Attractors and Methods: Volume 2: Attractors and Methods 9783110586992

Table of contents :
Preface
Contents
1. Attractor and its dimension estimation
2. Inertial manifold
3. The approximate inertial manifold
Bibliography
Index

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Boling Guo, Liming Ling, Yansheng Ma, and Hui Yang Infinite-Dimensional Dynamical Systems

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Boling Guo, Liming Ling, Yansheng Ma, and Hui Yang

Infinite-Dimensional Dynamical Systems |

Volume 1: Attractors and Inertial Manifolds

Mathematics Subject Classification 2010 76D03, 35Q35, 35E15, 35A01, 35A02 Authors Prof. Boling Guo Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics 6 Huayuan Road Haidian District 100088 Beijing People’s Republic of China [email protected] Prof. Liming Ling South China University of Technology School of Mathematics Wushan RD., Tianhe District 381 510640 Guangzhou People’s Republic of China [email protected]

Dr Yansheng Ma Northeast Normal University School of Mathematics and Statistics 5268 Renmin Street Jilin Province 130024 Changchun People’s Republic of China [email protected] Prof. Hui Yang Yunnan Normal University School of Mathematics 768 Junxian Road Yunnan Province 650500 Kuming People’s Republic of China [email protected]

ISBN 978-3-11-054925-6 e-ISBN (PDF) 978-3-11-054965-2 e-ISBN (EPUB) 978-3-11-054942-3 Library of Congress Control Number: 2018934551 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck Cover image: Chong Guo www.degruyter.com

Preface This book introduces the mathematical theory, research methods and results for infinite-dimensional dynamical systems. In 1993, the author gave a brief introduction to the conceptual framework, methods and research progress on infinite-dimensional dynamical systems in his monograph “Nonlinear evolution equations”. However he didn’t have an extensive and in-depth discussion on the topic due to limited space. As numerous new and important research achievements began to accumulate during the past decades, the authors have made a decision to write a monograph about infinite-dimensional dynamical systems. The aim of this book is to introduce the rudimentary knowledge, some interesting problems and important and new results including the authors’, cooperators’ and other scholars’ recent research on infinite-dimensional dynamical systems through some concise but heuristic methods. The main emphasis in the first volume is on the mathematical analysis of attractors and inertial manifolds. In Chapter 1 “Attractor and its dimension estimation” we mainly introduce the global attractor and estimation of Hausdorff and fractal dimensions for some dissipative nonlinear evolution equations in modern physics. Chapter 2 “Inertial manifold” deals with inertial manifolds for a range of generalized differential equations and the spectral gap conditions. Moreover, we study the existence, smoothness and normal hyperbolic properties of inertial manifolds. Chapter 3 “Approximate inertial manifold” constructs the inertial manifolds and investigates the convergence of approximate inertial manifolds, which provides a constructive method to establish existence of inertial manifolds. The second volume devotes to the modern analytical tools and methods in infinite dimensional dynamic system. In Chapter 1 “Discrete attractor and approximate calculation” we introduce results on the existence of discrete attractor and approximate calculation which are closely related to infinite-dimensional dynamical systems. Employing numerical calculations, we provide images of global attractors and approximate inertial manifolds. Chapter 2 “Some properties of a global attractor” introduces some properties of a global attractor, including oscillatory properties and asymptotic behavior. The asymptotic behavior of inertial manifolds can be determined only by the properties of a few points and it is closely related to the unstable manifold of the hyperbolic fixed point. We estimate an upper bound of Hausdorff length of level sets through the geometric measure method and provide a new method to give a lower bound on the dimensional estimate for the attractor. In Chapter 3 “Structures of small dissipative dynamical systems” we mainly introduce the structure of stable and unstable manifolds with small perturbations and the chaotic behaviors by employing the geometric singular perturbation theory, the center manifold theory in infinite dimensional setting and Melnikov method. The structure of stable and unstable manifolds is related to the first Chern number on the fiber bundle. Chapter 4 “Existence and stahttps://doi.org/10.1515/9783110549652-201

VI | Preface bility of solitary waves” uses the concentration-compactness principle to study the existence of solitary wave solutions and discusses the nonlinear stability, instability and asymptotic stability for the solitary waves by the energy functional method and spectral analysis. As the content of infinite-dimensional dynamical systems is quite rich and extensive, it is closely related to many subjects, such as fluid mechanics, functional analysis, topology, geometric measure theory, numerical mathematics, and so on. There are numerous new methods and results due to the quick developments of infinitedimensional dynamical systems. Owing to the limited time and knowledge of the authors, there must be some inadvertent errors and omissions in the book. Any suggestions and comments are welcomed. Beijing, China July 2017

Boling Guo

Contents Preface | V 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 2 2.1 2.2 2.3 2.4

3 3.1 3.2 3.3 3.4 3.5 3.6

Attractor and its dimension estimation | 1 Global attractor and estimation of Hausdorff and fractal dimensions | 1 Kuramoto–Sivashinsky equation | 6 A type of nonlinear viscoelastic wave equation | 25 Coupled KdV equations | 38 Davey–Stewartson equation | 51 Derivative Ginzburg–Landau equation | 62 Ginzburg–Landau model in superconductivity | 78 Landau–Lifshitz–Maxwell equation | 86 Nonlinear Schrödinger–Boussinesq equations | 108 A new method to prove existence of a strong topology attractor | 123 Nonlinear KdV–Schrödinger equation | 131 The Landau–Lifshitz equation on a Riemannian manifold | 145 The dissipation Klein–Gordon–Schrödinger equations on R3 | 165 Two-dimensional unbounded region derivative Ginzburg–Landau equation | 181 The relation between attractor and turbulence | 193 Inertial manifold | 201 The inertial manifold for a class of nonlinear evolution equations | 202 Inertial manifold and normal hyperbolicity property | 221 The finite-dimensional inertial form for the one-dimensional generalized Ginzburg–Landau equation | 258 The existence of inertial manifolds for the generalized KS equation | 276 The approximate inertial manifold | 307 Two-dimensional Navier–Stokes equation | 307 The Gevrey regularity of solutions | 316 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 323 Two-dimensional Ginzburg–Landau equation | 337 Bernard convection equation | 350 Long wave–short wave (LS) equation | 362

VIII | Contents 3.7 3.8 3.9

One-dimensional ferromagnetic chain equation | 375 Nonlinear Schrödinger equation | 383 The convergence of approximate inertial manifolds | 394

Bibliography | 419 Index | 429

1 Attractor and its dimension estimation 1.1 Global attractor and estimation of Hausdorff and fractal dimensions In this section, a very important concept of a global attractor is introduced in the infinite-dimensional dynamical system. Moreover, the existence theorem of a global attractor and the estimates of the Hausdorff and fractal dimensions are given. Definition 1.1.1. Assume that E is a Banach space and S(t) is a continuous semigroup of operators, i. e., S(t) : E → E, S(t + τ) = S(t) ⋅ S(τ), for any t, τ ≥ 0, S(0) = I (identity operator). If a compact set A ⊂ E is (i) invariant, i. e., S(t)A = A for any t ≥ 0; (ii) attractive, i. e., for any bounded set B ⊂ E, 󵄩 󵄩 dist(S(t)B, A ) = sup inf 󵄩󵄩󵄩S(t)x − y󵄩󵄩󵄩E → 0, y∈A x∈B

t → ∞;

and, in particular, if all the paths S(t)u0 departed from the initial data u0 converge to the set A , i. e., dist(S(t)u0 , A ) → 0,

t → ∞,

(1.1.1)

then the compact set A is called the global attractor. The structure of the global attractor is rather complicated. The semigroup S(t) generated from an initial problem of nonlinear evolution equation du(t) = F(u(t)), dt u(0) = u0 ,

(1.1.2) (1.1.3)

may involve, besides the simple equilibrium point (maybe a multiple solution), also a time-periodic orbit, quasi-periodic orbit, a fractal and strange attractor, and so on. The semigroup is likely not to be a smooth manifold, possessing non-integer dimension. To give the existence theorem of a global attractor, we need to introduce the concept of an attractive set. Definition 1.1.2. If a bounded set B0 ⊂ E is such that for any bounded set B ⊂ E there exists a t0 (B) > 0 such that S(t)B ⊂ B0 ,

∀t ≥ t0 (B),

then the set B0 is called an attractive set in the set E. https://doi.org/10.1515/9783110549652-001

(1.1.4)

2 | 1 Attractor and its dimension estimation Theorem 1.1.1. Suppose E is a Banach space and S(t), t ≥ 0 is a semigroup of operators, S(t) : E → E, S(t+τ) = S(t)S(τ), t, τ ≥ 0, S(0) = I, where I is the identity operator. Assume the following conditions: (i) The semigroup S(t) is uniformly bounded in the set E, i. e., for any R > 0, there exists a constant C(R) such that when ‖u‖E ≤ R, it follows that 󵄩 󵄩󵄩 󵄩󵄩S(t)u󵄩󵄩󵄩E ≤ C(R),

∀t ∈ [0, ∞).

(1.1.5)

(ii) There exists a bounded attractive set B0 in the set E. (iii) When t > 0, the operator S(t) is completely continuous. Then the semigroup S(t) possesses a compact global attractor A . Remark 1.1.1. If we replace the bounded attractive set B0 of the condition (ii) with the compact attractive set B0 , then the completely continuous property of semigroup S(t) can be replaced with that of being a continuous operator. Theorem 1.1.1 will remain valid. Remark 1.1.2. Furthermore, we can prove that the global attractor A is the ω limiting set of the attractive set B0 , i. e., A = ω(B0 ) = ⋂ ⋃ S(t)B0 s≥0 t≥s

(1.1.6)

where the closure is taken in the set E. Another frequently-used existence theorem of an attractor is the following: Theorem 1.1.2. Suppose E is a Banach space and a semigroup S(t) is continuous. Assume that there exists an open set U ⊂ E and a bounded set B in U such that the set B is absorptive in U . Also the following conditions are satisfied: (1) The operator S(t) is uniformly compact for any t large enough, i. e., for every bounded set B , there exists t = t0 (B ) such that the set ⋃ S(t)B

t≥t0

(1.1.7)

is relatively compact in the set E. Or (2) S(t) = S1 (t) + S2 (t), where the operator S1 (⋅) is uniformly compact for t large enough (i. e., satisfies condition (1.1.7)), the operator S2 (t) is a continuous map, S2 (t) : E → E, and for every bounded set B ⊂ E, 󵄩 󵄩 rB (t) = sup 󵄩󵄩󵄩S2 (t)ϕ󵄩󵄩󵄩E → 0, ϕ∈B

(1.1.8)

then the ω limiting set of B is a compact attractor, which attracts the bounded set in U . It is the largest attractor in U , and when U is convex and connected, A is connected.

1.1 Global attractor and estimation of Hausdorff and fractal dimensions | 3

Therefore, to prove the existence of a global attractor, we merely need to verify whether the conditions in Theorem 1.1.1 or 1.1.2 are valid. The most important is that: (i) The existence and continuity of semigroup S(t); (ii) There exists a bounded or compact absorbing set; (iii) The semigroup S(t) (t ≥ 0) is a completely continuous operator or satisfies condition (1.1.7) (or condition (1.1.8)). To portray the geometric property of a global attractor in the simplest way, we can estimate the Hausdorff and fractal dimension. Definition 1.1.3. The Hausdorff measure of a set X is μH (X, d) = lim μH (X, d, ϵ) = sup μH (X, d, ϵ)

(1.1.9)

μH (X, d, ϵ) = inf ∑ rid

(1.1.10)

ϵ→0

ϵ>0

where i

where inf is taken over all coverings of X by balls with radius ri ≤ ϵ. If there exists a number d = dH (X) ∈ [0, +∞] such that μH (X, d) = 0,

μH (X, d) = ∞,

d > dH (X),

d < dH (X),

(1.1.11) (1.1.12)

then the number dH (X) is called the Hausdorff dimension of the set X. Definition 1.1.4. The fractal dimension of a set X is dF (X) = lim sup ϵ>0

lg nX (ϵ) lg ϵ1

(1.1.13)

where nX (ϵ) is the smallest number of balls with radius ≤ ϵ in a covering of X. It is readily seen that dF (X) = inf{d > 0, μF (X, d) = 0}

(1.1.14)

μF (X, d) = lim sup ϵd nX (ϵ).

(1.1.15)

where ϵ→0

Since μF (X, d) ≥ μH (X, d), then dH (X) ≤ dF (X).

(1.1.16)

4 | 1 Attractor and its dimension estimation In the following, we consider the initial problem du(t) = F(u(t)), dt u(0) = u0 ,

t > 0,

(1.1.17) (1.1.18)

where F(u) is a determined function, F(u) : E → E, and E is a Hilbert space. Suppose that for any u0 ∈ E, there exists a global solution u(t) ∈ E, denoted by u(t) = S(t)u0 , where the map S(t) : E → E is the semigroup of the initial problem (1.1.17)–(1.1.18). Suppose F : E → E is Fréchet differentiable and the linear initial problem dU(t) = F 󸀠 (S(t)u0 )U(t), dt U(0) = ξ ,

(1.1.19) (1.1.20)

is solvable for every u0 and ξ ∈ E. Finally, suppose S(t) is differentiable, with the derivative L(t, u0 ), i. e., L(t, u0 )ξ = U(t),

∀ξ ∈ E,

(1.1.21)

and U(t) is the solution of (1.1.19) and (1.1.20). Since (1.1.19) is the first order variational equation of (1.1.17), the above assumptions are natural and can be readily verified. For a fixed u0 ∈ E, solutions ξ1 , ξ2 , . . . , ξJ are J elements of E, U1 (t), U2 (t), . . . , UJ (t) denote J solutions of linearized equation (1.1.19) with the initial data U1 (0) = ξ1 ,

U2 (0) = ξ2 ,

...,

UJ (0) = ξJ .

Through direct calculation we arrive at d 󵄩󵄩 󵄩2 󵄩 󵄩2 󸀠 󵄩U (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)󵄩󵄩󵄩∧E − 2 tr(F (u(t)) ⋅ QJ )󵄩󵄩󵄩U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)󵄩󵄩󵄩∧E = 0, dt 󵄩 1

(1.1.22)

where F 󸀠 (u(t)) = F 󸀠 S(t)u0 is a linear map U → F 󸀠 (u(t))U, u(t) = S(t)u0 is the solution of (1.1.17) and (1.1.18), ∧ denotes the wedge product, tr denotes the trace of operator, and QJ represents the orthogonal projection from E to the subspace spanned by U1 (t), U2 (t), . . . , UJ (t). The J-dimensional volume ⋀Jj=1 ξj is 󵄩 󵄩2 ωJ (t) = sup sup 󵄩󵄩󵄩U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)󵄩󵄩󵄩∧J E , u0 ∈A ξi ∈E

(1.1.23)

where A is the invariant set of the semigroup {S(t)}t≥0 . It is readily verified that ωj (t) is subexponential about t, i. e., ωj (t + t 󸀠 ) ≤ ωj (t)ωj (t 󸀠 ),

∀t, t 󸀠 ≥ 0.

(1.1.24)

Therefore, 1

lim ωj (t) t = Πj ,

t→∞

∀j, 1 ≤ j ≤ J

(1.1.25)

1.1 Global attractor and estimation of Hausdorff and fractal dimensions | 5

exists. By (1.1.22) we have ΠJ ≤ exp qJ

(1.1.26)

where qJ = lim sup qJ (t),

(1.1.27)

t→∞

qJ (t) =

t

1 sup { ∫ tr(F 󸀠 (S(τ)u0 )QJ (τ))dτ}. u0 ∈A, ξj ∈E ‖ξj ‖E ≤1 t sup

(1.1.28)

0

Definition 1.1.5. If a group of sequence Λ 1 , Λ 2 , . . . , Λ m is defined as Λ 1 = Π1 ,

Λ 1 Λ 2 = Π2 ,

...,

Λ 1 ⋅ ⋅ ⋅ Λ m = Πm ,

or Πm , Πm−1

m ≥ 2,

ω (t) t = lim ( m ) , t→∞ ωm−1 (t)

m ≥ 2,

Λ 1 = Π1 ,

Λm =

(1.1.29)

1

Λm

then Λ m is the global (or uniform) Lyapunov number in the set A and μm = lg Λ m ,

m>1

is the corresponding Lyapunov index. Due to equation (1.1.26), we have μ1 + μ2 + ⋅ ⋅ ⋅ + μJ ≤ qJ .

(1.1.30)

Theorem 1.1.3. Under the assumptions of initial problems (1.1.17)–(1.1.18) and (1.1.19)–(1.1.20), if we have qJ (t) ≤ −δ < 0,

∀t ≥ t0 ,

(1.1.31)

for some J and t0 > 0, then the volume element ‖U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)‖∧J E exponentially decays as t → ∞. For u0 ∈ A, ξ1 , ξ2 , . . . , ξJ ∈ E, it follows that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)󵄩󵄩󵄩∧m E ≤ 󵄩󵄩󵄩U1 (t0 ) ∧ ⋅ ⋅ ⋅ ∧ Um (t0 )󵄩󵄩󵄩∧J E exp(−δ(t − t0 )) uniformly. If A is the bounded functional invariant set of the semigroup S(t), then for some j the inequality qj < 0

(1.1.32)

6 | 1 Attractor and its dimension estimation is valid. From Πj = Λ 1 Λ 2 ⋅ ⋅ ⋅ Λ j < 1,

μ1 + μ2 + ⋅ ⋅ ⋅ + μj < 0,

it follows that Λ j < 1,

(1.1.33)

μj < 0.

(1.1.34)

i. e.,

Theorem 1.1.4. Assume that there exists a global attractor of the initial problem of the nonlinear evolution equation (1.1.17)–(1.1.18), which is bounded in H 1 (Ω). Suppose the linear initial problem (1.1.19)–(1.1.20) is solvable and the semigroup S(t) determined by the initial problem (1.1.17)–(1.1.18) is differentiable. If qj < 0

(1.1.35)

is determined by (1.1.27) for some j, then the Hausdorff dimension of the global attractor A is finite and ≤ j. Its fractal dimension is less than or equal to j(1 + max

1≤l≤j−1

(ql )+ ). |qj |

(1.1.36)

We can weaken the Fréchet differentiability condition of S(t), i. e., we can merely demand that the operator S(t) is uniformly differentiable in the compact invariant subset, i. e., for every u ∈ X, there exists a linear operator L(t, u) ∈ L (E ), and sup

u,v∈X, 0 0,

s where ωd (L) = ω1−s n (L)ωn+1 (L), d = n + s. Then we have

Theorem 1.1.5. Under the assumptions (1.1.37)–(1.1.39), the Hausdorff dimension of X is finite and less than or equal to d.

1.2 Kuramoto–Sivashinsky equation The Kuramoto–Sivashinsky equation (KS equation) [30, 181] 1 φt + (φx )2 + νφ + αφxx + γφxxxx = 0 2

(1.2.1)

1.2 Kuramoto–Sivashinsky equation

| 7

was proposed by Kuramoto [157] in the study of a reaction diffusion in 1978 and independently by Sivashinsky [193] who researched combustion flame propagation in 1977, where α, γ and ν are positive constants. This model also appears when studying a membrane vibration [3] and bifurcation solutions of Navier–Stokes equation [192]. Nicolaenko et al. [183] made an in-depth and systematic research on the global attractor and bifurcation solution of the one-dimensional KS equation. B. Nicolaenko [179] proposed a generalized KS-type equation (involving a high-dimensional KS equation). Guo et al. [101] studied the asymptotic behavior as t → ∞, the structure of the traveling solution, the similarity solution by Lie group and infinitesimal transformation, and the approximate solution by the spectral method. Guo [88] proved the existence of a global attractor of the KS equation and its finite dimension. In 1993, Guo and Su [109] gave the first proof of the existence of a global attractor of the high-dimensional KS equation and estimated the Hausdorff and fractal dimensions. Setting u = φx , equation (1.2.1) can be converted into ut + uux + νu + αuxx + γuxxxx = 0.

(1.2.2)

We consider the following generalized KS-type equation: ut + α△u + γ△2 u + ∇ ⋅ f (u) + △φ(u) = g(u) + h(x),

(1.2.3)

and periodic initial condition u(x, t) = u(x + 2dei , t), u(x, 0) = u0 (x),

x ∈ Ω, t ≥ 0, i = 1, 2, . . . , n,

x ∈ Ω,

(1.2.4) (1.2.5)

where Ω ⊂ Rn is the n dimensional cube with side length 2d, i. e., Ω = {x = (x1 , . . . , xn ) | |xi | ≥ d, i = 1, 2, . . . , n} where x + 2dei = (x1 , . . . , xi−1 , xi + 2d, xi+1 , . . . , xn ), i = 1, 2, . . . , n; α ≥ 0; γ > 0; ∇ ⋅ f (u) = 2 𝜕fk (u) ; △u = ∑nk=1 𝜕𝜕xu2 . If α = 0 and f (u) = 0, equation (1.2.3) is the inhomogeneous ∑nk=1 𝜕x k

k

Cahn–Hilliard equation. To prove the existence of a global attractor, the following priori estimates are given: Lemma 1.2.1. Suppose (1) φ󸀠 (u) ≤ φ0 ; α+φ (2) γ > 2 0 ; α+φ −1 (3) g(0) = 0, g 󸀠 (u) ≤ g0 , g0 < − 20 ; 2 2 (4) h(x) ∈ L (Ω), u0 (x) ∈ L (Ω).

8 | 1 Attractor and its dimension estimation Then the smooth solution of problem (1.2.3)–(1.2.5) satisfies the following estimate: 󵄩2 󵄩󵄩 (2g +α−1−φ0 )t 󵄩 󵄩󵄩u0 (x)󵄩󵄩󵄩2 󵄩󵄩u(⋅, t)󵄩󵄩󵄩 ≤ e 0 󵄩 󵄩 1 󵄩2 󵄩 + (1 − e(2g0 −α−1−φ0 )t )󵄩󵄩󵄩h(x)󵄩󵄩󵄩 , |2g0 + α + φ0 + 1|

0 < t < ∞.

(1.2.6)

Moreover, we have ‖h(x)‖2 󵄩2 󵄩 lim sup 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩 ≤ = E0 , t→∞ |2g0 + α + 1 + φ0 |

(1.2.7)

t

1 󵄩 1 󵄩 󵄩2 󵄩2 󵄩 󵄩2 lim sup ∫ 󵄩󵄩󵄩△u(⋅, t)󵄩󵄩󵄩 dt ≤ [󵄩󵄩u (x)󵄩󵄩 + 󵄩󵄩h(x)󵄩󵄩󵄩 ]. t→∞ t [2γ − (α + φ0 )] 󵄩 0 󵄩 󵄩

(1.2.8)

0

Proof. Taking the inner product of (1.2.3) and u, it follows that (u, ut + α△u + γ△2 u + ∇ ⋅ f (u) + △φ(u) − g(u) − h(x)) = 0,

(1.2.9)

where n

𝜕fk (u) udx 𝜕xk k=1

(∇ ⋅ f (u), u) = ∫ ∑ Ω

n

= − ∑ ∫ fk (u)uxk dx k=1 Ω n

= − ∑ ∫ Φk (u)xk dx = 0, k=1 Ω

u

Φk (u) = ∫ fk (u)du, 0

(△φ(u), u) = − (∇φ(u), ∇u) = −(φ󸀠 (u), |∇u|2 ) ≥ −φ0 ‖∇u‖2 . By Gagliardo–Nirenberg inequality, we obtain 1 ‖∇u‖2 ≤ ‖△u‖‖u‖ ≤ (‖△u‖2 + ‖u‖2 ). 2 It follows that

α 󵄨󵄨 󵄨 2 2 2 󵄨󵄨(u, α∇u)󵄨󵄨󵄨 = α‖∇u‖ ≤ (‖∇u‖ + ‖u‖ ), 2 (g(u), u) ≤ g0 ‖u‖2 , 1 (h(x), u) ≤ (‖u‖2 + ‖h(x)‖2 ). 2

Thus, through (1.2.9), we arrive at α + φ0 󵄩󵄩 1 d 󵄩󵄩 󵄩2 󵄩2 )󵄩󵄩△u(⋅, t)󵄩󵄩󵄩 󵄩u(⋅, t)󵄩󵄩󵄩 + (γ − 2 dt 󵄩 2 α + φ + 1 󵄩󵄩 󵄩2 1 󵄩 󵄩2 ≤ (g0 + )󵄩󵄩u(⋅, t)󵄩󵄩󵄩 + 󵄩󵄩󵄩h(x)󵄩󵄩󵄩 . 2 2

(1.2.10)

1.2 Kuramoto–Sivashinsky equation

| 9

Finally, equations (1.2.6), (1.2.7) and (1.2.8) can be deduced from (1.2.10) using the Gronwall inequality. Lemma 1.2.2. Under the conditions of Lemma 1.2.1, assume that (1) maxk=1,...,n |fk (u)| ≤ A|u|p , 1 ≤ p ≤ 1 + n6 ; (2) |φ󸀠 (u)| ≤ B|u|q , 0 ≤ q < 4n ; (3) h(x) ∈ L2 (Ω), u0 (x) ∈ H 1 (Ω), Ω ⊂ Rn , 1 ≤ n ≤ 6. Then the solution of problem (1.2.3)–(1.2.5) satisfies 󵄩󵄩 󵄩2 󵄩2 2g t 󵄩 󵄩󵄩∇u(⋅, t)󵄩󵄩󵄩 ≤ e 0 󵄩󵄩󵄩∇u0 (x)󵄩󵄩󵄩 1 󵄩 󵄩2 (1 − e2g0 t )(C6 󵄩󵄩󵄩h(x)󵄩󵄩󵄩 + C7 ), + |g0 |

0 ≤ t < ∞,

(1.2.11)

where functions C6 (⋅) and C7 (⋅) depend on ‖u(⋅, t)‖. Moreover, we have 󵄩 󵄩2 lim 󵄩󵄩∇u(⋅, t)󵄩󵄩󵄩

t→∞ 󵄩

1 󵄩 󵄩󵄩 󵄩2 󵄩 󵄩 [max C (󵄩󵄩u(⋅, t)󵄩󵄩󵄩)󵄩󵄩󵄩h(x)󵄩󵄩󵄩 + max C7 (󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩)] = E1 , t≥0 |g0 | t≥0 6 󵄩

(1.2.12)

6 󵄩 1 󵄩 󵄩2 󵄩2 lim ∫ 󵄩󵄩󵄩∇△u(⋅, t)󵄩󵄩󵄩 dt ≤ [󵄩󵄩󵄩u0 (x)󵄩󵄩󵄩H 1 + max C6 ‖h‖2 + max C7 ]. t→∞ t t≥0 t≥0 γ

(1.2.13)

≤

t

0

Proof. Taking the inner product of (1.2.3) and △u, we arrive at (△u, ut + α△u + γ△2 u + ∇ ⋅ f (u) + △φ(u) − g(u) − h(x)) = 0, where (△u, −g(u)) = (∇g(u), ∇u) = (g 󸀠 (u)∇u, ∇u) 󵄨 󵄨 ≤ g0 ‖∇u‖2 󵄨󵄨󵄨(△u, △φ(u))󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨(∇φ(u), ∇△u)󵄨󵄨󵄨 = 󵄨󵄨󵄨(φ󸀠 (u)∇u, ∇△u)󵄨󵄨󵄨 γ 3󵄩 󵄩2 ≤ ‖∇△u‖2 + 󵄩󵄩󵄩φ󸀠 (u)∇u󵄩󵄩󵄩 . 6 2γ By virtue of |φ󸀠 (u)| ≤ B|u|q , it follows that 3 󵄩󵄩 󸀠 3 󵄩 󸀠 󵄩2 3 2 2q 󵄩2 2 2 󵄩φ (u)∇u󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩φ (u)󵄩󵄩󵄩∞ ‖∇u‖ ≤ B ‖u‖∞ ‖∇u‖ . 2γ 󵄩 2γ 2γ Due to the Sobolev interpolation inequality, 11

6−n 6

1 3

2 3

‖u‖∞ ≤ C1 ‖∇△u‖ 6 ‖u‖

+ C1󸀠 ‖u‖,

‖∇u‖ ≤ C2 ‖∇△u‖ ‖u‖ + C2󸀠 ‖u‖,

1 ≤ n ≤ 6,

(1.2.14)

10 | 1 Attractor and its dimension estimation hence it follows that nq+2 3 2 2q 2 3 󵄩󵄩 󸀠 󵄩2 ((16−n)q+1)/3 󵄩󵄩φ (u)∇u󵄩󵄩󵄩 ≤ B C1 C2 ‖∇△u‖ 3 ‖u‖ 2γ 2γ γ 4 ≤ ‖∇△u‖2 + C3 (C1 , C2 , q, ‖u‖), 0 < q < , 6 n 󵄨󵄨 n 󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨(∇ ⋅ f (u), △u)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ∫ fk (u) 𝜕 △udx 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 𝜕x k 󵄨󵄨 k=1 Ω 󵄨󵄨

n

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩fk (u)󵄩󵄩󵄩‖∇△u‖ k=1

≤

γ 3 n 󵄩󵄩 󵄩2 ‖∇△u‖2 + ∑ 󵄩f (u)󵄩󵄩 , 6 2γ k=1 󵄩 k 󵄩

where 3 2 2p−2 2 3 2 2p 3 󵄩󵄩 󵄩2 󵄩f (u)󵄩󵄩 ≤ A ‖u‖p ≤ A ‖u‖∞ ‖u‖ 2γ 󵄩 k 󵄩 2γ 2γ n(p−1) n+(6−n)p 3 ≤ A2 C 2p−2 ‖∇△u‖ 3 ‖u‖ 3 , 2γ 6 3 1 ≤ p < 1 + ≤ ‖∇△u‖2 + C4 (γ, A, ‖u‖), n 2γ 1

2

α‖∇‖2 ≤ αC2 ‖∇△u‖ 3 ‖u‖ 3 γ ≤ ‖∇△u‖2 + C5 (α, C2 , ‖u‖), 6 (△u, −g(u)) = (g 󸀠 (u)∇u, ∇u) ≤ g0 ‖∇u‖2 γ 󵄩 󵄩2 (△u, −h(x)) ≤ ‖∇△u‖2 + C6 󵄩󵄩󵄩h(x)󵄩󵄩󵄩 . 12 Finally, due to equation (1.2.14), we obtain γ 1 d ‖∇u‖2 + ‖∇△u‖2 ≤ g0 ‖∇u‖2 + C6 ‖h‖2 + C7 (‖u‖), 2 dt 12 which implies 1 󵄩 󵄩2 ‖∇u‖2 ≤ e2g0 t 󵄩󵄩󵄩∇u0 (x)󵄩󵄩󵄩 + (1 − e−g0 t )(C6 ‖h‖2 + C8 ), |g0 | as well as (1.2.12) and (1.2.13). Lemma 1.2.3. Under the conditions of Lemma 1.2.2, suppose that (1) maxk=1,...,n ‖fk󸀠 (u)‖ ≤ A|u|p−1 , |φ󸀠󸀠 (u)| ≤ B|u|q−1 ; (2) |g 󸀠󸀠 (u)| ≤ C|u|l , 0 ≤ l < 40 − 43 ; 3n 2 (3) u0 (x) ∈ H (Ω).

1.2 Kuramoto–Sivashinsky equation

| 11

Then we have 1 3 󵄩2 󵄩2 󵄩󵄩 2g t 󵄩 (1 − e2g0 t )( ‖h‖2 + C12 ), 󵄩󵄩△u(⋅, t)󵄩󵄩󵄩 ≤ e 0 󵄩󵄩󵄩△u0 (x)󵄩󵄩󵄩 + |g0 | γ

(1.2.15)

where the function C12 depends on ‖u(⋅, t)‖H 1 . Moreover, we have 1 3 󵄩󵄩 󵄩2 󵄩2 󵄩 lim 󵄩󵄩△u(⋅, t)󵄩󵄩󵄩 ≤ [ 󵄩󵄩h(x)󵄩󵄩󵄩 + max C12 (‖u(⋅, t)‖H 1 )] = E2 , t≥0 |g0 | γ

t→∞ 󵄩 t

1 󵄩 2 󵄩 󵄩2 󵄩2 3 lim ∫ 󵄩󵄩󵄩∇△u(⋅, t)󵄩󵄩󵄩 dt ≤ [󵄩󵄩󵄩u0 (x)󵄩󵄩󵄩 + ‖h‖2 + max C12 ]. t→∞ t t≥0 γ γ

(1.2.16) (1.2.17)

0

Proof. Taking the inner product of (1.2.3) and △2 u, we arrive at (△2 u, ut + α△u + γ△2 u + ∇ ⋅ f (u) + △φ(u) − g(u) − h(x)) = 0,

(1.2.18)

where n

󵄨󵄨 2 󵄨 󵄩 󸀠 󵄩 󵄩 2 󵄩 󵄨󵄨(△ u, ∇ ⋅ f (u))󵄨󵄨󵄨 ≤ ∑ 󵄩󵄩󵄩fk (u)󵄩󵄩󵄩∞ ‖∇u‖󵄩󵄩󵄩△ u󵄩󵄩󵄩 k=1

(p−1)(6−n) (p−1)n 󵄩 󵄩 󵄩󵄩 2 󵄩󵄩 6 ‖u‖ 6 ‖∇u‖󵄩󵄩󵄩△2 u󵄩󵄩󵄩 ≤ nA‖u‖p−1 ∞ ‖∇u‖󵄩 󵄩△ u󵄩󵄩 ≤ nAC1 ‖∇△u‖

n(p−1) (p−1)n (p−1)(6−n) 󵄩 󵄩 (p−1)n 󵄩 󵄩 ≤ nAC1 C⋆ 6 ‖∇u‖ 18 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩 9 ‖u‖ 6 ‖∇u‖󵄩󵄩󵄩△2 u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 + C3 (‖u‖H 1 ). 6

In the above equation, we use the following Sobolev interpolation inequalities: 1 󵄩 󵄩2 ‖∇△u‖ ≤ C⋆ ‖∇u‖ 3 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 3 + C‖u‖, 󵄨󵄨 󵄩 󸀠 2 󵄨 󸀠󸀠 2 󵄩󵄩 2 󵄩 󵄨󵄨(△φ(u), △ u)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩φ (u)△u + φ (u)(∇u) 󵄩󵄩󵄩󵄩󵄩󵄩△ u󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ [󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖ + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖∞ ‖∇u‖],

11 2 󵄩 2 󵄩 󵄩󵄩 2 󵄩󵄩 󵄩 2 󵄩1 q−1 󵄩 2 󵄩 1− n q 󵄩󵄩△ u󵄩󵄩 ≤ (B‖u‖∞ C5 󵄩󵄩󵄩△ u󵄩󵄩󵄩 3 ‖u‖ 3 + B‖u‖∞ C6 󵄩󵄩󵄩△ u󵄩󵄩󵄩 6 ‖∇u‖ 6 ‖△u‖)󵄩󵄩󵄩△ u󵄩󵄩󵄩 n 2 󵄩 󵄩 󵄩 qn 󵄩1 ≤ [BC4q C5 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 8 ‖u‖q(1− 8 ) 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 3 ‖u‖ 3 n n 󵄩 󵄩 󵄩 (q−1)n 󵄩n + BC4q−1 C6 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 8 ‖u‖(q−1)(1− 8 ) 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 6 ‖∇u‖2− 6 ], 󵄩󵄩 2 󵄩󵄩 󵄩 2 󵄩 3nq+8 󵄩 2 󵄩 (3q+1)n 󵄩 2 󵄩 󵄩󵄩△ u󵄩󵄩 ≤ C7 (󵄩󵄩󵄩△ u󵄩󵄩󵄩 24 + 󵄩󵄩󵄩△ u󵄩󵄩󵄩 24 )󵄩󵄩󵄩△ u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 + C8 (‖u‖H 1 ). 6

In the above we used the following Sobolev interpolation inequalities: n 󵄩 󵄩n ‖u‖∞ ≤ C4 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 8 ‖u‖1− 8 + C1󸀠 ‖u‖,

12 | 1 Attractor and its dimension estimation 1

󵄩 󵄩 ‖△u‖ ≤ C5 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 3 ‖△u‖ 3 + C5󸀠 ‖u‖, n 󵄩n 󵄩 ‖∇u‖∞ ≤ C6 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 6 ‖∇u‖1− 6 + C6󸀠 ‖u‖, (△2 u, g(u)) = (△u, △g(u)) = (△u, g 󸀠 (u)△u + g 󸀠󸀠 (u)(∇u)2 ) 󵄩 󵄩 ≤ g0 ‖△u‖2 + 󵄩󵄩󵄩g 󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖‖∇u‖∞ ‖△u‖ ≤ g0 ‖△u‖2 + C‖u‖l∞ ‖∇u‖‖∇u‖∞ ‖△u‖ 2

n 8 n 󵄩n 󵄩 ln 󵄩 󵄩1 󵄩 󵄩 ≤ g0 ‖△u‖2 + CC4l 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 8 ‖u‖l(1− 8 ) C6 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 6 ‖∇u‖ 3 − 6 C5 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 3

󵄩 󵄩 (3l+4)n+8 ≤ g0 ‖△u‖2 + C9 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 24 γ󵄩 󵄩2 ≤ g0 ‖△u‖2 + 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 + C10 (‖u‖H 1 ), 6 󵄨󵄨 2 󵄨󵄨 γ 󵄩󵄩 2 󵄩󵄩2 3 2 󵄨󵄨(△ u, h)󵄨󵄨 ≤ 󵄩󵄩△ u󵄩󵄩 + ‖h‖ , 12 γ 1 󵄩 󵄩2 γ󵄩 󵄩2 α‖∇△u‖2 ≤ αC∗ ‖∇u‖ 3 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 3 ≤ 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 + C11 (‖u‖H 1 ). 6 Thus, by using (1.2.18), it follows that γ󵄩 1 d 󵄩2 ‖△u‖2 + 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 ≤ g0 ‖△u‖2 + 2 dt 4

3 󵄩󵄩 󵄩2 󵄩 󵄩 󵄩h(x)󵄩󵄩󵄩 + C12 (󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1 ), γ󵄩

which yields 3 1 󵄩󵄩 󵄩2 󵄩2 2g t 󵄩 (1 − e2g0 t )( ‖h‖2 + C13 ), 󵄩󵄩△u(⋅, t)󵄩󵄩󵄩 ≤ e 0 󵄩󵄩󵄩△u0 (x)󵄩󵄩󵄩 + |g0 | γ making equations (1.2.15), (1.2.16) and (1.2.17) valid. Lemma 1.2.4. Under the conditions of Lemma 1.2.3, suppose that (1) f (u) ∈ C 2 , φ(u) ∈ C 3 , g(u) ∈ C 1 , 󵄨󵄨 󸀠󸀠 󵄨󵄨 󵄨󵄨 󸀠󸀠󸀠 󵄨󵄨 󵄨󵄨φ (u)󵄨󵄨 + 󵄨󵄨φ (u)󵄨󵄨 ≤ K, K > 0, 4 ≤ n < 5, 󵄨󵄨󵄨g 󸀠 (u)󵄨󵄨󵄨 ≤ C|u|l−1 , 0 ≤ l < 40 − 4 ; 󵄨 󵄨 3n 3 (2) u0 (x) ∈ H 2 (Ω), h(x) ∈ H 1 (Ω). Then the smooth solution of the problem (1.2.3)–(1.2.5) satisfies the following estimate: 󵄩󵄩 󵄩 E 󵄩󵄩∇△u(⋅, t)󵄩󵄩󵄩 ≤ 3 , t

t > 0,

(1.2.19)

where the constant E3 depends on ‖u0 (x)‖H 2 and ‖h(x)‖H 1 . Proof. Taking the inner product of (1.2.3) and t 2 △3 u, we have (t 2 △3 u, ut + α△u + γ△2 u + ∇ ⋅ f (u) + △φ(u) − g(u) − h(u)) = 0.

(1.2.20)

1.2 Kuramoto–Sivashinsky equation

| 13

By direct calculation we have: 1 d 󵄩 1 󵄩2 ‖t∇△u‖2 + 󵄩󵄩󵄩t 2 ∇△u󵄩󵄩󵄩 , 2 dt 󵄩2 󵄩 (t 2 △3 u, α△u) = αt 2 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 , 󵄩2 󵄩 γ(t 2 △3 u, △2 u) = −γ 󵄩󵄩󵄩t∇△2 u󵄩󵄩󵄩 , 󵄨󵄨 2 󵄨 3 󵄨 󵄨 󵄨󵄨(∇ ⋅ f (u), △ u)󵄨󵄨󵄨 = 󵄨󵄨󵄨(∇(∇ ⋅ f (u)), ∇△ u)󵄨󵄨󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ C1 [ max 󵄩󵄩󵄩fk󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖∞ ‖∇u‖ + 󵄩󵄩󵄩fk󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖]󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩. k=1,...,n (t 2 △3 u, ut ) = −

When n < 4, using the following interpolation inequalities: n

n

‖u‖∞ ≤ C2 ‖△u‖ 4 ‖u‖1− 4 + C2󸀠 ≤ const,

n 󵄩 󵄩n ‖∇u‖∞ ≤ C3 ‖△u‖1− 8 󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 8 + C3󸀠 ,

we obtain γ󵄩 󵄨󵄨 2 3 󵄨 2 󵄩2 󵄨󵄨(∇ ⋅ f (u), t △ u)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩t ⋅ ∇△ u󵄩󵄩󵄩 + C4 . 6 When 4 ≤ n < 6, due to the Kato interpolation inequality [148], we obtain 10−n

‖u‖∞ ≤ C5 ‖u‖H 62 ‖∇△u‖ 8−n 6 H2

‖∇u‖∞ ≤ C6 ‖u‖

‖∇△u‖

n−4 6

n−2 6

,

.

Moreover, we have the following estimate: 󵄨󵄨 󵄩 2 󵄩 4n−12 󵄩 2 󵄩 n−4 󵄩 2 󵄩 2 3 󵄨 󵄨󵄨(∇ ⋅ f (u), t △ u)󵄨󵄨󵄨 ≤ C7 [󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 3n + 󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 4 ]󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 + C8 . 6 Using a similar procedure, when n < 4, we have 󵄨󵄨 󵄨 3 󵄨 2 󵄨 󵄨󵄨(△φ(u), △ u)󵄨󵄨󵄨 = 󵄨󵄨󵄨(∇△φ(u), ∇△ u)󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨(∇(φ󸀠 (u))△u + φ󸀠󸀠 (u)(∇u), ∇△2 u)󵄨󵄨󵄨 󵄩 󵄩 ≤ C[󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖∞ ‖△u‖ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ ‖∇△u‖ + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2∞ ‖∇u‖]󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 + C9 . 6 When 4 ≤ n < 5, we have 󵄨󵄨 󵄩 2 󵄩 n−2 󵄩 2 󵄩 7n−16 󵄩 2 󵄩 n−2 󵄩 2 󵄩 3 󵄨 󵄨󵄨(△φ(u), △ u)󵄨󵄨󵄨 ≤ C[󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 6 + 󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 6n + 󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 3 ]󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 + C10 , 6

14 | 1 Attractor and its dimension estimation 󵄩2 󵄩2 γ 󵄩 󵄩 αt 2 󵄩󵄩󵄩△2 u󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩t∇△2 u󵄩󵄩󵄩 + C11 , 6 󵄩 2 󵄩 󵄩 󸀠 󵄩 󵄨󵄨 2 󵄩 3 󵄨 󵄩 󸀠 󵄨󵄨(g(u), △ u)󵄨󵄨󵄨 = 󵄩󵄩󵄩(g (u)∇u, ∇△ u)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩g (u)󵄩󵄩󵄩∞ ‖∇u‖󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 + C12 , 6 󵄩 2 󵄩 (40−n)(n−4) 󵄩 2 󵄩 󵄨󵄨 3 󵄨 󵄨󵄨(g(u), △ u)󵄨󵄨󵄨 = C13 󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 18n 󵄩󵄩󵄩∇△ u󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩∇△2 u󵄩󵄩󵄩 + C14 , 6 󵄨󵄨󵄨(h(x), △3 u)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(∇h, ∇△2 u)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 γ 󵄩󵄩 2 󵄩󵄩2 3 ≤ 󵄩󵄩∇△ u󵄩󵄩 + ‖∇h‖2 . 6 2γ Therefore, from (1.2.20) we have 1 d ‖t∇△u‖2 + 2 dt

γ 󵄩󵄩 2 󵄩2 2 2 󵄩t∇△ u󵄩󵄩󵄩 ≤ C15 [‖∇△u‖ + ‖∇h‖ + 1]. 6󵄩

Using the above equation, we obtain 󵄩󵄩 󵄩 E 󵄩󵄩∇△u(⋅, t)󵄩󵄩󵄩 ≤ 3 , t

t > 0,

where the parameter E3 merely depends on ‖u0 (x)‖H 2 , ‖h(x)‖H 1 and t, 0 ≤ t ≤ T. Now we prove the existence and uniqueness of the global smooth solution of periodic initial problem (1.2.3)–(1.2.5) by Galerkin method. Suppose ωj (x), j = 1, 2, . . . , are the eigenfunctions of the equation △u + λu = 0 with periodic boundary condition at λ = λj ; {ωj } is the standard orthogonal basis. Suppose an approximate solution of problem (1.2.3)–(1.2.5) can be represented as N

uN (x, t) = ∑ αjN (t)ωj (x) j=1

(1.2.21)

where αjN (t), j = 1, 2, . . . , N; N = 1, 2, . . . , are undetermined function coefficients for t ∈ R+ . According to the Galerkin method, the coefficient αjN (t) should satisfy the following first order nonlinear ordinary differential equation [137] (uN,t + α△uN + γ△2 uN + △ ⋅ f (uN ) + △φ(uN ) − g(uN ) − h(x), ωj (x)) = 0,

(1.2.22)

j = 1, 2, . . . , N, with the initial value condition (uN (x, 0), ωj (x)) = (u0 (x), ωj (x)),

j = 1, 2, . . . , N.

Obviously, we obtain 󸀠 (uN,t (x, t), ωj (x)) = αj,N (t),

(uN,t (x, 0), ωj (x)) = αj,N (0),

(1.2.23)

1.2 Kuramoto–Sivashinsky equation

| 15

and u0,j (x) = (u0 (x), ωj (x)), j = 1, 2, . . . , N, are the coefficients of the approximate expansion ∑Nj=1 u0,j ωj of the function u0 (x). Similarly as in the proofs of Lemmas 1.2.1–1.2.4, we can establish uniform estimates of the approximate solutions by Galerkin method. These uniform estimates ensure the existence of the global solution αj,N (t), j = 1, 2, . . . , N; 0 ≤ t ≤ T, for the problem (1.2.22)–(1.2.23). And we can prove that the approximate solutions uN (x, t) of problem (1.2.22)–(1.2.23) converge to the global solution of problem (1.2.3)–(1.2.5). Lemma 1.2.5. Suppose the following conditions are valid: α+φ (1) φ󸀠 (u) ≤ φ0 , γ > 2 0 ; 󸀠 (2) g(0) = 0, g (u) ≤ g0 ; (3) |f (k) (u)| ≤ A|u|p−k , k = 0, 1, 1 ≤ p ≤ n6 + 1, |φ(k) (u)| ≤ B|u|q−k−1 , k = 1, 2, 0 ≤ q < 1 + 4n , n < 4, |φ󸀠 (u)| ≤ B|u|q , 0 ≤ q < 4n , n > 4, |φ󸀠󸀠 (u)| + |φ󸀠󸀠󸀠 (u)| ≤ K, K > 0, |g (k) (u)| ≤ C|u|l+2−k , k = 1, 2, 0 < l < 40 − 43 ; 3n 2 3 2 (4) f (u) ∈ C , φ(u) ∈ C , g(u) ∈ C ; (5) h(x) ∈ H 1 (Ω), u0 (x) ∈ H 3 (Ω). Then the solutions of problem (1.2.22)–(1.2.23) satisfy the following estimate: 󵄩 󵄩2 sup 󵄩󵄩󵄩uN (⋅, t)󵄩󵄩󵄩H 3 (Ω) ≤ K3 ,

0≤t≤T

(1.2.24)

where the constant K3 depends on ‖u0 (x)‖H 3 (Ω) and ‖h(x)‖H 1 (Ω) , and does not depend on N. Lemma 1.2.6. Suppose the conditions of Lemma 1.2.5 are satisfied and (1) f (u) ∈ C 2m−1 , φ(u) ∈ C 2m , g(u) ∈ C 2m , m > 2; (2) u0 (x) ∈ H 2m (Ω), h(x) ∈ H 2(m−1) (Ω). Then the solutions of problem (1.2.22)–(1.2.23) satisfy the following estimate: 󵄩 󵄩 sup 󵄩󵄩󵄩△m uN (⋅, t)󵄩󵄩󵄩 ≤ Km

0≤t≤T

(1.2.25)

where the constant Km depends on ‖u0 (x)‖H 2m and ‖h(x)‖H 2m−2 , and does not depend on N. Proof. From the equation △ωj + λj ωj = 0, it follows that △2m ωj − λj2m ωj = 0,

m > 2.

Multiplying (1.2.22) by −λj2m αjN (t), and summing over all j = 1, 2, . . . , N, (uN,t + α△uN + γ△2 uN + △ ⋅ f (uN ) + △φ(uN ) − g(uN ) − h(x), △2m uN ) = 0,

(1.2.26)

16 | 1 Attractor and its dimension estimation where 1 d 󵄩󵄩 m 󵄩󵄩2 󵄩△ uN 󵄩󵄩 , 2 dt 󵄩 󵄩2 󵄩 α(△uN , △2m uN ) = (−1)2m−1 α󵄩󵄩󵄩∇△m uN 󵄩󵄩󵄩 , 󵄩2 󵄩 γ(△2m uN , △2 uN ) = γ 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 . (uN,t , △2m uN ) =

By the Lemma 1.2.5 and Sobolev imbedding theorem, one deduces 󵄩 󵄩 sup 󵄩󵄩󵄩uN (⋅, t)󵄩󵄩󵄩∞ ≤ K4 ,

0≤t≤T

1 ≤ n < 5,

where the constant K4 does not depend on N. On account of the following equalities: 󵄩󵄩 s 󵄩 󵄩 s 󵄩 󵄩󵄩D f (u)󵄩󵄩󵄩 ≤ Cs (‖u‖∞ )󵄩󵄩󵄩D u(s)󵄩󵄩󵄩,

f (u) ∈ C s ,

one deduces that 󵄨󵄨 󵄨 󵄨 m+1 󵄨 2m m−1 󵄨󵄨(△ ⋅ f (uN ), △ uN )󵄨󵄨󵄨 = 󵄨󵄨󵄨(△ uN , △ ∇ ⋅ f (uN ))󵄨󵄨󵄨 γ󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩△m+1 uN 󵄩󵄩󵄩 + C2m−1 (‖uN ‖∞ 󵄩󵄩󵄩∇2m−1 uN 󵄩󵄩󵄩 ), 6 󵄨󵄨 2m 󵄨󵄨 󵄨󵄨 m+1 m−1 󸀠 󸀠󸀠 2 󵄨󵄨 󵄨󵄨(△ u2N , △φ(uN ))󵄨󵄨 = 󵄨󵄨(△ uN , △ (φ (uN )△uN + φ (uN )(∇uN ) ))󵄨󵄨 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 0 ≤ C2m−2 [󵄩󵄩󵄩△m−1 φ󸀠 (uN )󵄩󵄩󵄩‖△uN ‖ + 󵄩󵄩󵄩△m uN 󵄩󵄩󵄩󵄩󵄩󵄩φ󸀠 (uN )󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 +󵄩󵄩󵄩△m−1 φ󸀠󸀠 (uN )󵄩󵄩󵄩‖∇uN ‖2 + 󵄩󵄩󵄩△m−1 (∇uN )2 󵄩󵄩󵄩󵄩󵄩󵄩φ󸀠󸀠 (uN )󵄩󵄩󵄩]󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 󵄩󵄩 m−1 󵄩󵄩 󵄩󵄩 m 󵄩󵄩󵄩󵄩 󸀠 󵄩󵄩 0 1 ≤ C2m−2 [C2m−2 󵄩󵄩△ uN 󵄩󵄩‖△uN ‖ + 󵄩󵄩△ uN 󵄩󵄩󵄩󵄩φ (uN )󵄩󵄩 󵄩󵄩 m−1 󵄩󵄩 󵄩 2m−1 󵄩󵄩 2 +C2m−2 uN 󵄩󵄩‖∇uN ‖∞ 󵄩󵄩△ uN 󵄩󵄩‖∇uN ‖∞ ‖∇uN ‖ + 2(󵄩󵄩󵄩∇ 󵄩󵄩 2m−2 󵄩󵄩 󵄩󵄩 󸀠󸀠 󵄩󵄩 󵄩󵄩 m−1 󵄩󵄩 +‖△uN ‖∞ 󵄩󵄩∇ uN 󵄩󵄩)󵄩󵄩φ (uN )󵄩󵄩]󵄩󵄩△ uN 󵄩󵄩 n + 󵄩 󵄩 2m−5 + n 󵄩 󵄩 󵄩󵄩 m−1 󵄩󵄩 2m−5 ≤ C1 [󵄩󵄩△ uN 󵄩󵄩 2m−1 2(2m+1) + 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 2m−1 4m ]󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 γ󵄩 󵄩2 ≤ 󵄩󵄩󵄩△m+1 uN 󵄩󵄩󵄩 + C2 . 6 The above inequality utilizes the following Sobolev interpolation inequalities: n 󵄩 󵄩 n ‖∇uN ‖∞ ≤ C 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 2(2m+1) ‖∇uN ‖1− 2(2m−1) + C 󸀠 ,

󵄩󵄩 m−1 󵄩󵄩 󵄩 m+1 󵄩 2m−5 1− 2m−5 󸀠 󵄩󵄩△ uN 󵄩󵄩 ≤ C 󵄩󵄩󵄩△ uN 󵄩󵄩󵄩 2m−1 ‖∇△uN ‖ 2m−1 + C , n n 󵄩 󵄩 ‖△uN ‖ ≤ C 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 4m ‖△uN ‖1− 4m + C 󸀠 . In a similar way, we obtain

󵄨󵄨 2m 󵄨 󵄨 m 󵄨 m 󵄨󵄨(△ uN , g(uN ))󵄨󵄨󵄨 = 󵄨󵄨󵄨(△ uN , △ g(uN ))󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨(△m uN , g 󸀠 (uN )∇m uN )󵄨󵄨󵄨 󵄨 󵄨 +󵄨󵄨󵄨(△m uN , ∇2m−1 (g 󸀠 (uN )∇uN ) − g 󸀠 (uN )△m uN )󵄨󵄨󵄨 󵄩 󵄩2 γ 󵄩 󵄩2 ≤ g0 󵄩󵄩󵄩△m uN 󵄩󵄩󵄩 + 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 + C3 , 6

1.2 Kuramoto–Sivashinsky equation

| 17

󵄨 󵄨 m−1 󵄨󵄨 2m m−1 󵄨 󵄨󵄨(△ uN , h)󵄨󵄨󵄨 = 󵄨󵄨󵄨(△ uN , △ h)󵄨󵄨󵄨 γ󵄩 󵄩2 3 󵄩 󵄩2 ≤ 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 + 󵄩󵄩󵄩△m−1 h󵄩󵄩󵄩 , 12 γ 󵄩2 γ 󵄩 󵄩 󵄩2 α󵄩󵄩󵄩∇2m+1 uN 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩△m−1 uN 󵄩󵄩󵄩 + C4 , 6 then, on account of (1.2.26), we deduce 1 d 󵄩󵄩 m 󵄩󵄩2 γ 󵄩󵄩 m+1 󵄩󵄩2 󵄩 m 󵄩2 󵄩△ uN 󵄩󵄩 + 󵄩󵄩△ uN 󵄩󵄩 ≤ g0 󵄩󵄩󵄩△ uN 󵄩󵄩󵄩 + 2 dt 󵄩 4

3 󵄩󵄩 m−1 󵄩󵄩2 󵄩△ h󵄩󵄩 + C5 . γ󵄩

Moreover, we get 󵄩 󵄩2 sup 󵄩󵄩󵄩△m uN 󵄩󵄩󵄩 ≤ Km ,

0≤t≤T

where the constant Km does not depend on N. Theorem 1.2.1. Under the conditions of Lemma 1.2.6, there exists a unique global smooth solution u(x, t) of the periodic initial value problem (1.2.3)–(1.2.5) such that u(x, t) ∈ L∞ (0, T; H 2m (Ω)), ut (x, t) ∈ L∞ (0, T; H 2m−4 (Ω)),

m > 1.

Proof. On account of Lemmas 1.2.5 and 1.2.6, we have 󵄩 󵄩 sup 󵄩󵄩󵄩uN (⋅, t)󵄩󵄩󵄩H 2m (Ω) ≤ Km ,

0≤t≤T

where the constant Km does not depend on N. Thus we can choose a subsequence {uNi (x, t)} from the approximate sequence {uN (x, t)} such that there exists a function u(x, t) ∈ L∞ (0, T; H 2m (Ω)) which satisfies the following conditions: – uNi (x, t) → u(x, t) is weak ∗ convergent in L∞ (0, T; H 2m (Ω)), – △uNi (x, t) → △u(x, t) is strongly and almost surely convergent in L∞ (0, T; L2 (Ω)) as Ni → ∞. Using the equation (

𝜕uN 𝜕uN , + α△uN + γ△2 uN + ∇f (uN ) + △φ(uN ) − g(uN ) − h(x)) = 0, 𝜕t 𝜕t

we obtain 󵄩󵄩 𝜕u 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩 ≤ Km , 󵄩󵄩 𝜕t 󵄩󵄩 󸀠 where the constant Km does not depend on N. Therefore, we have

–

𝜕uNi 𝜕t

→

𝜕u 𝜕t

is weak ∗ convergent in L∞ (0, T; L2 (Ω)) as Ni → ∞.

18 | 1 Attractor and its dimension estimation It follows that the function u(x, t) almost always satisfies equation (1.2.3) and the periodic initial value conditions (1.2.4) and (1.2.5). So the existence of a smooth solution of problem (1.2.3)–(1.2.5) is proved. The uniqueness of the smooth solution can be readily obtained by the energy method. To prove the existence of a global attractor for periodic initial value problem (1.2.3)–(1.2.5), we use the following result by Babin–Vishik [7]: Theorem 1.2.2. Suppose E is a Banach space, {St }t≥0 is a semigroup of operators St : E → E: St ⋅ Sτ = St+τ ,

S0 = I,

where I is the identity operator, and St satisfies the following conditions: (1) Operator St is bounded uniformly, i. e., if ∀R > 0, ‖u‖E ≤ R, then there exists a constant C(R) such that ‖St u‖E ≤ C(R),

t ∈ [0, ∞);

(2) There exists a bounded absorbing set B0 ⊂ E, i. e., for any bounded set B ⊂ E, there exists a constant T such that St B ⊂ B0 ,

t ≥ T;

(3) St is a completely continuous operator for all t > 0. Then the semigroup of operators St possesses a compact global attractor. Theorem 1.2.3. Suppose the problem (1.2.3)–(1.2.5) possesses a global smooth solution and satisfies: (1) f (u) ∈ C 2 , φ(u) ∈ C 3 , g(u) ∈ C 2 , 6 , n 4 󵄨󵄨 (k) 󵄨󵄨 q−k+1 , k = 1, 2, 0 ≤ q < 1 + , n < 4, 󵄨󵄨φ (u)󵄨󵄨 ≤ B|u| n 4 󵄨󵄨 󸀠 󵄨󵄨 q 󵄨󵄨φ (u)󵄨󵄨 ≤ B|u| , 0 ≤ q < , n ≥ 4; n 󵄨󵄨 󸀠󸀠 󵄨󵄨 󵄨󵄨 󸀠󸀠󸀠 󵄨󵄨 󵄨󵄨φ (u)󵄨󵄨 + 󵄨󵄨φ (u)󵄨󵄨 ≤ K, K > 0; 󵄨󵄨 (k) 󵄨󵄨 p−k 󵄨󵄨f (u)󵄨󵄨 ≤ A|u| ,

k = 0, 1, 1 ≤ p < 1 +

α+φ

(2) φ󸀠 (u) ≤ φ0 , γ > 2 0 ; α+φ +1 (3) g(0) = 0, g 󸀠 (u) ≤ g0 , g0 < − 20 , 󵄨󵄨 (k) 󵄨󵄨 l+2−k , 󵄨󵄨g 󵄨󵄨 ≤ C|u|

k = 1, 2, 0 ≤ l
0,

where ‖u0 ‖H 2 ≤ R. By compact imbedding, H 3 (Ω) 󳨅→ H 2 (Ω), we obtain that the semigroup operator St is completely continuous when t > 0. So the theorem is proved. Remark 1.2.1. As stated in Theorem 1.1.1 of [197], the global attractor A obtained in Theorem 1.2.3 is the ω limit of absorbing set A,̄ i. e., A = ω(A)̄ = ⋂ ⋃ St A.̄ τ≥0 t≥τ

To establish estimates of the Hausdorff and fractal dimensions of the global attractor for the problem (1.2.3)–(1.2.5), we consider the following linear variational problem of (1.2.3)–(1.2.5): vt + L(u(t))v = 0,

v(0) = v0 ,

where L(u(t))v = α△v + γ△2 v + ∑nk=1 (fk (u)v)xk + △(φ󸀠 (u)v) − g 󸀠 (u)v.

(1.2.28) (1.2.29)

20 | 1 Attractor and its dimension estimation Since the solutions of problem (1.2.3)–(1.2.5) are smooth enough, one can readily prove that the linear problem (1.2.28)–(1.2.29) possesses a global smooth solution when the initial data v0 (x) is smooth enough, i. e., there exists a solution operator Gt such that v(t) = Gt v0 . Also it is readily proved that the semigroup operator St u0 is differentiable in L2 (Ω), i. e., the Fréchet differential St󸀠 u0 exists, and Gt v0 = St󸀠 u0 . Actually, setting ω(t) = St (u0 + v0 ) − St (u0 ) − Gt (u0 )v0 = u1 (t) − u(t) − v(t), we have 𝜕t ω(t) = L1 (u1 (t)) − L1 (u(t)) + L(u(t))v(t)

= L1 (u(t) + v(t) + ω(t)) − L1 (u(t)) + L(u(t))v(t),

ω(0) = 0,

(1.2.30) (1.2.31)

where ut = L1 (u) is the operator form of equation (1.2.3). Thus equation (1.2.30) can be rewritten as 𝜕t ω(t) + L(u(t))ω = Λ 0 (u, v, ω),

(1.2.32)

where Λ 0 (u, v, ω) = L1 (u(t) + v(t) + ω(t)) − L1 (u(t)) + L(u(t))(v + ω).

