1D Radiative Fluid and Liquid Crystal Equations 9782759829040

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Current Natural Sciences

Yuming QIN

1D Radiative Fluid and Liquid Crystal Equations

Printed in France

EDP Sciences – ISBN(print): 978-2-7598-2903-3 – ISBN(ebook): 978-2-7598-2904-0 DOI: 10.1051/978-2-7598-2903-3 All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. The printed edition is not for sale in Chinese mainland. Customers in Chinese mainland please order the print book from Science Press. ISBN of the China edition: Science Press 978-7-03-072137-2 Ó Science Press, EDP Sciences, 2022

In memory of my father, Zhenrong QIN and my mother, Xilan XIA To my wife, Yu YIN, my son, Jia QIN To my elder sister, Yujuan QIN, younger brother, Yuxing QIN and younger sister, Yuzhou QIN

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII CHAPTER 1 . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

1 1 1 5 6 7 8 9 12 13 14

Asymptotic Behavior of Solutions for the One-Dimensional Infrarelativistic Model of a Compressible Viscous Gas with Radiation . . . . . . . . . . . . . . . . 2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Global Existence and Uniform-in-Time Estimates in H1 . . . . . . . . . 2.3 Asymptotic Behavior of Solutions in H1 . . . . . . . . . . . . . . . . . . . . . 2.4 Global Existence and Uniform-in-Time Estimates in H2 . . . . . . . . . 2.5 Asymptotic Behavior of Solutions in H2 . . . . . . . . . . . . . . . . . . . . . 2.6 Global Existence and Uniform-in-Time Estimates in H4 . . . . . . . . . 2.7 Asymptotic Behavior of Solutions in H4 . . . . . . . . . . . . . . . . . . . . . 2.8 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

17 17 22 48 53 60 62 81 85

. . . .

. 89 . 89 . 91 . 100

Preliminary 1.1 Some 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.1.7 1.1.8 1.1.9

...................................... . . . . . . . . . .

Basic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . The Sobolev Inequalities . . . . . . . . . . . . . . . . . The Interpolation Inequalities . . . . . . . . . . . . . The Poincaré Inequality . . . . . . . . . . . . . . . . . . The Classical Bellman–Gronwall Inequality . . . The Generalized Bellman–Gronwall Inequalities The Uniform Bellman–Gronwall Inequality . . . . The Young Inequalities . . . . . . . . . . . . . . . . . . The Hölder Inequalities . . . . . . . . . . . . . . . . . . The Minkowski Inequalities . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

CHAPTER 2

CHAPTER 3 Global Existence and Regularity of a One-Dimensional Liquid Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Existence in H 1  H01  H 2 . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

VI

Contents

3.4 3.5

Proof of Theorem 3.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

CHAPTER 4 Large-time Behavior of Solutions to a One-Dimensional Liquid Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Uniform Estimates in H i  H0i  H i þ 1 ði ¼ 1; 2Þ and H 4  H04  H 4 . 4.3 Large-time Behavior in H i  H0i  H i þ 1 ði ¼ 1; 2Þ and H 4  H04  H 4 4.4 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

111 111 113 122 134

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Foreword

In this book, we shall present some recent results on the global well-posedness of strong solutions to 1D radiative fluid equations and liquid crystal equations. Most of the contents of this book are based on the research carried out by the authors and their collaborators in recent years, which have been previously published only in original papers; but some contents of the book have never been published until now. There are four chapters in this book. Chapter 1 will recall some basic properties of Sobolev spaces, some differential integral inequalities in analysis, some of which will be used in the subsequent chapters. In chapter 2, we shall study one-dimensional compressible infrarelativistic radiation equations and further prove the global existence and the large-time behavior of solutions to this system. Novelties of this chapter are: (1) Using a suitable expression of specific volume and the delicate priori estimates, we establish the positively lower bound and upper bound of the specific volume. (2) Using the embedding theorems and the delicate interpolation inequalities, we have overcome some mathematical difficulties caused by the higher order of partial derivatives to prove the global well-posedness of solutions in higher regular spaces. It is a remarkable fact that the difficulties we encounter in chapter 2 are how to deal with the radiative term, which makes the analysis in this book different from those in Qin [104], where the author studied some models without the radiative term. Chapters 3 and 4 will study one-dimensional compressible liquid crystal fluid equations. In chapter 3, we shall establish the existence of global solutions in Hi (i = 1, 2, 4) in Lagrangian coordinates. In chapter 4, we shall first establish the large-time behavior of solutions to one-dimensional compressible liquid crystal fluid equations. The novelty in this chapter is that using a suitable expression of the specific volume, we shall establish uniform bound of the specific volume by the embedding theorems and a sequence of delicate interpolation techniques and then prove the long-time behavior of solutions to the system using the Shen–Zheng inequality. DOI: 10.1051/978-2-7598-2903-3.c901 Ó Science Press, EDP Sciences, 2022

VIII

Foreword

For the contents of chapter 1, we refer the reader to [1, 2, 5–8, 37, 38, 40, 41, 45, 55, 75, 76, 95, 96, 105–108, 135, 137, 138, 141, 148, 149, 155]. For the theory of radiation hydrodynamical equations, we refer the reader to the monographs [12, 94, 99, 100] and [10, 17–21, 44, 59, 60, 74, 79, 80, 89, 111, 112, 116, 117, 127, 128]. For the theory of equations of liquid crystal, we refer the reader to [9, 11, 16, 23–25, 56, 57, 77, 78, 81–83, 85–88, 110, 118, 124, 143, 144]. Since the compressible Navier– Stokes equations are closely related to the radiation hydrodynamical equations and the liquid crystal equations under consideration of this book, we also refer the reader to related references of the compressible Navier–Stokes equations [3, 4, 13–15, 22, 26–36, 39, 42, 43, 46–50, 52–54, 58, 61–73, 84, 91–93, 97, 98, 102–105, 113, 114, 119–123, 125, 126, 129, 130–134, 136, 139, 140, 142, 146, 147, 150–154, 156], some techniques of which can be used to deal with our problems in this book. We sincerely wish that the reader will learn the essential ideas, basic theories and methods in deriving the global existence, asymptotic behavior and regularity of solutions for the systems considered in this book. We also wish that the reader can undertake the further research of these systems after having read this book. This book was financially supported in part by the NNSF of China with contract number 12171082, the Fundamental Research Funds for the Central Universities with contract number 2232022G-13 and by the Graduate Course (Textbook) Construction Project of Donghua University. We also take this opportunity to thank all the people who were once concerned about me. Last but not least, Yuming QIN hopes 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. Professor Yuming QIN Department of Mathematics College of Science Institute for Nonlinear Sciences Research Institute of Science and Technology Donghua University Shanghai 201620, P. R. China [email protected], [email protected]

Chapter 1 Preliminary This chapter will introduce some basic results, most of which will be used in the following chapters. First we shall recall some basic inequalities whose detailed proofs can be found in the related literature, see, e.g., Adams [1, 2], Friedman [37, 38], Gagliardo [40, 41], Nirenberg [95, 96], Yosida [148], etc.

1.1 1.1.1

Some Basic Inequalities The Sobolev Inequalities

We shall first introduce some basic concepts of Sobolev spaces. Definition 1.1.1. Assume X
Rn is a bounded or an unbounded domain with a smooth boundary Γ. For 1 ≤ p ≤ +∞ and m a non-negative integer, Wm,p(Ω) is defined to be the space of functions u in Lp(Ω) whose distribution derivatives of order up to m are also in Lp(Ω). That is, W m;p ðXÞ ¼ Lp ðXÞ \ fu : Da u 2 Lp ðXÞ; jaj  mg: The space Wm,p(Ω), called a Sobolev space, is equipped with a norm 0 11=p Z X kukm;p;X ¼ @ jDa ujp dx A ; if 1  p\ þ 1; or X jaj  m

kukm;p;X ¼ max esssup jDa uðxÞj; if p ¼ þ 1 jaj  m

ð1:1:1Þ

ð1:1:2Þ

x2X

DOI: 10.1051/978-2-7598-2903-3.c001 © Science Press, EDP Sciences, 2022

1D Radiative Fluid and Liquid Crystal Equations

2

which is clearly equivalent to

X

kDa ukp;X :

ð1:1:3Þ

jaj  m

If X ¼ Rn , we only denote kukm;p ¼ kukm;p;Rn ;

kuk0;p ¼ kukp :

Wm,p(Ω) is a Banach space. The space W0m;p ðXÞ is defined as the closure of C01 ðXÞ relative to the norm (1.1.3). Clearly, W 0;p ðXÞ ¼ Lp ðXÞ : with norm ∥⋅∥0,p,Ω ≡ ∥⋅∥p,Ω. For p = 2, W m,2(Ω) ¼ Hm(Ω), is a Hilbert space with respect to the scalar product X ðu; vÞm ¼ ðDa u; D a vÞL2 ðXÞ jaj  m

R

with ðf ; gÞL2 ðXÞ ¼ X f gdx, here g is the conjugate function of g. It is well-known that the Sobolev inequalities are important tools in the study of nonlinear evolutionary equations. First, we shall introduce these inequalities for functions in the space W01;p ðXÞ. Theorem 1.1.1 (The Sobolev Inequality). Assume that X
Rn ; n [ 1; is an open domain. There exists a constant C = C(n, p) > 0 such that 

(1) if n > p ≥ 1, and u 2 W01;p ðXÞ, then u 2 Lp ðXÞ and kukp ;X 

pðn  1Þ pffiffiffi kDukp;X 2ðn  pÞ n

ð1:1:4Þ

where p = np∕(n − p); (2) if p > n and Ω is bounded, and u 2 W01;p ðXÞ, then u 2 C ðXÞ and sup juj  C jXjðnpÞ kDukp;X : 1

1

X

ð1:1:5Þ

While, if X ¼ Rn , then 1

sup juj  C xn p kuk1;p;Rn Rn

n=2

ð1:1:6Þ

2p is the measure of the n-dimensional unit ball, Γ is the Euler where xn ¼ nCðn=2Þ     ðp1Þ=p p1 gamma function and C ¼ max 1; pn .

Preliminary

3

Remark 1.1.1. The Sobolev inequality (1.1.4) does not hold for p = n, p* = +∞. (1.1.4) was first proved by Sobolev [138] in 1938. Sobolev [138] stated that the Lp* norm of u can be estimated by ∥u∥1,p,Ω or ∥Du∥p,Ω, the Sobolev norm of u. However, we can bound a higher Lp norm of u by exploiting higher order derivatives of u as shown in the next theorem which generalizes theorem 1.1.1 from m = 1, p > n to m ≥ 1 an integer. Theorem 1.1.2. Assume X
Rn is an open domain. There exists a constant C = C(n, m, p) > 0 such that 

(1) if mp < n, p ≥ 1, and u 2 W0m;p ðXÞ, then u 2 Lp ðXÞ and kukp ;X  C kukm;p;X

ð1:1:7Þ

np ; where p ¼ nmp

(2) if mp > n, and u 2 W0m;p ðXÞ, then u 2 C ðXÞ and m 1 X 1 1 0 p sup juj  C jK j ðdiamK Þjaj kDa ukp;K a! X jaj¼0 1 ðm  n=pÞ1 kDm ukp;K þ ðdiamK Þ ðm  1Þ!



m

ð1:1:8Þ

where K = suppu, C = C(m, p, n) and diamK is the diameter of K. Remark 1.1.2. An important case considered in theorems 1.1.1 and 1.1.2 is X ¼ Rn . In this situation, W m;p ðRn Þ ¼ W0m;p ðRn Þ and therefore the results of theorems 1.1.1 and 1.1.2 apply to W m;p ðRn Þ. For p > n, the results of theorems 1.1.1 and 1.1.2 imply the fact that u is bounded. Indeed, u is Höder continuous, which we shall state as follows. Theorem 1.1.3. If u 2 W01;p ðXÞ; p [ n, then u 2 C 0;a ðXÞ where α = 1 − n∕p. Generally, the embedding theorems are closely related to the smoothness of the domain considered, which means that when we study the embedding theorems, we need some smoothness conditions for the domain. These conditions include that the domain Ω possesses the cone property, and it is a uniformly regular open set in Rn , etc. For example, when Ω 2 C1 or @Ω 2 Lip, Ω has the cone property. Mathematically, we need to define the special meaning of the word “embedding” or “compact embedding”. Definition 1.1.2. Assume A and B are two subsets of some function space. Set A is said to be embedded into B if and only if (1) A
B; (2) the identity mapping I: A → B is continuous, i.e., there exists a constant C > 0 such that for any x 2 A, there holds that kIxkB  C kxkA :

4

1D Radiative Fluid and Liquid Crystal Equations If A is embedded into B, then we simply denote by A ,! B. A is said to be compactly embedded into B if and only if

(1) A is embedded into B; (2) the identity mapping I: A ↦ B is a compact operator. If A is compactly embedded into B, then we simply denote by A ,!,! B. Now we draw some consequences from theorem 1.1.1. In fact, exploiting theorem 1.1.1, we have the following result which is an embedding theorem. np Corollary 1.1.1. If u 2 W01;p ðXÞ, then u 2 Lq(Ω) with p  q  np if 1 ≤ p < n, and p ≤ q < +∞ if p = n. Moreover, if p > n, u coincides a.e. in Ω with a (uniquely determined) function of C ðXÞ. Finally, there holds that np ; ð1:1:9Þ kukq;X  C kuk1;p;X if 1  p\n; p  q  np

kukq;X  C kuk1;p;X if p ¼ n; p  q\ þ 1;

ð1:1:10Þ

kukC  C kuk1;p;X if p [ n;

ð1:1:11Þ

where C = C(n, p, q) > 0 is a constant. We can generalize corollary 1.1.1 to functions from W0m;p ðXÞ which can be stated as the following embedding theorem. Theorem 1.1.4. Let u 2 W0m;p ðXÞ; p  1; m  0. Then h i np (1) if mp < n, then we have, for all q 2 p; nmp , W0m;p ðXÞ,!Lq ðXÞ;

ð1:1:12Þ

and hthere isi a constant C1 > 0 depending only on m, p, q and n such that for all np , q 2 p; nmp kukq;X  C1 kukm;p;X ;

ð1:1:13Þ

(2) if mp = n, then we have, for all q 2 [p, +∞), W0m;p ðXÞ,!Lq ðXÞ;

ð1:1:14Þ

and there is a constant C2 > 0 depending only on m, p, q and n such that for all q 2 [p, +∞), kukq;X  C2 kukm;p;X ;

ð1:1:15Þ

Preliminary

5

(3) if mp > n, each u 2 W0m;p ðXÞ is equal a.e. in Ω to a unique function in C k ðXÞ, for all k 2 [0, m − n∕p) and there is a constant C3 > 0 depending only on m, p, q and n such that kukC k  C3 kukm;p;X :

ð1:1:16Þ

Remark 1.1.3. In case (2) of theorem 1.1.4, the following exception case holds for m = n, p = 1, q = +∞: W n;1 ðXÞ,!L1 ðXÞ:

ð1:1:17Þ

Now we give the following compact embedding theorem. Theorem 1.1.5 (Embedding and Compact Embedding Theorem). Assume that Ω is a bounded domain of class Cm. Then we have (i) If mp < n, then Wm,p(Ω) is continuously embedded in Lq*(Ω) with 

W m;p ðXÞ,!Lq ðXÞ:

1 q

¼ p1  mn: ð1:1:18Þ

In addition, the embedding is compact for any q, 1 ≤ q < q*. (ii) If mp = n, then Wm,p(Ω) is continuously embedded in Lq(Ω), 8q; 1  q\ þ 1: W m;p ðXÞ,!Lq ðXÞ:

ð1:1:19Þ

In addition, the embedding is compact, 8q; 1  q\ þ 1. If p = 1, m = n, then the above still holds for q = +∞. (iii) If k þ 1 [ m  np [ k; k 2 N, then writing m  np ¼ k þ a; a 2 ð0; 1Þ; W m;p ðXÞ is continuously embedded in C k;a ðXÞ: W m;p ðXÞ,!C k;a ðXÞ;

ð1:1:20Þ

where C k;a ðXÞ is the space of functions in C k ðXÞ whose derivatives of order k are Hölder continuous with exponent α. Moreover, if n = m − k − 1, and α = 1, p = 1, then (1.1.20) holds for α = 1, and the embedding is compact from Wm,p(Ω) to C k;b ðXÞ, for all 0 ≤ β < α.

1.1.2

The Interpolation Inequalities

In this subsection, we shall present the Gagliardo–Nirenberg interpolation inequalities (see, e.g., Friedman [38] and Nirenberg [96]) which play a very important role in the theory of nonlinear evolutionary equations.

1D Radiative Fluid and Liquid Crystal Equations

6

h i For p > 0, jujp;X ¼ kukLp ðXÞ . For p < 0, set  np ¼ h þ a with h ¼  np and a 2 ½0; 1Þ. Define jujp;X ¼ sup jDh uj  X

jujp;X ¼ ½Dh ua;X  

X

supjD b uj; if a ¼ 0;

jbj¼h X

X

sup½D b ua

jbj¼h X

X

jD b uðxÞ  Db uðyÞj ; if a [ 0: jx  yja jbj¼h x;y2X;x6¼y sup

If X ¼ Rn , we simply write |u|p instead of |u|p,Ω. Theorem 1.1.6 (The Gagliardo–Nirenberg Interpolation Inequalities). Let j, m be any integers satisfying 0 ≤ j < m, and let 1  q; r  þ 1, and p 2 R; mj  a  1 such that

1 j 1 m 1a  ¼a  : þ p n r n q Then (i) For any u 2 W m;r ðRn Þ \ Lq ðRn Þ, there is a positive constant C = C(m, n, j, q, r, α) such that jDj ujp  C jDm ujar juj1a q

ð1:1:21Þ

with the following exception: if 1 < r < +∞ and m − j − n∕p is a non-negative integer, then (1.1.21) holds only for α satisfying j=m  a\1. (ii) For any u 2 Wm,r(Ω) \ Lq(Ω) where Ω is a bounded domain with smooth boundary, there are two positive constants C1, C2 such that jD j ujp;X  C1 jDm ujar;X juj1a q;X þ C2 jujq;X

ð1:1:22Þ

with the same exception as in (i). In particular, for any u 2 W0m;p ðXÞ \ Lq ðXÞ, the constant C2 in (1.1.22) can be taken as zero.

1.1.3

The Poincaré Inequality

In this subsection, we shall recall the Poincaré inequality in different forms. Theorem 1.1.7. Let Ω be a bounded domain in Rn and u 2 H01 ðXÞ. Then there is a positive constant C = C(Ω, n) such that for all u 2 H01 ðXÞ, kukL2 ðXÞ  C krukL2 ðXÞ :

ð1:1:23Þ

Preliminary

7

Theorem 1.1.8. Let X
Rn be a bounded domain of C1. Then there is a positive constant C = C(Ω, n) such that for any u 2 H1(Ω),

1 where u ¼ jXj

ku  u kL2 ðXÞ  C krukL2 ðXÞ

R X

ð1:1:24Þ

uðxÞdx is the integral average of u over Ω, and |Ω| is the volume of Ω.

Theorem 1.1.9. Under assumptions of theorem 1.1.8, for any u 2 H 1 ðXÞ, then Z 

   ð1:1:25Þ kukL2 ðXÞ  C krukL2 ðXÞ þ  udx  : X

1.1.4

The Classical Bellman–Gronwall Inequality

In this subsection, we shall give the following classical Bellman–Gronwall inequality which plays an important role in the study of global well-posedness of solutions to evolutionary differential equations. For more details, we can refer to Bellman [5–8] and Gronwall [45]. Theorem 1.1.10 (The Classical Bellman–Gronwall Inequality). If y(t) and g(t) are non-negative, continuous functions on 0 ≤ t ≤ τ, which satisfy the inequality Z t yðtÞ  g þ gðsÞyðsÞds; 0  t  s; ð1:1:26Þ 0

where η is a non-negative constant, then for all 0 ≤ t ≤ τ, Z t

yðtÞ  g exp gðsÞds :

ð1:1:27Þ

0

Gronwall [45] first proved the special case of (1.1.26) with g(t) = constant ≥ 0. Later on, Bellman [6] (see also Kuang [75]) extended this result to the form of theorem 1.1.10, which is a crucial tool in the analysis of differential equations. Until now, more and more improvements and generalizations of the classical Bellman– Gronwall inequality have been made. Specially, Bellman proved another inequality which can be stated as follows (see, e.g., Kuang [75]). Remark 1.1.4. Let u(t), b(t) be continuous on (α, β), and b(t) be non-negative. If for all t ≥ t0, t0, t 2 (α, β), Z t uðtÞ  uðt0 Þ þ bðsÞuðsÞds; t0

then for any t ≥ t0, Z t

Z t

uðt0 Þ exp  bðsÞuðsÞds  uðtÞ  uðt0 Þ exp bðsÞuðsÞds : t0

t0

1D Radiative Fluid and Liquid Crystal Equations

8

The above theorem gives bounds on solution of (1.1.26) in terms of the solution of a related linear integral equation Z t vðtÞ ¼ g þ gðsÞvðsÞds ð1:1:28Þ 0

and is one of the basic tools in the theory of differential equations. Based on the basis of various motivations, we know that it has been extended and used considerably in various contexts. For instance, in the Picard–Cauchy type of iteration for establishing existence and uniqueness of solutions, this inequality and its various variants play a significant role. Inequalities of this type (1.1.26) are also encountered frequently in the perturbation and stability theory of differential equations.

1.1.5

The Generalized Bellman–Gronwall Inequalities

In this subsection, we shall review the following generalized Bellman–Gronwall inequalities which can be found in Qin [104, 106, 123, 128]. Theorem 1.1.11 (The Generalized Bellman–Gronwall Inequality). Assume that f(t), g(t) and y(t) are non-negative integrable functions in [τ, T ] (τ < T) verifying the following integral inequality for all t 2 [τ, T], Z t yðtÞ  gðtÞ þ f ðsÞyðsÞds: s

Then it holds, for all t 2 [τ, T], Z t

Z t yðtÞ  gðtÞ þ exp f ðhÞdh f ðsÞgðsÞds: s

ð1:1:29Þ

s

In addition, if g(t) is a nondecreasing function in [τ, T], then, for all t 2 [τ, T],  Z t

 Z t yðtÞ  gðtÞ 1 þ exp f ðhÞdh f ðsÞds ð1:1:30Þ s



Z

 gðtÞ 1 þ If further T = +∞ and

where C ¼ 1 þ

R þ1 s

R þ1 s

s t

s

Z f ðsÞds exp

s

t

 f ðhÞdh :

ð1:1:31Þ

f ðsÞds\ þ 1, then

yðtÞ  CgðtÞ R þ1 f ðsÞds expð s f ðhÞdhÞ is a positive constant.

ð1:1:32Þ

The next result is a corollary of theorem 1.1.11, it can be found in Racke [135]. Corollary 1.1.2. Let a > 0, ϕ, h 2 C([0, a]), h ≥ 0 and g : ½0; a ! R is increasing. If for any t 2 [0, a],

Preliminary

9 Z

t

/ðtÞ  gðtÞ þ

ð1:1:33Þ

hðsÞ/ðsÞds; 0

then for all t 2 [0, a],

Z /ðtÞ  gðtÞ exp

t

hðsÞds :

ð1:1:34Þ

0

1.1.6

The Uniform Bellman–Gronwall Inequality

In this subsection, we shall introduce some uniform Gronwall inequalities which provide uniform bounds or decay rates. This type of integral inequalities plays a very crucial role in the study of the global existence and the large-time behavior of solutions to evolutionary equations. We start with the following theorem which is cited in Temam [141]. Theorem 1.1.12 (The Uniform Bellman–Gronwall Inequality). Assume that g(t), h(t) and y(t) are three positive locally integrable functions on (t0, +∞) such that y 0 ðtÞ is locally integrable on (t0, +∞) and there holds that for all t ≥ t0, Z t

t þr

dy  gy þ h; dt

Z

t þr

gðsÞds  a1 ;

Z

tþr

hðsÞds  a2 ;

t

yðsÞds  a3 ;

t

where r, ai(i = 1, 2, 3) are positive constants. Then, for all t ≥ t0, a  3 þ a 2 e a1 : yðt þ rÞ  r Next, we shall introduce some uniform generalizations which may provide some large-time behavior of functions. This class of inequalities is a very powerful tool in establishing the large-time behavior of solution when we use the energy methods to study problems of partial differential equations. We now give the familiar results in the classical calculus for the single real variable analysis. Lemma 1.1.1. (1) Assume y(t) 2 L1(0, +∞) with y(t) ≥ 0 for a.e. t  0, y 0 ðtÞ 2 L1(0, +∞). Then lim yðtÞ ¼ 0:

t! þ 1

(2) Assume y(t) 2 L1(0, +∞) with y(t) ≥ 0 for a.e. t  0, and limt→+∞y(t) exists. Then lim yðtÞ ¼ 0:

t! þ 1

1D Radiative Fluid and Liquid Crystal Equations

10

(3) Assume y(t) is uniformly continuous on [0, +∞), yðtÞ 2 L1 ð0; þ 1Þ. Then lim yðtÞ ¼ 0:

t! þ 1

(4) Assume y(t) is a monotone function on [0, +∞) and yðtÞ 2 L1 ð0; þ 1Þ. Then lim yðtÞ ¼ 0

t! þ 1

and yðtÞ ¼ oð1=tÞ as t ! þ 1: Obviously, the above lemma provides the asymptotic behavior of y(t) for the large time. The next theorem related to the uniform Gronwall inequality was first established by Shen and Zheng [137] in 1993 (see, e.g., Zheng [155]) which is very useful and powerful in dealing with the global well-posedness and asymptotic behavior of solutions to some evolutionary partial differential equations. We shall apply it frequently in the subsequent context of this book (see, chapters 2–4). Lemma 1.1.2 (The Shen–Zheng Inequality). Assume T is an arbitrarily given constant with 0 < T ≤ +∞, and y and h are non-negative continuous functions defined on [0, T] and satisfy the following conditions dy  A1 y 2 ðtÞ þ A2 þ hðtÞ; for all t  0; dt Z

T

Z yðsÞds  A3 ;

0

T

hðsÞds  A4 ; for all T [ 0;

ð1:1:35Þ

ð1:1:36Þ

0

where A1, A2, A3, A4 are given non-negative constants. Then for any r > 0, with 0 < r < T, for all t ≥ 0,

A3 þ A 2 r þ A 4 e A1 A3 : ð1:1:37Þ yðt þ rÞ  r Furthermore, if T = +∞, then lim yðtÞ ¼ 0:

t! þ 1

ð1:1:38Þ

Proof. We can find the proof in [137]. However, for reader’s convenience, we shall give the detailed proof. The proof is similar to that of the Uniform Gronwall Lemma (see lemma 1.1 in [141], p. 89, or theorem 1.1.12). R s Assume 0 ≤ t ≤ s ≤ t + r with any given r > 0. We multiply (1.1.35) by expð t A1 yðsÞdsÞ and obtain the relation

Preliminary

11

 Z s

 Z s

d yðsÞ exp  A1 yðsÞds  ðA2 þ hðsÞÞ exp  A1 yðsÞds  A2 þ hðsÞ: ds t t ð1:1:39Þ Then integrating it over [s, t + r] yields Z t þ r

Z yðt þ rÞ  yðsÞ exp A1 yðsÞds þ ðA2 r þ A4 Þ exp s



tþr

A1 yðsÞds t

 ðyðsÞ þ A2 r þ A4 ÞeA1 A3 :

ð1:1:40Þ

Integrating this inequality, with respect to s between t and t + r, gives us (1.1.37). From (1.1.35) and (1.1.37), it follows 

2 dy A3  A1 þ A2 r þ A4 eA1 A3 þ A2 þ hðtÞ dt r ð1:1:41Þ ¼ Ar þ hðtÞ; for all t  r; where 

2 A3 A1 A3 Ar :¼ A1 þ A2 r þ A 4 e þ A2 : r To prove (1.1.38), we use the contradiction argument. Assume it were not true. Then there would exist a monotone increasing sequence {tn} and a constant a > 0 such that for all n 2 N, a a tn  r þ ; tn þ 1  tn þ ; ð1:1:42Þ 4Ar 4Ar lim tn ¼ þ 1;

n! þ 1

yðtn Þ 

a [ 0: 2

On the other hand, from (1.1.41) we have Z tn a yðtn Þ  yðtÞ  Ar ðtn  tÞ þ hðsÞds; as tn   t\tn : 4A r t Combining (1.1.44) and (1.1.45) yields Z a a tn a  yðtÞ  yðtn Þ  yðtÞ  hðsÞds; as tn   t\tn : a 2 4 tn 4A 4Ar r

ð1:1:43Þ ð1:1:44Þ

ð1:1:45Þ

ð1:1:46Þ

1D Radiative Fluid and Liquid Crystal Equations

12

Therefore, Z yðtÞ þ Let

tn tn 4Aa r

hðsÞds 

a a ; as tn   t\tn : 4 4Ar

 a nT ¼ max n j n 2 N; r þ  tn \T : 4Ar

ð1:1:47Þ



ð1:1:48Þ

Thus, lim nT ¼ þ 1:

ð1:1:49Þ

T! þ 1

It turns out from (1.1.47) that for all T > 0, Z T Z T aA4 a A3 þ  yðsÞds þ yðsÞds 4Ar 0 4Ar 0 ! Z tn Z tn X a a2  hðsÞds þ hðsÞds  nT 4Ar tn 4Aa 16Ar tn 4Aa 1nn T

r

ð1:1:50Þ

r

which contradicts (1.1.36). Thus this completes the proof.

h

In the sequel, we shall collect other useful inequalities which play important roles in classical calculus. These inequalities include the Young inequality, the Höder inequality, and the Minkowski inequality.

1.1.7

The Young Inequalities

Theorem 1.1.13. Suppose f is a positive, real-valued, continuous and strictly increasing function on [0, c] with c > 0. If f (0) = 0, a 2 [0, c] and b 2 ½0; f ðcÞ, then Z a Z b f ðxÞdx þ f 1 ðxÞdx  ab ð1:1:51Þ 0 −1

with f

0

is the inverse function of f. Equality holds in (1.1.51) if and only if b = f(a).

This is a classical result called “the Young inequality” whose proof can be found in Young [149]. If we take f (x) = xp−1 with p > 1 in the above theorem, then we can conclude the following corollary. Corollary 1.1.3. There holds that ab 

a p bq þ p q

where a, b ≥ 0, p > 1 and 1∕p + 1∕q = 1.

ð1:1:52Þ

Preliminary

13

If 0 < p < 1, then ab 

a p bq þ : p q

ð1:1:53Þ

The equalities in (1.1.52) and (1.1.53) hold if and only if b = ap−1. In corollary 1.1.3, if we consider a and b as εa and ε−1b, respectively, we can get the next corollary. Corollary 1.1.4. For any ε > 0, there holds that ab  where a, b ≥ 0, p > 1 and

1 p

ep a p bq þ q p qe

þ 1q ¼ 1:

In fact, the Young inequality has the following several variants. Corollary 1.1.5. (1) Let a, b > 0,

1 p

þ 1q ¼ 1, 1 < p < +∞. Then

(i) a1∕pb1∕q ≤ a∕p + b∕q; (ii) a1∕pb1∕q ≤ a∕(pε1∕q) + bε1∕p∕q, for all ε > 0; (iii) a a b1a  aa þ ð1  aÞb; 0\a\1: P Q m p k Pm (2) Let ak  0; pk [ 0; m k¼1 pk ¼ 1. Then k¼1 ak  k¼1 pk ak :

1.1.8

The Hölder Inequalities

This subsection will introduce some Hölder inequalities. The following is the discrete Hölder inequality which was proved by Hölder in 1889 (see e.g., Hölder [55]). However, as pointed out by Lech [76] that in fact it should be called the Roger inequality or Roger–Hölder inequality since Roger established the inequality (1.1.54) in 1888 earlier than Hölder did in 1889. However, we still call it here the Hölder inequality. Theorem 1.1.14. If ak ≥ 0, bk ≥ 0 for k = 1, 2,…, n, and n X

ak bk 

n X

k¼1

k¼1

n X

n X

!p1 akp

n X

1 p

þ 1q ¼ 1 with p > 1, then

!1q bqk

:

ð1:1:54Þ

:

ð1:1:55Þ

k¼1

If 0 < p < 1, then

k¼1

ak bk 

k¼1

!p1 akp

n X

!1q bqk

k¼1

Here the equalities in (1.1.54) and (1.1.55) hold if and only if aakp ¼ bbqk for k = 1, 2,…, n where α and β are real non-negative constants with α2 + β2 > 0.

1D Radiative Fluid and Liquid Crystal Equations

14

Remark 1.1.5. If p = 1 or p = +∞, we have the trivial case. ! n n X X ak b k  ak sup bk ; if p ¼ 1; k¼1 n X

1k n

k¼1 n X

ak bk 

k¼1

!

sup ak ; if p ¼ þ 1:

bk

1k n

k¼1

Remark 1.1.6. When p = q = 2, (1.1.54) and (1.1.55) are called to be the Cauchy inequality, the Schwarz inequality, the Cauchy–Schwarz inequality or the Bunyakovskii inequality. By virtue of the discrete Hölder inequality (theorem 1.1.14), we can easily obtain the integral form of the Hölder inequality, namely. Theorem 1.1.15. If f 2 Lp(Ω), g 2 Lq(Ω) and X
Rn is a measurable set, then fg 2 L1 ðXÞ and kfgkL1 ðXÞ  kf kLp ðXÞ kgkLq ðXÞ

ð1:1:56Þ

with 1  p  þ 1; p1 þ q1 ¼ 1 and Z

1p

p

kf kLp ðXÞ ¼

X

jf ðxÞj dx

;

kf kL1 ðXÞ ¼ esssup jf ðxÞj: x2X

If 0 < p < 1, then kfgkL1 ðXÞ  kf kLp ðXÞ kgkLq ðXÞ :

ð1:1:57Þ

The equalities in (1.1.56) and (1.1.57) hold if and only if there exist b 2 R and real numbers C1, C2 which are not all zeros such that C1 jf ðxÞjp ¼ C2 jgðxÞjq and arg(f(x)g(x)) = β a.e. on Ω hold. Remark 1.1.7. We have the corresponding weighted Hölder inequality of the integral form. Let 1\p\ þ 1; f 2 Lp ðXÞ; g 2 Lq ðXÞ; p1 þ q1 ¼ 1, ω(x) > 0 on Ω. Then Z

Z X

1.1.9

jfgjxðxÞdx 

p

X

jf ðxÞj xðxÞdx

p1 Z X

q

jgðxÞj xðxÞdx

q1

:

The Minkowski Inequalities

Note that, in 1896, Minkowski established the following famous inequality.

Preliminary

15

Theorem 1.1.16. Let a = {a1,…, an} or a = {a1,…, an,…} be a real sequence or complex sequence. Define !p1 X kakp ¼ jak jp if 1  p\ þ 1; k

kak1 ¼ sup jak j if p ¼ þ 1: k

Then for 1 ≤ p ≤ +∞, ka þ bkp  kakp þ kbkp :

ð1:1:58Þ

ka þ bkp  kakp þ kbkp

ð1:1:59Þ

If 0 ≠ p < 1, then

where when p < 0, we require that ak, bk, ak + bk ≠ 0 (k = 1, 2,…). Moreover, when p ≠ 0, 1, the equality in (1.1.58) holds if the sequences a and b are proportional. When p = 1, the equalities in (1.1.58) and (1.1.59) hold if and only if argak ¼ argbk ; for all k: Remark 1.1.8. If we replace p by 1∕p in (1.1.58), we can obtain the following assertion: (1) if 1 ≤ p < +∞, then there holds !p !p !p X X X 1 1 1 jak þ bk jp  jak jp þ jbk jp ; k

k

k

(2) if 0 < p < 1, then there holds !p !p !p X X X 1 1 1 p p p jak þ bk j  jak j þ jbk j : k

k

k

In the applications, the following integral form of the Minkowski inequality is used frequently. Theorem 1.1.17. Assume that Ω is a smooth open set in Rn and f, g 2 Lp(Ω) with 1 ≤ p ≤ +∞. Then f þ g 2 Lp ðXÞ and kf þ gkLp ðXÞ  kf kLp ðXÞ þ kgkLp ðXÞ :

ð1:1:60Þ

1D Radiative Fluid and Liquid Crystal Equations

16

If 0 < p < 1, then kf þ gkLp ðXÞ  kf kLp ðXÞ þ kgkLp ðXÞ :

ð1:1:61Þ

If p > 1, the equality in (1.1.60) holds if and only if there exists a constant C1 ≠ 0 such that C1 f ðxÞ ¼ gðxÞ a.e. in Ω. If p = 1, then the equality in (1.1.60) holds if and only if argf(x) = argg(x) a.e. in Ω or there exists a non-negative measurable function h such that fh = g a.e. in the set A = {x 2 Ω | f(x)g(x) ≠ 0}.

