃(𝑥, 𝑡) ≥ 𝐶1 𝑡0 + max

1 𝑥∈[0,1] 𝜃0 (𝑥)

)−1

)−1

≥ 𝐶1−1 ,

.

0 ≤ 𝑡 ≤ 𝑡0

which, together with (5.2.104) and (5.2.89), gives (5.2.101).

□

156


5.3 Exponential Stability in 𝑯 1 1 In this section, we shall establish the exponential stability of solutions in 𝐻+ . 1 Let 𝜌 = 𝑢 , and we easily get that the specific entropy

𝜂 = 𝜂(𝑢, 𝜃) = 𝜂(𝜌, 𝜃) = 𝑅 log 𝑢 + satisfies

4𝑎 3 𝑢𝜃 + 𝐶𝑣 log 𝜃 3

𝑝𝜃 ∂𝜂 𝑒𝜃 ∂𝜂 = − 2, = ∂𝜌 𝜌 ∂𝜃 𝜃

(5.3.1)

(5.3.2)

with 𝑝 = 𝑅𝜌𝜃 + 𝑎3 𝜃4 . We consider the transform 𝒜 : (𝜌, 𝜃) ∈ 𝒟𝜌,𝜃 = {(𝜌, 𝜃) : 𝜌 > 0, 𝜃 > 0} → (𝑢, 𝜂) ∈ 𝒜𝒟𝜌,𝜃 , where 𝑢 = 1/𝜌 and 𝜂 = 𝜂(1/𝜌, 𝜃). Owing to the Jacobian ) 𝑒𝜃 1 ( ∂(𝑢, 𝜂) = − 2 = − 2 𝐶𝑣 𝜃−1 + 4𝑎𝜌−1 𝜃2 < 0, ∂(𝜌, 𝜃) 𝜌 𝜃 𝜌

(5.3.3)

there is a unique inverse function (𝑢, 𝜂) ∈ 𝒜𝒟𝜌,𝜃 . Thus the function 𝑒, 𝑝 can be also regarded as the smooth functions of (𝑢, 𝜂). We write 𝑒 = 𝑒(𝑢, 𝜂) :≡ 𝑒(𝑢, 𝜃(𝑢, 𝜂)) = 𝑒(1/𝜌, 𝜃), 𝑝 = 𝑝(𝑢, 𝜂) :≡ 𝑝(𝑢, 𝜃(𝑢, 𝜂)) = 𝑝(1/𝜌, 𝜃). Let ∂𝑒 ∂𝑒 𝑣2 +𝜆𝑍+𝑒(𝑢, 𝜂)−𝑒(¯ 𝑢, 𝜂¯)− (¯ 𝑢, 𝜂¯)(𝑢−¯ 𝑢)− (¯ 𝑢, 𝜂¯)(𝜂−¯ 𝜂 ), (5.3.4) 2 ∂𝑢 ∂𝜂 ∫1 where 𝑒(𝑢, 𝜂) = 𝑒(𝑢, 𝜃) = 𝐶𝑣 𝜃 + 𝑎𝜌−1 𝜃4 , 𝑢 ¯ = 0 𝑢0 𝑑𝑥 and 𝜃¯ > 0 is determined by ℰ(𝑢, 𝑣, 𝜂, 𝑍) =

¯ = 𝑒(¯ 𝑒(¯ 𝑢, 𝜃) 𝑢, 𝜂¯) = and

∫ 0

1

(

1 2 𝑣 + 𝑒(𝑢0 , 𝜃0 ) + 𝜆𝑍0 2 0

)

𝑑𝑥

¯ 𝜂¯ = 𝜂(¯ 𝑢, 𝜃).