(1.2.33)

Using the theory of linear partial differential equations, we can obtain the following L2 estimate: 󵄩󵄩 󵄩 2 󵄩󵄩ω(t)󵄩󵄩󵄩 ≤ C‖v0 ‖ ,

(1.2.34)

which infers that the semigroup operator St is differentiable in L2 (Ω). Let v1 (t), v2 (t), . . . , vJ (t) be the solutions of the linear equation corresponding to the initial values v1 (0) = ξ1 , . . . , vJ (0) = ξJ , where ξ = (ξ1 , ξ2 , . . . , ξJ ) ∈ L2 . Using the results of [197], we arrive at d 󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩v (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩 + 2 tr(L(u(t)) ⋅ QJ )󵄩󵄩󵄩v1 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩 = 0, dt 󵄩 1

(1.2.35)

where L(u(t)) = L(St u0 ) is the linear map v → L(u(t))v, “∧” represents the wedge, tr represents the trace of an operator, QJ (t) is the orthogonal projection from the space L2 (Ω) to the subspace span{v1 (t), . . . , vJ (t)}. Thus, by equation (1.2.31), we obtain that the volume of a J-dimensional cube is bounded by ωJ (t) = sup

sup

u0 ∈A ξj ∈L2 ,|ξj |≤1

󵄩󵄩 󵄩2 󵄩󵄩v1 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩ΛJ ,L2

t

≤ sup exp(− ∫ inf(tr L(Sτ u0 ) ⋅ QJ (τ))dτ). u0 ∈A

0

(1.2.36)

1.2 Kuramoto–Sivashinsky equation

| 21

Notice that by the results of [197], ωj (t) is log-subadditive, i. e., ωj (t + t 󸀠 ) ≤ ωj (t)ωj (t 󸀠 ),

t, t 󸀠 ≥ 0.

(1.2.37)

Therefore, we have 1

lim ωj (t) t = Πj ,

t→∞

1 ≤ j ≤ J,

Πj ≤ exp(−qJ ),

(1.2.38) (1.2.39)

where t

1 qJ = lim sup( inf ∫ inf(tr L(S(τ)u0 ) ⋅ QJ (τ))dτ). t→∞ u0 ∈A t

(1.2.40)

0

Definition 1.2.1. The Hausdorff measure [55, 56] of a set X is defined as nH (X, d) = lim nH (X, d, ϵ) = sup nH (X, d, ϵ) ϵ→0

ϵ>0

where nH (X, d, ϵ) = inf ∑ rid , i

and inf is taken over all coverings of X by balls with radius ri ≤ ϵ. If there exists a number d = dH (X) ∈ [0, +∞] such that μH (X, d) = 0,

d > dH (X),

μH (X, d) = ∞,

d < dH (X),

then the number dH (X) is called the Hausdorff dimension of the set X. Definition 1.2.2. The fractal dimension of a set X is defined as the number dF (X) = lim sup ϵ>0

lg nX (ϵ) lg ϵ1

where nX (ϵ) denotes the minimum number of balls with radius ≤ ϵ in a covering of the set X. By the results in [197], dF (X) = inf{d > 0, nF (X, d) = 0} where nF (X, d) = lim supϵ→0 ϵd nX (ϵ). Theorem 1.2.4 ([33]). Suppose A is the attractor of a nonlinear evolution equation (e. g., Navier–Stokes equation, equation (1.2.3), etc.). If it is bounded in H 1 (Ω), then for a certain J, the Hausdorff dimension of A ≤ J. Its fractal dimension is no more than J(1 + max 1≤i≤J

−qi ). qJ

(1.2.41)

22 | 1 Attractor and its dimension estimation Lemma 1.2.7 (Generalized Sobolev–Lieb–Thirring inequality, [197]). Suppose Ω ⊂ Rn is a bounded domain, {φ1 , φ2 , . . . }, φj ∈ H 1 (Ω) is an orthogonal basis in L2 (Ω). Then we have the estimate N

1+ n2

󵄨2

󵄨 ∫(∑ 󵄨󵄨󵄨φj (x)󵄨󵄨󵄨 ) Ω

j=1

N

dx ≤ k0 ∑ ∫ |grad φj |2 dx. j=1 Ω

(1.2.42)

Theorem 1.2.5. Under the conditions of Theorem 1.2.3, the Hausdorff and fractal dimensions of the global attractor for problem (1.2.3)–(1.2.5) is finite and dH (A ) ≤ J0 ,

dF (A ) ≤ 2J0 ,

where J0 is the minimum integer satisfying the following equations: J0 ≥

1 1 󵄩 󵄩 󵄩 󵄩 2 [(α + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ )c󸀠 2 + (α + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ ) c󸀠 󸀠 2γc 1 󵄩 󵄩 + 4γc󸀠 [k0 c󸀠 4 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ (‖∇u‖ + ‖∇u‖L 5 ) 2 1 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩φ󸀠󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2∞ + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖∞ ] 2 ],

J0 ≥ (

3 7

4ν1 ) , c󸀠 γ

n = 3, − 52

2 5c󸀠 γ ) ν1 = ( 7 28

(1.2.43) (1.2.44)

2 1 󵄩 󵄩 [c󸀠 2 (α + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ )] 7

− 95

9 5c󸀠 γ + ( ) 14 56 +

n = 2,

− 43

4 3c󸀠 γ ( ) 7 28

1

14 󵄩 󵄩 k0 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ c󸀠 4 (‖∇u‖ + ‖∇u‖L 5 ) 9 2

7 󵄩 󵄩 󵄩 󵄩 (󵄩󵄩󵄩φ󸀠󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2∞ + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖∞ ) 4 ,

(1.2.45)

where c󸀠 and k0 are absolute constants, and 󵄩󵄩 󸀠󸀠 󵄩󵄩 󵄩 󸀠󸀠 󵄩 󵄩󵄩f (u)󵄩󵄩∞ = max 󵄩󵄩󵄩fk (u)󵄩󵄩󵄩∞ . k=1,...,n Proof. Based on Theorem 1.2.4, we must estimate the lower bound of tr(L(u(t)) ⋅ QJ ), which we do as follows: tr(L(u(t)) ⋅ QJ ) J

n

j=1

k=1

= ∑[(α△φj + γ△2 φj + ∑ (fk󸀠 (u)φj )x − △(φ󸀠 (u)φj ) − g 󸀠 (u)φj , φj )] k

J

n

j=1

k=1

= ∑[(γ‖△φj ‖2 − α‖∇φj ‖2 + ( ∑ fk󸀠 (u)φj ) − △(φ󸀠 (u)φj ) − g 󸀠 (u)φj , φj )] (1.2.46) xk

1.2 Kuramoto–Sivashinsky equation

| 23

where 󵄩󵄩 󵄩󵄩 󵄩󵄩 J 󵄩󵄩 J n n 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑(( ∑ f 󸀠 (u)φj ) , φj )󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ ( ∑ f 󸀠 (u)φj ), φjx 󵄩󵄩󵄩 k k 󵄩󵄩 󵄩󵄩 󵄩󵄩 k󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=1 k=1 󵄩󵄩 j=1 k=1 󵄩 xk 󵄨󵄨 󵄨󵄨 n J 󵄨󵄨 1 󵄨󵄨 = 󵄨󵄨󵄨( ∑ fk󸀠󸀠 (u)uxk , ∑ φ2j )󵄨󵄨󵄨 = 2 󵄨󵄨󵄨 k=1 󵄨󵄨󵄨 j=1

󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 n 󸀠󸀠 󵄨󵄨( ∑ fk (u)uxk , ρ(x))󵄨󵄨󵄨, 󵄨󵄨 2 󵄨󵄨󵄨 k=1 󵄨

J

ρ(x) = ∑ φ2j . j=1

By Lemma 1.2.7, we have 󵄨󵄨 n 󵄨󵄨 󵄩󵄩 n 󵄩󵄩 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨 ∑ (f 󸀠󸀠 (u)ux , ρ(x))󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩 ∑ f 󸀠󸀠 (u)ux 󵄩󵄩󵄩󵄩󵄩󵄩ρ(x)󵄩󵄩󵄩 k k 󵄨󵄨 󵄩󵄩 󵄩 k k󵄩 󵄨󵄨󵄨 󵄩󵄩󵄩 󵄨󵄨 󵄩󵄩 k=1 󵄨 k=1 󵄩

1 2

J

󵄩 󵄩 ≤ 󵄩󵄩󵄩fk󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖k0 (∑ ∫ |∇φj |2 dx) ,

n = 2,

j=1

5

2

3 󵄨󵄨 n 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 n 5 5 5 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨( ∑ f 󸀠󸀠 (u)ux , ρ(x))󵄨󵄨󵄨 ≤ (∫ 󵄨󵄨󵄨 ∑ f 󸀠󸀠 (u)ux 󵄨󵄨󵄨 dx) (∫ 󵄨󵄨󵄨ρ(x)󵄨󵄨󵄨 3 dx) k k 󵄨 󵄨󵄨 󵄨 󵄨󵄨 k 󵄨󵄨 k 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 k=1 Ω Ω 󵄨 k=1 1 2

J

󵄩 󵄩 ≤ 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖L 5 k0 (∑ ∫ |∇φj |2 dx) , 2

j=1

n = 3,

󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠 󵄨󵄨−(△(φ (u)φj ), φj )󵄨󵄨󵄨 = 󵄨󵄨󵄨(∇φ (u)φj , ∇φj )󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨(φ󸀠󸀠 (u)∇uφj , ∇φj ) + (φ󸀠 (u)∇φj , ∇φj )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨 = 󵄨󵄨󵄨− (∇φ󸀠󸀠 (u)∇u, φ2j ) + (φ󸀠 (u)∇φj , ∇φj )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 1 1 󵄨 󵄨 = 󵄨󵄨󵄨− (φ󸀠󸀠󸀠 (u)(∇u)2 , φ2j ) − (φ󸀠󸀠 (u)△u, φ2j ) + (φ󸀠 (u)∇φj , ∇φj )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 󵄩󵄩 󸀠󸀠󸀠 󵄩󵄩 2 2 ≤ 󵄩󵄩φ (u)󵄩󵄩∞ ‖∇u‖∞ ‖φj ‖ 󵄩 󵄩 󵄩 󵄩 +󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖∞ ‖φj ‖2 + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ ‖φj ‖2 Now select φj (x)s which are the eigenfunctions corresponding to eigenvalues λj , j = 1, 2, . . . , for equation −△u = λu with the periodic boundary conditions and such that ‖∇φj ‖2 = λj ,

‖△φj ‖2 = λj2 ,

‖φj ‖2 = 1.

As in [197], we estimate the eigenvalues λj as follows: 1

λj ≥ [

2

2 (j − 1) n − 1] ∼ cj n . 2

24 | 1 Attractor and its dimension estimation Hence, from (1.2.46) tr(L(u(t)) ⋅ QJ ) ≥

J

λ ∑ λj2 j=1

J

J

󵄩 󵄩 − α ∑ λj − 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ (‖∇u‖L 5 + ‖∇u‖)k0 (∑ λj ) 2

j=1

1 2

j=1

J

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 −J(󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2∞ + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖∞ ) − 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ ∑ λj j=1

J

J

j=1

j=1

1 󵄩 󵄩 ≥ γ ∑ λj2 − (α + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ )J 2 (∑ λj2 )

1 2

J

1 󵄩 󵄩 −k0 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ (‖∇u‖ + ‖∇u‖L 5 )J 4 (∑ λj2 ) 2

1 4

j=1

󵄩 󵄩 󵄩 󵄩 −J(󵄩󵄩󵄩φ󸀠󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2 + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖∞ ). 2

From the inequality λj ≥ c󸀠 j n , when n = 2, and J≥

1 1 󵄩 󵄩 󵄩 󵄩 2 [(α + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ )c󸀠 2 + [(α + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ) c󸀠 󸀠 2γc 1 󵄩 󵄩 + 4γc󸀠 (k0 c󸀠 4 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ (‖∇u‖ + ‖∇u‖L 5 ) 2

1 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩φ󸀠󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2∞ + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖∞ )] 2 ] = J0 ,

we have tr(L(u(t)) ⋅ QJ ) > 0. By the generalized Young inequality p 󸀠 b ap + ϵ1−p 󸀠 , p p 󸀠

ab ≤ ϵ

1 1 + 󸀠 = 1, p p

we get − 52

1 5 c󸀠 γ 72 2 5c󸀠 γ 󵄩 󵄩 c󸀠 2 (α + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ )J 3 ≤ J + ( ) 4 7 28 5 1 󵄩 󵄩 k0 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ c󸀠 4 (‖∇u‖ + ‖∇u‖L 5 )J 6 2

7 1 󵄩 󵄩 [c󸀠 2 (α + 󵄩󵄩󵄩φ󸀠 (u)󵄩󵄩󵄩∞ )] 2 ,

−5

14 1 c󸀠 γ 37 9 5c󸀠 γ 9 󵄩 󵄩 ≤ J + ( ) [k0 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ c󸀠 4 (‖∇u‖ + ‖∇u‖L 5 )] 9 , 2 4 14 56 󵄩󵄩 󸀠󸀠󸀠 󵄩󵄩 󵄩󵄩 󸀠󸀠 󵄩󵄩 2 (󵄩󵄩φ (u)󵄩󵄩∞ ‖∇u‖∞ + 󵄩󵄩φ (u)󵄩󵄩∞ ‖△u‖∞ )J

− 43

c󸀠 γ 37 4 3c󸀠 γ ≤ J + ( ) 4 7 28

7 󵄩 󵄩 󵄩 󵄩 (󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2 + 󵄩󵄩󵄩φ󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖△u‖∞ ) 4 .

(1.2.47)

1.3 A type of nonlinear viscoelastic wave equation

| 25

So when n = 3 and 3

J>(

4ν1 7 ) , c󸀠 γ

we have tr(L(u(t)) ⋅ QJ ) > 0. From the definition of ql it is easy to verify that −

ql ≤ 1, qJ

l ≤ J0 − 1.

So by Theorem 1.2.4 we get dH (A ) ≤ J0 ,

dF (A ) ≤ 2J0 .

1.3 A type of nonlinear viscoelastic wave equation Consider the following nonlinear wave equation with viscoelastic item: utt = auxx + σ(ux )x − f (u) + g(x),

x ∈ (0, 1),

t ∈ (0, ∞),

(1.3.1)

with the initial conditions u(0) = u0 ,

ut (0) = u1 ,

(1.3.2)

and the boundary conditions u(0, t) = u(1, t) = 0.

(1.3.3)

The problem comes from the homogeneous beam longitudinal motion, u(x, t) describes the displacement of the beam cross-section at time t. If we use T(x, t) to express the cross-section stress at time t, then equation (1.3.1) can be written as ρ0 utt = Tx − f (u) + g(x),

x ∈ (0, 1),

t ∈ (0, ∞),

(1.3.4)

where ρ0 means the density of the beam. Denote ρ0 = 1. In equation (1.3.4), we make structural assumptions on the stress T(x, t), namely T(x, t) = αuxt (x, t) + σ(ux ).

(1.3.5)

Then we obtain equation (1.3.1), where σ(s) is a smooth function, having the property σ(0) = 0,

σ 󸀠 (s) ≥ γ0 > 0,

where α and γ0 are positive constants.

∀s ∈ R,

(1.3.6)

26 | 1 Attractor and its dimension estimation Problem (1.3.1)–(1.3.3) has been studied by many mathematicians and physicists. When f = g = 0, in 1968–1969 Greenberg et al. studied the existence and stability of the global classical solution in [81, 80]. In 1981 Fitzgibben proved the existence of a global solution (u, ut ) ∈ W 1,∞ × W 1,2 in [57]. In 1984 Chang and Guo constructed its difference scheme in [22] and proved the convergence of the difference solution and the existence of a global solution. When σ 󸀠 (s) = 1, σ(ux )x = uxx , g(x) = 0, f 󸀠 (u) ≤ C, Massat [175] proved the existence of a global attractor in X α × X β and its finite dimension in 1983, where X α = D(Aα ), 0 ≤ β ≤ α < 1, A = −𝜕xx . When σ(s) is nonlinear, Berhaliev [15, 16] proved the existence of a global attractor in (E0 , E1 ) in 1985, where E0 = E1 . In 1994, Guo et al. [110] proved the existence of a compact global attractor in the space H = D(A) × L2 (Ω) for the problem (1.3.1)–(1.3.3) and obtained dimensional estimates. Due to the semigroup method, we can readily establish the local existence of a global solution for the problem (1.3.1)–(1.3.3). Theorem 1.3.1. Suppose σ(s) and f (u) are smooth functions, σ(0) = 0, g(x) ∈ L2 (Ω), Ω = [0, 1]. Then for (u0 , u1 ) ∈ H = D(A) × L2 (Ω) there exist t0 = t0 (u0 , u1 ) > 0 and u(t, u0 , u1 ) such that (u(t), ut (t)) from [0, t0 ] to H is continuous, (u(t), ut (t)) ∈ D(A) × D(A), (ut , utt ) exists for almost all t ∈ (0, t0 ) and (u(t), ut (t)) satisfies (1.3.1)–(1.3.3). We provide a priori estimate in the following. For the nonlinear functions, we make the following assumptions: (i) lim inf

|s|→∞

F(s) ≥ 0; |s|2

(1.3.7)

(ii) There exists ω > 0 such that lim

|s|→∞

sf (s) − F(s) ≥ 0, |s|2

(1.3.8)

1

where F(s) = ∫0 f (s)ds. For (i), suppose 0 < ω ≤ 21 , and let 2

1 2

‖v‖ = (∫ |v| dx) , Ω

(u, v) = ∫ u ⋅ vdx,

1 󵄩󵄩 󵄩 2 2 󵄩󵄩(u, v)󵄩󵄩󵄩 = (‖u‖H 2 + ‖v‖ ) 2 .

Ω

Lemma 1.3.1. Suppose σ(s) satisfies condition (1.3.6), f (s) satisfies conditions (1.3.7) and (1.3.8). Then for a solution of problem (1.3.1)–(1.3.3) there exist constants C1 = C1 (‖u0 , u1 ‖) and C2 = C2 (‖g‖) such that 󵄩 󵄩 ‖ux ‖2 + ‖u‖2 + ‖ut ‖2 ≤ C1 (󵄩󵄩󵄩(u0 , u1 )󵄩󵄩󵄩)e−ωt + C2 (‖g‖) is valid.

(1.3.9)

1.3 A type of nonlinear viscoelastic wave equation

| 27

Proof. Let v = ut + ρu, where ρ is a sufficiently small undetermined positive number. Then (1.3.1) becomes vt − αvxx − ρv − σ(ux )x + αρuxx + ρ2 u + f (u) = g(x).

(1.3.10)

Taking the inner product of (1.3.10) and v, we have αρ 1 d 1 2 { ‖v‖ + ∫ E(ux )dx − ‖u ‖2 2 dt 2 2 x Ω

2

ρ ‖u‖2 + ∫ F(u)dx − (g, u)} 2

+

Ω

2

+ {α‖vx ‖ − ρ‖v‖2 + ρ ∫ σ(ux )ux dx − αρ2 ‖ux ‖2 Ω

+ ρ3 ‖u‖2 + ρ ∫ f (u)udx − ρ(g, u)} = 0,

(1.3.11)

Ω s

s

where E(s) = ∫0 σ(s)ds, F(s) = ∫0 f (s)ds. Let αρ ρ2 1 H1 (u, v) = ‖v‖2 + ∫ E(ux )dx − ‖ux ‖2 + ‖ux ‖2 + ∫ F(u)dx − (g, u), (1.3.12) 2 2 2 Ω

Ω

2

2

K1 (u, v) = α‖vx ‖ − ρ‖v‖ + ρ ∫ σ(ux )ux dx Ω 2

2

3

− αρ ‖ux ‖ + ρ ‖u‖2 + ρ ∫ f (u)udx − ρ(g, u).

(1.3.13)

Then (1.3.11) can be written as d H (u, v) + K1 (u, v) = 0. dt 1

(1.3.14)

Now we estimate H1 (u, v) and K1 (u, v). Select ρ sufficiently small such that γ 2 ρ < min{ π 2 α, 0 }. 3 8α

(1.3.15)

Since σ(s) satisfies (1.3.6), we have σ(s)s ≥ E(s) ≥

γ0 2 s. 2

(1.3.16)

On the other hand, since f (u) satisfies (1.3.7) and (1.3.8), there exist constants C11 and C12 such that ∫ F(u)dx ≥ − Ω

ρ2 ‖u‖2 − C11 , 2

(1.3.17)

28 | 1 Attractor and its dimension estimation ∫[f (u)u − ωF(u)]dx ≥ − Ω

πγ0 ‖u‖2 − C12 . 16

(1.3.18)

Hence, since 0 < ω ≤ 21 , using (1.3.16)–(1.3.18), we have K1 (u, v) − ωρH1 (u, v) = α‖vx ‖2 − (ρ +

ρω )‖v‖2 + ρ ∫[σ(ux )ux − ωE(ux )]dx 2 Ω

αωρ2 ρ3 ω − (αρ2 − )‖ux ‖2 + (ρ3 − )‖u‖2 2 2 + ρ ∫[f (u)u − ωF(u)]dx + ρω(g, u) ≥ α‖vx ‖2 − (ρ + − (αρ2 −

ωρ ρ )‖v‖2 + ∫ σ(ux )ux dx 2 2

αωρ2 )‖ux ‖2 2

π 2 ργ0 ρ3 ω )‖u‖2 − ‖u‖2 − C12 ρ − ρ‖g‖‖u‖ 2 16 γ ρ 3ρ ≥ (απ 2 − )‖v‖2 + 0 ‖ux ‖2 2 4 + (ρ3 −

π 2 ργ0 󵄩󵄩 2 󵄩󵄩 󵄩u 󵄩 − C12 ρ − C(‖g‖) ≥ −C(‖g‖), 8 󵄩 󵄩 γ αρ ρ2 1 ‖ux ‖2 + ‖u‖2 − C(‖g‖) H1 (u, v) ≥ ‖v‖2 + 0 ‖ux ‖2 − 2 2 2 4 2 ρ 1 2 γ0 ≥ ‖v‖ + ‖ux ‖2 + ‖u‖2 − C(‖g‖). 2 4 4 − αρ2 ‖ux ‖2 −

(1.3.19)

(1.3.20)

From (1.3.14) and (1.3.19) we have d H (u, v) + ρωH1 (u, v) ≤ C(‖g‖). dt 1

(1.3.21)

H1 (u, v) ≤ H1 (u, v)(0)e−ωρt + C(‖g‖)(1 − e−ωρt ).

(1.3.22)

Gronwall inequality yields

Since u0 ∈ H 2 ∩ H01 (Ω), σ(s) is continuous, by the definition of H1 (u, v) and the embedding theorem, we arrive at |H1 (u, v)| ≤ C(‖(u0 , u1 )‖) and from (1.3.20) obtain the conclusion of the lemma. Now we estimate the uniform boundedness of ‖uxx ‖. Taking the inner product of (1.3.1) and uxx , we get d α { ‖u ‖2 − (ut , uxx )} + (σ 󸀠 (ux )uxx , uxx ) = ‖uxt ‖2 + (f (u) − g(x), uxx ). dt 2 xx

(1.3.23)

1.3 A type of nonlinear viscoelastic wave equation

| 29

Multiplying (1.3.23) by α and adding equation (1.3.14), we notice that 1 ‖uxt + ρux ‖2 ≥ ‖uxt ‖2 − ρ2 ‖ux ‖2 . 2

The latter implies

d H (u, ut ) + K2 (u, ut ) ≤ 0, dt 2

where H2 (u, ut ) =

α2 ‖u ‖2 − α(ut , uxx ) + H1 (u, v) 2 xx

(1.3.24)

(1.3.25)

K2 (u, ut ) = α ∫ σ 󸀠 (ux )|uxx |2 dx − α(f (u) − g, uxx ) Ω

− αρ‖ux ‖2 − ρ‖v‖2 + ρ ∫ σ(ux )ux dx Ω 2

2

3

2

− αρ ‖ux ‖ + ρ ‖u‖ + ρ ∫ f (u)udx − ρ(q, u).

(1.3.26)

Ω

Choosing ρ as in (1.3.15), we now get αγ α2 ρ α2 ρ α ‖uxx ‖2 ≥ ( 0 − )‖uxx ‖2 ≥ 0, ∫ σ 󸀠 (ux )|uxx |2 dx − 2 2 2 2 Ω

󵄨 󵄨 αγ αρ󵄨󵄨󵄨(ut , uxx )󵄨󵄨󵄨 ≤ 0 ‖uxx ‖2 + C‖ut ‖2 , 8 󵄩 󵄩2 󵄨󵄨 󵄨󵄨 αγ0 ‖uxx ‖2 + C(󵄩󵄩󵄩f (u)󵄩󵄩󵄩 + ‖g‖2 ). α󵄨󵄨(f (u) − g, uxx )󵄨󵄨 ≤ 8

Thus there exist C21 and C22 such that

󵄩 󵄩 󵄩 󵄩 K2 (u, ut ) − ρH2 (u, ut ) ≥ −C21 (‖ut ‖2 + ‖u‖2 + ‖ux ‖2 + 󵄩󵄩󵄩F(u)󵄩󵄩󵄩∞ + 󵄩󵄩󵄩f (u)󵄩󵄩󵄩∞ ) − C22 (‖g‖) ≥ −C3 (‖(u0 , u1 ‖) − C4 (‖g‖).

(1.3.27)

By Lemma 1.3.1, (1.3.27) and (1.3.24), we get d 󵄩 󵄩 H (u, ut ) + ρH2 (u, ut ) ≤ C3 󵄩󵄩󵄩(u0 , u1 )󵄩󵄩󵄩 + C4 (‖g‖). dt 2

(1.3.28)

󵄩 󵄩 H2 (u, ut ) ≤ H2 (u, ut )(0)e−ρt + ρ−1 (C3 (󵄩󵄩󵄩(u0 , u1 )󵄩󵄩󵄩) + C4 ).

(1.3.29)

Hence

Since H2 (u, ut )(0) ≤ C(‖(u0 , u1 )‖), through formula (1.3.29) we know that H2 (u, v) is uniformly bounded. On the other hand, H2 (u, ut ) ≥

α2 󵄩 󵄩 ‖u ‖2 − C(‖ux ‖2 + ‖u‖2 + ‖ut ‖2 + 󵄩󵄩󵄩F(u)󵄩󵄩󵄩∞ + ‖g‖). 4 xx

(1.3.30)

By the embedding theorem and Lemma 1.3.1, we know that ‖uxx ‖ is uniformly bounded. Thus we have

30 | 1 Attractor and its dimension estimation Theorem 1.3.2. Suppose σ(s) and f (u) satisfy (1.3.6) and (1.3.8), respectively. Then for any (u0 , u1 ) ∈ H = D(A) × L2 (Ω), problem (1.3.1)–(1.3.3) has a global solution (u, ut ) ∈ C(0, ∞; H ). From Theorems 1.3.1 and 1.3.2 we know that problem (1.3.1)–(1.3.3) generates a nonlinear semigroup S(t) on H , S(t)(u0 , u1 ) = (u(t), ut (t)), where u(t) is the unique solution of problem (1.3.1)–(1.3.3), and semigroup S(t) is continuous in t and (u0 , u1 ). We prove that S(t) has bounded absorbing sets in H . Proposition 1.3.1. Under the assumption of Theorem 1.3.2, there exists a constant M0 such that for any solution (u(t), ut (t)) of problem (1.3.1)–(1.3.3) with the initial value (u0 , u1 ) ∈ H satisfying ‖(u0 , u1 )‖ ≤ R there exists a T = T(R) > 0 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩S(t)(u0 , u1 )󵄩󵄩󵄩 = 󵄩󵄩󵄩(u(t), ut (t))󵄩󵄩󵄩 ≤ M0 ,

t ≥ T(R).

In other words, B = {(u1 , u2 ) ∈ H | ‖(u1 , u2 )‖ ≤ M0 } is a bounded absorbing set of S(t) in H . Proof. From Lemma 1.3.1 we know that there exist constants ρ0,∞ = ρ0,∞ (‖g‖) and T0 = T0 (R) such that ‖ux ‖2 + ‖u‖2 + ‖ut ‖2 ≤ ρ20,∞ ,

t ≥ T0 (R).

(1.3.31)

By Sobolev embedding theorem, ‖u‖∞ ≤ ‖ux ‖ ≤ ρ0,∞ , t ≥ T0 (R). Via equation (1.3.29) we have K2 (u, ut ) − ρH2 (u, ut ) ≥ −Cρ20,∞ − C‖g‖2 ≡ ρ21,∞ ,

t ≥ T0 (R).

(1.3.32)

From equations (1.3.24) and (1.3.31), d H (u, ut ) + ρH2 (u, ut ) ≤ ρ21,∞ , dt 2

t ≥ T0 (R),

(1.3.33)

thus H2 (u, ut ) ≤ H2 (u, ut )(0)e−ρt + ρ−1 ρ21,∞ ,

t ≥ T0 (R).

(1.3.34)

From equation (1.3.30) we know that there exists a constant ρ2,∞ such that ‖uxx ‖2 ≤ ρ22,∞ ,

t ≥ T1 (R).

(1.3.35)

Using equations (1.3.31) and (1.3.35), we finish the proof. In order to prove the existence of a compact global attractor of S(t), we must decompose S(t).

1.3 A type of nonlinear viscoelastic wave equation

| 31

Proposition 1.3.2. The semigroup S(t) defined by the problem (1.3.1)–(1.3.3) can be decomposed as S(t) = C(t) + V(t),

(1.3.36)

where V(t), when t > t0 , is consistent relatively compact; C(t) is a continuous mapping from H to H , and for any bounded set B1 ⊂ H it satisfies 󵄩 󵄩 rc (t) = sup 󵄩󵄩󵄩C(t)ϕ󵄩󵄩󵄩 → 0, ϕ∈B1

t → ∞.

(1.3.37)

Proof. Suppose (u0 , u1 ) ∈ B1 = {(u0 , u1 ) | ‖(u0 , u1 )‖ ≤ R}. Let v(x, t) ∈ L∞ (0, ∞; L2 (Ω)) ∩ L2 (0, T; H01 (Ω)), ∀T > 0 be the unique solution of the following problem: (P1 )

{vt − αvxx = 0, { v(x, 0) = u1 , { { {v(0, t) = v(1, t) = 0.

Also let v(x, t) be the unique solution of the following problem: (P2 )

v − αvxx = (σ(ux ))x − f (u) + g, { { t v(x, 0) = 0, { { { v(0, t) = v(1, t) = 0.

By the uniqueness of the solution of (P2 ), we deduce that u1 = v +v. On the other hand, from (P1 ) we have 󵄩󵄩 󵄩2 2 −απ 2 t . 󵄩󵄩v(t)󵄩󵄩󵄩 ≤ ‖u1 ‖ e

(1.3.38)

Since (u, ut ) ∈ L∞ (0, ∞; L2 (Ω)), by Proposition 1.3.1 we know that G(x, t) = σ(ux )x − f (u) + g(x) ∈ L∞ (0, ∞; L2 (Ω)) and 󵄩󵄩 󵄩 󵄩 󸀠 󵄩 󵄩 󵄩 󵄩󵄩G(x, t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩σ (ux )󵄩󵄩󵄩∞ ‖uxx ‖ + 󵄩󵄩󵄩f (u)󵄩󵄩󵄩 + ‖g‖ ≤ C(M0 ),

t ≥ T0 (R).

From the problem (P2 ) we know that v(x, t) ∈ L∞ (0, ∞; H01 (Ω)) and ‖v‖2 + ‖vx ‖2 ≤ C‖G‖2 (1 − e−

απ 2 2

t

) ≤ C(R),

t > T.

Set ω(x, t) ∈ L∞ (0, ∞; H 2 ∩ H01 ) to be the solution of the following problem:

(P3 )

ωu − αωxxt + α−1 σ 󸀠 (ux )(ωt − αωxx ) { { { { = α−1 σ 󸀠 (ux )v(x, t) ≜ F(x, t), { { {ω(0) = u0 , ωt (0) = u1 , { {ω(0, t) = ω(1, t) = 0,

(1.3.39)

32 | 1 Attractor and its dimension estimation where v(x, t) is the solution of (P1 ). It is easy to prove that the solution of (P3 ) is unique. In fact, let θ(x, t) be the unique solution of the following problem: 𝜕θ { { + α−1 σ 󸀠 (ux )θ = F(x, t), 𝜕t { { θ(x, 0) = u1 − αu0xx ∈ L2 . {

(P4 ) Then θ can be expressed as θ(x, t) = e−α

−1

+e

t

∫0 σ 󸀠 (ux (x,τ)dτ) t

(u1 − αu0xx )

−α−1 ∫0 σ 󸀠 (ux (x,τ)dτ)

t

⋅ ∫ F(x, τ)eα

−1

t

∫0 σ 󸀠 (ux (x,s)ds)

dτ

(1.3.40)

0

From formula (1.3.40), we know that θ ∈ L∞ (0, ∞; L2 (Ω)), θt ∈ L∞ (0, ∞; L2 (Ω)), and there exist constants β > 0 and C > 0 such that ‖θ‖2 ≤ C(‖u1 ‖2 + ‖u0xx ‖2 )e−βt ,

‖θt ‖2 ≤ C(‖u1 ‖2 + ‖u0xx ‖2 )e−βt .

(1.3.41)

Now, we solve the following problem (P5 )

ω − αωxx = θ(x, t); { { t ω(x, 0) = u0 ; { { {ω(0, t) = ω(1, t) = 0.

then ω(x, t) ∈ L∞ (0, ∞; H 2 ∩ H01 ), ωt (x, t) ∈ L∞ (0, ∞; L2 (Ω)), ω(x, t) is the solution of problem (P3 ). Obviously, the problem (P3 ) has a unique solution. By the virtue of (1.3.41) and (P5 ), there exist constants β2 and C = C(‖(u0 , u1 )‖), s. t. 󵄩󵄩 󵄩 󵄩 󵄩 −β t −β t 󵄩󵄩(ω, ωt )󵄩󵄩󵄩 ≤ C(󵄩󵄩󵄩(u0 , u1 )󵄩󵄩󵄩)e 2 ≤ C(R)e 2 .

(1.3.42)

Finally, we consider the following problem: (P6 )

(ω − αωxx )t + α−1 σ 󸀠 (ux )(ωt − αωxx ) = F; { { t {ω(x, 0) = 0, ωt (x, 0) = 0; { {ω(0, t) = ω(1, t) = 0.

where F(x, t) = α−1 σ 󸀠 (ux )ut − α−1 σ 󸀠 (ux )v − f (u) + g = α−1 σ 󸀠 (ux )v(x, t) − f (u) + g Since (u, ut ) ∈ H , v(x, t) ∈ L∞ (0, ∞; H01 ), and it satisfies (1.3.39), we can get F(x, t) ∈ L∞ (0, ∞; H 1 ), and 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩F(x, t)󵄩󵄩󵄩 + 󵄩󵄩󵄩F(x, t)󵄩󵄩󵄩 ≤ C(R0 ).

(1.3.43)

1.3 A type of nonlinear viscoelastic wave equation

| 33

Defining θ(x, t) = e−α

−1

t

∫0 σ 󸀠 (ux (x,τ))dτ

t

⋅ ∫ F(x, t)e−α

−1

t

∫0 σ 󸀠 (ux (x,s))ds

dτ

(1.3.44)

0

we know θ(x, t) ∈ L∞ (0, ∞; H 1 ), θt (x, t) ∈ L∞ (0, ∞; H 1 ), and ‖θ‖2H 1 + ‖θt ‖2H 1 ≤ C(R). Now we solve the following problem ω − αωxx = θ(x, t); { { t ω(x, 0) = 0, { { {ω(0, t) = ω(1, t) = 0. Then ω(x, t) ∈ H 3 ∩ H01 is the unique solution of (P5 ), and we have 󵄩󵄩 󵄩 󵄩󵄩ω(x, t)󵄩󵄩󵄩H 3 ≤ C(R).

(1.3.45)

By the uniqueness of solution of problem (P6 ), we get u(x, t) = ω(x, t) + ω(x, t). Now we define C(t)(u0 , u1 ) = (ω, v),

U(t)(u0 , u1 ) = (ω, v). Then using (1.3.38), (1.3.39), (1.3.42) and (1.3.45), we obtain S(t) = C(t) + U(t), where C(t) and U(t) satisfy the required properties of Proposition 1.3.2. From Propositions 1.3.1, 1.3.2 and Theorem 1.1.1 in [197], we get Theorem 1.3.3. The semigroup operator S(t) defined by problem (1.3.1)–(1.3.3) has a global attractor in H , which attracts all the bounded sets of H . In the following, we estimate the dimension of the attractor A . Set ξ0 = (u0 , u1 ) ∈ H , and let u(t) be the solution of the corresponding problem (1.3.1)–(1.3.3), that is, S(t)ξ0 = (u(t), ut (t)). It is easy to prove the consistent differentiability of S(t) in a similar way.

Proposition 1.3.3. Set ξ01 = (u10 , u11 ), ξ02 = (u20 , u21 ) ∈ H , and ‖ξ01 ‖ ≤ R, ‖ξ02 ‖ ≤ R. Then for all T, R, 0 < T, R < +∞, there exists a constant C = C(R, T) such that 󵄩󵄩 󵄩 1 1 2 󵄩2 2󵄩 󵄩󵄩S(t)ξ0 − S(t)ξ0 󵄩󵄩󵄩 ≤ C(R, T)󵄩󵄩󵄩ξ0 − ξ0 󵄩󵄩󵄩,

0 ≤ t ≤ T.

(1.3.46)

Now we consider the linear variational problem for problem (1.3.1)–(1.3.3): Utt − Uxxt − (σ 󸀠 (ux )Ux )x + f 󸀠 (u)U = 0,

(1.3.47)

34 | 1 Attractor and its dimension estimation U(x, 0) = ω0 ,

Ut (x, 0) = ω1 ,

(1.3.48)

U(0, t) = U(1, t) = 0,

(1.3.49)

where (u, ut ) = S(t)ξ0 and η0 = (ω0 , ω1 ) ∈ H . Since S(t)ξ0 ∈ C(R+ ; H ), the linear problem (1.3.47)–(1.3.49) has a unique solution (U(t), Ut (t)) ∈ C(R+ ; H ). We can prove that (DS(t)ξ0 )η0 = (U(t), Ut (t)),

(1.3.50)

where S(t) is consistently differentiable at ξ0 . Proposition 1.3.4. For any R and T, 0 < R, T < ∞, there is a constant C(R, T) such that for any ξ0 = (u0 , u1 ), h0 = (h01 , h02 ) satisfying ‖ξ0 ‖ ≤ R, ‖ξ0 + h0 ‖ ≤ R, t ≤ T, we have 󵄩󵄩 󵄩 2 󵄩󵄩S(t)(ξ0 + h0 ) − S(t)ξ0 − (DS(t)ξ0 )h0 󵄩󵄩󵄩 ≤ C(R, T)‖h0 ‖ .

(1.3.51)

Take η10 , η20 , . . . , ηm 0 ∈ H and study the evolution of Gram determinant 󵄩󵄩 1 󵄩 2 m i j 󵄩󵄩η (t) ∧ η (t) ∧ ⋅ ⋅ ⋅ ∧ η (t)󵄩󵄩󵄩∧m (H ) = det ((η , η )), 1≤i,j≤m

(1.3.52)

j

where ηj (t) = (DS(t)ξ0 )η0 , and ((⋅, ⋅)) means the inner product in H . We will prove, for sufficiently large m and when t → ∞, that the determinant (1.3.52) exponentially decays to zero. Theorem 1.3.4. Set f 󸀠 (u) ≥ 0. If A is the global attractor of problem (1.3.1)–(1.3.3) then there are constants u > 0, C1 > 0 and C2 > 0 such that for any ξ0 ∈ A , m ≥ 1 and t ≥ 0, we have 󵄩󵄩 1 2 m󵄩 󵄩󵄩(DS(t)ξ0 )η ∧ (DS(t)ξ0 )η ∧ ⋅ ⋅ ⋅ ∧ (DS(t)ξ0 )η 󵄩󵄩󵄩∧m (H ) 󵄩 󵄩󵄩 m ≤ 󵄩󵄩󵄩η10 ∧ η20 ∧ ⋅ ⋅ ⋅ ∧ ηm 0󵄩 󵄩∧m (H ) C1 exp(C2 √m − μm)t,

∀ηi0 ∈ H .

(1.3.53)

Proof. Firstly, we consider an equivalent norm on H . Let ξ0 ∈ A , (u, ut ) = S(t)ξ0 and η(t) = (U(t), Ut (t)) = (DS(t)ξ0 )η0 , where η0 = (ω0 , ω1 ) ∈ H . Then U(t) satisfies (1.3.46)–(1.3.49), setting ω(t) = eμt U(t), where μ is a positive number to be determined, and ω(t) satisfies ωtt − αωxxt − 2μωt − (σ 󸀠 (ux )ωx )x + αμωxx + μ2 ω + f 󸀠 (u)ω = 0.

(1.3.54)

Multiplying (1.3.54) by ωt and integrating over Ω, we have αμ μ2 d 1 1 1 { ‖ωt ‖2 + ∫ σ 󸀠 (ux )|ωx |2 dx − ‖ωx ‖2 + ‖ω‖2 + ∫ f 󸀠 (u)|ω|2 dx} dt 2 2 2 2 2 Ω

Ω

1 1 = −α‖ωxt ‖2 + 2μ‖ωt ‖2 + ∫ σ 󸀠󸀠 (ux )uxx |ωx |2 dx − ∫ f 󸀠󸀠 (u)ut |ω|2 dx. 2 2 Ω

Ω

(1.3.55)

1.3 A type of nonlinear viscoelastic wave equation

| 35

Multiplying (1.3.54) by ωxx and integrating over Ω, we obtain μ d α { ‖ω ‖2 − (ωx , ωxx ) − ‖ωx ‖2 } dt 2 xx 2 = ‖ωxt ‖2 − ∫ σ 󸀠 (ux )|ωxx |2 dx − ∫ σ 󸀠󸀠 (ux )uxx ωx ωxx dx Ω

Ω 2

2

2

+ αμ‖ωxx ‖ − μ ‖ωx ‖ − ∫ f (u)|ωx |2 dx − ∫ f 󸀠󸀠 (u)ux ωωx dx. 󸀠

Ω

(1.3.56)

Ω

Multiplying (1.3.55) by a constant k and then adding (1.3.56) yields d d J(ξ (t)) = J(ω, ωt ) = K(ξ (t)) = K(ω, ωt ), dt dt

(1.3.57)

where ξ (t) = (ω, ωt ) ∈ H , J(ξ ) =

μ α ‖ξ ‖2 − (ξ2 , ξ1xx ) − ‖ξ1x ‖2 2 1xx 2 k k + ‖ξ2 ‖2 + [∫ σ 󸀠 (ux )ξ1x − αμ‖ξ1x ‖2 ] 2 2 +

Ω

2

μk k ‖ξ ‖2 + ∫ f 󸀠 (u)|ξ1 |2 dx, 2 1 2

(1.3.58)

Ω

2

K(ξ ) = (−αk + 1)‖ξ2x ‖ + 2μk‖ξ2 ‖2 − ∫ σ 󸀠 (ux )|ξ1xx |2 dx Ω

2 + αμ‖ξ1xx ‖ − μ ‖ξ1x ‖ + ∫ σ 󸀠󸀠 (ux )uxx |ξ1x |2 dx k 2

2

2

Ω

k + ∫ f 󸀠󸀠 (u)ut |ξ1 |2 dx − ∫ f 󸀠 (u)|ξ1x |2 dx 2 Ω

Ω

− ∫ f (u)ux ξ1 ξ1x dx − ∫ σ 󸀠󸀠 (ux )uxx ξ1x ξ1xx dx 󸀠󸀠

Ω

(1.3.59)

Ω

for ξ = (ξ1 , ξ2 ) ∈ H and ξ2x ∈ L2 (Ω). Now we take k = 0 < u < min(

1 α

and μ such that

γ0 απ 2 , , 1). 2α 4

(1.3.60)

Since f 󸀠 (u) ≥ 0, we have J(ξ ) ≥

α α 1 k ‖ξ ‖2 − ‖ξ1xx ‖2 − ‖ξ2 ‖2 + ‖ξ2 ‖2 2 1xx 4 2 2 kγ0 μ kαμ + ‖ξ ‖2 − ‖ξ1x ‖2 − ‖ξ ‖2 4 1x 2 4 1x

(1.3.61)

36 | 1 Attractor and its dimension estimation

+

μ2 k k ‖ξ ‖2 + ∫ f 󸀠 (u)|ξ1 |2 dx 2 1 2 Ω

γ 2μ2 1 α ‖ξ ‖2 . ≥ ‖ξ1xx ‖2 + ‖ξ2 ‖2 + 0 ‖ξ1x ‖2 + 4 α 4α α 1

(1.3.62)

Since S(t)ξ0 = (u(t), ut (t)) ∈ A , the norms ‖ux ‖∞ , ‖u‖∞ , ‖uxx ‖ and ‖u‖ are uniformly bounded. By the definitions (1.3.58) and (1.3.61), there exist positive constants k0 and k1 such that k0 ‖ξ0 ‖ ≤ J(ξ ) ≤ k1 ‖ξ ‖2 .

(1.3.63)

1

So J(ξ ) 2 is an equivalent norm of H . As for K(ξ ), we have 󵄨󵄨 k 󵄨󵄨 󵄨󵄨 󵄨 󸀠󸀠 2 󵄨󵄨 ∫ σ (ux )uxt |ξ1x | dx 󵄨󵄨󵄨 󵄨󵄨 2 󵄨󵄨 Ω

󵄨󵄨 󵄨󵄨 k 󵄨 󵄨 = 󵄨󵄨󵄨− ∫ σ 󸀠󸀠󸀠 (ux )uxx ut |ξ1x |2 dx − k ∫ σ 󸀠󸀠 (ux )ut ξ1x ξ1xx dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 Ω

Ω

≤ C(‖ξ1x ‖2∞ ) + ‖ξ1x ‖‖ξ1xx ‖ 3

1

≤ C(‖ξ1 ‖2H 2 + ‖ξ2 ‖2 ) 4 ‖ξ1x ‖ 2 . The three terms on the right-hand side of K(ξ ) can be estimated similarly. Since k and μ are selected in (1.3.63), the first three terms of K(ξ ) ≤ 0, i. e., (−αk + 1)‖ξ2x ‖2 + 2μk‖ξ2 ‖2 − ∫ σ 󸀠 (ux )|ξ1xx |2 dx + αμ|ξ1xx |2 dx ≤ 0. Ω

Thus we have 3

1

K(ξ ) ≤ C(‖ξ1 ‖2H 2 + ‖ξ2 ‖2 ) 4 ‖ξ1x ‖ 2 .

(1.3.64)

To estimate the variation of the m-dimensional volume, we need the following lemma: Lemma 1.3.2. Suppose ϕ(⋅, ⋅) and ϕ1 (⋅, ⋅) are two inner products in a Hilbert space, which are continuous and equivalent, i. e., αϕ(ξ , ξ ) ≤ ϕ1 (ξ , ξ ) ≤ βϕ(ξ , ξ ),

∀ξ ∈ H.

(1.3.65)

Then we have αm det ϕ(ξ i , ξ j ) ≤ det ϕ1 (ξ i , ξ j ) ≤ βm det ϕ(ξ i , ξ j ), 1≤i,j≤m

1≤i,j≤m

1≤i,j≤m

∀ξ i ∈ H, i = 1, 2, . . . , m. (1.3.66)

1.3 A type of nonlinear viscoelastic wave equation

| 37

Set ωi (t) = eμt U i (t), i = 1, 2, . . . , m. Then (U i (t), Uti (t)) = (DS(t)ξ0 )ηi0 . Let ηi (t) = (ω (t), ωit − μωi ), ξ i (t) = (ωi (t), ωit (t)), and then i

󵄩2 󵄩󵄩 l 2 m 󵄩󵄩η (t) ∧ η (t) ∧ ⋅ ⋅ ⋅ ∧ η (t)󵄩󵄩󵄩∧m (H ) 󵄩 󵄩 = e−2μmt 󵄩󵄩󵄩ηl (t) ∧ η2 (t) ∧ ⋅ ⋅ ⋅ ∧ ηm (t)󵄩󵄩󵄩∧m (H ) = e−2μmt det ((ηi (t), ηj (t))) 1≤i,j≤m

=e

−2μmt

det ϕ1 (ξ i (t), ξ j (t)),

(1.3.67)

1≤i,j≤m

where ϕ1 (ξ , η) = ((ξ1 , η1 ))2s + (ξ2 − μξ1 , η2 − μη1 ), ξ = (ξ1 , ξ2 ), η = (η1 , η2 ) ∈ H . Since μ < 1, we know that ϕ1 (⋅, ⋅) satisfies the condition of Lemma 1.3.2. Hence there exists a constant C such that det ϕ1 (ξ i (t), ξ j (t)) ≤ C det ((ξ i (t), ξ j (t))).

1≤i,j≤m

(1.3.68)

1≤i,j≤m

In order to estimate det1≤i,j≤m ((ξ i (t), ξ j (t))), we introduce the following bilinear form on H , for any ξ = (ξ1 , ξ2 ), η = (η1 , η2 ) ∈ H : α (ξ , η ) 2 1xx 1xx μ 1 = [(ξ2 , η1xx ) + (η2 , ξ1xx )] − (ξ1x , η1x ) 2 2 αμk k k 󸀠 + (η2 , ξ2 ) + ∫ σ (ux )ξ1x η1x dx − (ξ , η ) 2 2 2 1x 1x

ϕ(ξ , η) =

Ω

μ2 k k + (ξ , η ) + ∫ f 󸀠 (u)ξ1 η1 dx, 2 1 1 2

(1.3.69)

Ω

1

and then ϕ(η, η) = J(η). From equation (1.3.63), we know that ϕ(η, η) 2 is an equivalent norm. Hence, by Lemma 1.3.2, we have k0m det ((ξ i (t), ξ j (t))) ≤ det ϕ(ξ i (t), ξ j (t)) ≤ k1m det ((ξ i (t), ξ j (t))). 1≤i,j≤m

1≤i,j≤m

(1.3.70)

1≤i,j≤m

Moreover, we need to estimate Hm (t) = det1≤i,j≤m ϕ(ξ i (t), ξ j (t)). Similarly as in the proof in [74], we have j m K(∑m dHm (t) j=1 xj ξ (t)) = Hm (t) ∑ max min . m j ̸ J(∑ dt F⊂Rm ,dim F=l, x∈F,x=0 j=1 xj ξ (t)) l=1

(1.3.71)

From equations (1.3.63) and (1.3.64), we have 3

1

󵄩󵄩 2 j 2 󵄩 m 󵄩m ‖ ∑m 󵄩󵄩 dHm (t) C j=1 xj ξ (t)‖H 󵄩 󵄩󵄩 j 󵄩󵄩 ≤ Hm (t) ∑ max min x ξ (t) ∑ 󵄩 j m 󵄩 󵄩󵄩 2 j m 󵄩 ̸ ‖∑ dt k0 F⊂R ,dim F=l, x∈F,x =0 󵄩󵄩 j=1 xj ξ (t)‖H 󵄩 l=1 󵄩 j=1

38 | 1 Attractor and its dimension estimation 1

m

≤ CHm (t) ∑ l=1

max

m

1

l=1

λl4

≤ CHm (t) ∑

min

̸ F⊂Rm ,dim F=l, x∈F,x=0

1

j 2 ‖ ∑m j=1 xj ξ (t)‖ 1

2 j ‖ ∑m j=1 xj ξ (t)‖H

≤ C √mHm (t).

(1.3.72)

Since an eigenvalue of A is λl = π 2 l2 , we have Hm (t) ≤ Hm (0)eC√mt .

(1.3.73)

Now using (1.3.67), (1.3.70) and (1.3.72), we complete the proof of the theorem. As a corollary of Theorem 1.3.4, we have Theorem 1.3.5. If f 󸀠 (u) ≥ 0 then the attractor determined by Theorem 1.3.4 has finite fractal and Hausdorff dimensions. Proof. With the aid of Theorem 1.3.4 we know that ∀ξ0 ⊂ A , ωm (DS(t)ξ0 ) ≤ C1m exp(C2 √m − μm)t, where ωm is a Lyapunov index. So ωm (A ) < 1. If m is large enough, say m > ( Cμ2 )2 , then ωm (A ) is a consistent Lyapunov index of A . Using Lemma V.3.1 in [197], we can deduce that A has finite fractal and Hausdorff dimensions.

1.4 Coupled KdV equations Now we consider the following coupled Korteweg–de Vries (KdV) equations: ut = uxxx + 6uux + 2vvx ,

(1.4.1)

vt = 2(uv)x ,

(1.4.2)

which were proposed by Kupershmidt [156] in 1985 and could describe the interaction of two long waves. Ito proposed the operator method to give the infinite symmetries and motion constants in [144]. In 1991, Guo and Tan proved the existence and uniqueness of a global smooth solution for the above equation [113]. In 1996, Guo and Yang considered the corresponding dissipation equation and proved the existence and finiteness of dimension of the global attractor [128, 127]. And the other similar resutls on the well-posedness can refer to the reference [129, 130, 131, 132]. Now we consider the following dissipation coupled KdV equations with periodic initial value problem: ut + f (u)x − αuxx + βuxxx + 2vvx = G1 (u, v) + h1 (x),

vt − γvxx + 2(uv)x = G2 (u, v) + h2 (x),

x ∈ R,

t ≥ 0,

(1.4.3) (1.4.4)

1.4 Coupled KdV equations | 39

u(x + D, t) = u(x − D, t), u(x, 0) = u0 (x),

v(x + D, t) = v(x − D, t),

v(x, 0) = v0 (x),

∀x ∈ R,

t ≥ 0,

x ∈ R,

(1.4.5) (1.4.6)

where D > 0, α > 0, β ≠ 0 and γ > 0 are constants. We first make a uniform priori estimates for problem (1.4.3)–(1.4.6) which are independent of t, and then prove the global existence of the attractor. Finally, we estimate an upper bound of its dimension. Lemma 1.4.1. Suppose (1) Gi (0, 0) = 0, i = 1, 2, and for any (ξ , η) ∈ R2 , −G (ξ , η) ( 1u −G2u

−G1v ξ ) ( ) ≥ b0 (ξ 2 + η2 ); −G2v η

where b0 > 0 is a constant; (2) u0 (x), v0 (x) ∈ L2 (Ω), hi (x) ∈ L2 (Ω), i = 1, 2, Ω = (−D, D); then the following estimate for problem (1.4.3)–(1.4.6) holds: ‖u‖2 + ‖v‖2 ≤ e−b0 t (‖u0 ‖2 + ‖v0 ‖2 ) +

1 (1 − e−b0 t )(‖h1 ‖2 + ‖h2 ‖2 ). b20

(1.4.7)

Furthermore, we have 1 󵄩 󵄩2 󵄩 󵄩2 lim (󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩 + 󵄩󵄩󵄩v(⋅, t)󵄩󵄩󵄩 ) ≤ 2 (‖h1 ‖2 + ‖h2 ‖2 ) ≡ E0 , b0

t→∞ t

1 1 󵄩 󵄩2 󵄩 󵄩2 (‖h1 ‖2 + ‖h2 ‖2 ). lim ∫[α󵄩󵄩󵄩ux (⋅, τ)󵄩󵄩󵄩 + γ 󵄩󵄩󵄩vx (⋅, τ)󵄩󵄩󵄩 ]dτ ≤ t→∞ t 2b0

(1.4.8) (1.4.9)

0

Proof. Taking the inner product of equation (1.4.3) and u, as well as of (1.4.3) and ν, we get (u, ut + f (u)x − αuxx + βuxxx + 2vvx ) = (u, G1 (u, v) + h1 ),

(v, vt − γuxx + 2(uv)x ) = (v, G2 (u, v) + h2 ),

(1.4.10) (1.4.11)

D

where (u, ω) = ∫−D u(x, t)ω(x, t)dx and (u, f (u)x ) = 0,

(u, −αuxx ) = α‖ux ‖2 ,

(u, 2vvx ) + (u, 2(uv)x ) = 0,

(u, uxxx ) = 0,

(u, G1 (u, v)) + (v, G2 (u, v)) ≤ −b0 (‖u‖2 + ‖v‖2 ). From equations (1.4.10) and (1.4.11) we get b 1 d 1 (‖u‖2 + ‖v‖2 ) + α‖ux ‖2 + γ‖vx ‖2 + 0 (‖u‖2 + ‖v‖2 ) ≤ ((‖h1 ‖2 + ‖h2 ‖2 )) (1.4.12) 2 dt 2 2b0 Inequality (1.4.12) now yields (1.4.8) and (1.4.9).