Chapter 2 Asymptotic Behavior of Solutions for the One-Dimensional Infrarelativistic Model of a Compressible Viscous Gas with Radiation 2.1

Main Results

This chapter will be devoted to the study of the large-time behavior of global solutions to the one-dimensional infrarelativistic model of a compressible viscous gas with radiation. The content of this chapter is adopted from Qin et al. [109], which has improved the results of Qin et al. [111]. We note that the existence of global solutions to such a model has been proved by Ducomet and Nečasová [18] and Qin et al. [111]. It is well-known that the radiative model in the one-dimensional case can be reduced into the following equations (see, Ducomet and Nečasová [19–21]) 8 qs þ ðqvÞy ¼ 0; > > > < ðqvÞ þ ðqv 2 Þ þ p ¼ lv  ðS Þ ; 
s 1 2  y  y
yy1 2  F R  ð2:1:1Þ q e þ 2 v s þ qv e þ 2 v þ pv  jhy  lvvy y ¼ ðSE ÞR ; > > > :1 c It þ xIy ¼ S: Now we assume that the fluid motion is small enough with respect to the velocity of light c so that we can drop all the 1c factors in the previous formulation and then get an “infrarelativistic” model of a compressible Navier–Stokes system for a one-dimensional flow coupled to the radiative transfer equation given in the following system 8 qs þ ðqvÞy ¼ 0; > > < ðqvÞ þ ðqv 2 Þ þ p ¼ lv ; 
s 1 2  y  y
yy1 2   ð2:1:2Þ > q e þ 2 v s þ qv e þ 2 v þ pv  jhy  lvvy y ¼ ðSE ÞR ; > : xIy ¼ S:

DOI: 10.1051/978-2-7598-2903-3.c002 © Science Press, EDP Sciences, 2022

1D Radiative Fluid and Liquid Crystal Equations

18

Under the Lagrangian coordinates, i.e., Z y x¼ qðn; sÞdn;

t ¼ s;

0

system (2.1.2) reduces to the following system 8 g ¼ vx ; > > > t > > > < vt ¼ rx ;  1 2 > > e þ 2 v ¼ ðrv  QÞx  gðSE ÞR ; > > t > > : xIx ¼ gS;

ð2:1:3Þ ð2:1:4Þ ð2:1:5Þ ð2:1:6Þ

where x 2 [0, 1], η is the specific volume (i.e., g ¼ q1), v denotes the velocity, θ is the temperature, I represents the radiative intensity depending on the Lagrangian mass coordinates (x, t) and also on two extra variables: the radiation frequency m 2 R þ ¼ ð0; þ 1Þ and the angular variable x 2 S 1 :¼ ½1; 1; r :¼ p þ l vgx is the stress and Q :¼ j hgx is the heat flux with the heat conductivity κ and μ is the viscosity coefficient. Source term S in the last equation is expressed as Sðx; t; m; xÞ ¼ ra ðm; x; g; hÞ½Bðm; hÞ  I ðx; t; m; xÞ þ rs ðm; g; hÞ½I~ðx; t; mÞ  I ðx; t; m; xÞ;

ð2:1:7Þ

R1 where I~ðx; t; mÞ :¼ 12 1 I ðx; t; m; xÞdx and B is a function of temperature and frequency describing the equilibrium state. We define the radiative energy as Z 1 Z þ1 I ðx; t; m; xÞdmdx; ð2:1:8Þ ER ¼ 1

0

the radiative flux Z FR ¼

1 1

Z

þ1

xI ðx; t; m; xÞdmdx;

ð2:1:9Þ

0

and the radiative energy sourceradiative energy source Z 1 Z þ1 Sðx; t; m; xÞdmdx: ðSE ÞR ¼ 1

ð2:1:10Þ

0

We now consider a typical initial boundary value problem for (2.1.3)–(2.1.6) in the reference domain Ω × [0, +∞) = (0, 1) × [0, +∞) under the Dirichlet–Neumann boundary conditions for the fluid unknowns vð0; tÞ ¼ vð1; tÞ ¼ 0;

Qð0; tÞ ¼ Qð1; tÞ ¼ 0;

8 t  0;

ð2:1:11Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

and transparent boundary conditions for the radiative intensity  I ð0; t; m; xÞ ¼ 0 for x 2 ð0; 1Þ; 8 t  0; I ð1; t; m; xÞ ¼ 0 for x 2 ð1; 0Þ; 8 t  0;

19

ð2:1:12Þ

and initial conditions gðx; 0Þ ¼ g0 ðxÞ;

vðx; 0Þ ¼ v0 ðxÞ;

hðx; 0Þ ¼ h0 ðxÞ on X;

ð2:1:13Þ

and I ðx; 0; m; xÞ ¼ I0 ðx; m; xÞ on X  R þ  S 1 :

ð2:1:14Þ

Pressure and energy of the matter are related by the thermodynamical relation eg ðg; hÞ ¼ pðg; hÞ þ hph ðg; hÞ:

ð2:1:15Þ

For the system (2.1.3) and (2.1.14), Ducomet and Nečasová [18] proved the global existence of solutions in Hi ði ¼ 1; 2Þ. However, estimates obtained there depend on any given time T, so they could not establish the large-time behavior of global solutions in Hi ði ¼ 1; 2Þ based on their estimates. Moreover, in Ducomet and Nečasová [18], all estimates hold only for q ≥ 2r + 1. Later on, Qin et al. [111] had improved the results in [18]. Recently, Qin et al. [109] have improved the results in [111] by establishing the uniform-in-time estimates of (η(t), v(t), θ(t), I(t)) in Hi ði ¼ 1; 2; 4Þ, which hold for q and r satisfying (2.1.16). Furthermore, the system considered here is different from that in Qin [104], so our uniform-in-time estimates are also different from those in Qin [104]. We now assume that e, p, σ and κ are twice continuously differential on 0 < η < +∞ and 0 ≤ θ < +∞, and there are exponents q and r satisfying one of the following relations 8 1 1 > > \q; 0\r  ; > > 2 2 > > > >1 5 2r þ 1 > > \q; < \r  ; 2 2 4 ð2:1:16Þ 5 17 5r þ 1 > > > \r  ; \q; > > 2 5 9 > > > > 17 10r þ 4 > :  r; \q 5 19 and we suppose the following growth conditions: 8 eðg; 0Þ  0; c1 ð1 þ hr Þ  eh ðg; hÞ  C1 ð1 þ hr Þ; > > > > c2 g2 ð1 þ h1 þ r Þ  pg ðg; hÞ   C2 g2 ð1 þ h1 þ r Þ; > > > r > < jph ðg; hÞj  C3 g1 ð1 þ h Þ; 1þr c4 ð1 þ h Þ  gpðg; hÞ  C4 ð1 þ h1 þ r Þ; pg ðg; hÞ\0; > > > 0  pðg; hÞ  C5 ð1 þ h1 þ r Þ; > > > c ð1 þ hq Þ  jðg; hÞ  C6 ð1 þ hq Þ; > > : 6 jjg ðg; hÞj þ jjgg ðg; hÞj  C7 ð1 þ hq Þ;

ð2:1:17Þ

1D Radiative Fluid and Liquid Crystal Equations

20

and that the absorption–emission coefficient σa(ν, ω; η, θ) and the scattering coefficient σs(ν; η, θ) satisfy the following conditions: 8 > gra ðm; x; g; hÞB m ðm; hÞ  C8 jxjha þ 1 f ðm; xÞ for m ¼ 1; 2; > > > > 0\ra ðm; x; g; hÞ  C9 jxj2 gðm; xÞ; > > < ½r þ jðr Þ j þ jðr Þ jðm; x; g; hÞ½1 þ Bðm; hÞ þ jB ðm; hÞj þ jB ðm; hÞj  C jxjhðm; xÞ; a a g a h h hh 10 2 > ðm; g; hÞ  C jxj kðmÞ; 0\r s 11 > > > > ½jðra Þgg j þ jðra Þgh j þ jðra Þhh jðm; x; g; hÞð1 þ Bðm; hÞ þ jBh ðm; hÞjÞ  C12 jxjlðm; xÞ; > > : ½jðr Þ j þ jðr Þ j þ jðr Þ j þ jðr Þ j þ jðr Þ jðm; g; hÞ  C jxjMðm; xÞ; s g s h s gg s gh s hh 13 ð2:1:18Þ where 0 ≤ α ≤ r, the numbers ci ; Cj ; ði ¼ 1; . . .; 7; j ¼ 1; . . .; 13Þ are positive constants and the non-negative functions f ; g; h; k; l; M are such that f ; g; h; k; l; M 2 L1 ðR þ  S 1 Þ \ L1 ðR þ  S 1 Þ: We assume that the viscosity coefficient μ is a positive constant. In the following, we denote Z þ1 Z I ðx; tÞ :¼ I ðx; t; m; xÞdxdm S1

0

for the integrated radiative intensity. In particular, Z þ1 Z I ðx; 0Þ  I 0 ¼ I ðx; 0; m; xÞdxdm: 0

S1

We define H1 ¼ fðg; v; h; I Þ 2 H 1 ð0; 1Þ  H01 ð0; 1Þ  H 1 ð0; 1Þ  L1 ðR þ  S 1 ; H 2 ð0; 1ÞÞ : gðxÞ [ 0; hðxÞ [ 0; x 2 ½0; 1; vjx¼0;1 ¼ 0; I jx¼0 ¼ 0 for x 2 ð0; 1Þ; I jx¼1 ¼ 0 for x 2 ð1; 0Þg; Hi ¼ fðg; v; h; I Þ 2 H i ð0; 1Þ  H0i ð0; 1Þ  H i ð0; 1Þ  L1 ðR þ  S 1 ; H i þ 1 ð0; 1ÞÞ : gðxÞ [ 0; hðxÞ [ 0; x 2 ½0; 1; vjx¼0;1 ¼ 0; hx jx¼0;1 ¼ 0; I jx¼0 ¼ 0 for x 2 ð0; 1Þ; I jx¼1 ¼ 0 for x 2 ð1; 0Þg; i ¼ 2; 4:

The main aim of this chapter was to establish the global existence and the large-time behavior of solutions in Hi (i = 1, 2, 4) to the system (2.1.3) and (2.1.14). The notation in this chapter will be as follows: Lq ; 1  q  þ 1, Wm,q, m 2 N; H 1 ¼ W 1;2 ; H01 ¼ W01;2 denote the usual (Sobolev) spaces on [0, 1]. In addition, ∥⋅∥B denotes the norm in space B; we also put kk ¼ kkL2 ½0; 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 constants depending on the kðg0 ; v0 ; h0 ; I 0 ÞkHi ; minx2½0; 1 g0 ðxÞ, minx2½0; 1 h0 ðxÞ, but not depending on t.

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

21

Our main results read as follows (see also Qin et al. [109]), which has improved the result in [111]. The next result concerns the global existence and asymptotic behavior of solutions in H1 . Theorem 2.1.1. Suppose that ðg0 ; v0 ; h0 ; I0 Þ 2 H1 and the compatibility conditions hold. Under assumptions (2.1.15)–(2.1.18), there exists a unique global solution ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ 2 L1 ð½0; þ 1Þ; H1 Þ to the problem (2.1.3)–(2.1.14) such that for all ðx; tÞ 2 ½0; 1  ½0; þ 1Þ, 0\C11  gðx; tÞ  C1 ;

ð2:1:19Þ

and for all t > 0, 2 kgðtÞ  gk2H 1 þ kvðtÞk2H 1 þ hðtÞ  h H 1 þ kI ðtÞk2L1 ðR þS 1 ;H 2 ð0;1ÞÞ Z t 2 þ ðkg  gk2H 1 þ kv k2H 2 þ h  h H 2 þ kht k2 ÞðsÞds 0

þ

Z tZ 0

1

0

Z

þ1

Z

0

S1

It2 dxdmdxds  C1 :

Moreover, we have, as t → +∞, kgðtÞ  gkH 1 ! 0;

kvðtÞkH 1 ! 0;

R1

where g ¼ 0 gðx; tÞdx ¼ eðg0 ; h0 Þ þ FR ð0ÞÞdx.

R1 0

g0 dx,

hðtÞ  h 1 ! 0; H

ð2:1:20Þ

kI ðtÞkL1 ðR þ S 1 ;H 2 ð0;1ÞÞ ! 0;

ð2:1:21Þ R1 1 2 h [ 0 is determined by eðg; hÞ ¼ 0 ð2 v0 þ

In the next theorem, we shall establish the global existence and asymptotic behavior of solutions in H2 . Theorem 2.1.2. Suppose that ðg0 ; v0 ; h0 ; I0 Þ 2 H2 and the compatibility conditions hold. Under assumptions (2.1.15)–(2.1.18), there exists a unique global solution ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ 2 L1 ð½0; þ 1Þ; H2 Þ to the problem (2.1.3)–(2.1.14) satisfying for any t > 0, 2 kgðtÞ  gk2H 2 þ kvðtÞk2H 2 þ hðtÞ  h H 2 þ kI ðtÞk2L1 ðR þ S 1 ;H 3 ð0;1ÞÞ þ kvt ðtÞk2 Z t 2 2 þ kht ðtÞk þ ðkvxt k2 þ khxt k2 þ h  h H 3 þ kv k2H 3 0

þ kg  gk2H 2 ÞðsÞds  C2 : Moreover, we have, as t → +∞, kgðtÞ  gkH 2 ! 0;

kvðtÞkH 2 ! 0;

ð2:1:22Þ hðtÞ  h 2 ! 0; H

kI ðtÞkL1 ðR þ S 1 ;H 3 ð0;1ÞÞ ! 0: ð2:1:23Þ

1D Radiative Fluid and Liquid Crystal Equations

22

Theorem 2.1.3. Suppose that ðg0 ; v0 ; h0 ; I0 Þ 2 H4 and the compatibility conditions hold. Under assumptions (2.1.15)–(2.1.18), there exists a unique global solution ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ 2 L1 ð½0; þ 1Þ; H4 Þ to the problem (2.1.3) and (2.1.14) verifying that for any t > 0, kgðtÞ  gk2H 4 þ kgt ðtÞk2H 3 þ kgtt ðtÞk2H 1 þ kvðtÞk2H 4 þ kvtt ðtÞk2 2 þ hðtÞ  h H 4 þ kht ðtÞk2H 2 þ khtt ðtÞk2 þ kI k2L1 ðR þ S 1 ;H 5 ð0;1ÞÞ Z t 2 þ ðkg  gk2H 4 þ kv k2H 5 þ kvt k2H 3 þ kvtt k2H 1 þ h  h H 5 0

þ kht k3H 3 þ khtt k2H 1 ÞðsÞds  C4 ; Z

t 0

ðkgt k2H 4 þ kgtt k2H 2 þ kgttt k2 ÞðsÞds  C4 :

Moreover, we have as t → +∞, kgðtÞ  gkH 4 ! 0;

ð2:1:24Þ

kvðtÞkH 4 ! 0;

hðtÞ  h 4 ! 0; H

ð2:1:25Þ

kI ðtÞkL1 ðR þ S 1 ;H 5 ð0;1ÞÞ ! 0; ð2:1:26Þ

R1 R1 R1 where g ¼ 0 gðx; tÞdx ¼ 0 g0 dx, h [ 0 is determined by eðg; hÞ ¼ 0 ð12 v02 þ eðg0 ; h0 Þ þ FR ð0ÞÞdx. Corollary 2.1.1. The global solution ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ obtained in theorem 2.1.3 is, in fact, a classical solution such that as t → +∞, ðgðtÞ  g; vðtÞ; hðtÞ  hÞ 3 þ 1 ! 0: 4þ1 3 ! 0; kI ðtÞk 1 1 2 2 ðC

ð0;1ÞÞ

L ðR þ S ;C

ð0;1ÞÞ

Remark 2.1.1. Theorems 2.1.1–2.1.3 also hold for the boundary conditions (2.1.12) and vð0; tÞ ¼ vð1; tÞ ¼ 0;

hð0; tÞ ¼ hð1; tÞ ¼ T0 ¼ const: [ 0;

where h can be replaced by T0.

2.2

Global Existence and Uniform-in-Time Estimates in H1

We note that the global existence of solutions in H1 has been established in [18]. This section will study the global existence and asymptotic behavior of global solutions in H1 . To this end, we shall first establish some uniform-in-time estimates in H1 .

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

23

Lemma 2.2.1. Under assumptions in theorem 2.1.1, there holds that hðx; tÞ [ 0; Z

1

8 ðx; tÞ 2 ½0; 1  ½0; þ 1Þ; Z

gðx; tÞdx ¼

0

1

g0 ðxÞdx  g0 ;

8 t [ 0;

ð2:2:1Þ ð2:2:2Þ

0

Z

1

ðh þ h1 þ r Þðx; tÞdx  C1 ;

8 t [ 0;

ð2:2:3Þ

0

Z

1

½ðh  log h  1Þ þ h1 þ r þ v 2 ðx; tÞdx 0  Z t Z 1 ð1 þ hq Þh2x lvx2 þ þ ðx; sÞdxds  C1 : gh gh2 0 0

ð2:2:4Þ

Proof. Inequality (2.2.1) is a consequence of the generalized maximum principle [3] and one can find the proof in [19–21]. Integrating (2.1.3) over Qt = (0, 1) × (0, t), and using the boundary conditions, we can easily deduce (2.2.2). From (2.1.6), (2.1.9) and (2.1.10), we can infer ðFR Þx ¼ gðSE ÞR :

ð2:2:5Þ

Inserting (2.2.5) into (2.1.5), we arrive at   1 e þ v 2 ¼ ðrv  Q  FR Þx : 2 t

ð2:2:6Þ

Integrating (2.2.6) over Qt and using boundary conditions (2.1.11) and (2.1.12), we have   Z 1 Z 1 Z t 1 2 1 2 x¼1 e þ v ðx; tÞdx þ FR jx¼0 ds ¼ e0 þ v0 ðxÞdx: ð2:2:7Þ 2 2 0 0 0 Z 0

Using (2.1.12), the contribution of the radiation term reads (see, e.g., [18])  Z t Z þ 1 Z 1 Z þ1 Z 0 t FR jx¼1 ds ¼ xI ð1; t; m; xÞdxdm  xI ð0; t; m; xÞdxdm ds x¼0 0

0

0

0

1

 0; which, together with (2.2.7), implies  Z 1 1 e þ v 2 ðx; tÞdx  C1 : 2 0

ð2:2:8Þ

1D Radiative Fluid and Liquid Crystal Equations

24

Combining (2.2.8) with (2.1.17) yields (2.2.3). Noting that radiative term η(SE)R appears in (2.1.5), our estimate (2.2.8) is different from the one in [17] where there is no radiative term. We define the free energy ψ: = e − θS with ψθ = − S and ψη = − p with the specific entropy S. Let us consider the auxiliary function Eðg; hÞ :¼ wðg; hÞ  wð1; 1Þ  ðg  1Þwg ð1; 1Þ  ðh  1Þwh ðg; hÞ:

ð2:2:9Þ

We have the following estimate (see, e.g., Ducomet and Nečasová [18] for details)   Z 1 Z t Z 1 2 Z t Z 1 Z þ1 Z 1 lvx jh2 g þ 2x dxds þ E þ v 2 dx þ ra Idxdmdxds 2 gh gh 0 0 0 0 0 h 0 S1  Z t Z þ 1 Z 1 Z þ1 Z 0 þ xI ð1; t; m; xÞdxdm  xI ð0; t; m; xÞdxdm ds  C1 : 0

0

0

1

0

ð2:2:10Þ Using the Taylor theorem and the definition of E(η, θ), we can conclude Eðg; hÞ  wðg; hÞ þ wðg; 1Þ þ ðh  1Þwh ðg; hÞ ¼ wðg; 1Þ  wð1; 1Þ  wg ðg; hÞ Z 1 2 ¼ ðg  1Þ ð1  nÞwgg ð1 þ nðg  1Þ; 1Þdn  0: 0

Thus, Eðg; hÞ  wðg; hÞ  wðg; 1Þ  ðh  1Þwh ðg; hÞ Z 1 2 ð1  sÞwhh ðg; h þ sð1  hÞÞds ¼ ð1  hÞ 0

Z 1 ð1  sÞf1 þ ½h þ sð1  hÞr g ds  C11 ð1  hÞ2 h þ sð1  hÞ 0 ( C 1 ð1hr Þ C 1 ð1hr þ 1 Þ C11 ðh  log h  1Þ þ 1 r þ 1 r þ1 ; ¼ 1 2C1 ðh  log h  1Þ;  C11 ðh

 log h 

1Þ þ C11 hr þ 1



C11 :

for r [ 0; for r ¼ 0; ð2:2:11Þ

Combining (2.2.11) and (2.2.10), and using (2.1.17) yields (2.2.4). The proof is now complete. h The following two lemmas concerning the uniform-in-time estimate of specific volume η play a very crucial role in this chapter. The uniform-in-time estimate is different from the one in Ducomet and Nečasová [18], where estimates are dependent on any given time T > 0. Lemma 2.2.2. For any t ≥ 0, there exists one point x1 = x1(t) 2 [0, 1] such that the solution η(x, t) to the problem (2.1.3)–(2.1.6), (2.1.11)–(2.1.14) possesses the following expression:

Asymptotic Behavior of Solutions for the Compressible Viscous Gas 

1 gðx; tÞ ¼ Dðx; tÞZ ðtÞ 1 þ l where

Z

t

1

gðx; sÞpðx; sÞD ðx; sÞZ

1

25

ðsÞds

ð2:2:12Þ

0

 Z x Z x 1 vðy; tÞdy  v0 ðyÞdy l x1 ðtÞ 0  Z Z x 1 1 g ðxÞ v0 ðyÞdydx ; þ g0 0 0 0

Dðx; tÞ ¼ g0 ðxÞ exp



1 Z ðtÞ ¼ exp  lg0

Z tZ

1

ðv þ gpÞðy; sÞdyds : 2

0

ð2:2:13Þ

ð2:2:14Þ

0

Proof. The proof is the same as that of lemma 2.1.3 in Qin [104]. But for the book’s self-contained, we copy its proof here. Let Z x Z t hðx; tÞ ¼ v0 ðyÞdy þ rðx; sÞds: 0

0

The from (2.1.13), h(x, t) satisfies hx ¼ v;

ht ¼ r

ð2:2:15Þ

lhxx g

ð2:2:16Þ

and from (2.1.11) it solves the equation ht ¼ p þ with x ¼ 0; 1 : hx ¼ v ¼ 0:

ð2:2:17Þ

Hence we derive from (2.2.16) that ðghÞt ¼ hvx  gp þ lhxx :

ð2:2:18Þ

Integrating (2.2.18) over [0, 1] × [0, t] and using (2.2.17), we arrive at Z 1 Z 1 Z tZ 1 ghdx ¼ g0 h0 dx  ðgp þ v 2 Þdxds  /ðtÞ: ð2:2:19Þ 0

0

0

0

Then for any t ≥ 0, there exists one point x1 = x1(t) 2 [0, 1] such that Z 1 Z 1 ghdx ¼ gdx  hðx1 ðtÞ; tÞ ¼ g0  hðx1 ðtÞ; tÞ; /ðtÞ ¼ 0

0

i.e., Z

t 0

Z pðx1 ðtÞ; sÞds ¼ 0

x1 ðtÞ

v0 ðyÞdy þ l log

gðx1 ðtÞ; tÞ /ðtÞ  g0 ðx1 ðtÞÞ g0

ð2:2:20Þ

1D Radiative Fluid and Liquid Crystal Equations

26

with /ðtÞ ¼ 

Z tZ

1

Z

1

ðv þ gpÞðx; sÞdxds þ 2

0

0

Z g0 ðxÞ

0

x

v0 ðyÞdydx:

ð2:2:21Þ

0

Moreover, (2.1.4) can be rewritten as vt  lðlog gÞxt ¼ px ¼ p x with p ¼ p  we get

R1 0

ð2:2:22Þ

pðx; tÞdx: Integrating (2.2.22) over [x1(t), x] × [0, t] for fixed t > 0,

( "Z x g0 ðxÞgðx1 ðtÞ; tÞ 1 exp gðx; tÞ ¼ ðvðy; tÞ  v0 ðyÞÞdy g0 ðx1 ðtÞÞ g0 x1 ðtÞ #) Z t ðpðx; sÞ  pðx1 ðtÞ; sÞÞds : þ

ð2:2:23Þ

0

Inserting (2.2.20) into (2.2.23) and noting (2.2.13), (2.2.14) and (2.2.21), we conclude  Z t 1 g1 ðx; tÞ exp pðx; sÞds ¼ D 1 ðx; tÞZ 1 ðtÞ ð2:2:24Þ g0 0 which implies that  Z t Z 1 1 t 1 exp pðx; sÞds ¼ 1 þ D ðx; sÞZ 1 ðsÞgðx; sÞpðx; sÞds: g0 0 g0 0 Thus (2.2.12) follows from (2.2.24) and (2.2.25).

ð2:2:25Þ h

Lemma 2.2.3. There holds that 0\C11  gðx; tÞ  C1 ; Z 0

t

8 ðx; tÞ 2 ½0; 1  ½0; þ 1Þ;

kvðsÞk2L1 ds  C1 ;

8 t [ 0:

ð2:2:26Þ ð2:2:27Þ

Proof. Let Mg ðtÞ ¼ max gðx; tÞ: x2½0; 1

Using the Young inequality, the Hölder inequality and lemma 2.2.1, we get

Z Z Z x Z x

x 1 1

vðy; tÞdy  v0 ðyÞdy þ g0 ðxÞ v0 ðyÞdydx

x1 ðtÞ g0 0 0 0 Z 1 12 Z 1 12 Z 1 12 Z 1 1 2 2 2  v dy þ v0 dy þ g ðxÞ v0 dy dx g0 0 0 0 0 0 ð2:2:28Þ  C1 kv k2 þ C1  C1 :

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

27

Equations (2.2.28) and (2.2.13) yield the existence of some positive constant C1 > 0 such that 0\C11  Dðx; tÞ  C1 ;

8 ðx; tÞ 2 ½0; 1  ½0; þ 1Þ:

From (2.1.14) and (2.2.1), we deduce Z 1 Z 1 Z ðv 2 þ gpÞðx; tÞdx  gpðx; tÞdx  0

0

0

1

ðc4 þ h1 þ r Þdx  C11 ;

8 t [ 0: ð2:2:29Þ

Using lemma 2.2.1, we get Z 1 Z ðv 2 þ gpÞðx; tÞdx  kv k2 þ C4 0

1

ð1 þ h1 þ r Þdx  C1 ;

8 t [ 0:

ð2:2:30Þ

0

Thus, from (2.2.29) and (2.2.30) it follows that for all 0 ≤ s ≤ t, Z tZ 1 1 C1 ðt  sÞ  ðv 2 þ gpÞðx; sÞdxds  C1 ðt  sÞ; s

ð2:2:31Þ

0

which, together with (2.2.14), gives that for any 0 ≤ s ≤ t,  Z tZ 1 1 1 C1 ðtsÞ 1 2 e  Z ðtÞZ ðsÞ ¼ exp  ðv þ gpÞðy; sÞdyds  eC1 ðtsÞ : lg0 s 0 ð2:2:32Þ It thus derives from (2.2.4) and the convexity of the function  log y that Z 1 Z 1 Z 1 hdx  log hdx  1  ðh  log h  1Þdx  C1 0

0

0

which results in the existence of a(t) 2 [0, 1] and two positive roots of the equation y  log y  1 ¼ C1 such that Z 1 hðx; tÞdx ¼ hðaðtÞ; tÞ  r2 : 0\r1  0

This gives, for any t > 0, such that

Z Z 1

x

m1 m1 m1 jh ðx; tÞ  h ðaðtÞ; tÞj ¼ ðh ðx; tÞÞx dy  C1 hm1 1 jhx jdx

aðtÞ 0 Z  C1 0 1 2

1

1 þ hq 2 hx dx gh2

12 Z 0

1

h2m1 g dx 1 þ hq

1 2

 C1 V ðtÞMg ðtÞ where V ðtÞ ¼

R 1 1 þ hq 0

gh2

h2x dx and 0 ≤ m1 ≤ m = (q + r + 1)/2.

12 ð2:2:33Þ

1D Radiative Fluid and Liquid Crystal Equations

28

Then for any (x, t) 2 [0, 1] × [0, +∞), we get C11  C11 V ðtÞMg ðtÞ  h2m1 ðx; tÞ  C1 þ C1 V ðtÞMg ðtÞ:

ð2:2:34Þ

Thus we conclude from lemma 2.2.1 and (2.2.32)–(2.2.34)

 Z 1 t 1 1 gðx; tÞ ¼ Dðx; tÞ Z ðtÞ þ gðx; sÞpðx; sÞD ðx; sÞZ ðtÞZ ðsÞds l 0

 Z t C1 t C1 ðtsÞ  C1 e þ ð1 þ V ðsÞMg ðsÞÞe ds Z  C1 þ C1

0 t

Mg ðsÞV ðsÞds;

0

i.e., Z Mg ðtÞ  C1 þ C1

t

Mg ðsÞV ðsÞds;

0

which, with (2.2.4) and using the Gronwall inequality, yields Mg ðtÞ  C1 :

ð2:2:35Þ

Using (2.2.12) and (2.2.32), we find that there exists a large time t0 such that as t ≥ t0, x 2 [0, 1],  Z 1 t gðx; tÞ ¼ Dðx; tÞZ ðtÞ 1 þ gðx; sÞpðx; sÞD1 ðx; sÞZ 1 ðsÞds l 0 

Z t  C11 eC1 t þ eC1 ðtsÞ ds  C11

Z

0

t

e

C1 ðtsÞ

ds  ð2C1 Þ1 :

0

ð2:2:36Þ

Note now that Dðx; tÞ  C11 ; Z ðtÞ  expðC1 tÞ and infer that for any (x, t) 2 [0, 1] × [0, t0], gðx; tÞ  Dðx; tÞZ ðtÞ  C11 expðC1 tÞ  C11 expðC1 t0 Þ; which, together with (2.2.36), gives that for any (x, t) 2 [0, 1] × [0, +∞) gðx; tÞ  C11 :

ð2:2:37Þ

Combining (2.2.35) and (2.2.37) gives immediately (2.2.26). Using the Hölder inequality, (2.2.26) and lemma 2.2.1, we obtain for any t ≥ 0 2  Z t Z t Z 1 Z t Z 1 2 Z 1 vx dx jvx jdx ds  hdx ds  C1 : kvðsÞk2L1 ds  0 0 0 0 0 h 0 The proof is now complete.

h

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

29

Corollary 2.2.1. If assumptions in theorem 2.1.1 hold, then there holds C1  C1 V ðtÞ  h2m ðx; tÞ  C1 þ C1 V ðtÞ; ðx; tÞ 2 ½0; 1  ½0; 1Þ;

ð2:2:38Þ

where Z

1

V ðtÞ ¼ 0

ð1 þ hq Þh2x dx; h2

Z

1

V ðsÞds  C1 :

0

Corollary 2.2.2. If assumptions in theorem 2.1.1 hold, then there holds that Z tZ 1 ð1 þ hÞ2m v 2 dxds  C1 ; 8t [ 0: ð2:2:39Þ 0

0

Proof. Using the Poincaré inequality, lemma 2.2.1 and (2.2.38), we obtain Z tZ 1 Z tZ 1 Z tZ 1 ð1 þ hÞ2m v 2 dxds  C1 v 2 dxds þ C1 V ðsÞv 2 dxds 0 0 0 0 0 0 Z t V ðsÞds  C1 :  C1 þ C1 0

h

The proof is now complete. Set K ¼ sup khðsÞkL1 : 0st

Lemma 2.2.4. If assumptions in theorem 2.1.1 hold, then the following estimates hold for any t > 0, Z tZ 1 2 kgx ðtÞk þ ð1 þ h1 þ r Þg2x dxds  C1 ð1 þ KÞb ; ð2:2:40Þ 0

Z

t

0

kvx ðsÞk2 ds  C1 ð1 þ KÞb=2 ;

ð2:2:41Þ

0

with b ¼ maxðr þ 1  q; 0Þ: Proof. Obviously, equation (2.1.3) can be written as   g ð2:2:42Þ v  l x þ pg gx ¼ ph hx : g t   Multiplying (2.2.42) by v  l ggx and then integrating the result over [0, 1] × (0, t), we have

1D Radiative Fluid and Liquid Crystal Equations

30

2 Z t Z 1 1 lpg g2x v  l g x þ dxds 2 g g 0 0 2 Z t Z 1

  1 g0x gx ¼ v 0  l  p g gx v þ p h h x v  l dxds: 2 g0 g 0 0 Now using the Young inequality, (2.1.17) and lemmas 2.2.1–2.2.3, we can infer for any ϵ > 0, 2 Z t Z 1 1 v  l gx þ ð1 þ h1 þ r Þg2x dxds 2 g 0 0

   Z t Z 1

g ð1 þ h1 þ r Þjgx vj þ ð1 þ hr Þ

hx v  l x

dxds  C1 þ C1 g 0 0 Z tZ 1 ð1 þ h1 þ r Þðeg2x þ C1 v 2 Þdxds  C1 þ C1 Z

0

0

12  Z t Z

12 h2 ð1 þ hr Þ2 2 v dxds q ð1 þ h Þ 0 0 0 Z tZ 1 Z tZ 1 ð1 þ hr Þ2 h2x 1þr 2 þ ð1 þ h Þgx dxds þ C1 dxds 1 þ h1 þ r 0 0 0 0 Z tZ 1 d  C1 ð1 þ KÞmaxðb;2Þ þ  ð1 þ h1 þ r Þg2x dxds þ C1

t

V ðsÞds

Z  C1 ð1 þ KÞb þ  þ C1

Z tZ 0

0

0

t

1

0

V ðsÞkgx k2 ds þ 

0 1

ð1 þ hr Þjhx gx jdxds

Z tZ 0

0

1

hr þ 1 g2x dxds ð2:2:43Þ

with d ¼ maxðr þ 1  2q; 0Þ  b: Now we estimate the last term in (2.2.43). Using the Young inequality and lemmas 2.2.1–2.2.3, we can conclude for any ϵ > 0, Z tZ 1 Z Z Z tZ 1  t 1 ð1 þ hr Þ2 2 ð1 þ hr Þjhx gx j  ð1 þ h1 þ r Þg2x dxds þ C1 ðÞ h dxds 1þr x 2 0 0 0 0 0 0 1þh Z Z  t 1  ð1 þ h1 þ r Þg2x dxds þ C1 þ C1 Kb ; 2 0 0 which, together with (2.2.43), yields 2 Z t Z 1 Z tZ 1 1þr 2 v  l g x þ ð1 þ h Þgx dxds   ð1 þ h1 þ r Þg2x dxds þ C1 ð1 þ KÞb : g 0 0 0 0 Thus for small ϵ > 0 in the above inequality, and applying the generalized Bellman–Gronwall inequality, we conclude (2.2.40).

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

31

Multiplying (2.1.4) by v, integrating the result over Qt, and using the Young inequality, lemmas 2.2.1–2.2.3 and (2.2.38), we derive Z Z tZ 1 2 1 1 2 v v dx þ l x dxds 2 0 g 0 0 Z tZ 1 Z 1 1 ¼ v 2 dx  ðpg gx þ ph hx Þvdxds 2 0 0 0 0 Z tZ 1  C1 þ C1 ½ð1 þ h1 þ r Þjgx vj þ ð1 þ hr Þjhx vjdxds 0

 C1 þ C1

0

Z t Z 0

þ C1

0

 C1 K

ð1 þ h

1þr

0

Z t Z

b=2

1

1

0

þ C1 K

Þg2x dxds

ð1 þ hq Þh2x dxds h2 d=2

1=2 Z t Z

1=2 Z t Z

 C1 þ C1 K

0

0

0 1

1

ð1 þ h

1þr

1=2 Þv dxds 2

0

h2 ð1 þ hr Þ2 v 2 dxds 1 þ hq

!1=2

b=2

which yields (2.2.41). Thus this proves the proof. Lemma 2.2.5. There holds that for any t > 0, Z t kvx ðtÞk2 þ kvxx ðsÞk2 ds  C1 ð1 þ KÞb1 ;

h

ð2:2:44Þ

0

Z 0

t

kvx ðsÞk2L1 ds  C1 ð1 þ KÞb2 ; Z

kvx ðtÞk2 þ

t

kvt ðsÞk2 ds  C1 ð1 þ KÞb3

ð2:2:45Þ

ð2:2:46Þ

0

with



5b b b b1 ¼ max r þ 1 þ b; ; maxð2r þ 2  q; 0Þ ; b2 ¼ þ 1 ; 2 4 2  3 3 b3 ¼ max r þ 1 þ b; maxð2r þ 2  q; 0Þ; b1 þ b : 4 8

Proof. Multiplying (2.1.4) by vxx, and then integrating the resultants over [0, 1], we get  Z 1 Z 1  0 1d l l 2 px vxx dx  gx vx þ vxx vxx dx; ð2:2:47Þ kv x k ¼ 2 dt g g 0 0 i.e., for any ϵ > 0

1D Radiative Fluid and Liquid Crystal Equations

32 Z

t

2

kvx ðtÞk þ

kvxx ðsÞk2 ds

0

 C1 þ C1

Z tZ 0

 C1 þ C1

Z tZ

Z tZ 0

½jgx vx vxx j þ ð1 þ h1 þ r Þjgx vxx j þ ð1 þ hr Þjhx vxx jdxds

0

0

þ C1

1

0 1

0

1

vx2 g2x dxds þ C1

0

1

0

ð1 þ h1 þ r Þ2 g2x dxds

ð1 þ hÞ2r h2x dxds

 C1 þ C1 ð1 þ KÞr þ 1 Z tZ

Z tZ

Z tZ 0

1

ð1 þ h1 þ r Þg2x dxds þ C1

0

Z

t

kvx kkvxx kkgx k2 ds

0

ð1 þ hq Þh2x h2 ð1 þ hÞ2r dxds þ C1 ð1 þ hq Þ h2 0 0 Z t 1=2 Z t 1=2  C1 þ C1 ð1 þ KÞr þ 1 þ b þ C1 ð1 þ KÞb kvx ðsÞk2 ds kvxx ðsÞk2 ds 0 0 h2 ð1 þ hÞ2r þ C1 sup q 1 0  s  t 1 þ h 1

L

r þ1þb

 C1 þ C1 ð1 þ KÞ þ C1 ð1 þ KÞ2b þ C1 ð1 þ KÞmaxð2r þ 2q;0Þ Z t þ kvxx ðsÞk2 ds 0 Z t b1  C1 ð1 þ KÞ þ  kvxx ðsÞk2 ds; 5

0

which gives for small ϵ > 0,

Z

t

kvx ðtÞk2 þ

kvxx ðsÞk2 ds  C1 ð1 þ KÞb1 :

ð2:2:48Þ

0

Thus Z 0

t

Z kvx ðsÞk2L1 ds  C1

t

1=2 Z

t

kvx ðsÞk2 ds

0

 C1 ð1 þ KÞb2 :

1=2 kvxx ðsÞk2 ds

0

ð2:2:49Þ

Multiplying (2.1.4) by vt, integrating the resultants over [0, 1], and using lemmas 2.2.1–2.2.4, we get  Z 1 Z 1   0 l l kv t k2 ¼  px vt dx  gx vx þ vxx vt dx ð2:2:50Þ g g 0 0

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

33

Integrating (2.2.50) in t gives Z t kvx ðtÞk2 þ kvt ðsÞk2 ds 0 Z t Z t  C1 þ C1 kpx k2 ds þ kvx k3L3 dxds 0

Z tZ

h2 ð1 þ hÞ2r h2x ð1 þ hq Þ dxds 1 þ hq h2 0 0 0 0 Z t 3=4 Z t 1=4 sup kvx ðsÞk kvx k2 ds kvxx ðsÞk2 ds

 C1 þ C1 þ C1

0

Z tZ

1

ð1 þ h1 þ r Þ2 g2x dxds þ C1

0st

 C1 ð1 þ KÞ

0

r þ1þb

þ C1 ð1 þ KÞ

1

0 maxð2r þ 2q;0Þ

þ C1 ð1 þ KÞ4b1 þ 8b 3

3

 C1 ð1 þ KÞb3 : h Corollary 2.2.3. If assumptions in theorem 2.1.1 hold, then the following estimates hold for any t > 0, Z tZ 1 ð1 þ hÞ2m vx2 dxds  C1 ð1 þ KÞb1 ; ð2:2:51Þ 0

0

Z tZ 0

1

ð1 þ hÞ2m þ 1 vx2 dxds  C1 ð1 þ KÞb1 þ 1 ;

0

Z tZ 0

1

ð2:2:53Þ

0

Z tZ 0

where

ð1 þ hÞq þ 1 jvx j3 dxds  C1 ð1 þ KÞb4 ;

ð2:2:52Þ

0

1

ð1 þ hÞqr vx4 dxds  C1 ð1 þ KÞb6 ;

  1 q  3r  1 ;0 ; d1 ¼ maxðq  3r þ 1; 0Þ; d2 ¼ max 4 2 3b1 3b1 b b4 ¼ d1 þ ; b5 ¼ maxðq  r; 0Þ þ þ ; 4 2 2 b6 ¼ minðd2 þ 2b1 ; b5 Þ:

ð2:2:54Þ

1D Radiative Fluid and Liquid Crystal Equations

34

Proof. Using corollary 2.2.1 and lemmas 2.2.1–2.2.5, we can derive Z tZ 1 Z t Z t ð1 þ hÞ2m vx2 dxds  V ðsÞkvx k2 ds kvx k2 ds þ 0

0

0

0

b 2

 C1 ð1 þ KÞ þ C1 ð1 þ KÞb1  C1 ð1 þ KÞb1 : Note that Z tZ 0

1

0

ð1 þ hÞ2m þ 1 vx2 dxds  ð1 þ KÞ

Z tZ 0

 C1 ð1 þ KÞ and Z tZ 0

1

ð1 þ hÞq þ 1 jvx j3 dxds  C1 ð1 þ KÞd1

Z tZ

1

0 b1 þ 1

ð1 þ hÞ2m vx2 dxds

;

1

ð1 þ hÞ4ðq þ r þ 1Þ jvx j3 dxds 0 0

Z t  Z t 3  C1 ð1 þ KÞd1 V ðsÞ4 kvx k3L3 ds kvx k3L3 ds þ

0

3

0

 C1 ð1 þ KÞ

d1

0

Z

t

sup kvx ðsÞk s2½0; t

2

kvx k ds

34 Z

0

d1

þ C1 ð1 þ KÞ max kvx ðsÞk

5 2

2

kvxx k ds

0

Z

s2½0; t

t

t

2

kvxx k ds

14

0

 C1 ð1 þ KÞb4 where we have used Z

t 0

Z kvx k3L3 ds  C1

t

5 2

Z

1 2

kvx k kvxx k ds  C1

0

Z

 C1 sup kvx ðsÞk s2½0; t 3b1

0

0

1

3

10 3

kvx k ds

þ 3b 8

Z

V ðsÞ4 jvx j3 dxds  0

t

34 Z

0

t

2

kvx k ds

t

2

kvxx k ds

14

0

34 Z

0

 C1 ð1 þ KÞ 4 Z tZ

t

t

2

kvxx k ds

14

0

; Z

3

V ðsÞ4 kvx k3L3 ds  C1

Z

t

 C1

V ðsÞds

34 Z

0

 C1 sup kvx ðsÞk s2½0; t

5 2

Z

t

t

3

0 10

2

kvx k kvxx k ds

0 t

2

kvxx k ds

14

0

 C1 ð1 þ KÞ4b1 þ 4b1 ¼ C1 ð1 þ KÞ2b1 5

1

5

1

V ðsÞ4 kvx k2 kvxx k2 ds

3

14

14

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

and Z tZ 0

0

1

ð1 þ hÞqr vx4 dxds  C1 ð1 þ KÞd2  C1 ð1 þ KÞ

d2

Z tZ 0

ð1 þ hm Þvx4 dxds

0

Z tZ 0

1

1

ð1 þ V ðsÞ2 Þvx4 dxds

0

"Z  C1 ð1 þ KÞ

1

t

d2

Z

t



t

3

kvx k kvxx kds þ

0

Z

6

35

2

kvx k kvxx k ds

V ðsÞds

12

0

12 #

0

 C1 ð1 þ KÞ

d2

h

Z sup kvx ðsÞk Z

þ sup kvx ðsÞk s2½0; t

h

2

kvx k ds

s2½0; t 3

t

2

12 Z

0 t

2

kvxx k ds

2

kvxx k ds

12

0

12 i

0

 C1 ð1 þ KÞd2 ð1 þ KÞ4b þ 2b1 þ ð1 þ KÞ2b1 1

t

3

i

 C1 ð1 þ KÞd2 þ 2b1 : However, we also know that Z tZ 1 Z t ð1 þ hÞqr vx4 dxds  C1 ð1 þ KÞmaxðqr;0Þ kvx k4L4 ds 0 0 0 Z t maxðqr;0Þ  C1 ð1 þ KÞ kvx k3 kvxx kds 0