(5.3.5)

(5.3.6)

Lemma 5.3.1. Under assumptions of Theorem 5.1.1, there holds that, for any (𝑥, 𝑡) ∈ [0, 1] × [0, +∞), ( ) 𝑣2 + 𝜆𝑍 + 𝐶1−1 ∣𝑢 − 𝑢¯∣2 + ∣𝜂 − 𝜂¯∣2 ≤ ℰ(𝑢, 𝑣, 𝜂, 𝑍) 2 ( ) 𝑣2 + 𝜆𝑍 + 𝐶1 ∣𝑢 − 𝑢 ≤ ¯∣2 + ∣𝜂 − 𝜂¯∣2 . 2

(5.3.7)


157

Proof. The proof is similar to that of Lemmas 3.2.3 and 4.3.1, we omit the detail here. □ 1 Lemma 5.3.2. Under assumptions of Theorem 5.1.1, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ ′ ′ there are positive constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) < 𝛾0 /2 such that for any fixed 𝛾 ∈ (0, 𝛾1′ ], there holds for any 𝑡 > 0, ( 𝑒𝛾𝑡 ∥𝑣(𝑡)∥2 + ∥𝑍(𝑡)∥𝐿1 (0,1) + ∥𝑍(𝑡)∥2 + ∥𝑢(𝑡) − 𝑢¯∥2 ) + ∥𝜌(𝑡) − 𝜌¯∥2 + ∥𝜂(𝑡) − 𝜂¯∥2 + ∥𝜌𝑥 (𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2 ( ∫ 𝑡 𝑒𝛾𝑠 ∥𝑍𝑥 ∥2 + ∥𝜃𝑥 ∥2 + ∥𝑣𝑥 ∥2 + 0

∫ +

0

1

) 𝜙(𝜃, 𝑍) 𝑑𝑥 + ∥𝑢𝑥 ∥2 + ∥𝜌𝑥 ∥2 (𝑠) 𝑑𝑠 ≤ 𝐶1 .


(5.3.8) □ 1 𝐻+ ,

Lemma 5.3.3. Under assumptions of Theorem 5.1.1, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ there are positive constants 𝐶1 > 0 and 𝛾1 = 𝛾1 (𝐶1 ) ≤ 𝛾1′ such that for any fixed 𝛾 ∈ (0, 𝛾1 ], the following estimate holds: ( ) 𝑒𝛾𝑡 ∥𝑣𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2 + ∥𝑍𝑥 (𝑡)∥2 ∫ 𝑡 ( + 𝑒𝛾𝑠 ∥𝑣𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥 ∥2 + ∥𝑍𝑥𝑥 ∥2 0 ) + ∥𝑣𝑡 ∥2 + ∥𝜃𝑡 ∥2 + ∥𝑍𝑡 ∥2 (𝑠) 𝑑𝑠 ≤ 𝐶1 , ∀𝑡 > 0. (5.3.9) Proof. The proof is similar to that of Lemma 4.3.3.

□

5.4 Proof of Theorem 5.1.2 In this section, we shall sketch the proof of Theorem 5.1.2 just by giving a series of lemmas whose proofs will be omitted since similar arguments to those in Chapter 4 and [55] can be used. 2 , Lemma 5.4.1. Under assumptions of Theorem 5.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ we have, for any 𝑡 > 0, ∫ 𝑡 2 2 2 ∥𝑣𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , (5.4.1) ∥𝑣𝑥𝑥 (𝑡)∥ + ∥𝑣𝑥 (𝑡)∥𝐿∞ + ∥𝑣𝑡 (𝑡)∥ + 0 ∫ 𝑡 ∥𝜃𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 . (5.4.2) ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥 (𝑡)∥2𝐿∞ + ∥𝜃𝑡 (𝑡)∥2 + 0

Proof. We refer to Lemma 2.3.1 and Lemma 4.3.4 or see, e.g., Lemma 3.2 in [55]. □

158


2 Lemma 5.4.2. Under assumptions of Theorem 5.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ we have, for any 𝑡 > 0, ∫ 𝑡 ∥𝑢𝑥𝑥(𝑡)∥2 + ∥𝑢𝑥 (𝑡)∥2𝐿∞ + ∥𝑢𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , (5.4.3) 0 ∫ 𝑡 ∥𝑍𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶2 , (5.4.4) ∥𝑍𝑥𝑥(𝑡)∥2 + ∥𝑍𝑥 (𝑡)∥2𝐿∞ + ∥𝑍𝑡 (𝑡)∥2 + 0 ∫ 𝑡 (∥𝑣𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥 ∥2 + ∥𝑍𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 . (5.4.5) 0

Proof. We refer to the proofs of Lemmas 4.3.4–4.3.5.