40 | 1 Attractor and its dimension estimation Lemma 1.4.2. Under the conditions of Lemma 1.4.1, assume that (1) f ∈ C 2 , Gi ∈ C 1 , i = 1, 2 and 󵄨 󵄨󵄨 5−δ 󵄨󵄨f (u)󵄨󵄨󵄨 ≤ A|u| , δ > 0, A > 0, 󵄨 󵄨󵄨 5 5 󵄨󵄨Gi (u, v)󵄨󵄨󵄨 ≤ Bi (|u| + |v| ), Bi > 0, i = 1, 2; (2) if u0x , v0x ∈ L2 (Ω), then ‖ux ‖2 + ‖vx ‖2 ≤ 2e−2b0 t (‖u0x ‖2 + ‖v0x ‖2 − t

2 ∫ F(u0 (x))dx) β

+ 2e−2b0 t ∫ C1 e2b0 s ds 0

+

1 4 3 (1 − e−2b0 t )( ‖h2 ‖2 + ‖h1 ‖2 ) + C2 b0 γ α

(1.4.13)

where the functions C1 and C2 depend on ‖u‖ and ‖v‖. Moreover, we obtain that 󵄩 󵄩2 󵄩 󵄩2 lim (󵄩󵄩󵄩ux (⋅, t)󵄩󵄩󵄩 + 󵄩󵄩󵄩vx (⋅, t)󵄩󵄩󵄩 )

t→∞

≤

1 1 4 3 max C + ( ‖h ‖2 + ‖h1 ‖2 ) + max C2 ≜ E1 , t≥0 b0 t≥0 1 b0 γ 2 α

(1.4.14)

t

1 󵄩 󵄩2 󵄩 󵄩2 lim ∫[α󵄩󵄩󵄩uxx (⋅, τ)󵄩󵄩󵄩 + γ 󵄩󵄩󵄩vxx (⋅, τ)󵄩󵄩󵄩 ]dτ t→∞ t 0

≤ max(b0 C2 + C1 ) + t≥0

4 3 ‖h ‖2 + ‖h1 ‖2 . γ 2 α

(1.4.15)

Proof. Taking the inner product of (1.4.3) and uxx , we obtain (uxx , ut + f (u)x − αuxx + βuxx + 2vvx ) = (uxx , G1 (u, v) + h1 (x)), where (uxx , f (u)x ) = −(uxxx , f (u)) 1 = (ut + f (u)x − αuxx + 2vvx − G1 (u, v) − h1 , f (u)) β 1 d α = ∫ F(u(x, t))dx − (uxx , f (u)) β dt β 2 1 + (vvx , f (u)) − (G1 + h1 , f ) β β u

and F(u) = ∫0 f (s)ds. By Sobolev interpolation inequality, we have 󵄨󵄨 󵄨 󵄩 󵄩 5−δ 󵄨󵄨(uxx , f (ω))󵄨󵄨󵄨 ≤ ‖uxx ‖󵄩󵄩󵄩f (u)󵄩󵄩󵄩 ≤ A‖uxx ‖‖u‖2(5−δ) |β| ≤ ‖u ‖2 + C(|u‖), 12 xx

(1.4.16)

1.4 Coupled KdV equations | 41

󵄨󵄨 󵄨󵄨 2 2 󵄨 󵄩 󵄨󵄨 󵄩 ‖v‖ ‖v ‖ 󵄩󵄩f (u)󵄩󵄩󵄩2 󵄨󵄨 (vvx , f (u))󵄨󵄨󵄨 ≤ 󵄨󵄨 |β| 4 x 4 󵄩 󵄨󵄨 β γ α ≤ ‖uxx ‖2 + ‖vxx ‖2 + C(‖u‖, ‖v‖), 12 8 󵄨󵄨 1 󵄨󵄨 AB 󵄨󵄨 󵄨 5−δ 5 1 [‖u‖10−δ 󵄨󵄨 (G1 , f )󵄨󵄨󵄨 ≤ 10−δ + ‖u‖2(5−δ) ‖v‖10 ] 󵄨󵄨 β 󵄨󵄨 |β| γ α ≤ ‖uxx ‖2 + ‖vxx ‖2 + C(‖u‖, ‖v‖), 12 8 󵄨󵄨 1 󵄨󵄨 󵄨󵄨󵄨 (h1 , f (u))󵄨󵄨󵄨 ≤ 1 ‖h1 ‖‖f ‖ ≤ α ‖uxx ‖2 + C, 󵄨󵄨 β 󵄨󵄨 |β| 12 󵄨 󵄨 α 3 󵄨󵄨 󵄨󵄨 2 2 󵄨󵄨(uxx , h1 )󵄨󵄨 ≤ ‖uxx ‖‖h1 ‖ ≤ ‖uxx ‖ + ‖h1 ‖ . 12 α Due to equation (1.4.16), we have 2 7α 1 d [‖ux ‖2 − ∫ F(u)dx] + ‖uxx ‖2 − 2(uxx , vvx ) 2 dt β 12 γ 3 ≤ −(uxx , G1 (u, v)) + ‖vxx ‖2 − ‖h1 ‖2 + C, 4 α

(1.4.17)

where C depends on ‖u‖, ‖v‖ and ‖h1 ‖. Taking the inner product of (1.4.4) and vxx gives (vxx , vt − γvxx + 2(uv)x ) = (vxx , G2 (u, v) + h2 ), hence we obtain 1 d ‖v ‖2 + γ‖vxx ‖2 − 2(vxx , (uv)x ) = −(vxx , G2 ) − (vxx , h2 ). 2 dt x

(1.4.18)

From equations (1.4.17) and (1.4.18) it follows that 3γ 1 d 2 7α [‖ux ‖2 + ‖vx ‖2 − ∫ F(u)dt] + ‖uxx ‖2 + ‖vxx ‖2 2 dt β 12 4 ≤ −(uxx , G1 ) − (vxx , G2 ) + 2[(uxx , vvx ) + (vxx , (uv)x )] + (vxx , h2 ) +

3 ‖h ‖2 + C, α 1

where −(uxx , G1 ) − (vxx , G2 ) = ∫[ux (G1u ux + G1v vx ) + vx (G2u ux + G2v vx )]dx ≤ −b0 [‖ux ‖2 + ‖vx ‖2 ] 4 󵄨󵄨 󵄨 γ 2 2 󵄨󵄨(vxx , h2 )󵄨󵄨󵄨 ≤ ‖vxx ‖ + ‖h2 ‖ , 8 γ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 2 2 󵄨󵄨2(uxx , vvx ) + 2(vxx , (uv)x )󵄨󵄨󵄨 = 3󵄨󵄨󵄨 ∫ ux vx dx󵄨󵄨󵄨 ≤ 3‖ux ‖‖vx ‖4 󵄨󵄨 󵄨󵄨 γ α ≤ ‖uxx ‖2 + ‖vxx ‖2 + C, 24 8

42 | 1 Attractor and its dimension estimation 󵄨󵄨 󵄨󵄨 2b α 󵄨 󵄨󵄨 0 ‖u ‖2 + C. ∫ F(u)dx󵄨󵄨󵄨 ≤ C‖u‖6−σ 󵄨󵄨 6−σ ≤ 󵄨󵄨 24 xx 󵄨󵄨 β Let ϕ(t) = ‖ux ‖2 + ‖vx ‖2 −

2 β

∫ F(u)dx. Then we have

dϕ(t) 3 4 + α‖uxx ‖2 + γ‖vxx ‖2 + 2b0 ϕ(t) ≤ ‖h1 ‖2 + ‖h2 ‖2 + C. dt α γ Integrating the above inequality, we obtain t

1 4 3 (1 − e−2b0 t )( ‖h2 ‖2 + ‖h1 ‖2 ). 2b0 γ α

ϕ(t) ≤ e−2b0 t ϕ(0) + e−2b0 t ∫ Ce2b0 s ds + 0

Notice that

󵄨󵄨 2 󵄨󵄨 1 󵄨󵄨 󵄨 2 6−σ 󵄨󵄨 ∫ F(u)dx󵄨󵄨󵄨 ≤ A‖u‖6−σ ≤ ‖uxxx ‖ + C, 󵄨󵄨 β 󵄨󵄨 2

hence we get t

‖ux ‖2 + ‖vx ‖2 ≤ 2e−2b0 t ϕ(0) + 2e−2b0 t ∫ C1 e2b0 s ds 0

1 4 3 + (1 − e−2b0 t )( ‖h2 ‖2 + ‖h1 ‖2 ) + C2 , b0 γ α

(1.4.19)

where the functions C1 and C2 depend on ‖u‖ and ‖v‖. From equation (1.4.19), we get (1.4.14) and (1.4.15). Lemma 1.4.3. Under the conditions of Lemma 1.4.2, assume that (1) f (u) ∈ C 2 , Gi (u, v) ∈ C 2 , i = 1, 2; (2) u0 (x), v0 (x) ∈ H 2 (Ω), hi (x) ∈ H 1 (Ω), i = 1, 2. Then for the smooth solution of problem (1.4.3)–(1.4.6) we have ‖uxx ‖2 + ‖vxx ‖2 ≤ e−2b0 t [‖u0xx ‖2 + ‖v0xx ‖2 ] +

2 1 − e−2b0 t 5 [ ‖h1x ‖2 + ‖h2x ‖2 ] 2b0 α γ

+e

−2b0 t

t

∫ C3 e2b0 s ds,

(1.4.20)

0

where the function C3 depends on ‖u‖H 1 and ‖v‖H 1 . Furthermore, we have lim [‖uxx ‖2 + ‖vxx ‖2 ] ≤

t→∞ t

1 1 5 2 max C + [ ‖h ‖2 + ‖h2x ‖2 ] 2b0 t≥0 3 2b0 α 1x γ

1 5 2 lim ∫[α‖uxx ‖2 + ‖vxxx ‖2 ]dx ≤ max C3 + [ ‖h1x ‖2 + ‖h2x ‖2 ]. t→∞ t t≥0 α γ 0

(1.4.21) (1.4.22)

1.4 Coupled KdV equations | 43

Proof. First, by Lemmas 1.4.1, 1.4.2 and Sobolev embedding theorem, we have ‖u‖2∞ + ‖v‖2∞ ≤ C ∗ (‖u‖2H 1 + ‖v‖2H 1 ), where C ∗ is the Sobolev embedding constant. Moreover, 󵄩2 󵄩 󵄩2 󵄩 lim [󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩∞ + 󵄩󵄩󵄩v(⋅, t)󵄩󵄩󵄩∞ ] ≤ C ∗ (E0 + E1 ).

t→∞

Taking the inner product of (1.4.3) and uxxxx , we obtain (uxxxx , ut + f (u)x − αuxx + βuxxx + 2vvx ) = (uxxxx , G1 (u, v) + h1 ), where

(1.4.23)

󵄨󵄨 󵄨 󵄨 󵄨 󸀠󸀠 2 󸀠 󵄨󵄨(uxxxx , f (u)x )󵄨󵄨󵄨 = 󵄨󵄨󵄨(uxxx , f (u)ux + f (u)uxx )󵄨󵄨󵄨 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩f 󸀠 (u)󵄩󵄩󵄩∞ ‖uxxx ‖‖uxx ‖ + 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ ‖uxxx ‖‖ux ‖24 α ≤ ‖uxxx ‖2 + C, 10 󵄨󵄨 󵄨 󵄨 󵄨 2 󵄨󵄨(uxxxx , 2vvx )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(uxxx , 2vx + 2vvxx )󵄨󵄨󵄨 γ α ≤ ‖uxxx ‖2 + ‖vxxx ‖2 + C, 10 8 α 5 󵄨󵄨 󵄨 ‖u ‖2 + ‖h1x ‖2 . 󵄨󵄨(uxxxx , h1 )󵄨󵄨󵄨 ≤ ‖uxxx ‖‖h1x ‖ ≤ 10 xxx α

Taking the inner product of (1.4.3) and vxxxx , we have (vxxxx , vt − γvxx + (2uv)x ) = (vxxxx , G2 (u, v) + h2 ). Notice that

and

(1.4.24)

γ 󵄨󵄨 󵄨 α ‖u ‖2 + ‖vxxx ‖2 + C 󵄨󵄨(vxxxx , (2uv)x )󵄨󵄨󵄨 ≤ 10 xxx 8 (uxxxx , G1 (u, v)) + (vxxxx , G2 (u, v))

= (uxx , G1u uxx + G1v vxx + G1uu u2x + 2G1uv ux vx + G1vv vx2 )

+ (vxx , G2u uxx + G2v vxx + G2uu u2x + 2G2uv ux vx + G2vv vx2 ) γ α ≤ −b0 (‖uxx ‖2 + ‖vxx ‖2 ) + ‖uxxx ‖2 + ‖vxxx ‖2 + C, 10 8

where the function C depends on ‖u‖1H and ‖v‖1H . Moreover, from (1.4.23) and (1.4.24), we have d [‖u ‖2 + ‖vxx ‖2 ] + α‖uxxx ‖2 + γ‖vxxx ‖2 + 2b0 [‖uxx ‖2 + ‖vxx ‖2 ] dt xx 5 2 ≤ ‖h1x ‖2 + ‖h2x ‖2 + C. α γ

44 | 1 Attractor and its dimension estimation Thus we get t

‖uxx ‖2 + ‖vxx ‖2 ≤ e−2b0 t [‖u0xx ‖2 + ‖v0xx ‖2 ] + e−2b0 t ∫ C4 e2b0 s ds 0

5 2 1 (1 − e−2b0 t )( ‖h1x ‖2 + ‖h2x ‖2 ), + 2b0 α γ

lim [‖uxx ‖2 + ‖vxx ‖2 ] ≤

t→∞

(1.4.25)

1 5 2 [max C + ‖h ‖2 + ‖h2x ‖2 ] ≡ E2 . 2b0 t≥0 4 α 1x γ

Lemma 1.4.4. Under the conditions of Lemma 1.4.3, assume that (1) f (u) ∈ C 3 , Gi (u, v) ∈ C 2 , i = 1, 2; (2) u0 (x) ∈ H 2 (Ω), hi (x) ∈ H 2 (Ω), v0 (x) ∈ H 2 (Ω), i = 1, 2. Then for the smooth solution of problem (1.4.3)–(1.4.6), we have the following estimate: ‖uxxx ‖2 + ‖vxxx ‖2 ≤

E3 , t

t > 0,

(1.4.26)

where constant E3 depends on ‖u0 ‖H 2 , ‖v0 ‖H 2 , ‖hi ‖H 2 and t. 6

Proof. Taking the inner product of (1.4.3) and t 2 𝜕𝜕xu6 , and respectively the inner product 6

𝜕 v of (1.4.3) and t 2 𝜕x 6 , we have

(t 2

6 𝜕6 u 2𝜕 u , u − αu + βu + 2vv + f (u) ) = (t , G1 + h1 (x)), t xx xxx x x 𝜕x6 𝜕x 6 𝜕6 v 𝜕6 v ( 6 , vt − γvxx + 2(uv)x ) = (t 2 6 , G2 + h2 (x)). 𝜕x 𝜕x

Since (t 2

𝜕6 u 1 d , ut ) = − ‖tu ‖2 + ‖√tuxxx ‖3 , 6 2 dt xxx 𝜕x

𝜕6 u , −αuxx ) = −α‖tuxxxx ‖2 , 𝜕x6 󵄨󵄨 𝜕6 u 󵄨󵄨 󵄨󵄨 𝜕4 u 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 , f (u) ) , f (u) ) = (t 󵄨󵄨(t 󵄨 󵄨󵄨 󵄨 x xxx 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕x4 𝜕x6 α 2 ≤ ‖tuxxxx ‖ + C(‖tuxxx ‖2 + 1), 12 󵄨󵄨 𝜕6 u 󵄨󵄨 α 󵄨󵄨 2 󵄨 , 2vvx )󵄨󵄨󵄨 ≤ ‖tuxxxx ‖2 + C(‖tvxxx ‖2 + 1), 󵄨󵄨(t 󵄨󵄨 󵄨󵄨 12 𝜕x6 󵄨󵄨 𝜕6 u 󵄨󵄨 α 󵄨󵄨 2 󵄨 , G1 (u, v))󵄨󵄨󵄨 ≤ ‖tuxxxx ‖2 + C(‖tuxxx ‖2 + ‖tvxxx ‖2 + 1), 󵄨󵄨(t 󵄨󵄨 󵄨󵄨 12 𝜕x6 α ‖√tuxxx ‖2 = t‖uxxx ‖2 ≤ ‖tuxxxx ‖2 + C, 12 󵄨󵄨 𝜕6 u 󵄨󵄨 α 󵄨󵄨 2 󵄨 , h1 )󵄨󵄨󵄨 ≤ ‖tuxxxx ‖2 + C, 󵄨󵄨(t 6 𝜕x 󵄨󵄨 󵄨󵄨 12 (t 2

1.4 Coupled KdV equations | 45

where the constant C depends on ‖u0 ‖H 2 , ‖v0 ‖H 2 , and t, we obtain that d [‖tuxxx ‖2 + ‖tvxxx ‖2 ] + α‖tuxxxx ‖2 − γ‖tvxxxx ‖2 ≤ C(‖tuxxx ‖2 + ‖tvxxx ‖2 + 1). dt Hence ‖uxxx ‖2 + ‖vxxx ‖2 ≤ E3 /t,

t > 0.

By Galerkin method we prove the existence of a global smooth solution for problem (1.4.3)–(1.4.6). Set ωj (x), j = 1, 2, . . . to be the eigenfunctions for the equation uxx + αu = 0 with periodic boundary conditions with corresponding eigenvalues αj , j = 1, 2, . . . . Then {ωj } forms the standard orthonormal basis. The approximate solutions uN (x, t) and vN (x, t) of problem (1.4.3)–(1.4.6) possess the following form: N

uN (x, t) = ∑ αjN (t)ωj (t), j=1

N

vN (x, t) = ∑ βjN (t)ωj (t), j=1

(1.4.27)

where αjN , βjN , j = 1, 2, . . . , N; N = 1, 2, . . . , are the coefficient functions of t ∈ R+ . According to the Galerkin method, the coefficients αjN , βjN must satisfy the following first order nonlinear ordinary differential equations: (uNt + f (uN )x − αuNxx + βuNxxx − G1 (uN , vN ) − h1 , ωj ) = 0, (vNt − γvNxx + 2(uN vN )x − G2 (uN , vN ) − h2 , ωj ) = 0, j = 1, 2, . . . , N,

(1.4.28)

with the initial conditions αjN (0) = (uN (x, 0), ωj (x)) = (u0 (x), ωj (x)), βjN (0) = (vN (x, 0), ωj (x)) = (v0 (x), ωj (x)).

(1.4.29)

From the solution existence theory for the ordinary differential equations, we get the existence of a local smooth solution for problems (1.4.28)–(1.4.29). Similarly as in the proof of Lemmas 1.4.1–1.4.4, we can establish consistent integral estimates of approximate solutions {uN (x, t)} and {vN (x, t)}, which depend on N. The uniform priori estimates not only ensure the existence of a global solution αjN , βjN for the problem (1.4.28)–(1.4.29), but can also prove that the approximate solutions {uN (x, t)} and {vN (x, t)} converge to the global solution of problem (1.4.3)–(1.4.6). Then we have Theorem 1.4.1. Suppose the following conditions are satisfied: (1) f (u) ∈ C m , Gi ∈ C m−1 , i = 1, 2, and 󵄨󵄨 󵄨 5−σ 󵄨󵄨f (u)󵄨󵄨󵄨 ≤ A|u| ,

󵄨󵄨 󵄨 5 5 󵄨󵄨Gi (u, v)󵄨󵄨󵄨 ≤ B(|u| + |v| ),

B > 0, i = 1, 2;

46 | 1 Attractor and its dimension estimation (2) Gi (0, 0) = 0, i = 1, 2, and ∀(ξ , η) ∈ R2 , −G1u −G2u

(ξ , η) (

−G1v ξ ) ( ) ≥ −b0 (ξ 2 + η2 ), −G2v η

where b0 > 0 is a constant; (3) if u0 , v0 ∈ H m (Ω), hi (x) ∈ H m (Ω), i = 1, 2, then there exists a unique global solution for problem (1.4.3)–(1.4.6), u(x, t), v(x, t) ∈ L∞ (0, T; H m (Ω)). In order to prove the existence of a global attractor for the periodic initial value problem (1.4.3)–(1.4.6), we use the following theorem: Theorem 1.4.2 ([8]). Suppose E is a Banach space and {St , t ≤ 0} is a set of semigroup operators, St : E 󳨀→ E, St ⋅ Sτ = St+τ , S0 = I, where I is the identity operator. Assume that St satisfies: (1) St is bounded, i. e., for any R > 0, if ‖u‖E ≤ R, then there exists a constant C(R), such that ‖St u‖E ≤ C(R),

t ∈ [0, ∞);

(2) There exists a bounded absorbing set B0 ⊂ E, i. e., for any bounded set B ⊂ E, there exists a constant T such that St B ⊂ B0 ,

t ≥ T;

(3) St is a completely continuous operator for t > 0. Then the semigroup operator St possesses a compact global attractor. Theorem 1.4.3. Suppose there exists a unique global smooth solution for problem (1.4.3)–(1.4.6) and the conditions of Lemma 1.4.4 are satisfied. Then the periodic initial value problem (1.4.3)–(1.4.6) has a global attractor A , which possess the following properties: (1) St A = A , ∀t ∈ R+ . (2) limt→∞ dist(St B, A ) = 0, ∀ bounded set B ⊂ H 2 (Ω), where dist(X, Y) = sup inf ‖x − y‖E x∈X y∈Y

and St u0 is the semigroup generated by problem (1.4.3)–(1.4.6). Proof. By Theorem 1.4.2, we merely need to verify its conditions for the problem (1.4.3)–(1.4.6). Under the assumptions of Theorem 1.4.3, we know that problem (1.4.3)–(1.4.6) generates a semigroup St . Denote the Banach space E = H 2 × H 2 ,

1.4 Coupled KdV equations | 47

(u, v) ∈ E, ‖(u, v)‖2E = ‖u‖2H 2 + ‖v‖2H 2 , St : E 󳨀→ E. By Lemmas 1.4.1–1.4.3, if B ⊂ E is the ball {‖(u, v)‖E ≤ R}, then we have 󵄩2 󵄩2 󵄩 󵄩󵄩 2 2 󵄩󵄩St (u0 , v0 )󵄩󵄩󵄩E = 󵄩󵄩󵄩(u, v)󵄩󵄩󵄩E = ‖u‖H 2 + ‖v‖H 2 ≤ C(‖u0 ‖2H 2 + ‖v0 ‖2H 2 , ‖hi ‖2H 1 ) ≤ C(R2 , ‖h1 ‖2H 1 + ‖h2 ‖2H 1 ),

t ≥ 0, (u0 , v0 ) ∈ B.

This means that {St } is consistently bounded in E. Furthermore, by using the above lemmas, we have 󵄩󵄩 󵄩2 2 2 󵄩󵄩St (u0 , v0 )󵄩󵄩󵄩E = ‖u‖H 2 + ‖v‖H 2 ≤ 2(E0 + E1 + E2 ),

t ≥ t0 (R, ‖h1 ‖2H 1 + ‖h2 ‖2H 1 ).

(1.4.30)

Thus, 󵄩 󵄩2 A = {(u(⋅, t), v(⋅, t)) ∈ E, 󵄩󵄩󵄩(u, v)󵄩󵄩󵄩E ≤ 2(E0 + E1 + E2 )} is a bounded absorbing set of the semigroup operator St . From Lemma 1.4.4, we know ‖uxxx ‖2 + ‖vxxx ‖2 ≤

E3 , t

󵄩 󵄩 ∀t > 0, 󵄩󵄩󵄩(u0 , v0 )󵄩󵄩󵄩E ≤ R.

By use of the compact embedding H 3 (Ω) 󳨅→ H 2 Ω, we know that the semigroup operator St : E → E is continuous for t > 0. This proves the theorem. Remark 1.4.1. As [197] points out, we get the attractor A in Theorem 1.4.3 as the limit set of an absorbing set: A = ω(A) = ⋂ ⋃ St A. r≥0 t≥r

(1.4.31)

To establish upper bounds of Hausdorff and fractal dimensions of the global attractor for the initial value problem (1.4.3)–(1.4.6), we need to consider the variational problem for (1.4.3)–(1.4.6): vt + L(u, v)v = 0,

v(0) = v0 ,

(1.4.32) (1.4.33)

where η v = ( ), ξ

η0 ), ξ0

v0 = (

−αηxx + βηxxx + (f 󸀠 (u)η)x + 2(vξ )x − G1u η − G1v ξ ). L(u, v)v = ( −γξxx + 2(uξ + vη)x − G2u η − G2v ξ Since the solution of problem (1.4.3)–(1.4.6) is sufficiently smooth, we can prove that the linear problem (1.4.32)–(1.4.33) possesses a global smooth solution, as long as the

48 | 1 Attractor and its dimension estimation initial value v0 can be reasonably smooth, i. e., there exists Gt such that v(t) = Gt v0 . The semigroup operator St U0 is differentiable in L2 (Ω), which is equivalent to the existence of the Fréchet derivative St U0 , and Gt v0 = St U0 , U0 = (u0 , v0 )T . Indeed, denote ω(t) = St (U0 − v0 ) − St (U0 ) − Gt (U0 )v0 = U1 (t) − U(t) − v(t), where

u U(t) = ( ) . v

It follows that 𝜕t ω(t) = L1 (U1 ) − L1 (U) + L(U(t))v(t) = L1 (U + v + ω) − L1 (U) + L(U(t))v(t), ω(0) = 0,

(1.4.34) (1.4.35)

where U1 = L1 (U) is the subform of problem (1.4.3)–(1.4.6). Thus (1.4.34) can be written as 𝜕t ω − L(U)ω = Λ 0 (U, v, ω),

(1.4.36)

Λ 0 (U, v, ω) = L1 (U, v, ω) − L1 (U) + L(U)(v + ω)(t).

(1.4.37)

where

Applying the linear PDE theory, we have the following L2 estimate: 󵄩󵄩 󵄩 2 󵄩󵄩ω(t)󵄩󵄩󵄩 ≤ C‖v0 ‖ .

(1.4.38)

This yields that the semigroup operator St is differentiable in L2 (Ω). Denote by v1 (t), v2 (t), . . . , vJ (t) the solutions of the linear equation with initial values v1 (0) = ξ1 , v2 (0) = ξ2 , . . . , vJ (0) = ξJ , respectively. Here ξj ∈ L2 (Ω) × L2 (Ω), j = 1, 2, . . . , J. By a simple calculation, we have d 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩v (t) ∧ v2 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩 + 2 tr(L(U(t))QJ )󵄩󵄩󵄩v1 (t) ∧ v2 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩 = 0, (1.4.39) dt 󵄩 1 where L(U(t)) = L(St U0 ) is the linear mapping v 󳨀→ L(U(t))v, ∧ denotes the crossproduct, tr denotes the trace of an operator, Qj (t) means the representation space L2 (Ω) × L2 Ω) to v1 (t), v2 (t), . . . , vJ (t) into the subspace orthogonal projection. From formula (1.4.39), we can obtain the variation of volume of the J-dimensional cube ⋀Jj=1 ξj as ωJ (t) = sup

u0 ∈A

sup

ξ ∈L2 ,|ξ

j |≤1

‖v1 (t) ∧ v2 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)‖2L J L2 t

≤ sup exp(−2 ∫ tr(L(Sr U0 )QJ (r))dr), u0 ∈A

0

where A is an attractor.

1.4 Coupled KdV equations | 49

Noticing the result in [197], we know that ωJ (t) is log-subadditive, that is, ωJ (t + t1 ) ≤ ωJ (t)ωJ (t1 ),

t, t1 ≥ 0.

(1.4.40)

Therefore, we obtain 1

lim ωJ (t) t = πJ ≤ exp(−2qJ )

(1.4.41)

t→∞

where t

1 ∫ tr(L(Sτ U0 )QJ (τ))dτ). 2 ξ ∈L ,|ξj |≤1 t

(1.4.42)

qJ = lim sup ( sup t→∞ u ∈A 0

0

By use of the following theorem and lemma Theorem 1.4.4 ([33]). Let A be an attractor for a nonlinear evolution equation. Assume it is bounded in H 1 (Ω). If qJ > 0, for some J, then the Hausdorff dimension of the attractor q ≤ J and its fractal dimension ≤ J(1 + max1≤j≤J (− q j )). J

Theorem 1.4.5 (Generalized Sobolev–Lieb–Thirring inequality [197]). Let Ω ⊂ Rn be a bounded set, and {ϕ1 , ϕ2 , . . . , ϕN } an orthogonal basis in L2 (Ω), ϕi ∈ H m , i = 1, 2, . . . , N, and for almost all x ∈ Ω, ρ(x) = ∑Nj=1 |ϕj (x)|2 . Then for almost x ∈ Ω, ρ(x) = ∑Nj=1 |ϕj |2 , the following estimate holds: ∫ ρ(x)1+2m/n dx ≤

N k0 󵄨 󵄨2 ρ(x)dx + k ∑ ∫ ∫ 󵄨󵄨󵄨Dm ϕj 󵄨󵄨󵄨 dx, 0 |Ω|2m/n j=1

(1.4.43)

where the constant k0 depends on m, n and Ω, but is independent of N and ϕj . Theorem 1.4.6. Under the conditions of Theorems 1.4.1 and 1.4.3, the global attractor has finite Hausdorff and fractal dimensions for the periodic initial value problem (1.4.3)–(1.4.6), which can be bounded as dH (A ) ≤ J0 ,

dF (A ) ≤ J0 (1 +

2b√(b/(3a)) ), 3(aJ03 − bJ0 )

where J0 is the smallest integer such that 1

2 4k0 D2 1 1󵄩 󵄩 ( ( 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ + 1)(‖ux ‖ + ‖vx ‖) − b0 )] ≥ J0 − 1 J0 > [k0 + √ min{α, γ} 2D 2

and a= b=

min{α, γ} , 4k0 D2

min{α, γ} 1 1󵄩 󵄩 + ( 󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ + 1)(‖ux ‖ + ‖vx ‖) − b0 . √2D 2 󵄩 4D2

50 | 1 Attractor and its dimension estimation Proof. By Theorem 1.4.4, we only need to estimate the lower bound of tr(L(U)QJ ). Set {Φ1 , Φ2 , . . . , ΦJ }, Φj = (ϕj , ψj )T as the orthonormal basis of the subspace QJ (L2 × L2 ). We have J

tr(L(U(t))QJ ) = ∑ [(−αϕjxx − βϕjxxx + (f 󸀠 (u)ϕj )x j=1

+ (2vϕj )x − G1u (u, v)ϕj − G1v ψj , ϕj )

+ (−γψjxx + 2(uψj + vϕj )x − G2u ϕj − G2v ψj , ψj )] J

= ∑[α‖ϕjx ‖2 + γ‖ψjx ‖2 + j=1

1 ∫ f 󸀠󸀠 (u)ux ϕ2j dx + ∫ ux ψ2j dx 2

+ 2 ∫ vx ϕj ψj dx − ∫(G1u ϕ2j + G1v ϕj ψj + G2u ϕj ψj + G2v ψ2j )dx] J

≥ ∑[min{α, γ}(‖ϕjx ‖2 + ‖ψjx ‖2 ) j=1

1󵄨 󵄨 − ∫( 󵄨󵄨󵄨f 󸀠󸀠 (u)ux 󵄨󵄨󵄨 + |ux | + |vx |) 2 × (ϕ2j + ψ2j )dx + b0 ∫(ϕ2j + ψ2j )dx] 1 1 ≥ min{α, γ}[ ∫ ρ(x)3 dx − J] + b0 J k (2D)2 1 󵄩󵄩 1 󵄩󵄩 3 󵄩󵄩 󵄨󵄨 󸀠󸀠 󵄨󵄨 󵄩󵄩 3 − 󵄩󵄩( 󵄨󵄨f (u)ux 󵄨󵄨 + |ux | + |vx |)󵄩󵄩 (∫ ρ(x) dx) 󵄩󵄩 2 󵄩󵄩3,2 min{α, γ} 1 ≥ min{α, γ} ∫ ρ(x)3 dx + (b0 − )J k0 4D2

1 1 1󵄩 󵄩 − [( 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ + 1)(2D) 6 (‖ux ‖ + ‖vx ‖)]( ∫ ρ(x)3 dx) 3 . 2

(1.4.44)

Since 3

1 3

2

J = ∫ ρ(x)dx ≤ (∫ ρ(x) dx) (2D) 3 , Ω

it follows that ∫ ρ(x)3 dx ≥

1 3 J . (2D)2

Thus from equations (1.4.44) and (1.4.45), we get tr(U(t) ⋅ QJ ) ≥

min{α, γ} 3 min{α, γ} J + (b0 − )J 4k0 D2 4D2

1󵄩 1 󵄩 − ( 󵄩󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ + 1)(‖ux ‖ + ‖vx ‖) J > 0. √2D 2

(1.4.45)

1.5 Davey–Stewartson equation

If

| 51

1

b 2 J>( ) , a min{α, γ} , a= 4k0 D2 b=

min{α, γ} 1 1󵄩 󵄩 + ( 󵄩󵄩f 󸀠󸀠 (u)󵄩󵄩󵄩∞ + 1)(‖ux ‖ + ‖vx ‖) − b0 √2D 2 󵄩 4D2

and 1

b 2 J0 − 1 ≤ ( ) ≤ J0 , a −

ql bl − al3 2b√b/3a ≤ 3 ≤ , qJ0 aJ0 − bJ0 3(aJ03 − bJ0 )

then by Theorem 1.4.6, we have dH (A ) ≤ J0 , dF (A ) ≤ J0 (1 +

2b√b/3a ) 3(aJ03 − bJ0 )

and the claim is proved.

1.5 Davey–Stewartson equation We consider the following Davey–Stewartson (DS) equation [29, 62, 206]: 𝜕2 A 𝜕2 A 𝜕A − α 2 − b 2 = χA − β|A|2 A + γQA, 𝜕t 𝜕x 𝜕y

t > 0, (x, y) ∈ Ω,

𝜕2 Q 𝜕2 Q 𝜕2 + 2 = 2 (|A|2 ), 2 𝜕x 𝜕y 𝜕y

(1.5.1) (1.5.2)

with boundary condition A(t, x, y) = 0,

Q(t, x, y) = 0,

t ≥ 0, (x, y) ∈ 𝜕Ω

(1.5.3)

and initial value condition A(0, x, y) = A0 (x, y),

(x, y) ∈ Ω,

(1.5.4)

where a = a1 + ia2 , b = b1 + ib2 , β = β1 + iβ2 , γ = γ1 + iγ2 , χ = χ1 + iχ2 are complex numbers, Ω ⊂ R2 is a smooth bounded area. This system of equations was proposed by Davey et al. [39] while studying planar Poiseuille flow in a nonlinear threedimensional disturbance evolution. Here A(t, x, y) stands for complex amplitude, and Q(t, x, y) describes real speed disturbance.

52 | 1 Attractor and its dimension estimation If we set Q = |A|2 −

𝜕φ , 𝜕x

together with equation (1.5.2), we will have ΔQ = Δ|A|2 −

where Δ =

𝜕2 𝜕x2

+

𝜕2 , 𝜕y2

𝜕 𝜕2 Δφ = 2 |A|2 , 𝜕x 𝜕y

so we will obtain Δφ =

𝜕 |A|2 . 𝜕x

(1.5.5)

Hence the amplitude equation becomes 𝜕φ 𝜕2 A 𝜕2 A 𝜕A − a 2 − b 2 = χA − β|A|2 A − γA 𝜕t 𝜕x 𝜕x 𝜕y

(1.5.6)

where β = β + γ. The system of equations (1.5.5)–(1.5.6) was first proposed in [38, 48] by Davey and Stewartson. Ablowitz and Haberman [2] studied two-dimensional completely integrable nonlinear Schrödinger equation systems and obtained some special cases of the above equations. Many mathematical physicists did a series of studies about the equations, for instance, investigated the existence of a generalized local or global solution, the stability of a plane wave solution, studied the properties of the solitary wave solutions, solution of the singularity development, etc.; for details we could refer to Davey, Hocking and Stewartson [39], Holmes [142], Ghidaglia and Saut [75], Anker and Freeman [4], Ablowitz and Fokas [1], Tsutsumi [201], Hayashi and Saut [138], Linares and Ponce [165], and so on. In 1997, Guo and Yang [210] proved the existence of global smooth solutions for a class of DS equations, Guo and Li [105] proved the existence of the attractor for a class of DS equations, and showed finite fractal dimension. From equations (1.5.2) and (1.5.3) we can solve for Q as a function of A: Q = −(−Δ)−1

𝜕2 |A|2 ≜ E(|A|2 ), 𝜕y2

(1.5.7)

where (−Δ)−1 is the inverse of Laplace operator which has Dirichlet boundary conditions. We can reduce equations (1.5.1) and (1.5.2) into the following nonlinear Schrödinger (Ginzburg–Landau) equation: 𝜕A 𝜕2 A 𝜕2 A − a 2 − b 2 = χA − β|A|2 A − γAE(|A|2 ), 𝜕t 𝜕x 𝜕y A(t, x, y) = 0, t ≥ 0, (x, y) ∈ 𝜕Ω, A(0, x, y) = A0 (x, y),

t > 0, (x, y) ∈ Ω,

(x, y) ∈ Ω.

(1.5.9) (1.5.10)

From Sobolev inequality, there exists a C(p) > 0, 1 < p < ∞, such that 󵄩󵄩 𝜕2 u 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 2 󵄩󵄩󵄩 ≤ C(p)‖Δu‖p , 󵄩󵄩 𝜕y 󵄩󵄩p

(1.5.8)

u ∈ C0 (Ω),

1.5 Davey–Stewartson equation

| 53

where ‖ ⋅ ‖p denotes the norm of Lp (Ω). Hence 󵄩 󵄩󵄩 󵄩󵄩E(u)󵄩󵄩󵄩p ≤ C(p)‖u‖p ,

u ∈ C0∞ (Ω)

(1.5.11)

2

𝜕 p and E = (−Δ)−1 𝜕y 2 can be extended to a bounded linear operator (1 < p < ∞) in L (Ω) 2

𝜕 2,p (Ω), (−Δ)−1 is a linear bounded opwith norm C(p). Actually, 𝜕y 2 and Δ commute in W 2

𝜕 −1 ̄ A (Lp (Ω)) = erator in W 2,p (Ω), thus Ē = 𝜕y is the extension of E in Lp (Ω). And ‖E‖ 2 (−Δ) C(p). We will prove the following theorem:

Theorem 1.5.1. Suppose [H] a1 > 0, b1 > 0, β1 > 0, β1 + C(2)γ1 > 0, χ1 > 0. Then the semigroup operator generated by equations (1.5.8) and (1.5.9) has a global compact attractor in H01 (Ω), and its Hausdorff and fractal dimensions are finite. Lemma 1.5.1. Suppose condition [H] is satisfied, A0 ∈ L2 (Ω). Then A ∈ L∞ (R− , L2 (Ω)) satisfies 󵄩󵄩 󵄩2 2 −t 󵄩󵄩A(t)󵄩󵄩󵄩 ≤ ‖A0 ‖ e + C,

∀t ≥ 0.

Thus there exists a t1 (R) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩A(t)󵄩󵄩󵄩 ≤ C,

∀t ≥ t1 (R),

where ‖A0 ‖ ≤ R, C = C(β1 , γ1 , χ1 , Ω). Moreover, for all r > 0, we have t+r

󵄩 󵄩2 󵄩 󵄩4 ∫ (󵄩󵄩󵄩∇A(s)󵄩󵄩󵄩 + 󵄩󵄩󵄩A(s)󵄩󵄩󵄩4 )ds ≤ C(r, β1 , γ1 , χ1 , Ω),

∀t ≥ t1 (R).

t

Proof. Multiplying equation (1.5.8) by A, integrating over Ω, and taking its real component, we obtain 1 d ‖A‖2 + a1 ‖Ax ‖2 + b1 ‖Ay ‖2 2 dt

= χ1 ‖A‖2 − β1 ‖A‖44 − γ1 ∫ |A|2 E(|A|2 )dxdy.

(1.5.12)

Noticing that 󵄨 󵄨2 ∫ |A|2 E(|A|2 )dxdy = ∫ 󵄨󵄨󵄨[(−Δ) 2 |A|2 ]y 󵄨󵄨󵄨 dxdy ≥ 0, 1

󵄩 󵄩󵄩 󵄩 ∫ |A|2 E(|A|2 )dxdy ≤ 󵄩󵄩󵄩|A|2 󵄩󵄩󵄩󵄩󵄩󵄩E(|A|2 )󵄩󵄩󵄩 󵄩 󵄩2 ≤ C(2)󵄩󵄩󵄩|A|2 󵄩󵄩󵄩 = C(2)‖A‖44 ,

(1.5.13)

54 | 1 Attractor and its dimension estimation we have − β1 ‖A‖44 − γ1 ∫ |A|2 E(|A|2 )dxdy ≤ −K‖A‖44 ,

(1.5.14)

where K = β1 > 0 when γ1 > 0 and K = β1 + C(2)γ1 > 0 when γ1 < 0. By Hölder inequality, 1 χ1 ‖A‖2 ≤ K‖A‖44 + C, 2

(1.5.15)

where C only depends on β1 , γ1 , χ1 and Ω. Combining with equations (1.5.12)–(1.5.12), we obtain 1 1 d ‖A‖2 + α0 ‖∇A‖2 + K‖A‖44 ≤ C. 2 dt 2

(1.5.16)

Similar to equation (1.5.15), ‖A‖2 ≤ 21 K‖A‖44 + C, we have d ‖A‖2 + ‖A‖2 ≤ C. dt By Gronwall inequality we deduce that ‖A‖2 ≤ ‖A0 ‖2 e−t + C(1 − e−t ).

(1.5.17)

Hence there exists a t1 (R) > 0 such that ‖A‖2 ≤ 1+C, t ≥ t1 (R), ‖A0 ‖ ≤ R. From equations (1.5.16) and (1.5.17), we obtain t+r

t+r

t

t

󵄩 󵄩2 󵄩 󵄩4 2α0 ∫ 󵄩󵄩󵄩∇A(s)󵄩󵄩󵄩 ds + K ∫ 󵄩󵄩󵄩A(s)󵄩󵄩󵄩4 ds 󵄩 󵄩2 󵄨s=t+r ≤ Cr − 󵄩󵄩󵄩A(s)󵄩󵄩󵄩 󵄨󵄨󵄨s=t ≤ Cr + ‖A0 ‖2 e−t + C(1 − e−t ) ≤ Cr + C, ∀t > t1 (R), r > 0, and the lemma is proved. Lemma 1.5.2. Suppose condition [H] is satisfied, A0 ∈ H01 (Ω). Then there exist A ∈ L∞ (R+ , L10 (Ω)) and t2 (R) > 0, C > 0 such that 󵄩󵄩 󵄩2 󵄩󵄩∇A(t)󵄩󵄩󵄩 ≤ C,

∀t ≥ t2 (R),

where ‖A0 ‖H 1 ≤ R. Proof. Multiplying equation (1.5.8) by −△A, integrating over Ω, and taking its real component yields Re ∫ aAxx ΔAdxdy = a1 ‖Axx ‖2 + a1 ‖Axy ‖2 ,

1.5 Davey–Stewartson equation

| 55

Re ∫ aAyy △Adxdy = b1 ‖Ayy ‖2 + b1 ‖Axy ‖2 , and we have 1 d ‖∇A‖2 + a1 ‖Axx ‖2 + b1 ‖Ayy ‖2 + (a1 + b1 )‖Axy ‖2 2 dt

= χ1 ‖∇A‖2 − Re β ∫ |A|2 AΔAdxdy − Re γ ∫ AE(|A|2 )ΔAdxdy.

(1.5.18)

Noticing that a1 ‖Axx ‖2 + (a1 + b1 )‖Axy ‖2 + b1 ‖Ayy ‖2 ≥ α0 ‖ΔA‖2 , and using Gagliardo–Nirenberg inequality ‖u‖6 ≤ C‖u‖1/3 ‖∇u‖2/3 ,

∀u ∈ H01 (Ω),

we get the following estimates: 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 3 󵄨󵄨Re β ∫ |A| AΔAdxdy󵄨󵄨󵄨 ≤ |β|‖A‖6 ‖ΔA‖ 󵄨󵄨 󵄨󵄨 ≤ C‖A‖‖∇A‖2 ‖ΔA‖ 1 ≤ α0 ‖ΔA‖2 + C‖A‖2 ‖∇A‖4 , 6 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩 2 2 󵄩 󵄨󵄨Re γ ∫ AE(|A| )ΔAdxdy󵄨󵄨 ≤ |γ|‖A‖6 󵄩󵄩󵄩E(|A| )󵄩󵄩󵄩3 ‖ΔA‖ 󵄨󵄨 󵄨󵄨 ≤ |γ|‖A‖35 ‖ΔA‖ ≤ C‖A‖‖∇A‖2 ‖ΔA‖ 1 ≤ α0 ‖ΔA‖2 + C‖A‖2 ‖∇A‖4 , 6 1 χ1 ‖∇A‖2 ≤ α0 ‖ΔA‖2 + C‖A‖2 . 6 Consequently we obtain 1 d 1 ‖∇A‖2 + α0 ‖ΔA‖2 ≤ C‖A‖2 + C‖A‖2 ‖∇A‖4 . 2 dt 2

(1.5.19)

In order to prove the claim of the lemma, i. e., 󵄩󵄩 󵄩2 󵄩󵄩∇A(t)󵄩󵄩󵄩 ≤ C,

∀t ≥ t2 (R),

we need the following lemma Lemma 1.5.3 (Uniform Gronwall inequality). Let g(t), h(t) and y(t) ≥ 0 be such that y󸀠 (t) ≤ g(t)y(t) + h(t), t+r

If ∫t

t+r

g(s) ≤ k1 , ∫t

t+r

h(s)ds ≤ k2 , ∫t

∀t ≥ s.

y(s)ds ≤ k3 , r > 0, then

y(t + r) ≤ (

k3 + k2 )ek1 , r

∀t ≥ s.

56 | 1 Attractor and its dimension estimation Now let y(t) = ‖∇A(t)‖2 , g(t) = C‖A(t)‖2 ‖∇A(t)‖2 , h(t) = C‖A(t)‖2 . Then by Lemma 1.5.2, for any r > 0, t ≥ t1 (R), t+r

∫ y(s)ds ≤ Cr + C ≜ k3 , t

t+r

t+r

t

t

∫ g(s)ds ≤ C ∫ ‖∇A‖2 ds ≤ Cr + C ≜ k1 ,

t+r

∫ h(s)ds ≤ Cr ≜ k2 , t

where ‖A0 ‖H 1 (Ω) ≤ R. By the uniform Gronwall inequality, we have 0

C 󵄩 󵄩 y(t + r) = 󵄩󵄩󵄩∇A(t + r)󵄩󵄩󵄩 ≤ ( + C + Cr)eCr+C , r

t ≥ t1 (R),

r > 0.

Thus Lemma 1.5.3 is proved. Furthermore, from equation (1.5.19) we have t+r

∫ ‖ΔA‖2 ds ≤ C(r),

∀t ≥ 0.

t

Lemma 1.5.4. Suppose the conditions of Lemma 1.5.2 are satisfied. Then we get ‖ΔA‖2 ≤ C +

C , t

∀t > 0.

Proof. Acting Δ on both sides of equation (1.5.8), multiplying by −tΔA, integrating in Ω over x, and taking its real part, we obtain 1 1 d (t‖ΔA‖2 ) − ‖ΔA‖2 + t(a1 ‖ΔAx ‖2 + b1 ‖ΔAy ‖2 ) 2 dt 2

= χ1 t‖ΔA‖2 + Re βt ∫ Δ(|A|2 A)ΔAdxdy + Re γt ∫ Δ(AE(|A|2 ))ΔAdxdy.

Noticing that a1 ‖ΔAx ‖2 + b1 ‖ΔAy ‖2 ≥ α0 ‖∇ΔA‖2 , 3

2

1

4

‖∇A‖4 ≤ C‖A‖45 ‖∇ΔA‖ 3 , ‖ΔA‖4 ≤ C‖A‖45 ‖∇ΔA‖ 5 , we get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 󵄨󵄨Re β ∫ Δ(|A| A)ΔAdxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

≤ 2|β| ∫(|A|2 |ΔA|2 + |∇A|2 |A||ΔA|)dxdy

(1.5.20)

1.5 Davey–Stewartson equation

| 57

≤ C(‖A‖24 ‖ΔA‖24 + ‖∇A‖24 ‖A‖4 ‖ΔA‖4 ) 12

8

≤ C‖A‖45 ‖∇ΔA‖45 1 ≤ α0 ‖∇ΔA‖2 + C‖A‖12 H1 , 4

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 󵄨󵄨Re γ ∫ Δ(AE(|A| ))ΔAdxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩 󵄩 󵄩 󵄩 ≤ |γ|‖ΔA‖24 󵄩󵄩󵄩E(|A|2 )󵄩󵄩󵄩 + 2|γ|‖∇A‖4 󵄩󵄩󵄩∇E(|A|2 )󵄩󵄩󵄩‖ΔA‖4 󵄩 󵄩 + |γ|‖A‖4 ‖ΔA‖4 󵄩󵄩󵄩ΔE(|A|2 )󵄩󵄩󵄩

󵄩󵄩 𝜕2 |A|2 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ≤ C‖A‖24 ‖ΔA‖24 + C‖∇A‖4 󵄩󵄩󵄩∇|A|2 󵄩󵄩󵄩‖ΔA‖4 + C‖A‖4 ‖ΔA‖4 󵄩󵄩󵄩 󵄩 󵄩󵄩 𝜕y2 󵄩󵄩󵄩 󵄩󵄩 𝜕2 A 󵄩󵄩 󵄩󵄩 𝜕A 󵄩󵄩2 󵄩 󵄩 󵄩 󵄩 ≤ C‖A‖4 ‖ΔA‖24 + C‖∇A‖24 ‖A‖4 ‖ΔA‖4 + C‖A‖4 ‖ΔA‖4 (2‖A‖4 󵄩󵄩󵄩 2 󵄩󵄩󵄩 + 2󵄩󵄩󵄩 󵄩󵄩󵄩 ) 󵄩󵄩 𝜕y 󵄩󵄩4 󵄩󵄩 𝜕y 󵄩󵄩4 12

8

≤ C‖A‖45 ‖∇ΔA‖ 5 1 ≤ α0 ‖∇ΔA‖2 + C‖A‖12 H1 . 4 From equation (1.5.20) and the above estimate, we obtain 1 d 1 1 1 2 2 (t‖ΔA‖2 ) + α0 λ1 t‖ΔA‖2 ≤ Ct‖A‖12 H 1 + ‖ΔA‖ ≤ Ct + ‖ΔA‖ , 2 dt 2 2 2 where λ1 ‖u‖2 ≤ ‖∇u‖2 , u ∈ H01 (Ω) and λ1 is the first eigenvalue with Dirichlet boundary conditions of −Δ. Set α0 = min{a, b}. If α0 > 0, then a1 ‖ux ‖2 + b1 ‖uy ‖2 ≥ α0 λ1 ‖u‖2 . r

By ∫0 ‖ΔA(t)‖2 dt ≤ C(r), ∀r > 0, and Gronwall inequality, we get t

󵄩 󵄩2 t‖ΔA‖2 ≤ G(t) ≜ ∫ e−α0 λ1 (t−s) (Cs + 󵄩󵄩󵄩ΔA(s)󵄩󵄩󵄩 )ds.

(1.5.21)

0

Thus we have ‖ΔA‖2 ≤

C(r) , t

∀0 < t ≤ r.

(1.5.22)

∀t > 0.

(1.5.23)

From equation (1.5.21) we have 1 ‖ΔA‖2 ≤ G(t), t

Differentiating G(t) with respect to t, using formula (1.5.23), and inserting into the above equation, we have d 1 G(t) = Ct + ‖ΔA‖2 − α0 λ1 G(t) ≤ Ct − (α0 λ1 − )G(t). dt t

58 | 1 Attractor and its dimension estimation Taking r∗ >

2 , α0 λ1

we have d 1 G(t) + α0 λ1 G(t) ≤ Ct, dt 2

∀t ≥ r∗ .

From Gronwall inequality we obtain t

1

1

G(t) ≤ G(r∗ )e− 2 α0 λ1 (t−r∗ ) + ∫ e− 2 α0 λ1 (t−r) Csds r∗

t

1

≤ G(r∗ ) + C ∫ se− 2 α0 λ1 (t−s) ds 0

4C 2C t + 2 2, ≤ G(r∗ ) + α0 λ1 α0 λ1

∀t ≥ r∗ .

Hence ‖ΔA‖2 ≤

C(r∗ ) + C, t

∀t ≥ r∗ .

(1.5.24)

From equations (1.5.22) and (1.5.24) we get the claim. Now we rewrite (1.5.8) and (1.5.9) using a functional form At = F(A) ≜ LA + f (A), 2

A|t=0 = A0 ,

(1.5.25)

2

𝜕 𝜕 2 where L = P(D) = a 𝜕x 2 + b 𝜕y 2 is a differential operator on X = L (Ω), D(L) =

W 2,p ∩ W01,p (Ω), 1 < p < ∞, and the linear mapping f (A) = χA − β|A|2 A − γE(|A|2 ). Obviously 0 ∈ ρ(L), where ρ(L) is the resolvent equation of ρ. The symbol for L = P(D) is P(ξ ) = aξ12 + bξ22 , its real component Re P(ξ ) ≥ α0 |ξ |2 . So P(ξ ) is a strong elliptic polynomial, and L = P(D) can generate a bounded analytic semigroup on X. So we can define the fractal power of −L as (−L)α , its domain is D((−L)α ) = X α . In particular, 1 when p = 2α = 21 , we see D((−L) 2 ) = H01 (Ω), the nonlinear mapping f (A) is locally 1

1

continuous from X 2 to X 2 , that is,

󵄩󵄩 󵄩 󵄩󵄩f (A1 ) − f (A2 )󵄩󵄩󵄩X ≤ C(R)‖A1 − A2 ‖X 1/2 , 1

∀A1 , A2 ∈ X 2 = H01 (Ω),

‖Ak ‖ ≤ R,

k = 1, 2.

Hence, for A0 ∈ X 1/2 , and formula has the unique local solution A ∈ C([0, t0 ), X 1/2 ) ∩ C 1 ((0, t0 ), X). By the priori estimates of Lemmas 1.5.1–1.5.4, we know that the local solution can be extended to the global solution. So we have

1.5 Davey–Stewartson equation

| 59

Theorem 1.5.2. Suppose condition [H] is satisfied and A0 ∈ H01 (Ω). Then the system (1.5.8)–(1.5.9) has a unique global solution A ∈ C([0, +∞); H01 (Ω)) ∩ C 1 ((0, +∞), L2 (Ω)). The solution operator S(t) : A0 󳨃󳨀→ A(t) forms a continuous semigroup in H01 (Ω), and possesses an absorbing bounded set B ⊂ H01 (Ω). By Theorem 1.5.2 and Lemma 1.5.4, for any bounded set B ⊂ H01 (Ω), ⋃t≥1 S(t)B is bounded in H01 (Ω). That is, S(t) is compact for sufficiently large t. By the results in [197] we know Theorem 1.5.3. Suppose condition [H] is met, S(t) is the semigroup generated by problem (1.5.8)–(1.5.9) and B is the absorbing set of S(t) in H01 (Ω). Then the ω limit set of B is A = ⋂ ⋃ S(t)B , s≥0 t≥s

where closure is taken under the topology of H01 (Ω), and it satisfies: (1) S(t)A = A , ∀t ≥ 0 (invariance property). (2) limt→∞ dist(S(t)B, A ) ≜ limt→∞ supA0 ∈B dist(S(t)A0 , A ) = 0, ∀B ⊂ H01 (Ω) absorbing. (3) A is compact in H01 (Ω). We will prove that A has finite Hausdorff and fractal dimensions in the following. From Theorem 1.5.2 and Lemma 1.5.4 we know that for A0 ∈ H01 (Ω), equation (1.5.25) has a unique solution A ∈ C([0, +∞); H01 (Ω)) ∩ L2loc ([0, +∞); H 2 (Ω)) ∩ L∞ ((t∗ , +∞); H 2 (Ω)),

∀t∗ > 0.

Let U(t, ⋅, ⋅) is the solution of the following variational problem: dU = F 󸀠 (A(t))U dt = aUxx + bUyy + χU − 2β|A|2 U − βA2 U − γE(|A|2 )U − γE(AU + AU)A, t > 0, (x, y) ∈ Ω,

U(t, x, y) = 0,

t ≥ 0, (x, y) ∈ 𝜕Ω,

U(0, ⋅, ⋅) = U0 ∈ H01 (Ω),

(x, y) ∈ Ω.

It is easy to verify that the above problem has a unique solution U ∈ Cb (R+ ; H01 (Ω)) ∩ L2loc (R+ ; H 2 (Ω)),

60 | 1 Attractor and its dimension estimation this means that S(t) is Fréchet differentiable in H01 (Ω). Also U(t, ⋅, ⋅) = (DS(t)A0 )U0 is the Fréchet derivative of S(t) at A0 ∈ H01 (Ω). Then we have 󵄩 󵄩󵄩 2 󵄩󵄩S(t)(A0 + U0 ) − S(t)A0 − (DS(t)A0 )U0 󵄩󵄩󵄩H 1 (Ω) ≤ C(R, T)‖U0 ‖H 1 (Ω) . 0 0 For 0 ≤ t ≤ T, we have A0 , U0 ∈ H01 (Ω), ‖A0 ‖H 1 (Ω) ≤ R. Suppose that Uk0 ∈ H01 (Ω) is 0 linearly independent, Uk = (DS(t)A0 )Uk0 , 1 ≤ k ≤ m, then 󵄩󵄩 󵄩2 󵄩󵄩U1 (t) ∧ U2 (t) ∧ ⋅ ⋅ ⋅ ∧ Um (t)󵄩󵄩󵄩∧m H 1 (Ω) 0

t

󵄩 󵄩2 = 󵄩󵄩󵄩U10 (t) ∧ U20 (t) ∧ ⋅ ⋅ ⋅ ∧ Um0 (t)󵄩󵄩󵄩∧m H 1 (Ω) exp ∫ Re tr(F 󸀠 (S(τ)A0 ) ⋅ Qm (τ))dτ, 0 0

(1.5.26)

where (⋅, ⋅)H 1 (Ω) is the inner product in H01 (Ω) and (⋅, ⋅) is the inner product in L2 (Ω). We 0 omit the variable τ and get Re(F 󸀠 (A)ϕj , ϕj )H 1 (Ω) = −a1 ‖∇ϕjx ‖2 − b1 ‖∇ϕjy ‖2 + χ‖∇ϕj ‖ 0

− 2Re β(∇(|A|2 ϕj ), ∇ϕj ) − Re β(∇(A2 ϕj ), ∇ϕj )

− Re γ(∇(E(|A|2 )ϕj ), ∇ϕj ) − Re γ(∇(E(Aϕj + Aϕj )A), ∇ϕj ).

Using Gagliardo–Nirenberg inequality, we get 󵄨󵄨 󵄨 2 2 󵄨󵄨2 Re β(∇(|A| ϕj ), ∇ϕj )󵄨󵄨󵄨 ≤ C(‖A‖∞ ‖∇A‖3 ‖ϕj ‖6 + ‖A‖∞ ‖∇ϕj ‖)‖∇ϕj ‖

≤ C(‖∇A‖23 ‖∇ϕj ‖2 ) ≤ C‖A‖2/3 ‖ΔA‖4/3 ‖∇ϕj ‖2 ,

Similarly, we have 󵄨 󵄨󵄨 2/3 4/3 2 2 󵄨󵄨Re(∇(A ϕj ), ∇ϕj )󵄨󵄨󵄨 ≤ C‖A‖ ‖ΔA‖ ‖∇ϕj ‖ , 󵄨󵄨 󵄨 󵄩 󵄩 2 2 󵄩 2 2󵄩 󵄨󵄨Re γ(∇(E(|A| )ϕj ), ∇ϕj )󵄨󵄨󵄨 ≤ |γ|󵄩󵄩󵄩E(|A| )󵄩󵄩󵄩∞ ‖∇ϕj ‖ + C|γ|󵄩󵄩󵄩∇|A| 󵄩󵄩󵄩3 ‖ϕj ‖6 ‖∇ϕj ‖ 󵄩 󵄩 󵄩 󵄩 ≤ C|γ|󵄩󵄩󵄩∇E(|A|2 )󵄩󵄩󵄩3 ‖∇ϕj ‖2 + C|γ|‖A‖∞ 󵄩󵄩󵄩∇|A|󵄩󵄩󵄩3 ‖∇ϕj ‖2 ≤ C‖A‖2/3 ‖ΔA‖4/3 ‖∇ϕj ‖2 , 󵄨 󵄩 󵄩 󵄨󵄨 󵄨󵄨Re γ(∇E(Aϕj + Aϕj ), ∇ϕj )󵄨󵄨󵄨 ≤ |γ|󵄩󵄩󵄩E(Aϕj + Aϕj )󵄩󵄩󵄩6 ‖∇A‖3 ‖∇ϕj ‖ 󵄩 󵄩 + |γ|󵄩󵄩󵄩∇E(Aϕj + Aϕj 󵄩󵄩󵄩‖A‖∞ ‖∇ϕj ‖ ≤ C|γ|‖A‖∞ ‖ϕj ‖6 ‖∇A‖3 ‖∇ϕj ‖

≤ C‖A‖2/3 ‖ΔA‖4/3 ‖∇ϕj ‖2 . Thus we get

Re(F 󸀠 (A)ϕj , ϕj )H 1 (Ω) ≤ −α0 ‖Δϕj ‖2 + χ1 ‖∇ϕj ‖2 + C‖A‖2/3 ‖ΔA‖4/3 ‖∇ϕj ‖2 0

≤ −α0 ‖Δϕj ‖2 + χ1 ‖∇ϕj ‖2 + C0 (‖A‖2 + ‖ΔA‖2 )‖∇ϕj ‖2 ,

1.5 Davey–Stewartson equation

| 61

where constant C0 only depends on β, γ and Ω0 . Since ‖∇ϕj ‖ = 1, we have m

Re tr(F 󸀠 (A) ⋅ Qm ) ≤ −α0 ∑ ‖Δϕj ‖2 + m(χ1 + C0 ‖A‖2 ) + C0 m‖ΔA‖2 . j=1

(1.5.27)

m 2 2 Set ρm (x, y) = ∑m i=1 |∇ϕj (x, y)| , σm (x, y) = ∑j=1 |Δϕj (x, y)| . Then ∫Ω ρm (x, y)dxdy = m. By the generalized Sobolev–Lieb–Thirring inequality

∫ ρ2m (x, y)dxdy ≤ k0 ∫ σm (x, y)dxdy, Ω

Ω

where k0 only depends on Ω, so by Hölder inequality, we have 1/2

m = ∫ ρm (x, y)dxdy ≤ |Ω|1/2 (∫ ρ2m (x, y)dxdy) Ω

1/2

≤ k0 |Ω|1/2 (∫ σm (x, y)dxdy) .

Ω

Ω

Hence m

∑ ‖Δϕj ‖2 = ∫ σm (x, y)dxdy ≥ j=1

Ω

m2 . k02 |Ω|

(1.5.28)

For A0 ∈ A (global attractor), A(t) = S(t)A0 , and we have t

A(t) ≤ C1 ,

󵄩 󵄩2 ∫ 󵄩󵄩󵄩ΔA(s)󵄩󵄩󵄩 ds ≤ C2 t + C3 ,

∀t ≥ 0.

0

Hence, by equations (1.5.27) and (1.5.28), we obtain t

∫ Re tr(F 󸀠 (S(τ)A0 ) ⋅ Qm (τ))dτ 0

≤−

α0 m2 t + m(χ1 + C0 C1 )t + C0 C2 mt + C0 C3 m. k02 |Ω|

Thus we get t

1 qm = lim sup sup sup ∫ Re tr(F 󸀠 (S(τ)A0 ) ⋅ Qm (τ))dτ t→∞ A0 ∈A ‖Uj0 ‖≤1 t 0

α m2 ≤ − 20 + m(χ1 + C0 C1 + C0 C2 ). k0 |Ω| Set m0 to be the smallest integer satisfying m0 >

χ1 + C0 C1 − C0 C2 |Ω|k0 . α0

Then we have qm < 0, m ≥ m0 . Then using the results of [197] we obtain

(1.5.29)

62 | 1 Attractor and its dimension estimation Theorem 1.5.4. Suppose A is the attractor of problem (1.5.8)–(1.5.9) in H01 (Ω) and m0 is determined by (1.5.29). Then we have the following estimates: (1) the Hausdorff dimension of A ≤ m0 ; (2) the fractal dimension of A ≤ 2m0 .

1.6 Derivative Ginzburg–Landau equation Derivative Ginzburg–Landau equation appears in many physical problems, for example, Rayleigh–Benard convection, Taylor–Couette flow in fluid mechanics, the dissipation in the plasma drift flow, chemical reaction of turbulent flow, see [19, 46, 102, 103, 133, 134, 135]. The generalized (with a derivative term) derivative Ginzburg–Landau equation in a one dimensional space has the following form: 𝜕u = α0 u + α1 uxx + α2 |u|2 u + α3 |u|2 ux + α4 u2 u∗x + α5 |u|2σ u, 𝜕t

(1.6.1)

where σ > 0, αk = ak + ibk , α0 = a0 > 0, ak , bk are real constants and u∗x is the complex conjugate of ux . In 1992, Duan, Holmes and Titi [51] proved the global existence and uniqueness of a solution for equation (1.6.1) in a bounded area for σ = 2. In 1994, Duan, Holmes and Titi [50] proved the well-posedness of Cauchy problem (1.6.1). Guo and Gao proved the existence of the global attractor for the periodic initial value problem (1.6.1) and showed the finiteness of its fractal and Hausdorff dimensions [97]. If α2 = α3 = α4 = 0, then equation (1.6.1) reduces to 𝜕u = ρu + (1 + iv)Δu − (1 + iμ)|u|2σ u. 𝜕t

(1.6.2)

In 1990, Bartuccelli et al. [10] studied the “hard” and “soft” states. In 1994, Doering et al. [49] proved the existence of weak and strong solutions in a two-dimensional space. Now we consider the following generalized equation in a two-dimensional bounded area: 𝜕u = ρu + (1 + iv)Δu − (1 + iμ)|u|2σ u + αλ1 ⋅ ∇(|u|2 u) + β(λ2 ⋅ ∇u)|u|2 𝜕t

(1.6.3)

with initial conditions u(x, 0) = u0 (x),

x∈Ω

(1.6.4)

and boundary conditions Ω = (0, L1 ) × (0, L1 ),

u is periodic in Ω,

(1.6.5)

1.6 Derivative Ginzburg–Landau equation

| 63

where u(x, t) is an unknown complex variable function, σ > 0, ρ > 0, v, μ, α and β all are real constants and λ1 , λ2 are real vectors. 2 In 1996, Guo and Wang [114] proved that if u0 ∈ Hper (Ω) and ∃σ > 0 such that ϵ≤σ≤

1 √1 +

2 ( u−vδ )2 1+δ2

−1

(1.6.6)

,

then there exists a global unique solution u(x, t) of problem (1.6.1)–(1.6.5), u(x, t) ∈ L∞ (0, T; H 2 (Ω)) ∩ L2 (0, T; H 3 (Ω)),

∀T > 0,

and the equation has a global attractor having finite Hausdorff and fractal dimensions. Now we establish a consistent a priori estimate for problem (1.6.1)–(1.6.5). 2 Lemma 1.6.1. Suppose u0 ∈ Hper (Ω). Then for a solution of (1.6.1)–(1.6.5), we have

󵄩󵄩 󵄩2 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ C1 ,

∀t ≥ t1 .