Z  C1 ð1 þ KÞmaxðqr;0Þ

t

kvx k6 ds

12 Z

0

Z

 C1 ð1 þ KÞmaxðqr;0Þ sup kvx ðsÞk2 s2½0; t

Z

t



kvxx ðsÞk2 ds

t

kvxx k2 ds

12

0 t

kvx ðsÞk2 ds

12

0

12

0

 C1 ð1 þ KÞmaxðqr;0Þ þ 2b1 þ 4b ¼ C1 ð1 þ KÞb5 : 3

1

Therefore, Z tZ 0

0

1

ð1 þ hÞqr vx4 dxds  C1 ð1 þ KÞb6 :

ð2:2:55Þ h

1D Radiative Fluid and Liquid Crystal Equations

36

In the next lemma, we shall derive new uniform-in-time estimates on radiative term I(x, t; ν, ω) given in the following lemma, which are more complicated, delicate than and quite different from those in [111], where estimates are not uniform-in-time. Lemma 2.2.6. There holds that for any t > 0, Z tZ 1Z 1Z Z tZ 1Z 1Z vra I 2 dxdmdxds þ vrs ðI~  I Þ2 dxdmdxds 0

0

Z

þ

0 1Z S1

0

S1

0

0

1

vrs I 2 ðx; t; m; xÞdxdm  C1 Z

1

ð2:2:56Þ

h1 þ a dx  C1 ;

0

Z S1

0

S1

0

Z

I ðx; t; m; xÞdxdm  C1 :

ð2:2:57Þ

Proof. Multiplying (2.1.6) by I, integrating the result over (0, 1) × S1 × (0, ∞) and using boundary conditions (2.1.11) and (2.1.12), we get for any ϵ > 0, Z Z Z Z 1 1 1 1 xI 2 ð1; t; m; xÞdxdm  xI 2 ð0; t; m; xÞdxdm 2 0 2 0 S1 S1 Z 1Z 1Z Z 1Z 1Z þ gra I 2 dxdmdx þ grs ðI~  I Þ2 dxdmdx 0

0

Z

 C1 ðÞ

S1 1Z 1

0

Z  C1 ðÞ

S1

0 1

h

0

Z

1þa

Z

 C1 ðÞ

1

h1 þ a dx þ 

0

Z

1

 C1 þ  0

Z

1

Z

Z 0

1

Z

0

Z

dx 0

1

Z

gra B 2 dxdmdx þ 

0

Z

S1

0

S1 1

Z

f ðm; xÞdxdm þ 

Z 0

1

gra I 2 dxdmdx S1

1

Z

0

Z

0

1

Z gra I 2 dxdmdx S1

gra I 2 dxdmdx S1

gra I 2 dxdmdx: S1

0

0

Z

0

Similarly, we also deduce that Z 1Z 1Z Z gðra þ rs ÞI 2 dxdmdx  0

1

S1

1

h1 þ a dx  C1 :

ð2:2:58Þ

0

In order to derive (2.2.5), we consider now the following integro-differential equation 8 @ > > > x I ðx; t; m; xÞ ¼ gra ðm; x; g; hÞ½Bðm; hÞ  I ðx; m; xÞ þ grs ðm; g; hÞ > @x > > > > <  ½I~ðx; mÞ  I ðx; m; xÞ on X  ½0; t  R þ  S 1 ; ð2:2:59Þ I ð0; t; m; xÞ ¼ 0 for all x 2 ð0; 1Þ; > > > > > I ð1; t; m; xÞ ¼ 0 for all x 2 ð1; 0Þ; > > > : I ðx; 0; m; xÞ ¼ I0 ðx; m; xÞ on X  R þ  S 1 :

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

37

Solving explicitly the ordinary differential equation and using boundary condition (2.1.1), we arrive at (see [19] for details) 8 Z x R y gðra þ rs Þ dz g > ðra B þ rs I~Þdy for all x 2 ð0; 1Þ; > e x x < x 0 ð2:2:60Þ I ðx; t; m; xÞ ¼ Z 1 R y gðr þ r Þ a s > dz g > : ðra B þ rs I~Þdy for all x 2 ð1; 0Þ: e x x x x Using the Young inequality, (2.1.18), (2.2.3), and (2.2.56), we have for all ω 2 (0, 1), Z 1Z I dmdx 1 0 ZS 1 Z  Z x R y gðr þ r Þ  a s dz g ðra B þ rs I~Þdy dmdx ¼ e x x x S1 0 0

Z 1 Z Z 1


 g

~ ra B þ rs I dydmdx

 0 S1 0 x

Z 1 Z 1 Z

Z 1 Z 1Z

1 1 ~ ¼

ra Bdxdmdx þ rs ðI  I þ I Þdxdmdx

g g 0 0 0 0 S1 x S1 x

Z 1 Z 1 Z h

Z 1 i

1 2 2

~  C1 h1 þ a dx þ C1

g r þ r ð I  I Þ þ r I dxdmdx s s s

2 0 0 0 S1 x Z 1  C1 h1 þ a dx  C1 : ð2:2:61Þ 0

In the same manner, we have the same result for all ω 2 (−1, 0). This completes the proof. h Obviously, we can obtain the following result by lemmas 2.2.1 and 2.2.6. Lemma 2.2.7. If assumptions in theorem 2.1.1 hold, then there holds that for any t > 0, Z tZ 1 h þ h 1 þ r þ v 2 2 þ ð1 þ hÞq þ r h2x dxds  C1 ð1 þ KÞb8 ; ð2:2:62Þ 0

0

where b7 ¼ maxð2r þ 1  2q; 0Þ; n 1 3b b þ b þ r þ 1 ; b8 ¼ max b7 ; ðb þ maxð2r þ 2  q; 0ÞÞ; q þ 1 þ b; 1 ; 2 2 2 4 o b b maxðr  q; 0Þ þ 2 ; maxðq  r; 0Þ þ 2 : 2 2

1D Radiative Fluid and Liquid Crystal Equations

38

Proof. Multiplying (2.1.5) by e þ 12 v 2, integrating the resultant over [0, 1] × (0, t) and using lemmas 2.2.1–2.2.6 and (2.1.15), we get Z tZ 1 hðtÞ þ h1 þ r ðtÞ þ v 2 2 þ ð1 þ hÞq þ r h2x dxds  C1 þ C1

Z tZ 0

þ C1

Z tZ

0

ð1 þ hÞ2r þ 1 jvhx jdxds þ C1

0

Z tZ 0

0

1

1

Z tZ 0

ð1 þ hÞq þ r þ 1 jhx gx jdxds þ C1

0

Z tZ 0

1

ð1 þ hÞr þ 1 jvvx gx jdxds þ C1

Z tZ

1

ð1 þ hÞ2r þ 2 jvgx jdxds

0 1

ð1 þ hÞr þ 1 v 2 jvx jdxds

0 1

ð1 þ hÞr jvvx hx jdxds 0 0 0 0   Z tZ 1 Z t Z 1

q

gðSE Þ e þ 1 v 2 dxds ð1 þ hÞ jvvx hx jdxds þ C1 þ C1 R

2 0 0 0 0 8 X Di : :¼ C1 þ þ C1

ð2:2:63Þ

i¼1

Similarly to those in [104], we have estimates of D1, D2, and D3, for any ϵ > 0, Z tZ 1 ð1 þ hÞq þ r h2x dxds þ C1 ðÞð1 þ KÞb7 ; D1   0

0

D2  C1 ð1 þ KÞ2ðb þ maxð2r þ 2q;0ÞÞ ; Z tZ 1 ð1 þ hÞq þ r h2x dxds þ C1 ðÞð1 þ KÞb þ 1 þ q : D3   1

0

0

Here we only give estimates of Di, (i = 4, 5, 6, 7, 8), for any ϵ > 0, 12  Z t Z 1 12 Z t Z 1 ð1 þ hÞ2m vx2 dxds ð1 þ hÞr þ 1q v 4 dxds D4  C 1 0

0

 C1 ð1 þ KÞ

b1 2

sup kvðsÞkL1

s2½0; t

Z t Z 0

1

 C1 ð1 þ KÞ 2 sup kvx ðsÞk2 s2½0; t

0

0

ð1 þ hÞ2m v 2 dxds

12

0

Z t Z 1

b1

0 1

ð1 þ hÞ2m v 2 dxds

12

0

3b1 4

 C1 ð1 þ KÞ ; 12  Z t Z Z t Z 1 r þ1 2 ð1 þ hÞ gx dxds D5  C 1

1

Z

12

0

0

 C1 ð1 þ KÞ

b r þ1 2þ 2

0

 C1 ð1 þ KÞ

b þ b2 2

þ r þ2 1

;

t

Z kvx k2L1

0

0

1

v 2 dxds 0

ð1 þ hÞr þ 1 v 2 vx2 dxds

12

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

D6  

Z tZ 0

 D7  

1

0

Z tZ 0

1

0

Z tZ 0



0

Z tZ 0

1

0

1

ð1 þ hÞr þ q h2x dxds þ C1 ðÞ

Z tZ 0

1

0

39

ð1 þ hÞrq v 2 vx2 dxds b2

ð1 þ hÞr þ q h2x dxds þ C1 ðÞð1 þ KÞmaxðrq;0Þ þ 2 ; ð1 þ hÞr þ q h2x dxds þ C1 ðÞ

Z tZ 0

0

1

ð1 þ hÞqr v 2 vx2 dxds b2

ð1 þ hÞr þ q h2x dxds þ C1 ðÞð1 þ KÞmaxðqr;0Þ þ 2 :

At last, we use corollary 2.2.1, definitions of S and Ĩ, (2.1.17) and (2.2.61) to get

Z t Z 1

 1 

D8  gðSE ÞR e þ v 2 dxds 2 0 0 Z 1 Z 1 Z 1Z 1

Z t Z 1 

1 2

~ ¼ g eþ v ra ðB  I Þdmdx þ rs ðI  I Þdmdx dxds 2 0 0 1 0 1 0 Z 1 Z 1

Z t Z 1 

1

 g e þ v2 ra ðB  I Þdmdxdxds 2 0 0 1 0 Z 1 Z 1 Z t Z 1 

1

 g e þ v 2 ra ðB  I Þdmdx dxds 2 0 0 1 0 Z Z Z t Z 1  Z 1Z 1  1  1 1 1 þ a

 h fdmdx þ gra Idmdx dxds

e þ v2 2 0 0 1 0 1 0 Z t Z t Z 1  2 1þa  C1 V ðsÞds þ C1 kv kL1 h dx þ C1 ds 0 0 0 Z t Z t  2  C1 V ðsÞ þ kv kL1 ds  C1 þ kv kkvx kds 0 0 Z  t 12 b  C1 þ kvx k2 ds  C1 ð1 þ KÞ4 : 0

Inserting estimates of D1, D2, D3, D4, and D5 into (2.2.63), using the Young inequality and taking ε > 0 small enough, we conclude Z tZ 1 hðtÞ þ h1 þ r ðtÞ þ v 2 2 þ ð1 þ hÞq þ r h2x dxds  C1 Kb8 : 0

The proof is hence complete.

0

h

In the next lemma, we shall prove the new uniform-in-time upper bound on temperature θ(x, t). The difficulty of the proof is how to derive the uniform-in-time Rt R1 estimate on the last term 0 0 gðSE ÞR Kt dxds in (2.2.67) below.

1D Radiative Fluid and Liquid Crystal Equations

40

Lemma 2.2.8. If assumptions in theorem 2.1.1 hold, the there holds that for any t > 0, Z 1 Z tZ 1 ð1 þ h2q Þh2x ðx; tÞdx þ ð1 þ hq þ r Þh2t dxds  C1 ð1 þ KÞb13 ; ð2:2:64Þ 0

0

0

K  C1 ;

ð2:2:65Þ

where b9 ¼ f2 maxðq  r; 0Þ þ 2b þ b8 ; maxðq  r; 0Þ þ b þ ðb8 þ b1 þ 1Þ=2; b10

maxðq  r; 0Þ þ b þ ðb8 þ b6 Þ=2; maxðq  r; 0Þg; ¼ maxfmaxðq  r; 0Þ þ q þ 2 þ b; 2 maxðq  r; 0Þ þ r þ 2 þ 2b; maxðq  r; 0Þ þ b þ ðb1 þ r þ 3Þ=2; maxðq  r; 0Þ þ b þ ðb6 þ r þ 2Þ=2; ðmaxðq  rÞ; 0Þg;

b11 ¼ minðb9 ; b10 Þ;  maxð3q þ 2  r; 0Þ þ b1 þ b8 maxð3q þ 2  r; 0Þ þ b2 þ b þ b8 ; b12 ¼ ; 2 2  3q þ 4 þ b1 3q þ 4 þ b2 þ b ; ; b13 ¼ max 2 2 b14 ¼ minfb12 ; b13 g; b15 ¼ maxfb1 þ 1; b4 ; b6 ; b11 ; b14 ; maxð2q þ 2; b1 Þg: Proof. Let Z Kðg; hÞ ¼ XðtÞ ¼

0

Z tZ 0

1

ð1 þ h

0

q þr

h

jðg; uÞ du; g

Þh2t dxds;

Z

Y ðtÞ ¼

Then it is easy to verify that j Kt ¼ Kg vx þ ht ; g

0

1

ð1 þ h2q Þh2x dx:

    jht j Kxt ¼ þ Kgg vx gx þ g ht þ Kg vxx : g g x g t

We know from (2.1.18) that jKg j þ jKgg j  C1 ð1 þ hq þ 1 Þ:

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

41

The equation (2.1.5) can be rewritten as 

l eh ht þ hph vx  vx2 ¼ g

jhx g

 gðSE ÞR :

ð2:2:66Þ

x

Multiplying (2.2.66) by Kt and integrating the result over [0, 1] × (0, t), we arrive at Z tZ

Z t Z 1 h  l  jhx n jht  eh ht þ hph vx  vx2 Kt dxds þ g g g t 0 0 0 0 j o i þ Kgg vx gx þ Kg vxx þ g ht þ gðSE ÞR Kt dxds ¼ 0: g g x 1

ð2:2:67Þ

Now we estimate each term in (2.2.67). Similarly to those in [104], we have Z tZ 0

Z t Z

0

1



eh ht Kt dxds  C1 XðtÞ  C1 ð1 þ KÞb1 þ 1 ;

ð2:2:68Þ

0

hph vx 

0

1

C l0 vx2 

1 XðtÞ þ C1 ð1 þ KÞb1 þ 1 Kt dxds  g 8 ð2:2:69Þ

þ C1 ð1 þ KÞb4 þ C1 ð1 þ KÞb6 ; Z tZ 0

Z t Z

1 0

jhx  jhx  dxds  C1 Y ðtÞ  C1 ; g g t

ð2:2:70Þ

 jhx 

Kg vxx þ Kgg vx gx dxds g 0 0 Z tZ 1  C1 ð1 þ hÞ2q þ 1 jhx vxx þ hx vx gx jdxds 1

0

0

 C1 ð1 þ KÞ

maxð3q þ 2r;0Þ 2

Z t Z 0

maxð3q þ 2r;0Þ 2

þ C1 ð1 þ KÞ

0

maxð3q þ 2r;0Þ þ b1 þ b8 2

b12

 C1 ð1 þ KÞ :

ð1 þ hÞq þ r h2x dxds

Z t Z 0

 C1 ð1 þ KÞ

1

0

1

12  Z t Z

ð1 þ hÞq þ r h2x dxds

0

12  Z

maxð3q þ 2r;0Þ þ b2 þ b þ b8 2

þ C1 ð1 þ KÞ

0

0 t

1 2 vxx dxds

12

kvx k2L1 kgx k2 dxds

12

1D Radiative Fluid and Liquid Crystal Equations

42

On the other hand, it is easy to verify

Z t Z 1 jh

x ðKg vxx þ Kgg vx gx Þdxds

g 0 0 Z tZ 1  C1 ð1 þ hÞ2q þ 1 jhx vxx þ hx vx gx jdxds 0

0

12  Z t Z 1 12 ð1 þ hÞq h2x 2  C1 ð1 þ KÞ dxds vxx dxds 2 h 0 0 0 0 12  Z t 12  Z t Z 1 ð1 þ hÞq h2 3q þ 4 2 2 x þ C1 ð1 þ KÞ 2 dxds v k k 1 kgx k dxds x L h2 0 0 0 3q þ 4 2

Z t Z

3q þ 4 þ b1 2

 C1 ð1 þ KÞ

1

þ C1 ð1 þ KÞ

3q þ 4 þ b2 þ b 2

 C1 ð1 þ KÞb13 : Therefore

Z t Z

0

0

1

jhx

ðKg vxx þ Kgg vx ux Þdxds  C1 ð1 þ KÞb14 : g

ð2:2:71Þ

Now we estimate the last two terms in (2.2.67). First, by lemmas 2.2.1–2.2.7, we have Z tZ 1

Z t Z 1 jh  j 

jh



x

x gx ht dxds gx ht dxds  C1 ð1 þ hÞq

g g g g 0 0 0 0 Z tZ 1 C1 jhx 2  XðtÞ þ C1 ð1 þ hÞqr g2x dxds 8 g 0 0 Z t 2 jhx C1 maxðqr;0Þ þ b ds XðtÞ þ C1 ð1 þ KÞ  g 1 8 0 L Z t 2 jhx C1 maxðqr;0Þ þ b XðtÞ þ C1 ð1 þ KÞ  g 8 0    Z 1

jhx jhx þ

dx ðsÞds g g x 0 C1 XðtÞ þ C1 ð1 þ KÞmaxðqr;0Þ þ b  8 Z tZ 1 n  ð1 þ KÞmaxðqr;0Þ ð1 þ hÞq þ r h2x dxds þ

0



0

Z t Z

1

0

Z t Z 0

1 0

0

ð1 þ hÞq þ r h2x dxds ð1 þ hÞ

1=2

 

jhx

g

qr

x

2 1=2 o

dxds ;

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

43

which, along with (2.2.66), lemmas 2.2.4, 2.2.5 and corollary 2.2.2, leads to

Z t Z

0

0

1

jhx  j 

gx ht dxds g g g

C1 XðtÞ þ C1 ð1 þ KÞ2 maxðqr;0Þ þ b þ b8 þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ b8 =2  8  1=2 Z tZ 1h i q þr þ2 2 qr 4 qr 2 2 ð1 þ hÞ  XðtÞ þ vx þ ð1 þ hÞ vx þ ð1 þ hÞ g ðSE ÞR dxds 0

0

C1 XðtÞ þ C1 ð1 þ KÞ2 maxðqr;0Þ þ 2b þ b8 þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ ðb8 þ b1 þ 1Þ=2  4 þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ ðb8 þ b6 Þ=2 þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ b7 C1 XðtÞ þ C1 ð1 þ KÞb9 :  4

ð2:2:72Þ

However, we also know that

Z t Z 1 jh  j 

x gx ht dxds

g g g 0 0 Z tZ 1 C1 jhx 2  XðtÞ þ C1 ð1 þ hÞqr g2x dxds 8 g 0 0 C1  XðtÞ þ C1 ð1 þ KÞmaxðqr;0Þ þ b 8 Z t Z 1 Z tZ 1

 jh 

x 2q 2 q  ð1 þ hÞ hx dxds þ ð1 þ hÞ jhx j

dxds g x 0 0 0 0 Z t n C1  XðtÞ þ C1 ð1 þ KÞmaxðqr;0Þ þ b Kq þ 2 V ðsÞds 8 0

  2 Z t 1=2  Z t Z 1 1=2 o

jhx

dxds þ V ðsÞds h2 ð1 þ hÞq

g 0

0

0

x

C1 XðtÞ þ C1 ð1 þ KÞmaxðqr;0Þ þ q þ 2 þ b  8 Z t Z þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ ðr þ 2Þ=2 0

1 0

1=2

 jh  2

x ð1 þ hÞqr dxds

g x

C1 XðtÞ þ C1 ð1 þ KÞmaxðqr;0Þ þ q þ 2 þ b þ C1 ð1 þ KÞ2 maxðqr;0Þ þ 2b þ r þ 2  4 þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ ðr þ 3 þ b1 Þ=2 þ C1 ð1 þ KÞmaxðqr;0Þ þ b þ ðr þ 2 þ b6 Þ=2 þ C1 ð1 þ KÞmaxðqr;0Þ C1 XðtÞ þ C1 ð1 þ KÞb10 :  4

1D Radiative Fluid and Liquid Crystal Equations

44

Next, we estimate the last term,

Z t Z 1Z

Z t Z 1

gðSE ÞR Kt dxds 

0

0

0

þ

0

Z tZ 0

þ

0

Z

1

Z

0

Z

 gra Bdmdx jKt jdxds

S1 1Z

0

1

0

Z tZ

1

S1

0 1

Z

S1

0

 gra Idmdx jKt jdxds  grs jI~  I jdmdx jKt jdxds ð2:2:73Þ

¼: P þ Q þ R:

Using (2.1.18), lemma 2.2.6, the Young and the Hölder inequalities, we conclude

Z t Z 1Z 1Z 

j

gra Bdmdx Kg vx þ ht dxds jPj  g 0 0 0 S1 Z tZ 1 Z tZ 1 ð1 þ hq þ 1 þ a Þjvx jdxds þ C1 ð1 þ hq þ a Þjht jdxds  C1 0

0

0

0

1  XðtÞ þ C1 Kq þ 2ar ; 8 Z tZ

Z

Z

 j

gra Idmdx Kg vx þ ht dxds g S1 0 0 0 12 Z 1 Z 12 Z t Z 1 Z 1 Z 2  C1 gra dxdm gra I dxdm ð1 þ hq þ 1 Þjvx jdxds

Q

0

1

0

1

0

1 Z

S1

0

12 Z

S1

12 gra I 2 dxdm ð1 þ hq þ 1 Þjht jdxds 0 0 0 0 S1 S1  Z tZ 1 Z t Z 1 Z 1 Z 2q þ 2 2 2  C1 ð1 þ KÞ jvx j dxds þ XðtÞ þ C1 ðÞ gra I dxdm þ C1

Z tZ

1Z

1

Z

gra dxdm

0

 XðtÞ þ C1 ð1 þ KÞ

0 2q þ 2

0

0

0

S1

:

Using the same technique, we also get 1 R  XðtÞ þ C1 Kmaxðqr;0Þ : 8 Inserting estimates on P, Q and R into (2.2.73), using the Young inequality and taking  > 0 small enough, we derive XðtÞ þ Y ðtÞ  C1 ð1 þ KÞb15 :

ð2:2:74Þ

By lemmas 2.2.1–2.2.7 and the Hölder inequality, there exists a point a(t) 2 [0, 1] R1 such that for any t [ 0; 0 hðx; tÞdx ¼ hðaðtÞ; tÞ, we can hence deduce

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

jh

q þ r þ2 3

ðx; tÞ  h

q þ r þ2 3

Z

1

ðaðtÞ; tÞj  C1

hq þ

r þ3 2

jhx jdx

0

 C1

Z

1 0

45

ð1 þ h2q Þh2x dx

12  Z

1

hr þ 1 dx

12

0

1 2

 C1 Y ðtÞ: Then b15

1

K  C1 Y 2q þ r þ 3 þ C1  C1 ð1 þ KÞ2q þ r þ 3 :

ð2:2:75Þ

After a lengthy calculation, it is easy to verify that assumption (2.1.16) implies that β15 < 2q + r + 3. Therefore, by the Young inequality, it follows from (2.2.75) that K  C1 : h

This thus completes the proof.

The following lemmas are concerned with the new arguments on uniform-in-time estimates of radiative term I(x, t; ν, ω). Lemma 2.2.9. There holds Z 1 Z 2 I dxdm 1 Z

0

S

1

S1

0

Z tZ 0

Z

0

1

Z

L1 ðQt Þ

jIx jdxdm

1

Z S1

0

L1 ðQt Þ

 C1 ;

8 t [ 0;

ð2:2:76Þ

 C1 ;

8 t [ 0;

ð2:2:77Þ

It2 dxdmdxds  C1 ;

8 t [ 0:

ð2:2:78Þ

Proof. By (2.2.57) and (2.2.66), we can easily get (2.2.76). By the definition of Ĩ, we can derive from (2.2.76) that Z 1 Z ~  C1 : ð2:2:79Þ I dxdm 1 1 0

L ðQt Þ

S

From (2.2.60), using the Young and the Hölder inequalities, (2.1.18) and lemmas 2.2.7, 2.2.8, we derive Z 1Z Z 1 Z Z 1 2 g ðra B þ rs I~Þdx dmdx I 2 dmdx  0 0 0 x S1 S1 Z 1Z  2 Z 1  g  C1 r dx  ra B 2 dx dxdm a 2 S1 x 0 0 Z 1 Z Z 1 2 Z 1  g þ C1 rs dx  rs I~2 dx dxdm 2 0 0 x 0 S1 Z 1Z Z 1  C1 þ C1 ðrs ðI~  I Þ2 þ rs I 2 Þdxdxdm  C1 ; ð2:2:80Þ 0

S1

0

1D Radiative Fluid and Liquid Crystal Equations

46

which, by the definition of Ĩ, implies Z 1Z S1

0

I~2 dxdm  C1 :

ð2:2:81Þ

From (2.1.6), we get Ix ¼ 

g g ðra þ rs ÞI þ ðra B þ rs ~I Þ: x x

Integrating the above equality and using (2.1.18), we obtain Z 1Z Z 1Z 1 ðra þ rs ÞjI jdxdm jIx jdxdm  C1 1 jxj 0 0 S1 Z 1S Z 1 ðra B þ rs I~Þdxdm þ C1 jxj 1 0 S Z 1Z Z 1Z jI jdxdm þ C1 j~I jdxdm  C1 þ C1 0

S1

0

ð2:2:82Þ

S1

which, using (2.2.76) and (2.2.81), gives (2.2.77). By (2.2.60), we have for any ω 2 (0, 1), Z x Ry  g Z y g g ðr þ r Þdz ðra þ rs Þdz ðra B þ rs I~Þ It ¼ e xx a s tx 0 x x Z x Ry   g ðr þ r Þdz g ðra B þ rs I~Þ dy ¼: A2 þ B2 : þ e xx a s t x 0

ð2:2:83Þ

Using the Young inequality, (2.2.81), (2.1.18), lemmas 2.2.4 and 2.2.8, we deduce Z tZ 1Z A22 dxdmds 0 0 S1 Z t Z 1 Z hZ x Z y vx g ðra þ rs Þ þ ððra þ rs Þg vx  C1 x S1 0 0 0 x x g i2 ðra B þ rs I~Þdy dxdmds þ ðra þ rs Þh ht Þdz x Z t Z 1 Z Z 1 2 vx  C1 ðr þ rs þ ðra Þg þ ðrs Þg Þ2 2 a 0 0 0 x S1 Z 1 2 h2 g 2 2  þ t2 ððra Þh þ ðrs Þh Þ2 dx  r B dx dxdmds 2 a x 0 x Z t Z 1Z Z 1 2 vx þ C1 ðr þ rs þ ðra Þg þ ðrs Þg Þ2 2 a x 1 S 0 0 0 Z 1 2 h2t g 2 ~2  2 þ 2 ððra Þh þ ðrs Þh Þ dx  r I dx dxdmds 2 s x 0 x Z tZ 1 Z tZ 1Z Z 1  C1 ðvx2 þ h2t Þdxds þ C1 I~2 dxdxdmds 0

 C1 :

0

0

0

S1

0

ð2:2:84Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

Analogously, Z tZ 1Z Z tZ B22 dxdmds  C1 0

0

1

Z hZ

x

47

v

g ðra B þ rs I~Þ þ ððra Þg vx B x x 0 0 0 S1  i2 þ ðra Þh ht B þ rBh ht þ ðrs Þg I~vx þ ðrs Þh ht ~ I þ rs I~t Þ dx dxdmds Z tZ 1 Z xZ tZ 1Z 2 2  C1 ðvx þ ht Þdxds þ C1 It2 dxdmdsdy 0 0 0 0 0 S1 Z xZ tZ 1Z  C1 þ C1 It2 dxdmdsdy;

S1

0

0

0

x

S1

which, together with (2.2.84), implies Z Z tZ 1Z 2 It dxdmds  C1 þ C1 0

S1

0

0

x

Z tZ 0

1

0

Z S1

It2 dxdmdsdy:

Fixing t > 0 and using the Gronwall inequality, we get Z tZ 1Z It2 dxdmds  C1 eC1 x  C1 eC1  C1 ; 8 x 2 ½0; 1: 0

S1

0

h

This proves the proof. Lemma 2.2.10. If assumptions in theorem 2.1.1 hold, then there holds that kI xx ðtÞk  C2 ;

8 t [ 0:

ð2:2:85Þ

Proof. By virtue of the direct computation, we have Z 1Z 1 Z 2 Ixx dxdm dx kI xx ðtÞk2 ¼ Z

0

S1

0

Z

Z

2 1 ðgx S þ gSx Þdxdm dx S1 x 0 0 Z 1 h Z 1 Z 2  Z 1 Z 1 2 i 1 gx Sdxdm þ gSx dxdm dx  C1 S1 x S1 x 0 0 0 ¼: G þ H : ¼

1

1

Using (2.1.18), (2.2.40) and lemma 2.2.9, we see that Z 1Z 1Z 2 1 gx Sdxdm dx G¼ S1 x 0 0 Z 1 Z 1Z 2 1 ðra ðB  I Þ þ rs ðI~  I ÞÞdxdm dx ¼ g2x 0 0 S1 x Z 1  C1 g2x dx  C1 : 0

ð2:2:86Þ

ð2:2:87Þ

1D Radiative Fluid and Liquid Crystal Equations

48

Similarly, Z H ¼

1

Z

1

Z

g f½ðra Þg gx þ ðra Þh hx ðB  I Þ þ ra ðBh hx  Ix Þ S1 x 0 0 2 ~ ~ þ ½ðrs Þg gx þ ðrs Þh hx ðI  I Þ þ rs ðI  I Þx gdxdm dx Z

 C1 0

1

ðg2x þ h2x Þdx þ C1

 C1 :

ð2:2:88Þ

Plugging (2.2.87) and (2.2.88) into (2.2.86), we can get (2.2.85). The proof is complete. h

2.3

Asymptotic Behavior of Solutions in H1

This will establish the large-time behavior of global solutions in H1 , which completes the proof of theorem 2.1.1. Lemma 2.3.1. If assumptions in theorem 2.1.1 hold, then we have lim kgðtÞ  gkH 1 ¼ 0;

ð2:3:1Þ

lim kvðtÞkH 1 ¼ 0;

ð2:3:2Þ

t! þ 1

t! þ 1

where g ¼

R1 0

gðy; tÞdy ¼

R1 0

g0 ðyÞdy:

Proof. By lemmas 2.2.1–2.2.8, we can derive from (2.2.47) d kvx k2 þ C11 kvxx k2  C1 : dt

ð2:3:3Þ

Applying lemma 1.1.2 to (2.3.3), and using the Poincaré inequality, we obtain kvðtÞk2H 1  C1 kvx k2 ! 0 as t ! 1: Similarly, by lemmas 2.2.1–2.2.8, it follows from (2.2.42) that 2 Z 1 d v  l gx þ C 1 ð1 þ h1 þ r Þg2x dx  C1 : 1 dt g 0

ð2:3:4Þ

Applying lemma 1.1.2 to (2.3.4) again, and using lemmas 2.2.1–2.2.8 and (2.3.1), we conclude that as t → ∞, ! 2 gx 2 2 kgx k  C1 v  l þ kv k ! 0 g which gives (2.3.2).

h

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

49

Lemma 2.3.2. If assumptions in theorem 2.1.1 hold, then we have lim hðtÞ  h H 1 ¼ 0;

ð2:3:5Þ

t! þ 1

where h [ 0 is determined by eðg; hÞ ¼

R1 0

ð12 v02 þ eðg0 ; h0 Þ þ FR ð0ÞÞdx.

Proof. Equation (2.1.5) can be rewritten as eh ht þ ðp þ hph Þvx  rvx þ qx þ gðSE ÞR ¼ 0:

ð2:3:6Þ

Multiplying (2.3.6) by eh1 hxx , integrating the result over (0, 1) and using the Young inequality, the interpolation inequality and lemmas 2.2.1–2.2.8, we can conclude for any ε > 0, Z 1 2 d jhxx khx ðtÞk2 þ 2 dt 0 eh g Z 1h j gðSE ÞR i hph vx lvx2 ðgÞx hx   þ ¼2 hxx dx eh eh g eh eh 0 2 e  khxx ðtÞk2 þ C1 ðkvx k2 þ kvx k4L4 þ khx k4L4 þ kgx hx k2 þ ðSE ÞR Þ 2 e  khxx ðtÞk2 þ C1 ðkvx k2 þ kvx k3 kvxx k þ kvx k4 þ khx k3 khxx k þ khx k4 2 2 þ khx kL1 þ ðSE ÞR Þ 2 ð2:3:7Þ  ekhxx ðtÞk2 þ C1 ðkvx k2 þ kvxx k2 þ khx k2 þ ðSE ÞR Þ: Now we need to estimate the new radiative term η(SE)R in (2.3.7), which did not appear in the system considered in Qin [104] ((SE)R = 0). From (2.1.18), the Young and the Hölder inequalities and lemmas 2.2.7, 2.2.8, we derive Z 1  Z þ1 Z 2 ðSE Þ 2 ¼ ðra ðB  I Þ þ rs ðI~  I ÞÞdxdm dx R 0

1

 C1

h Z

0

þ

S1 þ1

0

Z

Z 0 1

Z  C1

Z S1

0

þ1

Z

h Z

ra ðB  I Þdxdm

rs ðI~  I Þdxdm

S1 þ1

2 i dx

 Z

Z

2

þ1

Z

ra dxdm 0

þ C1

0 þ1

Z

Z  C1 0

Z

S1

0 þ1

 Z

Z

rs dxdm 0 1

S1

ha þ 1 dx þ C1  C1 ;

0

S1

S1

ra ðB 2 þ I 2 Þdxdm

rs ðI~  I Þ2 dxdm



i dx ð2:3:8Þ

1D Radiative Fluid and Liquid Crystal Equations

50

which, together with (2.3.7), yields Z

t

2

khx ðtÞk þ

khxx ðsÞk2 ds  C1 :

0

Plugging (2.3.8) into (2.3.7), using lemmas 2.2.1–2.2.8 and taking ε > 0 small enough, we have Z 1 d ð1 þ hqr Þh2xx dx  C1 ðkvxx k2 þ 1Þ; ð2:3:9Þ khx ðtÞk2 þ C1 dt 0 which, together with lemma 1.1.2 and (2.2.44), (2.2.65), yields lim khx ðtÞk2 ¼ 0:

ð2:3:10Þ

t! þ 1

By the Poincaré inequality, we deduce hðtÞ  h 1  C1 khx ðtÞk; H which, combined with (2.3.10), gives (2.3.5). The proof is thus complete.

h

The following lemma is the new argument on the large-time behavior of radiative term I(x, t; ν, ω). Lemma 2.3.3. If assumptions in theorem 2.1.1 hold, then we have lim kI ðtÞkH 2 ¼ 0:

ð2:3:11Þ

t! þ 1

Proof. By (2.2.76) and (2.1.6), the direct computation yields Z Z Z 2 d d 1  þ1 kI x ðtÞk2 ¼ Ix dxdm dx dt dt 0 S1 0 Z 1  Z þ1 Z  Z þ 1 Z  ¼2 Ix dxdm Ixt dxdm dx 0

Z  C1 0

Z  C1

0

S1

0

1

1

Z Z

þ1

S1

0 þ1 0

Z



0

S1

jIxt jdxdm dx

Z

 1 jvx S þ gSt jdxdm dx ¼: A3 þ B3 : S1 x

We denote Z

1 A3 ¼

vx S dxdmdx x 1 0 0 S Z 1 Z þ1 Z

1 b þ D: b ¼

vx ðra ðB  I Þ þ rs ðI~  I ÞÞ dxdmdx ¼: C S1 x 0 0 Z

1

Z

þ1

ð2:3:12Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

51

Using (2.1.18), lemmas 2.2.7–2.2.9 and the Young inequality, we can derive Z 1 Z þ1 Z

1 b C

vx ra ðB  I Þ dxdmdx S1 x 0 0 Z 1 Z þ1 Z 1 ðjxjha þ 1 f ðm; xÞ þ jxjgðm; xÞI Þdxdmdx  C1 jvx j jxj 1 0 0 S Z 1 Z 1  C1 jvx jdx  C1 vx2 dx þ C1 ; 0

1

Z

0

Z

1 b D

vx rs ðI~  I Þ dxdmdx x 1 S 0 0 Z 1 Z þ1 Z 1 jxjkðm; xÞjI~  I jdxdmdx  C1 jvx j jxj 1 S 0 0 Z 1 Z 1 Z Z þ1 Z  C1 jvx j I dxdmdx  C1 jvx jdx  C1 Z

þ1

0

S1

0

0

1 0

vx2 dx þ C1 :

We denote Z

1

gSt dxdmdx x 1 S 0 0 Z 1 Z þ 1 Z nh i

g ¼ ðra Þg vx þ ðra Þh ht ðB  I Þ þ ra ðBh ht  It Þ

0 S1 x h 0 i o

þ ðrs Þg vx þ ðrs Þh ht ðI~  I Þ þ rs ðI~  I Þt dxdmdx Z

B3 ¼

1

Z

þ1

¼: E þ F; where Z E¼

Z nh 6 i o X

g

ðra Þg vx þ ðra Þh ht ðB  I Þ þ ra ðBh ht  It Þ dxdmdx ¼: Pi ;

0 0 S1 x i¼1 Z 1 Z þ 1 Z nh i o

g ðrs Þg vx þ ðrs Þh ht ðI~  I Þ þ rs ðI~  I Þt dxdmdx: F ¼:

x 1 0 0 S 1

Z

þ1

Using (2.1.18), lemmas 2.2.7–2.2.9 and the Young inequality, we can conclude Z þ1 Z Z 1 1 jðra Þg Bjdxdmdx jvx j jP1 j  C1 0 0 S 1 jxj Z 1 Z þ1 Z 1 jxjhðm; xÞdxdmdx  C1 jvx j jxj 1 S 0 0 Z 1 Z 1  C1 jvx jdx  C1 vx2 dx þ C1 : 0

0

1D Radiative Fluid and Liquid Crystal Equations

52

Analogously, by lemmas 2.2.7–2.2.9, Z 1 Z þ1 Z jvx j jP2 j  C1

1 jðra Þg I jdxdmdx S 1 jxj 0 0 Z 1 Z 1  C1 jvx jdx  C1 vx2 dx þ C1 ; 0

Z

Z

1

þ1

0

Z

1 jðra Þh jBdxdmdx jxj 0 0 Z 1 Z 1  C1 jht jdx  C1 h2t dx þ C1 ;

jP3 j  C1

ht

S1

0

Z

1

Z

Z

0

þ1

Z

Z

0

þ1

1 jðra Þh I jdxdmdx S 1 jxj 0 0 Z 1 Z 1  C1 jht jdx  C1 h2t dx þ C1 ;

jP4 j  C1

jht j

0

Z

1

1 jra Bh jdxdmdx 0 0 S 1 jxj Z 1 Z 1  C1 jht jdx  C1 h2t dx þ C1 ;

jP5 j  C1

0

Z

1

jht j

Z

þ1

0

0

0

Z

1 ra jIt jdxdmdx jP6 j  C1 jxj 1 S 0 0 Z 1 Z þ1 Z  C1 þ It2 dxdmdx: S1

Now we estimate F as follows, by lemmas 2.2.7–2.2.9, Z 1 Z þ1 Z Z 1 Z þ1 Z g g jðrs Þg vx kI~  I jdxdmdx þ jðrs Þh ht kI~  I jdxdmdx F jxj jxj 1 1 0 0 0 0 S S Z 1 Z þ1 Z 3 X g rs jðI~  I Þt jdxdmdx ¼: Mi : þ 0 0 S 1 jxj i¼1 By (2.1.18) and lemmas 2.2.7–2.2.9, we deduce Z 1 Z þ1 Z g jðrs Þg ~ jvx j M1  I  I jdxdmdx S 1 jxj 0 0 Z 1 Z 1 Z þ1 Z  C1 jvx j jI jdxdmdx  C1 vx2 dx þ C1 ; 0

Z M2  C1

1

Z

0

þ1

jht j

0

Z

M3  C1 þ C1 0

Z

S1

0 1

Z 0

S1

þ1

Z

0

jI jdxdmdx  C1 0

Z

S1

It2 dxdmdx:

1

h2t dx þ C1 ;

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

53

Inserting all previous estimates into (2.3.12) implies d kI x ðtÞk2  C1 ðkvx ðtÞk2 þ kht ðtÞk2 þ kIt k2L2 ðS 1 R þ XÞ Þ þ C1 ; dt

ð2:3:13Þ

which, together with lemma 1.1.2, (2.2.41), (2.2.64), (2.2.65) and (2.2.78), yields lim kI x ðtÞk2 ¼ 0:

t! þ 1

From (2.1.6), we derive Z þ 1 Z 2 Z ¼ I dxdm kI xx ðtÞk2 ¼ xx

2 1 ðgx S þ gSx Þdxdm S1 S1 x 0 0 Z þ 1 Z 2 Z þ 1 Z 2 1 1 gx Sdxdm gSx dxdm  C1 þ C1 x x 1 1 0 0 S S þ1

Z

¼: N1 þ N2 : Using (2.1.18) and lemmas 2.2.4–2.2.9, we deduce Z 1  Z þ1 Z 2 1 ½ra ðB  I Þ þ rs ð~I  I Þdxdm dx g2x N1  C 1 0 0 S1 x  C1 kgx ðtÞk2 ; Z

Z

ð2:3:14Þ

ð2:3:15Þ

ð2:3:16Þ

Z

1 ððr1 Þg gx þ ðra Þh hx ÞðB  I Þ þ ra ðBh hx  I Þ 0 0 S1 x  2 þ ððrs Þg gx þ ðrs Þh hx ÞðI~  I Þ þ rs ðI~  I Þx dxdm dx

N2  C 1

1

þ1

 C1 ðkgx ðtÞk2 þ khx ðtÞk2 þ kI x ðtÞk2 Þ:

ð2:3:17Þ

Inserting (2.3.16) and (2.3.17) into (2.3.15) and using (2.3.1), (2.3.10) and (2.3.14), we get lim kI xx ðtÞk ¼ 0:

t! þ 1

ð2:3:18Þ

Thus (2.3.11) follows from (2.3.14), (2.3.18).

h

Proof of Theorem 2.1.1. Combining lemmas 2.2.1–2.2.8 and 2.3.1–2.3.3, we can complete the proof of theorem 2.1.1. h

2.4

Global Existence and Uniform-in-Time Estimates in H2

This section will prove the global existence of solutions and uniform-in-time estimates in H2 . The next lemma concerns the uniform-in-time global (in time) positive lower bound (independent of t) of the absolute temperature θ.