□

2 Lemma 5.4.3. Under assumptions of Theorem 5.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ , ′ ′ ′ there exists a positive constant 𝛾2 = 𝛾2 (𝐶2 ) ≤ 𝛾1 such that for any fixed 𝛾 ∈ (0, 𝛾2 ], 2 to the problem (5.1.1)–(5.1.6) the generalized solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑍(𝑡)) in 𝐻+

satisfies the following estimate: ( ) 𝑒𝛾𝑡 ∥𝑣𝑡 (𝑡)∥2 + ∥𝜃𝑡 (𝑡)∥2 + ∥𝑍𝑡 (𝑡)∥2 + ∥𝑣𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑥𝑥 (𝑡)∥2 + ∥𝑍𝑥𝑥 (𝑡)∥2 ∫ 𝑡 + (∥𝑣𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 + ∥𝑍𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 > 0. (5.4.6) 0


□ 2 𝐻+ ,

Lemma 5.4.4. Under assumptions of Theorem 5.1.2, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ there exists a positive constant 𝛾2 = 𝛾2 (𝐶2 ) ≤ 𝛾2′ such that for any fixed 𝛾 ∈ (0, 𝛾2 ], 2 the generalized solution (𝑢(𝑡), 𝑣(𝑡), 𝜃(𝑡), 𝑍(𝑡)) in 𝐻+ to the problem (5.1.1)–(5.1.6) satisfies the following estimate for any 𝑡 > 0, ∫ 𝑡 𝑒𝛾𝜏 (∥𝑢𝑥𝑥 ∥2 + ∥𝑣𝑥𝑥𝑥 ∥2 + ∥𝜃𝑥𝑥𝑥∥2 + ∥𝑍𝑥𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶2 . (5.4.7) 𝑒𝛾𝑡 ∥𝑢𝑥𝑥 (𝑡)∥2 + 0

Proof. We refer to the proof of Lemma 4.3.5.

□

Proof of Theorem 5.1.2. By Lemmas 5.4.1–5.4.4 and Theorem 5.1.1, we complete the proof of Theorem 5.1.2. □


159

5.5 Proof of Theorem 5.1.3 In this section, we shall sketch the proof of Theorem 5.1.3 just by giving a series of lemmas whose proofs will be omitted since similar arguments to those in Chapter 4 and [57] can be used. 4 , Lemma 5.5.1. Under assumptions of Theorem 5.1.3, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ we have for 𝜀 small enough,

∥𝑣𝑡𝑥 (𝑥, 0)∥ + ∥𝜃𝑡𝑥 (𝑥, 0)∥ + ∥𝑍𝑡𝑥 (𝑥, 0)∥ ≤ 𝐶3 ,

(5.5.1)

∥𝑣𝑡𝑡 (𝑥, 0)∥ + ∥𝜃𝑡𝑡 (𝑥, 0)∥ + ∥𝑍𝑡𝑡 (𝑥, 0)∥ + ∥𝑣𝑡𝑥𝑥 (𝑥, 0)∥ (5.5.2) + ∥𝜃𝑡𝑥𝑥 (𝑥, 0)∥ + ∥𝑍𝑡𝑥𝑥 (𝑥, 0)∥ ≤ 𝐶4 , ∫ 𝑡 ∫ 𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, ∀𝑡 > 0, (5.5.3) 0 0 ∫ 𝑡 ∫ 𝑡 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀−1 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ∥𝜃𝑡𝑡 (𝑡)∥2 + 0 0 ∫ 𝑡 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, ∀𝑡 > 0, + 𝐶1 𝜀 0

∥𝑍𝑡𝑡 (𝑡)∥2 +

∫

𝑡

0

∥𝑍𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶4

∫ 0

𝑡

(5.5.4) (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏,


∀𝑡 > 0. (5.5.5) □

4 Lemma 5.5.2. Under assumptions of Theorem 5.1.3, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ , the following estimates hold for any 𝑡 > 0 and for 𝜀 ∈ (0, 1) small enough, 2