Proof. Taking the inner product of (1.6.3) and u in L2 (Ω), and then the real component, we obtain 1 d ‖u‖2 = ρ ∫ |u|2 dx − ‖∇u‖2 − ∫ |u|2σ+2 dx 2 dt Ω

Ω

2

+ α Re ∫(λ1 ⋅ 󳶚(|u| u))u∗ dx + β Re ∫(λ2 ⋅ 󳶚u)|u|2 u∗ dx. Ω

(1.6.7)

Ω

Denoting λ2 = (a, b), we have Re ∫(λ2 ⋅ 󳶚u)|u|2 u∗ dx Ω

= Re ∫(a Ω

𝜕u ∗ 𝜕u ∗ u +b u )|u|2 dx 𝜕x1 𝜕x2

𝜕 1 𝜕 1 |u|2 dx + ∫ b|u|2 |u|2 dx = ∫ a|u|2 2 𝜕x1 2 𝜕x2 Ω

L2

=

Ω

L1

L1

L2

1 𝜕 1 𝜕 |u|2 dx1 + ∫ dx1 ∫ b|u|2 |u|2 dx2 = 0, ∫ dx2 ∫ a|u|2 2 𝜕x1 2 𝜕x2 0

0

0

2

0

Re ∫(λ1 ⋅ 󳶚(|u| u))u dx ∗

Ω

= Re ∫(λ1 ⋅ 󳶚|u|2 )|u|2 dx + Re ∫(λ1 ⋅ 󳶚u)|u|2 u∗ dx Ω

Ω

(1.6.8)

64 | 1 Attractor and its dimension estimation = ∫(λ1 ⋅ 󳶚|u|2 )|u|2 dx +

1 ∫(λ1 ⋅ 󳶚|u|2 )|u|2 dx = 0, 2

(1.6.9)

Ω

Ω

thus from (1.6.7)–(1.6.9) we obtain 1 d ‖u‖2 + ‖∇u|2 + ∫ |u|2σ+2 dx = ρ ∫ |u|2 dx 2 dt Ω

1 ≤ ∫ |u|2σ+2 dx + C 2

Ω

(by Young inequality),

(1.6.10)

Ω

where the constant C depends on parameters σ and ρ. In the following, C will possibly denote a different constant and will depend on parameters (σ, ρ, v, μ). From equation (1.6.10), we have 1 d 1 ‖u‖2 + ‖∇u‖2 + ∫ |u|2σ+2 dx ≤ C. 2 dt 2 Ω

It follows that d ‖u‖2 + ∫ |u|2σ+2 dx ≤ C. dt

(1.6.11)

Ω

Using Young inequality we obtain |u|2 = |u|2 ⋅ 1 ≤ |u|2σ+2 + C.

(1.6.12)

Integrating equation (1.6.12), we get ‖u‖2 ≤ ∫ |u|2σ+2 dx + C.

(1.6.13)

Ω

From equations (1.6.11) and (1.6.13), we have d ‖u‖2 + ‖u‖2 ≤ C. dt Moreover, by Gronwall inequality we have 󵄩󵄩 󵄩2 2 −t 2 −t 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ ‖u0 ‖ e + C ≤ R e + C, ≤ 2C,

∀t ≥ 0,

∀t ≥ t∗ ,

2

where t∗ = ln RC . The lemma has been proved. Lemma 1.6.2. Suppose the conditions of Lemma 1.6.1 are satisfied, and there exists δ > 0 such that equation (1.6.6) is valid. Then ∀ε > 0 we have 1 d 1 ∫ |u|2σ+2 dx ≤ − ∫ |u|4σ+2 dx + ε‖Δu‖2 2(1 + σ) dt 2 Ω

Ω

1.6 Derivative Ginzburg–Landau equation

+ ε‖∇u‖4 − 2

| 65

1 󵄨 󵄨2 ∫ |u|2σ−2 ((1 + 2σ)󵄨󵄨󵄨∇|u|2 󵄨󵄨󵄨 4 Ω

− 2vσ∇|u| ⋅ i(u∇u∗ − u∗ ∇u) + |u∇u∗ − u∗ ∇u|2 )dx + C3 − C2 (ε)(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

,

∀t ≥ t2 ,

where the constant C3 depends on parameters (σ, ρ, ν, μ), C2 (ε) depends on ε, t2 depends on R, and ‖u0 ‖ ≤ R0 . Proof. Taking the inner product of (1.6.3) and |u|2σ u in L2 , we have ∫ Ω

𝜕u 2σ ∗ |u| u dx = ρ ∫ |u|2σ+2 dx + (1 + iv) ∫ Δu ⋅ |u|2σ u∗ dx − (1 + iμ) ∫ |u|4σ−2 dx 𝜕t Ω

Ω

2

Ω

2σ ∗

2σ+2 ∗

+ α ∫(λ1 ⋅ 󳶚(|u| u))|u| u dx + β ∫(λ2 ⋅ 󳶚u)|u| Ω

u dx. (1.6.14)

Ω

Since (1 + iv) ∫ Δu ⋅ |u|2σ u∗ dx = −(1 + iv) ∫ |∇u|2 |u|2σ dx Ω

Ω

− (1 + iv) ∫ σ|u|2σ−2 u∗ ∇u ⋅ ∇|u|2 dx,

(1.6.15)

Ω

taking the real component of equation (1.6.14), we get 1 d σ 󵄨 󵄨2 ∫ |u|2σ+2 dx = ρ ∫ |u|2σ−2 dx − ∫ |∇u|2 |u|2σ dx − ∫ |u|2σ−2 |󵄨󵄨󵄨∇u|2 󵄨󵄨󵄨 dx 2(1 + σ) dt 2 Ω

Ω

Ω

Ω

1 + vσ ∫ |u|2σ−2 ∇|u|2 ⋅ i(u∇u∗ − u∗ ∇u)dx 2 Ω

− ∫ |u|4σ+2 dx + Re α ∫(λ1 ⋅ (|u|2 u))|u|2σ u∗ dx Ω

Ω

+ Re β ∫(λ2 ⋅ ∇u)|u|2σ−2 u∗ dx.

(1.6.16)

Ω

Since |u|2 |∇u|2 =

1 󵄨󵄨 1 ∗ ∗ 2 2 󵄨2 󵄨∇|u| 󵄨󵄨󵄨 + |u∇u − u ∇u| , 4󵄨 4

(1.6.17)

we obtain − ∫ |∇u|2 |u|2σ dx − Ω

σ 1 󵄨 󵄨2 ∫ |u|2σ−2 󵄨󵄨󵄨|∇u|2 󵄨󵄨󵄨 dx + vσ ∫ |u|2σ−2 ∇|u|2 ⋅ i(u∇u∗ − u∗ ∇u)dx 2 2 Ω

Ω

66 | 1 Attractor and its dimension estimation 1 󵄨 󵄨2 = − ∫ |u|2σ−2 ((1 + 2σ)󵄨󵄨󵄨|∇u|2 󵄨󵄨󵄨 − 2vσ∇|u|2 ⋅ i(u∇u∗ − u∗ ∇u) 4 Ω

+ |u∇u∗ − u∗ ∇u|2 )dx

1 3 ρ ∫ |u|2σ+2 dx = ρ ∫ |u|2σ+1 ⋅ |u|dx ≤ ∫ |u|4σ+2 dx + ρ2 ∫ |u|2 dx 6 2 Ω

Ω

1 ≤ ∫ |u|4σ+2 dx + C, 6

Ω

(1.6.18)

Ω

∀t ≥ t1 ,

(1.6.19)

Ω

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2σ+2 ∗ u dx󵄨󵄨󵄨 ≤ |βλ2 | ∫ |∇u||u|2 |u|2σ+1 dx 󵄨󵄨Re β ∫(λ2 ⋅ ∇u)|u| 󵄨󵄨 󵄨󵄨 Ω

Ω

1 ≤ 3|βλ2 | ∫ |∇u| |u| dx + ∫ |u|4σ+2 dx 12 2

2

4

Ω

Ω


3 and 0 < γ ≤ 1, we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2σ+2 ∗ u dx󵄨󵄨󵄨 󵄨󵄨Re β ∫(λ2 ⋅ ∇u)|u| 󵄨󵄨 󵄨󵄨 Ω

4(1−θ) ≤ C|βλ2 |2 ‖u‖4θ+1 + H 2 ‖u‖H 1 ‖u‖q

≤

γ‖u‖2H 2

+ C(γ)|βλ2 |

4 1−4θ

2 1−4θ H1

‖u‖

1 ∫ |u|4σ+2 dx 12 Ω

8(1−θ) 1−4θ

‖u‖q

+

1 ∫ |u|4δ+2 dx. 12 Ω

When q >

14 , 3

we have

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2σ+2 ∗ u dx 󵄨󵄨󵄨 󵄨󵄨Re β ∫(λ2 ⋅ ∇u)|u| 󵄨󵄨 󵄨󵄨 Ω

8

16(1−θ)

≤ γ‖u‖2H 2 + γ‖u‖4H 1 + C(γ)|βλ2 | 1−8θ ‖u‖q 1−8θ +

1 ∫ |u|4σ+2 dx. 12 Ω

(1.6.23)

1.6 Derivative Ginzburg–Landau equation

When q >

34 , 3

| 67

we have

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2σ+2 ∗ u dx󵄨󵄨󵄨 󵄨󵄨Re β ∫(λ2 ⋅ ∇u)|u| 󵄨󵄨 󵄨󵄨 Ω

≤ γ‖u‖2H 2 + γ‖u‖4H 1 +

8q 1 1 ‖u‖qq + C(γ)|βλ2 | q−8qθ−16−16θ + ∫ |u|4σ−2 dx. 12 12

Ω

Since σ ≥ 3, we have q = 4σ + 2 >

34 3

and so

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2σ+2 ∗ u dx󵄨󵄨󵄨 󵄨󵄨Re β ∫(λ2 ⋅ ∇u)|u| 󵄨󵄨 󵄨󵄨 Ω

≤ γ‖u‖2H 2 + γ‖u‖4H 1 +

8(1+σ)(1+2σ) 1 1 6σ 2 −11σ−7 + ‖u‖4σ+2 ∫ |u|4σ−2 dx, 4σ+2 + C(γ)|βλ2 | 12 12

Ω

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2σ+2 ∗ u dx󵄨󵄨󵄨 󵄨󵄨Re β ∫(λ2 ⋅ ∇u)|u| 󵄨󵄨 󵄨󵄨 Ω

8(1+σ)(1+2σ) 1 ∫ Ω|u|4σ+2 dx + C(γ)|βλ2 | 6σ2 −11σ−7 6 8(1+σ)(1+2σ) 1 2 4 ≤ γC‖Δu‖ + 8γ‖∇u‖ + ∫ |u|4σ+2 dx + C + C(γ)|βλ2 | 6σ2 −11σ−7 . 6

≤ γ‖u‖2H 2 + γ‖u‖4H 1 +

(1.6.24)

Ω

Since ∇(|u|2 u) = |u|2 ∇u + u∇|u|2 = 2|u|2 ∇u + u2 ∇u,

(1.6.25)

we similarly have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2σ ∗ 󵄨󵄨Re α ∫(λ1 ⋅ ∇(|u| u))|u| u dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

≤ 3|αλ1 | ∫ |∇u||u|2 ⋅ |u|2σ+1 dx ≤ γC‖Δu‖2 Ω

+ 8γ‖∇u‖4 +

8(1+σ)(1+2σ) 1 ∫ |u|4σ+2 dx + C(γ)|αλ1 | 6σ2 −11σ−7 . 6

Ω

From equations (1.6.16), (1.6.18), (1.6.19), (1.6.24) and (1.6.26), we get 1 d 1 ∫ |u|2σ+2 dx ≤ − ∫ |u|4σ+2 dx + γC‖Δu‖2 2(1 + σ) dt 2 Ω

Ω

+ 16γ‖∇u‖4 + C + C(γ)(|αλ1 | + |βλ2 |) 1 󵄨 󵄨2 − ∫ |u|2σ−2 ((1 + 2σ)󵄨󵄨󵄨∇|u|2 󵄨󵄨󵄨 4 Ω

8(1+σ)(1+2σ) 6σ 2 −11σ−7

(1.6.26)

68 | 1 Attractor and its dimension estimation − 2vσ∇|u|2 ⋅ i(∇u∗ − u∗ ∇u) + |u∇u∗ − u∗ ∇u|2 )dx, when γ is small enough, and the lemma is proved. Lemma 1.6.3. Suppose the condition of Lemma 1.6.2 is satisfied. Then we have ‖∇u‖2 ≤ C3 + C3 (|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

,

∀t ≥ t3 ,

(1.6.27)

where constant C3 depends on data parameters; t3 depends on data parameters and R; ‖u0 ‖H 1 ≤ R. Proof. Taking the L2 inner product of equation (1.6.3) and Δu, we get 1 d ‖∇u‖2 + ‖Δu‖2 = ρ‖∇u‖2 + Re(1 + iμ) ∫ |u|2σ uΔu∗ dx − α Re ∫(λ1 ⋅ ∇(|u|2 u))Δu∗ dx 2 dt Ω 2

Ω

− β Re ∫(λ2 ⋅ ∇u)|u| Δu dx,

(1.6.28)

∗

Ω

∀ε > 0, and, by Young inequality, ρ‖∇u‖2 ≤ ε‖∇u‖4 + C(ε),

(1.6.29)

Re(1 + iμ) ∫ |u|2σ uΔu∗ dx Ω

= − Re(1 + iμ) ∫ |u|2σ |∇u|2 dx − Re(1 + iμ) ∫ σ|u|2σ−2 uΔu∗ ⋅ ∇|u|2 dx Ω

Ω

σ 󵄨 󵄨 = − ∫ |u| |∇u| dx − ∫ |u|2σ−2 󵄨󵄨󵄨∇|u|2 󵄨󵄨󵄨2 dx 2 2σ

2

Ω

Ω

1 + μσ ∫ |u|2σ−2 ∇|u|2 ⋅ i(u ⋅ ∇u − u∇u∗ )dx 2 Ω

1 󵄨 󵄨2 = − ∫ |u|2σ−2 ((1 + 2σ)󵄨󵄨󵄨∇|u|2 󵄨󵄨󵄨 4 Ω

− 2μσ∇|u|2 ⋅ i(u∗ ∇u − u∇ u∗ ) + |u∗ ∇u − u∇u∗ |2 )dx 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 ∗ 󵄨󵄨−β Re ∫(λ2 ⋅ ∇u)|u| Δu dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(by (1.6.25))

Ω

≤ |βλ2 | ∫ |∇u||u|2 |Δu|dx Ω

≤ |βλ2 |‖Δu‖‖∇u‖4 ‖u‖28 1 2

(by Hölder inequality) 1

2(1−θ) ≤ C|βλ2 |‖Δu‖‖∇u‖H 1 ‖∇u‖ 2 ‖u‖2θ H 2 ‖u‖q 2θ− 3

1

≤ C|βλ2 |‖u‖H 2 2 ‖∇u‖ 2 ‖u‖2(1−θ) q 4

2

8(1−θ)

≤ γ‖u‖2H 2 + C(γ)|βλ2 | 1−4θ ‖∇u‖ 1−4θ ‖u‖q1−4θ

(when q > 3, 0 < γ ≤ 1),

(1.6.30)

1.6 Derivative Ginzburg–Landau equation

| 69

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2 ∗ 󵄨󵄨−β Re ∫(λ2 ⋅ ∇u)|u| Δu dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

16(1−θ)

8

≤ γ‖u‖2H 2 + γ‖∇u‖4 + C(γ)|βλ2 | 1−8θ ‖u‖q 1−8θ

(when q >

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2 ∗ 󵄨󵄨−β Re ∫(λ2 ⋅ ∇u)|u| Δu dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

14 ), 3

Ω

8q

≤ γ‖u‖2H 2 + γ‖∇u‖4 + γ‖u‖qq + C(γ)|βλ2 | q−8qθ−16+16θ

(when q >

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 ∗ 󵄨󵄨−β Re ∫(λ2 ⋅ ∇u)|u| Δu dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

34 ), 3

8q

≤ γ‖△u‖2 + γ‖∇u‖4 + γ‖u‖qq + C(γ)|βλ2 | q−8qθ−16+16θ ≤ γ‖Δu‖2 + γ‖∇u‖4 + γ‖u‖4σ+2 4σ+2 + C(γ)|βλ2 |

8(1+σ)(1+2σ) 6σ 2 −11σ−7

(when σ ≥ 3, q = 4σ + 2 >

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 ∗ 󵄨󵄨−β Re ∫(λ2 ⋅ ∇u)|u| Δu dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

34 ), 3

Ω

8(1+σ)(1+2σ) 1 1 1 6σ 2 −11σ−7 . ≤ ε‖△u‖2 + ε‖∇u‖4 + ε‖u‖4σ+2 4σ+2 + C(ε)|βλ2 | 2 2 2

(1.6.31)

Similarly, by equation (1.6.25) we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 ∗ 󵄨󵄨−α Re ∫(λ1 ⋅ ∇(u|u| ))Δu dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

8(1+σ)(1+2σ) 1 1 1 6σ 2 −11σ−7 . ≤ ε‖Δu‖2 + ε‖∇u‖4 + ε‖u‖4σ+2 4σ+2 + C(ε)|αλ1 | 2 2 2

(1.6.32)

From equations (1.6.28)–(1.6.32), we have 1 d ‖∇u‖2 + ‖Δu‖2 ≤ ϵ‖Δu‖2 + 2ε‖∇u‖4 + ε‖u‖4σ+2 4σ+2 2 dt + C(ε) + C(ε)(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

−

1 ∫ |u|2σ−2 ((1 + 2σ)|∇|u|2 |2 4 Ω

2

− 2μσ∇|u| ⋅ i(u ⋅ ∇u − u∇u ) + |u ⋅ ∇u − u∇u∗ |2 )dx. ∗

∗

∗

Using equation (1.6.33) and Lemma 1.6.2, we arrive at 1 d δ2 δ2 (‖∇u‖2 + ∫ |u|2σ+2 dx) + ‖Δu‖2 + ∫ |u|4σ+2 dx 2 dt 1+σ 2 2

2

Ω

2

4

≤ ε(1 + δ )‖Δu‖ + ε(2 + δ )‖∇u‖

Ω

+ ε ∫ |u|4σ−2 dx + C(ε) + C(ε)(|αλ1 | + |βλ2 |) Ω

8(1+σ)(1+2σ) 6σ 2 −11σ−7

(1.6.33)

70 | 1 Attractor and its dimension estimation

−

1 󵄨 󵄨2 ∫ |u|2σ−2 ((1 + 2σ)(1 + δ2 )󵄨󵄨󵄨|∇u|2 󵄨󵄨󵄨 4 Ω

+ 2σ(vδ2 − μ)∇|u|2 ⋅ i(u∗ ∇u − u∇u∗ )

+ (1 + δ2 )|u∗ ∇u − u∇u∗ |2 )dx.

(1.6.34)

Since ‖∇u‖2 = (−Δu, u) ≤ ‖Δu‖‖u‖ ≤ K‖Δu‖,

(1.6.35)

where K depends only on data parameters, but not on ε, we have ε(1 + σ 2 )‖Δu‖2 + ε(2 + δ2 )‖∇u‖4 ≤ εK0 ‖Δu‖2 1 ≤ ‖Δu‖2 . 2

(K0 is independent of ε) (1.6.36)

Inspecting condition (1.6.6), we deduce that the matrix (1 + 2σ)(1 + σ 2 ) σ(vδ2 − μ)

σ(vδ2 − μ) ) 1 + σ2

(

2

is negative definite. So the last term on the right of (1.6.34) is positive. Choose ε ≤ δ4 small enough so that equation (1.6.34) is valid. By equations (1.6.34) and (1.6.36), we get δ2 1 1 d 1 (‖∇u‖2 + ∫ |u|2σ+2 dx) + ‖Δu‖2 + δ2 ∫ |u|4σ+2 dx 2 dt 1+σ 2 4 Ω

≤ C + C(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

Ω

(1.6.37)

.

From equation (1.6.35), we get ‖∇u‖2 ≤ K‖Δu‖ ≤ ‖Δu‖2 +

1 2 K , 4

(1.6.38)

δ2 δ2 ∫ |u|2σ+2 dx = ∫ |u|2σ+1 |u|dx 2(1 + σ) 2(1 + σ) Ω

≤

Ω

2

δ δ2 ∫ |u|4σ+2 dx + C ∫ |u|2 dx ≤ ∫ |u|4σ+2 dx + C. 4 4 Ω

Ω

Ω

From equations (1.6.37)–(1.6.39), we get d δ2 δ2 (‖∇u‖2 + ∫ |u|2σ+2 dx) + ‖∇u‖2 + ∫ |u|2σ+2 dx dt 1+σ 1+σ Ω

≤ C + C(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

Ω

,

∀t ≥ t∗ ,

where t∗ = max{t1 , t2 } and t1 , t2 are described in Lemmas 1.6.1 and 1.6.2.

(1.6.39)

| 71

1.6 Derivative Ginzburg–Landau equation

Using Gronwall inequality, we deduce δ2 󵄩2 󵄩󵄩 ∫ |u|2σ+2 dx 󵄩󵄩∇u(t)󵄩󵄩󵄩 + 1+σ Ω

δ2 󵄩2 󵄨2σ+2 󵄩 󵄨 dx)e−(t−t∗ ) + C ≤ (󵄩󵄩󵄩∇u(t∗ )󵄩󵄩󵄩 + ∫ 󵄨󵄨u(t )󵄨󵄨 1+σ 󵄨 ∗ 󵄨 Ω

+ C(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

∀t ≥ t∗ .

,

(1.6.40)

From the existence of a global solution for equation (1.6.3)–(1.6.5), we can readily obtain δ2 󵄩󵄩 󵄩2 󵄨 󵄨2σ+2 dx ≤ C(R), ∫ 󵄨󵄨u(t )󵄨󵄨 󵄩󵄩∇u(t∗ )󵄩󵄩󵄩 + 1+σ 󵄨 ∗ 󵄨

(1.6.41)

Ω

where C(R) depends on data parameters and R, and ‖u0 ‖H 2 ≤ R. Then, by equations (1.6.40)–(1.6.41), we get δ2 󵄩󵄩 󵄩2 󵄨 󵄨2σ−2 dx ∫ 󵄨󵄨u(t)󵄨󵄨󵄨 󵄩󵄩∇u(t)󵄩󵄩󵄩 + 1+σ 󵄨 Ω

≤ C(R)e

−(t−t∗ )

+ C + C(|αλ1 | + |βλ2 |)

≤ 2C + C(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

,

8(1+σ)(1+2σ) 6σ 2 −11σ−7

,

∀t ≥ t∗󸀠

∀t ≥ t∗󸀠 ,

where t∗󸀠 = max{t∗ , t∗ + ln C(R) − ln(C + C(|αλ1 | + |βλ2 |)

8(1+σ)(1+2σ) 6σ 2 −11σ−7

)}

which yields the claim of this lemma. Lemma 1.6.4. Suppose the condition of Lemma 1.6.2 is satisfied. Then we have 16(1+σ)(1+2σ)(7+8σ) 8+ 240(1+σ)(1+2σ) 󵄩󵄩 󵄩2 󵄩󵄩Δu(t)󵄩󵄩󵄩 ≤ C4 + C4 (|αλ1 | + |βλ2 |) 3(6σ2 −11σ−7) + C4 (|αλ1 | + |βλ2 |) 6σ2 −11σ−7 ,

∀t ≥ t4

where C4 depends on data parameters, t4 depends on data parameters and R, and ‖u0 ‖H 2 ≤ R. Proof. Taking the inner product equation (1.6.3) and Δ2 u, and then its the real component, we arrive at 1 d ‖Δu‖2 = ρ‖Δu‖2 − ‖∇Δu‖2 2 dt

− Re(1 + iμ) ∫ |u|2σ uΔ2 u∗ dx Ω

72 | 1 Attractor and its dimension estimation + α Re ∫(λ1 ⋅ ∇(|u|2 u))Δ2 u∗ dx Ω

+ β Re ∫(λ2 ⋅ ∇u)|u|2 Δ2 u∗ dx.

(1.6.42)

Ω

Applying Lemmas 1.6.1 and 1.6.3, we get ∀t ≥ t∗ , d ‖Δu‖2 + ‖∇Δu‖2 + ‖Δu‖2 dt ≤ C + C(|αλ1 | + |βλ2 |)

16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7)

8+ 240(1+σ)(1+2σ) 2

+ C(|αλ1 | + |βλ2 |)

6σ −11σ−7

(1.6.43)

.

Using Gronwall lemma, we obtain 󵄩󵄩 󵄩2 󵄩 󵄩2 −(t−t∗ ) +C 󵄩󵄩Δu(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Δu(t∗ )󵄩󵄩󵄩 e + C(|αλ1 | + |βλ2 |)

16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7)

8+ 240(1+σ)(1+2σ) 2

+ C(|αλ1 | + |βλ2 |)

6σ −11σ−7

,

∀t ≥ t∗ . (1.6.44)

With the aid of the priori estimate of the global solution, we have 󵄩󵄩 󵄩2 󵄩󵄩Δu(t∗ )󵄩󵄩󵄩 ≤ C(R), where C(R) depends on data parameters and R, and ‖u0 ‖H 2 ≤ R. When t is large enough, we get 16(1+σ)(1+2σ)(7+8σ) 8+ 240(1+σ)(1+2σ) 󵄩󵄩 󵄩2 󵄩󵄩Δu(t)󵄩󵄩󵄩 ≤ 2C + 2C(|αλ1 | + |βλ2 |) 3(6σ2 −11σ−7) + 2C(|αλ1 | + |βλ2 |) 6σ2 −11σ−7 ,

which proves the claim. Notice that 2 󵄩 󵄩 ‖u‖2H 2 ≤ C(󵄩󵄩󵄩Δu(t)󵄩󵄩󵄩 + ‖u‖)

≤ C + C(|αλ1 | + |βλ2 |)

16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7)

8+ 240(1+σ)(1+2σ) 2

+ C(|αλ1 | + |βλ2 |)

6σ −11σ−7

,

∀t ≥ t∗ , (1.6.45)

,

∀t ≥ t∗ . (1.6.46)

where C merely depends on data parameters and (σ, ρ, v, μ). Also ‖u‖2∞ ≤ C(‖u‖‖u‖H 2 ) ≤ C + C(|αλ1 | + |βλ2 |)

16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7)

8+ 240(1+σ)(1+2σ) 2

+ C(|αλ1 | + |βλ2 |)

6σ −11σ−7

Lemma 1.6.5. Suppose the condition of Lemma 1.6.2 is satisfied. Then we have 󵄩󵄩 󵄩2 󵄩󵄩∇Δu(t)󵄩󵄩󵄩 ≤ K,

∀t ≥ t∗ ,

where K depends on data parameters (σ, ρ, v, μ, α, β, λ1 , λ2 , Ω); t5 depends on data parameters (σ, ρ, v, μ, α, β, λ1 , λ2 , Ω) and R, and ‖u0 ‖H 2 ≤ R.

1.6 Derivative Ginzburg–Landau equation

| 73

Proof. Taking the inner product of equation (1.6.3) and Δ3 u, and then its the real component, we deduce 1 d 󵄩2 󵄩 ‖∇Δu‖2 = ρ‖∇Δu‖2 − 󵄩󵄩󵄩Δ2 u󵄩󵄩󵄩 2 dt

+ Re(1 + iμ) ∫ |u|2σ uΔ3 u∗ dx Ω

− Re α ∫(λ1 ⋅ ∇(|u|2 u))Δ3 u∗ dx Ω

− Re β ∫(λ2 ⋅ ∇u)|u|2 Δ3 u∗ dx.

(1.6.47)

Ω

Applying Lemmas 1.6.1, 1.6.3 and 1.6.4, after further complicated operations, we get 󵄩 󵄩 |Re(1 + iμ) ∫ |u|2σ uΔ3 u∗ dx| ≤ K 󵄩󵄩󵄩Δ2 u󵄩󵄩󵄩 Ω

1󵄩 󵄩2 ≤ 󵄩󵄩󵄩Δ2 u󵄩󵄩󵄩 + K 6 󵄨󵄨 󵄨󵄨 󵄨 󵄨 × 󵄨󵄨󵄨α ∫(λ1 ⋅ ∇(|u|2 u))Δ3 u∗ dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(1.6.48)

Ω

1󵄩 󵄩2 ≤ 󵄩󵄩󵄩Δ2 u󵄩󵄩󵄩 + K‖∇Δu‖2 + K 6 󵄨󵄨 󵄨󵄨 󵄨 󵄨 × 󵄨󵄨󵄨β ∫(λ2 ⋅ ∇u)|u|2 Δ3 u∗ dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(1.6.49)

Ω

1󵄩 󵄩2 ≤ 󵄩󵄩󵄩Δ2 u󵄩󵄩󵄩 + K‖∇Δu‖2 + K 6

(1.6.50)

From (1.6.47)–(1.6.50), we have 1 d 󵄩 󵄩2 ‖∇Δu‖2 ≤ ρ‖∇Δu‖2 − 󵄩󵄩󵄩Δ2 u󵄩󵄩󵄩 + 2 dt ≤ (K + ρ)‖∇Δu‖2 + K,

1 󵄩󵄩 2 󵄩󵄩2 2 󵄩Δ u󵄩󵄩 + K‖∇Δu‖ + K 2󵄩

thus arriving at d ‖∇Δu‖2 ≤ K‖∇Δu‖2 + K. dt

(1.6.51)

Due to equation (1.6.43), we have d ‖Δu‖2 + ‖∇Δu‖2 ≤ K, dt

∀t ≥ t∗ .

(1.6.52)

Integrating equation (1.6.52) from t to t + 1, we have t+1

󵄩󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩Δu(t + 1)󵄩󵄩󵄩 − 󵄩󵄩󵄩Δu(t)󵄩󵄩󵄩 + ∫ ‖∇Δu‖ dt ≤ K, t

∀t ≤ t∗ .

74 | 1 Attractor and its dimension estimation Applying Lemma 1.6.4, we obtain t+1

∫ ‖∇Δu‖2 dt ≤ K,

∀t ≥ t∗ .

t

(1.6.53)

Using equations (1.6.51) and (1.6.53) and consistent Gronwall lemma, we get 󵄩2 󵄩󵄩 󵄩󵄩∇Δu(t)󵄩󵄩󵄩 ≤ K,

t ≥ t∗ + 1,

which shows that the lemma has been proved. Now we establish the global existence of the attractor and its Hausdorff and fractal dimension estimates for the problem (1.6.3)–(1.6.5). From (1.6.45), we infer that the ball B = {u ∈ H 2 (Ω) : ‖u‖H 2 ≤ K0 } is the absorbing set of S(t) in H 2 (Ω). Lemma 1.6.5 shows that the semigroup S(t) in H 2 (Ω) is compact for sufficiently large t. So we can get the global existence of the attractor by using results of [197]. We get the following: Theorem 1.6.1. Suppose (1.6.6) is valid. Then the ω limit set A = ω(B) = ⋂ ⋃ S(t)B s≥0 t≥s

is the compact attractor of S(t) in H 2 (Ω). Here the closure is taken in H 2 (Ω). Next, we prove that the dimension of the global attractor A is finite. For this, rewrite equation (1.6.3) in the abstract form du = F(u), dt

(1.6.54)

where F(u) means the right-hand side of equation (1.6.3). We consider a variational equation for problem (1.6.3)–(1.6.5), namely vt = F 󸀠 (u(t))v

(1.6.55)

v(0) = v0 ∈ H,

(1.6.56)

with the initial value

where F 󸀠 (u(t))ν = ρν + (1 + iv)Δν − (1 + iμ)(1 + σ)|u|2σ v

− (1 + iμ)σ|u|2σ−2 u2 v∗ + 2α(λ1 ⋅ ∇(|u|2 v))

1.6 Derivative Ginzburg–Landau equation

| 75

+ α(λ1 ⋅ ∇(u2 v∗ )) + β(λ2 ⋅ ∇v)|u|2 + β(λ2 ⋅ ∇u)(vu∗ + uv∗ ),

u(t) = S(t)u0 is the solution of problem (1.6.3)–(1.6.5), and u0 ∈ A . We know that, for u0 ∈ A , we have S(t)u0 ∈ H 2 (Ω). Using standard methods, ∀v0 ∈ H we can prove that the linear initial value problem (1.6.55)–(1.6.56) has a unique solution v(t) such that v(t) ∈ L2 (0, T; H 1 (Ω)) ∩ L∞ (0, T; H),

∀T > 0.

(1.6.57)

For ∀v0 ∈ H, let G(t)v0 denote the solution of (1.6.55)–(1.6.56), and through complex calculations and energy estimation, we can prove that for ∀ω0 , u0 ∈ A , ‖S(t)ω0 − S(t)u0 − G(t)(ω0 − u0 )‖2 ≤ K‖ω0 − u0 ‖, ‖ω0 − u0 ‖2

∀0 ≤ t ≤ T,

where K depends on the data parameters (σ, ρ, v, μ, α, β, λ1 , λ2 , Ω), T and R; ‖u0 ‖2H ≤ R. This inequality shows that the semigroup S(t) is differentiable on A , and the differential operator L(t, u0 ) : v0 ∈ H → G(t)v0 ∈ H. We consider that v0 = v01 , . . . , v0m are m elements in H. The corresponding solutions of (1.6.55)–(1.6.56) are v(t) = v1 (t), . . . , vm (t). Then by the results of [197] we have t

󵄨󵄨 󵄨 󸀠 󵄨󵄨ν1 (t) ∧ ⋅ ⋅ ⋅ ∧ νm (t)󵄨󵄨󵄨Λm H = |v01 ∧ ⋅ ⋅ ⋅ ∧ v0m |Λm H exp ∫ Re tr F (u(τ)) ⋅ Qm (τ)dτ,

(1.6.58)

0

where Qm (τ) = Qm (τ, u0 , v01 , . . . , v0m ) is the orthogonal projection of {v1 (τ), . . . , vm (τ)}. Suppose for an given time τ, φj (τ), j ∈ N is the orthogonal basis of H, and Qm (τ)H = span{v1 (τ), . . . , vm (τ)}, vj (τ) ∈ H 1 (Ω). Then Re tr F 󸀠 (u(τ)) ⋅ Qm (τ) m

= ∑ Re(F 󸀠 (u(τ)) ⋅ Qm (τ)φj (τ), φj (τ)) j=1 m

= ∑ Re(F 󸀠 (u(τ)φj (τ)), φj (τ)). j=1

Omitting parameter τ, we get Re (F 󸀠 (u)φj , φj ) = ρ‖φj ‖2 − ‖∇φj ‖2 − (1 + σ) ∫ |u|2σ |φj |2 dx Ω

(1.6.59)

76 | 1 Attractor and its dimension estimation 2

− Re(1 + iμ)σ ∫ |u|2σ−2 u2 (φ∗j ) dx + Re 2α ∫(λ1 ⋅ ∇(|u|2 φj ))φ∗j dx Ω

+

Ω

Re α ∫(λ1 ⋅ ∇(|u|2 φ∗j ))φ∗j dx Ω

+ Re β ∫(λ2 ⋅ ∇φj )|u|2 φ∗j dx Ω

2

+ Re β ∫(λ2 ⋅ ∇u)(u |φj | + ∗

2 u(φ∗j ) )dx.

(1.6.60)

Ω

Now we estimate the right-hand side of (1.6.60): 2

− Re(1 + iμ)σ ∫ |u|2σ−2 u2 (φ∗j ) dx Ω

2 ≤ σ|1 + iμ|‖u‖2σ ∞ ‖φj ‖

≤ C‖φj ‖2 + C‖φj ‖2 (|αλ1 | + |βλ2 |)

8σ(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7)

4σ+ 120σ(1+σ)(1+2σ) 2

+ C‖φj ‖2 (|αλ1 | + |βλ2 |)

6σ −11σ−7

(1.6.61)

,

Re 2α ∫(λ1 ⋅ ∇(|u|2 φj ))φ∗j dx Ω

= −2α Re ∫(λ1 ⋅ ∇φ∗j )|u|2 φj dx Ω

≤ 2|αλ1 |‖u‖2∞ ‖∇φj ‖‖φj ‖ ≤ ≤

1 ‖∇φj ‖2 + C|αλ1 |‖u‖4∞ ‖φj ‖2 8

1 2 2+ 16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7) ‖∇φj ‖2 + C‖φj ‖2∞ (|αλ1 | + |βλ2 |) + C‖φj ‖2 (|αλ1 | − |βλ2 |) 8 10+ 240(1+σ)(1+2σ) 2

+ C‖φj ‖2 (|αλ1 | − |βλ2 |)

6σ −11σ−7

(1.6.62)

,

Re α ∫(λ1 ⋅ ∇(u2 φ∗j ))φ∗j dx = −α Re ∫(λ1 ⋅ ∇φ∗j )u2 φ∗j dx Ω

Ω

≤ |αλ1 |‖u‖2∞ ‖∇φj ‖‖φj ‖ ≤

1 2 2+ 16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7) ‖∇φj ‖2 + C‖φj ‖2 (|αλ1 | + |βλ2 |) + C‖φj ‖2 (|αλ1 | + |βλ2 |) 8 10+ 240(1+σ)(1+2σ) 2

+ C‖φj ‖2 (|αλ1 | + |βλ2 |)

6σ −11σ−7

,

(1.6.63)

Re β ∫(λ2 ⋅ ∇φj )|u|2 φ∗j dx Ω

≤ |βλ2 |‖u‖2∞ ‖∇φj ‖‖φj ‖ ≤

1 2 2+ 16(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7) ‖∇φj ‖2 + C‖φj ‖2 (|αλ1 | + |βλ2 |) + C‖φj ‖2 (|αλ1 | + |βλ2 |) 8 10+ 240(1+σ)(1+2σ) 2

+ C‖φj ‖2 (|αλ1 | + |βλ2 |)

6σ −11σ−7

,

(1.6.64)

1.6 Derivative Ginzburg–Landau equation

| 77

2

Re β ∫(λ2 ⋅ ∇u)(u∗ |φj |2 + u(φ∗j ) )dx Ω

≤ 2|βλ2 |‖u‖∞ ‖∇u‖‖φj ‖24

≤ C|βλ2 |‖u‖∞ ‖∇u‖‖φj ‖‖φj ‖H 1 1 ≤ ‖φj ‖2H 1 + C|βλ2 |2 ‖u‖2∞ ‖∇u‖2 ‖φj ‖2 8 1 ≤ ‖φj ‖2H 1 + C|βλ2 |2 ‖u‖2∞ (−Δu, u)‖φj ‖2 8 1 ≤ ‖φj ‖2H 1 + C|βλ2 |2 ‖u‖‖u‖H 2 ‖Δu‖‖u‖‖φj ‖2 8 1 ≤ ‖φj ‖2H 1 + C|βλ2 |2 ‖u‖2H 2 ‖φj ‖2 8 1 1 2 ≤ ‖∇φj ‖2 + ‖φj ‖2 + C‖φj ‖2 (|αλ1 | + |βλ2 |) 8 8 2+ 16(1+σ)(1+2σ)(7+8σ) 2

+ C‖φj ‖2 (|αλ1 | + |βλ2 |)

3(6σ −11σ−7)

10+ 240(1+σ)(1+2σ) 6σ 2 −11σ−7

+ C‖φj ‖2 (|αλ1 | + |βλ2 |)

(1.6.65)

.

By (1.6.61)–(1.6.65), we have m 1 m Re tr F 󸀠 (u(τ)) ⋅ Qm (τ) ≤ − ∑ ‖∇φj ‖2 + E ∑ ‖φj ‖2 , 2 j=1 j=1

where

(1.6.66)

2

E = C + C(|αλ1 | + |βλ2 |) + C(|αλ1 | + |βλ2 |)

8σ(1+σ)(1+2σ)(7+8σ) 3(6σ 2 −11σ−7)

4σ+ 120σ(1+σ)(1+2σ) 2

+ C(|αλ1 | + |βλ2 |)

6σ −11σ−7

2+ 16(1+σ)(1+2σ)(7+8σ) 2

+ C(|αλ1 | + |βλ2 |)

3(6σ −11σ−7)

10+ 240σ(1+σ)(1+2σ) 6σ 2 −11σ−7

+ C(|αλ1 | + |βλ2 |) Set

.

m

η = η(x, τ) = ∑ |φj |2 . j=1

(1.6.67) (1.6.68)

Since {φj ; j = 1, . . . , m} is an orthonormal set in H, we have m

∑ ‖φj ‖2 = ∫ ηdx = m. j=1

(1.6.69)

Ω

By Sobolev–Lieb–Thirring inequality, we have m

∫ η2 dx ≤ C0 ∫ ηdx + C0 ∑ ‖∇φj ‖2 , Ω

Ω

where C0 depends only on the shape of Ω.

j=1

(1.6.70)

78 | 1 Attractor and its dimension estimation By Hölder inequality we have 2

(∫ ηdx) ≤ |Ω| ∫ η2 dx. Ω

(1.6.71)

Ω

Due to equations (1.6.66)–(1.6.71), we have Re tr F 󸀠 (u(τ)) ⋅ Qm (τ) ≤ − ≤−

m m2 + + Em 2C0 |Ω| 2

2

1 m2 + C0 |Ω|(E + ) . 4C0 |Ω| 2

(1.6.72)

For i = 1, . . . , m and v0i ∈ H, we define t

1 qm (t) = sup sup ( ∫ Re tr F 󸀠 (S(τ)u0 ) ⋅ Qm (τ)dτ), u0 ∈A ‖ν0i ‖≤1 t qm (t) = lim sup qm (t).

0

t→∞

From (1.6.72), we obtain qm (t) ≤ −

2

m2 1 + C0 |Ω|(E + ) . 4C0 |Ω| 2

Hence, if m satisfies 1 m − 1 < √8C0 |Ω|(E + ) ≤ m, 2

(1.6.73)

then qm < 0. From these considerations we have Lemma 1.6.6. Suppose A is the global attractor of problem (1.6.3)–(1.6.5). Then the dimension of A ≤ m, and its fractal dimension ≤ 2m, where m is determined by (1.6.73).

1.7 Ginzburg–Landau model in superconductivity In this section, we consider the evolutional Ginzburg–Landau (GL) model in superconductivity 2

i ηΨt + iηkΦΨ + ( ∇ + A) Ψ − Ψ + |Ψ|2 Ψ = 0, (x, t) ∈ Ω × R+ , k i At + grad Φ + curl2 A + (Ψ∗ grad Ψ − Ψ grad Ψ∗ ) + |Ψ|2 A = 0, 2k

(1.7.1) (1.7.2)

with the boundary conditions ∇Ψ ⋅ n = 0,

i ( ∇Ψ + AΨ) × n = 0, k

(x, t) ∈ 𝜕Ω × R+

(1.7.3)

1.7 Ginzburg–Landau model in superconductivity | 79

and the initial conditions Ψ(x, 0) = Ψ0 (x),

A(x, 0) = A0 (x),

x ∈ Ω,

(1.7.4)

where Ω ⊂ RN , N = 2, 3, is a domain with a smooth boundary 𝜕Ω; R+ = [0, ∞); η, k are positive constants related to some physical quantities; i = −√−1, n is the unit outward normal vector of 𝜕Ω. In equations (1.7.1)–(1.7.2), Ψ, A, Φ are unknown functions, Ψ is a complex function and serves as an order parameter in GL theory, Ψ∗ denotes the complex conjugate of Ψ, |Ψ|2 means the density of superconducting charge carriers. Function Φ is scalar and real, representing the electrical potential. The vector function A is real and represents the magnetic potential, i. e., H = curl A is the magnetic field. In 1950, this model was first proposed by Ginzburg and Landau based on the second order phase transition theory of steady state conditions in fluids. In 1968, Gorkov and Eliashberg obtained equations (1.7.2)–(1.7.3) from Bardeen–Cooper–Schrieffer (BCS) theory. The discovery of high-temperature superconductivity has brought potential commercial use. The Ginzburg–Landau model of superconductivity has drawn interest of many people. Since 1984, Berger, Chapman and Yang, among others, did some studies on the existence, uniqueness and regularity of a global solution [14, 23, 208]. But regarding the asymptotical behavior when t → ∞, little research is done. In 1995, Guo and Wu [125] studied the long time asymptotic behavior and the existence of an attractor. The other results for the similar model can refer to the reference [58, 67, 72, 73, 92, 149, 150, 151, 153, 154, 155, 162, 164, 166, 167, 168, 176, 187, 200, 202, 204, 207]. From equations (1.7.2)–(1.7.3) we know that the number of unknown functions is larger than the number of equations. In order to guarantee the uniqueness of a solution, usually we must attach three types of gauge transformation: (1) The Coulomb gauge. For this gauge, A is taken to be divergence-free, that is, div A = 0, and it can be converted into 2

i ηΨt + iηkΦΨ + ( ∇ + A) Ψ − Ψ + |Ψ|2 Ψ = 0, (x, t) ∈ Ω × R+ , k i At + grad Φ + curl2 A + (Ψ∗ grad Ψ − Ψ grad Ψ∗ ) + |Ψ|2 A = 0, 2k i −ΔΦ = div[ (Ψ∗ grad Ψ∗ ) + |Ψ|2 A]. 2k

(1.7.5) (1.7.6)

(2) The instantaneous gauge, Φ = div A. In this case, this model becomes 2

i ηΨt + iηk div AΨ + ( ∇ + A) Ψ − Ψ + |Ψ|2 Ψ = 0, k i At + ΔA + (Ψ∗ grad Ψ − Ψ grad Ψ∗ ) + |Ψ|2 A = 0, 2k

(1.7.7) (x, t) ∈ Ω × R+ . (1.7.8)

80 | 1 Attractor and its dimension estimation (3) Zero potential gauge, Φ(x) = 0. Then the model can be rewritten as 2

i ηΨt + ( ∇ + A) Ψ − Ψ + |Ψ|2 Ψ = 0, k i At + curl2 A + (Ψ∗ grad Ψ) + |Ψ|2 A = 0, 2k

(x, t) ∈ Ω × R+ ,

(1.7.9)

(x, t) ∈ Ω × R+ .

(1.7.10)

The existence of a global solution for all kinds of gauge has been obtained. But regarding the asymptotical behavior, we cannot obtain satisfactory results under any gauges, since it is difficult to get the uniformly bounded estimate under various kinds of gauge. Here we get the H 2 -estimate of a solution under instantaneous gauge. Assume H m (Ω) is a standard Sobolev space and H m denotes the corresponding complex function space Hn1 (Ω) = {A ∈ H 1 (Ω), A ⋅ n = 0, x ∈ 𝜕Ω}

(1.7.11) 1

1

with the norm (‖div A‖2 + ‖ curl A‖2 ) 2 , A ∈ Hn1 (Ω), and the norm (‖div A‖2 + ‖curl A‖2 ) 2 is equivalent to the norm ‖∇A‖, i. e., ‖u‖ ≤ K1 ‖∇u‖,

u ∈ Hn1 (Ω).

(1.7.12)

In the following we establish an a priori estimate. Set DA = ki ∇+A. Then equations (1.7.1)–(1.7.2) can be written as for the gauge Φ = div A, namely ηΨt + iηk div AΨ + D2A Ψ − Ψ + |Ψ|2 Ψ = 0, Ω × R+ , 1 At − ΔA = [Ψ∗ DA Ψ + Ψ(DA Ψ)∗ ], Ω × R+ . 2

(1.7.13) (1.7.14)

It is easy to see that, for Ψ, Φ ∈ H 1 , A ∈ H 1 , DA satisfies (D2A Ψ, Φ∗ ) = (DA Ψ, DA Φ∗ ), (DA Ψ)t = DA Ψt + At Ψ.

(1.7.15) (1.7.16)

In the following, we always assume |Ψ0 (x)| ≤ 1, x ∈ Ω0 . Then we know |Ψ(x, t)| ≤ 1, (x, t) ∈ Ω × R+ . Multiplying equation (1.7.13) by Ψ∗ , integrating by parts, and taking the real part, we obtain η

d ‖Ψ‖2 + 2‖DA Ψ‖2 − ‖Ψ‖2 + ‖Ψ‖44 = 0. dt

(1.7.17)

Multiplying both sides of equation (1.7.13) by their conjugates, and integrating by parts, we arrive at η2 ‖Ψt ‖2 + ‖D2A Ψ‖2 − η

d ‖D Ψ‖2 = η(At Ψt , (DA Ψ)∗ ) + η(At Ψ∗ , DA Ψ) dt A ≤ 2‖DA Ψ‖‖At ‖ ≤ ‖DA Ψ‖2 + ‖At ‖2 .

(1.7.18)

1.7 Ginzburg–Landau model in superconductivity | 81

Multiplying both sides of equation (1.7.14) with themselves, and integrating by parts, we obtain ‖At ‖2 + ‖ΔA‖2 + =

d [‖div A‖2 + ‖curl A‖2 ] dt

1 2 ∫[Ψ∗ DA Ψ + Ψ(DA Ψ)∗ ] dx ≤ ‖DA Ψ‖2 . 4

(1.7.19)

Ω

The last inequality follows from |Ψ(0)| ≤ 1, x ∈ Ω. Then we have d [‖div A‖2 + ‖curl A‖2 + η‖Ψ‖2 ] + ‖At ‖2 + ‖ΔA‖2 + ‖DA Ψ‖2 ≤ C|Ω|. dt

(1.7.20)

Moreover, we multiply (1.7.13) by 2Ψt∗ , equation (1.7.14) by At , and add them together to get d 1 [‖ div A‖2 + ‖ curl A‖2 + ‖Ψ‖44 − ‖Ψ‖2 + ‖DA Ψ‖2 ] dt 2 ≤ −2η‖Ψt ‖2 − 2‖At ‖2 + iηk ∫ div A(Ψ∗ Ψt − ΨΨt∗ )dt Ω

4k 2 ≤ ‖div A‖2 − η‖Ψt ‖2 − 2‖At ‖2 . η

(1.7.21)

Noting that 1

1

1

1

‖∇A‖ ≤ C‖A‖ 2 ‖ΔA‖ 2 ≤ C‖∇A‖ 2 ‖ΔA‖ 2 ‖div A‖2 + ‖curl A‖2 ≤ C‖∇A‖, we get (‖div A‖2 + ‖curl A‖2 ) ≤ K2 ‖ΔA‖. This, together with equations (1.7.17), (1.7.19) and (1.7.21), implies that 1 d [Cη‖Ψ‖2 + C1 ‖div A‖2 + ‖curl A‖2 + ‖Ψ‖44 − ‖Ψ‖2 + ‖DA Ψ‖2 ] dt 2 1 + C2 [Cη‖Ψ‖2 + C1 ‖div A‖2 + ‖curl A‖2 + ‖Ψ‖44 − ‖Ψ‖2 + ‖DA Ψ‖2 ] 2 + C3 [η‖Ψt ‖2 + ‖At ‖2 ] ≤ C|Ω|.

(1.7.22)

By Gronwall inequality, we have 1 Cη‖Ψ‖2 + C1 ‖div A‖2 + ‖curl A‖2 + ‖Ψ‖44 − ‖Ψ‖2 + ‖DA Ψ‖2 2 1 ≤ e−C2 t [Cη‖Ψ0 ‖2 + C1 ‖∇A0 ‖2 + ‖Ψ0 ‖44 − ‖Ψ0 ‖2 + ‖DA Ψ0 ‖2 ] + C|Ω|(1 − e−C2 t ). 2 (1.7.23)

82 | 1 Attractor and its dimension estimation Choosing t0 large enough, for t ≥ t0 we have 1 Cη‖Ψ‖2 + C1 ‖div A‖2 + ‖curl A‖2 + ‖Ψ‖44 − ‖Ψ‖2 + ‖DA Ψ‖2 ≤ C|Ω|, 2

(1.7.24)

where the constant C is independent of Ψ0 and A0 . Hence, for t ≥ t0 , we get 1 (‖div A(t)‖2 + ‖curl A(t)‖2 ) 2 ≤ K3 . Noticing that ‖DA Ψ‖ ≥

1 ‖∇Ψ‖ − ‖A‖, k

we obtain 1 ‖∇Ψ‖ ≤ ‖DA Ψ‖ + ‖A‖, k

∀t ≥ t0 .

Thus ‖∇Ψ‖ ≤ K4 . Furthermore, from equation (1.7.22) we have t+1

󵄩 󵄩2 󵄩 󵄩2 ∫ [󵄩󵄩󵄩Ψs (s)󵄩󵄩󵄩 + 󵄩󵄩󵄩As (s)󵄩󵄩󵄩 ]ds ≤ K5 ,

t ≥ t0 .

(1.7.25)

t

Then we see that 1 󵄩 󵄩2 B = {(Ψ, A) ∈ H 1 × H 1 | ‖Ψ‖ ≤ 1, ‖∇Ψ‖ ≤ K4 , (󵄩󵄩󵄩div A(t)󵄩󵄩󵄩 + ‖curl A‖2 ) 2 ≤ K3 }

is an absorbing set of equation (1.7.13)–(1.7.14) in H 1 × H 1 . In order to prove that the ω-limit set of B is the attractor of (1.7.13)–(1.7.14) and (1.7.3)–(1.7.4) in H 1 × H 1 , |Ψ| ≤ 1, we must prove the compactness of this absorbing set. Noting that (D2A Ψ, Ψ∗ ) = (DA Ψ, (DA Ψ∗ )),

(D2A Ψ)t = D2A Ψt + DA (At Ψ) + At DA Ψ,

we differentiate (1.7.13) with respect to t, multiply it by Ψt∗ , and take the real part to get η d ‖Ψ ‖2 − ηk Im ∫(div At )ΨΨt∗ dx + ‖DA Ψt ‖2 2 dt t Ω

+ ∫ At Ψ(DA Ψt ) dx + ∫(DA Ψ)(At Ψt )∗ dx − ‖Ψt ‖2 + ∫ |Ψ|2 |Ψt |2 dx ≤ 0, ∗

Ω

Ω

(1.7.26)

Ω

which yields η d ‖Ψ ‖2 + ‖DA Ψt ‖2 + ∫ Ω|Ψ|2 |Ψt |2 dx 2 dt t ≤ ‖Ψt ‖2 + ηk‖∇At ‖‖Ψt ‖ − ∫ At Ψ(DA Ψt )∗ dx − ∫(DA Ψ)(At Ψt )∗ dx. Ω

Ω

(1.7.27)

1.7 Ginzburg–Landau model in superconductivity | 83

In the same way, we differentiate (1.7.14) with respect to t, multiply it by At , and integrate in by parts to obtain 1 d ‖A ‖2 + ‖∇At ‖2 = − Re ∫(DA Ψ)∗ (At Ψt )dx − Re ∫(At Ψ)∗ (DA Ψ + At Ψ)dx 2 dt t Ω

Ω

≤ ‖DA Ψ‖‖At Ψt ‖ + ‖At Ψ‖‖DA Ψt ‖ − ‖At Ψ‖2 ,

(1.7.28)

which implies 1 d ‖A ‖2 + ‖∇At ‖2 + ‖At Ψ‖2 ≤ ‖DA Ψ‖‖At Ψt ‖ + ‖At Ψ‖‖DA Ψt ‖, (1.7.29) 2 dt t d [η‖Ψt ‖2 + ‖At ‖2 ] + ‖DA Ψt ‖2 + ‖∇At ‖2 + ‖At Ψ‖2 ≤ C(‖At ‖2 + ‖Ψt ‖2 ). (1.7.30) dt Using the uniform Gronwall inequality and equation (1.7.20), we get ‖At ‖ ≤ K6 ,

‖Ψt ‖ ≤ K6 ,

t > 1,

(1.7.31)

which in turn ensures the bounds on the H 2 -norm ‖A‖H 2 ≤ K7 ,

‖Ψ‖H 2 ≤ K7 ,

t > 1.

(1.7.32)

Set the semigroup S(t) to be S(t) : H 1 × H 1 → H 1 × H 1

where S(t)(Ψ0 , A0 ) = (Ψ(t), A(t)),

(Ψ(t), A(t)) is the solution of problem (1.7.13)–(1.7.14) and (1.7.3)–(1.7.4) with the initial value (Ψ0 , A0 ). By the Sobolev embedding theorem, we know that ⋃t>1 S(t)B is compact in H 1 × H 1 , and the ω-limit sets of B, namely A = ω(B) = ⋂ ⋃ S(t)B, s≥0 t≥s

is the attractor of |Ψ| ≤ 1 in H 1 × H 1 . Then we have Theorem 1.7.1. Assume that |Ψ0 | ≤ 1, x ∈ Ω. Under the instantaneous gauge Φ = div A, the problem (1.7.1)–(1.7.4) possesses a unique attractor in H × H 1 . Now we estimate the dimension of the attractor A . Set u(t) = (Ψ(t), A(t)), u0 = (Ψ(0)0 , A(0)0 ). The variational equations for (1.7.13)–(1.7.14) are Vt + L(u)V = 0,

V(x, 0) = V0 (x),

(x, t) ∈ Ω × R+ ,

V ∈ H 1 × H1

(1.7.33)

with the initial condition V(x, 0) = V0 (x),

x ∈ Ω,

(1.7.34)

84 | 1 Attractor and its dimension estimation where V = (Ψ, E), L(u)V = (L1 , L2 ), L1 = ik(div A)Ψ − ik(div E)Ψ +

1 i i [D Ψ + ∇(EΨ) + E ⋅ Ψ + 2A ⋅ EΨ − Ψ + |Ψ|2 Ψ + 2Ψ2 Ψ∗ + 2|Ψ|2 Ψ], (1.7.35) η A k k

1 L2 = [Ψ∗ DA Ψ + Ψ∗ DA Ψ + EΨ + DA Ψ∗ + ΨD∗A Ψ + EΨ∗ ]. 2

(1.7.36)

We can see that, when V0 (x) ∈ H 1 × H 1 , there exists a unique global solution of problem (1.7.13) such that V(x, t) ∈ L∞ ([0, ∞), H 1 × H 1 ) ∩ L∞ ((0, ∞); H 2 × H 2 ). Let Vi (t) be the solution of equation (1.7.33) with the initial value Vi (0) = ξi , i = 1, 2, . . . , N, where ξ1 , ξ2 , . . . , ξN ∈ L2 are linearly independent. Also QN (t) means the orthogonal projection from L2 to the linear span of V1 (t), V2 (t), . . . , VN (t). Set t

1 qN = lim sup ( sup ∫ tr(L(u(s))QN (s))ds), t→∞ u ∈A ξ ∈L2 ,|ξ |=1 t 0 i j 0

where tr means the trace of an operator. To estimate the dimension of the global attractor A , we need the following lemma: Lemma 1.7.1 ([33]). Suppose A is the global attractor of (1.7.13), (1.7.3), and (1.7.4). If qN > 0 for some N, then the Hausdorff dimension of A satisfies dH (A ) ≤ N, while the fractal dimension of A satisfies dF (A ) ≤ N(1 + max − 1≤j≤N

qj

qN

).

Lemma 1.7.2 ([197]). Let Φj ∈ H m , 1 ≤ j ≤ N be orthogonal in L2 , for almost x ∈ Ω, and set N

󵄨 󵄨2 ρ(γ) = ∑ 󵄨󵄨󵄨Ψj (x)󵄨󵄨󵄨 . j=1

Then for any p satisfying 1

1 ‖∇Ψj ‖‖Ej ‖ − 2‖ϕj ‖4 ‖A‖4 ‖Ej ‖ − 2‖ϕj ‖‖A‖∞ ‖Ej ‖} k

1 N 1 ‖∇ϕj ‖2 + ‖div Ej ‖2 + ‖curl Ej ‖2 ) ∑( 2 j=1 ηk 2 N

− C ∑( j=1

1 ‖ϕ ‖2 + ‖Ej ‖2 ). ηk 2 j

(1.7.37)

Set ρ(x) = ∑Nj=1 ( ηk1 2 ‖Φj ‖2 + ‖Ej ‖2 ). Applying Lemma 1.7.2, for N = 3, we have 2

N

∫ ρ(x) 5 dx ≤ k1 {∑( j=1

Ω

1 ‖∇Φj ‖2 + ‖div Ej ‖2 + ‖curl Ej ‖2 )} ηk 2

(1.7.38)

and tr(L(u)QN (t)) ≥ ≥

1 N 1 ‖∇Ψj ‖2 + ‖div Ej ‖2 + ‖curl Ej ‖2 ) − C ∫ ρ(x)dx ∑( 2 j=1 ηk 2 N

Ω

1 1 ‖∇Ψj ‖2 + ‖ div Ej ‖2 + ‖ curl Ej ‖2 ) − C(η, k, A1 , A2 , A3 ) ∑( 4 j=1 ηk 2 N

= ∑( j=1

1 (α + α2 + ⋅ ⋅ ⋅ + αN ) + μ1 + μ2 + ⋅ ⋅ ⋅ + μN ) − C(η, k, A1 , A2 , A3 ), ηk 2 1 (1.7.39)

86 | 1 Attractor and its dimension estimation where αi , μi , i = 1, 2, . . . , N, are the eigenvalues in H and Hn1 , respectively. Choose N0 large enough so that N

tr(L(u)QN (t)) ≥ ∑( j=1

1 (α + α2 + ⋅ ⋅ ⋅ + αN ) + (μ1 + μ2 + ⋅ ⋅ ⋅ + μN )) ηk 2 1

− C(η, k, A1 , A2 , A3 ) ≥ 0.

(1.7.40)

Then by a theorem in [197], for the problem (1.7.1)–(1.7.4), its Hausdorff and fractal dimensions are bounded for the instantaneous gauge as follows: dH (ω(B)) ≤ Nc ,

dF (ω(B)) ≤ 2N0 .

(1.7.41)

Finally, we have the following theorem: Theorem 1.7.2. Under the assumptions of Theorem 1.7.1, the attractor for the problem (1.7.1)–(1.7.4) has bounded Hausdorff and fractal dimensions under the instantaneous gauge. In this case, the long time behavior of the solution of problem (1.7.1)–(1.7.4) is determined by finite many parameters.