1D Radiative Fluid and Liquid Crystal Equations

54

Lemma 2.4.1. If assumptions in theorem 2.1.1 hold, then the generalized global solution ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ to the problem (2.1.3)–(2.1.6) and (2.1.11)–(2.1.14) satisfies for all (x, t) 2 [0, 1] × [0, +∞), 0\C11  hðx; tÞ:

ð2:4:1Þ

Proof. We prove (2.4.1) by contradiction. If (2.4.1) is not true, that is, inf

ðx;tÞ2½0; 1½0; þ 1Þ

hðx; tÞ ¼ 0;

then there exists a sequence (xn, tn) 2 [0, 1] × [0, +∞) such that as n → +∞, hðxn ; tn Þ ! 0:

ð2:4:2Þ

If sequence {tn} has a subsequence, denoted also by tn, converging to +∞, then by the asymptotic behavior results in theorem 2.1.1, we know that as n → +∞, hðxn ; tn Þ ! h [ 0 which contradicts (2.4.2). If sequence {tn} is bounded, i.e., there exists constant M > 0, independent of n, such that for any n = 1, 2, 3,…, 0 < tn ≤ M. Thus there exists point (ðx ; t Þ, t*) 2 [0, 1] × [0, M] such that ðxn ; tn Þ ! ðx ; t Þ as n → +∞. On the other hand, by (2.4.2) and the continuity of solutions in lemmas 2.2.1 and 2.2.2, we conclude that hðxn ; tn Þ ! hðx ; t Þ ¼ 0 as n ! þ 1, which contradicts (2.2.1). Thus the proof is complete. h Lemma 2.4.2. If assumptions in theorem 2.1.2 hold, then the following estimates hold for all t > 0, Z t ðkvxt k2 þ khxt k2 ÞðsÞds  C2 ; ð2:4:3Þ kht ðtÞk2 þ kvt ðtÞk2 þ 0

Z 2

t

2

kvxx ðtÞk þ khxx ðtÞk þ

ðkvxxx k2 þ khxxx k2 ÞðsÞds  C2 ;

ð2:4:4Þ

kgxx ðsÞk2 ds  C2 :

ð2:4:5Þ

0

Z 2

kgxx ðtÞk þ

t

0

Proof. Differentiating (2.1.4) with respect to t, multiplying the result by vt and integrating over (0, 1), we infer that d 1 kvt ðtÞk2 þ C11 kvxt ðtÞk2  kvxt ðtÞk2 þ C1 ðkvx ðtÞk2 þ kvx ðtÞk4L4 þ kht ðtÞk2 Þ dt 2C1 1  kvxt ðtÞk2 þ C1 ðkvxx ðtÞk2 þ kht ðtÞk2 Þ 2C1

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

which, together with lemmas 2.2.1–2.2.8, yields Z t Z t 2 2 ðkvxx k2 þ kht k2 ÞðsÞds  C2 : kvxt k ðsÞds  C2 þ C1 kvt ðtÞk þ 0

55

ð2:4:6Þ

0

Differentiating (2.2.42) with respect to x, using (2.1.3) (ηtxx = vxxx), we see that @  gxx  l ð2:4:7Þ  pg gxx ¼ vtx þ Eðx; tÞ @t g with Eðx; tÞ ¼ ðpgg g2x þ 2phg hx gx þ phh h2x Þ þ ph hxx  2l0 vx g2x =g3 þ 2l0 gx vxx =g2 : Multiplying (2.4.7) by ηxx/η, and by the Young inequality, lemmas 2.2.1–2.2.3 and (2.1.18), we can deduce that 2 2 d gxx ðtÞ þ C 1 gxx ðtÞ 1 dt g g 2 1 gxx þ C1 ðkhx ðtÞk4 4 þ kgx ðtÞk4 4 þ kvxt ðtÞk2  L L 4C1 g 2 þ khxx ðtÞk2 þ vx g2x ðtÞ Þ 2 1 gxx ðtÞ þ C2 ðkhxx ðtÞk2 þ kgx ðtÞk2 þ kvxt ðtÞk2 Þ ð2:4:8Þ  2C1 g which, combined with lemmas 2.2.1–2.2.8, gives Z t kgxx ðtÞk2 þ kgxx ðsÞk2 ds  C2 ; 8t [ 0:

ð2:4:9Þ

0

By the embedding theorem, (2.1.4), (2.1.5) and (2.4.6), we conclude for all t > 0, kvx kL1  C1 kvxx k  C1 ðkvx k þ khx k þ kgx k þ kvx kÞ  C1 ; Z 0

t

Z 2

kvxxx k ds  C1

t

ðkvtx k2 þ khxx k2 þ kgxx k2 þ kvxx k2 Þds  C2 :

ð2:4:10Þ ð2:4:11Þ

0

Using equation (2.1.5), lemmas 2.2.1–2.2.8, (2.3.8), the Gagliardo–Nirenberg interpolation inequality and the Young inequality, we have ð2:4:12Þ khxx ðtÞk  C1 ðkht ðtÞk þ gðSE ÞR Þ  C1 ðkht ðtÞk þ 1Þ: Differentiating (2.1.5) with respect to t, multiplying the result by θt and integrating over (0, 1), we infer that for any ε > 0,

1D Radiative Fluid and Liquid Crystal Equations

56

d pffiffiffiffiffi 2 k eh ht ðtÞk þ C11 khxt ðtÞk2 dt n

 ekhxt ðtÞk2 þ C1 khx ðtÞk2 þ kvx ðtÞk2 þ kht ðtÞk3L3 þ kht ðtÞk2 1

1

þ kvxt ðtÞk2 þ ðkht ðtÞk þ kht ðtÞk2 khtx ðtÞk2 Þkhxt ðtÞk 2 o þ C1 ½gðSE ÞR t :

ð2:4:13Þ

Integrating (2.4.12) with respect to t and using lemmas 2.2.1–2.2.8 and the Young inequality, we derive for any ε > 0, Z t 2 kht ðtÞk þ khxt ðsÞk2 ds 0 Z t Z t 5 1 2  C2 þ e ðkht k2 khtx k2 þ kht k3 ÞðsÞds khxt ðsÞk ds þ C1 0 0 Z t 2 ½gðSE Þ  ðsÞds þ C1 R t 0 Z t Z t 4 2 ½gðSE Þ  2 ðsÞds 3 khxt ðsÞk ds þ C1 sup kht ðsÞk þ C1  C2 þ e R t 0st

0

Z  C2 þ e

0

t

1 khxt ðsÞk2 ds þ sup kht ðsÞk2 þ C1 2 0st

Z

0 t

0

½gðSE Þ  2 ðsÞds: R t

ð2:4:14Þ

2 Rt Noting that the new radiative term 0 ½gðSE ÞR t ðsÞds, we need to obtain the uniform-in-time estimate. From (2.2.41), (2.2.46), (2.2.65) and (2.2.78), we can derive Z t ½gðSE Þ  2 ðsÞds R t 0

Z tZ

1

½vx ðSE ÞR þ g½ðSE ÞR t 2 dxds 0 Z t Z 1  Z þ1 Z  C1 vx2 ra ðB  I Þ þ rs ðI~ S1 0 0 0 Z t Z 1 n Z þ1 Z ¼

0

þ C1

0

0

0

S1

2  I Þdxdm dxds

½ðra Þg vx þ ðra Þh ht ðB  I Þ þ ra ðBh ht  It Þ

o2 I  I Þ þ rs ðI~  I Þt dxdm dxds þ ½ðrs Þg vx þ ðrs Þh ht ð~ Z t Z t Z 1 Z þ1 Z  C1 ðkvx k2 þ kht k2 ÞðsÞds þ C1 It2 dxdmdxds 0

0

0

0

S1

ð2:4:15Þ

 C1 :

Inserting (2.2.15) into (2.2.14), then taking supremum in t on the left-hand side of (2.2.14), picking ε > 0 small enough, we can get for all t > 0, Z t ð2:4:16Þ kht ðtÞk2 þ khxt ðsÞk2 ds  C2 ; 0

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

57

which, together with (2.2.12), implies khxx ðtÞk  C2 :

ð2:4:17Þ

Differentiating (2.1.5) with respect to x, using the Young inequality, the Gagliardo–Nirenberg interpolation and the Poincaré inequality and lemmas 2.2.1– 2.2.8, (2.2.16), (2.2.17) and (2.4.5), we deduce Z t Z t ðkgx k2 þ kgxx k2 þ kvx k2 þ kvxx k2 þ kht k2 khxxx ðsÞk2 ds  C2 0 0 Z t ½gðSE Þ  2 ðsÞds þ khxt k2 ÞðsÞds þ C2 R x 0 Z t ½gðSE Þ  2 ðsÞds:  C2 þ C2 ð2:4:18Þ R x 0

The same estimate as (2.4.16) yields Z t Z tZ 1 ½gðSE Þ  2 ðsÞds ¼ ðgx ðSE ÞR þ g½ðSE ÞR x Þ2 dxds R x 0 0 0 Z t  C1 ðkgx k2 þ khx k2 ÞðsÞds 0

Z þ C1

þ1

Z

0

S1

2 jIx jdxdm

L1 ðQt Þ

 C1 ; which, together with (2.4.18), implies that for all t > 0, Z t khxxx ðsÞk2 dxds  C2 :

ð2:4:19Þ

ð2:4:20Þ

0

Thus (2.4.3), (2.4.4) follow from (2.4.6), (2.4.16), (2.4.17) and (2.4.20). The proof follows immediately. h The following two lemmas are the new arguments on the uniform-in-time estimates of radiative term I(x, t; ν, ω). Lemma 2.4.3. There holds that for all t > 0, Z þ 1 Z jIt jdxdm 1 0

S

L1 ðQt Þ

 C2 :

ð2:4:21Þ

Proof. Using (2.1.18), the Young inequality, lemmas 2.2.5 and 2.2.9, we derive from (2.2.83)

1D Radiative Fluid and Liquid Crystal Equations

58 Z

þ1

Z S1

0

Z

jA2 jdxdm þ1 Z Z

x

Z

y

vx ðra þ rs þ ðra þ rs Þg Þ 0 0 x x

g ht

ðra B þ rs I~Þdy dxdm þ ðra þ rs Þh dz x x Z þ 1 Z h Z x  Z y vx

ðra þ rs þ ðra þ rs Þg Þ  C1

0 0 x x S1 Z Z x 2 i x 2 ht g 2 2

2 ~2 þ ðra þ rs Þh dz dy þ r B dy þ r dy I

dxdm a s 2 x 0 x 0 Z 1  C1 ðvx2 þ h2t Þdx  C2 :

 C1

S1

ð2:4:22Þ

0

Z

Analogously, Z þ1 Z jB2 jdxdm  C1

0

Z h Z

g ðra Þg vx B x x 0 0 S1  i

þ ðra Þh ht B þ rBh ht þ ðrs Þg I~vx þ ðrs Þh ht ~I þ rs I~t dy dxdm Z 1 Z x Z þ1 Z

g ~ ðjvx j þ jht jÞdx þ C1  C1

rs It dxdmdy S1 x 0 0 0 Z x Z þ1 Z  C2 þ C1 jIt jdxdmdy;

S1

þ1

0

x

v

x

ðra B þ rs I~Þ þ

S1

0

which, together with (2.4.22) and (2.4.18) and using the Gronwall inequality, implies Z þ1 Z jIt jdxdm  C2 : S1

0

Thus the proof is hence complete.

h

Lemma 2.4.4. If assumptions in theorem 2.1.2 hold, then there holds that kI xxx ðtÞk  C2 :

ð2:4:23Þ

Proof. The elementary computation yields Z 1  Z þ1 Z 2 Ixxx dxdm dx kI xxx ðtÞk2 ¼ 0

Z

Z

0

Z

S1

2 1 ðgxx S þ 2gx Sx þ gSxx Þdxdm dx 0 0 S1 x Z 1 h Z þ 1 Z 2  Z þ 1 Z 1 2 1 gxx Sdxdm þ gx Sx dxdm  C1 S1 x S1 x 0 0 0 3  Z þ1 Z 1 2 i X gSxx dxdm dx ¼: þ Ji : 0 S1 x i¼1 ¼

1

þ1

ð2:4:24Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

59

Employing the Gagliardo–Nirenberg interpolation inequality and using (2.1.18), lemma 2.2.9 and (2.4.5), we conclude Z 1  Z þ1 Z 2 1 gxx Sdxdm dx J1  C1 S1 x 0 0 Z 1  Z þ1 Z 2 1 ½ra ðB  I Þ þ rs ðI~  I Þdxdm dx  C1 g2xx S1 x 0 0 Z 1 g2xx dx  C2 ;  C1 ð2:4:25Þ 0

Z

1

J2  C1

Z

0

þ1

0

Z

1 n gx ½ðra Þg gx þ ðra Þh hx ðB  I Þ S1 x

o 2 þ ra ðBh hx  Ix Þ þ ½ðrs Þg gx þ ðrs Þh hx ðI~  I Þ þ rs ðI~  I Þx dxdm dx Z 1 ðg4x þ g2x h2x þ g2x Þdx  C1 0 Z 1   C1 max g2x ðg2x þ h2x Þdx þ C1 X

0

 C1 ðkgx kkgxx k þ kgx k2 Þ þ C1

ð2:4:26Þ

 C2 :

Analogously, we infer from (2.1.18), (2.4.4), (2.4.5) and (2.2.85) that Z 1  Z þ1 Z gn ½ðra Þg gx þ ðra Þh hx ðB  I Þ J3  C1 S1 x 0 0 2 o þ ra ðBh hx  Ix Þ þ ½ðrs Þg gx þ ðrs Þh hx ð~ I  I Þ þ rs ðI~  I Þx dxdm dx x Z 1  C1 ðg4x þ h4x þ g2x h2x þ g2xx þ h2xx þ I 2xx Þdx 0

Z

 C1 ðkgx kkgxx k þ kgx k2 þ khx kkhxx k þ khx k2 Þ 0

1

ðg2x þ h2x Þdx þ C2

 C2 ; which, along with (2.4.25) and (2.4.26), gives (2.4.23). The proof is thus complete. h By lemmas 2.4.1–2.4.4, we conclude that the global existence of solutions ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ exists in H2 such that for all t > 0, kðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞkH2  C2 :

ð2:4:27Þ

1D Radiative Fluid and Liquid Crystal Equations

60

2.5

Asymptotic Behavior of Solutions in H2

This section will derive the asymptotic behavior of global solutions ðg; v; h; I Þ in H2 based on uniform-in-time estimates in section 2.4. Lemma 2.5.1. If assumptions in theorem 2.1.2 hold, then we have lim kgðtÞ  gkH 2 ¼ 0;

ð2:5:1Þ

lim kvðtÞkH 2 ¼ 0;

ð2:5:2Þ

t! þ 1

t! þ 1

where g ¼

R1 0

gðy; tÞdy ¼

R1 0

g0 ðyÞdy:

Proof. Applying lemma 1.1.2 to (2.4.8) and using lemmas 2.2.1–2.2.8 and 2.4.1, 2.4.2, we get, as t → ∞, kgxx k ! 0:

ð2:5:3Þ

Thus (2.5.1) follows from (2.5.3) and (2.3.1) in lemma 2.3.1. Applying lemma 1.1.2 to (2.4.6), using lemmas 2.2.1–2.2.8, we conclude, as t → ∞, kvt ðtÞk ! 0 which, with (2.4.10), gives, as t → ∞, kvxx ðtÞk ! 0:

ð2:5:4Þ

Then (2.5.2) follows from (2.5.3) and (2.3.2) in lemma 2.3.1. The proof is now complete. h Since radiative term (SE)R is present in our model, whose uniform-in-time estimates are more complicated than those in Qin [104], we have to estimate a new 2 radiative term ðSE ÞR in the next lemma. Lemma 2.5.2. If assumptions in theorem 2.1.2 hold, then we have lim hðtÞ  h H 2 ¼ 0; t! þ 1

where h [ 0 is determined by eðg; hÞ ¼

R1 0

ð2:5:5Þ

ð12 v02 þ eðg0 ; h0 Þ þ FR ð0ÞÞdx.

Proof. By (2.4.13), we can get d pffiffiffiffiffi 2 k eh ht ðtÞk þ C11 khxt ðtÞk2  C2 ðkhx ðtÞk2 þ kvx ðtÞk2 þ kht ðtÞk2 þ kvxt ðtÞk2 dt 2 þ ½gðSE ÞR t Þ; which, combined with (2.2.41), (2.2.62), (2.2.64), (2.2.65), (2.4.3), (2.4.15), and lemma 1.1.2, we can conclude lim kht ðtÞk2 ¼ 0:

t! þ 1

ð2:5:6Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

61

Using (2.1.18) and lemma 2.2.9, we can deduce Z 1  Z þ1 Z  Z þ 1 Z  d ðSE Þ 2 ¼ 2 Sdxdm S dxdm dx t R dt S1 S1 0 0 0  C2 ð1 þ kvx ðtÞk2 þ kht ðtÞk2 Þ; which, together with (2.2.41), (2.2.64) and lemma 1.1.2, implies 2 lim ðSE ÞR ¼ 0:

ð2:5:7Þ

t! þ 1

By equation (2.1.5), we see that

khxx ðtÞk  C2 ðkvx ðtÞk þ kht ðtÞk þ khx ðtÞk þ ðSE ÞR Þ;

which, along with (2.3.2), (2.3.10), (2.5.6) and (2.5.7), gives lim khxx ðtÞk ¼ 0:

ð2:5:8Þ

t! þ 1

Thus (2.5.5) follows from (2.5.7) and (2.5.8). The proof is then complete.

h

A new asymptotic behavior of solutions of radiative term I(x, t; ν, ω) in H will be given in the following lemma. 3

Lemma 2.5.3. If assumptions in theorem 2.1.2 hold, then we have lim kI ðtÞkH 3 ¼ 0:

t! þ 1

Proof. By (2.1.6), we denote Z þ 1 Z 2 Z þ 1 Z 2 1 1 2 gxx Sdxdm þ C1 gx Sx dxdm kI xxx ðtÞk  C1 x x 1 1 S S 0 0 Z þ 1 Z 2 1 gSxx dxdm þ ¼: R1 þ R2 þ R3 : 0 S1 x

ð2:5:9Þ

ð2:5:10Þ

Similarly, by (2.1.18), lemma 2.2.9 and the more delicated computation, we see that R1  C1 kgxx ðtÞk2 ;

ð2:5:11Þ

R2  C1 ðkgx ðtÞk2 þ khx ðtÞk2 Þ;

ð2:5:12Þ

R3  C1 ðkgxx ðtÞk2 þ khxx ðtÞk2 þ kI xx ðtÞk2 Þ:

ð2:5:13Þ

Plugging (2.5.11)–(2.5.13) into (2.5.10) and using (2.3.1), (2.3.5), (2.3.11), (2.5.1), (2.5.5), (2.5.8) and (2.3.18), we obtain

1D Radiative Fluid and Liquid Crystal Equations

62

lim kI xxx ðtÞk2 ¼ 0;

t! þ 1

h

which, combined with (2.3.11), yields (2.5.9).

Proof of Theorem 2.1.2. Combining lemmas 2.4.1–2.4.4 and 2.5.1–2.5.3, we can complete the proof of theorem 2.1.2. h

2.6

Global Existence and Uniform-in-Time Estimates in H4

In this section, we shall first establish the global existence solutions in H4 . Lemma 2.6.1. If assumptions in theorem 1.1.3 hold, then there holds that for any ðg0 ; v0 ; h0 ; I 0 Þ 2 H4 , the following estimates hold for any t > 0, kvxt ðx; 0Þk þ khxt ðx; 0Þk  C3 ;

ð2:6:1Þ

kvtt ðx; 0Þk þ khtt ðx; 0Þk þ kvxxt ðx; 0Þk þ khxxt ðx; 0Þk  C4 ;

ð2:6:2Þ

Z kvtt ðtÞk2 þ

t

Z kvxtt ðsÞk2 ds  C4 þ C4

0

Z 2

khtt ðtÞk þ 0

t

t

khxxt ðsÞk2 ds;

ð2:6:3Þ

0

2

3

1

Z

t

þ C2 e khxxt ðsÞk2 ds 0 Z t ðkvxtt k2 þ kvxxt k2 ÞðsÞds: þ C1 e

khxtt ðsÞk ds  C4 e

ð2:6:4Þ

0

Proof. Using theorems 2.1.1 and 2.1.2, we derive from (2.1.4) kvt ðtÞk  C1 ðkgx ðtÞk þ khx ðtÞk þ kvx ðtÞkL1 kgx ðtÞk þ kvxx ðtÞkÞ  C2 ðkvx ðtÞkH 1 þ kgx ðtÞk þ khx ðtÞkÞ:

ð2:6:5Þ

Differentiating (2.1.4) with respect to x and using theorems 2.1.1 and 2.1.2, we deduce kvxt ðtÞk  C2 ðkvx ðtÞkH 2 þ kgx ðtÞkH 1 þ khx ðtÞkH 1 Þ

ð2:6:6Þ

or kvxxx ðtÞk  C2 ðkvðtÞkH 2 þ kgx ðtÞkH 1 þ khx ðtÞkH 1 þ kvxt ðtÞkÞ:

ð2:6:7Þ

Similarly, differentiating (2.1.4) with respect to x twice, using theorems 2.1.1 and 2.1.2 and the embedding theorem, we get

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

63

kvxxt ðtÞk  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 3 þ khx ðtÞkH 2 þ kvx ðtÞkL1 kgxxx ðtÞk þ kgx ðtÞkL1 kvxxx ðtÞk þ kvxx ðtÞkL1 kgxx ðtÞkÞ  C2 ðkgx ðtÞkH 2 þ khx ðtÞkH 2 þ kvx ðtÞkH 3 Þ

ð2:6:8Þ

or kvxxxx ðtÞk  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ kvxxt kÞ:

ð2:6:9Þ

Using theorems 2.1.1, 2.1.2, (2.1.18) and the Hölder inequality, we have Z

1 0

Z ½gðSE ÞR 2x dx ¼

1

0

Z Z

2

Z

1

 C1  C1

gðSE ÞR þ g½ðSE ÞR x g2x ðSE Þ2R dx þ C1

0 1

kgx k2 0 Z 1Z

Z

þ C1

0

þ1

0 þ1

0

Z S1

Z n S1

0

1

dx

½ðSE ÞR 2x dx

2 ra ðB  I Þ þ rs ðI~  I Þdxdm dx

½ðra Þg gx þ ðra Þh hx ðB  I Þ þ ra ðBh hx  Ix Þ

o 2 I  I Þx dxdm dx þ ½ðrs Þg gx þ ðrs Þh hx ðI~  I Þ þ rs ð~ Z 1 Z þ1 Z  Z þ1 Z  ra B 2 dxdm þ I 2 dxdm dx  C1 kgx k2 0

S1

0

Z

1

þ C1

g2x

h Z

0 þ1

Z

þ1

Z

þ1

Z

S1

0

0

2  Z ðra Þg Bdxdm þ

S1 þ1

Z S1

0

2 ðra Þg Idxdm

Z 1 h Z þ 1 Z 2 i 2 þ ðra Þg ðI~  I Þdxdm dx þ C1 h2x ðra Þh Idxdm 0 0 0 S1 S1  Z þ1 Z 2  Z þ 1 Z 2 þ ðra Þh Bdxdm þ ra Bh dxdm Z

þ

Z

S1

0

S1

0

Z þ C1 0

1

Z

0

0

S1

Z 2 i ðrs Þh ðI~  I Þdxdm dx þ C1 þ1

Z S1

2 rs ðI~  I Þx dxdm dx

0

1

Z

þ1

0

Z

2 ra Ix dxdm dx S1

 C1 ðkgx ðtÞk2 þ khx ðtÞk2 þ kI x ðtÞk2 Þ;

i.e.,

½gðSE Þ   C1 ðkgx ðtÞk þ khx ðtÞk þ kI x ðtÞkÞ: R x

ð2:6:10Þ

Similarly, we can derive ½gðSE Þ   C2 ðkgx ðtÞk 1 þ khx ðtÞk 1 þ kI x ðtÞk 1 Þ R xx H H H

ð2:6:11Þ

and

½gðSE Þ   C2 ðkvx ðtÞk þ kht ðtÞk þ kI t ðtÞkÞ R t

ð2:6:12Þ

1D Radiative Fluid and Liquid Crystal Equations

64

or

½gðSE Þ   C2 ðkvx ðtÞk þ kht ðtÞk þ kIt ðtÞk 2 R t L ðXð1;1ÞR þ Þ Þ:

ð2:6:13Þ

From (2.1.5) and theorems 2.1.1 and 2.1.2 it follows that kht ðtÞk  C1 ðkvx ðtÞk þ kvx ðtÞkL1 kvx ðtÞk

þ ðkgx ðtÞk þ khx ðtÞkÞkhx ðtÞkL1 þ khxx ðtÞk þ ðSE ÞR Þ  C1 ðkhx ðtÞk 1 þ kvx ðtÞk 1 þ ðSE Þ Þ: H

H

R

ð2:6:14Þ

Differentiating (2.1.5) with respect to x, and using theorems 2.1.1 and 2.1.2, we arrive at khxt ðtÞk  C2 ðkht ðtÞk þ khx ðtÞkH 2 þ kgx ðtÞkH 1 þ kvxx ðtÞk þ ½gðSE ÞR x Þ  C2 ðkgx ðtÞkH 1 þ kvx ðtÞkH 1 þ khx ðtÞkH 2 þ kI x ðtÞkÞ

ð2:6:15Þ

or khxxx ðtÞk  C2 ðkgx ðtÞkH 1 þ khx ðtÞkH 1 þ kvx ðtÞkH 1 þ khxt ðtÞk þ kI x ðtÞkÞ:

ð2:6:16Þ

Differentiating (2.1.5) with respect to x twice, using theorems 2.1.1 and 2.1.2 and the embedding theorem, we conclude khxxt ðtÞk  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 3 þ ½gðSE ÞR xx Þ  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 3 þ kI x ðtÞkH 1 Þ;

ð2:6:17Þ

or khxxxx ðtÞk  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ khxxt ðtÞk þ kI x ðtÞkH 1 Þ: ð2:6:18Þ Differentiating (2.1.4) with respect to t, using (2.6.6), (2.6.8), (2.6.14) and (2.6.15), we have kvtt ðtÞk  C2 ðkvx ðtÞkH 1 þ kgx ðtÞk þ kht ðtÞk þ khxt ðtÞk þ kvxt ðtÞk þ kvxxt ðtÞkÞ  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 3 þ khx ðtÞkH 2 þ kI x ðtÞkÞ:

ð2:6:19Þ

Analogously, we get khtt ðtÞk  C2 ðkvx ðtÞk þ kgx ðtÞk þ kht ðtÞk þ khxt ðtÞk þ kvxt ðtÞk þ khx ðtÞkH 2 þ khxxt ðtÞk þ ½gðSE ÞR t Þ  C2 ðkgx ðtÞkH 2 þ kvx ðtÞkH 3 þ khx ðtÞkH 2 þ kI x ðtÞkH 1 þ kI t ðtÞkÞ:

ð2:6:20Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

65

Thus by theorem 2.1.1, estimates (2.6.1) and (2.6.2) follow from (2.6.6), (2.6.8), (2.6.15), (2.6.17), (2.6.19) and (2.6.20). Differentiating (2.1.4) with respect to t twice, multiplying the result by vtt in L2(0, 1), performing an integration by parts and using the Young inequality and using theorems 2.1.1 and 2.1.2, we obtain 1d kvtt ðtÞk2 þ C21 kvxtt k2  C2 ðkvx ðtÞk2 þ kht ðtÞk2 þ khxt ðtÞk2 2 dt þ kvxt ðtÞk2 þ khtt ðtÞk2 Þ:

ð2:6:21Þ

Thus, by theorems 2.1.1 and 2.1.2, Z t Z t kvtt ðtÞk2 þ kvxtt ðsÞk2 ds  C4 þ C2 khtt ðsÞk2 ds 0

0

which, together with (2.6.20), (2.6.13) and lemma 2.2.9, gives (2.6.3). Differentiating (2.1.6) with respect to t twice, multiplying the resulting equation by θtt in L2(0, 1), integrating by parts and using the Young inequality, we get Z 1 Z 1 Z 1d 1 jhx  2 eh htt dx ¼  hxtt dx  ðehtt ht þ egtt vx Þhtt dx 2 dt 0 g tt 0 0 Z Z 1 3 1 vx   eht h2tt dx  eg þ p  l vxtt htt dx 2 0 g 0 Z 1h  vx  i 2 egt   p þ l vxt htt dx g t 0 Z 1 vx  þ vx htt dx  pþl g tt 0 Z 1 7 X ðgðSE ÞR Þtt htt dx ¼: Ai :  ð2:6:22Þ 0

i¼1

Using theorems 2.1.1–2.1.3 and (2.6.1), (2.6.2), and the embedding theorem, we deduce for any ϵ 2 (0, 1), A1   C11 khttx ðtÞk2 þ C2 khtx ðtÞkL1 ðkvx ðtÞk þ kht ðtÞkÞhttx ðtÞ   k ðtÞ khx ðtÞkL1 khxx ðtÞk þ C2 u tt   ð2C1 Þ1 khttx ðtÞk2 þ C2 ðkvx ðtÞk2H 1 þ kht ðtÞk2 þ khtx ðtÞk2 þ kvtx ðtÞk2 þ khtt ðtÞk2 þ khtxx ðtÞk2 Þ;

ð2:6:23Þ

1D Radiative Fluid and Liquid Crystal Equations

66 Z

1

A2  C 1

½ðjvx j þ jht jÞ2 þ jvtx j þ jhtt jðjht j þ jvx jÞjhtt jdx

0

 C1 khtt ðtÞkL1 ðkht ðtÞk þ kvx ðtÞkÞfðkvx ðtÞkL1 þ kht ðtÞkL1 Þ  ðkvx ðtÞk þ kht ðtÞkÞ þ kvtx ðtÞk þ khtt ðtÞkg  C2 ðkhtt ðtÞk þ khttx ðtÞkÞðkvx ðtÞkH 1 þ kht ðtÞk þ khtx ðtÞk þ kvtx ðtÞk þ khtt ðtÞkÞ  khttx ðtÞk2 þ C2 1 ðkvx ðtÞk2H 1 þ kht ðtÞk2 þ khtx ðtÞk2 þ kvtx ðtÞk2 þ khtt ðtÞk2 Þ; Z

1

A3  C1 0

ð2:6:24Þ

ðjvx j þ jht jÞh2tt dx

 C1 ðkhtt ðtÞk þ khttx ðtÞkÞðkvx ðtÞk þ kht ðtÞkÞkhtt ðtÞk ð2:6:25Þ

 khttx ðtÞk2 þ C2 1 khtt ðtÞk2 ; A4  kvttx ðtÞk2 þ C2 1 khtt ðtÞk2 ;

ð2:6:26Þ

n A5  C2 kvx ðtÞkL1 khtt ðtÞk ðkvx ðtÞkL1 þ kht ðtÞkL1 Þðkvx ðtÞk þ kht ðtÞkÞ o þ kvtx ðtÞk þ khtt ðtÞk þ kvttx ðtÞk þ kvx ðtÞk  C2 khtt ðtÞkðkvx ðtÞkH 1 þ kht ðtÞk þ khtx ðtÞk þ kvtx ðtÞk þ khtt ðtÞk þ kvttx ðtÞkÞ  kvttx ðtÞk2 þ C2 1 ðkhtt ðtÞk2 þ kvx ðtÞk2H 1 þ kht ðtÞk2 þ kvtx ðtÞk2 þ khtx ðtÞk2 Þ

ð2:6:27Þ

and Z

1

A6  C 1

ðjvx j þ jht j þ jvtx j þ jvx j2 Þjvtx jjhtt jdx

0

 C2 kvtx ðtÞk1=2 kvtxx ðtÞk1=2 ðkvx ðtÞk þ kht ðtÞk þ kvtx ðtÞkÞkhtt ðtÞk

ð2:6:28Þ

which implies Z t Z t 1=4  Z t 1=4 A6 ds  C2 sup khtt ðsÞk kvtxx k2 ðsÞds kvtx k2 ðsÞds 0

0st

Z

0

1=2

t

0

 ðkvx k2 þ kht k2 þ kvtx k2 ÞðsÞds 0  Z t   sup khtt ðsÞk2 þ kvtxx k2 ðsÞds þ C2 3 : 0st

0

ð2:6:29Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

67

Now we estimate the last term A7. Using the Young inequality, we arrive at Z 1 ðgðSE ÞR Þtt htt dx A7 ¼  0

Z

1

¼

ðvxt þ 2vx ½ðSE ÞR t þ g½ðSE ÞR tt Þhtt dx

0

Z  C1

1

0

Z  C1

Z 2 vxt dx þ C1

1

0

2 vxt dx

1

0

Z vx2 ½ðSE ÞR 2t dx þ C1 Z

þ C1 kvx k2L1

0

1

½ðSE ÞR 2t dx

1

0

Z h2tt dx þ C1 Z

1

þ C1 0

h2tt dx

0

1

½ðSE ÞR 2tt dx Z

1

þ C1 0

½ðSE ÞR 2tt dx;

which, using the interpolation inequality, (2.6.13) and theorems 2.1.1 and 2.1.2, yields A7  C1 ðkvxt ðtÞk2 þ kvx ðtÞk2 þ kht ðtÞk2 þ khtt ðtÞk2 Z 1 2 þ kIt ðtÞkL2 ðXð1;1ÞR þ Þ Þ þ C1 ½ðSE ÞR 2tt dx: 0

ð2:6:30Þ

Performing the Hölder inequality and the interpolation inequality, using (2.1.18) and theorems 2.1.1 and 2.1.2, we have 2 Z 1 Z 1 Z þ 1 Z ½ðSE ÞR 2tt dx ¼ ra ðB  I Þ þ rs ðI~  I Þdxdm dx 0

0

Z ¼

S1

0

1

Z

0

þ1

tt

Z

ððra Þgg vx2 þ ðra Þhg ht vx þ ðra Þg vxt þ ðra Þhg vx ht

S1

0

þ ðra Þhh h2t þ ðra Þh htt ÞðB  I Þ þ 2ððra Þg vx þ ðra Þh ht ÞðBh ht  It Þ þ ra ðBhh h2t þ Bh htt  Itt Þ þ ððrs Þgg vx2 þ ðrs Þhg ht vx þ ðrs Þg vxt þ ðrs Þhg vx ht þ ðrs Þhh h2t þ ðrs Þh htt ÞðI~  I ÞÞ þ ððrs Þg vx þ 2ðrs Þh ht Þ 2  ðI~  I Þt þ rs ðI~  I Þtt dxdm dx Z 1 2  C1 ðvx4 þ h2t vx2 þ vxt þ h4t þ h2tt þ vx4 Þdx 0

Z

1

þ C1 0

Z

þ1

Z

0

S1

Itt2 dxdmdx

 C1 ðkvx ðtÞk2H 1 þ kvxt ðtÞk2 þ kht ðtÞk2H 1 þ khtt ðtÞk2 Z 1 Z þ1 Z þ C1 Itt2 dxdmdx: 0

Now we estimate we have

0

R1 R þ1 R 0

0



ð2:6:31Þ

S1

2 S 1 Itt dxdmdx.

Differentiating (2.2.83) with respect to t,

1D Radiative Fluid and Liquid Crystal Equations

68 Z

Itt ¼

Ry g

Z

hv

i 2 g g þ ðra þ rs Þ þ ððra þ rs Þg vx þ ðra þ rs Þh ht Þ dz x x x 0 x Z x Ry Z y v g g vx xt ðr þ r Þdz a s ðra þ rs Þ þ ððra þ rs Þg vx  ðra B þ rs I~Þdy þ e xx x x 0 x x vx þ ðra þ rs Þh ht Þ þ ððra þ rs Þgg vx þ ðra þ rs Þgh ht vx þ ðra þ rs Þg vxt x Z x Ry g g ðr þ r Þdz 2 ðra B þ rs I~Þdy þ þ ðra þ rs Þhh ht þ ðra þ rs Þh htt Þdz e xx a s x 0 v  v g x x ðra þ rs Þ þ ððra þ rs Þg vx þ ðra þ rs Þh ht Þ  ðra B þ rs I~Þ  x x x  g It Þ dy þ ððra Þg vx B þ ðra Þh ht B þ ra Bh ht þ ðrs Þg vx I~ þ ðra Þh ht I~ þ rs ~ x Z  x Ry g vx ðr þ r Þdz vxt þ ðra B þ rs I~Þ þ 2 ððra Þg vx B þ ðra Þh ht B þ ra Bh ht e xx a s x x 0 g ~ ~ ~ þ ðrs Þg vx I þ ðra Þh ht I þ rs It Þ þ ððra Þgg vx2 B þ ðra Þgh ht vx B þ ðra Þg vxt B x þ ðra Þg vx Bh ht þ ðra Þhg vx ht B þ ðra Þhh h2t B þ 2ðra Þh h2t Bh þ ðra Þg Bh ht þ ra Bhh h2t þ ra Bh htt þ ðrs Þ v 2 I~ þ 2ðrs Þ vx ht I~ þ ðrs Þ vxt I~ þ ðrs Þ h2 I~ þ 2ðrs Þ vx I~t x

e

ðra x x

þ rs Þdz

y

x

gg x

gh



þ 2ðrs Þh ht I~t þ rs I~tt Þ dy ¼:

4 X

g

hh t

g

ð2:6:32Þ

Di :

i¼1

Using the Hölder and the interpolation inequalities, (2.1.18) and theorems 2.1.1 and 2.1.2, we derive Z t Z þ1 Z Z tZ 1 2 D1 dxdmdx  C1 ðvx4 þ h4t Þdxds 0

S1

0

0

Z

0

t

 C1 0

Z

kvx k2L1

Z

þ C1

t

0

0

kht k2L1

1

vx2 dxds Z 0

1

h2t dxds

 C2 ; Z tZ 0

0

þ1

Z S1

D22 dxdmdx

 C1

Z tZ 0

0

ð2:6:33Þ 1

2 ðvxt þ vx4 þ vx2 h2t þ vx6 þ vx4 h2t

þ vx2 þ h4t þ vx2 h2tt Þdxds Z tZ 1 h2tt dxds;  C2 þ C2 0

0

ð2:6:34Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas Z tZ 0

þ1

Z S1

0

D32 dxdmdx

 C1

Z tZ 0

1

ðvx4 þ vx2 h2t þ h4t þ vx2 þ h2t Þdxds

0 tZ 1

Z

þ C2 0

Z

0

þ1

Z S1

0

It2 dxdmdxds ð2:6:35Þ

 C2 ; Z tZ 0

þ1

Z S1

0

D42 dxdmdx

 C1

69

Z tZ 0

Z

þ C2 0

1

2 ðvx4 þ vx2 h2t þ vxt þ h4t þ h2tt Þdxds Z þ1 Z Itt2 dxdmdyds

0 tZ x 0

Z tZ

 C2 þ C2 0 Z tZ þ C2 0

S1

0 1

h2tt dxds x Z þ1 Z

0

0

S1

0

ð2:6:36Þ

Itt2 dxdmdyds:

Squaring (2.6.32) in both sides and inserting (2.6.33)–(2.6.36) into the resulting equation, we obtain Z t Z x Z t Z þ1 Z Z t Z þ1 Z Itt2 dxdmds  C2 þ C2 Itt2 dxdmdsdy: khtt ðsÞk2 ds þ C2 0

0

S1

0

0

0

0

S1

ð2:6:37Þ For fixed t > 0, applying the Gronwall inequality to (2.6.37) implies Z t Z þ1 Z Z t   Itt2 dxdmds  C2 þ C2 khtt ðsÞk2 ds eC2 x S1 0 0 0 Z t  C2 þ C2 khtt ðsÞk2 ds: 0

Integrating (2.6.38) over Ω with respect to x, we have Z t Z 1 Z þ1 Z Z t 2 Itt dxdmds  C2 þ C2 khtt ðsÞk2 ds: 0

0

0

S1

ð2:6:38Þ

ð2:6:39Þ

0

Integrating (2.6.22) over (0, t), using (2.6.22), (2.6.39) and theorems 2.1.1 and 2.1.2, we obtain Z t Z t n o ðkvxxt k2 þ kvxtt k2 ÞðsÞds khtt ðtÞk2 þ khxtt ðsÞk2 ds  C1 e sup khtt ðsÞk2 þ 0

0st

þ C4 e3 þ C2 e1

Z

0 t

ðkhtt ðsÞk2 þ khxxt k2 ÞðsÞds;

0

which, together with (2.2.78), (2.6.20), (2.6.13), lemma 2.2.9 and theorem 2.1.1, implies (2.6.4). Similarly, we can derive the same result for ω 2 (−1, 0). The proof is now complete. h

1D Radiative Fluid and Liquid Crystal Equations

70

Lemma 2.6.2. If assumptions in theorem 2.1.3 hold, then for any ðg0 ; v0 ; h0 ; I 0 Þ 2 H, the following estimates hold for all t > 0, Z t Z t ðkhxtt k2 þ kvxtt k2 ÞðsÞds; ð2:6:40Þ kvxt ðtÞk2 þ kvxxt ðsÞk2 ds  C3 e6 þ C1 e2 0

0

Z

t

khxt ðtÞk2 þ

khxxt ðsÞk2 ds Z t  C3 e6 þ C1 e2 kvxxt k2 þ khxtt k2 þ khxxx k2 khxt k2 0  þ kIx k2L2 ðR þ S 1 ;L2 ð0;1ÞÞ þ kIt k2L2 ðR þ S 1 ;L2 ð0;1ÞÞ ðsÞds: 0

ð2:6:41Þ

Proof. Differentiating (2.1.4) with respect to x and t, multiplying the resulting equation by vtx in L2(0, 1), and integrating by parts, we arrive at 1d kvtx ðtÞk2 ¼ B0 ðx; tÞ þ B1 ðtÞ 2 dt

ð2:6:42Þ

with Z B0 ðx; tÞ ¼

rtx vtx jx¼1 x¼0 ;

B1 ðtÞ ¼ 

1

rtx vtxx dx: 0

Employing theorems 2.1.1 and 2.1.2, the interpolation inequality and the Poincaré inequality, we get B0  C1 ðkvx ðtÞkL1 þ kht ðtÞkL1 Þðkux ðtÞkL1 þ khx ðtÞkL1 Þ þ kvxx ðtÞkL1 þ khtx ðtÞkL1 þ kvtxx ðtÞkL1 þ kux ðtÞkL1 kvtx ðtÞkL1 þ kvx ðtÞkL1 kvxx ðtÞkL1 þ kvx ðtÞk2L1 Þkvtx ðtÞkL1  C2 ðB01 þ B02 Þkvtx ðtÞk1=2 kvtxx ðtÞk1=2

ð2:6:43Þ

where B01 ¼ kvx ðtÞkH 2 þ kht ðtÞk þ khtx ðtÞk and B02 ¼ khtx ðtÞk1=2 khtxx ðtÞk1=2 þ kvtxx ðtÞk1=2 kvtxxx ðtÞk1=2 þ kvtxx ðtÞk þ kvtx ðtÞk1=2 kvtxx ðtÞk1=2 : Exploiting the Young inequality several times, we have that for any ϵ 2 (0, 1), C2 B01 kvtx ðtÞk1=2 kvtxx ðtÞk1=2 

2 kvtxx ðtÞk2 þ C2 2=3 ðkvtx ðtÞk2 2 þ kvx ðtÞk2H 2 þ kht ðtÞk2 þ khtx ðtÞk2 Þ

ð2:6:44Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

71

and C2 B02 kvtx ðtÞk1=2 kvtxx ðtÞk1=2 

2 kvtxx ðtÞk2 þ 2 ðkhtxx ðtÞk2 þ kvtxxx ðtÞk2 Þ 2 þ C2 6 ðkhtx ðtÞk2 þ kvtx ðtÞk2 Þ:

ð2:6:45Þ

Thus we infer from (2.6.43)–(2.6.45), theorems 2.1.1, 2.1.2 and lemma 2.4.1 B0  2 ðkvtxx ðtÞk2 þ kvtxxx ðtÞk2 þ khtxx ðtÞk2 Þ þ C2 6 ðkvx ðtÞk2H 2 þ kht ðtÞk2 þ khtx ðtÞk2 þ kvtx ðtÞk2 Þ which, with theorems 2.1.1 and 2.1.2, further leads to Z t Z t B0 ds  2 ðkvtxx k2 þ kvtxxx k2 þ khtxx k2 ÞðsÞds þ C2 6 ; 8t [ 0: 0

ð2:6:46Þ

ð2:6:47Þ

0

In the same manner, using theorems 2.1.1, 2.1.2, the embedding theorem, we get that for any ϵ 2 (0, 1), Z 1 2 n vtxx B 1   l0 dx þ C1 ðkvx ðtÞk þ kht ðtÞkÞðkux ðtÞkL1 þ khx ðtÞkL1 Þ u 0 þ kvxx ðtÞk þ khtx ðtÞk þ kux ðtÞkL1 kvtx ðtÞk þ kvx ðtÞkL1 kvxx ðtÞk o þ kvx ðtÞk2L1 kux ðtÞk kvtxx ðtÞk   ð2C1 Þ1 kvtxx ðtÞk2 þ C2 ðkvx ðtÞk2H 1 þ kht ðtÞk2H 1 þ kvtx ðtÞk2 þ kux ðtÞk2 Þ

ð2:6:48Þ

which, with (2.6.42), (2.6.47), theorems 2.1.1, 2.1.2, and lemma 2.2.9, gives that for ϵ 2 (0, 1) small enough, Z t Z t ðkhtxx k2 þ kvtxxx k2 ÞðsÞds: ð2:6:49Þ kvtxx k2 ðsÞds  C3 6 þ C1 2 kvtx ðtÞk2 þ 0

0

Differentiating (2.1.4) with respect to x and t, and using theorems 2.1.1 and 2.1.2, we deduce kvxxxt ðtÞk  C1 kvttx ðtÞk þ C2 ðkvxx ðtÞkH 1 þ khx ðtÞkH 1 þ kgx ðtÞkH 1 þ kht ðtÞkH 2 Þ: ð2:6:50Þ Differentiating (2.1.5) with respect to x and t, multiplying the resulting equation by θxt, and integrating by parts, we get Z 1d 1 eh h2xt dx ¼ D0 þ D1 þ D2 þ D3 þ D4 þ D5 ð2:6:51Þ 2 dt 0

1D Radiative Fluid and Liquid Crystal Equations

72

where D0 ¼

 jh  x

D2 ¼ 

g Z 1 0

Z

Z

x¼1

hxt x¼0 ; D1 ¼  xt

1

0

 jh  x

g

xt

hxxt dx;

ðeg vx þ rvx Þxt hxt dx;

 1 eht þ ehx htt hxt dx; 2 0 Z 1 Z 1 D4 ¼  ðgðSE ÞR Þxt hxt dx ¼ ½gðSE ÞR t hxxt dx  ðgðSE ÞR Þt hxt jx¼1 x¼0 :

D3 ¼ 

1



ehxt ht þ

0

0

Similarly to (2.6.42)–(2.6.49), we infer D0  C2 ðkvx ðtÞkL1 þ kht ðtÞkL1 þ kvxx ðtÞkL1 þ khtx ðtÞkL1 þ kht ðtÞkL1 khxx ðtÞkL1 þ khtxx ðtÞkL1 þ khxx ðtÞkL1 Þkhtx ðtÞkL1  C2 ðD01 þ D02 ÞðD03 þ D04 Þ where 2 khtxx ðtÞk2 þ C2 2 ðkvx ðtÞk2H 2 þ khx ðtÞk2H 2 þ kht ðtÞk2H 1 Þ; 3 2 C2 D02 D03  ðkhtxx ðtÞk2 þ khtxxx ðtÞk2 Þ þ C2 6 khtx ðtÞk2 ; 3 C2 D01 D04  C2 ðkvx ðtÞk2H 2 þ kht ðtÞk2H 1 þ khx ðtÞk2H 2 Þ;

C2 D01 D03 

and C2 D02 D04 

2 ðkhtxx ðtÞk2 þ khtxxx ðtÞk2 Þ þ C2 2 khtx ðtÞk2 : 3

That is, D0  2 ðkhtxx ðtÞk2 þ khtxxx ðtÞk2 Þ þ C2 6 ðkvx ðtÞk2H 2 þ khx ðtÞk2H 2 þ kht ðtÞk2H 1 Þ: ð2:6:52Þ Similarly, D1   ð2C1 Þ1 khtxx ðtÞk2 þ C2 ðkvx ðtÞk2H 1 þ khx ðtÞk2H 2 þ kht ðtÞk2H 1 Þ;

ð2:6:53Þ

D2  2 kvtxx ðtÞk2 þ C2 2 ðkvx ðtÞk2H 2 þ kht ðtÞk2H 1 þ kvtx ðtÞk2 Þ;

ð2:6:54Þ

D3  2 khtxx ðtÞk2 þ C2 2 ðkvx ðtÞk2H 1 þ kht ðtÞk2H 1 þ khx ðtÞk2H 2 þ kvtx ðtÞk2 þ kux ðtÞk2 Þ:

ð2:6:55Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

73

Using the Young inequality and (2.6.13), (2.6.14), we derive D4  C1 khk1L1þ a khxt kL1 þ C1 kI kL2 ðR þS 1 ;L1 Þ khxt k þ e2 khxxt ðtÞk2 Z 1 2 þ C1 e ½gðSE ÞR 2t dx 0

 e2 khxxt ðtÞk2 þ C2 e2 ðkvx ðtÞk2 þ khx k2 þ kht ðtÞk2 þ kIt ðtÞk2L2 ðR þS 1 ;L2 ð0;1ÞÞ þ kIx ðtÞk2L2 ðR þS 1 ;L2 ð0;1ÞÞ Exploiting lemma 2.6.1, theorems 2.1.1, 2.1.2 and the embedding theorem result in

  k ðtÞ u txx  C2 ðkux ðtÞkH 1 þ kvx ðtÞkH 2 þ khx ðtÞkH 1 þ kht ðtÞkH 2 Þ;     k k þ ðtÞ ðtÞ u t u tx  C2 ðkvx ðtÞkH 1 þ kht ðtÞkH 1 Þ

and

    k k u x ðtÞ 1 þ u xx ðtÞ  C2 ðkux ðtÞkH 1 þ khx ðtÞkH 1 Þ  C2 L

which imply     k k u txx hx ðtÞ  C2 u txx ðtÞ  C2 ðkux ðtÞkH 1 þ kvx ðtÞkH 2 þ khx ðtÞkH 1 þ kht ðtÞkH 2 Þ;

ð2:6:56Þ

    k k u tx hxx ðtÞ  C2 u tx 1 L  C2 ðkux ðtÞkH 1 þ kvx ðtÞkH 2 þ khx ðtÞkH 1 þ kht ðtÞkH 2 Þ;

ð2:6:57Þ

      k k  C h ðtÞ khxxx ðtÞk 1 u t xxx u t L1  C2 ð1 þ khtx ðtÞkÞkhxxx ðtÞk;     k k u xx htx ðtÞ þ u x htxx ðtÞ  C2 khtx ðtÞkH 1 :

ð2:6:58Þ

ð2:6:59Þ

Differentiating (2.1.5) with respect to x and t, using estimates (2.6.57)–(2.6.59) and theorems 2.1.1, 2.1.2, we conclude

1D Radiative Fluid and Liquid Crystal Equations

74

      jhx  j j ðtÞ þ hxx ðtÞ þ hxxx ðtÞ khxxxt ðtÞk  C1 g xxt g t g xxt       j j j þ g xt hxx ðtÞ þ g xx hxt ðtÞ þ g xx hxt ðtÞ    j þ hxxt ðtÞ g x  C1 ðkgx ðtÞkH 1 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ kht ðtÞkH 2 þ khtt ðtÞkH 1 þ kvxt ðtÞkH 1 þ khxt ðtÞkkhxxx ðtÞk þ ½gðSE ÞR xt ðtÞ Þ: ð2:6:60Þ Now we estimate ½gðSE ÞR xt ðtÞ . Z 0

1

Z ½gðSE ÞR 2xt dx  ¼:

1

2 ðvxx ðSE Þ2R þ vx2 ½ðSE ÞR 2x þ g2x ½ðSE ÞR 2t þ ½ðSE ÞR 2xt Þdx

0 4 X

ð2:6:61Þ

Fi :

i¼1

Using the Hölder inequality, (2.1.18) and theorems 2.1.1, 2.1.2, we derive 2 Z 1 Z þ 1 Z 2 ~ F1 ¼ vxx ra ðB  I Þ þ rs ðI  I Þdxdm dx S1

0

0 2  C1 kvx ðtÞkH 1 ;

Z F2  0

1

vx2

Z

þ1 0

ð2:6:62Þ

Z S1

ððra Þg gx þ ðra Þh hx ÞðB  I Þ þ ra ðBh hx  Ix Þ

2 I  I Þ þ rs ð ~ I  I Þx dxdm dx þ ððrs Þg gx þ ðrs Þh hx Þð~ Z 1 ðvx2 g2x þ vx2 h2x þ vx2 Þdx  C1 ðkgx ðtÞk2H 1 þ khx ðtÞk2H 1 þ kvx ðtÞk2 Þ: ð2:6:63Þ  C1 0

Similarly, we have F3  C1 ðkvx ðtÞk2H 1 þ kht ðtÞk2H 1 þ kgx ðtÞk2 Þ

ð2:6:64Þ

and F4  C1 ðkvx ðtÞk2H 1 þ kgx ðtÞk2H 1 þ khx ðtÞk2H 1 þ kht ðtÞk2H 1 Þ Z 1 Z þ1 Z Ixt2 dxdmdx; þ C1 0

0

S1

which, together with (2.6.62)–(2.6.64), gives khxxxt ðtÞk  C1 ðkgx ðtÞkH 1 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ kht ðtÞkH 2 þ khtt ðtÞkH 1 þ kvxt ðtÞkH 1 þ khxt ðtÞkkhxxx ðtÞk þ kIxt ðtÞkL2 ðXð1;1ÞR þ Þ Þ: ð2:6:65Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

75

Differentiating (2.1.6) with respect to t, using (2.1.18), the Hölder inequality and theorem 2.1.1, we obtain Z t Z 1 Z þ1 Z Ixt2 dxdmdxds 1

0

S1

0

Z tZ

1

Z

þ1

Z

1 2 2 ðv S þ g2 St2 Þdxdmdxds 2 x x 1 0 0 Z tZ 1 Z t Z 1 Z þ1 Z ðvx2 þ h2t Þdxds þ C1 It2 dxdmdxds  C1 ¼

S1

1

0

1

0

S1

0

 C1 :

ð2:6:66Þ

Integrating (2.6.32) over (0, t), using (2.6.52)–(2.6.56), (2.6.66) and theorems 2.1.1, 2.1.2, we easily get (2.6.41). h Lemma 2.6.3. If assumptions in theorem 2.1.3 hold, then ðg0 ; v0 ; h0 ; I0 Þ 2 H4 , the following estimates hold, for all t > 0, Z t kvtt ðtÞk2 þ kvxt ðtÞk2 þ khtt ðtÞk2 þ khxt ðtÞk2 þ ðkvxtt k2 þ kvxtt k2

for

any

0

þ khxtt k2 þ khxxt k2 ÞðsÞds  C4 ; Z kgxxx ðtÞk2H 1 þ kgxx ðtÞk2W 1;1 þ

0

t

ðkgxxx k2H 1 þ kgxx k2W 1;1 ÞðsÞds  C4 ;

ð2:6:67Þ

ð2:6:68Þ

kvxxx ðtÞk2H 1 þ kvxx ðtÞk2W 1;1 þ khxxx ðtÞk2 þ khxx k2W 1;1 þ kgxxxx ðtÞk2 Z t þ kvxxt ðtÞk2 þ khxxt ðtÞk2 þ ðkvtt k2 þ khtt k2 þ kvxx k2W 2;1 þ khxx k2W 2;1 0

þ khxxt k2H 1 þ kvxxt k2H 1 þ khxt k2W 1;1 þ kvxt k2W 1;1 þ kgxxxt k2H 1 ÞðsÞds  C4 ; Z 0

t

ðkvxxxx k2H 1 þ khxxxx k2H 1 ÞðsÞds  C4 :

ð2:6:69Þ ð2:6:70Þ

Proof. Adding up (2.6.40) and (2.6.41), picking ϵ 2 (0, 1) small enough, we arrive at Z t 2 2 v ðkvtxx k2 þ khtxx k2 ÞðsÞ k tx ðtÞk þ khtx ðtÞk þ 0 Z t ð2:6:71Þ  C3 6 þ C2 2 ðkvttx k2 þ khttx k2 þ khxxx k2 khtx k2 ÞðsÞds: 0

Now multiplying (2.6.3) and (2.6.4) by ϵ and ϵ3∕2, respectively, then adding the resultant to (2.6.72), and choosing ϵ 2 (0, 1) small enough, we obtain

1D Radiative Fluid and Liquid Crystal Equations

76

Z 2

2

2

2

t

ðkhtxx k2 þ kvtxx k2 kvtx ðtÞk þ khtx ðtÞk þ kvtt ðtÞk þ khtt ðtÞk þ 0 Z t 2 2 6 2 þ kvttx k þ khttx k ÞðsÞds  C4  þ C2  khxxx k2 khtx k2 ðsÞds 0

which, with lemma 2.3.2 and the Gronwall inequality, gives estimate (2.6.68). Differentiating (2.4.7) with respect to x, and using (2.1.3), we get @  gxxx  l ð2:6:72Þ  pg gxxx ¼ E1 ðx; tÞ @t g with

g g  xx x : g2 t

E1 ðx; tÞ ¼ vtxx þ Ex ðx; tÞ þ pgx gxx þ l

Obviously, we can infer from theorems 2.1.1, 2.1.2 and lemmas 2.6.1, 2.6.2 that kE1 ðtÞk  C2 ðkgx ðtÞkH 1 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ kvtxx ðtÞkÞ

ð2:6:73Þ

leading to Z

t

kE1 k2 ðsÞds  C4 ;

8t [ 0:

ð2:6:74Þ

0 2 Multiplying (2.6.72) by gxxx g in L (0, 1), we obtain 2 2 d gxxx ðtÞ þ C 1 gxxx ðtÞ  C1 kE1 ðtÞk2 1 dt g g

ð2:6:75Þ

which, combined with (2.6.74) and theorems 2.1.1, 2.1.2 and lemmas 2.6.1, 2.6.2 gives, for all t > 0, Z t ð2:6:76Þ kgxxx ðtÞk2 þ kgxxx ðsÞk2 ds  C1 : 0

Using (2.1.18), the Hölder inequality and theorem 2.1.1, we derive from (2.1.6) that for any t > 0, Z t Z 1 Z þ1 Z Z t Z 1 Z þ1 Z 1 2 Ix2 dxdmdxds  C1 S dxdmdxds  C1 ; ð2:6:77Þ 2 x 1 1 0 0 0 0 0 0 S S Z tZ 0

1 0

Z 0

þ1

Z S1

2 Ixx dxdmdxds  C1

þ

Z tZ 0

Z tZ 0

 C1 ;

1

0

0

1

ðg2x þ h2x Þdxds Z 0

þ1

Z S1

Ix2 dxdmdxds ð2:6:78Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

Z tZ 0

1

0

 C1

Z

þ1

0

 C1

Z S1

0

Z tZ

0

þ C1

Z

1

0

Z tZ

0

2 Ixxx dxdmdxds þ1

Z S1

0 1

1 2 2 2 ðg S þ g2x Sx2 þ Sxx Þdxdmdxds x2 xx

ðg2xx þ h2xx þ g2x þ h2x Þdxds

Z tZ 0

77

0

1

Z 0

þ1

Z S1

2 Ixx dxdmdxds

 C2 :

ð2:6:79Þ

By (2.6.7), (2.6.9), (2.6.16), (2.6.18), (2.6.72)–(2.6.79) and theorem 2.1.1, and using the interpolation inequality, we obtain for any t > 0, kvxxx ðtÞk2 þ khxxx ðtÞk2 þ kvxx ðtÞk2L1 þ khxx ðtÞk2L1 Z t þ ðkvxxxx k2H 1 þ khxxx k2H 1 þ kvxx k2W 1;1 þ khxx k2W 1;1 ÞðsÞds  C4 :

ð2:6:80Þ

0

Differentiating (2.1.4) and (2.1.5) with respect to t, using (2.6.69), (2.6.12) and theorems 2.1.1 and 2.1.2, we derive that for any t > 0, kvxxt ðtÞk  C1 kvtt ðtÞk þ C2 ðkvx ðtÞkH 1 þ kvxt ðtÞk þ kht ðtÞkH 1 Þ  C4 ;

ð2:6:81Þ

khxxt ðtÞk  C1 ðkhtt ðtÞk þ kI t ðtÞkÞ þ C2 ðkvx ðtÞkH 1 þ kvxt ðtÞk þ kht ðtÞkH 1 þ khx ðtÞkH 1 Þ  C4

ð2:6:82Þ

which, together with (2.6.9), (2.6.18), (2.6.80) and theorem 2.1.1, yields for all t > 0, kvxxxx ðtÞk2 þ khxxxx ðtÞk2 Z t þ ðkhxxt k2 þ khxxxx k2 þ kvxxt k2 þ kvxxxx k2 ÞðsÞds  C4 :

ð2:6:83Þ

0

Employing the interpolation inequality and (2.6.83), we get for any t > 0, Z t ðkvxxx k2L1 þ khxxx k2L1 ÞðsÞds  C4 : ð2:6:84Þ kvxxx ðtÞk2L1 þ khxxx ðtÞk2L1 þ 0

Now differentiating (2.6.72) with respect to x, we find @  gxxxx  l  pg gxxxx ¼ E2 ðx; tÞ @t g where E2 ðx; tÞ ¼ E1x ðx; tÞ þ pgx gxxx þ l

@  gxxx gx  : @t g2

ð2:6:85Þ

1D Radiative Fluid and Liquid Crystal Equations

78

Using the embedding theorem, lemmas 2.6.1 and 2.6.2 and theorems 2.1.1, 2.1.2, (2.6.68) and (2.6.82)–(2.6.85), we can conclude that kExx ðtÞk  C4 ðkhx ðtÞkH 3 þ kgx ðtÞkH 2 þ kvx ðtÞkH 2 Þ;     gxx gx ðtÞ kE1x ðtÞk  C1 kExx ðtÞk þ kvtxxx ðtÞk þ ðpgx gxx Þx ðtÞ þ g2 tx    C1 kvtxxx ðtÞk þ C4 khx ðtÞkH 3 þ kgx ðtÞkH 2 þ kvx ðtÞkH 3 which imply kE2 ðtÞk  C1 kvtxxx ðtÞk þ C4 ðkhx ðtÞkH 3 þ kgx ðtÞkH 2 þ kvx ðtÞkH 3 Þ: We infer from (2.6.19) and (2.6.20) that Z t ðkvtt k2 þ khtt k2 ÞðsÞds  C4 ; 8t [ 0

ð2:6:86Þ

ð2:6:87Þ

0

which, together with (2.6.48), (2.6.50) and (2.6.66), gives Z t ðkvtxxx k2 þ khtxxx k2 ÞðsÞds  C4 ; 8t [ 0:

ð2:6:88Þ

0

Thus it follows from (2.6.68), (2.6.88) and lemmas 2.6.1, 2.6.2 and theorems 2.1.1, 2.1.2 that Z t kE2 k2 ðsÞds  C4 ; 8t [ 0: ð2:6:89Þ 0 gxxxx g

in L2(0, 1), we get Multiplying (2.6.85) by 2 d gxxxx ðtÞ þ C 1 gxxxx ðtÞ  C1 kE2 ðtÞk2 : 1 dt g g Thus it follows from (2.6.89) and (2.6.90) that for all t > 0, Z t kgxxxx ðtÞk2 þ kgxxxx ðsÞk2 ds  C4 :

ð2:6:90Þ

ð2:6:91Þ

0

Differentiating (2.1.4) with respect to x three times, using lemmas 2.6.1, 2.6.2 and theorems 2.1.1, 2.1.2 and the Poincaré inequality, we can derive for all t > 0, we infer kvxxxxx ðtÞk  C1 kvtxxx ðtÞk þ C2 ðkux ðtÞkH 3 þ kvx ðtÞkH 3 þ khx ðtÞkH 3 Þ: Thus it follows from (2.6.89)–(2.6.92) that Z t ðkvxxxxx k2 þ kgxxxt k2H 1 ÞðsÞds  C4 :

ð2:6:92Þ

ð2:6:93Þ

0

Differentiating (2.1.5) with respect to x three times, using (2.6.67), (2.6.68) and theorems 2.1.1, 2.1.2 and the Poincaré inequality, we infer

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

79

khxxxxx ðtÞk  C4 ðkgx ðtÞkH 3 þ kvx ðtÞkH 3 þ khx ðtÞkH 3 þ khxxt ðtÞkH 1 þ kIxxx ðtÞkL2 ðXR þð1;1ÞÞ Þ:

ð2:6:94Þ

From (2.6.80), (2.6.84), (2.6.92) and (2.6.94), we conclude for all t > 0, Z t ð2:6:95Þ khxxxxx ðsÞk2 ds  C4 ; 0

which, together with (2.6.81) and (2.6.94), gives for all t > 0, Z t ðkvxx k2W 2;1 þ khxx k2W 2;1 ÞðsÞds  C4 :

ð2:6:96Þ

0

At last, using all previous estimates and the interpolation inequality, we can easily derive desired estimates (2.6.67)–(2.6.70). The proof is complete. h Lemma 2.6.4. If assumptions in theorem 2.1.3 hold, then for any ðg0 ; v0 ; h0 ; I 0 Þ 2 H, the following estimate holds for all t > 0, kI ðtÞkH 5  C4 :

ð2:6:97Þ

Proof. By (2.1.6), we have Z 1  Z þ1 Z 2 1 ðgSÞxxx dxdm dx kI xxxx ðtÞk2 ¼ 0 0 S1 x Z 1  Z þ1 Z 2  Z þ 1 Z 1 2 1 gxxx Sdxdm þ gxx Sx dxdm dx  C1 S1 x S1 x 0 0 0  Z þ1 Z 1 2  Z þ 1 Z 1 2 gx Sxx dxdm þ Sxxx dxdm dx þ S1 x S1 x 0 0 4 X Gi : ¼: i¼1

ð2:6:98Þ

Applying the Hölder inequality and the interpolation inequality, (2.1.18) and theorems 2.1.1, 2.1.2 and lemma 2.6.3, we get Z 1   Z þ1 Z 1 1 G1  C 1 g2xxx ra þ ra ðB 2 þ I 2 Þ þ 2 rs þ rs ðI~  I Þ2 dxdm dx 2 x 0 0 S1 x Z 1 ð2:6:99Þ g2xxx dx  C4 ;  C1 0

Z

Z

1

G2  C1 0

g2xx ðg2x þ h2x Þdx þ C1

Z

þ C1 0

1

0

1

g2xx

Z 0

þ1

Z S1

2 ðra þ rs ÞIx dxdm dx

ðg2x þ h2x Þdx

 C1 ðkgxx k2H 1 þ khx k2 Þ  C4 :

ð2:6:100Þ

1D Radiative Fluid and Liquid Crystal Equations

80

Analogously, we have Z G3  C1 ðkgx k2H 1 þ khx k2H 1 Þ þ C1

Z

1

0

þ1

Z

2 ðra þ rs ÞIxx dxdm dx

S1

0

ð2:6:101Þ

 C4 ; Z G4  C1 ðkgxx k2H 1 þ khxx k2H 1 Þ þ C1

Z

1

0

þ1

Z

2 ðra þ rs ÞIxxx dxdm dx

S1

0

 C4 :

ð2:6:102Þ

Inserting (2.6.99)–(2.6.102) into (2.6.98), we obtain for all t > 0, kI xxxx ðtÞk2  C4 :

ð2:6:103Þ

Differentiating (2.1.6) with respect to x four times, we arrive at 2 2  Z þ1 Z 1 1 gxxxx Sdxdm þ C1 gxxx Sx dxdm 0 0 0 S1 x S1 x 2 2  Z þ1 Z 1  Z þ1 Z 1 þ C1 gxx Sxx dxdm þ C1 gx Sxxx dxdm S1 x S1 x 0 0 5 2  Z þ1 Z 1 X ð2:6:104Þ þ C1 Hi : Sxxxx dxdm dx ¼: S1 x 0 i¼1 Z

1

kI xxxxx ðtÞk2  C1

Z

þ1

Z

Similarly, we can deduce from lemmas 2.2.1–2.2.9, 2.3.1 and 2.3.2, Z 1 g2xxxx dx  C4 ; H1  C 1

ð2:6:105Þ

0

Z

1

H2  C 1 0

Z g2xxx ðg2x þ h2x Þ þ C1

1

Z

1

Z

0

0 1

Z

1

þ khxx k2H 2 Þ þ C1

0

1

2 Ix dxdm dx

S1

S1 þ1

Z

Z S1

0 þ1 0

Z S1

ð2:6:106Þ

2 Ixx dxdm dx  C4 ;

ð2:6:107Þ

2 Ixxx dxdm dx  C4 ;

ð2:6:108Þ

2 Ixxxx dxdm dx  C4 :

ð2:6:109Þ

S1 þ1

Z

2 Ix dxdm dx  C4 ;

Z

0

0

Z H4  C1 ðkgxx k2H 2

Z

Z

0

H4  C1 ðkgxx k2H 1 þ khxx k2H 1 Þ þ C1

þ1 0

Z H3  C1 ðkgxx k2H 1 þ khxx k2H 1 Þ þ C1

Z

0

Z

 C1 ðkgx k2H 3 þ khx k2 Þ þ C1

þ1

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

81

Inserting (2.6.105)–(2.6.109) into (2.6.104), we get for all t > 0, kI xxxxx ðtÞk2  C4 ; which, together with (2.6.103) and theorems 2.1.1, 2.1.2, implies (2.6.97). The proof is now complete. h

2.7

Asymptotic Behavior of Solutions in H4

This section will be devoted to the study of the asymptotic behavior in H4 . Lemma 2.7.1. If assumptions in theorem 2.1.3 hold, then we have lim kgðtÞ  gkH 4 ¼ 0;

t! þ 1

where g ¼

R1 0

ð2:7:1Þ

gðy; tÞdy:

Proof. Using (2.6.68), (2.6.75) and lemma 1.1.2, we get lim kgxxx ðtÞk ¼ 0:

t! þ 1

ð2:7:2Þ

Recalling (2.6.91) and (2.6.92) and lemma 1.1.2, we obtain lim kgxxxx ðtÞk ¼ 0;

t! þ 1

which, together with (2.7.2), (2.5.1) in lemma 2.5.1 and the Poincaré inequality, yields (2.7.1). The proof is now complete. h Lemma 2.7.2. Under assumptions in theorem 2.1.3, we have lim kvðtÞkH 4 ¼ 0:

t! þ 1

ð2:7:3Þ

Proof. From (2.6.42)–(2.6.48), we can estimate for any ε > 0, d kvxt ðtÞk2 þ C11 kvxxt ðtÞk2  eðkvxxxt ðtÞk2 þ khxxt ðtÞk2 Þ þ C4 ðeÞðkvx ðtÞk2H 2 dt þ kht ðtÞk2H 1 þ khxt ðtÞk2 þ kvxt ðtÞk2 þ kgx ðtÞk2 Þ:

ð2:7:4Þ

Using (2.7.4), theorem 2.1.2, lemmas 2.6.1–2.6.3, we obtain lim kvxt ðtÞk2 ¼ 0;

t! þ 1

ð2:7:5Þ

which, along with (2.6.7) and theorem 2.1.2, implies lim kvxxx ðtÞk ¼ 0:

t! þ 1

ð2:7:6Þ

1D Radiative Fluid and Liquid Crystal Equations

82

By theorems 2.1.1, 2.1.2 and lemma 2.6.3, and using the interpolation inequality, we obtain kpxxt ðtÞk  C2 ðkvx ðtÞkH 2 þ kgx ðtÞkH 2 þ kht ðtÞkH 2 þ khx ðtÞkH 2 Þ:

ð2:7:7Þ

Differentiating (2.1.4) with respect to t once and x twice, multiplying the resulting by vxxt in L2(0, 1) and using the Young inequality and theorems 2.1.1, 2.1.2, we dedive d kvxxt ðtÞk2 þ kvxxxt ðtÞk2  C1 kpxxt ðtÞk2 þ C2 ðkvxt ðtÞk2H 2 þ kvx ðtÞk2H 2 þ kgx ðtÞk2H 2 Þ; dt which, together with (2.7.7), theorems 2.1.1, 2.1.2, lemmas 2.6.3 and 1.1.2, gives lim kvxxt ðtÞk2 ¼ 0:

ð2:7:8Þ

t! þ 1

From (2.6.51)–(2.6.55), we have derived for small ε > 0, d pffiffiffiffiffi 2 k eh hxt ðtÞk þ C21 khxxt ðtÞk2 dt  eðkhxxxt ðtÞk2 þ kvxxt ðtÞk2 Þ þ C2 ðeÞðkvx ðtÞk2H 2 þ khx ðtÞk2H 2 þ kht ðtÞk2H 1 þ kgx ðtÞk2 þ kvxt ðtÞk2 þ kIt ðtÞk2L2 ðXS 1 R þ Þ þ kIx k2L2 ðXS 1 R þ Þ Þ; which, combined with lemmas 2.2.9, 2.6.3 and 1.1.2, implies lim khxt ðtÞk ¼ 0:

ð2:7:9Þ

t! þ 1

By (2.7.9), (2.6.16), theorems 2.1.1, 2.1.2 and the fact limt! þ 1 kI x ðtÞk2 ¼ 0, we arrive at lim khxxx ðtÞk ¼ 0:

ð2:7:10Þ

t! þ 1

Thus, by (2.7.2), (2.7.6), (2.7.8), (2.7.10) and theorems 2.1.1, 2.1.2, we obtain lim kvxxxx ðtÞk ¼ 0;

t! þ 1

which, together with (2.7.6) and theorems 2.1.1, 2.1.2, yields (2.7.3). This proves the proof. h Lemma 2.7.3. If assumptions in theorem 2.1.3 hold, then we have lim hðtÞ  h H 4 ¼ 0;

ð2:7:11Þ

t! þ 1

where h [ 0 is determined by eðg; hÞ ¼

R1 0

ð12 v02 þ eðg0 ; h0 Þ þ FR ð0ÞÞdx.