∫

𝑡

∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ∫ 𝑡 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, ≤ 𝐶4 + 𝐶1 𝜀2 0 ∫ 𝑡 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ∥𝜃𝑡𝑥 (𝑡)∥2 + 0 ∫ 𝑡 (∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥 ∥2 ∥𝜃𝑥𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏, ≤ 𝐶4 + 𝐶2 𝜀2 0 ∫ 𝑡 ∥𝑍𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 . ∥𝑍𝑡𝑥 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥ +

0

0


(5.5.6)

(5.5.7) (5.5.8) □

160


4 Lemma 5.5.3. Under assumptions of Theorem 5.1.3, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ , we have for any 𝑡 > 0,

∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑍𝑡𝑡 (𝑡)∥2 + ∥𝑍𝑡𝑥 (𝑡)∥2 ∫ 𝑡( ∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 + 0 ) + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑡𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 ,

(5.5.9)

∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝑍𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 + ∥𝑍𝑡𝑥𝑥(𝑡)∥2 ∫ 𝑡( ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑍𝑡𝑡 ∥2 + 0 ) + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝑍𝑡𝑥𝑥 ∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , (5.5.10) ∫ 𝑡( ) ∥𝑣𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝑍𝑥𝑥𝑥𝑥∥2𝐻 1 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (5.5.11) 0


□

4 Lemma 5.5.4. Under assumptions of Theorem 5.1.3, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ , (1) (1) there exists a positive constant 𝛾4 = 𝛾4 (𝐶4 ) ≤ 𝛾2 (𝐶2 ) such that for any fixed (1) 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0 and 𝜀 ∈ (0, 1) small enough: ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏, (5.5.12) 𝑒𝛾𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + 0

𝑒𝛾𝑡 ∥𝜃𝑡𝑡 (𝑡)∥2 +

∫

𝑡

0

𝑒𝛾𝜏 ∥𝜃𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ∫ 𝑡 𝑒𝛾𝜏 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏 ≤ 𝐶4 𝜀−3 + 𝐶2 𝜀−1 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝑣𝑡𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏, (5.5.13) +𝜀 0 ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑍𝑡𝑡𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝑒𝛾𝜏 (∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏. 𝑒𝛾𝑡 ∥𝑍𝑡𝑡 (𝑡)∥2 + 0

0

0

(5.5.14)


□

4 Lemma 5.5.5. Under assumptions of Theorem 5.1.3, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ , (2) (1) (2) there exists a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0 and 𝜀 ∈ (0, 1) small enough: ∫ 𝑡 ∫ 𝑡 𝑒𝛾𝜏 ∥𝑣𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝐶2 𝜀2 𝑒𝛾𝜏 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, 𝑒𝛾𝑡 ∥𝑣𝑡𝑥 (𝑡)∥2 + 0

0

(5.5.15)



𝑒𝛾𝑡 ∥𝑍𝑡𝑥 (𝑡)∥2 +

∫

𝑡

0

∫

𝑒𝛾𝜏 ∥𝜃𝑡𝑥𝑥 (𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 + 𝜀2

161

∫ 0

𝑡

𝑒𝛾𝜏 (∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 )(𝜏 ) 𝑑𝜏, (5.5.16)

𝑡

𝑒𝛾𝜏 ∥𝑍𝑡𝑥𝑥(𝜏 )∥2 𝑑𝜏 ≤ 𝐶4 , (5.5.17) ∫ 𝑡 𝑒𝛾𝑡 (∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑍𝑡𝑥 (𝑡)∥2 ) + 𝑒𝛾𝜏 (∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥∥2 )(𝜏 ) 𝑑𝜏 0 ∫ 𝑡 𝑒𝛾𝜏 (∥𝑣𝑡𝑡𝑥 ∥2 + ∥𝜃𝑡𝑡𝑥 ∥2 )(𝜏 ) 𝑑𝜏. (5.5.18) ≤ 𝐶4 + 𝐶2 𝜀2 0