1.8 Landau–Lifshitz–Maxwell equation In 1935, Landau and Lifshitz [160] put forward the following ferromagnetic chain coupling equations of electromagnetic field [110]: zt = λ1 z × (Δz + H) − λ2 z × (z × (Δz + H)), 𝜕E ∇×H = + σE, 𝜕t 𝜕H 𝜕z ∇×E =− −β , 𝜕t 𝜕t ∇ ⋅ H + β∇ ⋅ z = 0, ∇ ⋅ E = 0,

(1.8.1) (1.8.2) (1.8.3) (1.8.4)

where λ1 , λ2 , σ, β are constants and λ2 ≥ 0, σ ≥ 0. The unknown vector-valued function z(x, t) = (z1 (x, t), z2 (x, t), z3 (x, t)) models magnetization, H(x, t) = (H1 (x, t), H2 (x, t), H3 (x, t)) means the magnetic field, E(x, t) = (E1 (x, t), E2 (x, t), E3 (x, t)) is the electromagnetic field, H e = Δz + H represents the effective magnetic field, Δ = ∑ni=1

𝜕2 , 𝜕xi2

∇ = ( 𝜕x𝜕 , 𝜕x𝜕 , . . . , 𝜕x𝜕 ), and × means vector cross-product. 1 2 n If H = 0, E = 0, then we can get Landau–Lifshitz–Maxwell equation system with Gilbert term: zt = λ1 z × Δz − λ2 z × z × (z × Δz)

(1.8.5)

where λ2 > 0 is Gilbert damping constant. Guo and Hong in 1993, Guo and Wang in 1995, Chen and Guo in 1996, Guo and Ding in 1997 systematically researched the properties of the solutions of equation (1.8.5); see respective references [100, 116, 24, 95, 96].

1.8 Landau–Lifshitz–Maxwell equation

| 87

Especially, in [100] the authors found a close relationship between the harmonic mapping on a Riemann manifold and the solutions of (1.8.5). When λ2 = 0, equation (1.8.5) can be reduced to zt = λ1 z × Δz.

(1.8.6)

In a one-dimensional space, it is an integrable system, and has the soliton solution. Nakamura, Lakshmanan, and Zakharov, among others, studied the interaction of soliton, infinite conservation law, and inverse scattering method in detail, respectively in [178, 159, 211]. Since 1982, Zhou and Guo [216] did a systematic and deep research on the kinds of determining solution problems (including the space of one-dimensional and multidimensional initial value problem, linear and nonlinear boundary value problems) for the equation (1.8.6). In particular, in 1991, Zhou, Guo and Tan [217] proved the existence and uniqueness of a smooth solution (in one spacial dimension). Here we consider the global existence of an attractor and its dimension estimation problem with periodic initial conditions of the system (1.8.1)–(1.8.4) as in [110], z(x + 2Dei , t) = z(x, t),

H(x + 2Dei , t) = H(x, t),

E(x + 2Dei , t) = E(x, t),

x ∈ Ω, t ≥ 0, i = 1, 2, . . . , n,

z(x, 0) = z0 (x),

H(x, 0) = H0 (x),

E(x, 0) = E0 (x),

x ∈ Ω,

(1.8.7) (1.8.8)

where x + 2Dei = (x1 , . . . , xi−1 , xi + 2D, xi+1 , . . . , xn ), i = 1, 2, . . . , n, D > 0, Ω ⊂ Rn is an n-dimensional cube with side length 2D. In the following we establish a priori estimate Lemma 1.8.1. Let |z0 (x) = 1|. Then for the smooth solution of the periodic initial value problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8), we have 󵄨󵄨 󵄨 󵄨󵄨z(x, t)󵄨󵄨󵄨 = 1,

x ∈ Ω, t ≥ 0.

(1.8.9)

Proof. Taking the dot product of equation (1.8.1) with z, we arrive at 𝜕 󵄨󵄨 󵄨2 󵄨z(x, t)󵄨󵄨󵄨 = 0, 𝜕t 󵄨 which yields the claim. Lemma 1.8.2. Let λ2 > 0, β > 0, σ ≥ 0, ∇z0 (x) ∈ L2 (Ω), E0 (x) ∈ L2 (Ω), H0 (x) ∈ L2 (Ω). Then we have the following estimates: 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 sup [󵄩󵄩󵄩∇z(⋅, t)󵄩󵄩󵄩2 + 󵄩󵄩󵄩E(⋅, t)󵄩󵄩󵄩2 + 󵄩󵄩󵄩H(⋅, t)󵄩󵄩󵄩2 ] ≤ K1 ,

0≤t 0, K are independent of t. This gives a priori estimate of e(t), and then we get the estimate of (H, E, z) ∈ (H 1 , H 1 , H 1 ). In fact, from (1.8.18)–(1.8.19) we have de(t) = (Ht , Htt ) + (∇H, ∇Ht ) + (Et , Ett ) + (∇E, ∇Et ) + (Δz, Δzt ) + η1 (Ht , Ht ) dt + η1 (H, Htt ) + η2 (Et , Et ) + η2 (E, Ett ),

(1.8.22)

90 | 1 Attractor and its dimension estimation where (Ht , Htt ) = (Ht , ΔH) − β(Ht , ztt ) − σ(Ht , Ht ) − σβ(Ht , zt ) + β(Ht , ∇(∇ ⋅ z)),

(∇H, ∇Ht ) = −(ΔH, Ht ),

(Et , Ett ) = (Et , ΔE) − σ(Et , Et ) − β(Et , (∇ × z)t ),

(∇E, ∇Et ) = (−ΔE, Et ),

(Δz, Δzt ) = λ1 (Δ(z × Δz), Δz) + λ1 (Δ(z × H), Δz) − λ2 (Δ(|∇z|2 z), Δz) + λ2 (Δ2 z, Δz) + λ2 (ΔH, Δz) − λ2 (Δ(z ⋅ H)z, Δz),

η1 (H, Htt ) = −η1 ‖∇H‖22 − βη1 (H, ztt ) − ση1 (H, Ht ) − σβη1 (H, zt ) + η1 β(H, ∇(∇ ⋅ z)),

η2 (E, Ett ) = −η2 ‖∇E‖22 − η2 σ(E, Et ) − η2 β(E, (∇ × z)t ), thus we obtain de(t) = −(σ − η1 )‖Ht ‖22 − (σ − η2 )‖Et ‖22 dt − η1 ‖∇H‖22 − η2 ‖∇E‖22 − λ2 ‖∇Δz‖22

− βη1 (H, ztt ) − η1 σ(H, Ht ) − σβη1 (H, zt )

− η2 σ(E, Et ) − η2 β(E, (∇ × z)t )

+ β(H, ∇(∇ ⋅ z)) + η1 β(H, ∇(∇ ⋅ z))

− β(Ht , ztt ) − σβ(Ht , zt ) − β(Et , (∇ × z)t )

+ λ1 (Δ(z × Δz), Δz) + λ1 (Δ(z × H), Δz)

+ λ2 (Δ(|∇z|2 z), Δz) + λ2 (Δz, ΔH) − λ2 (Δ(z ⋅ H)z, Δz),

(1.8.23)

where (H, ztt ) = (H, zt )t − (Ht , zt ),

(Ht , ztt ) = (Ht , zt )t − (zt , ΔH − βztt − σHt − σβzt + β∇(∇ ⋅ z))

β = (Ht , zt )t + (zt , zt )t − (zt , ΔH) + σ(zt , Ht ) + σβ‖zt ‖22 − β(zt , ∇(∇ ⋅ z)), 2 β2 β(Et , ∇ × z)t ) = β(Et , ∇ × z)t + (∇ × z, ∇ × z)t + σβ(Et , ∇ × z) + β(∇E, ∇(∇ × z)), 2 η2 β(E, (∇ × z)t ) = η2 β(E, ∇ × z)t − η2 β(Et , ∇ × z). Let

1 e1 (t) = G(t) + R(t), 2

where G(t) ≜ ‖Et ‖22 + ‖Ht ‖22 + ‖∇E‖22 + ‖∇H‖22 + ‖Δz‖22 ≜ 2e(t) − 2η1 (H, Ht ) − 2η2 (E, Et ),

1.8 Landau–Lifshitz–Maxwell equation

β2 ‖z ‖2 + β(Et , ∇ × z) 2 t β2 1 + ‖∇ × z‖22 − η2 β(E, ∇ × z) + ση1 ‖H‖22 2 2 1 2 + ση2 ‖E‖2 + η2 β(E, ∇ × z) + η2 (E, Et ) + η1 (H, vht ), 2

| 91

R(t) ≜ β(zt , Ht ) +

(1.8.24)

which yields de1 (t) + (σ − η1 )‖Ht ‖22 + (σ − η2 )‖Et ‖22 + η1 ‖∇H‖22 + η2 ‖∇E‖22 + λ2 ‖∇Δz‖22 + σβ2 ‖zt ‖2 dt = −(2σβ − η1 β)(zt , Ht ) + β(zt , ΔH) − (σ − η2 )β(Et , ∇ × z) − β(∇E, ∇(∇ × z)) + λ1 (Δ(z × Δz), Δz) + λ1 (Δ(z × H), Δz)

+ λ2 (Δ(|∇z|2 z), Δz) + λ2 (Δz, ΔH) − λ2 (Δ(z ⋅ H)z, Δz)

− σβη1 (H, zt ) + β(Ht , ∇(∇ ⋅ z)) + η1 β(H, ∇(∇ ⋅ z)) + β2 (zt , ∇(∇ ⋅ z)).

(1.8.25)

To estimate the right-hand side terms of (1.8.25), we need to use Sobolev interpolation inequality repeatedly. (1) Through (1.8.25), we have 󵄨 󵄨2 󵄨 󵄨2 |zt |2 = λ12 󵄨󵄨󵄨z × (Δz + H)󵄨󵄨󵄨 + λ22 󵄨󵄨󵄨z × (z × (Δz + H))󵄨󵄨󵄨 ≤ (λ12 + λ22 )|Δz + H|2

≤ 2(λ12 + λ22 )(|Δz|2 + |H|2 ). With the aid of Sobolev interpolation inequality and Hölder inequality, we obtain 󵄨󵄨 󵄨 2 2 󵄨󵄨−(2σβ + η1 β)(zt , Ht )󵄨󵄨󵄨 ≤ ε1 ‖Ht ‖2 + C(ε1 , σ, β, η1 )‖zt ‖2 ≤ ε1 ‖Ht ‖22 + 2C(λ12 + λ22 )‖Δz‖22 + d1 ≤ ε1 ‖H‖22 +

λ2 ‖∇Δz‖22 + d2 , l

where ε1 , l are undetermined positive coefficients. (2) Acting with ∇ on (1.8.1), we get ∇zt = λ1 ∇z × (Δz + H) + λ1 z × ∇Δz

− λ2 ∇z × (z × (Δz + H)) − λ2 z × (∇z × (Δz + H)) − λ2 z × (z × ∇Δz) − λ2 z × (z × ∇H),

β(zt , ΔH) = −β(∇zt , ∇H)

= −βλ1 (∇z × (Δz + H), ∇H) − βλ1 (z × ∇Δz, ∇H)

+ λ2 β(∇z × (z × Δz), ∇H) + λ2 β(∇z × (z × H), ∇H) + λ2 β(z × (∇z × Δz), ∇H) + λ2 β(z × (∇z × H), ∇H) + λ2 β(z × (z × ∇Δz), ∇H) − λ2 β‖z × ∇H‖22

92 | 1 Attractor and its dimension estimation ≤ −λ2 β‖z × ∇H‖22 + (|λ1 | + 2λ2 )β‖∇z‖∞ ‖Δz‖2 ‖∇H‖2 + (|λ1 | + 2λ2 )β‖∇z‖∞ ‖H‖2 ‖∇H‖2 ‖∇H‖2 + (|λ1 | + λ2 )β‖∇Δz‖2 ‖∇H‖2 3

≤ C1 (|λ1 | + 2λ2 )β‖∇z‖22

− 4n

1

‖∇Δz‖22 n 4

+ 4n

‖∇H‖2

+ C2 (|λ1 | + 2λ2 )β‖∇Δz‖2 ‖∇H‖2 (|λ | + |λ2 |) + 1 ‖∇Δz‖22 + (|λ1 | + λ2 )β2 ‖∇H‖22 4 n n C2 3− n ≤ 0 (|λ1 | + 2λ2 )2 ‖∇z‖2 2 ‖∇Δz‖1+ 2 + C3 ‖∇Δz‖ 2 λ2 (|λ | + λ2 ) + 1 ‖∇Δz‖22 + 2(|λ1 | + λ2 )β2 ‖∇H‖22 4 2

2) ( λl + (|λ1 |+λ )‖∇Δz‖22 { 4 { { { { + 2(|λ1 | + λ2 )β2 ‖∇H‖22 + C, when n = 1, ≤{ λ C2 (|λ |+λ ) 2 2 2 { {( l2 + 1 4 2 ) + λ20 (|λ1 | + 2λ2 ) ‖∇z‖2 ‖∇Δz‖2 { { 2 2 { + 2(|λ1 | + λ2 )β ‖∇H‖2 + C, when n = 2,

(3) 󵄨󵄨 󵄨 2 2 2 󵄨󵄨−(σ + η2 )β(Et , ∇ × z)󵄨󵄨󵄨 ≤ ε2 ‖Et ‖2 + C1 ‖∇z‖2 ≤ ε2 ‖Et ‖2 + C2 , (4) λ 󵄨󵄨 󵄨 λ 2 2 2 󵄨󵄨−β(∇E, ∇(∇ × z))󵄨󵄨󵄨 ≤ 2 ‖∇Δz‖2 + C1 ‖E‖2 ≤ 2 ‖∇Δz‖2 + C2 , l l (5) 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨λ1 (Δ(z × Δz), Δz)󵄨󵄨󵄨 ≤ |λ1 |󵄨󵄨󵄨(∇z × Δz, ∇Δz)󵄨󵄨󵄨 ≤ |λ1 |‖Δz‖∞ ‖Δz‖2 ‖∇Δz‖22 3

≤ C0 |λ1 |‖∇z‖22

− 4n

3

‖∇Δz‖22

λ2 ‖∇Δz‖22 + C, ≤{ l C0 |λ1 |‖∇z‖2 ‖∇Δz‖22 ,

+ 4n

n = 1, n = 2,

(6) 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨λ1 (Δ(z × H), Δz)󵄨󵄨󵄨 = 󵄨󵄨󵄨λ1 (∇(z × H), ∇Δz)󵄨󵄨󵄨 ≤ ‖∇z‖∞ |λ1 |‖H‖2 ‖∇Δz‖2 + |λ1 |‖∇H‖2 ‖∇Δz‖2 ≤

2λ2 λ2 ‖∇Δz‖22 + 1 ‖∇H‖22 + C, 4 λ2

(7) 󵄨󵄨 󵄨 󵄨 󵄨 2 2 󵄨󵄨λ2 (Δ(|∇z| z), Δz)󵄨󵄨󵄨 = 󵄨󵄨󵄨−λ2 (∇(|∇z| z), ∇Δz)󵄨󵄨󵄨

| 93

1.8 Landau–Lifshitz–Maxwell equation

≤ λ2 ‖∇z‖36 ‖∇△z‖2 + Cλ2 ‖∇z‖∞ ‖∇Δz‖2 n

1+ 4n

≤ Cλ2 (‖∇z‖3− 4 ‖∇Δz‖2 ≤

3

+ ‖∇z‖22

λ2 ‖∇Δz‖22 + C, {l Cλ2 (‖Δz‖22 + ‖∇z‖2 )‖∇Δz‖22 ,

− 4n

3

‖∇Δz‖22

+ 4n

)

n = 1, n = 2,

(8) 󵄨󵄨 󵄨 λ 2 2 󵄨󵄨λ2 (Δz, ΔH)󵄨󵄨󵄨 ≤ 2 ‖∇Δz‖2 + λ2 ‖∇H‖2 , 4 (9) 󵄨 󵄨󵄨 󵄨 3 󵄨 󵄨󵄨λ2 (Δ(z ⋅ H)z, Δz)󵄨󵄨󵄨 = λ2 󵄨󵄨󵄨(∇(z ⋅ H)z, ∇ z)󵄨󵄨󵄨

≤ λ2 (2‖H‖2 ‖∇z‖∞ ‖∇Δz‖2 + ‖∇H‖2 ‖∇Δz‖2 ) ≤

λ2 ‖∇Δz‖22 + 3λ2 ‖∇H‖22 + C, 8

(10) λ 󵄨󵄨 󵄨 2 󵄨󵄨−σβη1 (H, zt )󵄨󵄨󵄨 ≤ C1 ‖H‖2 ‖zt ‖2 ≤ C2 ‖Δz‖2 + d1 ≤ 2 ‖∇Δz‖2 + d2 , l (11) λ 󵄨󵄨 󵄨 2 󵄨󵄨β(Ht , ∇(∇ ⋅ z))󵄨󵄨󵄨 ≤ β‖Ht ‖2 ‖Δz‖2 ≤ ε3 ‖Ht ‖2 + 2 ‖∇Δz‖2 + C, l (12) λ 󵄨󵄨 󵄨 2 󵄨󵄨η1 β(H, ∇(∇ ⋅ z))󵄨󵄨󵄨 ≤ η1 β‖H‖2 ‖Δz‖2 ≤ 2 ‖∇Δz‖2 + C. l Combining (1)–(12), and using equation (1.8.25), we arrive at the following bounds: – when n = 1, de1 (t) + (σ − η1 − ε1 − ε3 )‖Ht ‖22 + (σ − η2 − ε2 )‖Et ‖22 dt (4 + 3β2 )(λ22 + 2λ12 ) + η2 ‖∇E‖22 + (η1 − )‖∇H‖22 λ2 +( –

1 7 − )λ ‖∇Δz‖22 ≤ C1 ; 8 l 2

when n = 2, de1 (t) + (σ − η1 − ε1 − ε3 )‖Ht ‖22 + (σ − η2 − ε2 )‖Et ‖22 dt (4 + 3β2 )(λ22 + 2λ12 ) + η2 ‖Et ‖22 + (η1 − )‖∇H‖22 λ2

(1.8.26)

94 | 1 Attractor and its dimension estimation 1 6 − )λ − C(|λ1 | + λ2 )‖∇z‖2 8 l 2 C 2 − (λ22 + (|λ1 | + 2λ2 ) ‖∇z‖22 )]‖∇Δz‖22 ≤ C2 , λ2

+ [(

(1.8.27)

where constants C1 , C2 are independent of t, and all the parameters ε1 , ε2 , ε3 , η1 , η2 and l can be selected as follows: σ>

(4 + 3β2 )(λ22 + 2λ12 ) λ2

(1.8.28)

and (4+3β2 )(λ22 +2λ12 ) (i) < η1 < σ, λ2 (ii) η2 = ε2 = σ4 , (iii) n = 1, l > 56; n = 2, l = 60. Suppose ‖∇z0 (x)‖, ‖E0 (x)‖ and ‖H0 (x)‖ are sufficiently small. Using Lemma 1.8.2 and C 2 2 [λ + (|λ1 | + 2λ22 ) ]‖∇z‖22 λ2 2 1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ C(|λ1 | + λ2 ) [󵄩E0 (x)󵄩󵄩󵄩 + 󵄩󵄩󵄩H0 (x)󵄩󵄩󵄩 + √2σ 󵄩󵄩󵄩E0 (x)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇z0 (x)󵄩󵄩󵄩] √β 󵄩

C(|λ1 | + λ2 )‖∇z‖2 +

+

C 2 σ 󵄩 21 󵄩2 {λ2 + (|λ1 | + 2λ2 ) [(1 + )󵄩󵄩󵄩E0 (x)󵄩󵄩󵄩 λ2 β β

λ 󵄩 󵄩2 󵄩 󵄩2 + 󵄩󵄩󵄩H0 (x)󵄩󵄩󵄩 + β󵄩󵄩󵄩∇z0 (x)󵄩󵄩󵄩 ]} < 2 , 40

(1.8.29)

there exists a constant a > 0 such that de1 (t) + a(‖Ht ‖22 + ‖Et ‖22 + ‖∇H‖22 + ‖∇E‖22 + ‖∇Δz‖22 ) ≤ C, dt

(1.8.30)

where the constant C is independent of t, for 1 ≤ n ≤ 2. 2D Since Δz(x, t) is a periodic function of x, ∫0 Δzdx = 0. Using Poincaré inequality, we have ‖Δz‖22 ≤ δ‖∇Δ‖22 , where we choose δ0 = min{a, aδ }. From equation (1.8.30), we have de1 (t) + δ0 (‖Ht ‖22 + ‖Et ‖22 + ‖∇H‖22 + ‖∇E‖22 + ‖Δz‖22 ) ≤ C. dt

(1.8.31)

Inequality (1.8.31) can be written as de1 (t) 󵄨 󵄨 + 2δ0 e1 (t) ≤ C + 2δ0 R(t) ≤ C + 2δ0 sup 󵄨󵄨󵄨R(t)󵄨󵄨󵄨. dt t

(1.8.32)

1.8 Landau–Lifshitz–Maxwell equation

| 95

Hence C 󵄨 󵄨 + sup 󵄨󵄨󵄨R(t)󵄨󵄨󵄨)(e2δ0 t − 1), 2δ0 t C 󵄨 󵄨 e1 (t) ≤ e1 (0) + + sup 󵄨󵄨󵄨R(t)󵄨󵄨󵄨, 2δ0 t

e1 (t)e2δ0 t ≤ e1 (0) + (

that is, 1 C 󵄨 󵄨 G(t) + R(t) ≤ C0 + sup 󵄨󵄨󵄨R(t)󵄨󵄨󵄨, C0 ≜ e1 (0) + , 2 2δ t 0 󵄨 󵄨 󵄨 󵄨 G(t) ≤ 2C0 + 2(sup 󵄨󵄨󵄨R(t)󵄨󵄨󵄨 − R(t)) ≤ 2C0 + 4 sup 󵄨󵄨󵄨R(t)󵄨󵄨󵄨. t

t

(1.8.33)

By equation (1.8.24), we have β2 󵄨󵄨 󵄨 2 󵄨󵄨R(t)󵄨󵄨󵄨 ≤ β‖zt ‖‖Ht ‖ + ‖zt ‖2 + β‖zt ‖2 ‖∇z‖2 2 β2 + η1 ‖H‖2 ‖∇E‖2 + η2 ‖E‖2 ‖Et ‖2 + ‖∇z‖22 2 1 1 2 2 + ση1 ‖H‖2 + ση2 ‖E‖2 + η2 β‖E‖2 ‖∇z‖2 2 2 β + β2 β ≤ ‖zt ‖22 + (‖Ht ‖22 + ‖Et ‖22 + ‖∇E‖22 ) + C1 2 2 β 2 2 ≤ (β + β )(λ1 + λ22 )‖Δz‖22 + (‖Ht ‖22 + ‖Et ‖22 + ‖∇E‖22 ) + C2 . 2

(1.8.34)

Set β < 21 , (β + β2 )(λ12 + λ22 ) < 41 , then we can take β a0 = 4 max{ , (λ12 + λ22 )(β + β2 )} < 1. 2 Due to equation (1.8.34), we have 1 󵄨󵄨 󵄨 1 2 2 2 2 󵄨󵄨R(t)󵄨󵄨󵄨 ≤ a0 (‖Ht ‖2 + ‖Et ‖2 + ‖∇E‖2 + ‖Δz‖2 ) ≤ a0 G(t). 4 4

(1.8.35)

Inserting equation (1.8.35) into equation (1.8.33), it follows that G(t) ≤ 2C0 + a0 sup G(t), t

2C0 sup G(t) ≤ ≜ d0 . 1 − a0 t

(1.8.36)

Thus we have the following result: Lemma 1.8.3. Suppose (z(x, t), H(x, t), E(x, t)) is a smooth solution of periodic initial value problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8). If the initial data satisfies (z0 (x), H0 (x), E0 (x)) ∈ (H 2 (Ω), H 1 (Ω), H 1 (Ω)), Ω ⊂ Rn , 1 ≤ n ≤ 2, |z0 (x)| = 1, and the following conditions:

96 | 1 Attractor and its dimension estimation (4+3β2 )λ22 +2λ12 ; λ2 1 2 2 , (β + β )(λ1 + λ22 ) 2

(1) λ2 > 0, σ >

(2) 0 < β < (3) when n = 2,

≤ 41 ;

‖∇z0 ‖2L2 (Ω) + ‖H0 ‖2L2 (Ω) + ‖E0 ‖2L2 (Ω) ≤ λ, where λ = λ(λ1 , λ2 , β) is an appropriately small constant, then we have the estimate 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 sup (󵄩󵄩󵄩z(⋅, t)󵄩󵄩󵄩H 2 (Ω) + 󵄩󵄩󵄩H(⋅, t)󵄩󵄩󵄩H 1 (Ω) + 󵄩󵄩󵄩E(⋅, t)󵄩󵄩󵄩H 1 (Ω) ) ≤ K,

t∈[0,∞)

where the constant K depends on ‖z0 (x)‖H 2 (Ω) , ‖H0 (x)‖H 1 (Ω) and ‖E0 (x)‖H 1 (Ω) . Using the theorems from [194], we obtain the following theorem: Theorem 1.8.1. Assume that the constants λ2 > 0, β ≥ 0, σ ≥ 0, and the initial function (z0 (x), H0 (x), E0 (x)) ∈ (H k (Ω), H k−1 (Ω), H k+1 (Ω)) for k ≥ 1 + [ n2 ], where Ω ⊂ Rn , 1 ≤ n ≤ 2, is a bounded domain, and |z0 (x)| = 1, ∇ ⋅ E0 = 0, ∇ ⋅ (H0 + βz0 ) = 0, when n = 2, are such that ‖∇z0 ‖ + ‖E0 ‖ + ‖H0 ‖ ≤ δ, where δ is an appropriately small constant. Then the periodic initial value problem (1.8.7)–(1.8.8) for the Landau–Lifshitz–Maxwell equation system (1.8.1)–(1.8.4) has a unique global smooth solution. Moreover, 󵄨󵄨 󵄨 󵄨󵄨z(x, t)󵄨󵄨󵄨 = 1,

x ∈ Ω, t ∈ R+ ,

[ k2 ]

s (0, T; H k−2s (Ω)), z(x, t) ∈ ⋂ W∞ s=0

k−1

s H(x, t) ∈ ⋂ W∞ (0, T; H k−1−s (Ω)), s=0

k−1

s E(x, t) ∈ ⋂ W∞ (0, T; H k−1−s (Ω)), s=0

Together with the above established a priori estimate, we get Theorem 1.8.2. Under the conditions of Theorem 1.8.1 and Lemma 1.8.3, the problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) has a unique attractor A , that is, the set A has the following properties: (1) A is weak compact in H 2 (Ω) × H 1 (Ω) × H 1 (Ω);

1.8 Landau–Lifshitz–Maxwell equation

| 97

(2) S(t)A = A ; (3) limt→∞ dist(S(t)B, A ) = 0, ∀B ⊂ H 2 × H 1 × H 1 bounded, where dist(X, Y) = sup inf ‖x − y‖, x∈X y∈Y

where S(t)(z0 , H0 , E0 ) is the semigroup operator generated by problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8). Proof. Using Theorem 1.8.1, the problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) generates a continuous semigroup operator S(t)(z0 , H0 , E0 ). Taking 󵄨 󵄨 E = {(z, H, E) ∈ H 2 (Ω) × H 1 (Ω) × H 1 (Ω) : 󵄨󵄨󵄨z(x, t)󵄨󵄨󵄨 = 1, ∇ ⋅ E = 0, ∇ ⋅ (H + βz) = 0} and its subset 󵄨 󵄨 B = {󵄨󵄨󵄨z(x, t)󵄨󵄨󵄨 = 1, ∇ ⋅ E = 0, ∇(H + βz) = 0,

z(⋅, t) ∈ H 2 (Ω), H(⋅, t) ∈ H 1 (Ω), E(⋅, t) ∈ H 1 (Ω) 󵄩󵄩󵄩z(⋅, t)󵄩󵄩󵄩2 2 + 󵄩󵄩󵄩H(⋅, t)󵄩󵄩󵄩2 1 + 󵄩󵄩󵄩E(⋅, t)󵄩󵄩󵄩2 1 ≤ ε0 + δ0 } 󵄩 󵄩H (Ω) 󵄩 󵄩H (Ω) 󵄩 󵄩H (Ω)

as a bounded absorbing set in E, we see that B is weakly compact in E. Then we know that the set A = ω(B) is the weakly compact attractor of the periodic initial value problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8). In the following, we estimate the upper bound of Hausdorff and fractal dimensions of the attractor A . Now consider the linear variational problem corresponding to (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8): zt = λ2 Δz + 2λ2 (∇z, ∇z)z + λ2 |∇z|2 z

+ λ1 z × Δz + λ1 z × h − λ1 (Δz + H) × z

+ λ2 h − λ2 (z ⋅ H)z − λ2 (z ⋅ h)z − λ2 (z ⋅ H)z,

et = ∇ × h − σe,

ht = −∇ × e − βzt ,

z(0) = z0 ,

z(x, t),

h(0) = h0 ,

e(x, t),

(1.8.37) (1.8.38) (1.8.39)

e(0) = e0 ,

h(x, t) are periodic in x with period 2D.

(1.8.40) (1.8.41)

Then equations (1.8.37)–(1.8.41) can be written in operator form as vt = −L(u)v,

v0 = v(0),

(1.8.42)

where u = (z, e, h) is the solution of problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) with v = (z, E, H) and v0 = (z0 , e0 , h0 ). Since the problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) possesses a smooth solution, the coefficients of the linear variational equations (1.8.37)–(1.8.41) are smooth. When

98 | 1 Attractor and its dimension estimation the initial value v0 is sufficiently smooth, equations (1.8.37)–(1.8.41) possess a unique smooth solution, i. e., there exists an operator G(t) such that vt = G(t)v0 . Furthermore, we prove that the semigroup operator S(t) is differentiable in L2 (Ω), and its Fréchet derivative is S󸀠 (t)u0 = G(t)v0 . Lemma 1.8.4. The smooth solution of the periodic initial value problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) continuously depends on the initial conditions. Proof. Set (zi (x, t), Hi (x, t), Ei (x, t)), i = 1, 2, to be the smooth solution of problem (1.8.1)–(1.8.4) with initial values zi (x, 0) = z0i (x), Hi (x, 0) = H0i (x), Ei (x, 0) = E0i (x), i = 1, 2. Set z(x, t) = z2 (x, t)−z1 (x, t), H(x, t) = H2 (x, t)−H1 (x, t) and E(x, t) = E2 (x, t)−E1 (x, t). Then (z(x, t), H(x, t), E(x, t)) satisfies zt = λ1 z × Δz + λ1 z1 × Δz + λ1 z × H2 + λ1 z1 × H + λ2 Δz + λ2 |Δz2 |2 z + λ2 (∇z, ∇(z1 + z2 ))z1

+ λ2 H − λ2 (z2 ⋅ H2 )z − λ2 (z2 ⋅ H + H1 ⋅ z)z1 ,

Et + σE = ∇ × H,

Ht + βzt = −∇ × E,

∇ ⋅ (H + βz) = 0, z(x, t),

(1.8.43) (1.8.44) (1.8.45)

∇ ⋅ E = 0,

H(x, t),

E(x, t)

are periodic in x with period 2D,

z(x, 0) = z0 (x),

H(x, 0) = H0 (x), 󵄨 󵄨 E(x, 0) = E0 (x), 󵄨󵄨󵄨z0 (x)󵄨󵄨󵄨 = 1, ∇ ⋅ (H0 + βz0 ) = 0, ∇E0 = 0. Then we can establish the following inequality: 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 sup [󵄩󵄩󵄩∇z(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩H(⋅, t)󵄩󵄩󵄩L2 + 󵄩󵄩󵄩E(⋅, t)󵄩󵄩󵄩L2 ]

0≤t≤T

󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 ≤ C(󵄩󵄩󵄩z0 (x)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩H0 (x)󵄩󵄩󵄩L2 + 󵄩󵄩󵄩E0 (x)󵄩󵄩󵄩L2 ),

where C is an absolute constant. Obviously, if inequality (1.8.44) is established, then Lemma 1.8.4 is proved. Indeed, taking the inner product of (1.8.43) and z, we get 1 d ∫ |z|2 dx + λ2 ‖∇z‖2 ≤ C1 [‖∇z‖2 + ‖H‖2 + ‖E‖2 ]. 2 dt Ω

Multiplying (1.8.43) by Δz, and integrating it with respect to x over Ω, we arrive at 1 d ∫ |∇z|2 dx + λ2 ‖Δz‖2 = −λ1 ∫ z × Δz ⋅ Δzdx 2 dt Ω

Ω

− λ1 ∫ z × H2 ⋅ Δzdx − λ1 ∫ z2 × H ⋅ Δzdx Ω

Ω

1.8 Landau–Lifshitz–Maxwell equation

| 99

− λ2 ∫ |∇z2 |2 z ⋅ Δzdx − λ2 ∫(∇z ⋅ ∇(z1 + z2 )z1 ) ⋅ Δzdx Ω

Ω

+ λ2 ∫ H ⋅ Δzdx + λ2 ∫(z2 ⋅ H2 )z ⋅ Δzdx Ω

Ω

+ λ2 ∫(z2 ⋅ H + H1 ⋅ z)z1 ⋅ Δzdx,

(1.8.46)

Ω

where 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨−λ1 ∫ z × Δz2 ⋅ Δzdx󵄨󵄨󵄨 = 󵄨󵄨󵄨λ1 ∫ z × ∇Δz2 ⋅ ∇zdx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

Ω

≤ |λ1 |‖∇Δz2 ‖∞ ‖z‖‖∇z‖ ≤ C|λ1 |(‖z‖2 + ‖∇z‖2 ), 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨−λ1 ∫ z × H2 ⋅ Δzdx󵄨󵄨󵄨 = 󵄨󵄨󵄨λ1 ∫ z × ∇H2 ⋅ Δzdx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

Ω

≤ |λ1 |‖∇H2 ‖∞ ‖z‖‖∇z‖ ≤ C|λ1 |(‖z‖2 + ‖∇z‖2 ),

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 󵄨󵄨−λ2 ∫ |∇z2 | z ⋅ Δzdx󵄨󵄨󵄨 ≤ λ2 ‖∇z2 ‖∞ ‖z‖‖Δz‖ 󵄨󵄨 󵄨󵄨 Ω

≤

λ2 ‖Δz‖2 + C(K)λ2 ‖z‖2 , K

where K is an unknown constant. Furthermore, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨−λ2 ∫ ∇z ⋅ ∇(z1 + z2 )z ⋅ Δzdx󵄨󵄨󵄨 ≤ λ2 󵄩󵄩󵄩∇(z1 + z2 )󵄩󵄩󵄩∞ ‖∇z‖‖Δz‖ 󵄨󵄨 󵄨󵄨 Ω

λ2 ‖Δz‖2 + C(K)λ2 ‖∇z‖2 , K 󵄨󵄨 󵄨󵄨 λ 󵄨󵄨 󵄨 2 2 󵄨󵄨λ2 ∫ H ⋅ Δzdx󵄨󵄨󵄨 ≤ 2 ‖Δz‖ + C(K)λ2 ‖H‖ , 󵄨󵄨 󵄨󵄨 K ≤

Ω

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨λ2 ∫(z2 ⋅ H2 )z ⋅ Δzdx󵄨󵄨󵄨 ≤ λ2 ‖H‖‖Δz‖ 󵄨󵄨 󵄨󵄨 Ω

≤ λ2 ‖H2 ‖∞ ‖z‖‖Δz‖ ≤

λ2 ‖Δz‖2 + C(K)λ2 ‖z‖2 , K

󵄨󵄨 󵄨󵄨 λ 󵄨󵄨 󵄨 2 2 2 󵄨󵄨λ2 ∫(z2 ⋅ H + H1 ⋅ z)z1 ⋅ Δzdx󵄨󵄨󵄨 ≤ 2 ‖Δz‖ + C(K)λ2 (‖H‖ + ‖z‖ ), 󵄨󵄨 󵄨󵄨 K Ω

and from equation (1.8.46) we have 1 d 5 ‖∇z‖2 +λ2 ‖Δz‖2 +λ1 ∫ z1 ×H ⋅Δzdx ≤ λ2 ‖Δz‖2 +C(K)(‖∇z‖2 +‖H‖2 +‖z‖2 ). (1.8.47) 2 dt K Ω

100 | 1 Attractor and its dimension estimation Multiplying equations (1.8.10) and (1.8.11) by E and H, respectively, and integrating in Ω (for β > 0), we get 1 d σ ∫(|E|2 + |H|2 )dx + ‖z‖2 = − ∫ zt ⋅ Hdx. 2β dt β Ω

(1.8.48)

Ω

From equations (1.8.47) and (1.8.48), we get 1 1 d ∫(|E|2 + |H|2 + (|E|2 + |H|2 ))dx + λ2 ‖Δz‖2 2 dt β Ω

≤ − ∫(zt ⋅ H + λ1 z1 × H ⋅ Δz)dx Ω

+

5 λ ‖Δz‖2 + C(K)(‖∇z‖2 ‖z‖2 + ‖H‖2 ). K2 2

(1.8.49)

Taking the inner product of equation (1.8.43) and H, we have ∫(zt ⋅ H + λ1 z1 × H ⋅ Δz)dx Ω

󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨 ≤ |λ1 |󵄨󵄨󵄨 ∫(z × Δz2 + z × H2 ) ⋅ Hdx󵄨󵄨󵄨 + λ2 󵄨󵄨󵄨 ∫ Δz ⋅ Hdx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

Ω

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 + λ2 󵄨󵄨󵄨 ∫ |∇z|2 z ⋅ Hdx󵄨󵄨󵄨 + λ2 󵄨󵄨󵄨 ∫ ∇z ⋅ ∇(z + z2 )z1 ⋅ Hdx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

Ω

󵄨󵄨 󵄨󵄨 󵄨 󵄨 + λ2 ‖H‖2 + λ2 󵄨󵄨󵄨 ∫(z2 ⋅ H2 )z ⋅ Hdx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 + λ2 󵄨󵄨󵄨 ∫(z2 ⋅ H)(z1 ⋅ H)dx󵄨󵄨󵄨 + λ2 󵄨󵄨󵄨 ∫(H1 ⋅ z)(z1 ⋅ H)dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

Ω

λ ≤ 2 ‖Δz‖2 + C(K)(‖∇z‖2 + ‖z‖2 + ‖H‖2 ). K

(1.8.50)

From equation (1.8.47)–(1.8.50), taking K ≥ 6, we get d 1 ∫[|∇z|2 + |z|2 + (|E|2 + |H|2 )]dx ≤ C(K, λ1 , λ2 )[‖∇z‖2 + ‖z‖2 + ‖H‖2 ], dt β Ω

which yields the claim. In order to prove that the semigroup S(t) is Fréchet differentiable, now we consider the linear variational problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8). Set DS(t)(z01 , H01 , E01 ) = (ω(t), I(t), F(t)) to be Fréchet differentiable at (z0 , H0 , E0 ) of semigroup operator. We have ωt (t) = λ1 ω × (Δz1 + H1 ) + λ1 z1 × (Δω + I)

1.8 Landau–Lifshitz–Maxwell equation

− λ2 ω × (z1 × (Δz1 + H1 )) − λ2 z1 × (ω × (Δz1 + H1 ))

− λ2 z1 × (z1 × (Δω + I)),

| 101

(1.8.51)

∇ × I = F + σF,

(1.8.52)

∇ × f = −it − βωt ,

(1.8.53)

∇ ⋅ (i + ρω) = 0,

∇f = 0,

(1.8.54)

(ω, I(t), F(t))t=0 = (z0 , H0 , E0 ),

(1.8.55)

where (z1 , H1 , E1 ) = S(t)(z01 , H01 , E01 ) is the solution having initial value (z01 , H01 , E01 ) of problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8). Let ̃ E) ̃ = (z, H, E) − (ω, I, F) (z̃, H, = S(t)(z01 , H01 , E01 ) − DS(t)(z01 , H01 , E01 )(z0 , H0 , E0 ).

(1.8.56)

Then we have zt = λ1 [z + (Δz2 + H2 ) − ω × (Δz1 + H1 )]

+ λ1 [z1 × (Δz + H) − z1 × (Δω + I)]

− λ2 [z × (z2 × (Δz2 + H2 )) − z × (z1 × (Δz1 + H1 )]

− λ2 [z1 × (z × (Δz2 + H2 )) − z1 × (ω × (Δz1 + H1 )]

− λ2 [z1 × (z1 × (Δz + H)) − z1 × (z1 × (Δω + I)], ̃ ̃ ̃ ∇ × H = Et + σ E, ̃t − βZ ̃t , ∇ × Ẽ = −H

̃ + βZ) ̃ = 0, ∇ ⋅ (H ̃ H, ̃ E)| ̃ t=0 = 0. (Z,

∇ ⋅ Ẽ = 0,

(1.8.57) (1.8.58) (1.8.59) (1.8.60) (1.8.61)

Thus equation (1.8.57) can be rewritten as ̃t = λ1 [Z̃ + (Δz1 + H1 ) + z × (Δz + H)] Z ̃ − λ2 [z × (z × (Δz2 + H2 )] + λ1 [z1 × (ΔZ̃ + H)] + z × (z1 × (Δz + H)) + Z̃ × [z1 × (Δz1 + H1 )]

− λ2 [z1 × (z̃ × (Δz1 + H1 ))] + z1 × (z × (Δz + H)) ̃ − λ2 [z1 × (z1 × (ΔZ̃ + H))]. With the aid of equation (1.8.62), we have 1 d ̃ 2 ̃ 2 + C2 (‖z‖ + ‖H‖ + ‖E‖)4 , ‖Z‖ ≤ C1 ‖Z‖ 2 dt t

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩4 󵄩󵄩 ̃ 󵄩󵄩2 󵄩󵄩 ̃ 󵄩󵄩 C1 t C (t−s) C2 (󵄩󵄩󵄩z(s)󵄩󵄩󵄩 + 󵄩󵄩󵄩H(s)󵄩󵄩󵄩 + 󵄩󵄩󵄩E(s)󵄩󵄩󵄩) ds 󵄩󵄩Z(t)󵄩󵄩 ≤ 󵄩󵄩Z(0)󵄩󵄩e + ∫ e 1 0

(1.8.62)

102 | 1 Attractor and its dimension estimation t

󵄩4 󵄩 󵄩 󵄩 󵄩 󵄩 = ∫ eC1 (t−s) C2 (󵄩󵄩󵄩z(s)󵄩󵄩󵄩 + 󵄩󵄩󵄩H(s)󵄩󵄩󵄩 + 󵄩󵄩󵄩E(s)󵄩󵄩󵄩) ds. 0

Using Lemma 1.8.4, we obtain 2 󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩Z(t)󵄩󵄩 ≤ C(T, K)(‖z0 ‖ + ‖H0 ‖ + ‖E0 ‖) ,

0 ≤ t ≤ T.

We can get similar estimates of ‖H(t)‖ and ‖E(t)‖. Finally, we have Lemma 1.8.5. If the solutions for problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) are sufficiently smooth, then S(t) : (z0 , H0 , E0 ) → {z(t), H(t), E(t)} is uniformly differentiable. If it is differentiable at (z0 , H0 , E0 ) ∈ A , then the mapping DS(t)(z0 , H0 , E0 ) = (ω(t), I(t), F(t)) is the solution of problem (1.8.51)–(1.8.55). Let v1 (t), v2 (t), . . . , vJ (t) be the solutions of problem (1.8.37)–(1.8.41) with the initial values v1 (0) = ξ1 , v2 (0) = ξ2 , . . . , vJ (0) = ξJ , respectively, where ξi ∈ L2 (Ω), i = 1, 2, . . . , J. By explicit calculation, we arrive at d 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩v1 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩2 + 2 tr(L(u(t)) ⋅ QJ )󵄩󵄩󵄩v1 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩2 = 0, dt

(1.8.63)

where L(u(t)) = L(S(t)u0 ) is the linear mapping v → L(u(t))ν; ∧ means the crossproduct, tr is the trace of an operator. Also QJ is the orthogonal projection from L2 (R) to the subspace spanned by {v1 (t), v2 (t), . . . , vJ (t)}. From equation (1.8.63) we get the variation of the volume for the J-dimensional cube ⋀Jj=0 ξj : ωJ (t) = sup

sup

u0 ∈A ξ ∈L2 ,|ξj |≤1

󵄩󵄩 󵄩2 󵄩󵄩v1 (t) ∧ ⋅ ⋅ ⋅ ∧ vJ (t)󵄩󵄩󵄩L2 (Ω) t

≤ sup exp (− ∫ inf(tr(L(S(τ)u0 ) ⋅ QJ (τ)))dτ). u0 ∈A

(1.8.64)

0

Now we rewrite equations (1.8.37)–(1.8.39) as zt + f (z, ∇z, Δz, h; Z, ∇Z, ΔZ, H) = 0, et + σe − ∇ × h = 0,

ht + βzt + ∇ × e = 0, where f (z, ∇z, Δz, h; Z, ∇Z, ΔZ, H)

= −λ2 Δz − 2λ2 (∇z ⋅ ∇Z)Z − λ2 |∇Z|2 z − λ1 Z × Δz

(1.8.65) (1.8.66) (1.8.67)

1.8 Landau–Lifshitz–Maxwell equation

− λ1 Z × h + λ1 (ΔZ + H) × z − λ2 h

+ λ2 (Z ⋅ H)z + λ2 (Z ⋅ h)zZ + λ2 (z ⋅ H)Z.

| 103

(1.8.68)

Now selecting a periodical orthogonal basis {φj , ej (t), hj (t)} which satisfies: (1) Δφj = −λj2 φj ; (2) ‖φj ‖2 = ‖ej ‖ = ‖hj ‖2 = 1; we obtain that ‖∇φj ‖2 = |λj |, ‖Δφj ‖2 = λj2 . By definition, it follows that J

tr{L(u(t)) ⋅ QJ (t)} = ∑[(f (φj , ∇φj , Δφj , hj ; Z, ∇Z, ΔZ, H), φj ) j=1

+ σ(ej , ej ) − (∇ × hj , ej ) + (∇ × ej , hj )

− β(f (φj , ∇φj , Δφj , hj ; Z, ∇Z, ΔZ, H), hj )].

(1.8.69)

Since (∇ × ej , hj ) − (∇ × hj , ej ) = ∫ ∇ ⋅ (ej × hj )dx = 0, Ω

we only need to estimate in (1.8.69) the quantity (f (φj , ∇φj , hj ; H), φj ) and −β(f (φj , ∇φj , Δφj , hj ; Z, ∇Z, ΔZ, H), φj ). By equation (1.8.68), (f (φj , ∇φj , Δφj , hj ; Z, ∇Z, ΔZ, H), φj )

= −λ2 (Δφj , φj ) − 2λ2 ((∇Z, ∇φj )Z, φj ) − λ2 (|∇Z|2 φj , φj ) − λ1 (Z × Δφj , φj ) − λ1 (Z × hj , φj ) − λ2 (hj , φj ) + λ2 ((φj ⋅ H)Z, φj )

+ λ2 ((Z ⋅ hj )Z, φj ) + λ2 ((Z ⋅ H)φj , φj ), where

−λ2 (Δφj , φj ) = λ2 λj2 , 󵄨󵄨 󵄨 󵄨󵄨−2λ2 ((∇Z ⋅ ∇φj )Z, φj )󵄨󵄨󵄨 ≤ 2λ2 ‖∇φj ‖2 ‖φj ‖2 ‖∇Z‖∞ = 2λ2 |λj |‖∇Z‖∞ , 󵄨󵄨 󵄨 2 2 2 2 󵄨󵄨−λ2 (|∇Z| φj , φj )󵄨󵄨󵄨 ≤ λ2 |∇Z|∞ ‖φj ‖2 = λ2 |∇Z|∞ , 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨−λ1 (Z × Δφj , φj )󵄨󵄨󵄨 = 󵄨󵄨󵄨λ1 (∇φj , ∇Z × φj )󵄨󵄨󵄨 ≤ |λ1 |‖∇φj ‖2 ‖φj ‖2 |∇Z|∞ = |λ1 λj |‖∇Z‖∞ , 󵄨󵄨 󵄨󵄨 󵄨󵄨−λ1 (Z × hj , φj )󵄨󵄨 ≤ |λ1 |‖hj ‖2 ‖φj ‖2 = |λ1 |,

104 | 1 Attractor and its dimension estimation 󵄨 󵄨󵄨 󵄨󵄨−λ2 (hj , φj )󵄨󵄨󵄨 ≤ λ2 , 󵄨 󵄨󵄨 2 󵄨󵄨λ2 ((φj ⋅ H)Z, φj 󵄨󵄨󵄨 ≤ λ2 ‖H‖∞ ‖φj ‖2 = λ2 ‖H‖∞ , 󵄨󵄨 󵄨󵄨 󵄨󵄨λ2 ((Z ⋅ hj )Z, φj 󵄨󵄨 ≤ λ2 ‖hj ‖2 ‖φj ‖2 = λ2 , 󵄨 󵄨󵄨 󵄨󵄨λ2 ((Z ⋅ H)φj , φj 󵄨󵄨󵄨 ≤ λ2 ‖H‖∞ , thus (f (φj , ∇φj , △φj , hj ; Z, ∇Z, ΔZ, H), φj )

≥ λ2 λj2 − (2λ2 + |λ1 |)|λj |‖∇Z‖∞ − λ2 ‖∇Z‖2∞ − (2λ2 + |λ1 |) − 2λ2 ‖H‖∞ .

(1.8.70)

On account of equation (1.8.70) − β(f (φj , ∇φj , △φj , hj ; Z, ∇Z, ΔZ, H), φj ) + λ2 β(Δφj , hj ) + 2λ2 β((∇Z ⋅ ∇φj )Z, hj ) + λ2 β(|∇Z|2 φj , hj ) + λ1 β(Z × Δφj , hj ) − λ1 β((ΔZ + H) × φj , hj ) + λ2 β(hj , hj ) − λ2 β((φj ⋅ H)Z, hj ) − λ2 β((Z ⋅ hj )Z, hj − λ2 β((Z ⋅ H)φj , hj ),

where 󵄨󵄨 󵄨 2 󵄨󵄨λ2 β(Δφj , hj )󵄨󵄨󵄨 ≤ λ2 β‖Δφj ‖2 ‖hj ‖2 = λ2 βλj , 󵄨󵄨 󵄨 󵄨󵄨2λ2 β((∇Z ⋅ ∇φj )Z, hj )󵄨󵄨󵄨 ≤ 2λ2 β‖∇φj ‖2 ‖hj ‖2 ‖∇Z‖∞ = 2λ2 β|λj |‖∇Z‖∞ , 󵄨󵄨 󵄨 2 2 2 󵄨󵄨λ2 β(|∇Z| φj , hj )󵄨󵄨󵄨 ≤ λ2 β‖∇Z‖∞ ‖hj ‖2 ‖φj ‖2 = λ2 β‖∇Z‖∞ , 󵄨󵄨 󵄨󵄨 2 󵄨󵄨λ1 β(Z × Δφj , hj )󵄨󵄨 ≤ |λ1 |‖β‖‖Δφj ‖2 ‖hj ‖2 = |λ1 |βλj , 󵄨󵄨 󵄨 󵄨󵄨−λ1 β((ΔZ + H) × φj , hj )󵄨󵄨󵄨 ≤ |λ1 |β|ΔZ + H|∞ ‖φj ‖2 ‖hj ‖2 = |λ1 |β‖ΔZ + H‖∞ , λ2 β(hj , hj ) = λ2 β, 󵄨󵄨 󵄨 󵄨󵄨−λ2 β((φj ⋅ H)Z, hj )󵄨󵄨󵄨 ≤ λ2 ‖H‖∞ , 󵄨󵄨 󵄨󵄨 󵄨󵄨−λ2 β((Z ⋅ hj )Z, hj )󵄨󵄨 ≤ λ2 β, 󵄨󵄨 󵄨 󵄨󵄨−λ2 β((Z ⋅ H)φj , hj )󵄨󵄨󵄨 ≤ λ2 ‖H‖∞ , we then have − β(f (φj , ∇φj , Δφj , hj ; Z, ∇Z, ΔZ, H), φj ) ≥ −|λ|βλj2 − λ2 βλj2 − 2λ2 β|λj |‖∇Z‖∞

− [λ2 β‖∇Z‖2∞ + |λ1 |β‖ΔZ + H‖∞ + 2λ2 β‖H‖∞ ].

(1.8.71)

1.8 Landau–Lifshitz–Maxwell equation

| 105

Substituting equations (1.8.70) and (1.8.71) into (1.8.69), we arrive at J

tr{L(u(t)) ⋅ QJ (t)} ≥ (λ2 − (λ2 + |λ1 |β)) ∑ λj2 j=1

J

− (2λ2 + 2λ2 β + |λ1 |) ∑ |λj |‖∇Z‖∞ j=1

+ (σ − λ2 (1 + β))‖∇Z‖2∞ − 2λ2 (1 + β)‖H‖∞ − |λ1 |β‖ΔZ + H‖∞ J.

(1.8.72)

Choose parameter β sufficiently small so that λ2 > (λ2 + |λ1 |)β, that is, 0

a + √a2 − 4δb 21 J 2δ

(1.8.75)

106 | 1 Attractor and its dimension estimation for λj , we have the estimate λj2

2

1

2 1 1 (j − 1) n − 1] = (j − 1) n − (j − 1) n + 1, ≥[ 2 4

that is, 1

λj2 ≥ { 41

(j − 1)2 − j + 2 = 41 j2 + 32 j + 49 , 1

(j − 1) − (j − 1) 2 + 1 = 41 j + 4

3 4

1

− (j − 1) 2 ,

when n = 1,

when n = 2.

Now consider several cases: (i) When n = 1, J

∑ λj2 ≥ j=1

9 1 J 2 3 J ∑j − ∑j + J 4 1 2 1 4

1 3 9 (J + 1)(2J + 1)J − (J + 1)J + J 24 4 4 1 3 5 2 37 = J − J + J. 12 8 24

=

Therefore this ensures that equation (1.8.75) is valid. Choose J0 such that 1 3 5 2 37 (a + √a2 − 4δb)2 )J0 > 0, J0 − J0 + ( − 12 8 24 4δ2 6 2 2J02 − 15J0 + 37 − 2 (a + √a2 − 4δb) > 0, δ 2

(J0 −

15 71 3 2 ) + − (a + √a2 − 4δb) > 0. 4 16 δ2

When a + √a2 − 4δb < 2δ, we can choose J0 = 1. When a + √a2 − 4δb ≥ 2δ, we can choose J0 > √

3(a + √a2 − 4δb) 71 15 − + . 16 4 δ2

(ii) When n = 2, J

∑ λj2 ≥ j=1

J J−1 1 1 1 J 3 1 3 ∑ j + J − ∑(j − 1) 2 = (J − 1)J + J − ∑ j 2 4 j=1 4 8 4 j=1 j=1

1.8 Landau–Lifshitz–Maxwell equation

| 107

1 2

J−1 J 2 + 7J ≥ − (∑ j) √J − 1 8 j=1

≥

J 2 + 7J 1 32 − J . √2 8

To guarantee that (1.8.75) is true, we choose J0 such that J0 + 7J 1 21 (a + √a2 − 4δb)2 , − J > √2 0 8 4δ2 that is, J0 > {2√2 + [1 + 2(

2

1

2

2 a + √a2 − 4δb )] }. δ

Lastly, we get the following theorem: Theorem 1.8.3. Assume that Ω ⊂ Rn , 1 ≤ n ≤ 2, is a bounded domain, and the following conditions are satisfied: (4+3β2 )λ22 +2λ12 ; λ2 λ2 1 0 < β < min{ 2 , λ +|λ | }; 2 1 (β + β2 )(λ12 + λ22 ) < 41 ,

(1) λ2 > 0, σ > (2)

(3) (4) when n = 2,

‖∇Z0 ‖2 + ‖H0 ‖2 + ‖E0 ‖2 ≤ v where v = v(λ1 , λ2 , β) is a sufficiently small constant. Then the periodic initial value problem (1.8.1)–(1.8.4) and (1.8.7)–(1.8.8) possesses a unique attractor A = ω(B), where B = {(Z, H, E) ∈ (H 2 (Ω), H 1 (Ω), H 1 (Ω)) | ‖Z‖2H 2 + ‖E‖2H 1 + ‖H‖2H 2 ≤ ε0 + d0 } is an absorbing set. The Hausdorff and fractal dimensions of the attractor A are bounded and satisfy: (1) if a2 − 4δb < 0, dH (A ) ≤ 1,

dF (A ) ≤ 2;

(2) if a2 − 4δb ≥ 0, (i) for n = 1, if a + √a2 − 4δb < 2δ, then dH (A ) ≤ 1,

dF (A ) ≤ 2;

108 | 1 Attractor and its dimension estimation If a + √a2 − 4δb ≥ 2δ, then dH (A ) ≤ J1 ,

dF (A ) ≤ 2J1 ,

where J1 is the smallest integer, satisfying J1 > √

3(a + √a2 − 4δb) 71 15 + ; − 16 4 δ2

(ii) for n = 2, dH (A ) ≤ J2 ,

dF (A ) ≤ 2J2 ,

where J1 is the smallest integer, satisfying 2

1

2

2 a + √a2 − 4δb J2 > {2√2 + [1 + 2( )] }, δ

thus δ = λ2 − (λ2 + |λ1 |)β,

a = (2λ2 + 2λ2 β + |λ1 |)‖∇Z‖∞ ,

b = σ − λ2 (1 + β)‖∇Z‖2∞ + 2λ2 (1 + β)‖H‖∞ − |λ1 β|‖ΔZ + H‖∞ .

1.9 Nonlinear Schrödinger–Boussinesq equations Consider the initial boundary value problem for the following dissipation of nonlinear Schrödinger–Boussinesq equations [99] iεt + Δε − nε − β|ε|2 ε + iγε = g(x),

nt = Δφ,

n(0) = n0 ,

ε |𝜕Ω = n |𝜕Ω = φ |𝜕Ω = 0,

(1.9.2)

2

(1.9.3)

φ(0) = φ0 ,

(1.9.4)

φt = n + f (n) + μnt − λΔn + |ε| − αφ, ε(0) = ε0 ,

(1.9.1)

(1.9.5)

where x ∈ Ω ⊂ RN , t ∈ R+ , α, β, γ, μ, λ > 0 are constants, i = √−1, ε(x, t) = (ε1 (x, t), ε2 (x, t), . . . , εJ (x, t)) is an unknown complex function vector, n(x, t), φ(x, t) are unknown real functions. This system of equations appears in the nonlinear interaction of laser and plasma, where ε represents electric field, n denotes density

1.9 Nonlinear Schrödinger–Boussinesq equations | 109

disturbance, φ denotes potential function; see [169, 184, 212]. Constants γ > 0, μ > 0, α > 0 describe the dissipation effect. When γ = μ = α = 0, this system is integrable and has the soliton solution. The global existence and uniqueness of a smooth solution for equation (1.9.1)–(1.9.5) was first obtained by Guo [83, 108] in 1983. In what follows, we study the existence of a global attractor and the estimation of dimension for the problem (1.9.1)–(1.9.5). For simplicity, we consider one-dimensional space, Ω = [0, L]. Suppose f ∈ C ∞ (R) and it satisfies (i) lim inf

|s|→∞

F(s) ≥ 0; |s|2

(1.9.6)

(ii) there exists ω ≥ 0 such that lim

|s|→∞

sf (s) − ωF(s) ≥ 0, |s|2

(1.9.7)

s

where F(s) = ∫0 f (δ)dδ. Without loss of generality, we set ω ≤ 1. 1

Denote the norm ‖u‖ = ‖u‖L2 = (∫ |u|2 dx) 2 , H m (Ω) = W m,2 (Ω) A = −𝜕xx . Lemma 1.9.1. Assume that ε0 (x) ∈ L2 (Ω), g(x) ∈ L2 (Ω). Then for a solution of (1.9.1)–(1.9.5), we have ‖g‖2 󵄩󵄩 󵄩2 2 −γt −γt 󵄩󵄩ε(t)󵄩󵄩󵄩 ≤ ‖ε0 ‖ e + 2 (1 − e ). γ

(1.9.8)

Proof. Multiplying the equation (1.9.1) by ε and integrating it over Ω, we get (iεt , ε) + (εxx , ε) − (nε, ε) − β(|ε|2 ε, ε) + iγ(ε, ε) = (g, ε),

(1.9.9)

where (⋅, ⋅) means the inner product in L2 (Ω). Taking the imaginary part of equation (1.9.9) yields 1 d ‖ε‖2 + γ‖ε‖2 = Im(g, ε), 2 dt then, using Cauchy and Gronwall inequalities, we get (1.9.8). Taking the real part of equation (1.9.9), we obtain − Im(εt , ε) − ‖εx ‖2 − ∫ n|ε|2 dx − β ∫ |ε|4 dx = Re(g, ε).

(1.9.10)

110 | 1 Attractor and its dimension estimation Multiplying equation (1.9.1) by εt and integrating it over Ω, we get i(εt , εt ) + (εxx , εt ) − (nε, εt ) − β(|ε|2 ε, εt ) + iγ(ε, εt ) = (g, εt ).