Proof. Obviously, the Young inequality gives Z 1 Z þ1 Z d It  Itt dxdmdx kIt ðtÞk2L2 ðXð1;1ÞR þ Þ ¼ 2 dt S1 0 0 Z 1 Z þ1 Z Z 1Z  It2 dxdmdx þ 0

0

S1

0

þ1 0

Z S1

Itt2 dxdmdx;

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

83

which, together with (2.6.39), (2.6.69) and lemma 1.1.2, gives lim kIt ðtÞk2L2 ðXð1;1ÞR þ Þ ¼ 0:

ð2:7:12Þ

t! þ 1

Thus it follows from (2.6.13), (2.7.12) and theorems 2.1.1 and 2.1.2 that lim ½gðSE ÞR t ¼ 0: ð2:7:13Þ t! þ 1

Similarly, from theorems 2.1.1 and 2.1.2, we derive for small ϵ > 0, d khtt ðtÞk2 þ C1 khxxt ðtÞk2  eðkhxtt ðtÞk2 þ kvxtt ðtÞk2 Þ þ C2 ðkvx ðtÞk2H 1 þ khtt ðtÞk2 dt þ kht ðtÞk2H 1 þ kvxt ðtÞk2 þ khxt ðtÞk2 þ kIt ðtÞkL2 ðXð1;1ÞR þ Þ þ kItt ðtÞkL2 ðXð1;1ÞR þ Þ Þ;

which, together with (2.6.39), theorems 2.1.1, 2.1.2, lemmas 2.6.3 and 1.1.2, implies lim khtt ðtÞk ¼ 0:

ð2:7:14Þ

t! þ 1

We can derive from the similar estimate as (2.6.21) khxxt ðtÞk  C2 ðkhtt ðtÞk þ kvx ðtÞk þ kgx ðtÞk þ kht ðtÞk þ khxt ðtÞk þ khx ðtÞkH 2 þ ½gðSE ÞR t Þ whence, by (2.7.9), (2.7.10) and (2.7.12)–(2.7.14), lim khxxt ðtÞk ¼ 0;

t! þ 1

which, combined with (2.1.25), limt! þ 1 kI xx ðtÞk2 ¼ 0, yields

(2.7.2),

(2.7.6),

(2.7.10)

and

lim khxxxx ðtÞk2 ¼ 0:

the

fact

ð2:7:15Þ

t! þ 1

Thus (2.7.11) follows from (2.7.10) and (2.7.15). The proof is thus complete. h Lemma 2.7.4. If assumptions in theorem 2.1.3 hold, then we have lim kI ðtÞkH 5 ¼ 0:

ð2:7:16Þ

t! þ 1

Proof. From (2.1.6), it follows Z þ 1 Z 2 Z þ 1 Z 2 1 1 2 gxxx Sdxdm þ C1 gxx Sx dxdm kI xxxx ðtÞk  C1 x x 1 1 0 0 S S Z þ 1 Z 2 Z þ 1 Z 2 1 1 g gS þ C1 S dxdm þ C dxdm 1 xxx x xx 1 x 1 x 0

¼:

4 X i¼1

Mi :

S

0

S

ð2:7:17Þ

1D Radiative Fluid and Liquid Crystal Equations

84

Using (2.1.18), theorems 2.1.1, 2.1.2 and lemma 2.6.3, we deduce Z þ 1 Z 2 Z 1 2 ~ M1  C1 gxxx ðr1 ðB  I Þ þ rs ðI  I ÞÞdxdm dx 0

0

S1

ð2:7:18Þ

2

 C1 kgxxx ðtÞk ; Z M2  C1 0

1

g2xx ðg2x þ h2x þ 1Þdx  C1 ðkgx ðtÞk2H 1 þ khx ðtÞk2H 1 Þ:

ð2:7:19Þ

Analogously, M3  C1 ðkgx ðtÞk2H 1 þ khx ðtÞk2H 1 þ kI xx ðtÞk2 Þ; M4  C1 ðkgx ðtÞk2H 2 þ khx ðtÞk2H 2 þ kI xx ðtÞk2 þ kI xxx ðtÞk2 Þ:

ð2:7:20Þ ð2:7:21Þ

Inserting (2.7.18)–(2.7.21) into (2.7.17) and using theorems 2.1.1, 2.1.2, (2.7.2) and (2.7.15), we have lim kI xxxx ðtÞk2 ¼ 0:

t! þ 1

ð2:7:22Þ

Similarly, we get Z þ 1 Z 2 kI xxxxx ðtÞk  C1

2 Z þ 1 Z 2 1 1 gxxxx Sdxdm þ C1 gxxx Sx dxdm 0 0 S1 x S1 x Z þ 1 Z 2 Z þ 1 Z 2 1 1 g g þ C1 S dxdm þ C S dxdm 1 xx xx x xxx 0 0 S1 x S1 x Z þ 1 Z 2 5 X 1 ¼: gS þ C1 dxdm Ni : ð2:7:23Þ xxxx 1 x 0

S

i¼1

Similarly to (2.7.17) and by a more delicated computation, we can derive N1  C1 kgx ðtÞk2H 3 ;

ð2:7:24Þ

N2  C4 ðkgx ðtÞk2H 1 þ khx ðtÞk2H 1 Þ;

ð2:7:25Þ

N3  C3 ðkgx ðtÞk2H 1 þ khx ðtÞk2H 1 þ kI xx ðtÞk2 Þ;

ð2:7:26Þ

N4  C1 ðkgx ðtÞk2H 2 þ khx ðtÞk2H 2 þ kI xx ðtÞk2 þ kI xxx ðtÞk2 Þ;

ð2:7:27Þ

N5  C1 ðkgx ðtÞk2H 3 þ khx ðtÞk2H 3 þ kI xx ðtÞk2 þ kI xxx ðtÞk2 þ kI xxxx ðtÞk2 Þ:

ð2:7:28Þ

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

85

Plugging (2.7.24)–(2.7.28) into (2.7.23) and using theorems 2.1.1, 2.1.2, (2.7.1), (2.7.11) and (2.7.22), we get lim kI xxxxx ðtÞk2 ¼ 0;

t! þ 1

which gives (2.7.16). Thus this completes the proof.

h

Proof of Theorem 2.1.3. Combining lemmas 2.6.1–2.6.4 and 2.7.1–2.7.4, we can complete the proof of theorem 2.1.3. h

2.8

Bibliographic Comments

It is well-known that radiation dynamics includes the radiative effects into the hydrodynamical framework. When equilibrium holds between matter and radiation, a simple way to do that is to include local radiation terms into state functions and transport coefficients. From quantum mechanics, we know that radiation can be described by its quanta, the photons, which are massless particles traveling at the speed c of light, characterized by their frequency ν, their energy E = hν (where h is ~ ~ Planck’s constant), and their momentum ~ p ¼ hm c X with X is a vector of the 2-unit sphere. Moreover, from statistical mechanics, we can describe macroscopically an assembly of massless photons of energy E and momentum ~ p using a distribution ~ mÞ. Using this fundamental quantity, function: the radiative density I ðr; t; X; we can derive global quantities by integrating with respect to the angular and frequency variables: the spectral radiative energy density ER(r, t) per unit volume RR ~ mÞdXdm, and the spectral radiative flux ! I ðr; t; X; FR ¼ is then ER ðr; tÞ :¼ 1c RR ~ ~ XI ðr; t; X; mÞ dXdm. If the matter is in thermodynamic equilibrium at constant temperature T and if radiation is also in thermodynamic equilibrium at matter, its temperature is also T and statistical mechanics tells us that the distribution function for photons is given by the Bose–Einstein statistics with zero chemical potential. When there are no radiative effects, the complete hydrodynamical system can be derived from the standard conservation laws of mass, momentum and energy using Boltzmann’s equation satisfied by the fm ðr; ~ v; tÞ and the Chapman-Enskog expansion Gallavotti [43]. Then this reduces to the compressible Navier–Stokes system 8 u Þ ¼ 0; < qt þ r  ðq~ ~ þ~ ðq~ u Þt þ r  ðq~ u ~ u Þ ¼ r  P f; ð2:8:1Þ : ~ ~ þ g; u Þ ¼ r~ qD :P ðqeÞt þ r  ðqe~ ~ ¼ pðq; T Þ~ where P I þ~ p is the material stress tensor for a newtonian fluid with the ~ þ kr  ~ viscous contribution ~ p ¼ 2lD u~ I with 3λ + 2μ ≥ 0 and μ > 0, and the strain @uj 1 @ui ~ ~ þ tensor D such that Dij ¼ .~ q is the thermal heat flux and ~ f and g are 2

@xj

@xi

external force and source terms. There are many mathematical researchers who studied such models and related models. We can refer to Antontsev et al. [3],

1D Radiative Fluid and Liquid Crystal Equations

86

Batchelor [4], Choe and Kim [13, 14], Constantin et al. [15], Ducomet and Zlotnik [22], Feireisl [28], Feireisl and Novotný [29], Feireisl and Petzeltova [31–33], Feireisl et al. [30], Fang and Zhang [26, 27], Foias and Temam [34, 35], Frid and Shelukhin [36], Fujita and Kato [39], Galdi [42], Hoff [46–49], Hoff and Serre [51], Hoff and Smoller [52], Hoff and Ziane [53, 54], Huang et al. [58], Jiang [61–67], Kawashima [68], Kawashima and Nishida [69], Nishibata and Zhu [70], Kazhikhov [71, 72], Kazhikov and Shelukhin [73], Lions [84], Matsumura and Nishida [90–93], Okada [97], Paicu and Zhang [98], Qin [101–105], Qin and Hu [113, 114], Qin et al. [115, 116, 118–120, 126, 127, 131], Qin and Huang [121, 122], Qin and Jiang [125], Qin and Rivera [129], Qin and Song [130], Qin and Wen [132], Qin and Zhao [134], Serrin [136], Temam [139, 140], Xin [146], Xin and Yan [147], Zhang and Fang [150–154], Zheng and Qin [156], and references therein. In the framework of special relativity, the foundations of radiative fluids have been described by Pomraning [99, 100] and Mihalas et al. [94]. Later on, Buet et al. [10] and Lowrie et al. [89] studied in the inviscid case. Dubroca et al. [17], Lin [79] and Lin et al. [80] investigated for numerical aspects. For more results, we can refer to Chandrasekhar [12], Gallavotti [43], Jiang [61], Lowrie et al. [89] and Zhong and Jiang [157]. When radiation is present, Chandrasekhar [12] investigated the radiation integro-differential equation: terms ~ f and g include the terms for the coupling between the matter and the radiation, depending on I, and I is driven by a transport equation. If the matter is at local thermodynamics equilibrium (LTE), the coupled system reads (see, e.g., Mihalas and Weibel-Mihalas [94] and Pomraning [99] for details) 8 u Þ ¼ 0; qt þ r  ðq~ > > > < ðq~ ~ þ ! u ~ u Þ ¼ r  P SF ; u Þt þ r  ðq~ ð2:8:2Þ ~:P ~ þ SE ; > þ r  ðqe~ u Þ ¼ r~ q  D ðqeÞ > t > :1 @ ~ ~ ~ ~ c @t I ðr; t; X; mÞ þ X  rI ðr; t; X; mÞ ¼ St ðr; t; X; mÞ; where qðx; tÞ; ~ u ðx; tÞ; hðx; tÞ represent the density, velocity and temperature, respectively, the coupling terms are  i ~ ~ X u h ~ mÞ ¼ ra m; ~ ~ mÞ St ðr; t; X; X; q; T ; Bðm; T Þ  I ðr; t; X; c Z Z ~0  X; ~ m0 ! mÞ þ rs ðr; t; q; X nm ~0 ; m0 ÞI ðr; t; X; ~ mÞ  rs ðr; t; q; X ~0  ~  0 I ðr; t; X X; m0 ! mÞ m o ~ mÞ dX0 dm0 ; ~0 ; m0 ÞI ðr; t; X;  I ðr; t; X the radiative energy source SE ðr; tÞ :¼

Z Z

~ mÞdXdm; St ðr; t; X;

Asymptotic Behavior of Solutions for the Compressible Viscous Gas

the radiative flux 1 ! SF ðr; tÞ :¼ c

Z Z

87

~ mÞdXdm; ~ t ðr; t; X; XS

the functions σa and σs describe in a phenomenological way the absorption–emission and scattering properties of the photon-matter interaction, and Planck function B(ν, θ) describes the frequency-temperature black body distribution. We also note some results in Buet and Després [10], Dubroca and Feugeas [17], Jiang [61], Lin [79], Lin, Coulombel and Goudon [80], Lowrie, Morel and Hittinger [89] and Zhong and Jiang [157]. Let us now recall some previous works concerning the one-dimensional radiative fluids. In Ducomet and Nečasová [19], Ducomet and Nečasová considered the following system 8 g ¼ vx ; > > < t vt ¼ rx  gðSF ÞR ; ð2:8:3Þ ðe þ 12 v 2 Þ ¼ ðrv  QÞx  gðSE ÞR ; > > : 1 It þ g ðcx  vÞIx ¼ cS R1 R þ1 with ðSF ÞR ¼ 1c 1 0 xSðx; t; m; xÞdmdx in the domain ð0; M Þ  R þ subjected to the Dirichlet–Neumann boundary conditions vjx¼0;M ¼ 0;

Qjx¼0;M ¼ 0;

ð2:8:4Þ

and I jx¼0 ¼ 0 for x 2 ð0; 1Þ;

I jx¼M ¼ 0 for x 2 ð1; 0Þ:

ð2:8:5Þ

For q ≥ r + 1 with some suitable assumptions, they proved the existence and uniqueness of weak solutions. However, all estimates depended on any given time T > 0. So they could not study the large-time behavior of problem (2.8.3)–(2.8.5) based on their estimates. Ducomet and Nečasová [20] investigated the problem (2.8.3) with the different boundary conditions from Ducomet and Nečasová [19] vjx¼0;M ¼ 0;

Qjx¼0;M ¼ 0;

ð2:8:6Þ

and I jx¼0 ¼ Ib ðmÞ for x 2 ð0; 1Þ;

I jx¼M ¼ Ib ðmÞ for x 2 ð1; 0Þ

ð2:8:7Þ

and proved that the unique strong solutions of problem (2.8.3)–(2.8.6) converge to a well-determined equilibrium state at exponential rate in H1(0,M) for the fluid variables η, v, θ and in L2(0, M) for the radiative intensity R þ1 R I¼ 0 S 1 I ðx; t; m; xÞdxdm. Ducomet and Nečasová [18] established the global existence of solutions to the system (2.1.3) and (2.1.14) in Hi ði ¼ 1; 2Þ. However, estimates obtained there depend on any given time T, so they also could not investigate the large-time behavior of global solutions in Hi ði ¼ 1; 2Þ based on their estimates. Moreover, in

88

1D Radiative Fluid and Liquid Crystal Equations

Ducomet and Nečasová [18], all estimates hold only for q ≥ 2r + 1. In [109], we have established uniform-in-time estimates of ðgðtÞ; vðtÞ; hðtÞ; I ðtÞÞ in Hi ði ¼ 1; 2; 4Þ, which hold for q ≥ r + 1. For q ≥ r + 1, α ≥ 0, Qin et al. [111] established the gobal existence of solutions in Hi ði ¼ 1; 2Þ. Hence our results in this chapter have improved those in Ducomet and Nečasová [18] and Qin et al. [111]. Furthermore, the system considered here is quite different from that in Qin [104], so our uniform-in-time estimates are also quite different from those in Qin [104]. Remark 2.8.1. The multi-dimensional viscous situation has been poorly understood even at the formal level. Since the one-dimensional model possesses the special constitutive state equations, which the multi-dimensional model do not have, to our knowledge, we have not found any results on the global existence and asymptotic behavior of solutions to system (2.8.2), i.e., the multi-dimensional case of (2.1.3)–(2.1.6). Moreover, some Sobolev embedding inequalities and interpolation inequalities involved in our arguments heavily depend on the dimension, hence this may bring about some difficulties in deriving uniform-in-time estimates. In a word, the method we deal with the one-dimensional case can not be applied directly to the multi-dimensional case, which depends on the special constitutive relations of state functions, and so on. However, we can refer to Kippenhahn and Weigert [74] for a macroscopic treatment of radiation in the astrophysical context, and Feireisl [28] and Qin [104] for the associated mathematical treatment.

Chapter 3 Global Existence and Regularity of a One-Dimensional Liquid Crystal System 3.1

Main Results

This chapter will establish the global existence and regularity of solutions to the following system 8 ð3:1:1Þ < qt þ ðquÞx ¼ 0; ð3:1:2Þ ðquÞt þ ðqu 2 Þx þ ðPðqÞÞx ¼ luxx  kðjnx j2 Þx ; : ð3:1:3Þ nt þ unx ¼ hðnxx þ jnx j2 nÞ where x 2 [0, 1]. Here, ρ > 0 is the density function, u denotes the velocity, n represents the optical director of the molecules, P(ρ) = ργ(γ ≥ 1) is the pressure, and λ, μ, θ are positive constants. For simplicity, we assume λ = μ = θ = 1 in this chapter. The results of this chapter are chosen from [117]. We consider the following initial-boundary value problem for (3.1.1)–(3.1.3) in the reference domain {(x, t): 0 < x < 1, t 2 [0,T ]}. for any given T > 0 under the initial conditions and boundary conditions qðx; 0Þ ¼ q0 ðxÞ; uðx; 0Þ ¼ u0 ðxÞ; nðx; 0Þ ¼ n0 ðxÞ;

ð3:1:4Þ

uð0; tÞ ¼ uð1; tÞ ¼ 0; nx ð0; tÞ ¼ nx ð1; tÞ ¼ 0:

ð3:1:5Þ

Equations (3.1.1)–(3.1.3) reveal system modeling the nematic liquid crystal flow which consists of subsystem of the compressible Navier–Stokes equations coupling with a subsystem including the heat flow equation for harmonic maps. It was derived from the theory of hydrodynamics motivated by the Ericksen–Leslie system (Ericksen [23] and Lesile [77]) for the nematic liquid crystal flow. Equation (3.1.1) represents the transporting relation (conservation of mass), equation (3.1.2) is the conservation of the linear momentum and equation (3.1.3) is the heat flow of harmonic map equation. DOI: 10.1051/978-2-7598-2903-3.c003 © Science Press, EDP Sciences, 2022

1D Radiative Fluid and Liquid Crystal Equations

90

The notation in this chapter is standard. We put kk ¼ kkL2 ½0;1 : Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use Ci (i = 1, 2) to denote the generic positive constant depending on the kðq0 ; u0 ; n0 ÞkH i H i H i þ 1 ði ¼ 1; 2Þ; minx2½0;1 u0 ðxÞ; minx2½0;1 n0 ðxÞ and time T, and Cj (j = 3, 4) depending on Hj[0, 1] norm of initial data (ρ0, u0, n0), minx2½0;1 u0 ðxÞ; minx2½0;1 n0 ðxÞ and time T. R1 Without loss of generality, we may assume 0 q0 ðxÞdx ¼ 1. Under the Lagrangian coordinates, a.e., Z x y¼ q0 ðn; sÞ dn; t ¼ s; 0

system (3.1.1)–(3.1.5) is transformed into the following system vt ¼ uy ; uy jny j2  2 P þ v v

ut ¼

nt ¼

ð3:1:6Þ ! ;

ð3:1:7Þ

y

1 ny  jny j2 þ 2 n; v v y v

ð3:1:8Þ

ðv; u; nÞjt¼0 ¼ ðv0 ; u0 ; n0 Þ;

ð3:1:9Þ

ujy¼0;1 ¼ 0; ny jy¼0;1 ¼ 0

ð3:1:10Þ

where v ¼ q1 and v0 ¼ q1 . 0 In the sequel, we shall only consider the following case: jnj2 ¼ ni ni ¼ 1:

ð3:1:11Þ

Our main results in this chapter will read as follows (see also Qin and Huang [124]). Theorem 3.1.1. Suppose that ðv0 ; u0 ; n0 Þ 2 H 1 ½0; 1  H01 ½0; 1  H 2 ½0; 1 and the compatibility conditions hold. Then there exists a unique global solution ðvðtÞ; uðtÞ; nðtÞÞ 2 H 1 ½0; 1  H01 ½0; 1  H 2 ½0; 1 to the problem (3.1.6)–(3.1.10) such that for any ðy; tÞ 2 ½0; 1  ½0; T  (for all T > 0), 0\C11  vðy; tÞ  C1 ;

ð3:1:12Þ

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 91

and kvðtÞk2H 1 þ kuðtÞk2H 1 þ knðtÞk2H 2 þ knt ðtÞk2 Z t   2 þ ku k2H 2 þ nty  þ kut k2 ðsÞ ds  C1 : 0

ð3:1:13Þ

Theorem 3.1.2. Suppose that ðv0 ; u0 ; n0 Þ 2 H 2 ½0; 1  H02 ½0; 1  H 3 ½0; 1 and the compatibility conditions hold. Then there exists a unique global solution ðvðtÞ; uðtÞ; nðtÞÞ 2 H 2 ½0; 1  H02 ½0; 1  H 3 ½0; 1 to the problem (3.1.6)–(3.1.10) such that for any ðy; tÞ 2 ½0; 1  ½0; T  (for all T > 0), kvðtÞk2H 2 þ kuðtÞk2H 2 þ knðtÞk2H 3 þ knt ðtÞk2H 1 þ kut ðtÞk2 Z t  2  2  2 þ ðuty  þ ntyy  þ uyyy  ÞðsÞ ds  C2 : 0

ð3:1:14Þ

Theorem 3.1.3. Suppose that ðv0 ; u0 ; n0 Þ 2 H 4 ½0; 1  H04 ½0; 1  H 4 ½0; 1 and the compatibility conditions hold. Then there exists a unique global solution ðvðtÞ; uðtÞ; nðtÞÞ 2 H 4 ½0; 1  H04 ½0; 1  H 4 ½0; 1Þ to the problem (3.1.6)–(3.1.10) such that for any (y, t) 2 [0, 1] × [0, T] (for all T > 0), kvðtÞk2H 4 þ kuðtÞk2H 4 þ knðtÞk2H 4 þ knt ðtÞk2H 2 þ kut ðtÞk2H 2 þ kntt ðtÞk2 þ kutt ðtÞk2 Z t  2  2  2  2   2  2 þ ðuy H 4 þ ny H 4 þ uty H 2 þ nty H 2 þ utty  þ ntty  ÞðsÞ ds  C4 : 0

ð3:1:15Þ Remark 3.1.1. It is worthy to point out here that the solution (v(t), u(t), n(t)) obtained in theorem 3.1.3 is, in fact, a classical solution such that kðvðtÞ; uðtÞ; nðtÞÞk

1

1

1

C 3 þ 2 ð0;1ÞC 3 þ 2 ð0;1ÞC 3 þ 2 ð0;1Þ

3.2

 C4 :

ð3:1:16Þ

Global Existence in H 1  H01  H 2

In this section, we shall prove theorem 3.1.1 by establishing a series of lemmas. Lemma 3.2.1. If assumptions in theorem 3.1.1 hold, then the following estimates are valid in the Euler coordinates, Z TZ 1 Z 1 ðqu 2 þ pðqÞ þ 2jnx j2 Þdx þ ðux2 þ jn  nxx j2 Þdxdt  C1 ; ð3:2:1Þ 0

0

0

1D Radiative Fluid and Liquid Crystal Equations

92 Z  where pðqÞ ¼

1

Z 2

jnx j dx þ

0 qc c1 ;

0

q log q  q;

T

Z

1

jnxx j2 dxdt  C1

ð3:2:2Þ

0

c [ 1; c ¼ 1:

Proof. Multiplying (3.1.2) by u, using (3.1.1) and integrating the resulting equality by parts, we have Z 1 Z 1 Z 1d 1 2 2 ðqu þ pðqÞÞdx þ ux dx ¼ jnx j2 ux dx: ð3:2:3Þ 2 dt 0 0 0 Thus we derive from (3.1.11) that nxx þ jnx j2 n ¼ n  ðn  nxx Þ:

ð3:2:4Þ

Multiplying (3.2.4) by nxx, we obtain Z Z 1 Z 1 d 1 jnx j2 dx þ jnx j2 ux dx þ 2 jn  nxx j2 dx ¼ 0 dt 0 0 0 which, along with (3.2.3), gives (3.2.1). Multiplying (3.1.3) by nxx in L2[0, 1], integrating the resultant by parts and using the Gagliardo–Nirenberg interpolation inequality, we derive Z Z 1 Z 1 Z 1 d 1 jnx j2 dx þ jnxx j2 dx ¼ unx  nxx dx þ jnx j4 dx dt 0 0 0 0 Z Z 1 1 1 ¼ jnxx j4 dx  jnx j2 ux dx 2 0 0 Z 1 Z 1 4  C1 jnx j dx þ C1 ux2 dx 0

0

 C1 knx k3 knxx k þ C1 kux k2 1  knxx k2 þ C1 ðknx k6 þ kux k2 Þ 2 which, together with (3.2.1), gives (3.2.2). Thus this proves the lemma.

h

Lemma 3.2.2. If assumptions in theorem 3.1.1 hold, then the following estimates are valid for any T > 0 in the Lagrangian coordinates, and for all t 2 [0, T], ! !  Z 1 Z TZ 1 2 2 n  2  jn j ju j 1 y y y  dydt  C1 ; þ n  u 2 þ p þ ðvÞ þ dy þ v v v v y 0 0 0 ð3:2:5Þ

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 93 Z  where p þ ðvÞ ¼

1

0

jny j2 dy þ v

v 1c c1 ;

v  log v  1;

Z

T

Z

0

1 0

  1 ny  2 dydt  C1 v v y

ð3:2:6Þ

c [ 1; c ¼ 1:

Proof. Using lemma 3.2.1 and the total mass conversation easily derive (3.2.5) and (3.2.6).

R1 0

vdy ¼

R1 0

v0 dy, we h

Lemma 3.2.3. For any t 2 [0, T ] (for all T > 0), there exists one point y1 = y1(t) 2 [0, 1] such that the solution v(y, t) to problem (3.1.6)–(3.1.10) possesses the following expression " # ! Z t jny j2 1 1 P þ 2 vD ðy; sÞZ ðsÞds ð3:2:7Þ vðy; tÞ ¼ Dðy; tÞZ ðtÞ 1 þ v 0 where 8 Z y

Z y Z Z 1 1 y > > > udn  u0 ðnÞdn þ u0 ðzÞdzdy ; Dðy; tÞ ¼ v0 ðyÞ exp > < v0 0 0 y1 0 ! ! Z Z Z 1 > 1 t 1 jny j2 > 2 > vP þ u þ v0 dy: dyds ; v 0 ¼ > : Z ðtÞ ¼ exp  v v 0 0 0 0

Proof. The proof is standard, we refer to lemma 2.2.2.

ð3:2:8Þ ð3:2:9Þ

h

Lemma 3.2.4. For any T > 0, we have 0\C11  vðy; tÞ  C1 ; Z 0

t

for all ðy; tÞ 2 ½0; 1  ½0; T ;

  ny ðsÞ2 1 ds  C1 ; L

for all t 2 ½0; T :

ð3:2:10Þ ð3:2:11Þ

Proof. The present proof is more delicate than that of lemma 4.2.5 of this book. Obviously, we derive from (3.2.3) that 0\C 1  Dðy; tÞ  C1 ; 0\C11  Z ðtÞ  1:   jn j2 Noting that v P þ vy2 [ 0, using (3.2.7) and (3.2.12), we have vðy; tÞ  Dðy; tÞZ ðtÞ  C11 [ 0:

ð3:2:12Þ

ð3:2:13Þ

1D Radiative Fluid and Liquid Crystal Equations

94

Employing W 1;1 ,!L1 , lemmas 3.2.2, 3.2.3 and (3.2.12), (3.2.13), we deduce ! Z t jny j2 vP þ vðy; tÞ  C1 þ C1 ds v 0  Z t jny j2    C1 þ C1  v  1 vds 0 L

 #2 Z 1  Z t "Z 1 jny j  jny j  dy þ  C1 þ C1 dy vðy; sÞds  v v y 0 0 0  !1=2 Z

1=2 Z t" Z 1 1 jny j2 1 dy  C1 þ C1 dy v 0 0 0 v 0 1 1=2

1=2 #

2 Z 1 Z 1  1 jn j   y þ@ vdy vðy; sÞds  dy A  v y 0 v 0 Z 1 1 jny j2 dy þ dy v 0 0 v 0 !

2 Z 1  Z 1 1  jny j  þ vdydy vðy; sÞds  dy þ  v y 0 v 0 0 1

2 Z t Z 1  1  jny j  A @1 þ  C1 þ C1   dy vðy; sÞds v y 0 0 v Z

 C1 þ C1

t

Z

1

ð3:2:14Þ

which, using the Gronwall inequality and (3.2.6), (3.2.13), gives (3.2.10). The estimate (3.2.11) hence follows from (3.2.10) and the proof of (3.2.14). Thus this completes the proof. h Lemma 3.2.5. If assumptions in theorem 3.1.1 hold, then the following estimate is valid for any T > 0, and for all t 2 [0, T], Z 0

1

vy2 dy þ

Z tZ 0

1 0

ðny4 þ jnyy j2 þ jnt j2 Þdyds  C1 :

ð3:2:15Þ

Proof. It clearly follows from (3.1.7) that  vy  2ny  nyy 2jny j2 vy cvy u ¼ þ þ cþ1 : 2 3 v t v v v v

ð3:2:16Þ

Multiplying (3.2.16) by u  vy and then integrating the resulting equation over Qt = [0, 1] × [0, t ],t 2 [0,T ] (for all T > 0), we have for any ε > 0,

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 95 Z tZ 1  vy   2 ðvy2 þ jny j2 vy2 Þdyds u   þ v 0 0  2 Z tZ 1   v0y  2   u  þ C jv uj þ jn  n uj þ jn  n v j þ jn j jv uj dyds 1 y y yy y yy y y y  0 v  0 0 0 Z tZ 1 ðvy2 þ jny j2 vy2 Þdyds  C1 þ e 0 t

Z þ C1

0

 2 ð1 þ ny  1 Þ L

0

Z

1

u dyds þ C1

Z tZ

2

0

0

1

jnyy j2 dyds

0

which, by taking ε > 0 small enough and using (3.2.5) and (3.2.11), gives Z tZ 1 Z tZ 1  vy   2 2 2 2 þ ðv þ jn j v Þdyds  C þ C jnyy j2 dyds: ð3:2:17Þ u    y 1 1 y y v 0 0 0 0 By (3.2.6), we derive ! Z tZ 1 Z tZ 1 Z t Z 1   2 2 jnyy j2 jny j vy2 1 ny ny  nyy vy þ dydt þ 2 dyds dyds ¼ v3 v5 v y v4 0 0 0 0 v 0 0 Z Z Z tZ 1 jny j2 vy2 1 t 1 jnyy j2 dyds þ C dyds:  C1 þ 1 2 0 0 v3 v5 0 0 Thus

Z tZ 0

1

Z jnyy j2 dyds  C1 þ C1

0

0

t

  ny ðsÞ2 1 L

Z 0

1

vy2 dyds

2

which, with (3.2.17) and ku k  C1 in (3.2.5), gives Z t Z Z tZ 1    2 2 2 2 vy ðtÞ2 þ   ðvy þ jny j vy Þdyds  C1 þ C1 ny ðsÞ L1 0

0

0

0

1

vy2 dyds: ð3:2:18Þ

Applying the Gronwall inequality to (3.2.18), and using (3.2.6) and (3.2.11), we conclude Z tZ 1   vy ðtÞ2 þ ðjny j4 þ jny j2 vy2 þ jnyy j2 þ jvy j2 Þdyds  C1 : ð3:2:19Þ 0

or

0

From (3.1.8) and (3.2.19) it follows that     knt ðtÞk  C1 ðny ðtÞ þ nyy ðtÞÞ

ð3:2:20Þ

    nyy ðtÞ  C1 ðny ðtÞ þ knt ðtÞkÞ:

ð3:2:21Þ

Thus we deduce from (3.2.11), (3.2.19) and (3.2.20) that for all t 2 [0, T], Z tZ 1 jnt j2 dyds  C1 ; 0

0

which, with (3.2.19) and (3.2.21), gives us (3.2.15). Hence this proves the lemma. h

1D Radiative Fluid and Liquid Crystal Equations

96

Lemma 3.2.6. If assumptions in theorem 3.1.1 hold, then the following estimates are valid for any T > 0 and for all t 2 [0, T],  2  2 knt ðtÞk2 þ nyy ðtÞ þ uy ðtÞ Z t  2  2 þ ðkut k2 þ uyy  þ nty  ÞðsÞds  C1 : ð3:2:22Þ 0

Proof. Multiplying (3.1.7) by uyy and then integrating the resulting equation over Qt, using the Poincaré inequality and (3.2.15), we obtain Z t     uy ðtÞ2 þ uyy ðsÞ2 ds 0 Z tZ 1   C1 þ C1 jvy j þ juy vy j þ jny  nyy j þ jny j2 jvy j juyy jdyds 0 0 Z Z th   2  4  2  2  2 i 1 t uyy ðsÞ2 ds þ C1 ðuy L1 þ ny L1 Þvy  þ ny L1 nyy  ðsÞds  C1 þ 2 0 0 Z Z t 2     2  2 1 t uyy ðsÞ ds þ C1  C1 þ ðuy uyy  þ ny L1 nyy  ÞðsÞds 2 0 0 Z Z t 2     3 t uyy ðsÞ ds þ C1 ny ðsÞ2 1 nyy ðsÞ2 ds;  C1 þ L 4 0 0 which implies   uy ðtÞ2 þ

Z

t

  uyy ðsÞ2 ds  C1 þ C1

0

Z

t

    ny ðsÞ2 1 nyy ðsÞ2 ds: L

0

ð3:2:23Þ

Differentiating (3.1.8) with respect to t, multiplying the resulting equation by nt, integrating it by parts and using lemmas 3.2.2–3.2.5, (3.2.20), (3.2.21) and (3.2.23), we arrive at Z tZ 1 Z tX 5 jnty j2 2 dyds ¼ n ðy; 0Þ þ 2 Ai ds knt ðtÞk2 þ 2 k k t v2 0 0 0 i¼1 Z tX 5  C1 þ 2 Ai ds ð3:2:24Þ 0

i¼1

where Z A1 ¼

ny ðuyy nt þ uy nty Þ dy; v3

1

0

Z A3 ¼

0

1

Z

Z A2 ¼ 3

nty  ny uy þ nty  nt vy dy; A4 ¼ v3

A5 ¼  0

1

2jny j2 n  nt uy dy: v3

Z 0

0 1

1

ny  nt uy vy dy; v4

jny j2 jnt j2 þ 2ny  nty n  nt dy; v2

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 97

Using lemmas 3.2.2–3.2.5, the embedding theorem and (3.2.23), we derive that for any ε 2 (0, 1), Z tZ 1 A1 dyds 0 0 Z tZ 1 ny ðuyy nt þ uy nty Þ dyds ¼ v3 0 0 Z t Z t    2  2    nty ðsÞ2 ds þ C1 ðeÞ e ðuy L1 ny  þ knt kL1 ny uyy ÞðsÞds 0 0 Z t Z t   2  1=2  nty ðsÞ ds þ C1 e knt ðsÞk1=2 nty ðsÞ þ knt ðsÞk uyy ðsÞds 0 0 " Z #

1=2 Z t

1=2 Z t t 2  2  2       þ C1 ðeÞ uy ðsÞ ds uyy ðsÞ ds þ uy ðsÞ ds 0

Z

t

e

0

  nty ðsÞ2 ds þ C1

0

Z

0

  nty ðsÞ2 ds

t

1=4 Z

0

t

þ C1

0

 2e 0

0

  uyy ðsÞ2 ds

t

t

  nty ðsÞ2 ds þ C1 ðeÞ

Z

t

Z

t

t

knt k2 ds t

 2 uyy  ds

0

1=3 Z

  uyy ðsÞ2 ds

0

 C1 þ 2e

  nty ðsÞ2 ds þ C1 ðeÞ

Z

0

t

0

 2e

  uyy ðsÞ2 ds

 2e

t

1=2

2=3 þ C1 ðeÞ

  uyy ðsÞ2 ds

2=3

0

 2=3 Z t       nty ðsÞ2 ds þ C1 ðeÞ 1 þ ny ðsÞ2 1 nyy ðsÞ2 ds L 0



Z

t

  nty ðsÞ2 ds þ C1 ðeÞ 1 þ

t

  nty ðsÞ2 ds þ C1 ðeÞ þ C1 ðeÞ sup knt ðsÞk43

0

Z

1=2

 4 knt ðsÞk uyy ðsÞ3 ds þ C1 ðeÞ

Z

Z

t

2 3

0 t

Z 0

Z

0

 2e

þ C1 ðeÞ

  nty ðsÞ2 ds þ C1 ðeÞ

þ C1

Z

1=2

0

þ C1 Z

3=4

0

0

Z

 2e

4 1 ðknt ðsÞk2 uyy ðsÞÞ3 ds

Z t 2 !1=2 Z t

1=2     uyy ðsÞ2 ds   ds n ðsÞ t  

þ C1

Z

t

0

0

Z

 C1 ðeÞ þ e sup knt ðsÞk2 þ 2e 0st

0

t

  ny ðsÞ2 1 ð1 þ knt ðsÞk2 Þds L

0st

t

  nty ðsÞ2 ds;

Z

t 0

2=3

  ny ðsÞ2 1 ds L

2=3

ð3:2:25Þ

1D Radiative Fluid and Liquid Crystal Equations

98

and Z t Z tZ A2 ds  C1 0

0

Z  C1

1

jny  nt uy vy jdyds

0 t

2

knt ðsÞk ds

12 Z

0

 C1

  sup ny ðsÞL1

0st

Z þ

t

  uy ðsÞ2 ds

      ny ðsÞ2 1 uy ðsÞ2 1 vy ðsÞ2 ds L L

t

  uy ðsÞ2 ds

0

" Z

12 #

t

14 Z

0

  uyy ðsÞ2 ds

t

12

14

0

0

 C1

 14 Z t  1  1   2 2 2      sup ny ðsÞ nyy ðsÞ uyy ðsÞ ds 1þ

0st

1 2



Z

 C1 sup ð1 þ knt ðsÞk Þ 1 þ 0st

0 t

  uyy ðsÞ2 ds

14

0

Z

t

2

 2e sup knt ðsÞk þ C1 þ C1 0st

  uyy ðsÞ2 ds

13

0

 13 Z t  2  2 2 ny ðsÞ 1 nyy ðsÞ ds  2e sup knt ðsÞk þ C1 1 þ L 0st

0

2

Z

 2e sup knt ðsÞk2 þ C1 þ C1 sup knt ðsÞk3 0st

0st

0

t

  ny ðsÞ2 1 ds L

2

 3e sup knt ðsÞk þ C1 :

1=3

ð3:2:26Þ

0st

Similarly, we conclude that for any ε > 0, Z t Z tZ 1  A3 ds  C1 jnty  ny uy j þ jnty  nt vy j dyds 0 0 0 Z t Z t 2 e knty ðsÞk ds þ C1 ðeÞ ðkny k2L1 kuy k2 þ knt k2L1 kvy k2 ÞðsÞds 0 0 Z t Z t 2 ð3:2:27Þ  2e knty ðsÞk ds þ C1 ðeÞ þ C1 ðeÞ kny ðsÞk2L1 kuy ðsÞk2 ds; 0

Z 0

t

0

Z tZ

1

A4 ds  C1 ðjnt j2 jny j2 þ jny  nty n  nt jÞdyds 0 0 Z t Z t 2 e knty ðsÞk ds þ C1 knt ðsÞk2L1 kny ðsÞk2 ds 0 0 Z t  2e knty ðsÞk2 ds þ C1 ðeÞ; 0

ð3:2:28Þ

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 99 Z

t

Z tZ

A5 ds  C1

0

0

Z  C1 Z e

jny j2 jn  nt uy jdyds

0 t

    ny ðsÞ2 1 uy ðsÞ2 ds

12 Z

L

0 t

1

  nty ðsÞ2 ds þ C1 ðeÞ þ C1 ðeÞ

Z

0

t

0 t

0

 2 knt ðsÞk2L1 ny ðsÞ ds

12

    ny ðsÞ2 1 uy ðsÞ2 ds: L

ð3:2:29Þ

Inserting (3.2.25)–(3.2.29) into (3.2.24), then taking supremum in t on the left-hand side of (3.2.24), picking ε > 0 small enough, we finally derive Z t Z t       nty ðsÞ2 ds  C1 þ C1 ny ðsÞ2 1 uy ðsÞ2 ds: ð3:2:30Þ knt ðtÞk2 þ L 0

0

Using (3.2.21), (3.2.23) and (3.2.30), we deduce Z Z t       t  uy ðtÞ2 þ uyy ðsÞ2 ds  C1 þ C1 sup nyy ðsÞ2 ny ðsÞ2 1 ds L 0st

0

0

 C1 þ C1 sup ðknt ðtÞk2 þ 1Þ  C1 þ C1

0st Z t  0

   ny ðsÞ2 1 uy ðsÞ2 ds L

which, using the Gronwall inequality and (3.2.11), implies Z t     uy ðtÞ2 þ uyy ðsÞ2 ds  C1 : 0

Thus it follows from (3.2.21) and (3.2.30) that Z t  2  2  2  2 2     ðuyy  þ nty  ÞðsÞds  C1 : knt ðtÞk þ uy ðtÞ þ nyy ðtÞ þ

ð3:2:31Þ

0

or

Moreover, we can derive from (3.1.7) that           kut ðtÞk  C1 ðuy ðtÞ þ uyy ðtÞ þ vy ðtÞ þ ny ðtÞ þ nyy ðtÞÞ

ð3:2:32Þ

          uyy ðtÞ  C1 ðuy ðtÞ þ kut ðtÞk þ vy ðtÞ þ ny ðtÞ þ nyy ðtÞÞ:

ð3:2:33Þ

Thus we deduce from (3.2.31) and (3.2.32) that Z t kut ðsÞk2 ds  C1 0

which, along with (3.2.31), gives us (3.2.22). The proof is complete.

h

Proof of Theorem 3.1.1. Using lemmas 3.2.2–3.2.6, we readily complete the proof of theorem 3.1.1. h

1D Radiative Fluid and Liquid Crystal Equations

100

3.3

Proof of Theorem 3.1.2

This section will establish the regularity in H 2  H02  H 3 . Lemma 3.3.1. Suppose assumptions in theorem 3.1.2 are valid, then the following estimates hold for any T > 0 and for all t 2 [0,T], Z t  2   uty ðsÞ2 ds  C1 ; ð3:3:1Þ kut ðtÞk2 þ uyy ðtÞ þ 0

  vyy ðtÞ2 þ

Z

t

 2  2 ðnyyy  þ uyyy  ÞðsÞds  C1 :

ð3:3:2Þ

0

Proof. Differentiating (3.1.7) with respect to t, multiplying the resulting equation by ut, integrating it by parts and using lemmas 3.2.2–3.2.6, (3.2.32), we deduce that for any ε > 0, Z t   uty ðsÞ2 ds kut ðtÞk2 þ 0