0


□

4 Lemma 5.5.6. Under assumptions of Theorem 5.1.3, for any (𝑢0 , 𝑣0 , 𝜃0 , 𝑍0 ) ∈ 𝐻+ , (2) there is a positive constant 𝛾4 ≤ 𝛾4 such that for any fixed 𝛾 ∈ (0, 𝛾4 ], the following estimates hold for any 𝑡 > 0, ( ) 𝑒𝛾𝑡 ∥𝑣𝑡𝑡 (𝑡)∥2 + ∥𝜃𝑡𝑡 (𝑡)∥2 + ∥𝑍𝑡𝑡 (𝑡)∥2 + ∥𝑣𝑡𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥 (𝑡)∥2 + ∥𝑍𝑡𝑥 (𝑡)∥2 ∫ 𝑡 ( ) + 𝑒𝛾𝜏 ∥𝑣𝑡𝑡𝑥 ∥2 ∥𝜃𝑡𝑡𝑥 ∥2 + ∥𝑍𝑡𝑡𝑥 ∥2 + ∥𝑣𝑡𝑥𝑥 ∥2 + ∥𝜃𝑡𝑥𝑥 ∥2 + ∥𝑍𝑡𝑥𝑥 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 , 0

∫

(5.5.19) 𝑡

𝑒𝛾𝑡 ∥𝑢𝑥𝑥𝑥(𝑡)∥2𝐻 1 + 𝑒𝛾𝜏 ∥𝑢𝑥𝑥𝑥(𝜏 )∥2𝐻 1 𝑑𝜏 ≤ 𝐶4 , 0 ( 𝑒𝛾𝑡 ∥𝑣𝑥𝑥𝑥 (𝑡)∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥(𝑡)∥2𝐻 1 + ∥𝑍𝑥𝑥𝑥(𝑡)∥2𝐻 1 ∫ +

0

+ ∥𝑣𝑡𝑥𝑥 (𝑡)∥2 + ∥𝜃𝑡𝑥𝑥 (𝑡)∥2 + ∥𝑍𝑡𝑥𝑥 (𝑡)∥2 𝑡

(5.5.20) )

( 𝑒𝛾𝜏 ∥𝑣𝑥𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑥𝑥𝑥𝑥 ∥2𝐻 1 + ∥𝑍𝑥𝑥𝑥𝑥∥2𝐻 1 + ∥𝑣𝑡𝑥𝑥 ∥2𝐻 1 + ∥𝜃𝑡𝑥𝑥 ∥2𝐻 1 ) + ∥𝑍𝑡𝑥𝑥∥2𝐻 1 + ∥𝑣𝑡𝑡 ∥2 + ∥𝜃𝑡𝑡 ∥2 + ∥𝑍𝑡𝑡 ∥2 (𝜏 ) 𝑑𝜏 ≤ 𝐶4 . (5.5.21)


□

Proof of Theorem 5.1.3. By Lemmas 5.5.1–5.5.6 and Theorems 5.1.1–5.1.2, we complete the proof of Theorem 5.1.3. □

5.6 Bibliographic Comments Radiation hydrodynamics (see, e.g., Mihalas and Weibel-Mihalas [44], Pomraning [48], Williams [77]) describes the propagation of thermal radiation through a fluid or gas. Similarly to ordinary fluid mechanics, the equations of motion are derived from conservation laws for macroscopic quantities. However, when radiation is