(1.9.11)

On account of d ‖ε ‖2 = −2 Re(εxx , εt ), dt x

d ∫ n|ε|2 dx = 2 Re(nε, εt ) + ∫ nt |ε|2 dx, dt 1 d 4 |ε| dx = Re(|ε|2 ε, εt ), 4 dt and after taking the real part of equation (1.9.11), we arrive at β d 1 1 [ ‖ε ‖2 + ∫ n|ε|4 dx + ∫ |ε|2 dx] dt 2 x 2 4 1 − ∫ nt |ε|2 dx + Im γ(ε, εt ) = − Re(g, εt ). 2

(1.9.12)

Combining equations (1.9.10) and (1.9.12), we have β 1 d 1 [ ‖εx ‖2 + ∫ n|ε|2 dx + ∫ |ε|2 dx + Re(g, ε)] dt 2 2 4 + γ[‖εx ‖2 + ∫ n|ε|2 dx + β ∫ |ε|4 dx + Re(g, ε)] =

1 ∫ nt |ε|2 dx. 2

(1.9.13)

Now we estimate 21 ∫ nt |ε|2 dx. Setting m = nt + ρn, where ρ is small enough, taking the inner product of equation (1.9.3) with m, we arrive at (φt , m) = (n + f (n) + μnt − λnxx + |ε|2 − αφ, m). Since Aφt = −ntt , we obtain (mt + (α − ρ)m + μAm + [λA2 + A − μρA − ρ(α − ρ)]n + Af (n) + A|ε|2 , A−1 m) = 0. Then we have d 1 󵄩󵄩 − 21 󵄩󵄩2 1 󵄩󵄩 21 󵄩󵄩2 1 { 󵄩A m󵄩󵄩 + λ󵄩󵄩A n󵄩󵄩 + ‖n‖2 + ∫ F(n)dx} dt 2 󵄩 2 2 󵄩󵄩 − 21 󵄩󵄩2 󵄩󵄩 21 󵄩󵄩2 + (α − ρ)󵄩󵄩A m󵄩󵄩 + λρ󵄩󵄩A m󵄩󵄩 + ρ‖n‖2 + μ‖m‖2 + ρ ∫ f (n)ndx − μρ(n, m) − ρ(α − ρ)(n, A−1 m) + ρ ∫ n|ε|2 dx + ∫ nt |ε|2 dx = 0. On account of 1 󵄨 󵄨 μρ󵄨󵄨󵄨(n, m)󵄨󵄨󵄨 ≤ μρ‖n‖‖m‖ ≤ ρ‖n‖2 + 2μ2 ‖m‖2 , 8

(1.9.14)

1.9 Nonlinear Schrödinger–Boussinesq equations | 111

󵄨 󵄨 1 󵄩2 󵄩 ρ(α − ρ)󵄨󵄨󵄨(n, A−1 m)󵄨󵄨󵄨 ≤ ρ‖n‖2 + 2ρ(α − ρ)2 󵄩󵄩󵄩A−1 m󵄩󵄩󵄩 8 1 ≤ ρ‖n‖2 + 2ρ(α − ρ)2 c0 ‖m‖2 , 8 where c0 depends on the first eigenvalue of A, if we choose ρ small enough so that ρ ≤ γ,

1 ρ ≤ α, 2

1 2μ2 ρ + 2ρ(α − ρ)2 c0 < μ, 2

(1.9.15)

then equation (1.9.14) becomes d 1 󵄩󵄩 − 21 󵄩󵄩2 1 󵄩󵄩 21 󵄩󵄩2 1 { 󵄩A m󵄩󵄩 + λ󵄩󵄩A n󵄩󵄩 + ‖n‖2 + ∫ F(n)dx} dt 2 󵄩 2 2 1 󵄩󵄩 − 21 󵄩󵄩2 1 󵄩󵄩 21 󵄩󵄩2 3 + α󵄩󵄩A m󵄩󵄩 + λρ󵄩󵄩A n󵄩󵄩 + ρ‖n‖2 + μ‖m‖2 2 4 2 + ρ ∫ f (n)ndx + ρ ∫ n|ε|2 dx + ∫ nt |ε|2 dx ≤ 0.

(1.9.16)

Denote 1 󵄩 1 󵄩2 H(ε, n, φ) = 󵄩󵄩󵄩A 2 ε󵄩󵄩󵄩 + ∫ n|ε|2 dx + β ∫ |ε|4 dx 2 1 󵄩󵄩 − 21 󵄩󵄩2 1 󵄩󵄩 21 󵄩󵄩2 1 + 2 Re(g, ε) + 󵄩󵄩A m󵄩󵄩 + λ󵄩󵄩A n󵄩󵄩 + ‖n‖2 + ∫ F(n)dx, 2 2 2

(1.9.17)

󵄩 1 󵄩2 K(ε, n, φ) = 2γ[󵄩󵄩󵄩A 2 ε󵄩󵄩󵄩 + ∫ n|ε|2 dx + β ∫ |ε|4 dx + Re(g, ε)] 1 󵄩 1 󵄩2 󵄩 1 󵄩2 + α󵄩󵄩󵄩A 2 ε󵄩󵄩󵄩 + λρ󵄩󵄩󵄩A 2 n󵄩󵄩󵄩 2 3 + ρ‖n‖2 + ρ ∫ f (n)ndx + ρ ∫ n|ε|2 dx, 4

(1.9.18)

then from equations (1.9.14) and (1.9.15), we get 1 d H(ε, n, φ) + K(ε, n, φ) + μ‖m‖2 ≤ 0. dt 2

(1.9.19)

Now we estimate the terms H(ε, n, φ) and K(ε, n, φ). Lemma 1.9.2. For any small θ > 0, there exist constants c1 and c2 , which only depend on θ, such that 󵄩 1 󵄩2 1 H(ε, n, φ) ≥ (1 − θ)󵄩󵄩󵄩A 2 ε󵄩󵄩󵄩 + β ∫ |ε|4 dx 2 1 󵄩󵄩 − 21 󵄩󵄩2 1 󵄩󵄩 21 󵄩󵄩2 + 󵄩󵄩A m󵄩󵄩 + λ󵄩󵄩A n󵄩󵄩 2 2 1 2 2 + ‖n‖ − ‖ε‖ − c1 ‖ε‖6 − ‖g‖2 − c2 . 4

(1.9.20)

112 | 1 Attractor and its dimension estimation Proof. By Young and Sobolev inequalities, ∀θi > 0, i = 1, 2, 3, there exist c(θi ), i = 1, 2, 3, such that 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2 4 2 2 6 2 󵄨󵄨 ∫ n|ε| dx 󵄨󵄨󵄨 ≤ θ1 ‖n‖ + c(θ1 )‖ε‖L4 ≤ θ1 ‖n‖ + θ2 ‖εx ‖ + c(θ1 )c(θ2 )‖ε‖ , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 ∫ F(n)dx 󵄨󵄨󵄨 ≤ θ3 ‖n‖2 + c(θ3 ). 󵄨󵄨󵄨 󵄨󵄨󵄨 By the definition of H, and choosing θ1 = θ3 = 81 , θ2 = 0 ∈ (0, 1), inequality (1.9.20) follows. Lemma 1.9.3. If ρ is small enough and such that (1.9.16) is valid, then there exist constants c3 , c4 such that ρωH − K ≤ ρ(‖g‖2 + ‖ε‖2 ) + c3 ‖ε‖6 + c4 .

(1.9.21)

Proof. Firstly, we have ρωH(ε, n, φ) − K(ε, n, φ) 1 󵄩 1 󵄩2 = −(2γ − ωρ)󵄩󵄩󵄩A 2 ε󵄩󵄩󵄩 − (2γ − ρω)β ∫ |ε|4 dx 2 1 1 1 󵄩 1 󵄩2 󵄩 1 󵄩2 − ( α − ρω)󵄩󵄩󵄩A− 2 m󵄩󵄩󵄩 − (1 − ω)λρ󵄩󵄩󵄩A− 2 n󵄩󵄩󵄩 2 2 2 3 1 − ( − ω)ρ‖n‖2 − ρ(ω − 1) ∫ n|ε|2 dx 4 2 − 2(γ − ωρ)(g, ω) − ρ ∫[f (n)n − ωF(n)]dx.

(1.9.22)

With the aid of equation (1.9.7), f (n)n − ωF(n) ≥ − 81 n2 − c, and then we have − ∫[f (n)n − ωF(n)]dx ≤

1 ‖n‖2 + c. 8

(1.9.23)

From equations (1.9.22) and (1.9.23), together with ω ≤ 1 and ρ, which satisfies equation (1.9.15), we get 󵄩 1 󵄩2 ρωH(ε, n, φ) − K(ε, n, φ) ≤ −(2γ − 2ρ)󵄩󵄩󵄩A 2 ε󵄩󵄩󵄩

1 1 󵄩 1 󵄩2 1 − (2γ − ρ)β ∫ |ε|4 dx − ( α − ρ)󵄩󵄩󵄩A− 2 m󵄩󵄩󵄩 2 2 2 1 1 2 2 6 + ρ‖g‖ − ρ‖ε‖ + c3 ‖ε‖ + c4 2 2 ≤ ρ‖g‖2 + ρ‖ε‖2 + c3 ‖ε‖6 + c4 .

It follows that d 1 H(ε, n, φ) + ωρH(ε, n, φ) + μ‖m‖2 ≤ ρ‖g‖2 + ρ‖ε‖2 + c3 ‖ε‖6 + c4 . dt 2

1.9 Nonlinear Schrödinger–Boussinesq equations | 113

By Gronwall inequality, we have H(ε, n, φ) ≤ H(ε0 , n0 , φ0 )e−ρωt t

+e

−ρωt

∫(ρ‖g‖2 + ρ‖ε‖2 + c3 ‖ε‖6 + c4 )eρωτ dτ,

(1.9.24)

0

then Lemma 1.9.1 yields Proposition 1.9.1. Let ε(t), n(⋅, t), φ(⋅, t) ∈ H01 (Ω) be the solution of problem (1.9.1)– (1.9.5). Suppose equations (1.9.6)–(1.9.15) are valid, then we have H(ε, n, φ) ≤ H(ε0 , n0 , φ0 )e−ρωt + a∞ (1 − e−ρωt ) + e−ρωt c(‖g‖2 , ‖ε0 ‖2 ),

∀t ≥ 0, (1.9.25)

where a∞ depends on ‖g0 ‖2 , c(‖g‖2 , ‖ε0 ‖2 ), ‖g‖2 and ‖ε0 ‖2 . By use of Galerkin method and the above a priori estimate, it is easy to prove Theorem 1.9.1. Suppose ε0 (x), n0 (x), φ0 (x) ∈ H01 (Ω), g(x) ∈ L2 (Ω), and f satisfies equations (1.9.6)–(1.9.7). Then there exists a unique global solution (ε(⋅, t), n(⋅, t), φ(⋅, t)) ∈ L∞ (0, ∞; H01 (Ω)) ∩ C(0, ∞; H01 (Ω)), nt ∈ L2 (0, ∞; L2 (Ω)) of problem (1.9.1)–(1.9.5). When t > 0, the map (ε0 , n0 , φ0 ) → (ε(t), n(t), φ(t)) is continuous in H01 × H01 × H01 . Let S(t) be the operator semigroup generated by problem (1.9.1)–(1.9.5), that is, S(t)u0 = S(t)(ε0 , n0 , φ0 ) = (ε(⋅, t), n(⋅, t), φ(⋅, t)). By Theorem 1.9.1, we know that S(t)u0 is continuous in H01 × H01 × H01 . Through Lemma 1.9.2, we have Proposition 1.9.2. Suppose (1.9.6) and (1.9.7) are valid, and there exists a constant ρ1 such that for any R > 0, there exists t1 (R) > 0 such that for any t ≥ t1 (R), whenever ε0 , n0 , φ0 ∈ H01 and ‖ε0 ‖2H 1 + ‖n0 ‖2H 1 + ‖φ0 ‖2H 1 ≤ R2 0

0

0

(1.9.26)

for the initial value (ε0 , n0 , φ0 ), the solution (ε(⋅, t), n(⋅, t), φ(⋅, t)) of problem (1.9.1)–(1.9.5) satisfies 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩ε(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩n(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩φ(⋅, t)󵄩󵄩󵄩H 1 ≤ ρ1 , 0 0 0

(1.9.27)

that is, there exists a bounded absorbing set for the problem (1.9.1)–(1.9.5) in H 2 . Now we consider the bounded absorbing set in H 2 . Proposition 1.9.3. Assume that (1.9.6)–(1.9.7) are valid and there exists a constant ρ2 ≥ 0 such that for any R > 0, there exists t2 = t2 (R) > 0 such that for any ε0 , n0 , φ0 ∈ D(A), 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩ε0 (x)󵄩󵄩󵄩H 2 (Ω) + 󵄩󵄩󵄩n0 (x)󵄩󵄩󵄩H 2 (Ω) + 󵄩󵄩󵄩φ0 (x)󵄩󵄩󵄩H 2 (Ω) ≤ R .

(1.9.28)

114 | 1 Attractor and its dimension estimation Then solutions of problem (1.9.1)–(1.9.5) satisfy 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩 2 󵄩󵄩ε(⋅, t)󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩n(⋅, t)󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩φ(⋅, t)󵄩󵄩󵄩H 2 ≤ ρ2 ,

t ≥ t2 .

(1.9.29)

Proof. We have a uniform estimate of solution using H 1 norm. Now for t large enough, we estimate ‖εxx ‖2 , ‖nxx ‖2 and ‖φxx ‖2 . Taking the inner product of(1.9.1) with Aεt + γAε, integrating it and taking real part, we arrive at Re(εxx , Aεt + γAε) − Re(nε, Aεt + γAε) − Re(β|ε|2 + g, Aεt + γAϵ) = 0.

(1.9.30)

Since Re(β|ε|2 ε, εxxt ) =

d 1 2 Re ∫[β(|ε|2x )εx + |ε|2 |εx |2 + (Re(ε ⋅ εx )) ]dx dt 2 1 + Re ∫ β[(|ε|2x )εt − |εx |2 |ε|2t 2 + 2 Re(εx , εt ) Re(ε, εx ) + 2 Re(ε, εt )|εx |2 ]dx

= where

dh1 + h2 , dt

(1.9.31)

1 2 h1 = Re β ∫[(|ε|2x )εx + |ε|2 |εx |2 + (Re(ε ⋅ εtx )) ]dx 2 1 h2 = β ∫[Re(|ε|2 )x εt − |εx |2 |ε|2t + 2 Re(εx εt ) + 2 Re(εεx )|εx |2 ]dx 2

and (g, Aεt ) =

(1.9.32) (1.9.33)

d (g, Aε), dt

using equation (1.9.30), we obtain d 1 [ |Aε|2 − h1 + Re(g, Aε)] + γ|Aε|2 − h2 dt 2 + βγ Re ∫(|ε|2 ε)x εx dx + Re γ(g, Aε) + γ Re ∫(nε)x εx dx + Re(nε, Aεt ) = 0.

(1.9.34)

On the other hand, setting m = nt + ρn, and taking the inner product of (1.9.3) with m, we have d 1 1 1 󵄩 1 󵄩2 1 󵄨 1 󵄨2 [ ‖m‖2 + λ‖An‖2 + 󵄩󵄩󵄩A 2 n󵄩󵄩󵄩 + ∫ f 󸀠 (n)󵄨󵄨󵄨A 2 n󵄨󵄨󵄨 dx] dt 2 2 2 2 󵄩󵄩 21 󵄩󵄩2 󵄩 1 󵄩2 2 + (α − ρ)‖m‖ + μ󵄩󵄩A m󵄩󵄩 − ρ(α − ρ)(n, m) − μρ(An, m) + ρ󵄩󵄩󵄩A 2 n󵄩󵄩󵄩 + ρλ‖An‖2

1.9 Nonlinear Schrödinger–Boussinesq equations | 115

1 + ∫[ρ|ε|2x nx − f 󸀠󸀠 (n)nt |nx |2 + ρf 󸀠 (n)|nx |2 ]dx = −(A|ε|2 , nt ). 2

(1.9.35)

Choosing ρ small enough, such that (1.9.15) is valid, we then get d 1 2 1 1 󵄩 1 󵄩2 1 󵄨 1 󵄨2 [ |m| + λ‖An‖2 + 󵄩󵄩󵄩A 2 n󵄩󵄩󵄩 + ∫ f 󸀠 (n)󵄨󵄨󵄨A 2 n󵄨󵄨󵄨 dx] dt 2 2 2 2 3 󵄩󵄩 21 󵄩󵄩2 1 󵄩 1 󵄩2 2 + (α − ρ)‖m‖ + ρ󵄩󵄩A n󵄩󵄩 + ρλ‖An‖2 + μ󵄩󵄩󵄩A 2 m󵄩󵄩󵄩 4 2 1 + ∫[ρ|ε|2x nx − f 󸀠󸀠 (n)nt |nx |2 + ρf 󸀠 (n)|nx |2 ]dx ≤ −(A|ε|2 , nt ). 2

(1.9.36)

Since (−A|ε|2 , nt ) − 2 Re(nε, Aεt ) =

d Re ∫[−n|εx |2 − nx εεx ]dx + 2 Re ∫ nx εt εx dx + nt |εx |2 dx, dt

(1.9.37)

taking (1.9.34) multiplied by 2, together with (1.9.36), we get d 1 1 [‖Aε‖2 + ‖m‖2 + λ‖An‖2 − 2 Re(g, Aε) − 2h1 + h3 ] dt 2 2 1 󵄩 1 󵄩2 2 + 2γ‖λε‖ + (α − ρ)‖m‖2 + ρλ‖An‖2 + μ󵄩󵄩󵄩A 2 m󵄩󵄩󵄩 2 − 2h2 + h1 + 2 Re(g, Aε) ≤ 0,

(1.9.38)

where h1 and h2 are defined by (1.9.32) and (1.9.33), respectively, 1 󵄩 1 󵄩2 1 󵄨 1 󵄨2 h3 = 󵄩󵄩󵄩A 2 n󵄩󵄩󵄩 + ∫ f 󸀠 (n)󵄨󵄨󵄨A 2 n󵄨󵄨󵄨 dx + Re ∫[n|εx |2 + 2nx εεx ]dx, 2 2 3 󵄩 1 󵄩2 h4 = ρ󵄩󵄩󵄩A 2 n󵄩󵄩󵄩 + 2βγ Re ∫(|ε|2 ε)x εx dx 4 + 2γ Re ∫(nε)x εx dx − 2 Re ∫ nx εt εx dx + ∫ nt |εx |2 dx.

(1.9.39)

(1.9.40)

Set ̃ n, φ) = ‖Aε‖2 + 1 ‖m‖2 + 1 λ‖An‖2 − 2 Re(g, Aε) − 2h1 + h3 , H(ε, (1.9.41) 2 2 ̃ n, φ) = 2γ‖Aε‖2 + (α − ρ)‖m‖2 − ρλ‖An‖2 + 2 Re(g, Aε) − 2h2 + h4 . (1.9.42) K(ε, Then equation (1.9.38) can be converted into 1 d̃ ̃ n, φ) + 1 μ󵄩󵄩󵄩A 2 m󵄩󵄩󵄩2 ≤ 0. H(ε, n, φ) + K(ε, 󵄩 󵄩 dt 2

(1.9.43)

̃ n, φ) and K(ε, ̃ n, φ). By Proposition 1.9.2, there exists a constant Now we estimate H(ε, 󸀠 ρ1 such that 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󸀠2 󵄩󵄩ε(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩n(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩φ(⋅, t)󵄩󵄩󵄩H 1 ≤ ρ1 . 0 0 0

116 | 1 Attractor and its dimension estimation On account of the embedding theorem, there exists a constant ρ󸀠󸀠 1 , which depends on ‖g‖, such that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󸀠󸀠 󵄩󵄩ε(⋅, t)󵄩󵄩󵄩L∞ (Ω) , 󵄩󵄩󵄩n(⋅, t)󵄩󵄩󵄩L∞ (Ω) , 󵄩󵄩󵄩φ(⋅, t)󵄩󵄩󵄩L∞ (Ω) ≤ ρ1 . By the definition of h1 and h3 , we have 2 (3) |h1 | ≤ Cρ󸀠󸀠 1 ∫ |εx | dx ≤ ρ1 ,

t ≥ T1 (R),

(4) |h3 | ≤ Cρ󸀠1 + C(ρ󸀠󸀠 1 ) ≤ ρ1 ,

(1.9.44)

t ≥ T2 (R),

(1.9.45)

(4) where ρ(3) 1 and ρ1 only depend on ‖g‖. As with h2 and h4 , we have 2 |h2 | ≤ Cρ󸀠󸀠 1 ∫ |εx | |εt |dx,

t ≥ T(R),

(1.9.46)

|h4 | ≤ Cρ󸀠1 + C2 ∫ |εx ||nx ||εt |dx + C3 ∫ |nt ||εx |2 dx.

(1.9.47)

Denote εt = iΔε + b1 for b1 = −inε − iβ|ε|2 − γε − ig ∈ L∞ (0, ∞; L2 ). Substituting εt into (1.9.46), we have 2 󸀠󸀠 2 |h2 | ≤ Cρ󸀠󸀠 1 ∫ |εx | |Δε|dx + Cρ1 ∫ |b1 ||Δεx | dx 3

3

1

3

≤ C‖Δε‖‖εx ‖2L4 + C‖b1 ‖‖εx ‖2L4 ≤ C‖Δε‖ 2 ‖εx ‖ 2 + C‖b1 ‖‖Δε‖ 2 ‖εx ‖ 2 ≤ θ‖Δε‖2 + C(θ),

∀θ ≥ 0,

(1.9.48)

where we use the embedding theorem, and C depends only on ‖g‖. Similarly, since m = nt + ρn, we have |h4 | ≤ C + C ∫ |εx |‖nx ‖2 |Δε|dx + C ∫ |εx ||nx ||b1 |dx + C ∫ |m||εx |2 dx + C ∫ |n||εx |2 dx ≤ C + C‖Aε‖‖εx ‖L4 ‖nx ‖L4 + C‖b1 ‖‖εx ‖L4 ‖nx ‖L4 + C‖m‖‖εx ‖2L4 1

1

3

3

≤ C + C‖Aε‖‖Aε‖ 4 ‖An‖ 4 ‖εx ‖ 4 ‖nx ‖ 4 1

1

3

3

1

3

+ C|b1 |‖Aε‖ 4 ‖An‖ 4 ‖εx ‖ 4 ‖nx ‖ 4 + C‖m‖‖Aε‖ 2 ‖εx ‖ 2

≤ C + θ1 ‖Aε‖2 + θ2 ‖m‖2 + θ3 ‖An‖2 ,

∀θ1 , θ2 , θ3 ≥ 0,

(1.9.49)

where C depends only on θi , i = 1, 2, 3, and ‖g‖. If we select θ, θi , i = 1, 2, 3, small ̃ and K, ̃ there exist constants C0 ≥ 0 and C1 ≥ 0 enough, then by the definition of H ̃−K ̃ ≤ C1 . From equation (1.9.43), we have such that C0 H ̃ dH ̃ + 1 μ‖Am‖2 ≤ C1 , + C0 K dt 2

∀t ≥ T1 (R).

(1.9.50)

1.9 Nonlinear Schrödinger–Boussinesq equations | 117

By Gronwall inequality, we have ̃ n, φ) ≤ e−C0 (t−T1 ) H(ε, ̃ n, φ)(T1 ) + C1 (1 − e−C0 (t−T1 ) ). H(ε,

(1.9.51)

Since also ̃ ≥ (1 − θ)‖Aε‖2 + 1 ‖m‖2 + λ ‖An‖2 − C, H 2 2 where 0 < θ < 1 and C depends only on ‖g‖ and ρ󸀠1 , this yields the claim. Now consider the long time behavior of the semigroup S(t)u0 . As a corollary of Proposition 1.9.2, we have Corollary 1.9.1. The set 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 B1 = {(ε, n, φ) ∈ V 3 = H01 × H01 × H01 , 󵄩󵄩󵄩ε(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩n(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩φ(⋅, t)󵄩󵄩󵄩H 1 ≤ ρ21 } 0 0 0 is the bounded absorbing set of S(t) in V 3 , and the set 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 B2 = {(ε, n, φ) ∈ D(A), 󵄩󵄩󵄩ε(⋅, t)󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩n(⋅, t)󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩φ(⋅, t)󵄩󵄩󵄩H 2 ≤ ρ22 } is the bounded absorbing set of S(t) in D(A). Now we consider the ω-limit set of B1 in V 3 and B2 in D(A). Set ωω (B2 ) = ⋂ ⋃ S(t)B2 s≥0 t≥s

where the closures are taken in weak topology of D(A)3 . Let ωω (B1 ) = ⋂ ⋃ S(t)B1 s≥0 t≥s

where the closures are in the weak topology of V 3 . Based on the related theorems in [197], we have Theorem 1.9.2. The set A = ωω (B2 ) satisfies: (1) A is weakly compact and is bounded in D(A). Also S(t)A = A ,

∀t ∈ R.

(2) For any bounded set in D(A), it follows that lim dω (S(t)B, A ) t→∞ H

= lim sup inf dω (S(t)x, y) = 0. t→∞ x∈B y∈A

(3) Under the weak topology of D(A), A is connected.

118 | 1 Attractor and its dimension estimation Corollary 1.9.2. For any bounded set B in D(A), the set S(t)B converges to A in V 3 . We now investigate the dimension of the invariant set in D(A). Assume that ε0 = (ξ0 , n0 , φ0 ) ∈ V × V × V is given, and let S(t)ξ0 = (ε(⋅, t), n(⋅, t), φ(⋅, t)) be the semigroup generated by the solution of problem (1.9.1)–(1.9.5). We consider the following variational equations: iUt + Uxx − nU − Vε − β|ε|2 U − 2β Re(εU)ε + iγU = 0, Vt = Wxx ,

Wt = V + f (n)V + μV − λΔV − αW + 2 Re(εU) 󸀠

(1.9.52) (1.9.53) (1.9.54)

with boundary values U(0) = u0 ,

V(0) = v0 ,

W(0) = ω0 ,

U(0, t) = U(L, t) = V(0, t) = V(L, t) = W(0, t) = W(L, t) = 0.

(1.9.55) (1.9.56)

It is easy to prove that, when (u0 , v0 , ω0 ) ∈ V ×V ×V, the linear problem (1.9.52)–(1.9.56) ̃ = (U(t), V(t), W(t)) ∈ C(R, V 3 ). By using the energy possesses a unique solution U(t) estimation method, we can prove Proposition 1.9.4. The semigroup operator S(t)ξ0 = (ε(t), n(t), φ(t)) is differentiable in ̃ = V × V × V, and its differential is the linear mapping η0 = (u0 , v0 , ω0 ) ∈ V 3 → U(t) ̃ (U(t), V(t), W(t)), that is, DS(t)ξ0 = U(t). Set u = eγt U, v = eγt V, ω = eγt W. Then (1.9.52)–(1.9.54) can be rewritten as iut + uxx − vε − nv − β|ε|2 u − 2β(εu)ε = 0,

vt = ωxx + γv,

ωt = v + f (n)v − μvt − μγv − λΔv − (α − λ)ω + 2 Re(εu). 󸀠

(1.9.57) (1.9.58) (1.9.59)

Multiplying equation (1.9.57) by u, integrating it over Ω, and taking the real part, we obtain d 1 1 2 ̄ [ ‖u ‖2 + ∫ n|u|2 dx + β ∫ |ε|2 |u|2 dx + β ∫[Re(εu)] dx + ∫ vεudx] dt 2 x 2 1 − ∫ nt |u|2 dx − β ∫(|ε|2t )|u|2 dx 2 − β ∫ Re(εu) Re(εt u)dx − Re ∫[vεt u + εuvt ]dx = 0.

(1.9.60)

Multiplying equation (1.9.58) by vt , equation (1.9.59) by ωt , integrating them over Ω, and adding together, produces d 1 1 1 [ λ‖v ‖2 + ‖ωx ‖2 + ∫( (1 + f 󸀠 (n))|v|2 − γ(v, ω))dx] dt 2 x 2 2 1 − ∫ f 󸀠󸀠 (n)nt |v|2 dx + μ‖vt ‖2 − (α − 2γ)(ω, νt ) + 2 Re ∫ vt εudx = 0. (1.9.61) 2

1.9 Nonlinear Schrödinger–Boussinesq equations | 119

Through (1.9.60) and (1.9.61), we obtain d 1 1 1 2 {‖ux ‖2 + λ‖vx ‖2 + ‖ωx ‖2 − ∫[n|u|2 + 2β|ε|2 |u|2 + 2β(Re(εu)) + 2 Re vεu]̄ dt 2 2 2 1 1 + ∫[ (1 + f 󸀠 (n))|v|2 − μγ|v|2 − γ(v, ω)]dx} 2 2 1 + ∫[−nt |u|2 − 2β(|ε|2 )t |u|2 − 2β Re(εt u) − 2vεt u − f 󸀠󸀠 (n)nt |v|2 ]dx 2 + μ‖vt ‖2 − (α − 2γ)(ω, vt ) = 0.

(1.9.62)

Let 1 1 1 J(η) = J(u, v, ω) = ‖ux ‖2 + λ‖vx ‖2 + ‖ωx ‖2 + J1 , 2 2 2

(1.9.63)

where 1 2 J1 = ∫{ n|u|2 + β|ε|2 |u|2 + β(Re(εu)) + Re(vεu)+ 2 1 1 + (1 + f 󸀠 (n))|v|2 − μγ|v|2 − γ(v, ω)}dx. 2 2

(1.9.64)

Setting I(η) = I(u, v, ω) = −μ‖vt ‖2 + (α − 2λ)(ω, vt ) + I1 ,

(1.9.65)

where 1 I1 (η) = ∫{nt |u|2 +2β(|ε|2 )t |u|2 +2β Re(εu) Re(εt u)+2 Re(vεt u)+ f 󸀠󸀠 (n)nt |v|2 }dx, (1.9.66) 2 equation (1.9.62) becomes d J(η) = I(η). dt

(1.9.67)

On the other hand, d ‖u‖2 = 2 Im ∫ εuvdx + 2β ∫ Re(εu) Im(εu)dx dt d 1 1 ( λ‖v‖2 + ‖ω‖2 ) = −μ‖ωx |2 + λγ‖v‖2 dt 2 2

(1.9.68)

− (α − γ)‖ω‖2 − 2 Re(εu, ω) + ∫(I + f 󸀠 (n))vωdx. (1.9.69) Letting Jσ (η) = Jσ (u, v, ω) = J(η) + σ‖u‖2 + σ(λ‖v‖2 + ‖ω‖2 ),

(1.9.70)

120 | 1 Attractor and its dimension estimation Iσ (η) = Iσ (u, v, ω) = I(η) − σI2 − 2σμ‖σω‖2 ,

(1.9.71)

where I2 = 2((1 + f 󸀠 (n))v, ω) + 2λγ‖ν‖2 − 2(α − λ)‖ω‖2 − 4 Re(εu, ω) + 2 Im ∫ εuvdx + 2β ∫ Re(εu) Im(εu)dx,

(1.9.72)

we get the energy equality dJσ (η) = Iσ (η). dt

(1.9.73)

Assume that X is a bounded invariant set of S(t) in D(A)3 , that is, S(t)X = X,

∀t ≥ 0.

Also X is bounded in D(A)3 , ξ0 ∈ X, S(t)ξ0 = (ε(t), n(t), φ(t)) ∈ X and 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |X|∞ = sup[󵄨󵄨󵄨ε(t)󵄨󵄨󵄨∞ + 󵄨󵄨󵄨n(t)󵄨󵄨󵄨∞ + 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨∞ ] < ∞, ξ0 ∈X

where | ⋅ |∞ = supt ‖ ⋅ ‖L∞ (Ω) . By the definition of Jσ in equation (1.9.71), we know that there exist constants σ ≥ 0, C0 , C1 ≥ 0 such that C0 (‖u‖2V + ‖v‖2V + ‖ω‖2V ) ≤ Jσ (u, v, ω)

≤ C1 (‖u‖2V + ‖v‖2V + ‖ω‖2V ),

∀(u, v, ω) ∈ V × V × V.

(1.9.74)

Thus Jσ (u, v, ω) is an equivalent norm in V × V × V. Now we introduce R-linear inner product in V 3 . Suppose that η = (η1 , η2 , η3 ), ξ = (ξ1 , ξ2 , ξ3 ) ∈ V × V × V. Define 1 1 1 Ψ(η, ξ ) = Re ∫[η1x ξ1x + λη2x ξ2x + η3x ξ3x + nη1 ξ1 2 2 2 1 1 2 + β‖ε‖ η1 ξ1 + β Re(ϵη1 ) Re(εξ1 ) + (1 + f 󸀠 (n))η2 ξ2 − μγη2 ξ2 ]dx 2 2 1 1 + ∫ Re(η2 εξ2 + η1 εξ1 )dx − γ ∫(η2 ξ3 + η3 ξ2 )dx 2 2 + σ Re ∫ η1 ξ1 dx + σ ∫ Re[λη2 ξ2 + η3 ξ3 ]dx.

(1.9.75)

It is easy to verify that Ψ(η, ξ ) is bilinear in V 3 × V 3 , and Ψ(η, η) = Jσ (η1 , η2 , η3 ). By 1 equation (1.9.74), we know that it is equivalent. Hence Ψ(η, ξ ) 2 is an equivalent norm in V × V × V. j Suppose that ξ0 = (u0j , v0j , ω0j ), j = 1, 2, . . . , m, are m elements in V × V × V. Assume ξ j (t), j = 1, 2, . . . , m, are solutions of problem (1.9.52)–(1.9.56), that is, j ξ j (t) = (uj (t), vj (t), ωj (t)) = (DS(t)ξ0 )ξ0 . Let ηi (t) = eγt ξ i (t), i = 1, 2, . . . , m. Then

1.9 Nonlinear Schrödinger–Boussinesq equations | 121

ηi (t) = (ui (t), vi (t), ωi (t)) is the solution of (1.9.57)–(1.9.59) with the initial value ηi (0) = (ui (0), vi (0), ωi (0)) = ξ0i . Now we consider the variation of the volume: 󵄨 󵄨󵄨 1 i j 2 m 󵄨󵄨ξ (t) ∧ ξ (t) ∧ ⋅ ⋅ ⋅ ∧ ξ (t)󵄨󵄨󵄨∧m (V 3 ) = det (ξ , ξ )V 3 1≤i,j≤m

(1.9.76)

where (⋅, ⋅)V 3 is the inner product of ψ(⋅, ⋅) determined by equation (1.9.75). Thus we have the following theorem: Theorem 1.9.3. Assume that X is a bounded invariant set in D(A)3 , then there exist constants C1 , C2 ≥ 0 such that for any ξ0 = (ε0 , n0 , φ0 ) ∈ X, ξ i (t) = (DS(t)ξ1 )ξ0i satisfies 󵄨󵄨 1 󵄨 2 m 󵄨󵄨ξ (t) ∧ ξ (t) ∧ ⋅ ⋅ ⋅ ∧ ξ (t)󵄨󵄨󵄨∧m (V 3 ) 󵄨 󵄨 ≤ 󵄨󵄨󵄨ξ01 (t) ∧ ξ02 (t) ∧ ⋅ ⋅ ⋅ ∧ ξ0m (t)󵄨󵄨󵄨∧m (V 3 ) ⋅ C1m e(C2 −mγ)t

(1.9.77)

where constants C1 , C2 depend on coefficients α, β, γ, μ and λ. Also m ≥ 1, t ≥ 0. Proof. First, notice that 󵄨󵄨 1 󵄨2 2 m 󵄨󵄨ξ (t) ∧ ξ (t) ∧ ⋅ ⋅ ⋅ ∧ ξ (t)󵄨󵄨󵄨∧m (V 3 ) 󵄨 󵄨 = e−2γmt 󵄨󵄨󵄨η1 (t) ∧ η2 (t) ∧ ⋅ ⋅ ⋅ ∧ ηm (t)󵄨󵄨󵄨∧m (V 3 ) = e−2γmt det Ψ(ηi (t), ηj (t)). 1≤i,j≤m

(1.9.78)

So we need to estimate det1≤i,j≤m Ψ(ηi (t), ηj (t)). Let Hm (t) = det Ψ(ηi (t), ηj (t)), 1≤i,j≤m

m d d Hm (t) = det Ψ(ηi (t), ηj (t)) − ∑ det Ψ(ηi (t), ηj (t))l , 1≤i,j≤m dt dt 1≤i,j≤m l=1

(1.9.79)

where Ψ(ηi (t), ηj (t))l = (1 − δjl )Ψ(ηi , ηj ) + δjl

d Ψ(ηi (t), ηj (t)). dt

The mapping Ψ(⋅, ⋅) is an R-linear inner product in V 3 . Hence Ψ(ηi (t), ηj (t)) =

1 [Ψ(ηi + ηj , ηi + ηj ) − Ψ(ηi − ηj , ηi − ηj )], 4

and then, by equation (1.9.75), we have d 1 d Ψ(ηi (t), ηj (t)) = J (ηi (t) + ηj (t)) dt 4 dt σ 1 d 1 − J (ηi (t) − ηj (t)) = [Iσ (ηi + ηj ) − Iσ (ηi − ηj )]. 4 dt σ 4

(1.9.80)

122 | 1 Attractor and its dimension estimation By Lemma 1.3.2 in [74], from (1.9.79)–(1.9.80), we have j m Iσ (∑m dHm (t) j=1 xj η (t)) . = Hm (t) ∑ max min j ̸ dt F⊂Rm ,dim F=l x=0,x∈F Jσ (∑m l=1 xj η (t)) l=1

(1.9.81)

Using the definition in (1.9.71), we obtain Iσ (∑ xj ηj (t)) = Iσ (∑ xj uj , ∑ xj vj , ∑ xj ωj ) m 󵄩󵄩 󵄩󵄩2 = −μ󵄩󵄩󵄩 ∑ xj vjt 󵄩󵄩󵄩 + (α − 2γ)(∑ xj ωj , ∑ xj vjt ) 󵄩 󵄩 j=1

󵄩󵄩 󵄩󵄩2 − I1 (∑ xj ηj (t)) − μσ 󵄩󵄩󵄩 ∑ xj ωjt (t)󵄩󵄩󵄩 + σI2 (∑ xj ηj (t)) 󵄩 󵄩 󵄩󵄩 󵄩󵄩2 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≤ C 󵄩󵄩󵄩 ∑ xj ωj (t)󵄩󵄩󵄩 + 󵄨󵄨󵄨I1 (∑ xj ηj (t))󵄨󵄨󵄨 + σ 󵄨󵄨󵄨I2 (∑ xj ηj (t))󵄨󵄨󵄨. 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 On account of equations (1.9.74) and (1.9.72), there exists C3 ≥ 0 such that 3

1

󵄩󵄩 2 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 Iσ (∑ xj ηj (t)) ≤ C3 󵄩󵄩󵄩 ∑ xj ηj (t)󵄩󵄩󵄩 3 󵄩󵄩󵄩 ∑ xj ηj (t)󵄩󵄩󵄩 . 󵄩 󵄩 󵄩V 󵄩

(1.9.82)

On the other hand, by (1.9.74), we have 󵄩󵄩 󵄩󵄩2 Iσ (∑ xj ηj (t)) ≥ C0 󵄩󵄩󵄩 ∑ xj ηj (t)󵄩󵄩󵄩 3 . 󵄩 󵄩V

(1.9.83)

By (1.9.81), (1.9.82) and (1.9.83), we have 3

m ‖ ∑ xj ηj (t)‖ 2 C d Hm (t) ≤ 3 Hm (t) ∑ max min 3 ̸ dt C0 F⊂Rm ,dim F=t x=0,x∈F l=1 ‖ ∑ xj ηj (t)‖V2 3

≤

3

‖ ∑ xj uj ‖ ‖ ∑ xj vj ‖ ‖ ∑ xj ωj ‖ 2 C3 Hm (t) max min ( + + ) m ̸ C0 ‖ ∑ xj uj ‖V ‖ ∑ xj vj ‖V ∑ xj ωj ‖V F⊂R ,dim F=t x=0,x∈F 3

m C 3 2 ≤ 3 Hm (t) ∑( ) C0 l=1 √λl

≤

√27 m 1 C ∑ H (t), C0 3 l=1 43 m λl

(1.9.84)

where we use the maximum and minimum principles, λl is the lth eigenvalue of A = −𝜕xx . We know that λl ∼ Cl2 as l → ∞. Hence, there exists constant a C2 such that dHm (t) ≤ C2 Hm (t). dt

(1.9.85)

Hence, using Hm (t) ≤ e2C2 t Hm (0), t ≥ 0, together with equation (1.9.78), we get equation (1.9.77).

1.10 A new method to prove existence of a strong topology attractor

| 123

Using Theorem 1.9.3, we have Theorem 1.9.4. The global attractor A determined by Theorem 1.9.2 possesses bounded Hausdorff and fractal dimensions. Proof. If we introduce the inner product Ψ(⋅, ⋅) in V 3 , then by Theorem 1.9.3, we get ω(DS(t)ξ0 ) ≤ C m exp(C2 − γm)t,

∀ξ0 ∈ A ,

where ωm is the Lyapunov index. When m is large enough, ωm (A ) < 1, here ω(A ) is the uniform Lyapunov index of A . Using Theorem V.31 of [197], we know that A possesses bounded Hausdorff and fractal dimensions. Since the strongness of equivalent, the attractor has the same Hausdorff and fractal dimension. Theorem 1.9.4 is proved.

1.10 A new method to prove existence of a strong topology attractor In 1995, Guo and Wu [124] studied the Benjamin–Ono (BO) global attractor in an unbounded region with the linear dissipation equation, proving the existence of a strongly compact attractor on H 1 (R). An important method which can make a weakly compact attractor become a strong one was proposed, i. e., for a class of nonlinear evolution equations with a generalized energy conservation integral, they proved that the weak convergence of a solution sequence is actually equivalent to the convergence in norm; moreover, the existence of a strong compact attractor of BO equation was proved. This is consistent with the idea that Ball and Ghidaglia [27] used to prove the existence of a strongly compact attractor of KdV equation, and it was found almost at the same time. By now this method has received much attention and is widely used, for instance, for the two-dimensional Davey–Stewartson equations some very good results have been obtained. Now we consider the following Cauchy problem for Benjamin–Ono equation (BO) with weak dissipation equation [98]: ut + Huxx + (u2 )x + C0 u = f , u(x, 0) = u0 (x),

x ∈ R, t > 0,

x ∈ R,

(1.10.1) (1.10.2)

where C0 > 0 is a constant, C0 u means the zeroth order dissipation effect, f is the external force, which is independent of t, and H is Hilbert transform, that is, ∞

ϕ(y) 1 H(ϕ) = P ∫ dy, π y−x −∞

where P denotes Cauchy integral principal value. If C0 = 0, equation (1.10.1) is the well-known Benjamin–Ono equation, which can be used to describe the propagation

124 | 1 Attractor and its dimension estimation of inner wave of long wavelength in the stream physically, it has a soliton solution and lots of important features, see [147]. For the BO problem and its generalization, the existence and uniqueness of a global smooth solution has been established, see, e. g., [215, 143]. In order to prove the existence of a weakly compact attractor for the BO problem, we must now establish a uniform a priori estimate. First, for the Hilbert transform and for any f , g ∈ L2 (R), we have: H 2 f = −f ,

H(fg) = H(HfHg) + fHg + gHf ,

(f , Hg) = −(Hf , g), (Hf , Hg) = (f , g), Hfx = (Hf )x ,

(Hf , f ) = 0,

(1.10.3) (1.10.4)

‖Hf ‖ = ‖f ‖,

(1.10.5)

∀f ∈ H (R).

(1.10.6)

1

Multiplying (1.10.1) with u, and integrating by parts over R, we get 1 d ‖u‖2 + C0 ‖u‖2 = (f , u). 2 dt

(1.10.7)

1 󵄩󵄩 󵄩2 −C t 2 −C t 2 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ e 0 ‖u0 ‖ − 2 (1 − e 0 )‖f ‖ . C0

(1.10.8)

Moreover, we obtain

Hence, for t ≥ t0 =

2 C0

ln

C0 ‖u0 ‖ , ‖f ‖

we have 2 󵄩󵄩 󵄩2 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ 2 = A1 . C0

(1.10.9)

In order to get an estimate of the H 1 norm, we multiply (1.10.1) by uxx , and integrate by parts, and then get 1 d ‖u ‖2 + C0 ‖ux ‖2 = (fx , ux ) + ∫(u2 )x uxx dx. 2 dt x

(1.10.10)

Multiplying (1.10.1) by u3 and integrating over R with respect to x, we have 1 d ‖u‖4L4 + C0 ‖u‖4L4 + ∫ u3 Huxx dx + ∫(u2 )x u3 dx = (f , u3 ). 4 dt

(1.10.11)

By (1.10.4), the above equation can be reduced to 1 d ‖u‖4L4 + C0 ‖u‖4L4 = 3 ∫ u2 ux Hux dx + (f , u3 ). 4 dt Furthermore, d ∫ u2 Hux dx = 2 ∫ uut Hux dx + u2 Huxx dx dt = −2 ∫ uHux [Huxx + C0 u + (u2 )x − f ]dx − ∫ Hu2 ux dx

(1.10.12)

1.10 A new method to prove existence of a strong topology attractor

| 125

= 2 ∫ fuHux dx − 2C0 ∫ u2 Hux dx − 2 ∫ u(u2 )x Hux dx − 2 ∫ uHux Huxx dx + ∫ Hu2 [Huxx + (u2 )xx + C0 ux − fx ]dx = 2 ∫ fuHux dx − 3C0 ∫ u2 Hux dx − 4 ∫ u2 Hux dx + ∫(u)2x uxx dx + ∫ ux (Hux )2 dx + ∫ u2 Hfx dx.

(1.10.13)

Noting that H(ux Hux ) = H(Hux H 2 ux ) − ux H 2 ux + (Hux )2 ,

(1.10.14)

(Hux )2 = u2x + 2H(ux Hux ).

(1.10.15)

we get

Hence ∫ ux (Hux )2 dx = 2 ∫ ux H(ux Hux )dx + ∫ u3x dx = −2 ∫ ux (Hux )2 dx − 2 ∫ uux uxx dx,

(1.10.16)

through which we arrive at 1 ∫ ux (Hux )2 dx = − ∫(u2 )x uxx dx, 3 d ∫ u2 Hux dx = 2 ∫ fuHux dx − 3C0 ∫ u2 Hux dx dt

(1.10.17)

2 ∫(u2 )x uxx dx. 3

− 4 ∫ u2 ux Hux dx + ∫ u2 Hfx dx +

(1.10.18)

Combining with equations (1.10.9), (1.10.10) and (1.10.18), we have d [‖ux ‖2 − 3 ∫ u2 Hux dx − ‖u‖4L4 + 2C0 ‖ux ‖2 − 4C0 ‖u‖4L4 ] dt = 2(fx , ux ) − 6 ∫ fuHux dx + 9C0 ∫ u2 Hux dx − 3 ∫ u2 Hfx dx − 4(f , u3 ).

(1.10.19)

Using Gagliardo–Nirenberg inequality, we have 3

1

‖u‖4L4 ≤ C‖u‖ 4 ‖ux ‖ 4 ,

2

1

∀u ∈ H 1 (R).

‖u‖L6 ≤ C‖u‖ 3 ‖ux ‖ 3 ,

(1.10.20)

Hence 3

3

∫ u2 Hux dx ≤ ‖u‖2L4 ‖ux ‖ ≤ ‖u‖ 2 ‖ux ‖ 2 ,

(1.10.21) 1

5

∫ fuHux dx ≤ ‖f ‖L4 ‖u‖L4 ‖ux ‖ ≤ C‖f ‖H 1 ‖u‖ 4 ‖ux ‖ 4 ,

(1.10.22)

126 | 1 Attractor and its dimension estimation 1

3

∫ u2 Hfx dx ≤ C‖fx ‖‖u‖ 2 ‖ux ‖ 2 .

(1.10.23)

Therefore d [‖ux ‖2 − 3 ∫ u2 Hux dx − ‖u‖4L4 ] + 2C0 [‖ux ‖2 − 3 ∫ u2 Hux dx − ‖u‖4L4 ] dt = 2(fx , ux ) + 2C0 ‖u‖4L4 + 3C0 ∫ u2 Hux dx − 3 ∫ u2 Hfx dx − 6 ∫ fuHux dx − 4(f , u3 ) 4

8

≤ C[‖fx ‖2 + ‖u‖6 + ‖fx ‖ 3 ‖u‖2 + ‖f ‖ 3 ‖u‖2 + ‖f ‖2 ‖u‖2 ],

(1.10.24)

which yields ‖ux ‖2 − 3 ∫ u2 Hux dx − ‖u‖4L4 ≤ (‖u0x ‖2 − 3 ∫ u20 Hu0x dx − ‖u0 ‖4L4 )e−C0 t t

4

8

+ C ∫[‖fx ‖2 + ‖u‖6 + ‖fx ‖ 3 ‖u‖2 + ‖f ‖ 3 ‖u‖2 + ‖f ‖2 ‖u‖4 ]e−C0 (t−s) ds.

(1.10.25)

0

For t ≤ t0 , we have ‖ux ‖2 − 3 ∫ u2 Hux dx − ‖u‖4L4 ≤ (‖u0x ‖2 − 3 ∫ u20 Hu0x dx − ‖u0 ‖4L4 ) t0

4

8

+ C ∫[‖fx ‖2 + ‖u‖6 + ‖fx ‖ 3 ‖u‖2 + ‖f ‖ 3 ‖u‖2 + ‖f ‖2 ‖u‖4 ]e−C0 (t−s) ds 0

≤ C(t0 , ‖u0 ‖H 1 , ‖f ‖H 1 ).

(1.10.26)

For t ≥ t0 , we have ‖ux ‖2 − 3 ∫ u2 Hux dx − ‖u‖4L4 󵄩 󵄩2 󵄩 󵄩4 ≤ (󵄩󵄩󵄩ux (t0 )󵄩󵄩󵄩 − 3 ∫ u2 Hux (t0 )dx − 󵄩󵄩󵄩u(t0 )󵄩󵄩󵄩L4 )e−C0 (t−t0 ) + C[‖fx ‖2 +

1 14 2 2 4 2 ‖f ‖2 + 2 ‖f ‖2 ‖fx ‖ 3 + 2 ‖f ‖ 3 + 2 ‖f ‖4 ] 2 C0 C0 C0 C0

󵄩 󵄩2 󵄩 󵄩4 ≤ C‖f ‖6H 1 + (󵄩󵄩󵄩ux (t0 )󵄩󵄩󵄩 − 3 ∫ u2 Hux (t0 )dx − 󵄩󵄩󵄩u(t0 )󵄩󵄩󵄩L4 )e−C0 (t−t0 ) .

(1.10.27)

Therefore there exists t1 > t0 such that 󵄩󵄩 󵄩2 6 󵄩󵄩ux (t)󵄩󵄩󵄩 ≤ C‖f ‖H 1 .

(1.10.28)

1.10 A new method to prove existence of a strong topology attractor

| 127

Thus ‖ux (t)‖2 ≤ C‖f ‖6H 1 + 3 ∫ u2 Hux dx + ‖u‖4L4 3 3 1 ≤ C‖f ‖6H 1 + C[‖u‖ 2 ‖ux ‖ 2 + ‖u‖3 ‖ux ‖] ≤ C‖f ‖6H 1 + ‖ux ‖2 + C‖u‖6 . 2

When t > t1 , we obtain that ‖ux ‖2 ≤ C‖f ‖6H 1 = A2 .

(1.10.29)

Setting B = {ϕ(x) | ϕ(x) ∈ H 1 , ‖ϕ‖ ≤ A1 , ‖ϕx ‖ ≤ A2 }, it is easy to see that B is the absorbing set of the BO problem. Define the ω limit set by ωω (B) = ⋂ ⋃ S(t)Bω s>0 t≥s

(1.10.30)

where the closure is taken with respect to the weak topology of H 1 (R). Since B is a bounded set in H 1 (R), it is weakly compact in H 1 (R). Since H 1 (R) is a separable topological space, we know that B is metrizable. Define dω to be its distance. Under the effect of semigroup S(t), a bounded set A attracts a bounded set B relative to the measure dω if and only if lim dω 1 (S(t)B, A) t→∞ H

= lim sup inf dω (S(t)x, y). t→∞ x∈B y∈A

(1.10.31)

Following the idea in [197], we have Proposition 1.10.1. The set ωω (B) is the nonempty weakly compact attractor of the BO problem. Now we prove that this global weakly compact attractor ωω (B) is also a global strongly compact attractor in H 1 . Since ωω (B) is invariant under the action of the semigroup S(t), we prove that for any bounded set A ∈ H 1 (R), we have lim dH 1 (S(t)A, ωω (B)) = 0.

t→∞

(1.10.32)

To do so, we only need to prove that if ∀tn → ∞, ϕn ∈ A such that S(tn )ϕn → u(x) weakly in H 1 (R)

(1.10.33)

where u(x) ∈ ωω (B), then S(tn )ϕn → u(x)

strongly in H 1 (R).

(1.10.34)

128 | 1 Attractor and its dimension estimation In fact, ∀T > 0, ∃nj → ∞, v(x) ∈ ωω (B) such that for tnj > T we have weakly in H 1 (R).

vnj (x) = S(tnj − T)ϕnj → v(x)

Set Wnj = S(tnj + t − T)ϕnj . Since S(t) : H 1 (R) → H 1 (R) is bounded, we have S(t) : H 1 (R) → L2 (R) 1

1

is continuous,

S(t) : H (R) → H (R) is weakly continuous. Hence, we have Wnj (t) → S(t)v

weakly in H 1 (R),

S(T)v = u.

Since 1 d 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩W (t)󵄩󵄩 + C0 󵄩󵄩󵄩Wnj (t)󵄩󵄩󵄩 = (f , Wnj (t)), 2 dt 󵄩 nj 󵄩 we have t

1 󵄩󵄩 󵄩2 󵄩 󵄩2 −2C t ∫(f , Wnj (s))e−2C0 (t−s) ds, 󵄩󵄩Wnj (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩vnj (t)󵄩󵄩󵄩 e 0 + 2C0 0

t

1 󵄩󵄩 󵄩2 󵄩2 −2C t 󵄩 ∫(f , Wnj (s))e−2C0 (T−s) ds. 󵄩󵄩Wnj (t)󵄩󵄩󵄩 = e 0 󵄩󵄩󵄩vnj (t)󵄩󵄩󵄩 + 2C0 0

Since ϕnj ∈ B, where B is the absorbing set in H 1 (R), we can find C > 0 such that (f , Wnj (s)e−2C0 (t−s) + C) > 0. Therefore 󵄩 󵄩2 CT ≤ e−2C0 T lim sup ‖vnj ‖2 lim sup 󵄩󵄩󵄩Wnj (T)󵄩󵄩󵄩 + 2C0 nj →∞ nj →∞ T

+ lim sup nj →∞

1 ∫[(f , Wnj (s))e−2C0 (T−s) + C]ds. 2C0 0

Noting that Wnj (t) → S(t)v

weakly in H 1 (R),

∀t > 0,

we have (f , Wnj (s)) → (f , S(t)v),

∀t > 0.

(1.10.35)

1.10 A new method to prove existence of a strong topology attractor

| 129

Together with Fatou lemma, this implies T

lim sup ∫[(f , Wnj (s))e−2C0 (T−s) + C]ds nj →∞

0

T

≤ ∫[(f , S(s)v)e−2C0 (T−s) + C]ds

(1.10.36)

0

and 󵄩 󵄩2 CT lim sup 󵄩󵄩󵄩Wnj (T)󵄩󵄩󵄩 + 2C0 nj →∞ ≤ e2C0 T lim sup ‖vnj ‖2 + nj →∞

T

1 CT . ∫(f , S(s)v)e−2C0 (T−s) ds + 2C0 2C0

(1.10.37)

0

On the other hand, since u = S(T)v, we have T

1 ‖v‖ + ∫(f , S(s)v)e−2C0 (T−s) ds. 2C0

(1.10.38)

󵄩 󵄩2 lim sup 󵄩󵄩󵄩Wnj (T)󵄩󵄩󵄩 ≤ ‖u‖2 + e−2C0 T [lim sup ‖vnj ‖2 − ‖v‖2 ].

(1.10.39)

2

‖u‖ = e

−2C0 T

2

0

Hence nj →∞

nj →∞

Noting that B is the bounded absorbing set in H 1 (R), for tj > T, we have {unj ⊂ A},

‖unj ‖2 ≤ C, ‖vnj ‖2 ≤ C, where C is independent of T and nj . So

󵄩 󵄩2 lim sup 󵄩󵄩󵄩Wnj (T)󵄩󵄩󵄩 ≤ ‖u‖2 + 2Ce−2C0 T . nj →∞

(1.10.40)

On account of Wnj (T) = S(tnj )ϕnj , we have lim sup ‖Snj ϕnj ‖2 ≤ ‖u‖2 − 2Ce−2C0 T . nj →∞

Letting T → ∞, we get lim sup ‖S(tnj )ϕnj ‖2 ≤ ‖u‖2 . nj →∞

(1.10.41)

On the other hand, since Snj ϕnj → u weakly in H 1 (R), lim sup ‖S(tnj )ϕnj ‖2 ≥ ‖u‖2 . nj →∞

(1.10.42)

130 | 1 Attractor and its dimension estimation From equations (1.10.41) and (1.10.42), we have lim sup ‖S(tnj )ϕnj ‖2 = ‖μ‖2 . nj →∞

(1.10.43)

Together with the fact that S(tnj )ϕnj → u (weakly), we get S(tnj )ϕnj → u

strongly in L2 (R).

Since for any subsequences tnj and ϕnj the above limit is unique, we obtain S(tnj )ϕn → u

strongly in L2 (R).

Noting that, if un → u weakly in H 1 (R), and un → u strongly in L2 (R), then un → u strongly in Lp (R) for any p > 2. From equation (1.10.24), we get 󵄩󵄩 󵄩2 󵄩 󵄩4 2 󵄩󵄩ux (t)󵄩󵄩󵄩 − 3 ∫ n Hux dx − 󵄩󵄩󵄩u(t)󵄩󵄩󵄩L4 = e−2C0 t [‖u0x ‖2 − 3 ∫ u20 Hu0x dx − ‖u0 ‖4L4 ] t

1 󵄩 󵄩4 + [∫ e−2C0 (t−s) [2(fx , ux (s)) + 2C0 󵄩󵄩󵄩u(s)󵄩󵄩󵄩L4 ] 2C0 0

+ 3C0 ∫ u2 Hux dx − 3 ∫ u2 Hfx dx − 4(f , u3 )]ds.

(1.10.44)

Similar to the proof of the L2 -convergence, we can prove S(tn )ϕn → u strongly in H 1 (R). Hence, we have Theorem 1.10.1. Assume that u0 ∈ H 1 , f ∈ H 1 , and C0 > 0 is a constant, then the BO problem has a unique strongly compact global attractor in H 1 (R). We can also prove that the semigroup S(t) is continuous in H 1 . From the above proof, we know that on the global attractor, the weak H 1 -convergence induces the strong H 1 -convergence. Thus we get Theorem 1.10.2. The semigroup operator S(t) is H 1 -continuous on the global attractor. From the dimension estimation of the global attractor, we get Theorem 1.10.3. Suppose the assumption of Theorem 1.10.1 is valid. Then the attractor has bounded fractal dimension.

1.11 Nonlinear KdV–Schrödinger equation

| 131

1.11 Nonlinear KdV–Schrödinger equation We consider the following periodic initial problem of nonlinear KdV–Schrödinger equation with weak damping iεt + εxx − bnε + iγε = g1 (x), β 1 1 nt + f (n)x + nxxx + vn + |ε|2x = g2 (x), 2 2 2 ε(x, t) = ε(x + L, t), n(x, t) = n(x + L, t), ε(x, 0) = ε0 (x),

(1.11.1) (1.11.2) ∀x ∈ R, t ≥ 0,

n(x, 0) = n0 (x),

(1.11.3) (1.11.4)

where ε(x, t) = (ε1 (x, t), ε2 (x, t), . . . , εN (x, t)) is an unknown complex vector function; n(x, t) is a real function; f (n) is a nonlinear function; b, γ, β, v are constants; L > 0. The problem (1.11.1)–(1.11.4) appears in the studies of laser and plasma physics, including electric field ε, where n denotes a density disturbance. When there is no damping, γ = ν = 0, g1 = g2 = 0, Appert, Vaclavik, Makhankov, Gibbons, among others, [5, 170, 78] did lots of explicit studies on the soliton solution of equations (1.11.1)–(1.11.4). In 1983, Guo [84] first proved the existence and uniqueness of a global solution for the periodic initial value problem (1.11.1)–(1.11.4). In 1997, Guo and Chen [93] proved the existence and boundedness of dimension of the global attractor for the problem (1.11.1)–(1.11.4). The other related problems on the well-posedness can refer to the reference [111, 112]. In the following, suppose that ν > γ > 0, bβ < 0 and f (s) ∈ C (4) (R) is such that there exist A0 and B > 0 so that 󵄨󵄨 󵄨 2 󵄨󵄨f (n)󵄨󵄨󵄨 ≤ A0 n + B,

∀n ∈ R.

(1.11.5)

Suppose that m is a non-negative integer, L is a positive real number, HLm means the Sobolev space H m with period L. Also denote IHLm = HLm × HLm × ⋅ ⋅ ⋅ × HLm , ‖v‖2m

m

= ∑L

2k

k=0

N

|v|2k ,

|v|2k 1 2

‖ε‖m = [∑ ‖εj ‖2m ] , j=1

L

= ∫| 0

dk v 2 | dx, dxk

ε = (ε1 , . . . , εN ) ∈ IHLm .

Letting Xk = IHLk × HLk ,

k = 1, 2, . . . , m, 1

‖η‖k = (‖η1 ‖2k + ‖η2 ‖2k ) 2 , η = (η1 , η2 ) ∈ Xk ,

k = 1, 2, . . . , m,

132 | 1 Attractor and its dimension estimation we obtain the Sobolev inequality 1

|v|L∞

1 2 1 󵄨 󵄨 = sup 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 ≤ |v|02 (2|v|1 + |v|0 ) , L 0≤x≤L

∀v ∈ HL1 .

(1.11.6)

Following the method in [84], we have Theorem 1.11.1. Suppose f (n) satisfies condition (1.11.5). Let bβ < 0, ν > γ > 0, g = (g1 , g2 ) ∈ Xm , (ε0 (x), n0 (x)) ∈ Xm , m ≥ 2. Then for any T, 0 < T < ∞, there exists a global unique solution (ε(x, t), n(x, t)) ∈ C(0, T; Xm ) for the problem (1.11.1)–(1.11.4), and it is continuous with respect to the initial value. The semigroup S(t)ζ0 = S(t)(ε0 , n0 ) = (ε(t), n(t)) is generated by the solution of problem (1.11.1)–(1.11.4), where S(t) is defined on R+ , and is continuous in t. But from the bound on the solution, we know that for any R > 0, 0 < T < ∞, there exists a constant C0 (R, T) such that 󵄩 󵄩 sup 󵄩󵄩󵄩S(t)(ε0 , n0 )󵄩󵄩󵄩2 ≤ C0 (R, T),

t∈(0,T]

󵄩 󵄩 provided 󵄩󵄩󵄩(ε0 , n0 )󵄩󵄩󵄩2 ≤ R.