Z tZ 1h i  kut ðy; 0Þk2 þ C1 juy j þ juy j2 þ jny  nty j þ jny j2 juy j juty jdydt 0 0 Z t Z t  2  2  2  2 uty ðsÞ ds þ C1 uy  þ uy  1 uy   C2 þ e L 0 0   4  2  2  2 þ ny L1 uy  þ ny L1 nty  ðsÞds Z  C2 þ e

t

  uty ðsÞ2 ds þ C1

t

  uty ðsÞ2 ds:

0

Z  C2 þ e

Z th  2  2  2 i uy  þ uyy  þ nty  ðsÞds 0

0

Now taking ε 2 (0, 1) small enough and using (3.2.33), we obtain (3.3.1). Differentiating (3.1.8) with respect to y, using lemmas 3.2.2–3.2.6 and the embedding theorem, we deduce       nty ðtÞ  C1 ðny ðtÞ 2 þ vy ðtÞ 1 Þ ð3:3:3Þ H H or

      nyyy ðtÞ  C1 ðnty ðtÞ þ ny ðtÞ

  þ vy ðtÞH 1 Þ:

ð3:3:4Þ

Similarly, we infer from (3.1.7),         uty ðtÞ  C1 ðvy ðtÞ 1 þ uy ðtÞ 2 þ ny ðtÞ 2 Þ H H H

ð3:3:5Þ

H1

or

    uyyy ðtÞ  C1 ðvy ðtÞ

H1

      þ uy ðtÞH 1 þ ny ðtÞH 2 þ uty ðtÞÞ:

ð3:3:6Þ

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 101

Now differentiating (3.1.7) with respect to y, using (3.1.6) (vtyy = uyyy), we derive v  vyy yy þ c c þ 1 ¼ uty þ Eðy; tÞ ð3:3:7Þ v t v where ðc þ 1Þvy2 2vy uyy 2vy2 uy 2jnyy j2 þ 2ny  nyyy þ  þ vc þ 2 v2 v3 v2 2 2 2 6jny j vy 8ny  nyy vy þ 2jny j vyy  þ : 3 v v4

Eðy; tÞ ¼ c

v

Multiplying (3.3.7) by vyy, integrating the resulting equation over Qt = [0, 1] × [0, t] and using the Young inequality, lemmas 3.2.2–3.2.6 and (3.3.1), we conclude Z t Z t Z t         2 vyy ðtÞ2 þ vyy ðsÞ2 ds  C2 þ e vyy ðsÞ2 ds þ C1 ðeÞ uty  þ kE k2 ðsÞds 0

0

0

ð3:3:8Þ where Z t Z t  2   2   2   2   2  4 2 vy  1 vy  þ vy  1 uyy  þ uy  1 vy  4 kEðsÞk ds  C1 L L L L 0

0

 2  2   2  2  2  2  2 þ nyy L1 nyy  þ ny L1 nyyy  þ ny L1 vy L1 nyy    4  2  4  4 þ ny L1 vyy  þ ny L1 vy L4 ðsÞds  C1

Z t  2    vy  1 þ ny 2 2 ðsÞds: 0

H

ð3:3:9Þ

H

Inserting (3.3.9) into (3.3.8), picking ε 2 (0, 1) small enough, and using lemmas 3.2.2–3.2.6, (3.3.1) and (3.3.4), we conclude Z t Z t          vyy ðtÞ2 þ vyy ðsÞ2 ds  C2 þ C1 ðeÞ nyyy 2 þ vyy 2 ðsÞds 0

0

Z  C2 þ C1 ðeÞ

t

  vyy ðsÞ2 ds

0

which, using the Gronwall inequality and estimates (3.3.1), (3.3.4) and (3.3.6), gives us (3.3.2). The proof is complete. h Lemma 3.3.2. Under assumptions in theorem 3.1.2, the following estimate holds for any T > 0 and for all t 2 [0, T], Z t       nty ðtÞ2 þ nyyy ðtÞ2 þ ntyy ðsÞ2 ds  C1 : ð3:3:10Þ 0

102

1D Radiative Fluid and Liquid Crystal Equations

Proof. Differentiating (3.1.8) with respect to t and y, multiplying the resulting equation by nty in L2(0, 1) and integrating by parts, we arrive at Z 1  1d jntyy j2 nty ðtÞ2 þ dy ¼ B0 ðtÞ þ B1 ðtÞ ð3:3:11Þ 2 dt v2 0 where

8

Z 1 2nyy uy nty vy þ ny uyy ny vy uy > > > þ 3 ntyy dy; B ðtÞ ¼ > < 0 v3 v3 v4 0 ! Z 1 > jny j2 n > > ðtÞ ¼ nty dy: B > : 1 v2 0 ty

We now employ lemmas 3.2.2–3.2.6, the Gagliardo–Nirenberg interpolation inequality and the Poincaré inequality to get Z 1  2  2  2  2 jntyy j2 B0  e dy þ C1 ðeÞðuy ðtÞL1 nyy ðtÞ þ ny ðtÞL1 uyy ðtÞ 2 v 0  2  2  2  2  2  þ vy ðtÞL1 nty ðtÞ þ ny ðtÞL1 vy ðtÞL1 uy ðtÞ Þ Z 1     jntyy j2 ð3:3:12Þ uy ðtÞ2 1 þ nty ðtÞ2 : e dy þ C ðeÞ 1 H v2 0 Similarly, using lemmas 3.2.2–3.2.6, 3.3.1 and the embedding theorem, we derive that for any small ε 2 (0, 1), Z 1     B1  C1 ½ðjnty  nyy j þ jny  ntyy j þ jny  nyy nt j þ jny  nyy uy j þ jnty ny j2 þ jny uy ny j2 0      þ jnty  ny vy j þ jnt vy ny j2 þ juyy ny j2 þ jvy uy ny j2 Þjnty jdy Z 1   2  2  2  2  jntyy j2 e dy þ C2 knt ðtÞk2H 1 þ nty ðtÞL1 nyy ðtÞ þ uy ðtÞH 1 þ ny ðtÞH 1 2 v 0 Z 1   2    jntyy j2 2   1 þ ny ðtÞ2 1 dy þ C n ðtÞ  2e k k 1 þ uy ðtÞ 2 t H H H v2 0 which, combined with (3.3.11), (3.3.12), (3.3.1)–(3.3.3) and lemmas 3.2.2–3.2.6, gives that for ε 2 (0, 1) small enough, Z t     nty ðtÞ2 þ ntyy ðsÞ2 ds  C2 : ð3:3:13Þ 0

By (3.3.3) and (3.3.13), we deduce   nyyy ðtÞ  C2 which, with (3.3.13), gives (3.3.10). The proof is complete.

h

Proof of Theorem 3.1.2. Using lemmas 3.3.1 and 3.3.2, we readily complete the proof of theorem 3.1.2. h

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 103

3.4

Proof of Theorem 3.1.3

This section will establish the regularity in H 4  H04  H 4 . Lemma 3.4.1. If assumptions in theorem 3.1.3 are valid, then the following estimates hold for all t 2 [0, T],       ntyy ðy; 0Þ2 þ uty ðy; 0Þ2 þ utyy ðy; 0Þ2 þ kntt ðy; 0Þk2 þ kutt ðy; 0Þk2  C4 ;  2 kutt ðtÞk2 þ kntt ðtÞk2 þ ntyy ðtÞ þ

Z

 2  2 þ ntty  þ ntyyy  ÞðsÞds  C4 ;

t

ð3:4:1Þ

 2  2 ðutty  þ utyy  þ kntt k2

0

ð3:4:2Þ Z

      uty ðtÞ2 þ uyyy ðtÞ2 þ utyy ðtÞ2 þ

t

  utyyy ðsÞ2 ds  C4 :

ð3:4:3Þ

0

Proof. Differentiating (3.1.7) and (3.1.8) with respect to y twice, using theorems 3.1.1, 3.1.2 and the embedding theorem, we deduce         utyy ðtÞ  C2 ðuy ðtÞ 3 þ vy ðtÞ 2 þ ny ðtÞ 3 Þ; ð3:4:4Þ H H H     ntyy ðtÞ  C2 ðny ðtÞ

H3

or

    uyyyy ðtÞ  C2 ðuy ðtÞ

H2

  þ vy ðtÞH 2 Þ

      þ vy ðtÞH 2 þ ny ðtÞH 3 þ utyy ðtÞÞ;

    nyyyy ðtÞ  C2 ðny ðtÞ

H2

    þ vy ðtÞH 2 þ ntyy ðtÞÞ:

ð3:4:5Þ

ð3:4:6Þ ð3:4:7Þ

Differentiating (3.1.7), (3.1.8) with respect to t, respectively, using theorems 3.1.1, 3.1.2, (3.3.3), (3.3.5), (3.4.4) and (3.4.5), we obtain           ð3:4:8Þ kutt ðtÞk  C2 ðutyy ðtÞ þ uty ðtÞ þ nty ðtÞ þ ntyy ðtÞ þ uy ðtÞH 1 Þ        C2 ðuy ðtÞH 3 þ ny ðtÞH 3 þ vy ðtÞH 2 Þ;

ð3:4:9Þ

      kntt ðtÞk  C2 ðknt ðtÞk þ nty ðtÞ þ ntyy ðtÞ þ uy ðtÞH 1 Þ

ð3:4:10Þ

       C2 ðny ðtÞH 3 þ vy ðtÞH 2 þ uy ðtÞH 1 Þ

ð3:4:11Þ

1D Radiative Fluid and Liquid Crystal Equations

104

or

          utyy ðtÞ  C1 kutt ðtÞk þ C2 ðuty ðtÞ þ nty ðtÞ þ ntyy ðtÞ þ uy ðtÞ 1 Þ; ð3:4:12Þ H       ntyy ðtÞ  C1 kntt ðtÞk þ C2 ðknt ðtÞk þ nty ðtÞ þ uy ðtÞ 1 Þ: H

ð3:4:13Þ

Differentiating (3.1.7) with respect to y and t, using theorems 3.1.1 and 3.1.2, we derive             utyyy ðtÞ  C2 ðutty ðtÞ þ uty ðtÞ 1 þ nty ðtÞ 2 þ ny ðtÞ 2 þ uy ðtÞ 2 Þ: H H H H ð3:4:14Þ Similarly, we get             ntyyy ðtÞ  C2 ðntty ðtÞ þ uy ðtÞ 2 þ vy ðtÞ 1 þ nty ðtÞ 1 þ ny ðtÞ 2 Þ: H H H H ð3:4:15Þ Thus estimate (3.4.1) follows from (3.3.5), (3.4.4), (3.4.5), (3.4.9) and (3.4.11). Differentiating (3.1.7) with respect to t twice, multiplying the resulting equation by utt in L2(0, 1), performing an integration by parts, using theorems 3.1.1, 3.1.2 and (3.4.1), we have ! 1 uy jny j2  cþ  2 utty ds v v v 0 tt "   !  # Z 1 2         1   utty jny j2    uty uy  þ uy uty  þ uy 3 6 utty  þ  dy þ C2  þ    vc   v2 L  v 0 tt tt  2  2  2  2  2   C11 utty ðtÞ þ C2 ðuy ðtÞH 1 þ uty ðtÞ þ nty ðtÞH 1 þ ntty ðtÞ Þ

1d kutt ðtÞk2 ¼  2 dt

Z

1

which thus, by theorems 3.1.1 and 3.1.2, implies Z t Z t     utty ðsÞ2 ds  C4 þ C2 ntty ðsÞ2 ds: kutt ðtÞk2 þ 0

ð3:4:16Þ

0

By (3.3.10) and (3.4.10), we further have Z t kntt ðsÞk2 ds  C2 :

ð3:4:17Þ

0

Similarly, differentiating (3.1.8) with respect to t twice, multiplying the resulting equation by ntt in L2(0, 1) and integrating it by parts, we arrive at 1d kntt ðtÞk2 ¼ D0 þ D1 þ D2 2 dt

ð3:4:18Þ

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 105

where Z

1

D0 ¼  0

Z

1

D2 ¼ 0

n  n  y tt dy; v tt v y ! jny j2 n ntt dy: v2

Z

1

D1 ¼ 0

      1 ny 2 ny þ ntt dy; v tt v y v t v ty

tt

Employing theorems 3.1.1 and 3.1.2, the Gagliardo–Nirenberg interpolation inequality and the Poincaré inequality, we conclude that for any ε > 0, Z 1  2  2  4 jntty j2 D0   dy þ C2 ½kntt ðtÞk2 þ nty ðtÞ þ uty ðtÞ þ uy ðtÞL4 2 v 0      2    þ ð nty ðtÞ þ uty ðtÞ þ uy ðtÞL4 þ kntt ðtÞkÞntty ðtÞ  2  2  2  2   C11 ntty ðtÞ þ C2 ðnty ðtÞ þ uty ðtÞ þ uy ðtÞH 1 þ kntt ðtÞk2 Þ; ð3:4:19Þ Z D1  C1

1

(

h

i ðjuty j þ juy j2 Þðjnyy j þ jny vy jÞ jntt j

0

) þ jntyy j þ jnyy uy j þ jnty vy j þ jny uyy j þ jny vy uy j juy kntt j dy 

 2  2  2  2  C2 ðkntt ðtÞk2 þ nty ðtÞH 1 þ uy ðtÞH 1 þ ny ðtÞH 2 þ uty ðtÞ Þ; Z D2  C 1

1

ð3:4:20Þ

ðjnty j2 þ jny  ntty j þ jny  nty knt j þ jny j2 jntt j

0

þ jny  nty uy j þ jny j2 jnt uy j þ jny j2 juty j þ jny j2 juy j2 Þjntt jdy  2  2  2  2  entty ðtÞ þ C2 ðkntt ðtÞk2 þ nty ðtÞH 1 þ uty ðtÞ þ uy ðtÞH 1 Þ which, along with (3.4.17)–(3.4.20), (3.4.1) and theorem 3.1.2, leads to that for ε 2 (0, 1) small enough, Z t   ntty ðsÞ2 ds  C4 : ð3:4:21Þ kntt ðtÞk2 þ 0

On the other hand, we derive from (3.4.16) and (3.4.21) that Z t   2 utty ðsÞ2 ds  C4 : kutt ðtÞk þ

ð3:4:22Þ

0

Exploiting theorems 3.1.1, 3.1.2, (3.4.12), (3.4.13), (3.4.15), (3.4.21) and (3.4.22), we obtain (3.4.2).

1D Radiative Fluid and Liquid Crystal Equations

106

Differentiating (3.1.7) with respect to t and y, multiplying the resulting equation by uty in L2(0, 1) and integrating by parts, we arrive at  1d uty ðtÞ2 ¼ F0 ðtÞ þ F1 ðtÞ 2 dt where F0 ðy; tÞ ¼

uy jny j2  2 P þ v v

!

Z uty j10 ; ty

F1 ðtÞ ¼ 0

1

uy jny j2  2 P þ v v

ð3:4:23Þ ! utyy dy: ty

Employing (3.4.1) and (3.4.2), theorems 3.1.1 and 3.1.2, the Gagliardo–Nirenberg interpolation inequality and the Poincaré inequality and the Young inequality, we get         F0  C1 ðuyy ðtÞL1 þ vy ðtÞL1 uy ðtÞL1 þ utyy ðtÞL1         þ uyy ðtÞL1 uy ðtÞL1 þ uty ðtÞL1 vy ðtÞL1  2     þ uy ðtÞL1 vy ðtÞL1 Þuty ðtÞL1    1  1    C1 ðuy ðtÞH 2 þ utyy ðtÞ2 utyyy ðtÞ2 þ utyy ðtÞ  1  1  1  1 þ uty ðtÞ2 utyy ðtÞ2 Þuty ðtÞ2 utyy ðtÞ2  2  2  2  2  C2 ðutyy ðtÞ þ utyyy ðtÞ þ uy ðtÞH 2 þ uty ðtÞ Þ ð3:4:24Þ which, together with (3.4.2), (3.4.14) and theorem 3.1.2, further leads to Z t Z t  2  2 F0 ds  C2 ðutyy  þ utyyy  ÞðsÞds  C4 : ð3:4:25Þ 0

0

Similarly, using theorems 3.1.1 and 3.1.2, (3.4.1), (3.4.2) and the embedding theorem, we conclude that for any small ε 2 (0, 1), Z 1 F1  C1 ðjuyy j þ jvy uy j þ juyy uy j þ juty vy j þ juy2 vy j þ jnty  nyy j þ jny  ntyy j 0    þ jny  nyy uy j þ jny  nty vy j þ juyy ny j2 þ jny j2 juy vy j jutyy jdy  2  2  2  2  2  C2 ðuty ðtÞH 1 þ nty ðtÞH 1 þ uy ðtÞH 1 þ ny ðtÞH 1 þ vy ðtÞH 1 Þ which, combined with (3.4.23), (3.4.25), (3.4.1), (3.4.2) and theorems 3.1.1 and 3.1.2, gives us Z t     uty ðtÞ2 þ utyy ðsÞ2 ds  C4 : ð3:4:26Þ 0

Therefore, by (3.3.6), (3.4.1), (3.4.2), (3.4.12), (3.4.14) and (3.4.26), we can obtain (3.4.3). This completes the proof. h Lemma 3.4.2. If assumptions in theorem 3.1.3 are valid, then the following estimates hold for any t 2 [0, T](T > 0),

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 107       vyyy ðtÞ2 þ nyyyy ðtÞ2 þ uyyyy ðtÞ2  C4 ;   vyyyy ðtÞ2 þ

Z

t

ð3:4:27Þ

 2  2 ðnyyyyy  þ uyyyyy  ÞðsÞds  C4 :

ð3:4:28Þ

0

Proof. Differentiating (3.3.7) with respect to y, we arrive at v  vyyy yyy þ c c þ 1 ¼ E1 ðy; tÞ v t v

ð3:4:29Þ

with E1 ðy; tÞ ¼ utyy þ Ey ðy; tÞ þ

v v  ðc þ 1Þvyy vy yy y þc : 2 v vc þ 2 t

Using theorems 3.1.1 and 3.1.2 and lemma 3.4.1, we have  2  2  2  2 kE1 ðtÞk2  C2 ðutyy ðtÞ þ vy ðtÞH 2 þ uy ðtÞH 2 þ ny ðtÞH 3 Þ leading to Z

t

Z

t

kE1 ðsÞk2 ds  C4 þ C2

0

 2  2 ðvyyy  þ nyyyy  ÞðsÞds:

ð3:4:30Þ

0

Thus it follows from (3.4.2), (3.4.7) and (3.4.30) that for all t 2 [0, T], Z t Z t   2 vyyy ðsÞ2 ds: kE1 ðsÞk ds  C4 þ C2 ð3:4:31Þ 0

0 vyyy v ,

Multiplying (3.4.29) by integrating the result over Qt, and using the Young inequality, (3.4.1) and (3.4.31), we infer that Z t Z t Z t       vyyy ðtÞ2 þ vyyy ðsÞ2 ds  C4 þ C2 vyyy ðsÞ2 ds kE1 ðsÞk2 ds  C4 þ C2 0

0

which, using the Gronwall inequality, gives us for all t 2 [0, T], Z t     vyyy ðtÞ2 þ vyyy ðsÞ2 ds  C4 :

0

ð3:4:32Þ

0

Thus from (3.4.2), (3.4.3), (3.4.6), (3.4.7), (3.4.32) and theorem 3.1.2 it follows that     nyyyy ðtÞ þ uyyyy ðtÞ  C4 which, together with (3.4.32), gives us estimate (3.4.27). Now differentiating (3.4.29) with respect to y, we arrive at v  vyyyy yyyy þ c c þ 1 ¼ E2 ðy; tÞ v t v

ð3:4:33Þ

1D Radiative Fluid and Liquid Crystal Equations

108

with E2 ðy; tÞ ¼ E1y ðy; tÞ þ

v



yyy vy v2 t

þ cðc þ 1Þ

vyyy vy : vc þ 2

Exploiting the embedding theorem, lemma 3.4.1 and theorem 3.1.2 and (3.4.27), we can derive         ð3:4:34Þ kE2 ðtÞk  C2 ðutyyy ðtÞ þ uy ðtÞH 3 þ vy ðtÞH 3 þ ny ðtÞH 4 Þ: Differentiating (3.1.7), (3.1.8) with respect to y three times, respectively, using theorems 3.1.1 and 3.1.2, the Poincaré inequality and the Young inequality, we derive           uyyyyy ðtÞ  C1 utyyy ðtÞ þ C2 ðvy ðtÞ 3 þ uy ðtÞ 3 þ ny ðtÞ 4 Þ; ð3:4:35Þ H H H       nyyyyy ðtÞ  C1 ntyyy ðtÞ þ C2 ðny ðtÞ

H3

  þ vy ðtÞH 3 Þ

ð3:4:36Þ

which, along with (3.4.2), (3.4.3), (3.4.27), (3.4.34) and theorem 3.1.2, implies Z t Z t   vyyyy ðsÞ2 ds: ð3:4:37Þ kE2 ðsÞk2 ds  C4 þ C2 0

0 vyyyy v ,

Multiplying (3.4.33) by integrating the resulting equation over Qt, and using the Young inequality, (3.4.1) and (3.4.37), we derive Z t Z t     vyyyy ðtÞ2 þ vyyyy ðsÞ2 ds  C4 þ C2 kE2 ðsÞk2 ds 0 0 Z t   vyyyy ðsÞ2 ds  C4 þ C2 0

which, using the Gronwall inequality, gives us for all t 2 [0, T], Z t     vyyyy ðtÞ2 þ vyyyy ðsÞ2 ds  C4 :

ð3:4:38Þ

0

Therefore, it follows from (3.4.2), (3.4.3), (3.4.27), (3.4.35), (3.4.36), (3.4.38) and theorem 3.1.2 that for all t 2 [0, T], Z t  2  2 ðnyyyyy  þ uyyyyy  ÞðsÞds  C4 ; 0

which, together with (3.4.38), gives us estimate (3.4.28). This readily completes the proof. h Proof of Theorem 3.1.3. Using lemmas 3.4.1, 3.4.2, we can complete the proof of theorem 3.1.3. h

Global Existence and Regularity of a One-Dimensional Liquid Crystal System 109

3.5

Bibliographic Comments

In this section, we briefly review some related literature. Note that the dynamic theory of nematic liquid crystals was established by Ericksen [23] and Leslie [78] in the 1960s. This theory is in fact derived from the macroscopic point of view; which helps in understanding the coupling between the director field and the velocity field, and also gives a tool to describe the motion of the defects in the molecule configurations under the influence of the flow velocity. In order to describe the dynamic property of nematic materials, Ericksen [24] and Leslie [77] established a system consisting of equations for the conversation of the mass, the linear momentum and an extra equation for the conversation of the momentum due to vector field n. The Ericksen–Leslie system is well studied for describing many special flows for the materials, especially for those with small molecules, and is widely accepted in engineering and mathematical communities studying liquid crystals. For the following simplified Ericksen–Leslie equation 8 < vt þ ðv  rÞv  mDv þ rP ¼ kr  ðrd  rdÞ; d þ ðv  rÞd ¼ cðDd  f ðdÞÞ; ð3:5:1Þ : t r  v ¼ 0; where v is the flow velocity and d is the relaxation of the molecule direction. The 2

in the stress tenterm kr  ðrd  rdÞ ¼ krj ðri d k rj d k Þ ¼ kri d k rj d k þ k2 ri jrdj 2 sor represents the anisotropic feature of the system. When the system (3.5.1) is subjected to Dirichlet boundary conditions, Lin and Liu [81] proved the global existence of weak solutions and classical solutions, and discussed the uniqueness and some stability properties of the system. Later on, they established the partial regularity results in [82], and most results were extended to the general Ericksen–Leslie equations in [83]. When the system (3.5.1) is subjected to free-slip boundary condition for v and Neumann boundary for d (i.e., v  m ¼ 0; ðr  vÞ  m ¼ 0; @d @m ¼ 0 on @Ω), Liu and Shen [86] proved the local classical solutions and global weak solutions in 2D and 3D cases. When the system (3.5.1) is subjected to the Dirichlet boundary condition and reproductivity conditions (i.e., v(x, 0) = v(x, T), d(x, 0) = d(x, T)), Blanca et al. [9] showed the existence of weak solutions with the reproductivity in time property in 2D and 3D cases. Fan and Ozawa [25] proved some regularity criteria for this simplified Ericksen–Leslie system and also obtained the existence and uniqueness of global smooth solutions for a regularization model of this simplified system. Hu and Wang [57] studied three-dimensional case of (3.5.1) in a smooth bounded domain, and obtained the existence and uniqueness of the global strong solutions with small initial data and also proved that when the strong solution exists, all the global weak solutions constructed in Lin and Liu [81] must be equal to the unique strong solution. Hong [56] studied the system in 2D, and proved global existence of solutions with initial data, where the solutions are regular except for at a finite number of singular times. Some of the numerical experiments to the system (3.5.1) were performed in Liu and Walkington [87, 88] who also demonstrated the coupling between the fluid field and the director field. In this direction, we also mention the works by Calderer et al. [11] and Liu [85] for the nematic liquid

110

1D Radiative Fluid and Liquid Crystal Equations

crystal model. Wen and Ding [143] studied the incompressible hydrodynamic flow of the nematic liquid crystals in dimension N (N = 2 or 3): 8 q þ r  ðquÞ ¼ 0; > > < t ðquÞt þ r  ðqu  uÞ þ rP ¼ cDu  kr  ðrd  rdÞ; ð3:5:2Þ r  u ¼ 0; > > : dt þ ðu  rÞd ¼ hðDd þ jrnj2 nÞ; where ρ denotes density, u is the flow velocity and d is the relaxation of the molecule direction. Under the assumption ρ0 > 0, the authors obtained the local existence and uniqueness of the solutions. In addition, if ρ0 had a positive bound from below, and N = 2, they obtained the global existence and uniqueness of solutions with small initial data. We also note that for the system (3.1.1)–(3.1.5), Ding [16] proved the existence of global strong solutions under assumptions for initial data ðq0 ; u0 ; n0 Þ 2 H 1 ð0; 1Þ  H01 ð0; 1Þ  H 2 ð0; 1Þ with 0\c01  q0  c0 , and global smooth (classical) solutions under assumptions for initial data ðq0 ; u0 ; n0 Þ 2 C 1 þ a ½0; 1  C 2 þ a ½0; 1  C 2 þ a ½0; 1 with 0\a\1; 0\c01  q0  c0 . It is different from our result. First, Ding [16] studied the existence of global strong solutions to problem (3.1.1)–(3.1.5) in Euler coordinates, while we have established the existence of global solutions in Hi (i = 1, 2, 4) in Lagrangian coordinates in this chapter. Second, we have established the regularity in H 2  H02  H 3 and H 4  H04  H 4 , while Ding [16] proved the existence smooth solutions in C1+α × C2+α × C2+α, which is a direct consequence of our results.

Chapter 4 Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 4.1

Introduction

In this chapter, we shall continue to study the large-time behavior of solutions in H i
H0i
H i þ 1 ði ¼ 1; 2Þ and H 4
H04
H 4 to the following 1D liquid crystal system based on the results in chapter 3: 8 vt ¼ u y ; ð4:1:1Þ > > > ! > > > uy jny j2 > >  2 ; ð4:1:2Þ > ut ¼ P þ > > v v > < y 1 ny  jny j2 > > ¼ þ n; ð4:1:3Þ n > t > > v v y v2 > > > > ð4:1:4Þ ðv; u; nÞjt¼0 ¼ ðv0 ; u0 ; n0 Þ; > > > : ujy¼0;1 ¼ 0; ny jy¼0;1 ¼ 0 ð4:1:5Þ where v ¼ q1 ; v0 ¼ q1 and P ¼ q ¼ 1v (i.e., γ = 1 in chapter 3). 0 In this chapter, we still restrict ourselves to the following case: jnj2 ¼ ni ni ¼ 1:

ð4:1:6Þ

In this chapter, we use Ci (i = 1, 2) to denote the generic positive constant depending on the kðq0 ; u0 ; n0 ÞkH i
H i
H i þ 1 ði ¼ 1; 2Þ; minx2½0;1 u0 ðxÞ; minx2½0;1 n0 ðxÞ, but not depending on time T, and Cj (j = 3, 4) depending on Hj[0, 1] norm of initial data (ρ0, u0, n0), minx2½0;1 u0 ðxÞ; minx2½0;1 n0 ðxÞ, but not depending on T. The results of this chapter are selected from Qin and Feng [110].

DOI: 10.1051/978-2-7598-2903-3.c004 © Science Press, EDP Sciences, 2022

1D Radiative Fluid and Liquid Crystal Equations

112

Theorem 4.1.1. Suppose that ðv0 ; u0 ; n0 Þ 2 H 1 ð0; 1Þ
H01 ð0; 1Þ
H 2 ð0; 1Þ and the compatibility conditions are valid. Then there exists a unique global solution ðvðtÞ; uðtÞ; nðtÞÞ 2 H 1 ð0; 1Þ
H01 ð0; 1Þ
H 2 ð0; 1Þ to the problem (4.1.1)–(4.1.5) such that for all ðy; tÞ 2 ½0; 1
½0; þ 1Þ, 0\C11  vðy; tÞ  C1 ;

ð4:1:7Þ

kvðtÞk2H 1 þ kuðtÞk2H 1 þ knðtÞk2H 2 þ knt ðtÞk2 Z t   2 þ ku k2H 2 þ nty  þ kut k2 ðsÞds  C1 :

ð4:1:8Þ

and for all t > 0,

0

Moreover, we have as t → +∞,  kH 2 ! 0 kvðtÞ  v kH 1 ! 0; kuðtÞkH 1 ! 0; knðtÞ  n R1 R1 R1  ¼ 0 nðy; tÞdy: where  v ¼ 0 vðy; tÞdy ¼ 0 v0 dy, n

ð4:1:9Þ

Theorem 4.1.2. Suppose that ðv0 ; u0 ; n0 Þ 2 H 2 ð0; 1Þ
H02 ð0; 1Þ
H 3 ð0; 1Þ and the compatibility conditions are valid. Then there exists a unique global solution ðvðtÞ; uðtÞ; nðtÞÞ 2 H 2 ð0; 1Þ
H02 ð0; 1Þ
H 3 ð0; 1Þ to the problem (4.1.1)–(4.1.5) such that for all t > 0, kvðtÞk2H 2 þ kuðtÞk2H 2 þ knðtÞk2H 3 þ knt ðtÞk2H 1 þ kut k2 Z t  2      uty  2 þ ntyy 2 þ uyyy 2 ðsÞds  C2 : þ H 0

ð4:1:10Þ

Moreover, we have as t → +∞, kvðtÞ  v kH 2 ! 0;

kuðtÞkH 2 ! 0;

 kH 3 ! 0: kn  n

ð4:1:11Þ

Theorem 4.1.3. Suppose that ðv0 ; u0 ; n0 Þ 2 H 4 ð0; 1Þ
H04 ð0; 1Þ
H 4 ð0; 1Þ and the compatibility conditions are valid. Then there exists a unique global solution ðvðtÞ; uðtÞ; nðtÞÞ 2 H 4 ð0; 1Þ
H04 ð0; 1Þ
H 4 ð0; 1Þ to the problem (4.1.1)–(4.1.5) such that for any t > 0, kvðtÞk2H 4 þ kuðtÞk2H 4 þ knðtÞk2H 4 þ knt ðtÞk2H 2 þ kut ðtÞk2H 2 þ kntt ðtÞk2 þ kutt ðtÞk2 Z t  2           uy  4 þ ny 2 4 þ uty 2 2 þ nty 2 2 þ utty 2 þ ntty 2 ðsÞ ds  C4 : þ H H H H 0

ð4:1:12Þ Moreover, we have, as t → +∞, v kH 4 ! 0; kvðtÞ  

kuðtÞkH 4 ! 0;

 kH 4 ! 0: knðtÞ  n

ð4:1:13Þ

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 113

Uniform Estimates in H i
H0i
H i þ 1 ði ¼ 1; 2Þ and H 4
H04
H 4

4.2

The global existence of solutions in H i
H0i
H i þ 1 ði ¼ 1; 2Þ and H 4
H04
H 4 has been established in Qin and Huang [124] and also in chapter 3. In this section, we shall derive some uniform estimates in H i
H0i
H i þ 1 ði ¼ 1; 2Þ and H 4
H04
H 4 by establishing a series of lemmas. First, we shall establish some uniform estimates in H 1
H01
H 2 . Lemma 4.2.1. If assumptions in theorem 4.1.1 are valid, then the following estimates hold in Euler coordinates for any t > 0, Z 1 Z tZ 1 2 2 ðqu þ pðqÞ þ 2jnx j Þdx þ ðux2 þ jn
nxx j2 Þdxds  C1 ; ð4:2:1Þ 0

0

Z

1

jnx j2 dx þ

0

Z tZ

0

0

1

jnt j2 dxds  C1

ð4:2:2Þ

0

where pðqÞ ¼ q log q  q: Proof. Multiplying (4.1.2) by u, using (4.1.1) and integrating the result by parts, we have Z Z 1 Z 1 1d 1 2 2 ðqu þ pðqÞÞdx þ ux dx ¼ jnx j2 ux dx: ð4:2:3Þ 2 dt 0 0 0 We derive from |n|2 = 1 that nxx þ jnx j2 n ¼ n
ðn
nxx Þ: Multiplying the above equation by nxx, we obtain Z Z 1 Z 1 d 1 jnx j2 dx þ jnx j2 ux dx þ 2 jn
nxx j2 dx ¼ 0; dt 0 0 0 which, along with (4.2.3), gives (4.2.1). Using (4.1.6), we easily get n  nt ¼ 0: Multiplying (4.1.3) by nt, integrating the result over [0, 1] × [0, t], applying the Young inequality and the Poincaré inequality, we have for any ε > 0, Z 1 Z tZ 1 Z tZ 1 jnx j2 dx þ jnt j2 dxds ¼  unx nt dxds 0

0

0

0

e

Z tZ 0

e e

0

Z tZ 0

1

Z jnt j2 dxds þ C1

0

Z tZ 0

jnt j2 dxds þ C1

0

Z tZ 0

0 1

Z 2

jnt j dxds þ C1 0

u 2 jnx j2 dxds

0 t

0

1

1

t

Z

ku k2L1 Z ku x k

2

1

0 1

0

jnx j2 dxds

jnx j2 dxds:

1D Radiative Fluid and Liquid Crystal Equations

114

Choosing ε > 0 small enough, performing the Gronwall inequality, and using (4.2.1), we obtain Z 1 Z tZ 1 jnx j2 dx þ jnt j2 dxds  C1 : 0

0

0

h

This proves the lemma.

Lemma 4.2.2. If assumptions in theorem 4.1.1 are valid, then the following estimates hold in Euler coordinates for any t > 0, Z tZ 1 ðjnx j4 þ jnxx j2 Þdxds  C1 : ð4:2:4Þ 0

0

Proof. Squaring (4.1.3) in both sides, integrating the result over [0, 1] × [0, t], employing integration by parts, the Young inequality and the Poincaré inequality, and using (4.2.1) and (4.2.2), we conclude for any ε > 0, Z tZ 1 ðjnxx j2 þ jnx j4 Þdxds 0

0

¼

Z tZ 0

1

ðjnt j2 þ u 2 jnx j2 þ 2nt unx Þdxds  2

0

Z tZ 0

1

nxx jnx j2 ndxds

0

Z Z 2 t 1 ¼ ðjnt j þ u jnx j þ 2nt unx Þdxds þ jnx j4 dxds 3 0 0 0 0 Z tZ 1 Z tZ 1 2 2 2 4 2 ðjnt j þ u 2 jnx j Þdxds þ jnx j dxds 3 0 0 0 0 Z tZ 1 Z t 2  C1 þ jnx j4 dxds þ C1 ku k2L1 knx k2 dxds 3 0 0 0 Z Z Z t 2 t 1 4  C1 þ jnx j dxds þ C1 kux k2 knx k2 dxds 3 0 0 0 Z Z Z t 2 t 1  C1 þ jnx j4 dxds þ C1 kux k2 dxds 3 0 0 0 Z Z 2 t 1  C1 þ jnx j4 dxds: 3 0 0 Z tZ

1

2

2

2

Therefore (4.2.4) follows from (4.2.5). This completes the proof.

ð4:2:5Þ h

Lemma 4.2.3. If assumptions in theorem 4.1.1 are valid, then the following estimates hold in Lagrangian coordinates, for any t > 0, ! #  Z tZ 1 Z 1" n  2 jny j2 juy j2 1  y 2  dyds  C1 ; ðy; tÞdy þ þ n
u þ ðv  log v  1Þ þ v v v v y 0 0 0 ð4:2:6Þ

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 115

Z

1

0

jny j2 dy þ v

Z tZ 0

0

1

  1 ny  2 dyds  C1 : v v y

Proof. Using lemmas 4.2.1–4.2.2 and the total mass conservation we easily derive (4.2.6) and (4.2.7).

ð4:2:7Þ R1 0

vdy ¼

R1 0

v0 dy; h

Lemma 4.2.4. For any t ≥ 0, there exists one point y1 = y1(t) 2 [0, 1] such that the solution v(y, t) to problem (4.1.1)–(4.1.5) possesses the following expression: " # ! Z t jny j2 1 1 1 v þ 2 vD ðy; sÞZ ðsÞds ð4:2:8Þ vðy; tÞ ¼ Dðy; tÞZ ðtÞ 1 þ v 0 where 8 Z y  Z Z Z y 1 1 y > > > udn  u0 ðnÞdn þ uðzÞdzdy ; Dðy; tÞ ¼ v0 ðyÞ exp > <  v0 0 0 y1 0 ! ! Z Z Z 1 > 1 t 1 jny j2 > 2 > 1þu þ v0 dy: dyds ; v0 ¼ > Z ðtÞ ¼ exp  :  v0 0 0 v 0

ð4:2:9Þ ð4:2:10Þ

h

Proof. See, e.g., lemma 2.2.2 or lemma 3.2.3. Lemma 4.2.5. There holds 0\C11  vðy; tÞ  C1 ; Z 0

t

for all ðy; tÞ 2 ½0; 1
½0; þ 1Þ;

  ny ðsÞ2 1 ds  C1 ; L

for all t [ 0:

ð4:2:11Þ ð4:2:12Þ

Proof. Obviously, using the Young inequality, the Hölder inequality and lemma 4.2.1, we have Z y  Z Z Z y   1 1 y  udn  u0 ðnÞdn þ u0 ðzÞdzdy   v0 0 0 y1 0 Z 1 12 Z 1 12 12 Z Z 1 1 1 2 2 2  u dy þ u0 dy þ u0 dy dy  v0 0 0 0 0  C 1 ku k2 þ C 1  C 1 :

ð4:2:13Þ

Equations (4.2.9) and (4.2.6) imply that there exists some positive constant C1 > 0 such that for all (y, t) 2 [0, 1] × [0, +∞),

1D Radiative Fluid and Liquid Crystal Equations

116

0\C11  Dðy; tÞ  C1 : Noting that for all t > 0, Z 1

! jny j2 1þu þ ðy; tÞdy  C1 ; v

1

0

then for 0 ≤ s ≤ t, we get t s

Z tZ s

ð4:2:14Þ

2

0

1

! 2 jn j y 1 þ u2 þ dy  C1 ðt  sÞ: v

ð4:2:15Þ

Therefore, (4.2.10) and (4.2.15) imply that for any 0 ≤ s ≤ t, ! ! Z Z 1 t 1 jny j2 1 C1 ðtsÞ 1 2  Z ðtÞZ ðsÞ ¼ exp  1þu þ e dyds  eC1 ðtsÞ :  v0 s 0 v ð4:2:16Þ Using (4.2.8) and (4.2.16), we derive that there exists a large time t0 such that as t ≥ t0, y 2 ½0; 1, " # ! Z t 2 jn j y v 1 þ 2 vD 1 ðy; sÞZ 1 ðsÞds vðy; tÞ ¼ Dðy; tÞZ ðtÞ 1 þ v 0 " # ! Z t jny j2 C1 ðtsÞ 1 C1 t  C1 e þ 1þ ds e v 0 Z t  C11 eC1 ðtsÞ ds 0

 ð2C1 Þ1 :

ð4:2:17Þ

Noting that Dðy; tÞ  C11 ; Z ðtÞ  expðC1 tÞ, ðy; tÞ 2 ½0; 1
½0; t0 ,

we

infer

that

for

any

vðy; tÞ  Dðy; tÞZ ðtÞ  C11 expðC1 tÞ  C11 expðC1 t0 Þ; which, together with (4.2.17), implies that for any (y, t) 2 [0, 1] × [0, +∞), vðy; tÞ  C11 :

ð4:2:18Þ

Employing W1,1 ,!L∞, by the Hölder inequality, lemmas 4.2.3–4.2.4 and (4.2.18), we deduce

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 117 "

# ! jny j2 C1 ðtsÞ vðy; tÞ  C1 e þ 1þ e v 0  Z t Z t jny j2 C1 ðtsÞ C1 ðtsÞ    C1 þ C1 e ds þ C1 ds  v  1 ve 0 0 L !12 12 Z 1 Z t Z 1 1 jny j2 dy dy  C1 þ C1 v 0 0 v 0 12 2 Z 1   2 !12 Z 1 1  ny  þ vdy veC1 ðtsÞ ds  v y  dy v 0 0 ! Z 1 Z t Z 1 1 jny j2 dy dy  C1 þ C1 v 0 0 v 0  Z 1   2 !Z 1 1  ny  þ dy vdy veC1 ðtsÞ ds  v y 0 v 0 Z t" Z 1   2 # 1  ny  eC1 ðtsÞ þ  C1 þ C1  v y  dy vds 0 0 v Z

C1 t

t

ð4:2:19Þ

which, using the Gronwall inequality and (4.2.7), (4.2.18), gives (4.2.11). Using (4.2.7) and the Poincaré inequality, we obtain  Z t Z t jny j2     ny ðsÞ2 1 ds  C1  v  1 ds L 0 0 L  Z t jn j 2   y  C1  ds   v y 0  C1 :

ð4:2:20Þ h

This proves the lemma.