162


present, the classical “material” flow has to be coupled with the radiation which is an assembly of photons and needs a priori a relativistic treatment (the photons are massless particles travelling at the speed of light). The whole problem under consideration when the matter is in local thermodynamical equilibrium (LTE) is thus a coupling between standard hydrodynamics for the matter and a radiative transfer equation for the photon distribution, through a suitable description, such as in plasma when the radiation is LTE with matter and velocities are not too large, a non-relativistic one temperature description is possible [44, 77]. Moreover, if the matter is extremely radiative opaque, so that the matter free-path of photons is much smaller than the typical length of the flow, we obtain a simplified description (radiation hydrodynamics in the diffusion limit) which amounts to solving a standard hydrodynamical (compressible Navier-Stokes) problem system with additional correction terms in the pressure, the internal energy and the thermal conduction. To describe richer physical processes, for simplicity we may consider the fluid as reactive and couple the dynamics with the first-order chemical kinetics of combustion type, namely the one-order Arrhenius kinetics. Although it is simplified, this model can be proved to model correctly some astrophysical situations of interest such as stellar evolution or interstellar medium dynamics (see, e.g., Chen [3], Ducomet [11]). For a viscous heat-conducting real gas with (5.1.10) and (5.1.14)–(5.1.16) without radiative effect (𝑍 ≡ 0, 𝜙(𝜃, 𝑍) ≡ 0), there are many results on the global existence and large-time behavior of solutions; we refer to Hsiao and Luo [24], Luo [39], Matsumara and Nishida [43], Kawohl [32], Jiang [26, 27] and Qin [49, 50, 51, 52, 54, 55, 57, 59], etc. In recent years, heat-conducting radiative viscous gas has drawn the attention of a number of mathematicians (see, e.g., Donatelli and Triosia [10], Ducomet [11, 12], Ducomet, Feireisl, Petzeltov and Stra˜skraba [13], Ducomet and Feireisl [14, 15], Ducomet and Zlotnik [16, 17, 18, 19], Secchi [74], Umehara and Tani [75]). Among them, we would like to mention the work by Ducomet [12] with constitutive relations (5.1.14)–(5.1.16) in which the global existence and exponential decay in 𝐻 1 of smooth solutions to the 1D viscous reactive and radiative gas for problem (5.1.1)–(5.1.6) were established. However, we should note that there indeed exist some defects and even mistakes in the argument in [12], for example, in Lemmas 2-3, 13, etc. More precisely, first, in Lemma 2 (concerning the representation of the ∫1 specific volume 𝑢), the term 𝑢 ¯0 = 0 𝑢0 (𝑥)𝑑𝑥 in 𝐷(𝑥, 𝑡) and 𝐵(𝑡) was missed (cf., Lemma 5.2.3 in this chapter); second, in the derivation of (19) in Lemma 3 in [12], ∫1 we only know the estimate 0 𝑢𝜃4 𝑑𝑥 ≤ 𝐶 = const . > 0 due to Lemma 5.2.1; the ∫1 author used the wrong estimate 0 𝜃4 𝑑𝑥 ≤ 𝐶 = const . > 0, so the range of 𝑞 (2𝑟− 𝑞 = 4) in the derivation of (19) in [12] is indeed wrong; third, in the proof of positive lower and upper uniform bounds of the specific volume 𝑢 on page 1340 in Lemma 13 in [12], the wrong estimate 𝑢𝑝 ≤ 𝐶(1 + 𝜃4 ) was used, while the correct estimate should be 𝑢𝑝 ≤ 𝐶(𝜃+𝑢𝜃4 ), this estimate is very crucial for the proof of the positive upper bound of 𝑢 (see, e.g., Lemma 5.2.4); fourth, the proof in [12] of Theorem 3,