(1.11.7)

Lemma 1.11.1. For t ∈ R, the map S(t) is continuous with respect to X1 on the bounded set X2 . Proof. Assume that (ε10 , n10 ), (ε20 , n20 ) ∈ X2 , ‖εk0 ‖2 + ‖nk0 ‖2 ≤ R, k = 1, 2, and (ε1 (t), n1 (t)), (ε2 (t), n2 (t)) stand for the corresponding solutions (εk (t), nk (t)) = S(t)(εk0 , nk0 ), k = 1, 2, respectively. Let ε(t) = ε1 (t) − ε2 (t), n(t) = n1 (t) − n2 (t). Then (ε(t), n(t)) satisfies iεt + εxx − b(n1 ε1 − n2 ε2 ) + iγε = 0, β 1 1 nt + (f (n1 ) − f (n2 ))x + nxxx + νn + (|ε1 |2x − |ε2 |2x ) = 0. 2 2 2

(1.11.8) (1.11.9)

Multiplying equation (1.11.8) by ε and integrating over (0, L), we get L

(iεt , ε)t + (εxx , ε) − b ∫(n1 |ε|2 + n1 ε2 ε)dx + iγ|ε|20 = 0.

(1.11.10)

0

Taking the imaginary part of the above equation, we obtain L

1 d 2 |ε| + γ|ε|20 − b Im ∫ ε2 nεdx = 0. 2 dt 0

(1.11.11)

0

On the other hand, taking the real part of equation (1.11.10), we get Im(εt , ε) +

|εx |20

t

2

L

+ b ∫ n1 |ε| dx + b Re ∫ nεεdx = 0. 0

0

(1.11.12)

1.11 Nonlinear KdV–Schrödinger equation

| 133

Multiplying equation (1.11.8) by ε, integrating over (0, L), and taking its real part, we get d |ε |2 + Re b(n1 ε + ε2 n, εt ) + γ Im(ε, εt ) = 0. (1.11.13) dt x 0 Combining with equations (1.11.12) and (1.11.13), we have L

L

1 d |ε |2 + γ|εx |20 + γb ∫ n1 |ε|2 dx + γb Re ∫ nε2 εdx + Re b(n1 ε + ε2 n, ε) = 0. 2 dt x 0 0

(1.11.14)

0

Multiplying equation (1.11.9) by n and integrating over [0, L], we get 1

1 d 2 1 1 |n| + ν|n|20 − ∫ f 󸀠 (τn + (1 − τ)n2 )dτ ⋅ nnx − (ε1 ε + ε2 ε, nx ) = 0. 2 dt 0 2 2

(1.11.15)

0

Multiplying equation (1.11.9) by nxx and integrating over [0, L], we arrive at d 1 1 { |n |2 − (ε ε + ε2 ε, n)} + ϕ(ε, n) = 0, dt 2 x 0 β 1 where

L

1

0

0

(1.11.16)

3 ϕ(ε, n) = − ∫(∫ f 󸀠 (τn1 + (1 − τ)n2 )dτ) n2x dx 4 x L

1

1 − ∫(∫ f 󸀠 (τn1 + (1 − τ)n2 )dτ) nnx dx 2 xx 0

0

1 1 + (ε1 ε + ε2 εt , n) + (ε1t ε + ε2t ε, n) β β 1

1 1 − (ε1 ε + ε2 ε, (∫ f 󸀠 (τn1 + (1 − τ)n2 )dτ ⋅ n) + vn). β 2 x

(1.11.17)

0

Together with equation (1.11.16) and (1.11.11), (1.11.14), (1.11.15), we have 1 1 d 1 { |n |2 + |ε |2 + σ|n|20 + σ|ε|20 − (ε1 ε + ε2 ε, n)} dt 2 x 0 2 x 0 β +

v|εx |20

L

L

2

+ γb ∫ n1 |ε| dx + γb Re ∫ nε2 εdx 0

0

+ Re b(n1 ε + ε2 n, εt ) + + σν|n|20 −

σγ|ε|20

L

− bσ Im ∫ ε2 nεdx

1

0

σ (∫ f 󸀠 (τn1 + (1 − τ)n2 )dτ ⋅ n, nx ) 2 0

σ − (ε1 ε + ε2 ε, nx ) + ϕ(ε, n) = 0, 2

(1.11.18)

134 | 1 Attractor and its dimension estimation , |εk (t)|L∞ ≤ C(R, T), |nk (t)|L∞ ≤ C(R, T), k = 1, 2. Using equation (1.11.1) where σ ≥ 2C β and integrating by parts, from equation (1.11.18) we have d 1 1 1 { |n |2 + |ε |2 + σ|n|20 + σ|ε|20 − (ε1 ε + ε2 ε, n)} dt 2 x 0 2 x 0 β

1 1 1 ≤ C1 { |nx |20 + |εx |20 + σ|n|20 + σ|ε|20 − (ε1 ε + ε2 ε, n)}. 2 2 β

Hence, we get 1 1 1 󵄩 󵄩2 |nx |20 + |εx |20 + σ|n|20 + σ|ε|20 − (ε1 ε + ε2 ε, n) ≤ C2 (󵄩󵄩󵄩(n(0), ε(0))󵄩󵄩󵄩1 )eC1 t , 2 2 β proving the lemma. Proposition 1.11.1. For any t ∈ R+ , the semigroup operator S(t) determined by problem (1.11.1)–(1.11.4) is weakly continuous from X2 to X2 . Proof. Let ζ n be a weakly convergent sequence in X2 , and let ξ be its limit. For a fixed t ∈ R+ , from equation (1.11.7) we can infer that {S(t)ζ n } is bounded in X2 . Since it is weakly compact, we can extract a subsequence {S(t)un }, which weakly converges to v ∈ X2 . On the other hand, by the compactness of X2 embedded into X1 , {un } strongly converges to u in X1 . In virtue of Lemma 1.11.1, {S(t)un } strongly converges to S(t)u, v = S(t)u. Hence, the sequence {S(t)un } weakly converges to S(t)u. The proposition has been proved. In the following we prove the existence of an absorbing set in X1 and X2 . Proposition 1.11.2. Assume that v > γ > 0, bβ < 0 and let f (n) satisfy equation (1.11.5), where g = (g1 , g2 ) ∈ X1 . Then there exists a constant ρ1 = ρ1 (L, ν, γ, ‖g‖1 ) such that for any R > 0, there is T1 = T1 (R) > 0 such that for every (ε0 , n0 ) in X1 , ‖(ε0 , n0 )‖21 ≤ R2 , we have 󵄩󵄩 󵄩2 2 󵄩󵄩S(t)(ε0 , n0 )󵄩󵄩󵄩1 ≤ ρ1 ,

∀t ≥ T1 (R).

(1.11.19)

To prove Proposition 1.11.2, we need the following two lemmas. Lemma 1.11.2. Assume that ε0 (x) ∈ L2 (Ω), Ω = [0, L]. Then for any solution (ε(t), n(t)) of problem (1.11.1)–(1.11.4), we have |g1 |2 󵄩󵄩 󵄩2 −γt 2 −γt 󵄩󵄩ε(t)󵄩󵄩󵄩0 ≤ |ε|0 e + 2 0 (1 − e ). γ

(1.11.20)

Proof. Taking the inner product of equation (1.11.4) with ε, and then its imaginary part, we obtain the result.

1.11 Nonlinear KdV–Schrödinger equation

| 135

Lemma 1.11.3. Assume that v > γ, (ε0 , n0 ) ∈ X1 . Then for any solution (ε(t), n(t)) of problem (1.11.1)–(1.11.4), there exist constants C1 = C1 (|n0 |20 , |ε0 |21 ), C2 = C2 (‖g1 ‖21 , |g2 |0 ) such that ‖n‖20 ≤ |b|−1 |ε|0 |εx |0 + C1 e−γt + C2 (1 − e−2γt ).

(1.11.21)

Proof. From equations (1.11.1) and (1.11.2), we deduce that L

1 d ∫[|b|n2 + Sign b Im(εεx )]dx + v|b||n|20 2 dt 0

L

+ γ ∫ Sign b Im(εεx )dx − |b|(g2 , n) + Sign b(g1 x, ε) = 0.

(1.11.22)

0

From equation (1.11.22), when v > γ we know that L

d ∫[|b|n2 + Sign b Im(εεx )]dx dt 0

L

+ 2γ ∫[|b|n2 + Sign b Im(εεx )]dx ≤ 0

|b||g2 |20 + 2|g1x |0 |ε|0 . 2(v − γ)

Using Gronwall inequality and Lemma 1.11.2, we deduce that ‖n0 ‖20 ≤ |b|−1 |ε|0 |εx |0 + C1 e−γt + C2 (1 − e−γt ), where

C1 = |n0 |20 + |b|−1 |ε|0 |ε0x |0 + γ −1 |ε|20 , C2 = |b|−1 (

|g1x |20 |g|20 |b||g2 |20 + 3 + ). 2γ 4γ(v − γ) 2γ

Using Lemmas 1.11.2 and 1.11.3, we now prove Proposition 1.11.2. Taking the inner product of (1.11.2) with n, we have 1 d 2 1 |n| + (|ε|2 , n) + ν|n|20 + (g2 , n) = 0. 2 dt 0 2 x

(1.11.23)

On the other hand, by direct calculation we deduce from equation (1.11.1) that L

L

bβ d {|εx |20 + b ∫ n|ε|2 dx + 2 Re(g1 , ε) − |n |2 + b ∫ F(n)dx} dt 2 x0 0

0

L

+ 2γ[|εx |20 + b ∫ n|ε|2 dx + 2 Re(g1 , ε)] − vbβ|nx |20 L

0

L

+ vb ∫ f (n)ndx + bv ∫ n|ε|2 dx − b(g2 , f (n) − nx − |ε|2 ) = 0, 0

0

(1.11.24)

136 | 1 Attractor and its dimension estimation n

where F(n) = ∫0 f (s)ds. Combining equations (1.11.22) and (1.11.2), we get d H (ε, n) + K1 (ε, n) = 0, dt 1

(1.11.25)

where H1 (ε, n) =

|εx |20

L

L

bβ + b ∫ n|ε| dx + 2 Re(g1 , ε) − |n |2 + b ∫ F(n)dx + λ|n|20 , (1.11.26) 2 x0 0

2

0

L

K1 (ε, n) = 2γ[|εx |20 + b ∫ n|ε|2 dx + Re(g1 , ε)] 0

L

L

0

0

− vbβ|nx |20 + vb ∫ f (n)ndx + bβ ∫ n|ε|2 dx 2

− b(g2 , f (n) − nx − |ε| ) +

2vλ|n|20

L

+ λ ∫ n|ε|2x dx − λ(g2 , n),

(1.11.27)

0

and λ is an unknown positive constant. According to Lemmas 1.11.2 and 1.11.3, there exists a constant ρ0 = ρ0 (‖g1 ‖22 , ‖g2 ‖20 ) such that for any solution (ε(t), n(t)) of the problem (1.11.1)–(1.11.4) with the initial value (ε0 , n0 ) ∈ X1 , ‖(ε0 , n0 )‖21 ≤ R2 , there exists T0 = T0 (R) > 0 such that |ε|20 ≤

2 |g |2 , γ2 1 0

t ≥ T0 (R),

|n|20 ≤ |b|−1 |ε0 ||εx |0 + ρ0 ,

(1.11.28) t ≥ T0 (R).

(1.11.29)

Hence, when t ≤ T0 (R), we have L

∫ |n|3 dx ≤ |n|∞ |n|20 ≤ (2|n|1 + 0

5 1 |n|0 )|n|02 L

1

5

2 4 √2|g1 |0 1 |n|0 ) ( |ε|1 + ρ0 ) L γ|b| θ ≤ θ1 ‖εx ‖20 + θ2 |nx |20 + 2 |n|20 + C(θ1 , θ2 , |g|21 ), 2L

≤ (2|n|1 +

where C(θ1 , θ2 , ‖g1 ‖21 ) =

γ 2 |b|2 ρ20 55 |g1 |10 0 θ + , 1 2|g1 |20 28 γ 10 |b|10 θ15 θ22

θ1 , θ2 > 0.

(1.11.30)

1.11 Nonlinear KdV–Schrödinger equation

| 137

Hence, by assumption (1.11.5) on f (n), we have L

L

L

0

0

󵄨󵄨 |b|A 󵄨󵄨 󵄨 󵄨 0 |b|󵄨󵄨󵄨 ∫ F(n)dx󵄨󵄨󵄨 ≤ ∫ |n|3 dx + |b|B ∫ |n|dx 󵄨󵄨 󵄨󵄨 3 0

|bβ| |bβ| |b| 1 |n |2 + ( + )|n|20 + C(L, v, γ, ‖g1 ‖1 ). ≤ |εx |20 + 4 8 x0 16L 2

(1.11.31)

On the other hand, 󵄨󵄨 L 󵄨󵄨 |b| 1 󵄨󵄨 󵄨 2 |n|20 + |ε|20 + C(|g1 |20 ), 󵄨󵄨b ∫ n|ε| dx󵄨󵄨󵄨 ≤ 󵄨󵄨 󵄨󵄨 2 4

t ≥ T0 (R).

(1.11.32)

0

Hence, when t ≥ T0 (R), there exists a constant C3 (‖g‖21 ) such that |bβ| |bβ| 1 H1 (ε, n) ≥ |εx |20 + |nx |20 + (λ − − |b|)|n|20 − C3 (‖g‖21 ), 2 4 16L 3|bβ| |bβ| 3 |nx |20 + (λ + + |b|)|n|20 + C3 (‖g‖21 ). H1 (ε, n) ≤ |εx |20 + 2 4 16L

(1.11.33) (1.11.34)

As to K1 (ε, n), we have the following lemma: Lemma 1.11.4. Suppose the conditions of Proposition 1.11.1 are satisfied. Then there exists a constant C4 = C1 (‖g1 ‖) such that |bγ|γ 1 K1 (ε, n) − γH1 (ε, n) ≥ (γλ − (2v + γ)|b| − − 2|b||g|L∞ A0 )|n|20 − C4 . 2 2

(1.11.35)

Proof. First, K1 (ε, n) − γH1 (ε, n) = γ|εx |20 + (2v − γ)|bβ||nx |20 + (2v −

γ)λ|n|20

L

L

+ (γ + v)b ∫ n|ε|2 dx 0

+ vb ∫[f (n)n − F(n)]dx − b(g2 , f (n) − nx − |ε|2 + λn) 0

L

+ λ ∫ n|ε|2x dx.

(1.11.36)

0

Similar to equations (1.11.30) and (1.11.32), when t ≥ T0 (R), we have L

(γ + v)b ∫ n|ε|2 dx ≤ 0

(v + γ)|b| 2 γ |n0 + |εx |20 + C(|g1 |20 ) 2 4

(1.11.37)

138 | 1 Attractor and its dimension estimation L 󵄨󵄨 󵄨󵄨 L 4A 󵄨󵄨 󵄨󵄨 3 󵄨󵄨vb ∫[f (n)n − F(n)]dx󵄨󵄨 ≤ |vb| ∫ 0 |n| + B0 |n|dx 󵄨󵄨 󵄨󵄨 3 0

0

|bβ|γ |bβ|γ |vb| γ |nx |20 + ( + )|n|20 ≤ |εx |20 + 4 4 8L 2

+ C(‖g‖21 ), λγ 󵄨 |bβ|γ 󵄨󵄨 2 |nx |20 + |n|20 󵄨󵄨b(g2 , f (n) − nx − |ε| + λn)󵄨󵄨󵄨 ≤ 4 4 + |b||g2 |∞ A0 |n|20 + C(‖g‖20 ).

(1.11.38)

(1.11.39)

L

For λ ∫0 n|ε|2x dx we have 󵄨󵄨 L 󵄨󵄨 󵄨󵄨 L 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 2 2 2 󵄨󵄨λ ∫ n|ε|x dx󵄨󵄨󵄨 = 󵄨󵄨󵄨λ ∫ nx |ε| dx󵄨󵄨󵄨 ≤ λ|nx |0 ‖ε‖L4 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0

0

|bβ|γ λγ |εx |20 + |n|20 − C(‖g1 ‖20 ). ≤ 4 4

(1.11.40)

From equations (1.11.36)–(1.11.40) we know that there exists a constant C4 = C4 (‖g‖1 ) such that equation (1.11.35) is valid. Now if we choose λ large enough, namely λ > max{

|bβ|γ |bβ| + |b|, (2v + γ + − 2|g2 |∞ |b|γ −1 )}, 16L 4

(1.11.41)

then from equations (1.11.25) and (1.11.35) we get d H (ε, n) + γH1 (ε, n) ≤ C4 , dt 1

t ≥ T0 (R).

(1.11.42)

Hence, by Gronwall inequality, we have H1 (ε, n) ≤ H1 (ε, n)(T0 )e−γ(t−T0 ) +

C4 (1 − e−γ(t−T0 ) ), γ

t ≥ T0 .

(1.11.43)

From equations (1.11.33), (1.11.34) and (1.11.43) we can deduce the claim of Theorem 1.11.2. Proposition 1.11.3. Assume that v > γ > 0, bβ < 0, f satisfies equation (1.11.5) and g = (g1 , g2 ) ∈ X. Then there exists a constant ρ2 = ρ2 (L, ν, γ, ‖g‖2 ) such that for any R > 0, there is T2 (R) > 0 such that for any (ε0 , n0 ) ∈ X, ‖(ε0 , n0 )‖22 ≤ R2 , we have 󵄩󵄩 󵄩2 2 󵄩󵄩S(t)(ε0 , n0 )󵄩󵄩󵄩2 ≤ ρ2 ,

∀t ≥ T2 (R).

(1.11.44)

In other words, in the closed ball in X2 , B2 = {ξ = (ξ1 , ξ2 ) ∈ X | ‖ξ1 ‖22 + ‖ξ2 ‖22 ≤ ρ22 } is the bounded set of the semigroup S(t).

(1.11.45)

1.11 Nonlinear KdV–Schrödinger equation

| 139

Proof. By Proposition 1.11.2, we only need to estimate the uniform upper bound of |εxx |0 and |nxx |0 . Denote A = −𝜕xx . Taking the inner product of (1.11.1) with Aεt + γAε, and then its real part, we obtain L

d 1 [ |Aε|20 + Re(g1 , Aε)] + bγ Re ∫(nε)x εx dx + γ|Aε|20 dt 2 0

+ Re(g1 , Aε) + b Re(Aε, Aεt ) = 0.

(1.11.46)

On the other hand, by equation (1.11.2) we get L

d (3β|An|20 − 5 ∫ f 󸀠 (n)n2x dx) = h1 − 6(|ε|2xx , nt ) − 6vβ|An|20 , dt

(1.11.47)

0

where L

h1 = 6β(g2xx , nxx ) + 2 ∫ f 󸀠 (n)nx |ε|2xx dx L

0

+ 10v ∫ f 󸀠 (n)n2x dx − 10(f 󸀠 (n)n2x , g2x ) L

0

L

2 + ∫ f 󸀠 (n)n2x |ε|2xx dx + 5v ∫ f 󸀠󸀠 (n)nn2x dx 5 0

− 5(f 󸀠󸀠 (n)n2x , gx ) + L

+

L

0

5 ∫ g 󸀠󸀠 (n)f 󸀠 (n)n3x dx 2 0

β ∫ f (4) (n)n5x dx − 6(|ε|2xx , n − g2 ). 8

(1.11.48)

0

Since (|ε|2xx , nt ) =

d (|ε|2xx , nt ) − (|ε|2t , nxx ) dt L

d = (|ε|2xx , n) − 2 Re ∫ εεt nxx dx − Re(nε, Aεt ) dt 0

= 2 Re(nx εx , εt ) + Re(nxx ε, εt ) + Re(nεxx , εt ),

(1.11.49)

from equations (1.11.46), (1.11.48) and (1.11.49) we have d {6|Aε|20 + 3|bβ||An|20 + h2 } + 12γ|Aε|20 + 6v|bβ||An|20 + h3 + h4 + bh1 = 0, dt

(1.11.50)

140 | 1 Attractor and its dimension estimation where L

h2 = 12 Re(g1 , Aε) + 5b ∫ f (n)n2x dx + 6b(|ε|2x , nx ), 0

(1.11.51)

L

h3 = 12 Re(g1 , Aε) + 12bγ Re ∫(εn)x εx dx − 24b Re(nx εx , εt ),

(1.11.52)

0

h4 = 12b Im((n2 ε)x , εx ) − 12b Im((iγεn − g1 n)x , εx ).

(1.11.53)

By Proposition 1.11.2 we know, when t ≥ T1 (R), 󵄩󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩ε(t)󵄩󵄩󵄩1 + 󵄩󵄩󵄩n(t)󵄩󵄩󵄩1 ≤ ρ1 , 󵄩󵄩 󵄩 󵄩 󵄩2 2 󵄩󵄩ε(t)󵄩󵄩󵄩L∞ + 󵄩󵄩󵄩n(t)󵄩󵄩󵄩L∞ ≤ Cρ1 . Hence |f (k) (n)|∞ ≤ C(ρ1 ). Then we know that |h2 | ≤ |Aε|20 + C(ρ1 , ‖g‖21 ).

(1.11.54)

Since |Re(nx , εx , εt )| ≤ θ|Aε|20 + C(θ, ρ, ‖g‖21 ), ∀θ > 0, we have |h3 | ≤ γ|Aε|20 + C(ρ1 , ‖g‖21 ), |h4 | ≤ |bh1 | ≤

C(ρ1 , ‖g‖21 ), |βb|v|nxx |20 +

C(ρ1 , ‖g‖22 ).

(1.11.55) (1.11.56) (1.11.57)

From equations (1.11.50) and (1.11.54)–(1.11.57), when t ≤ T1 (R), we get d {6|Aε|20 + 3|bβ|An|20 + h2 } + γ{6|Aε|20 + 3|bβ||An|20 + h2 } ≤ C(ρ1 , ‖g‖22 ). dt

(1.11.58)

Notice that H2 satisfies equation (1.11.54). By Gronwall inequality, we obtain the claim. Define ω

A = ω (B2 ) = ⋂ ⋃ S(t)B2 s≥0 t≥s

where the closure is taken with respect to the weak topology of X2 and B2 is the absorbing set defined by equation (1.11.45). Using Proposition 1.11.3 and Lemma 1.11.1, we can prove Theorem 1.11.2. The set A = ωω (B2 ) has the following properties: (1) A is weakly compact in X, S(t)A = A , ∀t ∈ R+ ;

1.11 Nonlinear KdV–Schrödinger equation

| 141

(2) for any bounded set B ⊂ X2 , lim dω (S(t)B, A ) t→∞ H

= 0;

(3) A is connected in the weak sense in X2 . In other words, A is the weak global attractor of S(t) in X2 . Corollary 1.11.1. For any bounded set B in X2 , S(t)B converges to A when t → ∞ with respect to the norm of X1 . Now we estimate the dimension of A . Let ζ0 = (ε0 , n0 ) ∈ X2 and let S(t)ζ0 = (ε(t), n(t)) be the semigroup defined by problem (1.11.1)–(1.11.4) with the initial value (ε0 , n0 ). Consider iUt + Uxx − bVε − bnU + iγU = 0, β 1 1 Vt + (f 󸀠 (n)V)x + Vxxx + vV + (εU + Uε)x = 0, 2 2 2 U(x, t) = U(x + L, t), V(x, t) = V(x + L, t), ∀x ∈ R, U(0) = u0 ∈

IHL1 ,

V(0) = v0 ∈

IHL1 .

(1.11.59) (1.11.60) t ≥ 0,

(1.11.61) (1.11.62)

Since S(t)ζ0 ∈ C(R+ ; X2 ), it is easy to see that the linear problem (1.11.59)–(1.11.62) has a unique solution (U(t), V(t)) ∈ C(R+ ; X1 ). It is easy to prove that the linear mapping (DS(t)ζ0 )(u0 , v0 ) = (U(t), V(t))

(1.11.63)

is the “uniform differential” of S(t). Then we have Proposition 1.11.4. For any R, T, 0 < R, T < ∞, there exists a constant C = C(R, T) such that for any ζ0 = (ε0 , n0 ), η0 = (h0 , k0 ), ‖ξ0 ‖1 ≤ R1 , ‖ζ0 + η0 ‖1 ≤ R, t < T, we have 󵄩󵄩 󵄩 2 󵄩󵄩S(t)(ζ0 + η0 ) − S(t)ζ0 − (DS(t)ζ0 )η0 󵄩󵄩󵄩1 ≤ C‖η0 ‖1 ,

(1.11.64)

where (DS(t)ζ0 )η0 = (U(t), V(t)) is the solution of problem (1.11.59)–(1.11.62) with the initial value U(0) = h0 , V(0) = k0 . In order to study the variation of the m-dimensional volumes of DS(t), we introduce the energy equation of problem (1.11.59)–(1.11.62). Let (ε0 , n0 ) ∈ X2 , (ε(t), n(t)) be the solution of problem (1.11.1)–(1.11.4) with the initial value (ε0 , n0 ). Then (U(t), V(t)) is the solution of problem (1.11.59)–(1.11.62) with the initial value η0 = (u0 , v0 ). Set u = eγt U, v = eγt V. Then (u, v) satisfies iut + vxx − bvε − bnu = 0,

(1.11.65)

142 | 1 Attractor and its dimension estimation β 1 1 vt + (f 󸀠 (n)v)x + vxxx + vu + (εu + uε)x = 0. 2 2 2

(1.11.66)

Similar to equations (1.11.1) and (1.11.14)–(1.11.16), we have 1 d 2 |u| = Im b(vε, u), 2 dt 0

(1.11.67)

d 1 ( |u |2 + k1 ) = k2 + Re b ∫ εuvt dx, dt 2 x 0

(1.11.68)

L

0

d|v|20 = k3 , dt

L

(1.11.69)

d β 1 ( |v |2 − ∫ f 󸀠 (n)|v|2 dx) = −(v − γ)β|vx |20 + 2(Re εu, vt ) + k4 , dt 2 x 0 2

(1.11.70)

0

where L

k1 =

L

b ∫ n|u|2 dx + b Re ∫ vεudx, 2 0

(1.11.71)

0

L

L

bβ b 1 1 k2 = − ∫( f (n)x + vn + |ε|2x − g2 )|u|2 dx + ∫ nxx Re(u, ux )dx, 2 2 2 2 L

0

(1.11.72)

0

k3 = − ∫(f 󸀠 (n)v)x vdx − 2(v − γ)|v|20 − 2(Re(εu)x , ν),

(1.11.73)

0

1 k4 = (2 Re(εu), (f 󸀠 (n)v)x + (ν − γ)v) 2 β 1 󸀠 1 1 + (f (n)[ f (n)x + vn + |ε|2x − g2 ] − f 󸀠󸀠󸀠 (n)nx nxx , v2 ) 2 2 2 2 (v − γ)(v, f 󸀠 (n)v) − ((f 󸀠 (n)v)x , 2 Re(εu)).

(1.11.74)

Define Jμ (u, v) =

|ux |20

L

|bβ| b + |v |2 + 2k1 + ∫ f 󸀠 (n)|v|2 dx + μ|u|20 + μ|v|20 , 2 x0 2

(1.11.75)

0

Iμ (u, v) = 2k2 + 2μ Im(vε, u) − μk3 − bk4 + (v − γ)bβ|vx |20 .

(1.11.76)

Then from equations (1.11.67)–(1.11.74), we have dJμ (u, v) dt

= Iμ (u, v),

(1.11.77)

where μ is a large enough positive constant and Jμ (u, v) is an equivalent norm in X1 .

1.11 Nonlinear KdV–Schrödinger equation

| 143

Assume that X is a bounded invariant set in X2 , that is, S(t)X = X,

∀t ≥ 0.

Set ζ0 ∈ X. Then S(t)ζ0 = S(t)(ε0 , n0 ) = (ε(t), n(t)) ∈ X. Hence 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |X|∞ = sup sup{󵄨󵄨󵄨ε(t)󵄨󵄨󵄨L∞ + 󵄨󵄨󵄨n(t)󵄨󵄨󵄨L∞ + 󵄨󵄨󵄨nx (t)󵄨󵄨󵄨L∞ } < ∞. ζ0 ∈X t≥0

(1.11.78)

By equation (1.11.75) and from the definition of Ju , we know that there exist μ > 0 and M0 , M1 such that M0 (‖u‖21 + ‖v‖21 ) ≤ Ju (u, v) ≤ M1 (‖u‖21 + ‖v‖21 ),

∀(u, v) ∈ X1 .

(1.11.79)

So Jμ (u, v) is an equivalent norm in X1 . Now we introduce an R-linear inner product in X1 . Let η = (η1 , η2 ), ζ = (ζ1 , ζ2 ) ∈ X1 . Define L

Ψ(η, ξ ) = Re ∫[η1x ξ1x + 0

|bβ| b η ξ + f 󸀠 (n)η2 ξ2 + μη1 ξ1 + μη2 ξ2 + bnη1 ξ 1 ]dx 2 2x 2x 2

L

+ b Re ∫(ξ2 εη1 + η1 εξ1 )dx.

(1.11.80)

0

It is easy to prove that Ψ(η, ξ ) is an R-linear symmetric form in X1 , and Ψ(η, η) = Jμ (η, η) 1

is coercive by (1.11.79). So Ψ(η, η) 2 is an equivalent norm in X1 . j Now let ξ0j = (u0j , ν0 ), j = 1, 2, . . . , m be m elements in X1 , and ξ j (t) = (DS(t)ξ0 )ξ 0 be the corresponding solutions of (1.11.59)–(1.11.62). Let ηj (t) = eγt ζ j (t), j = 1, 2, . . . , m. Then ηj (t) = (uj (t), vj (t)) satisfies equation (1.11.66) and has initial value ηj (0) = (u0j , v0j ). We study the following evolution of volume quantities: 󵄨󵄨 1 󵄨 2 m i j 󵄨󵄨ξ (t) ∧ ξ (t) ∧ ⋅ ⋅ ⋅ ∧ ξ (t)󵄨󵄨󵄨∧m (X1 ) = det (ξ , ξ ). 1≤i,j≤m

(1.11.81)

Theorem 1.11.3. Let X be the bounded invariant set in X2 . Then there exist constants C1 and C2 such that for any ζ0 (ε0 , n0 ) ∈ X2 , m ≥ 1, t ≥ 0, we have 󵄨󵄨 1 󵄨 2 m 󵄨󵄨ξ (t) ∧ ξ (t) ∧ ⋅ ⋅ ⋅ ∧ ξ (t)󵄨󵄨󵄨∧m (X1 ) √m−γm)t 󵄨 󵄨 , ≤ 󵄨󵄨󵄨ξ01 (t) ∧ ξ02 (t) ∧ ⋅ ⋅ ⋅ ∧ ξ0m (t)󵄨󵄨󵄨∧m (X ) C1 e(C2 1

j

∀ξ0 ∈ X1 .

(1.11.82)

Proof. Firstly we note that 󵄨󵄨 1 󵄨2 2 m 󵄨󵄨ξ (t) ∧ ξ (t) ∧ ⋅ ⋅ ⋅ ∧ ξ (t)󵄨󵄨󵄨∧m (X1 ) 󵄨 󵄨 = e−2γmt 󵄨󵄨󵄨η1 ∧ η2 ∧ ⋅ ⋅ ⋅ ∧ ηm 󵄨󵄨󵄨∧m (X ) 1 = e−2γmt det Ψ(ηj (t)). 1≤i,j≤m

(1.11.83)

144 | 1 Attractor and its dimension estimation So we only need to estimate Hm (t) = det1≤i,j≤m Ψ(ηi (t), ηj (t)). We have j m Iμ (∑m dHm (t) j=1 xj η (t)) = Hm (t) ∑ max min m j ̸ Jμ (∑ dt F⊂Rm ,dim F=l, x∈F,x =0 j=1 xj η (t)) l=1

(1.11.84)

by equation (1.11.78). The norms |n|L∞ , |nx |L∞ , |ε|L∞ , |εx |L∞ are consistently bounded and, for ζ0 = (ε0 , n0 ) ∈ X, we have L

|k2 | ≤ C|u|20 + C ∫ |nxx ||u||ux |dx, 0

3

1

C|u|20 + C‖n‖2 |u|L∞ |nx |0 ≤ C|u|12 |u|02 , L

μ|k3 | = |μ| − ∫(f 󸀠 (n)v)x vdx − 2(v − γ)|v|20 + 2(Re(εu), vx ) 0

1 ≤ (v − γ)|bβ||vx |20 + C(|v|20 + |vx |0 |u|0 ) 8 3 1 3 1 1 ≤ (v − γ)|bβ||vx |20 + C(‖v‖12 |v|02 + ‖u‖12 |u|02 ), 8 |bk4 | ≤ C(|vx |0 |u|0 + |u|20 + |v|20 + ‖v‖2L4 ) 3 1 3 1 1 ≤ (v − γ)|bβ||vx |20 + C(‖u‖12 |u|02 + ‖v‖12 |v|02 ). 4 Hence, we note that Iμ (u, v) = 2k2 + 2μ Im(vε, u) − μk3 − bk4 + (v − γ)bβ|vx |20 3 1 3 1 1 ≤ (v − γ)bβ|vx |20 + C(‖u‖12 |u|02 + ‖v‖12 |v|02 ) 2 3

1

3

1

(1.11.85)

≤ M(‖u‖12 |u|02 + ‖v‖12 |v|02 ),

since v > γ, bβ < 0, where M is a positive constant. On the other hand, Jμ = Jμ (u, v) ≥ M0 (‖u‖21 + ‖v‖21 ). From (1.11.83) and (1.11.84), we have 3

1

m 2 2 m ‖ ∑m dHm (t) M j=1 xj uj (t)‖1 | ∑j=1 xj uj (t)|0 ≤ Hm (t) ∑ max min m m 2 2 ̸ ‖∑ dt M0 F⊂Rm ,dim F=l, x∈F,x=0 j=1 xj uj (t)‖1 + ‖ ∑j=1 xj vj (t)‖1 l=1 3

+

1

m 2 2 ‖ ∑m j=1 xj vj (t)‖1 | ∑j=1 xj vj (t)|0

m 2 2 ‖ ∑m j=1 xj uj (t)‖1 + ‖ ∑j=1 xj vj (t)‖1

1

1

2 2 m | ∑m | ∑m M j=1 xj uj (t)|0 j=1 xj vj (t)|0 ≤ max min { + } Hm (t) ∑ 1 1 ̸ M0 F⊂Rm ,dim F=l, x∈F,x=0 2 2 l=1 ‖ ∑m ‖ ∑m j=1 xj uj (t)‖1 j=1 xj vj (t)‖1

≤

m 2M 1 Hm (t) ∑ 1 , M0 l=1 λ 4 l

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 145

where we use Max–Min theorem and λj is the jth eigenvalue of A = −𝜕xx . We know that λj ∼ cj2 as j → ∞, so there exists a constant C2 such that dHm (t) ≤ 2C2 √mHm (t). dt

(1.11.86)

Hence, Hm (t) ≤ e2C2 √mt Hm (0), t > 0. Combining with equation (1.11.83), we get equation (1.11.82). As a corollary of Theorem 1.11.3, we have Theorem 1.11.4. The global attractor A defined by Theorem 1.11.2 in X1 possesses bounded fractal and Hausdorff dimensions. Proof. By Theorem 1.11.3 we know that, for χ0 ∈ A , ωm (DS(t)χ0 ) ≤ C m exp(C2 √m − γm)t, where ωm is the Lyapunov index. When m is large enough, ωm (A ) < 1, where ωm (A ) is the uniform Lyapunov index of A . By Theorem V.3.1 in [197], we know that A in X1 has bounded fractal and Hausdorff dimensions.

1.12 The Landau–Lifshitz equation on a Riemannian manifold Assume that (M, γ) and (N, g) are two Riemann manifolds, M is boundless and N is S2 . We will prove the existence of an attractor for the Landau–Lifshitz (LL) [25] equation in a Riemann manifold and will give upper and lower bounds on the attractor dimension. For this purpose, we must establish an a priori estimate to prove the existence of a global solution and of the attractor. Consider the following Landau–Lifshitz equation on a Riemann manifold: 𝜕t u = −α1 u × (u × Δ M u) + α2 u × Δ M u

(1.12.1)

with the initial value u|t=0 = u0 (x),

󵄨󵄨 󵄨2 󵄨󵄨u0 (x)󵄨󵄨󵄨 = 1,

x = (x1 , . . . , xn ) ∈ M,

(1.12.2)

where u : M → S2 and ΔM is the Laplace–Beltrami operator, ΔM =

𝜕 𝜕2 1 𝜕 αβ αβ k 𝜕 (γ ) = γ − Γαβ . √γ α 𝜕x √γ 𝜕xβ 𝜕x k 𝜕x β 𝜕x β

In 1993, Guo and Hong [100] proved in the classical sense that u is the solution of problem (1.12.1)–(1.12.2) if and only if u is the solution of equation 𝜕t u = α1

1 𝜕 𝜕u 1 𝜕 𝜕u (γ αβ √γ β ) + α1 |∇u|2 u + α2 u × (γ αβ √γ α ) α β 𝜕x √γ 𝜕x √γ 𝜕x 𝜕x

(1.12.3)

146 | 1 Attractor and its dimension estimation with the same initial value as in (1.12.2). Meanwhile, they proved, also in the classical sense, that u : M → S2 is a harmonic mapping if and only if u satisfies (1.12.2), 𝜕t u(x, t) = 0, ∀t ≥ 0, and heat flow equation for a harmonic mapping M → S2 is 𝜕t u = Δ M u + |∇u|2 u.

(1.12.4)

Now we establish a uniform a priori estimate. Let (M, γ) be a bounded or unbounded compact Riemannian manifold and let ∇ denote the contact (covariant derivative) of γ, i i 󵄨󵄨 󵄨2 αβ 𝜕u 𝜕u . 󵄨󵄨∇u(x)󵄨󵄨󵄨 = ∑ ∑ γ 𝜕x α 𝜕x β αβ i

(1.12.5)

For the real function φ ∈ C k (M) (k ≥ 0 is an integer), define 󵄨󵄨 k 󵄨󵄨2 σ σ σ 󵄨󵄨∇ φ󵄨󵄨 = ∇ 1 ⋅ ∇ 2 ⋅ ⋅ ⋅ ∇ k ⋅ ∇σ1 ⋅ ∇σ2 ⋅ ⋅ ⋅ ∇σk φ In particular, |∇1 φ| = |∇φ|, |∇1 φ|2 = |∇φ|2 = ∇v φ∇v φ, ∇k φ is the kth covariant derivative of φ. We consider the vector space Lpk of C ∞ functions φ, |∇l φ| ∈ Lp (M), ∀0 ≤ l ≤ k, where k and l are integers, and p ≥ 1 is a real number. Sobolev space Wpk (M) is the completion of space Lpk (M) with respect to the norm k

󵄩 󵄩 ‖φ‖W k (M) = ∑ 󵄩󵄩󵄩∇i φ󵄩󵄩󵄩p . p

i0

In particular, W2k (M) = H k (M), ‖ ⋅ ‖2 = ‖ ⋅ ‖. Lemma 1.12.1. Let |u0 (x)|2 = 1. Then for a smooth solution of the initial value problem (1.12.1)–(1.12.2) we have 󵄨󵄨 󵄨2 󵄨󵄨u(x, t)󵄨󵄨󵄨 = 1,

∀(x, t) ∈ M × [0, ∞).

(1.12.6)

Proof. Multiplying equation (1.12.2) by u, we get u ⋅ 𝜕t u = 0,

∀(x, t) ∈ M × [0, ∞).

Then through (1.12.2) we get (1.12.6). Lemma 1.12.2. Suppose the condition of Lemma 1.12.1 is satisfied, and ‖∇u0 ‖ < ∞. Then we have t

󵄨󵄨 󵄨2 2 󵄨󵄨∇u(⋅, t)󵄨󵄨󵄨 ≤ ‖∇u0 ‖ ,

2α1 ∫ ‖u × Δ M u‖2 dt ≤ ‖∇u0 ‖2 , 0

(1.12.7) ∀0 ≤ t < ∞.

(1.12.8)

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 147

Proof. Multiplying equation (1.12.1) by Δ M u, we get Δ M u ⋅ ut = −α1 Δ M u ⋅ (u × (u × Δ M u))

= −α1 (u × Δ M u) ⋅ (Δ M u × u) = α1 |u × Δ M u|2 , 𝜕u 𝜕u 𝜕u ∫ Δ M u ⋅ ut dM = − ∫ γ αβ √γ α βt dx + ∫ ut γ αβ √γ α cos(v, x β )ds 𝜕x 𝜕x 𝜕x

M

M

𝜕ui 𝜕ui 1 d =− ∫ γ αβ α β dM 2 dt 𝜕x 𝜕x

M

M

1 d 󵄩󵄩 󵄩2 =− 󵄩∇u(⋅, t)󵄩󵄩󵄩 . 2 dt 󵄩 Hence d 󵄩󵄩 󵄩2 2 󵄩󵄩∇u(⋅, t)󵄩󵄩󵄩 + 2α1 |u × Δ M u| dM = 0, dt which yields d 󵄩󵄩 󵄩2 󵄩∇u(⋅, t)󵄩󵄩󵄩 ≤ 0, dt 󵄩 proving the lemma. Lemma 1.12.3 (Sobolev interpolation inequality on a compact Riemann manifold). Let M be a Riemannian manifold with smooth boundary. Assume q, r are real numbers, 1 ≤ q, r ≤ ∞, while j, m are integers, 0 ≤ j < m. Then there exists a constant C, depending on n, m, j, q, r and a, such that for any f ∈ Wrm (M) ∩ Lq (M), we have 󵄩󵄩 j 󵄩󵄩 a 1−a 󵄩󵄩∇ f 󵄩󵄩p ≤ C‖f ‖Wrm (M) ‖f ‖q ,

(1.12.9)

where j 1 m 1 1 = + a( − ) + (1 − a) p n r n q for any a such that

j m

≤ a ≤ 1; p is a non-negative integer.

Proof. From Theorem 3.70 in [6] we have 󵄩󵄩 i 󵄩󵄩 󵄩 m 󵄩a 1−a 󵄩󵄩∇ F 󵄩󵄩p ≤ C 󵄩󵄩󵄩∇ F 󵄩󵄩󵄩r ‖F‖q , where F = f − f,

f =

1 ∫ f dM, vol M M

1 i 1 m 1 = + a( − ) + (1 − a) . p n r n q Now consider several cases:

(1.12.10)

148 | 1 Attractor and its dimension estimation (1) When i > 0, by equation (1.12.10) we get 1−a 󵄩 m 󵄩a 󵄩󵄩 i 󵄩󵄩 󵄩󵄩∇ f 󵄩󵄩p ≤ C 󵄩󵄩󵄩∇ f 󵄩󵄩󵄩r (‖f ‖q + ‖f ‖q ) 󵄩a 󵄩 󸀠 a 1−a ≤ C 󸀠 󵄩󵄩󵄩∇m f 󵄩󵄩󵄩r ‖f ‖1−a q ≤ C ‖f ‖Wrm (M) ‖f ‖q .

(2) When i = 0, using Hölder inequality, we have p

α

β

αl

1 l

βl󸀠

1 l󸀠

∫ |f | dx = ∫ |f | |f | dx ≤ (∫ |f | dx) (∫ |f | dx) , where α + β = p,

αl = r,

βl󸀠 = q,

1 1 + = 1. l l󸀠

Then 1 (p − q) r =( ) , l (q − r) p

1 1 =1− l󸀠 l

and from equations (1.12.1) and (1.12.2) we get (1.12.9). Remark 1.12.1. Instead of the space Wkp (M), Aubin introduced the space Vkp (M) in [6], which is the completion of space Skp (M) with respect to the norm 󵄩 󵄩 ‖φ‖V p (M) = ∑ 󵄩󵄩󵄩ΔlM φ󵄩󵄩󵄩p + k 0≤t≤ k2

󵄩 󵄩 ∑ 󵄩󵄩󵄩∇ΔlM φ󵄩󵄩󵄩p ,

0≤l≤ k−1 2

where the vector space Skp comprises functions φ ∈ C ∞ (M), ΔlM φ ∈ Lp (M), 0 ≤ l ≤ k2 , . such that |∇ΔlM φ| ∈ Lp (M), 0 ≤ l ≤ k−1 2 Lemma 1.12.4. Suppose the condition of Lemma 1.12.3 is satisfied. Let 󵄩󵄩 󵄩 󵄩󵄩∇u0 (x)󵄩󵄩󵄩 ≤ λ,

n = 2,

(1.12.11)

where the constant λ is small enough. Then we have 󵄩󵄩 󵄩 E 󵄩󵄩Δ M u(⋅, t)󵄩󵄩󵄩 ≤ 1 , t

∀x = (x1 , . . . , xn ), t > 0, n ≤ 2,

(1.12.12)

where the constant E1 depends only on ‖∇u0 (x)‖H 1 (M) , 0 < T ≤ T. Proof. Acting with Δ M on equation (1.12.2), then taking the inner product with tΔ M u, we get (tΔ M u, tΔ M ut − α1 Δ2M ut − α1 Δ M (|∇u|2 u) − α2 Δ M (u × Δ M u)) = 0, where t(Δ M u, Δ M ut ) =

1 d 1 (t‖Δ M u‖2 ) − ‖Δ M u‖2 , 2 dt 2

(1.12.13)

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 149

𝜕Δ u 1 𝜕 (γ αβ √γ Mα )√γdx 𝜕x √γ 𝜕x β 𝜕Δ u 𝜕Δ u M = −α1 ∫ γ αβ √γ Mα dx 𝜕x 𝜕x β = −α1 ‖∇Δ M u‖2 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕 𝜕 󵄨 󵄨 󵄨 󵄨󵄨 αβ 󵄨󵄨 ∫ Δ M u ⋅ Δ M (u × Δ M u)√γdx󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ Δ M u ⋅ β (γ √γ α (u × Δ M u))dx 󵄨󵄨󵄨 󵄨󵄨 𝜕x 𝜕x 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕Δ u 𝜕(u × Δ u) 󵄨 󵄨 M = 󵄨󵄨󵄨 ∫ γ αβ Mβ √γdx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕xα 𝜕x 󵄨󵄨 󵄨󵄨 𝜕Δ M u 𝜕Δ u 𝜕u 󵄨󵄨 󵄨 ]√γdx = 󵄨󵄨󵄨 ∫ γ αβ Mβ [ α × Δ m u + u × 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕x 𝜕x α 𝜕x ≤ C1 ‖∇Δ M u‖‖∇u‖∞ ‖Δ M u‖. (1.12.14) (Δ M u, α1 Δ2M u) = α1 ∫ Δ M u ⋅

In the above equation, we used ∫ γ αβ

𝜕Δ M u 𝜕Δ M u (u × )√γdx = 0. 𝜕xα 𝜕xβ

(1.12.15)

By Lemma 1.12.2, we get 1 󵄩 󵄩1 ‖∇u‖∞ ≤ C2 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 2 ‖∇u‖ 2 + C3 , 1 2

n = 2,

󵄩 󵄩 ‖Δ M u‖ ≤ C4 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 ‖∇u‖ + C5 , 1 2

(1.12.16)

where constants C3 and C5 depend on ‖∇u0 ‖. Substituting equation (1.12.16) into equation (1.12.14), we get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 3 2 󸀠 󵄨󵄨 ∫ Δ M ⋅ Δ M (u × Δ M )dM 󵄨󵄨󵄨 ≤ 2C1 C2 C4 C6 ‖∇u‖‖∇ u‖ + C6 , 󵄨󵄨 󵄨󵄨

(1.12.17)

M

where 󵄩 󵄩 ‖∇Δ M u‖ ≤ C6󸀠 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 and constant C6󸀠 depends on ‖∇u0 ‖. Now we estimate the term (Δ M (|∇u|u), Δ2M u) of equation (1.12.13) as follows: 󵄨 𝜕 αβ 𝜕|∇u|2 u 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2 dx󵄨󵄨 󵄨󵄨(Δ M u, Δ M (|∇u| u))󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ Δ M μ ⋅ β γ √γ 󵄨󵄨 󵄨󵄨 𝜕xα 𝜕x M

󵄨󵄨 󵄨󵄨 𝜕 𝜕 󵄨 󵄨 = 󵄨󵄨󵄨 ∫ γ αβ √γ α |∇u|2 u ⋅ β Δ M udx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕x 𝜕x M

󵄨󵄨 󵄨󵄨 𝜕 𝜕u 𝜕 󵄨 󵄨 = 󵄨󵄨󵄨 ∫ γ αβ √γ(( α |∇u|2 u)u + |∇u|2 α ) β Δ M udx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕x 𝜕x 𝜕x M

150 | 1 Attractor and its dimension estimation 󵄨󵄨 𝜕2 u 𝜕u 𝜕u 𝜕u 󵄨 = 󵄨󵄨󵄨 ∫ γ αβ [2γ lδ (x) α l δ u + γ lδ (x)󸀠 l δ 󵄨󵄨 𝜕x 𝜕x 𝜕x 𝜕x 𝜕x M

󵄨󵄨 𝜕u 𝜕Δ M u 󵄨󵄨 ] dM 󵄨󵄨 󵄨󵄨 𝜕xα 𝜕x β 󵄩󵄩 2 󵄩󵄩 ≤ C7 [‖∇u‖∞ ‖u‖∞ 󵄩󵄩∇ u󵄩󵄩‖∇Δ M u‖ + ‖∇u‖2 ‖u‖∞ ‖∇Δ M u‖ + ‖∇u‖2∞ ‖∇u‖‖∇Δ M u‖] 󵄩 󵄩 (1.12.18) ≤ C7 [2C2 C4 ‖∇u‖ + (2C22 + 1)‖∇u‖2 ]󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C8 , + |∇u|2

where constants C7 , C8 depend on ‖∇u0 ‖ and supx∈M (|γ αβ (x)|, |γ αβ (x)|). Hence, from equations (1.12.8), (1.12.12) and (1.12.13) we get 󸀠

1 d 1 t‖Δ M u‖2 − ‖Δ M u‖2 + α1 t‖Δ M u‖2 2 dt 2 󵄩 󵄩 ≤ 2t[|α2 |C1 C2 C4 C6 + α1 C7 C2 C4 + α1 (C22 + 1)C7 ‖∇u‖]‖∇u‖󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C9 .

(1.12.19)

Now we estimate a lower bound of ‖∇Δ M u‖2 . By Ricci formula, k−1

Δ(∇k f ) = ∇k (Δf ) + ∑ Ski (∇k−i f ),

(1.12.20)

i=0

where Ski is a linear functional which depends on the tensor covariant derivative ∇i R curve. Moreover, we get 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ‖∇Δ M u‖2 = 󵄩󵄩󵄩Δ M (∇u) − S10 (∇u)󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩Δ M (∇u)󵄩󵄩󵄩 − S10 󵄩󵄩󵄩(∇u)󵄩󵄩󵄩.

(1.12.21)

Since 󵄩󵄩 󵄩2 󵄩󵄩Δ M (∇u)󵄩󵄩󵄩 = ∫ ∑(∑ ujii )(∑ ujkk )dM M j

i

k

= ∑ ∫(∑ ujii )(∑ ujkk )dM = − ∑ ∫(∑ uji )(∑ ujkki )dM j M

i

j M

k

i

k

= − ∑ ∫(∑ uji ) ∑(ujkik + ujl Rlkki + ulk Rljki )dM j M

i

k

= − ∑ ∫(∑ uji ) ∑(ujikk + ul Rljkik + ulk Rljki + ujl Rlkki + ulk Rijki )dM j M

i

k

3 󵄨󵄨2

󵄨 ≥ ∫ 󵄨󵄨󵄨∇ u󵄨󵄨 dM − C11 ∫ |∇u|2 dM, M

(1.12.22)

M

equations (1.12.19), (1.12.21) and (1.12.22) yield d t‖Δ M u‖2 − ‖Δ M u‖2 + 2t[α1 − (2α2 C1 C2 C4 C6 + 2α1 C7 C2 C4 dt 󵄩 󵄩2 + 2α1 (C22 + 1)C7 ‖∇u‖ + C12 ‖∇u‖)]‖∇u‖󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 ≤ C13 .

(1.12.23)

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 151

Choose ‖∇u0 ‖ small enough so that α1 − (2|α2 |C1 C2 C4 C6 − 2α1 C7 C2 C4 + 2α1 (C22 + 1)C7 ‖∇u0 ‖ + C12 ‖∇u0 ‖)‖∇u0 ‖ >

α > 0. 2

From equation (1.12.23), we get t

t

1 󵄩 󵄩2 t‖Δ M u‖ − ∫ ‖Δ M u‖ dt + α ∫ t 󵄩󵄩󵄩Δ3M u󵄩󵄩󵄩 dt ≤ C13 t. 2 2

2

(1.12.24)

0

0

t

In order to estimate the middle term ∫0 ‖Δ M u‖2 dt on the left of equation (1.12.24), we need the following lemma: Lemma 1.12.5. Under the conditions of Lemma 1.12.2, we get t

󵄩󵄩 𝜕u 󵄩󵄩2 (α2 + α22 ) 󵄩 󵄩 ‖∇u0 ‖2 , ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 dt ≤ 1 󵄩󵄩 𝜕t 󵄩󵄩 α1 t

∀t ∈ R+ ,

(1.12.25)

0

∫ ‖Δ M u‖2 dt ≤ C14 ,

0 ≤ t ≤ T,

(1.12.26)

0

where the constant C14 depends on ‖∇u0 ‖ and T. Proof. Multiplying equation (1.12.3) by 𝜕t u, and integrating with respect to (x, t) ∈ M × [0, t), we get t

t

t

α d ∫ ∫ |𝜕u| dMdt + 1 ∫ ‖∇u‖2 dt − α2 ∫ ∫ 𝜕t u ⋅ (u × Δ M u)dMdt = 0. 2 dt 2

0 M

0

(1.12.27)

0 M

Taking the cross-product of (1.12.3), we get u × 𝜕t u = α1 u × (Δ M u) + α2 (u × (u × Δ M u))

= α1 (u × Δ M u) − α2 Δ M u − α2 |∇u|2 u.

Since −Δ M u − |∇u|2 u = −

α 1 𝜕 u + 2 (u × Δ M u), α1 t α1

the latter equation yields u × 𝜕t u +

α2 α2 𝜕t u = (α1 + 2 )(u × Δ M u). α1 α1

Multiplying equation (1.12.28) by 𝜕t u, we have 𝜕t u ⋅ (u × Δ M u) = α2 (α12 + α22 ) |𝜕t u|2 , −1

(1.12.28)

152 | 1 Attractor and its dimension estimation t

t

α2 α2 ∫ ∫ 𝜕t u ⋅ (u × Δ M u)dMdt = 2 2 2 ∫ ∫ |𝜕t u|2 dMdt. α1 + α2 0 M

0 M

By equation (1.12.27) we have t

α12 α 󵄩 󵄩2 ∫ ∫ |𝜕t u|2 dMdt + 1 (󵄩󵄩󵄩∇u(⋅, t)󵄩󵄩󵄩 − ‖∇u0 ‖2 ) = 0, 2 2 α1 + α22 0 M

i. e., t

∫ ∫ |𝜕t u|2 dMdt ≤ 0 M

α12 + α22 ‖∇u0 ‖2 , 2α1

∀t ∈ R+ .

(1.12.29)

From equation (1.12.3), we have t

t

2

t

t

∫ ‖Δ M u‖ dt ≤ C15 (∫ ‖ut ‖ dt + ∫ ‖u × Δ M u‖ dt + ∫ ∫ |∇u|4 dMdt). 0

2

0

2

0

(1.12.30)

0 M

With the aid of Sobolev inequality (1.12.9), we have ∫ |∇u|4 dM ≤ C16 ‖∇2 u‖2 ‖∇u‖2 + C17 ,

∀0 ≤ t ≤ T,

(1.12.31)

M

where the constant C17 depends on ‖∇u0 ‖ and 0 ≤ t ≤ T. By the definition of Laplace–Beltrami operator, we get 󵄩 󵄩2 󵄩 󵄩2 C18 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩 + C19 ‖∇u‖2 ≥ ‖∇M u‖2 ≥ C18 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩 − C19 ‖∇u‖2 , αβ

(1.12.32)

k supx∈M |Γαβ |.

Hence, from

t 󵄩󵄩 𝜕u 󵄩󵄩2 󵄩󵄩 󵄩󵄩 ≤ C15 (∫ 󵄩󵄩 󵄩󵄩 dt + ∫ ‖u × Δ M u‖2 dt + C17 ) + C19 ‖∇u0 ‖2 t. 󵄩󵄩 𝜕t 󵄩󵄩

(1.12.33)

where the constants C18 , C19 depend on supx∈M |γ (x)| and equation (1.12.9), we get t

t

󵄩 󵄩2 󵄩2 󵄩 C18 ∫ 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩 dt − C15 C16 ∫ 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩 dt ⋅ ‖∇u0 ‖2 0

0

t

0

0

Choose ‖∇u0 ‖ small enough so that C18 − C15 C16 ‖∇u0 ‖2 ≥

C18 . 2

(1.12.34)

From equations (1.12.7), (1.12.8), (1.12.31) and (1.12.32), we have t

∫ ‖Δ M u‖2 dt ≤ C14 , 0

This completes the proof.

0 ≤ t ≤ T.

(1.12.35)

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 153

Using equations (1.12.34) and (1.12.24), we get ‖Δ M u‖2 ≤

El , t

∀x = (x1 , x2 , . . . , xn ), n ≤ 2, t > 0,

(1.12.36)

where the constant El only depends on ‖∇u0 ‖ and T. This completes the proof of the lemma. For the inequality (1.12.12) when n = 1, we do not need the restriction ‖∇u0 ‖ ≤ λ. In fact, by Lemma 1.12.2, we have 3 󵄩 󵄩1 ‖∇u‖∞ ≤ C2 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 4 ‖∇u‖ 4 + C3 ,

1 󵄩 󵄩1 ‖Δ M u‖ ≤ C4 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 2 ‖∇u‖ 2 + C5 (n = 1), 󵄩 󵄩 ∫ |∇u|4 dM ≤ C16 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩‖∇u‖3 + C17 ,

M

which yield inequality (1.12.12). Now we prove the existence of a global unique smooth solution of problem (1.12.1)–(1.12.2) Theorem 1.12.1 ([100]). Let u0 : M ε ≡ M → S2 be a smooth mapping. Then there exists a constant ε ≥ 0 and a unique smooth mapping u : M × [0, ε] → S2 , u ∈ Lp2 (M ε ), such that 𝜕t u = α1 Δ M u + α1 |∇u|2 u + α2 u × Δ M u,

(x, t) ∈ M × [0, ε],

u = u0 ,

x ∈ M × {0}.

Lemma 1.12.6. Let u0 (x) ∈ H 1 (M), |u0 (x)|2 = 1. Then for a smooth solution of problem (1.12.1)–(1.12.2) we have 󵄩 󵄩 sup 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1 (M) ≤ C1 ,

0≤t≤T

(1.12.37)

where the constant C1 depends on ‖u0 (x)‖H 1 (M) . Proof. Similar to that of Lemmas 1.12.1 and 1.12.2. Lemma 1.12.7. Suppose the condition of Lemma 1.12.6 is satisfied, and ‖∇u0 ‖ ≤ λ,

n = 2,

(1.12.38)

where constant λ is small enough. Then we have 󵄩 󵄩 sup 󵄩󵄩󵄩∇2 u(⋅, t)󵄩󵄩󵄩 ≤ C2 ,

0≤t≤T

where constant C2 depends on ‖u0 (x)‖H 2 (M) .

(1.12.39)

154 | 1 Attractor and its dimension estimation Proof. Acting with Δ M on equation (1.12.3), multiplying by Δ M u, and integrating with respect to x ∈ M, we get (Δ M u, Δ M ut − α1 Δ2M u − α1 Δ M (|∇u|2 u) − α2 Δ M (u × Δ M u)) = 0,

(1.12.40)

where 1 d ‖Δ u‖2 , 2 dt M (Δ M u, α1 Δ2M u) = −α1 ‖∇Δ M u‖2 . (Δ M u, Δ M ut ) =

Similarly as in the proof of Lemma 1.12.3, we get 1 d 󵄩 󵄩2 ‖Δ M u‖2 + [α1 − (C3 + C4 ‖∇u‖) × ‖∇u‖]󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 ≤ C3 , 2 dt

(1.12.41)

where constant C3 depends on ‖∇u0 ‖. Select ‖∇u0 ‖ small enough so that α1 − (C3 + C4 ‖∇u0 ‖)‖∇u0 ‖ ≥

α1 . 2

By equation (1.12.41), this implies sup ‖Δ M u‖2 ≤ C4󸀠 .

0≤t≤T

(1.12.42)

Lemma 1.12.8. Suppose the condition of Lemma 1.12.7 is satisfied. Set u0 (x) ∈ H 3 (M). Then we have 󵄩 󵄩2 sup 󵄩󵄩󵄩∇Δ M u(⋅, t)󵄩󵄩󵄩 ≤ C5 ,

0≤t≤T

(1.12.43)

where constant C5 depends on ‖u0 (x)‖H 3 (M) . Proof. Acting with ∇Δ M on equation (1.12.3), and integrating with ∇Δ M u, we get (∇Δ M u, ∇Δ M ut − α1 ∇Δ2M u − α1 ∇Δ M (|∇u|2 u) − α2 ∇Δ M (u × Δ M u)) = 0, where 1 d ‖∇Δ M u‖2 , 2 dt 𝜕 𝜕 (∇Δ2M u, ∇Δ M u) = ∫ γ δl δ Δ2M u ⋅ l Δ M udM 𝜕x 𝜕x

(∇Δ M u, ∇Δ M ut ) =

M

= − ∫ Δ2M u ⋅ M

𝜕 𝜕 (γ δl √γ l Δ M u)dx δ 𝜕x 𝜕x

2 󵄩 󵄩2 = − ∫(Δ2M u) dM = −󵄩󵄩󵄩Δ2M u󵄩󵄩󵄩 , M

(1.12.44)

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 155

(∇Δ M (|∇u|2 u), ∇Δ M u) = ∫ γ δl M

𝜕 𝜕 Δ M (|∇u|2 u) l Δ M u ⋅ √γdx δ 𝜕x 𝜕x

= − ∫ Δ M (|∇u|2 u) M

𝜕 𝜕 √γγ αβ l Δ M udx δ 𝜕x 𝜕x

= − ∫ Δ M (|∇u|2 u)Δ2M udM.