Lemma 4.2.6. If assumptions in theorem 4.1.1 are valid, then the following estimates hold for any t > 0, Z tZ 1 Z 1 2 vy dy þ ðjny j4 þ jnyy j2 þ jnt j2 þ vy2 Þdyds  C1 ; ð4:2:21Þ 0

0

0

knt ðy; 0Þk  C1 :

ð4:2:22Þ

Proof. Using (4.1.1), we may rewrite (4.1.2) as 

u

vy  vy 2jny j2 vy 2ny  nyy ¼ 2þ  : v t v v3 v2

ð4:2:23Þ

1D Radiative Fluid and Liquid Crystal Equations

118 v

Multiplying (4.2.23) by u  vy, integrating the resulting equation over [0, 1] × [0, t], and using the Young and the Poincaré inequalities, we deduce for any ε > 0, Z tZ 1  vy  2  u  þ ðvy2 þ jny j2 vy2 Þdyds   v 0 0 2  Z tZ 1    v 0y  þ C1  jvy uj þ jny  nyy uj þ jny  nyy vy j þ jny j2 jvy uj dyds  u 0   v0 0 0 Z tZ 1 Z t    uy ðsÞ2 ds vy2 þ jny j2 vy2 dyds þ C1  C1 þ e Z þ C1 0

0 0 t 

ny 2 1 L

Z

1

u dyds þ C1

Z tZ

2

0

0

0 1

jnyy j2 dyds:

ð4:2:24Þ

0

Taking ε small enough in (4.2.24), and using (4.2.6), (4.2.12) and (4.2.24), we can obtain Z tZ 1 Z tZ 1  vy   2 ðvy2 þ jny j2 vy2 Þdyds  C1 þ C1 jnyy j2 dyds: ð4:2:25Þ u   þ v 0 0 0 0 By (4.2.7), we have ! Z tZ 1 Z tZ 1 Z t Z 1   2 2 jnyy j2 jny j vy2 1 ny ny  nyy vy þ dydt þ 2 dyds dyds ¼ 3 5 v v v y v4 0 0 0 0 v 0 0 Z Z Z tZ 1 jny j2 vy2 1 t 1 jnyy j2 dyds þ C dyds:  C1 þ 1 2 0 0 v3 v5 0 0 Thus Z tZ 0

1

Z jnyy j2 dyds  C1 þ C1

0

0

t

 2 ny  1 L

Z 0

1

vy2 dyds

 2 which, with (4.2.25) and the fact ku k2  C1 uy   C1 , gives Z t Z Z tZ 1    2 2 2 2 vy ðtÞ2 þ   ðvy þ jny j vy Þdyds  C1 þ C1 ny L1 0

0

0

0

1

vy2 dyds:

ð4:2:26Þ

Applying the Gronwall inequality to (4.2.26), and using (4.2.7) and (4.2.12), we obtain Z tZ 1   vy ðtÞ2 þ ðjny j2 vy2 þ jnyy j2 þ jvy j2 Þdyds  C1 : ð4:2:27Þ 0

0

Thus from (4.2.7) and (4.2.12) it follows that Z tZ 1 Z t Z  2 4   jny j dyds  C1 ny L1 0

0

0

1 0

jny j2 dyds  C1 :

ð4:2:28Þ

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 119

By (4.1.3) and (4.2.27), we can derive   

  nyy ðtÞ  C1 ny ðtÞ þ knt ðtÞk ;

ð4:2:29Þ

  

 knt ðtÞk  C1 ny ðtÞ þ nyy ðtÞ

ð4:2:30Þ

or

which implies (4.2.22). Thus, using the Poincaré inequality and (4.2.27), we can infer Z t Z t  2  2  2 ny  þ nyy  ðsÞds knt ðsÞk ds  C1 0 0 Z t   nyy ðsÞ2 ds  C1 ;  C1 0

which, with (4.2.27) and (4.2.28), yields (4.2.21). This proves the lemma.

h

Lemma 4.2.7. If assumptions in theorem 4.1.1 hold, then the following estimates are valid for any t > 0,  2  2 knt ðtÞk2 þ nyy ðtÞ þ uy ðtÞ Z t  2  2  þ kut k2 þ uyy  þ nty  ðsÞds  C1 : ð4:2:31Þ 0

Proof. Obviously, using lemma 3.2.6 in chapter 3 and the uniform estimate of v, we may complete the proof. h In what follows, we shall derive some uniform estimates in H 2
H02
H 3 . The proof of the following is different from lemma 3.3.1 in chapter 3 to some extent. Lemma 4.2.8. If assumptions in theorems 4.1.2 hold, the following estimates are valid for any t > 0, Z t  2   uty ðsÞ2 ds  C1 ; ð4:2:32Þ kut ðtÞk2 þ uyy ðtÞ þ 0

  vyy ðtÞ2 þ

Z t  2     vyy  þ nyyy 2 þ uyyy 2 ðsÞds  C1 :

ð4:2:33Þ

0

Proof. Differentiating (4.1.2) with respect to t, multiplying the resulting equation by ut, integrating it by parts, and using (3.2.32) and lemmas 4.2.3–4.2.7, we deduce for any ε > 0,

1D Radiative Fluid and Liquid Crystal Equations

120 Z

t

2

kut ðtÞk þ

  uty ðsÞ2 ds

0

Z tZ 1h i  kut ðy; 0Þk2 þ C1 juy j þ juy j2 þ jny  nty j þ jny j2 juy j juty jdydt 0 0 Z t Z t  2  2  2  2 uy  þ uy  1 uy     C2 þ e uty ðsÞ ds þ C1 L 0 0   4  2  2  2 þ ny L1 uy  þ ny L1 nty  ðsÞds Z  C2 þ e

t

  uty ðsÞ2 ds þ C1

t

  uty ðsÞ2 ds

0

Z  C2 þ e

Z th  2  2  2 i uy  þ uyy  þ nty  ðsÞds 0

0

which, by taking ε 2 (0, 1) small enough and using (4.2.31), yields (4.2.32). Differentiating (4.1.2) with respect to y, using (4.1.1) (vtyy = uyyy), we arrive at v  vyy yy þ 2 ¼ uty þ Eðy; tÞ ð4:2:34Þ v t v where Eðy; tÞ ¼

2vy2 2vy uyy 2vy2 uy 2jnyy j2 þ 2ny  nyyy þ  þ v3 v2 v3 v2 2 2 2 6jny j vy 8ny  nyy vy þ 2jny j vyy  þ : 3 v v4 v

Multiplying (4.2.34) by vyy, integrating the result over [0, 1] × [0, t] and using the Young inequality, we have for any ε > 0, Z th Z t Z t i        2 vyy ðtÞ2 þ vyy ðsÞ2 ds  C1 þ e vyy ðsÞ2 ds þ C1 uty  þ kE k2 ðsÞds 0

0

0

ð4:2:35Þ where Z

t 0

2

kEðsÞk ds  C1

Z tZ 0

1 0



2 vy4 þ vy2 uyy þ vy4 uy2 þ jnyy j4 þ jny j2 jnyyy j2



2 þ jny j2 jnyy j2 vy2 þ jny j4 vyy þ jny j4 vy4 Þdyds Z t  2  2  2  2  2  4 vy  1 vy  þ vy  1 uyy  þ uy  1 vy  4  C1 L L L L 0

 2  2  2  2  2  2  2 þ nyy L1 nyy  þ ny L1 nyyy  þ ny L1 vy L1 nyy   4  2  2  4  þ ny L1 vyy  þ ny L1 vy L4 ds:

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 121

Using the interpolation inequality, the Young inequality and lemmas 4.2.3–4.2.7, we conclude Z t Z t  2         vy  1 þ vy 4 4 þ ny 2 1 vyy 2 þ nyyy 2 ds kEðsÞk2 ds  C1 L L L 0 0 Z t      2   3    4 ðvy vyy  þ vy  þ vy  vyy  þ vy   C1 0

 2  2  2 þ ny L1 vyy  þ nyyy  Þds Z t Z t  2  2  2   vyy ðsÞ ds þ C1 ðny ðsÞL1 vyy ðsÞ  C1 þ e 0

0

 2 þ nyyy ðsÞ Þds:

ð4:2:36Þ

Differentiating (4.1.3) with respect to y, using the interpolation inequality, the Young inequality and lemmas 4.2.3–4.2.7, we can conclude Z t Z t  2     2   2  nyyy ðsÞ2 ds  C1 nty  þ vy nyy 2 þ  ny vy2  þ ny vyy  0 0     2   2 2  3 2   þ ny  nyy þ ny vy  þ ny  ds Z t  2   2   2   2   2 nyy  1 vy  þ ny  1 vy  1 vy   C1 þ C1 L L L 0

 2  2  2  2  4  2 þ ny L1 vyy  þ ny L1 nyy  þ ny L1 vy   4  2 þ ny L1 ny  ds Z

t

 C1 þ e

  nyyy ðsÞ2 ds þ C1

0

Z 0

t

    ny ðsÞ2 1 vyy ðsÞ2 ds: L

Picking ε 2 (0, 1) small enough, we obtain Z t Z t       nyyy ðsÞ2 ds  C1 þ C1 ny ðsÞ2 1 vyy ðsÞ2 ds: L 0

ð4:2:37Þ

0

Inserting (4.2.36) and (4.2.37) into (4.2.35), and taking ε small enough, we can obtain Z t Z t         vyy ðtÞ2 þ vyy ðsÞ2 ds  C1 þ C1 ny ðsÞ2 1 vyy ðsÞ2 ds: L 0

0

Performing the Gronwall inequality, and using (4.2.12), we conclude that for all t > 0, Z t    2    vyy ðtÞ2 þ vyy  þ nyyy 2 ðsÞds  C1 : ð4:2:38Þ 0

1D Radiative Fluid and Liquid Crystal Equations

122

Differentiating (4.1.2) with respect to y, by the imbedding theorem and lemmas 4.2.3–4.2.7, we can obtain          uty ðtÞ  C1 vy ðtÞ 1 þ uy ðtÞ 2 þ ny ðtÞ 2 ð4:2:39Þ H H H or

    uyyy ðtÞ  C1 vy ðtÞ

H1

      þ uy ðtÞH 1 þ ny ðtÞH 2 þ uty 

ð4:2:40Þ

which, with (4.2.31), (4.2.32) and (4.2.38), gives (4.2.33). This proves the lemma. h Using the same estimate as in chapter 3, we can obtain the following two lemmas concerning uniform estimates in H i
H0i
H i þ 1 ði ¼ 1; 2Þ and H 4
H04
H 4 using uniform estimates of specific volume v in lemma 4.2.5. Lemma 4.2.9. If assumptions in theorem 4.1.2 are valid, then the following estimate holds for any t > 0, Z t       nty ðtÞ2 þ nyyy ðtÞ2 þ ntyy ðsÞ2  C2 : ð4:2:41Þ 0

Lemma 4.2.10. If assumptions in theorem 4.1.3 are valid, then the following estimates hold for any t > 0, Z t  2  2   2 2   utty  þ utyy 2 kutt ðtÞk þ kntt ðtÞk þ ntyy ðtÞ þ 0  2  2  2     þ kntt k þ ntty þ ntyyy ðsÞds  C4 ; ð4:2:42Þ       uty ðtÞ2 þ uyyy ðtÞ2 þ utyy ðtÞ2 þ

Z

  utyyy ðsÞ2 ds  C4 ;

t

ð4:2:43Þ

0

      vyyy ðtÞ2 þ nyyyy ðtÞ2 þ uyyyy ðtÞ2 þ

Z

  vyyy ðsÞ2 ds  C4 ;

ð4:2:44Þ

Z t       vyyyy 2 þ nyyyyy 2 þ uyyyyy 2 ðsÞds  C4 :

ð4:2:45Þ

t

0

  vyyyy ðtÞ2 þ

0

4.3

Large-Time Behavior in H i
H0i
H i þ 1 ði ¼ 1; 2Þ and H 4
H04
H 4

In this section, we shall complete the proofs of theorems 4.1.1–4.1.3.

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 123 Lemma 4.3.1. If assumptions in theorem 4.1.1 are valid, then we have lim kvðtÞ   v kH 1 ¼ 0

ð4:3:1Þ

t! þ 1

where  v¼

R1 0

vðy; tÞdy ¼

R1 0

v0 dy:

Proof. Differentiating (4.1.1) with respect to y, multiplying the result by vy, then integrating the result over [0, 1], using the Young inequality, we can deduce          d vy ðtÞ2  vy ðtÞ2 þ uyy ðtÞ2  1 þ 1 vy ðtÞ4 þ uyy ðtÞ2 dt 2 2 which, along with (4.2.21), (4.2.31) and lemma 1.1.2, leads to  2 lim vy ðtÞ ¼ 0: t! þ 1

ð4:3:2Þ

Moreover, using the embedding theorem, we can deduce   v k  C1 vy  kv   which, together with (4.3.2), gives (4.3.1). This proves the lemma.

h

Lemma 4.3.2. If assumptions in theorem 4.1.1 are valid, then we have lim kuðtÞkH 1 ¼ 0:

ð4:3:3Þ

t! þ 1

Proof. Equation (4.1.1) can be rewritten as ut ¼

jny j2 v 1  2 v

! þ

u 

y

y

v

y

:

ð4:3:4Þ

Let b p¼b p ðy; tÞ ¼ Then

1 jny j2 þ 2 ; v v

1 jny j2 uy b r¼b r ðy; tÞ ¼   2 þ : v v v

 uy  ut ¼ b pþ ¼b ry : v y

ð4:3:5Þ

Put b p  ðy; tÞ ¼ b p ðy; tÞ  p ¼ b

Z

1

b p ðy; tÞdy;

ð4:3:6Þ

b r ðy; tÞdy:

ð4:3:7Þ

0

b  ðy; tÞ ¼ b b r ðy; tÞ  r ¼ r

Z 0

1

1D Radiative Fluid and Liquid Crystal Equations

124

Then Z

1

b p  ðy; tÞdy ¼ 0;

0

Z

1

b r  ðy; tÞdy ¼ 0;

ð4:3:8Þ

0

ðb p  Þy ¼ b py ;

ðb r  Þy ¼ b ry :

ð4:3:9Þ

Integrating (4.3.9) by parts, using (4.3.5), lemma 4.2.3 and the Young inequality, we can infer that for any ε > 0,  Z y  2     b p k ¼ ðb p ;b p Þ ¼ b py ; p dx kb  0 Z y u  Z y y   b b ¼ b py ; ; p dx ¼ ut  p dx v y 0  Z y0   Z y uy b b ¼ ut ; p  dx  p  dx ; v y 0 0 Z 1 12 Z 1 Z y 2 !12 Z 1 uy  2  b b  p dy ut dy p dx dy þ 0 0 0 0 v   2  p  k2 þ C ðeÞuy   ek b p  k2 þ C ðeÞkut k2 þ C1 ekb   2  ð4:3:10Þ  ðC1 þ 1Þek b p  k2 þ C1 kut k2 þ uy  : Choosing ε > 0 so small that (C1 + 1)ε < 1, integrating the result over (0, t), we obtain Z t ð4:3:11Þ p  ðsÞk2 ds  C1 : kb 0

Note that

 Z y d  b p ðtÞk2 ¼ 2ðb p; b p t Þ ¼ 2 b p y ;  p t dy kb dt 0   Z y uy b ¼2 ut ;  p t dy v y  Z y 0   Z y uy b b ¼ 2 ut ; ; p t dy  2 p t dy v y 0 0   2   2  2  C1 kut ðtÞk þ uy ðtÞ þ  b p ðtÞ : t

ð4:3:12Þ

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 125

From (4.2.21) and (4.2.30), it follows that  2   2  2   2  2 b    p t ðtÞ  C1 kvt ðtÞk þ ny ðtÞ  nty ðtÞ þ ny ðtÞ  vt ðtÞ  2  2  2  4  2   C1 uy ðtÞ þ ny ðtÞL1 nty ðtÞ þ ny ðtÞL1 uy ðtÞ  2  2  2  4  2   C1 uy ðtÞ þ nyy ðtÞ nty ðtÞ þ nyy ðtÞ uy ðtÞ  2  2   C1 uy ðtÞ þ nty ðtÞ : Combining (4.3.12) and (4.3.13), we get   2  2  d  p ðtÞk2  C1 kut ðtÞk2 þ uy ðtÞ þ nty ðtÞ : kb dt

ð4:3:13Þ

ð4:3:14Þ

Using lemmas 4.2.7 and 1.1.2, we can derive p  ðtÞk2 ¼ 0: lim kb

t! þ 1

Hence from (4.3.4), (4.3.5), (4.3.7) and (4.3.11), it follows Z t Z t  2  r  ðsÞk2 ds  C1 p  k þ uy  ðsÞds  C1 : kb kb 0

ð4:3:15Þ

ð4:3:16Þ

0

By (4.3.4), (4.3.5), and (4.3.7), and integrating by parts, we derive  Z y d b r; b r t Þ ¼ 2 b r y ;  r t dx r  ðtÞk2 ¼ 2ðb kb dt  0 Z y  b ¼ 2 ut ;  r t dx 0 Z y 2 !   2   b  C1 kut ðtÞk þ  r t dx   0

ð4:3:17Þ

where Z 0

and

y

b r t dy

¼

Z y

Z

1

b rt 

0

b r t ðn; tÞdn dx

0

! 1 jny j2 uy b r t ðy; tÞ ¼   2 þ v v v t ! 2   ut vy  uy2 vt jny j ut ¼ 2 þ þ : v v2 v y v2 t

ð4:3:18Þ

1D Radiative Fluid and Liquid Crystal Equations

126

Noting that for all t > 0, ut ð0; tÞ ¼ ut ð1; tÞ; we derive that for any y 2 [0, 1], Z y  Z 1   2 2   b  C r ðx; tÞdx ju þ n n þ n u þ u v þ u jdy þ ju j t 1 y y ty t y t y y y   0 h 0             C1 ðuy  þ ny   nty  þ ny2   uy  i    2 þ kut k  vy  þ uy  Þ þ jut j

      C1 uy  þ nyt  þ kut k þ jut j : Analogously,

Z   

1

y

 

     b r t ðx; tÞdx   C1 uy  þ nyt  þ kut k þ jut j :

Combining (4.3.18)–(4.3.20), we obtain Z y Z y Z b b r t dy ¼ rt  0

0

Z

y

1

1

Z

1

b b r t dx  r t dxdn 0 0 y

      C1 uy  þ nyt  þ kut k þ jut j ¼

ð4:3:20Þ

b r t ðn; tÞdn dx

0

Z

ð4:3:19Þ

ð4:3:21Þ

which gives Z   

0

y

2 

b r t dx  

  2  2  C1 uy ðtÞ þ nyt ðtÞ þ kut ðtÞk2 :

ð4:3:22Þ

By (4.3.17) and (4.3.22), we can infer   2  2 d r  ðtÞk2  C1 uy ðtÞ þ nyt ðtÞ þ kut ðtÞk2 kb dt    2  C1 1 þ nyt ðtÞ þ kut ðtÞk2 ; which, along with (4.3.16) and lemma 1.1.2, yields r  ðtÞk2 ¼ 0: lim k b

t! þ 1

Noting that Z 0

1

u  ðy; tÞdy ¼ 0; v y

ð4:3:23Þ

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 127

we arrive at uy uy  ¼ þ v v

Z 0

1

uy dy ¼ ðb r þ b pÞ þ v

Z 0

1

uvy dy v2

which, together with lemma 4.2.3, gives u     

 y uy ðtÞ  C1  p  ðtÞk þ kuðtÞkvy ðtÞ : r ðtÞk þ kb    C1 k b v Thus from (4.3.15), (4.3.23), (4.3.2) and (4.3.24) it follows that   lim uy ðtÞ ¼ 0: t! þ 1

ð4:3:24Þ

ð4:3:25Þ

By the Poincaré inequality, we get

  kuðtÞkH 1  C1 uy ðtÞ

which, together with (4.3.25), gives (4.3.3). This completes the proof.

h

Lemma 4.3.3. If assumptions in theorem 4.1.1 are valid, then we have  kH 2 ¼ 0 lim knðtÞ  n

t! þ 1

¼ where n

R1 0

ð4:3:26Þ

nðy; tÞdy:

Proof. Multiplying equation (4.1.3) by nt, integrating the result over [0, 1] with respect to y and using (4.1.6), we obtain Z 1 Z 1   nt ny jnt j2 dy ¼ dy v y 0 0 v Z 1  ny nty nt vy  ð4:3:27Þ ¼  2 dy: v v 0 v By the Young inequality, we deduce that for any ε > 0, Z Z 1 Z 1 Z 1 1d 1 jny j2 dy þ jnt j2 dy  e jnt j2 dy þ C1 jny j2 vy2 dy: 2 dt 0 0 0 0 Taking ε small enough and using (4.2.21), we have    d ny ðtÞ2 þ knt ðtÞk2  C1 ny ðtÞ2 1 L dt which, together with lemma 1.1.2 and (4.2.12), results in  2 lim ny ðtÞ ¼ 0: t! þ 1

By the Poincaré inequality, we derive

   kH 1  C1 ny ðtÞ knðtÞ  n

ð4:3:28Þ

ð4:3:29Þ

1D Radiative Fluid and Liquid Crystal Equations

128

which, together with (4.3.28), leads to  kH 1 ¼ 0: lim knðtÞ  n

t! þ 1

ð4:3:30Þ

By lemma 4.2.7, we can obtain  2  2  2 d knt ðtÞk2 þ nty ðtÞ  ny ðtÞL1 uy ðtÞ dt which, combined with (4.2.12), (4.2.31) and lemma 1.1.2, gives lim knt ðtÞk ¼ 0:

t! þ 1

ð4:3:31Þ

From (4.1.3) and (4.2.31), and using the Poincaré inequality, we can infer   2      nyy ðtÞ2  C1 knt ðtÞk2 þ ny ðtÞvy ðtÞ2 þ  ny2 ðtÞ   2  2  2  2   C1 knt ðtÞk2 þ ny ðtÞL1 vy ðtÞ þ ny ðtÞL1 ny ðtÞ   2  2  2  2   C1 knt ðtÞk2 þ nyy ðtÞ vy ðtÞ þ nyy ðtÞ ny ðtÞ   2  2   C1 knt ðtÞk2 þ vy ðtÞ þ ny ðtÞ ; which, along with (4.3.2), (4.3.30) and (4.3.31), gives  2 lim nyy ðtÞ ¼ 0: t! þ 1

Thus (4.3.25) follows from (4.3.32). This completes the proof.

ð4:3:32Þ h

Proof of Theorem 4.1.1. Combining lemmas 4.2.3–4.2.7 and lemmas 4.3.1–4.3.3, we complete the proof of theorem 4.1.1. h Lemma 4.3.4. If assumptions in theorem 4.1.2 are valid, then there holds v kH 2 ¼ 0 lim kvðtÞ  

t! þ 1

where  v¼

R1 0

vðy; tÞdy ¼

R1 0

ð4:3:33Þ

v0 ðyÞdy:

Proof. Differentiating (4.1.1) with respect to twice y, multiplying the resulting equation by vyy, then integrating it over [0, 1], by the Young inequality, we can deduce Z 1  1d vyy ðtÞ2 ¼ uyyy vyy dy 2 dt 0 2 1  2 1 1  4 1  2 1  uyyy ðtÞ þ vyy ðtÞ  þ vyy ðtÞ þ uyyy ðtÞ 2 2 4 4 2 which, together with lemmas 4.2.8 and 1.1.2, implies  2 lim vyy ðtÞ ¼ 0: t! þ 1

ð4:3:34Þ

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 129

By the embedding theorem, we have lim kvðtÞ   v kH 2 ¼ 0:

t! þ 1

h

This proves the proof.

Lemma 4.3.5. If assumptions in theorem 4.1.2 are valid, then the following estimate holds lim kuðtÞkH 2 ¼ 0:

t! þ 1

Proof. By (4.3.5), we have 1 jny j2 þ 2 v v

b py ¼ ¼

! y

vy 2ny  nyy 2jny j2 vy þ  2 v v2 v3

¼ ut þ

u  y

v

ð4:3:35Þ

y

:

ð4:3:36Þ

ð4:3:37Þ

Hence, by lemmas 4.2.3–4.2.9, and using the interpolation inequality, we obtain         b p y ðtÞ  C1 vy ðtÞ þ ny ðtÞ  nyy ðtÞ þ jny ðtÞj2 vy ðtÞ        2    C1 vy ðtÞ þ nyy ðtÞL1 ny ðtÞ þ ny ðtÞL1 vy ðtÞ h   1  1     2  i  C1 vy ðtÞ þ nyy ðtÞ2 nyyy ðtÞ2 þ nyy ðtÞ ny ðtÞ þ nyy ðtÞ vy ðtÞ   

  C1 vy ðtÞ þ ny ðtÞ which gives

  lim b p y ðtÞ ¼ 0:

t! þ 1

ð4:3:38Þ

Differentiating (4.3.37) with respect to t, multiplying the result by ut, then integrating it over [0, 1], employing an integration by parts, and using the Young inequality, we can obtain for any ε > 0, Z 1 Z 1 2 Z 1 2 uty uy uty 1d 2 b dy ¼ p t uty dy þ dy kut ðtÞk þ 2 dt v v2 0 0 0   2  4   euty ðtÞ þ C1 kb p t ðtÞk2 þ uy ðtÞL4 : ð4:3:39Þ Choosing ε > 0 small enough, using (4.2.21), (4.2.31), (4.2.32) and the interpolation inequality, we can conclude

1D Radiative Fluid and Liquid Crystal Equations

130

 2 d kut ðtÞk2 þ uty ðtÞ dt  2  2  2  C1 ðuy ðtÞ þ ny ðtÞL1 nty ðtÞ  2  2  3    4 þ ny ðtÞL1 vy ðtÞ þ uy ðtÞ uyy ðtÞ þ uy ðtÞ Þ  2  2  2   C1 uy ðtÞ þ nty ðtÞ þ ny ðtÞL1 :

ð4:3:40Þ

Hence we infer from (4.3.40), (4.2.6), (4.2.12), (4.2.31) and lemma 1.1.2 lim kut ðtÞk2 ¼ 0:

t! þ 1

ð4:3:41Þ

By (4.3.37), (4.1.8) and the interpolation inequality, we derive     

 uyy ðtÞ  C1  b p y ðtÞ þ kut ðtÞk þ uy ðtÞvy ðtÞ    1  1     C1 ð b p y ðtÞ þ kut ðtÞk þ vy ðtÞ2 vyy ðtÞ2 þ vy ðtÞ uy ðtÞÞ   

 p y ðtÞ þ kut ðtÞk þ uy ðtÞ  C1  b which, along with (4.3.25), (4.3.37), (4.3.40), gives  2 lim uyy ðtÞ ¼ 0: t! þ 1

ð4:3:42Þ

Thus (4.3.35) follows from (4.3.42) and (4.1.9). This completes the proof.

h

Lemma 4.3.6. If assumptions in theorem 4.1.2 are valid, then there holds that  kH 3 ¼ 0 lim knðtÞ  n

t! þ 1

¼ where n

R1 0

ð4:3:43Þ

nðy; tÞdy:

Proof. By lemma 3.3.2, we can derive      d nty ðtÞ2  C1 nty ðtÞ2 þ nyy ðtÞ2 dt which, along with (4.2.21), (4.2.31) and lemma 1.1.2, leads to  2 lim nty ðtÞ ¼ 0: t! þ 1

Hence it follows from (4.2.41), (4.3.32), (4.3.34) and (4.3.44) that  2 lim nyyy ðtÞ ¼ 0; t! þ 1

ð4:3:44Þ

ð4:3:45Þ

which, using the embedding theorem and (4.3.26), gives (4.3.43). This proves the lemma. h

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 131 Proof of Theorem 4.1.2. Combining lemmas 4.2.8–4.2.9 and lemmas 4.3.4–4.3.6, we can complete the proof of theorem 4.1.2. h Lemma 4.3.7. If assumptions in theorem 4.1.3 are valid, then there holds that lim kvðtÞ   v kH 4 ¼ 0

t! þ 1

where  v¼

R1 0

vðy; tÞdy ¼

R1 0

ð4:3:46Þ

v0 ðyÞdy:

Proof. Differentiating (4.1.1) with respect to y three times, multiplying the resulting equation by vyyy, integrating it over [0, 1], and then using the Young inequality, we can derive      d vyyy ðtÞ2  uyyyy ðtÞ2 þ vyyy ðtÞ2 dt 4  2 1 1  þ vyyy ðtÞ þ uyyyy ðtÞ 2 2 which, together with lemma 4.2.10 and lemma 1.1.2, yields  2 lim vyyy ðtÞ ¼ 0: t! þ 1

ð4:3:47Þ

Differentiating (4.1.1) with respect to y four times, multiplying the resulting equation by vyyyy, integrating it over [0, 1], and then using the Young inequality, we can derive      d vyyyy ðtÞ2  uyyyyy ðtÞ2 þ vyyyy ðtÞ2 dt 4  2 1 1  þ vyyyy ðtÞ þ uyyyyy ðtÞ 2 2 which, along with lemma 4.2.10 and lemma 1.1.2, leads to  2 lim vyyyy ðtÞ ¼ 0: t! þ 1

ð4:3:48Þ

Therefore, using the embedding theorem, we conclude from (4.3.33), (4.3.47) and (4.3.48), lim kvðtÞ   v kH 4 ¼ 0:

t! þ 1

h

This completes the proof. Lemma 4.3.8. If assumptions in theorem 4.1.3 are valid, then there holds that lim kuðtÞkH 4 ¼ 0:

t! þ 1

ð4:3:49Þ

132

1D Radiative Fluid and Liquid Crystal Equations

Proof. Clearly, using lemma 4.2.10 and the Young inequality, we can derive          d uty ðtÞ2  C2 ðuty ðtÞ2 þ uty ðtÞ2 2 þ uy ðtÞ2 2 þ ny ðtÞ2 1 H H H dt  2  2 þ nty ðtÞH 1 þ vy ðtÞH 1 Þ  4  2  2  2  C2 þ C2 uty ðtÞ þ C2 ðuty ðtÞH 2 þ uy ðtÞH 2 þ ny ðtÞH 1  2  2 þ nty ðtÞH 1 þ vy ðtÞH 1 Þ which, along with lemmas 4.2.3–4.2.10 and lemma 1.1.2, leads to  2 lim uty ðtÞ ¼ 0: t! þ 1

ð4:3:50Þ

Differentiating (4.3.36) with respect to y, using the interpolation inequality and lemmas 4.2.3–4.2.10, we infer             b p yy ðtÞ  C1 ðvyy ðtÞ þ vy ðtÞL1 vy ðtÞ þ nyy ðtÞL1 nyy ðtÞ           þ ny ðtÞL1 nyyy ðtÞ þ ny ðtÞL1 vy ðtÞL1 nyy ðtÞ  2    2     þ ny ðtÞL1 vyy ðtÞ þ vy ðtÞL1 ny ðtÞL1 ny ðtÞÞ         

  C1 vy ðtÞ þ vyy ðtÞ þ ny ðtÞ þ nyy ðtÞ þ nyyy ðtÞ which, combined with (4.3.2), (4.3.29), (4.3.34), (4.3.32) and (4.3.45), implies   lim b ð4:3:51Þ p yy ðtÞ ¼ 0: t! þ 1

Differentiating (4.3.37) with respect to y, using the interpolation inequality and lemmas 4.2.3–4.2.9, we derive               uyyy ðtÞ  C1 uty ðtÞ þ  b p yy ðtÞ þ uyy ðtÞvy ðtÞ þ uy ðtÞvyy ðtÞ þ uy ðtÞvy2 ðtÞ           C1 uty ðtÞ þ  b p yy ðtÞ þ uy ðtÞH 1 þ vy ðtÞH 1 which, together with (4.3.50), (4.3.51), (4.3.25), (4.3.34), (4.3.2) and (4.3.42), leads to   lim uyyy ðtÞ ¼ 0: ð4:3:52Þ t! þ 1

Differentiating (4.3.36) with respect to y twice, using the interpolation inequality and lemmas 4.2.3–4.2.10, leads to           

 b p yyy ðtÞ  C4 vyyy ðtÞ þ vyy ðtÞ þ nyyy ðtÞ þ vy ðtÞ þ ny ðtÞ which, along with (4.3.2), (4.3.29), (4.3.34), (4.3.45) and (4.3.47), results in   lim b ð4:3:53Þ p yyy ðtÞ ¼ 0: t! þ 1

Large-Time Behavior of Solutions to a One-Dimensional Liquid Crystal System 133 Differentiating (4.3.36) with respect to t and y, using the Poincaré and the Gagliardo–Nirenberg interpolation inequalities, and using lemmas 4.2.3–4.2.10, we can obtain   2 2  2  2  2  b p yyt ðtÞ  C2 uy ðtÞH 2 þ nty ðtÞH 2 þ ny ðtÞH 2 þ vy ðtÞH 2 : ð4:3:54Þ Differentiating (4.3.37) with respect to t once and y twice, multiplying the resulting equation by utyy, employing an integration by parts, and using the Young inequality, we can conclude that for any ε > 0, Z 1 2    2  2 2 utyyy 1d utyy ðtÞ2 þ dy  eutyyy ðtÞ þ C1 ðeÞb p yyt ðtÞ þ C1 utyy vy  2 dt v 0 2    2   2 4   þ uty vyy  þ uty vy2  þ uyy L4 þ uy uyyy  2   2  4    þ uy uyy vy  þ uy2 vyy  þ uy vy L4 : Choosing ε 2 (0, 1) small enough, using the interpolation inequality and lemmas 4.2.3–4.2.9, we deduce     2  2 d utyy ðtÞ2 þ utyyy ðtÞ2  C1 b p yyt ðtÞ þ C2 ðuty ðtÞH 2 dt  2  2 þ uy ðtÞ 2 þ vy ðtÞ 2 Þ: H

H

ð4:3:55Þ

Inserting (4.3.54) into (4.3.55), using lemmas 4.2.3–4.2.10 and lemma 1.1.2, we can derive  2 lim utyy ðtÞ ¼ 0: ð4:3:56Þ t! þ 1

Differentiating (4.3.37) with respect to y twice, using the Gagliardo–Nirenberg interpolation inequality and lemmas 4.2.3–4.2.10, we obtain               uyyyy ðtÞ  C1  b p yyy ðtÞ þ utyy ðtÞ þ uyyy vy  þ uyy vy  þ uyy vy2            þ uyy vyy  þ uy vyyy  þ uy vyy vy  þ uy vy3               C1  b p yyy ðtÞ þ utyy ðtÞ þ uy ðtÞ þ uyy  þ vy ðtÞ þ vyyy ðtÞ   þ C4 vyy ðtÞ which, together with (4.3.2), (4.3.25), (4.3.34), (4.3.42), (4.3.47), (4.3.53) and (4.3.56), gives   lim uyyyy ðtÞ ¼ 0: ð4:3:57Þ t! þ 1

Therefore, it follows from (4.3.33), (4.3.52) and (4.3.57) that lim kuðtÞkH 4 ¼ 0:

t! þ 1

Thus the proof follows immediately.

h

1D Radiative Fluid and Liquid Crystal Equations

134

Lemma 4.3.9. If assumptions in theorem 4.1.3 are valid, then the following estimate holds  kH 4 ¼ 0 lim knðtÞ  n

t! þ 1

¼ where n

R1 0

ð4:3:58Þ

nðy; tÞdy:

Proof. Exploiting lemma 4.2.10 and the Young inequality, we easily infer that   2  2  2  2  d kntt ðtÞk2 þ ntty ðtÞ  C2 kntt ðtÞk2 þ nty ðtÞH 1 þ uy ðtÞH 1 þ uty ðtÞ dt  2  C2 þ C2 kntt ðtÞk4 þ C2 ðnty ðtÞH 1  2  2 þ uy ðtÞH 1 þ uty ðtÞ Þ which, along with (4.2.33), (4.2.31), (4.2.41) and lemma 1.1.2, implies lim kntt ðtÞk2 ¼ 0:

t! þ 1

ð4:3:59Þ

Then (4.3.55) follows from lemma 4.2.10, (4.3.31), (4.3.43), (4.3.47), (4.3.59) and lemmas 4.3.1–4.3.6 immediately. Thus the proof follows readily. h Proof of Theorem 4.1.3. Using lemma 4.2.10 and lemmas 4.3.7–4.3.9, we can finally complete the proof of theorem 4.1.3. h

4.4

Bibliographic Comments

In addition to the comments in section 3.5, we would like to mention here that the global existence and regularity of solutions to (3.1.1)–(3.1.5) or (3.1.6)–(3.1.10) have been established for the pressure P ¼ qc ¼ v1c ðc  1Þ, while the large-time behavior of global solutions to (3.1.6)–(3.1.10) (i.e., (4.1.1)–(4.1.5)) has only been proved for γ = 1 in P and it is still open for γ > 1. For the large-time behavior of solutions for incompressible liquid crystal system, we would like to refer to Wu [144] and the references therein.

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Index

A Angular Variable, 18 Asymptotic Behavior, 17, 81 Asymptotic Behavior of Solutions in H1 , 48 Asymptotic Behavior of Solutions in H2 , 60 Asymptotic Behavior of Solutions in H4 , 81 C Cauchy–Schwarz Inequality, 14 Compactness Theorem, 5 Compressible Navier–Stokes Equations, 89 Conservation of Linear Momentum, 89 Conservation of Mass, 89

I Infrarelativistic Model, 17 Initial Boundary Value Problem, 89 L Lagrangian Coordinates, 18 Large-time Behavior, 111 Liquid Crystal Flow, 89 Liquid Crystal System, 111 Local Thermodynamics Equilibrium, 86 M Minkowski Inequality, 15

D Density Function, 89

O Optical Director of the Molecules, 89

E Embedding and Compactness Theorem, 5 Embedding Theorem, 5 Ericksen–Leslie System, 89

P Planck Function, 87

G Gagliardo–Nirenberg interpolation inequalities, 6 Global Existence, 53, 89 Global Existence and Uniform-in-time Estimates, 22, 62 H Harmonic Map Equation, 89 Heat Conductivity, 18 Heat Flow Equation for Harmonic Maps, 89 Heat Flux, 18 Hölder Inequality, 13

R Radiation Frequency, 18 Radiative Energy, 18 Radiative Energy Source, 18, 86 Radiative Flux, 18, 87 Radiative Intensity, 18 Regularity, 89 S Schwarz Inequality, 14 Shen–Zheng Inequality, 10 Sobolev Inequality, 2 Sobolev Space, 1 Source Term, 18 Specific Volume, 18 Stress, 18

Index

144

T The Classical Bellman–Gronwall Inequality, 7 The Generalized Bellman–Gronwall Inequality, 8 The Minkowski Inequalities, 14 The Poincaré Inequality, 6 The Uniform Bellman–Gronwall Inequality, 9 The Young Inequalities, 12

Thermodynamical Relation, 19 Transporting Relation, 89 U Uniform Estimates, 113 Uniform-in-time Estimates, 19, 53 V Viscosity Coefficient, 20