163

which describes the exponential decay of global solutions in 𝐻 1 for a large time, was not given, in fact its detailed proof does not merely follow as the author indicated in [12]; for a 1𝐷 viscous heat-conducting real gas, some new techniques are needed. Just for the above reasons, it is necessary for us to rephrase first the detailed proof of the positive lower and upper bounds of the specific volume 𝑢 in Lemma 5.2.4, and then to give a detailed proof of Theorem 5.1.1. To prove Lemma 5.2.4, we have to modify the representation of the specific volume 𝑢 in Lemma 5.2.3 (in fact, the proof in [12] has some defects, we have to rephrase Lemma 5.2.3 by introducing a parameter 𝛿 ≥ 0). On the other hand, Theorem 5.1.1 has improved Theorem 3 in [12] which only stated that the global solutions exponentially decay for a large time, that is, our results in Theorem 5.1.1 indicate the global solutions are exponentially stable for all 𝑡 > 0 (not merely for a large time). Moreover, the range of (𝛽, 𝑞) ∈ {𝑞 ≥ 4, 𝛽 > 0} was considered in [12], while the larger range of (𝛽, 𝑞) ∈ 𝐸 (see Theorem 5.1.1) is considered in this chapter. We should also note that the exponential decay of ∥𝑍∥ (see (5.3.8)) plays a crucial role in establishing exponential stability of solutions in 𝐻 1 (see the proof of Lemma 5.3.2). In this chapter, we first correct some defects in [12] and establish the existence of global solution in 𝐻 𝑖 (𝑖 = 1, 2, 4) for problem (5.1.1)–(5.1.6) with constitutive relations (5.1.7)–(5.1.8). Note that the ranges of exponents 𝑞 and 𝑟 in Qin [51, 52, 54] cover those of 𝑞 and 𝑟 = 3 in (5.1.7)–(5.1.8); however, 𝑝(𝑢, 𝜃), 𝑒(𝑢, 𝜃) and 𝑘(𝑢, 𝜃) in (5.1.7)–(5.1.8) can not satisfy assumptions in Qin [51, 52, 54] completely. Therefore it is necessary for us to investigate problem (5.1.1)–(5.1.6). There are two difficulties to overcome in proving our results. The first arises from the higher order nonlinearities of temperature 𝜃 in 𝑝(𝑢, 𝜃), 𝑒(𝑢, 𝜃) and 𝑘(𝑢, 𝜃) in (5.1.7)–(5.1.8). To overcome this difficulty, we make use of Corollary 5.2.1 and the interpolation techniques to reduce the higher order of 𝜃. The second is that in order to study the exponential stability, we need to establish uniform estimates depending only on the initial data, but independent of any length of time. The main novelties of this chapter are as follows: (1) We have corrected some defects in [12] and improved the results in [12] for the global solutions in 𝐻 1 . (2) We bound the norms of (𝑢, 𝑣, 𝜃, 𝑍) and their derivatives in terms of an expression of the form (1 + sup0≤𝑠≤𝑡 ∥𝜃∥𝐿∞ )Λ with Λ being a positive constant only depending on 𝑞 and 𝛽. (3) We first establish the global existence and exponential stability of global solutions in 𝐻 2 and 𝐻 4 for boundary condition (5.1.5) for this model. (4) We first establish the global existence and exponential stability of classical solutions for boundary condition (5.1.5) for this model.

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Index absolute temperature, 2 annular domains, 1 classical solutions, 3 compressible heat-conducting flow, 33 conservation of mass, 94 dissipative processes, 33 energy method, 3 exponential stability, 33, 93, 127 external force, 1

shear viscosity, 33 specific heat capacity, 2 spherically symmetric solutions, 1 Stefan-Boltzmann model, 127 stress tensor, 33 symmetric motion, 2 transverse velocity, 34 uniform estimates, 32, 163 viscosity coefficients, 2, 33 viscous polytropic ideal gas, 2

global existence, 1, 33, 93, 127 heat source, 1 Helmholtz free energy function, 128 ideal gas, 128 initial boundary value problem, 1, 34, 128 kinetic energy, 33 large-time behavior , 162 longitudinal velocity, 34 Navier-Stokes equations, 33, 93 Newtonian fluid, 33 non-autonomous compressible Navier-Stokes equations, 1 radiative gas, 127 real gas, 128

171

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Integrability of Nonlinear Systems

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Nonlinear Systems

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Nonlinear Systems

Nonlinear Systems

Foundations of global nonlinear analysis

Approximation of nonlinear evolution systems

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Nonlinear systems analysis

Nonlinear systems analysis

Nonlinear Control Systems

Nonlinear Control Systems

Nonlinear Dynamical Control Systems

Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations

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Nonlinear Systems, Second Edition

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Semiclassical states for weakly coupled nonlinear Schrodinger systems

Integrability of Nonlinear Systems

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Nonlinear Systems

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Nonlinear Systems

Nonlinear Systems

Foundations of global nonlinear analysis

Approximation of nonlinear evolution systems

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Nonlinear Dynamics of Production Systems

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Nonlinear control systems 2

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