(1.12.45)

M

Noting the following equations: n

Δ M (|∇u|2 u) = ∑(|∇u|2 u)ii i=1 n

= ∑(|∇u|2i u + |∇u|2 ui )i i=1 n

= ∑(|∇u|2ii + 2|∇u|2i ui + |∇u|2 uii ) i=1

= Δ M (|∇u|2 ) + 2(|∇u|2 )∇u + |∇u|2 Δ M u,

Δ M (|∇u|2 ) = 2 ∑ u2ij + 2 ∑ ui uijj

= 2 ∑ u2ji + 2 ∑ ui (Δ M u)i + 2 Ric(∇u, ∇u)

(see [190, p. 129]), from the equation (1.12.45) we get ∫ Δ M (|∇u|2 u)Δ2M udM

M

= ∫[2∇2 u + 2∇u ⋅ ∇Δ M u + 2 Ric(∇u, ∇u)]Δ2M udM, M

where 󵄨󵄨 󵄨󵄨 α 󵄨󵄨 󵄩 2 󵄩2 󵄨 2 2 󵄨󵄨2 ∫ ∇ uΔ M udM 󵄨󵄨󵄨 ≤ 1 󵄩󵄩󵄩Δ M u󵄩󵄩󵄩 + C7 , 󵄨󵄨 󵄨󵄨 8 M

2 󵄩 󵄩1 ‖∇u‖∞ ≤ C 󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 3 ‖∇u‖ 3 , 󵄩 󵄩 󵄩 󵄩 ‖∇Δ M u‖ = 󵄩󵄩󵄩Δ M (∇u) − S10 (∇u)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Δ M (∇u)󵄩󵄩󵄩 + C‖∇u‖ 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩 + C‖∇u‖ 󵄩 󵄩 ≤ 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄩 2 󵄩 2 󵄨󵄨2 ∫ ∇u ⋅ ∇Δ M u ⋅ Δ M udM 󵄨󵄨󵄨 ≤ 2‖∇u‖∞ ‖∇Δ M u‖󵄩󵄩󵄩Δ M u󵄩󵄩󵄩 󵄨󵄨 󵄨󵄨

M

α1 󵄩󵄩 4 󵄩󵄩 󵄩Δ u󵄩󵄩 + C8 , 8󵄩 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄩 2 󵄩 2 󵄨󵄨 ∫ 2 Ric(∇u, ∇u)Δ M udM 󵄨󵄨󵄨 ≤ 2‖∇u‖∞ ‖∇u‖󵄩󵄩󵄩Δ M u󵄩󵄩󵄩 󵄨󵄨 󵄨󵄨 ≤

M

(1.12.46)

156 | 1 Attractor and its dimension estimation α1 󵄩󵄩 4 󵄩󵄩2 󵄩∇ u󵄩󵄩 + C, 8󵄩 󵄩 2 󵄩2 󵄩 3 󵄩2 󵄩󵄩 4 󵄩󵄩2 󵄩󵄩∇ u󵄩󵄩 − C(󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + 1) ≤ 󵄩󵄩󵄩Δ M u󵄩󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 ≤ 󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 + C[󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + 1], ≤

where we use Ricci formula. Then we have 󵄨󵄨 󵄨󵄨 3α 󵄩 3 󵄩2 󵄨󵄨 󵄨 󵄩 4 󵄩2 2 2 󵄨󵄨 ∫ Δ M (|∇u| u)Δ M udM 󵄨󵄨󵄨 ≤ 1 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + C10 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + C11 . 󵄨󵄨 󵄨󵄨 8

(1.12.47)

M

Then, we estimate the terms in equation (1.12.44), namely α2 ∫ ∇Δ M u ⋅ ∇Δ M (u × Δ M u)dM M

and ∫ ∇Δ M u ⋅ ∇Δ M (u × Δ M u)dM = − ∫ Δ2M u ⋅ Δ M (u × Δ M u)dM,

M

M

where Δ M (u × Δ M u) = ∑(u × ∑ ujj ) i

ii

j

= ∑[(ui × ∑ ujj ) + (u × ∑ ujji )] i

j

j

i

= ∑(uii × ∑ ujj ) + ∑ ui × ∑ vjji + ∑(ui × ∑ ujji ) + ∑ uj × ∑ ujjii i

j=1

i

= 2 ∑ ui × ∑ ujji + u × i=1

j=1

j=1

i

j=1

j

j=1

Δ2M u,

which implies 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ ∇Δ M u ⋅ ∇Δ M (u × ∇Δ M u)dM 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 M

󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 ≤ 2 ∫ 󵄨󵄨󵄨Δ2M u󵄨󵄨󵄨 ⋅ |∇u|󵄨󵄨󵄨∇Δ M u󵄨󵄨󵄨dM ≤ 2‖∇u‖∞ 󵄩󵄩󵄩Δ2M u󵄩󵄩󵄩‖∇Δ M u‖ 󵄨󵄨 󵄨󵄨 M

2 󵄩 󵄩 󵄩 󵄩1 ≤ C 󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 3 ‖∇u‖ 3 󵄩󵄩󵄩Δ2M u󵄩󵄩󵄩‖∇Δ M u‖ + C

󵄩 󵄩1 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 3 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 + C 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C α 󵄩 󵄩2 󵄩 󵄩2 ≤ 1 󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 + C12 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C13 , 4|α2 |

(1.12.48)

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 157

where we used the following Sobolev inequality: 󵄩 4 󵄩1 󵄩 2 󵄩1 󵄩󵄩 3 󵄩󵄩 󵄩󵄩∇ u󵄩󵄩 ≤ C 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 2 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 2 + C. Due to equations (1.12.47) and (1.12.48), we get 3 󵄩 d 󵄩2 󵄩 󵄩 ‖∇Δ M u‖2 + α1 󵄩󵄩󵄩∇4 u󵄩󵄩󵄩 ≤ C14 󵄩󵄩󵄩∇3 u󵄩󵄩󵄩 + C15 . dt 4

(1.12.49)

To the above inequalities, we use the inequality 󵄩󵄩 4 󵄩󵄩2 󵄩 3 󵄩2 󵄩 2 󵄩2 󵄩 4 󵄩2 󵄩 3 󵄩2 󵄩󵄩∇ u󵄩󵄩 − C(󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + 1) ≤ 󵄩󵄩󵄩Δ M u󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + C(󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + 1). Integrating equation (1.12.49) with respect to t ∈ [0, T], we get t

t

󵄩󵄩 󵄩2 3 󵄩 4 󵄩 󵄩 3 󵄩2 2 󵄩󵄩∇Δ M u(⋅, t)󵄩󵄩󵄩 + α1 ∫ 󵄩󵄩󵄩∇ u󵄩󵄩󵄩dt ≤ C14 ∫ 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 dt + C15 t + ‖∇Δ M u0 ‖ . 4 0

(1.12.50)

0

Using the inequality 󵄩󵄩 3 󵄩󵄩2 󵄩 3 󵄩2 2 󵄩󵄩∇ u󵄩󵄩 − C ≤ ‖∇Δ M u‖ ≤ 󵄩󵄩󵄩∇ u󵄩󵄩󵄩 + C, and also Gronwall inequality, from equation (1.12.50) we have ‖∇Δ M u‖2 ≤ C16, where constant C16 depends on ‖u0 (x)‖H 3 . Using Lemmas 1.12.6, 1.12.2 and 1.12.8, we get 󵄩 󵄩 sup 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 3 (M) ≤ C17,

0≤t≤T

where constant C17 depends on ‖u0 (x)‖H 3 (M) . Lemma 1.12.9. Suppose the condition of Lemma 1.12.8 is satisfied, and u0 (x) ∈ H m (M), m ≥ 4. Then we have 󵄩 󵄩 sup 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H m (M) ≤ C18

0≤t≤T

(1.12.51)

where constant C18 depends on ‖u0 (x)‖H m (M) . Proof. Firstly, letting m = 2n, acting with Δ2n M on equation (1.12.3), and taking the inner product with Δ2n u, we get M 2n 2n+1 2n 2 2n (Δ2n M u, Δ M ut − α1 Δ M u − α1 Δ M (|∇u| u) − α2 Δ M (u × Δ M u)) = 0.

158 | 1 Attractor and its dimension estimation By induction, Sobolev interpolation inequality, Ricci formula, and 󵄩 󵄩 󵄩󵄩 󵄩󵄩 4n 󵄩󵄩 󵄩 󵄩 󵄩 󸀠󵄩 󵄩 4n−1 u󵄩󵄩󵄩 + C 󸀠 , C3󸀠 󵄩󵄩󵄩∇4n u󵄩󵄩󵄩 − C4󸀠 󵄩󵄩󵄩∇4n−1 u󵄩󵄩󵄩 − C5󸀠 ≤ 󵄩󵄩󵄩Δ2n 2 M u󵄩 󵄩 󵄩 ≤ 󵄩󵄩∇ u󵄩󵄩 + C1 󵄩󵄩∇ 󵄩 󵄩 󵄩󵄩 󵄩󵄩 4n+1 󵄩󵄩 󵄩 󵄩 󵄩 󸀠󸀠 󵄩 󵄩󵄩∇4n u󵄩󵄩󵄩 + C 󸀠󸀠 , + C ∇ u ≤ u C3󸀠󸀠 󵄩󵄩󵄩∇4n+1 u󵄩󵄩󵄩 − C4󸀠󸀠 󵄩󵄩󵄩∇4n u󵄩󵄩󵄩 − C5󸀠󸀠 ≤ 󵄩󵄩󵄩∇Δ2n 󵄩 󵄩 󵄩 2 1 󵄩 M 󵄩 󵄩 󵄩 󵄩 we get d 󵄩󵄩 2n 󵄩󵄩2 󵄩 4n−1 󵄩󵄩2 u󵄩󵄩 ≤ C20 . 󵄩󵄩∇Δ M u󵄩󵄩 + C19 α1 󵄩󵄩󵄩∇ dt

(1.12.52)

Secondly, for m = 2n + 1 we have d 󵄩󵄩 2n+1 󵄩󵄩2 󵄩 4n+2 󵄩󵄩2 u󵄩󵄩 ≤ C22 . 󵄩󵄩Δ M u󵄩󵄩 + C21 α1 󵄩󵄩󵄩∇ dt

(1.12.53)

From equations (1.12.52) and (1.12.53), we get inequality (1.12.51). From Lemmas 1.12.6–1.12.10 and Theorem 1.12.7, we get the existence of a smooth solution for problem (1.12.1)–(1.12.2). As for the uniqueness of a smooth solution, it is easy to obtain. Theorem 1.12.2. Let M be a boundless Riemannian manifold, which satisfies the following conditions: (1) u0 (x) ∈ H m (M), m ≥ 2, |u0 (x)|2 = 1, x = (x1 , . . . , xn ) ∈ M,

1 ≤ n ≤ 2;

(2) when n = 2, 󵄩󵄩 󵄩 󵄩󵄩∇u0 (x)󵄩󵄩󵄩 ≤ λ, where the constant λ is small enough. Then there exists a unique global smooth solution u(x, t) : M × [0, ∞) → S2 for the initial value problem (1.12.1)–(1.12.2), u(x, t) ∈ L∞ (0, ∞; H m (M)). Using Theorem 1.12.2 and Lemmas 1.12.1–1.12.2, we will prove that problem (1.12.1)–(1.12.2) possess an attractor. Theorem 1.12.3. Let M be a boundless Riemannian manifold (n ≤ 2), which satisfies the following conditions: (1) α1 > 0, |u0 (x)| = 1, u0 (x) ∈ H 1 (M), x = (x1 , . . . , xn ) ∈ M, n ≤ 2; (2) ‖∇u0 (x)‖ ≤ λ, x ∈ M, n = 2, where constant λ is small enough. Then on the manifold M the initial value problem (1.12.1)–(1.12.2) of Landau–Lifshitz equation has a unique attractor A . It is compact in H 1 (M), and A = ω(B1 ) = ⋂ ⋂ S(t)B1 s≥0 t≥s

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 159

where 󵄩 󵄩 󵄨 󵄨 B1 = {u ∈ H 1 (M), 󵄨󵄨󵄨u(⋅, t)󵄨󵄨󵄨 = 1, 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1 (M) ≤ ρ1 } is a bounded absorbing subset of E = {u ∈ H 1 (M) | |u(⋅, t)| = 1} for S(t) in H 1 (M), ρ1 is a positive constant, and S(t)u0 is the semigroup operator generated by problem (1.12.1)–(1.12.2). Proof. By Theorem 1.12.2, we can get the existence of a global smooth solution for problem (1.12.1)–(1.12.2). Moreover, it forms a semigroup. From Lemmas 1.12.1–1.12.2, we know that 󵄩 󵄩 B1 = {u ∈ H 1 (M), |u| = 1, ‖u‖H 1 (M) ≤ 󵄩󵄩󵄩∇u0 (x)󵄩󵄩󵄩 + vol M = ρ1 } is a bounded subset of the set E in H 1 (M) for S(t). By Lemma 1.12.4, we have E 󵄩󵄩 󵄩 󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 2 (M) ≤ 1 , t

t > 0,

where constant E1 depends on ‖∇u0 (x)‖H 1 (M) . This implies that the semigroup operator S(t) is completely continuous in H 1 (M), t > 0. Then by a theorem in [197], we know that the semigroup S(t) produces a compact attractor in H 1 (M), A = ⋂ ⋃ S(t)B1 = ω(B1 ). s≥0 t≥s

Now we estimate the upper and lower Hausdorff and fractal dimensions of the attractor A . We consider the linear variational problem of (1.12.1)–(1.12.2): v(t) + L(u(t))v = 0, v(0) = v0 (x),

(1.12.54) (1.12.55)

where L(u(t))v = −α1 Δ M v − α1 |∇u|2 v − 2α1 ∇uu∇v − α2 A󸀠 (u)Δ M uv − α2 A(u)Δ M v, 0 A(u) = ( u2 −u2

−u3 0 u1

u2 −u1 ) , 0

(1.12.56)

u1 u = ( u2 ) . u3

Since the solution of problem (1.12.1)–(1.12.2) is sufficiently smooth, it is easy to prove that the linear problem (1.12.54)–(1.12.55) has a unique global solution v(x, t) ∈ L∞ (0, ∞; H 2 (M)), provided that the initial value v0 (x) is sufficiently smooth. In fact, for the linear equation (1.12.54), its main part is m

m

α,β=1

α,β=1

∑ Dα (aαβ (x, t, u)Dβ u) = ∑ Dα √γ(γ αβ Dβ u + u × γ αβ Dβ u),

(1.12.57)

160 | 1 Attractor and its dimension estimation ij

the corresponding system aα,β is ij

aα,β = √γγ αβ gij A#, where α1 g = (gij ) = ( α2 u3 −α2 u2

−α2 u3 α1 α2 u1

α2 u2 −α2 u1 ) . α1

(1.12.58)

Thus we have s

m

m

ij

∑ ∑ aα,β (x, t, η)ξ α ξ β ζi ζj = α1 |ζ |2 ∑ γ αβ √γξ α ξ β > 0

i,j=1 α,β=1

α,β=1

(1.12.59)

for any (x, t, η) ∈ M × [0, T] × R3 , ξ = (ξ 1 , . . . , ξ m ) ∈ Rm \ {0}, ζ = (ζ1 , ζ2 , ζ3 ) ∈ R3 \ {0}. Thus the problem (1.12.54)–(1.12.55) is solvable. Let Gt be the solution operator, v(t) = Gt v0 . It is easy to prove that the semigroup St v0 is differentiable in L2 (M). The Fréchet derivative of St v0 exists. And Gt v0 = St󸀠 v0 . In order to estimate the dimension of A , we need the following theorems. Theorem 1.12.4 ([77]). Let (M, g) be an n-dimensional Riemannian manifold. For any p such that max{1,

n n } 0, j=1

m 4

−

C1 H] − C2 H α1

162 | 1 Attractor and its dimension estimation where 󵄩 󵄩 󵄩 󵄩 H = 2α1 ‖∇u‖∞ ‖∇u‖2 + 2α1 ‖u‖∞ 󵄩󵄩󵄩∇2 u󵄩󵄩󵄩2 + α2 󵄩󵄩󵄩A󸀠 (u)󵄩󵄩󵄩∞ ‖∇u‖2 n

4n

n

2− 4n

2−n (4−n)(2−n) 2+n C C 2 2 J ≥ max{(C3 ( )) ⋅ [ 1 H + 1] , (C3 ( )) [ 2 H] n α1 n α1

},

and where we used the following inequality [197]: J

2

∑ j m ≥ C3 ( j=1

2 m2 +1 )J , m

with C1 , C2 depending on the constant of manifold M. Now we estimate a lower bound of dimension for the attractor A . Consider a Banach space E and a continuous operator semigroup S(t) : E → E, S(t + s) = S(t)S(s), S(0) = I

∀s, t > 0,

(the identity operator on E).

(1.12.64) (1.12.65)

Assume the semigroup (t, u0 ) → S(t)u0 : R+ × E → E is continuous. Let Z be a fixed point of S(t), that is, S(t)Z = Z,

∀t ∈ R+ .

(1.12.66)

Assume the mapping u → S(t)u is Fréchet differentiable in a neighborhood of Z, say Θ, and its differential S󸀠 (t) satisfies the Hölder condition 󵄩󵄩 󸀠 󵄩 󸀠 α 󵄩󵄩S (t)u1 − S (t)u2 󵄩󵄩󵄩 ≤ C3 (T)‖u1 − u2 ‖ ,

0 < α ≤ 1, ∀u1 , u2 ∈ Θ, ∀t ∈ [0, T],

(1.12.67)

where constant C3 depends on T, but not on u1 , u2 . Definition 1.12.1. We say that the fixed point Z is hyperbolic, provided that the following conditions are satisfied: (1) The spectrum σ(S󸀠 (t)) of S(t) does not intersect the circle {λ ∈ C, |λ| = 1}. (2) E+ has finite dimension, where E+ = E+ (Z) and E− = E− (Z) are the linear invariant subspaces of E, which correspond to the subsets of S󸀠 (t) included in {λ ∈ C, |λ| > 1} and {λ ∈ C, |λ| < 1}, respectively. E+ and E− are independent of time t. Definition 1.12.2. The unstable manifold μ+ (Z) of Z is the set of points u∗ ∈ H (may be empty), which totally belongs to the complete path {u(t), t ∈ R} and satisfies u(t) → u0

as t → −∞,

μ+ (Z) = {u0 ∈ E, ∀t ≤ 0, ∃u(t) ∈ S(−t)−1 u0 , and u(t) → Z, when t → −∞}.

1.12 The Landau–Lifshitz equation on a Riemannian manifold | 163

The stable manifold μ− (Z) of Z is the set of points u∗ ∈ H (may be empty), which belongs to the complete path {u(t), t ∈ R}, u∗ = u(t0 ) and satisfies u(t) = S(t − t0 )u∗ → u0 ,

μ− (Z) = {u0 ∈ E,

t → ∞,

∀t ≤ 0, ∃u(t) ∈ S(−t)−1 u0 , and S(t)u0 → Z, when t → ∞}.

From the definition, we directly deduce that S(t)μ+ (Z) = μ+ (Z),

S(t)μ− (Z) = μ− (Z),

∀t ≥ 0.

Definition 1.12.3. A heteroclinic orbit is an orbit which connects the unstable manifold of a steady point u∗ to another stable manifold of a steady solution u∗∗ , u∗∗ ≠ u∗ ; while if u∗∗ = u∗ , such an orbit is called a homoclinic orbit. The points belonging to a heteroclinic (homoclinic) orbit are called heteroclinic (homoclinic) points. Consider the ball with radius R > 0, ΘR (Z) = {y ∈ E, ‖y − Z‖0 ≤ R}, and let μR− (Z) = {u0 ∈ Θ(Z), ∀n ∈ N, ∃un ∈ Sn (u0 ) ∩ ΘR (Z), and u(t) → Z when n → ∞}. (1.12.68) Theorem 1.12.6 ([51]). Let E be a Banach space, and suppose S(t) is a semigroup operator, t ∈ R+ , which satisfies assumptions (1.12.64), (1.12.65) and (1.12.67). Suppose S(t) possesses a unique global attractor A , and Z ∈ A is the hyperbolic fixed point of S(t). Then we have A ⊃ μ− (Z) ⊃ μR+ (Z),

(1.12.69)

for R > 0 small enough. Now we consider the following initial value problem for the Landau–Lifshitz equation: Zt = −α1 (Z × (Z × Δ M Z)) + Z × JZ, 𝜕Z 󵄨󵄨󵄨󵄨 Z|t = 0 = Z0 (x), x ∈ M, 󵄨 = 0, 𝜕v 󵄨󵄨󵄨𝜕M

(1.12.70) (1.12.71)

where M is an unbounded compact Riemannian manifold and J = diag(J1 , J2 , 0), α1 > 0. By Theorem 1.12.6, we get a lower bound of the dimension for the attractor of problem (1.12.70)–(1.12.71). Theorem 1.12.7. Assume that A is the global attractor of problem (1.12.70)–(1.12.71), J1 , J2 < 0, then we have n

dim A ≥ Cα− 2 ,

(1.12.72)

where dim A is Hausdorff or fractal dimension of A and C is a positive constant.

164 | 1 Attractor and its dimension estimation Proof. It is easy to see that Z = (0, 0, 1) is the fixed point of S(t) generated by problem (1.12.70)–(1.12.71), that is, Z = (0, 0, 1) satisfies the equation − A1 (Z) = −α1 (Z × (Z × Δ M Z)) + Z × JZ = 0.

(1.12.73)

The variational problem for equation (1.12.73) is − A󸀠1 (Z)v = α1 Δ M v + Z × Jv = 0.

(1.12.74)

Let ζ be the eigenvalue of matrix B(Z)ν = Z × Jv = ζv, where 0 B(Z) = (J1 0

−J2 0 0

0 0) . 0

Hence ζ 2 + J1 J2 = 0,

ζ 2 = −√−J1 J2 .

ζ1 = √−J1 J2 > 0,

Let λk , k ∈ N be the eigenvalues of the operator −Δ M of M, that is, 𝜕ψk = 0, 𝜕v 𝜕M 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ , λk → ∞,

−Δ M ψk = λk ψk ,

(1.12.75) k → ∞.

Let μk , k ∈ N stand for the characteristic value sequence of the linear operator − A󸀠1 (Z)ωk = α1 Δ M ωk + B(Z)ωk = μk ωk ,

(1.12.76)

where ωk (x) = ψk (x)pk , pk ∈ C 3 . Then we have (α1 λk + B(Z))pk = μk pk .

(1.12.77)

If μk is the root of equation det(α1 λk + B(Z) − μk I) = 0,

Re μk > 0,

(1.12.78)

then there exists a nonzero solution pk . By the assumption of the theorem, when α1 ≠ 0, there exists at least one root ζ1 = √−J1 J2 > 0. Hence when α1 λk < δ, there exists a root of equation (1.12.78) μk , Re μk > 0, where δ is a small enough constant. n Then due to the asymptotic behavior of characteristic values λk , λk ∼ Ck − 2 , we get the inequality n

− n2

1 ≤ k ≤ C1 δ 2 α 1

−n

= C2 α1 2 .

By Theorem 1.12.6, we get −n

dim A ≥ Cα1 2 .

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

|

165

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3 We consider the following Klein–Gordon–Schrödinger (KGS) equations [104, 136] iψt + Δψ = −ϕψ, 2

(1.13.1)

2

ϕtt − Δϕ + μ ϕ = |ψ| ,

(1.13.2)

where ψ(x, t) is a complex nuclear field; ϕ is a real meson field; and μ is the quality of a meson. The Cauchy and initial–boundary value problems of KGS equations have been studied by many authors, e. g., in [9, 70, 106, 139], and so on. In [9], the existence of a global solution is obtained by use of the Lp –Lq estimates for Schrödinger equations. In [70], the asymptotic behavior of multidimensional KGS equations has been studied. The authors considered the initial–boundary value problem and the existence of a three-dimensional strong solution for KGS equations in [106], which was improved in [139]. When we consider the effect of damping, we have the following dissipation KGS equations: iψt + Δψ + iαψ + ϕψ = f ,

(1.13.3)

2

ϕtt + (I − Δ)ϕ + βϕt = |ψ| + g,

(1.13.4)

where α, β are positive; f (x), g(x) are known functions; f is complex; and g is real. The long time asymptotic behavior for equations (1.13.3)–(1.13.4) in a bounded domain Ω was obtained in [18, 161]. In [18], Biler proved the existence of a global attractor in the weak topology of H01 × H01 (Ω) and the finiteness of Hausdorff dimension. In [161], the authors proved the existence of a finite dimensional global attractor in H 2 ∩ H01 (Ω) × H 2 ∩ H01 (Ω). Here we consider the KGS equations (1.13.3)–(1.13.4) as in [8] in the whole space R3 and the initial conditions ϕ(0, x) = ψ0 (x),

ϕ(0, x) = ϕ0 (x),

ϕt (0, x) = ψ1 (x),

x ∈ R3 .

(1.13.5)

We prove that there exists a global attractor in H 2 × H 2 × H 1 (R3 ) for problem (1.13.3)–(1.13.5), which attracts bounded sets in H 3 × H 3 × H 2 (R3 ) with respect to H 2 × H 2 × H 1 (R3 ). 󸀠 Because the domain is unbounded, the embedding of H s (R3 ) into H s (R3 ) for s > s󸀠 is not compact. In order to overcome this difficulty, we use the noncompact Kuratowski α measure to prove the asymptotic smoothness of the semigroup S(t), and then reuse the theory in [137] to prove the existence of the attractor. Let θ = ϕt + δϕ, with δ being an unknown positive constant. Equations (1.13.3)–(1.13.5) are equivalent to iψt + Δψ + iαψ + ϕψ = f ,

ϕt + βϕ = θ,

(1.13.6) (1.13.7)

166 | 1 Attractor and its dimension estimation θt + (β − δ)θ + (1 − δ(β − δ) − Δ)ϕ = |ψ|2 + g,

(1.13.8)

with the initial condition (ψ, ϕ, θ)(0, x) = (ψ0 , ϕ0 , θ0 )(x),

x ∈ R3 ,

(1.13.9)

β

where θ0 = δϕ0 + ϕ1 . If δ ≤ min{ 2 , 2β1 }, then A = 1 − δ(β − δ) − Δ is a positive, self-adjoint second-order elliptic operator. Denote 1

1

H = L2 × H 2 × H − 2 (R3 ), V = H 1 × H 1 × L2 (R3 ),

X = H 2 × H 2 × H 1 (R3 ),

Y = H 3 × H 3 × H 2 (R3 ). Then Y 󳨅→ X 󳨅→ V is a continuous embedding. Lemma 1.13.1. Let f ∈ L∞ (R+ ; L2 (R3 )). Then ψ ∈ L∞ (R+ ; L2 (R3 )), and it satisfies ‖|f |‖2 󵄩󵄩 󵄩2 2 (1 − exp(−αt)). 󵄩󵄩ψ(t)󵄩󵄩󵄩 ≤ ‖ψ0 ‖ exp(−αt) + α2 Hence, there exists t1 (R) > 0 such that ‖|f |‖2 󵄩󵄩 󵄩2 , 󵄩󵄩ψ(t)󵄩󵄩󵄩 ≤ 1 + α2

∀t ≥ t1 (R),

provided ‖ψ0 ‖ < R, where ‖|f |‖ means the norm of f in L∞ (R− ; L2 (R3 )). Proof. Taking the inner product of (1.13.6) and ψ in L2 (R3 ), then its imaginary part, we obtain α ‖f ‖2 1 d ‖ψ‖2 + α‖ψ‖2 = Im(f , ψ) ≤ ‖f ‖‖ψ‖ ≤ ‖ψ‖2 + . 2 dt 2 2α By Gronwall inequality, we get the claim of the lemma. Lemma 1.13.2. Let f , g ∈ L∞ (R+ ; L2 (R3 )). Then for (ψ0 , ϕ0 , θ0 ) ∈ V, solution (ψ, ϕ, θ) ∈ L∞ (R+ , V). Furthermore, there exists t2 (R) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩(ψ, ϕ, θ)󵄩󵄩󵄩V ≤ C,

∀t ≥ t2 (R),

whenever ‖(ψ0 , ϕ0 , θ0 )‖V ≤ R. Proof. Taking the inner product of (1.13.6) and −(ψt + αψ) in L2 , then its real part, we obtain 1 d ‖∇ψ‖2 + α‖∇ψ‖2 − Re(ϕψ, ψt ) − α Re(ϕψ, ψ) 2 dt d = − Re(f , ψ) − Re(ft , ψ) − α Re(f , ψ). dt

(1.13.10)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

|

167

Noting that − Re(ϕψ, ψt ) = −

1 1 d (ϕ, |ψ|2 ) + (ϕt , |ψ|2 ), 2 dt 2

from equation (1.13.10) we get 1 d (‖∇ψ‖2 − ∫ ϕ|ψ|2 dx + 2 Re ∫ f ψdx) + α‖∇ψ‖2 2 dt 1 − α ∫ ϕ|ψ|2 dx + α ∫ f ψdx − ∫ ϕt |ψ|2 dx = 0. 2

(1.13.11)

Taking the inner product of (1.13.8) and θ in L2 , then using equation (1.13.7), we get 1 d (‖θ‖2 + (1 − δ(β − δ))‖ϕ‖2 + ‖∇ϕ‖2 ) 2 dt + (β − δ)‖θ‖2 + δ(1 − δ(β − δ))‖ϕ‖2 + δ‖∇ϕ‖2 = ∫ ϕt |ψ|2 dx + δ ∫ ϕ|ψ|2 dx + ∫ gθdx.

(1.13.12)

Then multiplying (1.13.11) by 2 and adding (1.13.12), we get d H (t) + I1 (t) = 0, dt 1

(1.13.13)

where H1 (t) = 2‖∇ψ‖2 − 2 ∫ ϕ|ψ|2 dx + 2 Re ∫ f ψdx + ‖θ‖2 + (1 − δ(β − δ))‖ϕ‖2 + ‖∇ϕ‖2 , I1 (t) = 4‖∇ψ‖2 − 2(2α + δ) ∫ ϕ|ψ|2 dx + 4 Re ∫ f ψdx + 2(β − δ)‖θ‖2 + 2δ(1 − δ(β − δ))‖ϕ‖2 + 2δ‖∇ϕ‖2 − 2 ∫ gθdx.

(1.13.14)

Since H 1 (R3 ) 󳨅→ L6 (R3 ) 1

1

‖ψ‖3 ≤ C‖ψ‖ 2 ‖∇ψ‖ 2 , for any ε1 , ε2 > 0 we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 󵄨󵄨 ∫ ϕ|ψ| dx󵄨󵄨󵄨 ≤ C‖ψ‖6 ‖ψ‖3 ‖ψ‖ 󵄨󵄨 󵄨󵄨 3

1

≤ C‖∇ϕ‖‖ψ‖ 2 ‖∇ψ‖ 2 ≤ ε1 ‖∇ψ‖2 + ε2 ‖∇ϕ‖2 + C(ε1 , ε2 )‖ψ‖6 , (1.13.15)

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ f ψdx󵄨󵄨󵄨 ≤ ‖f ‖‖ψ‖. 󵄨󵄨 󵄨󵄨

168 | 1 Attractor and its dimension estimation In equation (1.13.15), we take ε1 = 21 , ε2 = 41 , and then obtain H1 (t) ≥ ‖∇ψ‖2 + ‖θ‖2 + (1 − δ(β − δ))‖ϕ‖2 1 + ‖∇ϕ‖2 − C‖ψ‖6 − 2‖f ‖‖ψ‖, 2 H1 (t) ≤ 3‖∇ψ‖2 + ‖θ‖2 + (1 − δ(β − δ))‖ϕ‖2 3 + ‖∇ϕ‖2 + C‖ψ‖6 + 2‖f ‖‖ψ‖. 2 Taking ε1 =

α ,ε 2α+δ 2

=

δ 2(2α+δ)

(1.13.16)

in equation (1.13.15), together with

󵄨󵄨 󵄨󵄨 β−δ 2 1 󵄨󵄨 󵄨 ‖θ‖ + ‖g‖2 , 󵄨󵄨 ∫ gθdx 󵄨󵄨󵄨 ≤ ‖g‖‖θ‖ ≤ 󵄨󵄨 󵄨󵄨 2 2(β − δ) we know that I1 (t) ≥ 2α‖ψ‖2 + (β − δ)‖θ‖2 + δ(1 − δ(β − δ))‖ϕ‖2 1 + δ‖∇ψ‖2 − C‖ψ‖6 − 4‖f ‖‖ψ‖ − ‖g‖2 . β−δ

(1.13.17)

From equations (1.13.16) and (1.13.17), we can find β1 > 0 such that β1 H1 (t) ≤ I1 (t) + C‖ψ‖6 + C‖f ‖‖ψ‖ + C‖g‖2 .

(1.13.18)

Hence, from equations (1.13.13) and (1.13.18), we get d H (t) + β1 H1 (t) ≤ C‖ψ‖6 + C‖f ‖‖ψ‖2 + C‖g‖2 ≜ K1 . dt 1 By Gronwall inequality, we get H1 (t) ≤ H1 (0)e−β1 t +

K1 (1 − e−β1 t ), β1

(1.13.19)

and then from equations (1.13.16) and (1.13.19) obtain the claim of the lemma. Lemma 1.13.3. Let f , g ∈ L∞ (R+ ; H 1 (R3 )). Then for (ψ0 , ϕ0 , θ0 ) ∈ X, solution (ψ, ϕ, θ) ∈ L∞ (R+ , X). Furthermore, there exists t3 (R) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩(ψ, ϕ, θ)󵄩󵄩󵄩X ≤ C,

∀t ≥ t2 (R),

whenever ‖(ψ0 , ϕ0 , θ0 )‖X ≤ R. Proof. Taking the inner product of (1.13.6) and Δψt + αΔψ, then its real part, we get 1 d ‖Δψ‖2 + α‖Δψ‖2 + Re ∫ ϕψΔψt dx + α Re ∫ ϕψΔψdx 2 dt d = − Re ∫ ∇f ∇ψdx − α Re ∫ ∇f ∇ψdx. dt

(1.13.20)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

|

169

Noting that Re ∫ ϕψΔψt dx =

d ∫ Re ϕψΔψdx − Re ∫ ϕt ψΔψdx − Re ∫ ϕψt Δψdx, dt

from equation (1.13.7) we get − Re ∫ ϕt ψΔψdx = − Re ∫ θψΔψdx + δ Re ∫ ϕψΔψdx. Through equation (1.13.6), we have ψt = −i(f − Δψ − iαψ − ϕψ), − Re ∫ ϕψt Δψdx = Re ∫ iϕ[f − Δψ − iαψ − ϕψ]Δψdx = − Im ∫ ϕf Δψdx + α Re ∫ ϕψΔψdx + Im ∫ ϕ2 ψΔψdx. From equation (1.13.20) we get 1 d (‖Δψ‖2 + 2 Re ∫ ϕψΔψdx + 2 Re ∫ ∇f ∇ψdx) 2 dt + α‖Δψ‖2 + (2α + δ) Re ∫ ϕψΔψdx + α Re ∫ ∇f ∇ψdx − Re ∫ θψΔψdx − Im ∫ ϕf Δψdx + Im ∫ ϕ2 ψΔψdx = 0.

(1.13.21)

Taking the inner product of (1.13.8) and −Δθ, we get 1 d (‖∇θ‖2 + (1 − δ(β − δ))‖∇ϕ‖2 + ‖Δϕ‖2 ) 2 dt + (β − δ)‖∇θ‖2 + δ(1 − δ(β − δ))‖∇ϕ‖2 + δ‖Δϕ‖2 = − ∫ θΔ|ψ|2 dx + ∫ ∇g∇θdx = −2 Re ∫ θψΔψdx − 2 ∫ θ|Δψ|2 dx + ∫ ∇g∇θdx.

(1.13.22)

Let H2 (t) = ‖Δψ‖2 + 2 Re ∫ ϕψΔψdx + 2 Re ∫ ∇f ∇ψdx 1 1 1 + ‖∇θ‖2 + (1 − δ(β − δ))‖∇ϕ‖2 + ‖Δϕ‖2 2 2 2

I2 (t) = 2α‖Δψ‖2 + 2(α + δ) Re ∫ ϕψΔψdx

+ 2α Re ∫ ∇f ∇ψdx − 2 Im ∫ ϕf Δψdx + 2 Im ∫ ϕ2 ψΔψdx + (β − δ)‖∇θ‖2

(1.13.23)

170 | 1 Attractor and its dimension estimation + δ(1 − δ(β − δ))‖∇ϕ‖2 + δ‖Δϕ‖2 + 2 ∫ θ|∇ψ|2 dx − ∫ ∇g∇θdx.

(1.13.24)

Then multiplying (1.13.21) by 2 and adding (1.13.22), we get d H (t) + I2 (t) = 0. dt 2

(1.13.25)

We estimate the terms of H2 (t) and I2 (t) with undetermined signs as follows: 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2 2 2 󵄨󵄨 Re ∫ ϕψΔψdx 󵄨󵄨󵄨 ≤ ‖ϕ‖4 ‖ψ‖4 ‖Δψ‖ ≤ ε3 ‖Δψ‖ + C(ε3 )‖∇ϕ‖ ‖∇ψ‖ , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 Re ∫ ∇f ∇ψdx 󵄨󵄨󵄨 ≤ ‖∇f ‖‖∇ψ‖, 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Im ∫ ϕf Δψdx 󵄨󵄨󵄨 ≤ ‖ϕ‖4 ‖f ‖4 ‖Δψ‖ ≤ 1 ‖Δψ‖2 + C‖∇f ‖2 ‖∇ϕ‖2 , 󵄨󵄨 󵄨󵄨 4 󵄨 󵄨 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨󵄨 2 4 2 2 2 󵄨󵄨 Im ∫ ϕ ψΔψdx 󵄨󵄨󵄨 ≤ ‖ϕ‖6 ‖ψ‖6 ‖Δψ‖ ≤ ‖Δψ‖ + C‖∇ϕ‖ ‖∇ϕ‖ , 󵄨󵄨 󵄨󵄨 4 󵄨󵄨 󵄨󵄨 3 1 󵄨󵄨 󵄨 2 2 󵄨󵄨 ∫ θ|∇ψ| dx 󵄨󵄨󵄨 ≤ ‖θ‖‖∇ϕ‖4 ≤ C‖θ‖‖∇ψ‖ 2 ‖Δψ‖ 2 󵄨󵄨 󵄨󵄨 1 ≤ ‖Δψ‖2 + C‖θ‖4 ‖∇ψ‖2 , 4 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨 ∫ ∇g∇θdx 󵄨󵄨󵄨 ≤ ‖∇θ‖‖∇g‖ ≤ β − δ ‖∇θ‖2 + ‖∇g‖2 . 󵄨󵄨 󵄨󵄨 2 2(β − δ) 󵄨 󵄨 When estimating H2 (t), we take ε3 = 41 ; and when estimating I2 (t), we take ε3 = Then from the above inequalities, we have 1 1 H2 (t) ≥ ‖Δψ‖2 + [‖∇θ‖2 + ‖Δϕ‖2 2 4 + (1 − δ(β − δ))‖∇ϕ‖2 − C‖∇ϕ‖2 ‖∇ψ‖2 − 2‖∇f ‖‖∇ψ‖], 3 H2 (t) ≤ ‖Δψ‖2 + ‖∇θ‖2 + ‖Δϕ‖2 2 + (1 − δ(β − δ))‖∇ϕ‖2 + C‖∇ϕ‖2 ‖∇ψ‖2 + 2‖∇f ‖‖∇ψ‖, 1 δ I2 (t) ≥ α‖Δψ‖2 + (β − δ)‖∇θ‖2 + ‖Δϕ‖2 2 2 + δ(1 − δ(β − δ))‖∇ϕ‖2 − C(ε3 )‖∇ϕ‖2 ‖∇ψ‖2

α . 2(2α+δ)

(1.13.26)

− C‖∇f ‖‖∇ψ‖ − C‖∇f ‖2 ‖∇ϕ‖2

− C‖∇ϕ‖4 ‖∇ψ‖2 − C‖θ‖4 ‖∇ψ‖2 − C‖g‖2 . Hence, there exists a constant β2 > 0 such that β2 H2 (t) ≤ I2 (t) + C(‖ϕ‖H 1 , ‖ψ‖H 1 , ‖θ‖, ‖f ‖H 1 , ‖g‖H 1 )

(1.13.27)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

and

| 171

d H (t) + β2 H2 (t) ≤ K2 , dt 2

where K2 ≜ C(‖ϕ‖H 1 , ‖ψ‖H 1 , ‖θ‖, ‖f ‖H 1 , ‖g‖H 1 ). Hence by Gronwall inequality, we have H2 (t) ≤ H2 (0)e−β2 t +

K2 (1 − e−βt ). β2

(1.13.28)

Now from equations (1.13.26) and (1.13.28), we get the claim. Corollary 1.13.1. Let f , g ∈ L∞ (R+ ; H 1 (R3 )). Then for (ψ0 , ϕ0 , θ0 ) ∈ X, solution (ψ, ϕ, θ) ∈ L∞ (R+ × R3 ). Lemma 1.13.4. Let f , g ∈ L∞ (R+ ; H 2 (R3 )). Then for (ψ0 , ϕ0 , θ0 ) ∈ Y, solution (ψ, ϕ, θ) ∈ L∞ (R+ , Y). Furthermore, there exists t4 (R) > 0 such that for any t ≥ t4 (R), 󵄩󵄩 󵄩 󵄩󵄩(ψ, ϕ, θ)󵄩󵄩󵄩Y ≤ C, whenever ‖(ψ0 , ϕ0 , θ0 )‖Y ≤ R. Proof. The claim follows using similar arguments as in the proof of Lemma 1.13.3. From the estimates above, we get the following results: Theorem 1.13.1. Let f , g ∈ L∞ (R+ ; H 1 (R3 )). Then for (ψ0 , ϕ0 , θ0 ) ∈ X, there exist a unique solution (ψ, ϕ, θ) ∈ L∞ (R+ , X) for problem (1.13.6)–(1.13.9). Furthermore, the solution operator S(t) from X to X is continuous, and it has a bounded absorbing set B1 ⊂ X. Proof. We firstly prove the existence of a local solution by the standard iterative procedure. Then with the aid of the a priori estimate, by Theorems 1.13.1–1.13.3, we may extent the local solution to the global one. The uniqueness of the solution can be deduced from the continuity of S(t) : X → X. In fact, let (ψk , ϕk , θk ), k = 1, 2, be two solutions of problem (1.13.6)–(1.13.9) with the initial value (ψ0k , ϕ0k , θ0k ). Let (ψ, ϕ, θ) = (ψ1 − ψ2 , ϕ1 − ϕ2 , θ1 − θ2 ), (ψ0 , ϕ0 , θ0 ) = (ψ01 − ψ02 , ϕ01 − ϕ02 , θ01 − θ02 ). Then (ψ, ϕ, θ) satisfies iψt + Δψ + iαψ = −ϕ1 ψ − ϕψ2 , ϕt + δϕ = θ,

θt + (β − δ)θ + (1 − δ(β − δ) − Δ)ϕ = ψ1 ψ + ψψ2 ,

(ψ, ϕ, θ)|t=0 = (ψ0 , ϕ0 , θ0 )(x),

x ∈ R3 .

It is easily proved that ‖ψ‖2 + ‖Δψ‖2 + ‖θ‖2 + ‖∇θ‖2 + (1 − δ(β − δ))‖ϕ‖2 + (2 − δ(β − δ))‖∇θ‖2 + ‖Δθ‖2 ≤ C(‖ψ0 ‖2 + ‖Δψ0 ‖2 + ‖θ0 ‖2 + ‖∇θ0 ‖2 + ‖ϕ0 ‖2 + ‖Δϕ0 ‖2 )eCt . From this we get the continuity of S(t). The existence of a bounded absorbing set has been proved in Lemma 1.13.3.

172 | 1 Attractor and its dimension estimation Remark 1.13.1. We can prove using the approximation method that the problem (1.13.6)–(1.13.9) has the solution (ψ, ϕ, θ) ∈ V when (ψ0 , ϕ0 , θ0 ) ∈ V. But the continuity of S(t) in V is unknown. However, S(t) is continuous from V to H. Hence, the solution is unique in V. Theorem 1.13.2. Let f , g ∈ L∞ (R+ ; H 2 (R3 )). Then for any (ψ0 , ϕ0 , θ0 ) ∈ Y, there exists a unique solution (ψ, ϕ, θ) ∈ L∞ (R+ , Y) for problem (1.13.6)–(1.13.9). Furthermore, the solution operator S(t) from Y to Y is continuous, and has a bounded absorbing set B2 ⊂ Y. Proof. The argument is similar to the proof of Theorem 1.13.1, hence omitted. In the following, assume that f , g ∈ H 2 (R3 ) are independent of t. Then S(t) generates a semigroup. Let B ⊂ Y be a bounded set, then S(t)B ⊂ Y is bounded, too. We decompose S(t), so that we can use the Kuratowski noncompact α measure to prove the asymptotic smoothness of S(t). In other words, we decompose S(t) into two parts, S1 (t) and S2 (t), where, when t → ∞, α(S1 (t)B) → 0 and S2 (t) is relatively compact in X. For a set A ⊂ X, its noncompact α measure is defined as α(A) = inf{d | there exists a cover of A with finitely many balls of radius < d}. Hence α(S(t)B) ≤ α(S1 (t)B) + α(S2 (t)B) = α(S1 (t)B) → 0,

t → ∞.

Let B ⊂ Y, supξ ∈B ‖ξ ‖Y ≤ R, and assume (ψ, ϕ, θ) = S(t)(ψ0 , ϕ0 , θ0 ) ∈ B is the solution of problem (1.13.6)–(1.13.9) with the initial value (ψ0 , ϕ0 , θ0 ) ∈ B. We have shown that (ψ, ϕ, θ) is uniformly bounded in Y. Let χL(x) ∈ C0∞ (R3 ), 0 ≤ χL ≤ 1 satisfy χL(x) = {

1, 0,

|x| ≤ L, |x| ≥ 1 + L.

Then for any η ∈ (0, 1), there exists L(η) > 0 (large enough) such that ‖f − fη ‖2H 2 (R3 ) ≤ η,

󵄩󵄩 2 󵄩󵄩|ψ| −

‖g − gη ‖2H 2 (R3 ) 󵄩2 |ψ|2 χL(η) 󵄩󵄩󵄩H 2 (R3 )

≤ η,

fη = fχL(η) , gη = gχL(η) ,

≤ η.

Suppose (ψη , ϕη , θη ) is the solution of the following problem: iψηt + Δψη + iαψη − iηΔψη + ϕψη = f − fη − iηΔψ,

(1.13.29)

ψηt + (I − Δ)ϕη + βψηt = (|ψ| + g)(I − χL(η) ),

(1.13.30)

2

ψη (0, x) = ψ0 (x),

ϕη (0, x) = ϕ0 (x),

ϕηt (0, x) = ϕ1 (x) = θ0 − δϕ0 ,

x ∈ R3 ,

(1.13.31)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

| 173

θη = ϕηt + δϕη . Let S1η (t)(ψ0 , ϕ0 , θ0 ) = (ψη , ϕη , θη ). Then (uη , vη , ωη ) = S2η (t)(ψ0 , ϕ0 , θ0 ) = S(t)(ψ0 , ϕ0 , θ0 ) − S1η (t)(ψ0 , ϕ0 , θ0 ) = (ψ − ψη , ϕ − ϕη , θ − θη ) is the solution of problem iuηt + Δuη + iαuη − iηΔuη + ϕψη = fη (x),

(1.13.32)

vηtt + (I − Δ)vη + βvηt = |ψ| χL(η) + gη (x),

(1.13.33)

2

uη (0, x) = 0,

vη (0, x) = vηt (0, x) = 0,

x ∈ R3

(1.13.34)

and ωη = vηt + δvη . We need the following lemma: Lemma 1.13.5. There exist a constant C > 0 and an increasing function ω(η) (ω(0) = 0) such that the solution of (1.13.29)–(1.13.31) satisfies ‖ψη ‖H 2 , ‖ϕη ‖H 2 , ‖ϕηt ‖H 1 ≤ C,

∀0 < η ≤ 1, t ≥ 0,

‖ψη ‖H 2 , ‖ϕη ‖H 2 , ‖ϕηt ‖H 1 ≤ ω(η),

∀0 < η ≤ 1, t ≥ t∗ (∃t∗ > 0).

Proof. Taking the inner product of (1.13.29) and 2ψη , and taking its imaginary part, we obtain d ‖ψ ‖2 + 2α‖ψη ‖ + 2η‖∇ψη ‖2 dt η = 2 Im(f − fη , ψη ) + 2η Im(∇ψ, ∇ψη )

≤ C‖f − fη ‖2 + α‖ψη ‖2 + η‖∇ψη ‖2 + η‖∇ψ‖2 .

Hence

d ‖ψ ‖2 + α‖ψη ‖2 + η‖∇ψη ‖2 ≤ Cη, dt η

and by Gronwall inequality we have ‖ψη ‖ ≤ ‖ψ0 ‖2 e−αt + Therefore

‖ψη ‖2 ≤ C,

Cη (1 − e−αt ). α

∀t ≥ 0, 0 < η ≤ 1.

(1.13.35)

Taking t1 (R) > 0 such that e−αt1 ‖ψ0 ‖2 ≤ e−αt1 R2 < η, we deduce that ‖ψη ‖2 ≤ Cη, Thus ‖ψη ‖ ≤ C √η, t ≥ t1 .

∀t ≥ t1 .

(1.13.36)

174 | 1 Attractor and its dimension estimation Taking the inner product of (1.13.29) and Δ2 ψη , and taking its real part, we have 1 d ‖Δψη ‖2 + α‖Δψη ‖2 + η‖∇Δψη ‖2 2 dt = − Im(Δϕψη , Δψη ) − 2 Im(∇ϕ∇ψη , Δψη ) + Im(Δ(f − fη ), Δψη ) + η Im(∇Δψ, ∇Δψη ) ≤ C‖Δϕ‖‖ψη ‖∞ ‖Δψη ‖ + 2‖∇ϕ‖4 ‖∇ψη ‖4 ‖Δψη ‖ 󵄩 󵄩 + 󵄩󵄩󵄩Δ(f − fη )󵄩󵄩󵄩‖Δψη ‖ + η‖∇Δψ‖‖∇Δψη ‖ 7

1

1

15

≤ C‖Δϕ‖‖ψη ‖ 4 ‖Δψη ‖ 4 + C‖ϕ‖H 2 ‖ψη ‖ 8 ‖Δψη ‖ 8 3 1 α + ‖Δψη ‖2 + ‖Δ(f − fη )‖2 + η‖∇Δψη ‖2 + η‖∇Δψ‖2 6 2α 4 η α 2 2 ≤ ‖Δψη ‖ + ‖∇Δψη ‖ + C‖Δϕ‖8 ‖ψη ‖2 2 2 16 + C‖ϕ‖H 2 ‖ψη ‖2 + C‖f − fη ‖2H 2 + η‖∇Δψ‖2 . Hence d 2 ‖Δψη ‖2 + α‖Δψη ‖2 + η‖∇Δψη ‖2 ≤ Cη + C(‖ϕ‖16 H 2 + 1)‖ψη ‖ . dt By Gronwall inequality, we have ‖Δψη ‖ ≤ ‖Δψ0 ‖2 e−αt +

C(‖ϕ‖16 + 1) H2 2

‖ψη ‖2 (1 − e−αt ).

(1.13.37)

From equations (1.13.35), (1.13.36) and (1.13.37), ‖Δψη ‖2 ≤ C,

∀t ≥ 0, 0 < η ≤ 1.

Take t2 = t2 (R) ≥ t1 such that e−αt2 ‖Δψ0 ‖2 ≤ e−αt2 R2 < η. Then by equations (1.13.37) and (1.13.36), we have ‖Δψη ‖2 ≤ Cη,

t ≥ t2 .

Hence ‖Δψη ‖ ≤ C, t ≥ t2 . Now we estimate ϕη . Taking the inner product of (1.13.30) and 2ϕηt , we arrive at 1 d (‖ϕηt ‖2 + ‖ϕη ‖ + ‖∇ϕη ‖2 ) + 2β‖ϕηt ‖2 2 dt = 2((|ψ|2 + g)(1 − χL(η) ), ϕηt ) 󵄩 󵄩2 ≤ β‖ϕηt ‖2 + C(󵄩󵄩󵄩|ϕ|2 (1 − χL(η) )󵄩󵄩󵄩 + ‖g − gη ‖2 ), which yields d (‖ϕηt ‖2 + ‖ϕη ‖ + ‖∇ϕη ‖2 ) + β‖ϕηt ‖2 dt 󵄩 󵄩2 ≤ C 󵄩󵄩󵄩|ψ|2 (1 − χL(η) )󵄩󵄩󵄩 + C‖g − gη ‖2 ≤ Cη.

(1.13.38)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

| 175

Taking the inner product of (1.13.30) and ϕη in L2 , we get d 1 d ∫ ϕη ϕηt dx − ‖ϕηt ‖2 + ‖ϕη ‖2 + β ‖ϕη ‖2 dt 2 dt 1 = ((|ϕ|2 + g)(1 − χL(η) ), ϕη ) ≤ ‖ϕη ‖2 + Cη. 2

(1.13.39)

Then multiplying (1.13.39) by δ and adding (1.13.38), we obtain d H + I ≤ Cη, dt η η

(1.13.40)

where 1 Hη = ‖ϕηt ‖2 + (1 + δβ)‖ϕη ‖2 + ‖∇ϕη ‖2 + δ ∫ ϕη ϕηt dx, 2 δ Iη = (β − δ)‖ϕηt ‖2 + ‖ϕη ‖2 + δ‖∇ϕη ‖2 . 2 Note that 󵄨󵄨 󵄨󵄨 δ 1 󵄨󵄨 󵄨 2 2 󵄨󵄨δ ∫ ϕη ϕηt dx󵄨󵄨󵄨 ≤ δ‖ϕη ‖δ‖ϕηt ‖ ≤ δβ‖ϕη ‖ + ‖ϕηt ‖ . 󵄨󵄨 󵄨󵄨 2 2β If δ ≤ 21 β, then for large enough L1 > 0, we have 3 1 ‖ϕηt ‖2 + (1 + δβ)‖ϕη ‖2 + ‖∇ϕη ‖2 ≤ Hη (t) 4 4 5 3 2 ≤ ‖ϕηt ‖ + (1 + δβ)‖ϕη ‖2 + ‖∇ϕη ‖2 ≤ L1 Iη (t). 4 4

(1.13.41)

Taking β3 = L−1 1 , then from equations (1.13.40) and (1.13.41) we have d H (t) + β3 H3 (t) ≤ Cη. dt η Gronwall inequality now gives Hη (t) ≤ Hη (0)e−β3 t +

Cη (1 − e−β3 t ). β3

(1.13.42)

From equations (1.13.41) and (1.13.42), we have 3 1 ‖ϕ ‖2 + (1 + δβ)‖ϕη ‖2 + ‖∇ϕη ‖2 ≤ C, 4 ηt 4

∀t ≥ 0, 0 < η ≤ 1.

Take t3 = t3 (R) such that Hη (0)e−β3 t3 ≤ CR2 e−β3 t3 ≤ η. Then from equations (1.13.41)–(1.13.42) we get ‖ϕηt ‖2 + ‖ϕη ‖2 + ‖∇ϕη ‖2 ≤ Cη.

176 | 1 Attractor and its dimension estimation Multiplying (1.13.30) by −2Δϕηt , and integrate it by parts over R3 , we get d (‖∇ϕηt ‖2 + ‖∇ϕη ‖2 + ‖Δϕη ‖2 ) + 2β‖∇ϕηt ‖2 dt = 2(∇((|ψ|2 + g)(1 − χL(η) )), ∇ϕηt ) 󵄩2 󵄩 ≤ β‖∇ϕηt ‖2 + C 󵄩󵄩󵄩∇((|ψ|2 + g)(1 − χL(η) ))󵄩󵄩󵄩 . Hence d (‖∇ϕηt ‖2 + ‖∇ϕη ‖2 + ‖Δϕη ‖2 ) + β‖∇ϕηt ‖2 ≤ Cη. dt

(1.13.43)

Taking the inner product of (1.13.30) and −Δϕη , we get −

d 1 d ∫ ψηt Δϕη dx − ‖∇ϕηt ‖2 + ‖∇ϕη ‖2 + ‖Δϕη ‖2 + β ‖∇ϕηt ‖2 dt 2 dt = ((|ψ|2 + g)(1 − χL(η) ), Δϕη ) 1 ≤ ‖Δϕη ‖2 + Cη. 2

Adding equation (1.13.43) to equation (1.13.44) multiplied by δ, we deduce d 1 (‖∇ϕηt ‖2 + (1 + δβ)‖∇ϕη ‖2 + ‖Δϕη ‖2 − δ ∫ ϕηt Δϕη dx) dt 2 + (β − δ)‖∇ϕηt ‖2 + δ‖∇ϕη ‖2 + δ‖Δϕη ‖2 ≤ Cη.

Let 1 Jη (t) = ‖∇ϕηt ‖2 + (1 + δβ)‖∇ϕη ‖2 + ‖Δϕη ‖2 − δ ∫ ϕηt Δϕη dx. 2 Then Jη (t) ≤ Jη (0)e−β4 t +

Cη (1 − e−β4 t ), β4

‖∇ϕηt ‖2 + ‖Δϕη ‖2 ≤ C,

∀t ≥ 0, 0 < η ≤ 1.

Take t4 = t4 (R) ≥ t3 such that Jη (0)e−β4 t ≤ CRe−β1 t ≤ η. Then we have ‖∇ϕηt ‖2 + ‖Δϕη ‖2 ≤ Cη, which proves the lemma.

t ≥ t4 ,

(1.13.44)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

| 177

Lemma 1.13.6. There exist constants C1 (η), C2 (η), C3 (η), C4 (η) such that 󵄩 󵄩󵄩 󵄩󵄩|x|uη 󵄩󵄩󵄩 ≤ C1 (η), 󵄩 󵄩󵄩 󵄩󵄩 2 󵄩󵄩|x|∇uη 󵄩󵄩󵄩, 󵄩󵄩󵄩|x|Djk uη 󵄩󵄩󵄩 ≤ C2 (η), 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩|x|vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vηt 󵄩󵄩󵄩 ≤ C3 (η), 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 2 󵄩󵄩|x|Djk vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vηt 󵄩󵄩󵄩 ≤ C4 (η), where uη , vη form the solution of problem (1.13.32), (1.13.33) and (1.13.34). Proof. First we consider uη . Taking the inner product of (1.13.32) and 2|x|2 uη , and then taking its imaginary part, we get d ∫ |x|2 |uη |2 dx + 2α ∫ |x|2 |uη |2 dx + 2η ∫ |x|2 |∇uη |2 dx dt = −2 Im ∫ fη |x|2 uη dx + 4η Re ∫ x∇uη uη dx + 4 Im ∫ x∇uη uη dx.

(1.13.45)

Then the right-hand side of equation (1.13.45) is bounded as 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 (4 + 4η)󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩‖uη ‖ + 2󵄩󵄩󵄩|x|fη 󵄩󵄩󵄩󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩

4 󵄩2 󵄩 󵄩2 1 󵄩 󵄩 󵄩2 ≤ η󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 + (4 + )‖uη ‖2 + α󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|fη 󵄩󵄩󵄩 . η α

Hence d 󵄩󵄩 1 󵄩󵄩 4 󵄩2 󵄩2 󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩|x|uη 󵄩󵄩󵄩 + α󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩 + η󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 ≤ (4 + )‖uη ‖ + 󵄩󵄩|x|fη 󵄩󵄩󵄩 . dt η 4α

(1.13.46)

Since fη has compact support, ‖|x|fη ‖ is finite. By Gronwall inequality, we get 󵄩󵄩 󵄩2 󵄩󵄩|x|uη 󵄩󵄩󵄩 ≤ C1 (η),

∀t ≥ 0.

Letting D2jk = D2xj xk act on the two sides of equation (1.13.32), we get iD2jk uηt + ΔD2jk uη + iαD2jk uη − iηΔD2jk uη = Fη (x),

(1.13.47)

where Fη = D2jk fη − D2jk ϕuη − Dj ϕDk uη − Dk ϕDj uη . Then we know that equation (1.13.45) still holds. If Fη is replaced by fη , through integration by parts we get ∫ D2jk ϕuη |x|2 D2jk uη dx = − ∫ Dj ϕ[Dk uη |x|2 D2jk uη + 2uη xk D2jk uη + uη |x|2 Dk D2jk uη ]dx,

178 | 1 Attractor and its dimension estimation d ∫ |x|2 |D2jk uη |2 dx + 2α ∫ |x|2 |D2jk uη |2 dx + 2η ∫ |x|2 |∇D2jk uη |2 dx dt = −2 Im ∫[D2jk fη − Dj ϕDk uη − Dk ϕDj uη ]|x|2 D2jk uη dx − 2 Im ∫ Dj ϕ[Dk uη |x|2 D2jk uη + 2uη xk D2jk uη + uη |x|2 Dk D2jk uη ]dx + 4η Re ∫ x∇D2jk uη D2jk uη dx + 4 Im ∫ x∇D2jk uη D2jk uη dx.

(1.13.48)

Since ψ, ϕ are bounded in H 3 (R3 ), and ψη , ϕη are bounded in H 2 (R3 ), uη = ψ − ψη , vη = ϕ − ϕη are bounded in H 2 (R3 ). Thus ‖uη ‖H 2 , ‖|x|uη ‖, ‖fη ‖, ‖|x|fη ‖ ≤ C(η), ‖∇ϕ‖∞ ≤ C‖ϕ‖H 3 ≤ C. Then the right-hand side of (1.13.48) 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ 2󵄩󵄩󵄩|x|D2jk fη 󵄩󵄩󵄩󵄩󵄩󵄩|x|D2jk uη 󵄩󵄩󵄩 + 6‖∇ϕ‖∞ 󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩󵄩󵄩󵄩|x|D2jk uη 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 + 4‖Dj ϕ‖∞ 󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩‖D2jk uη ‖ + 2‖Dj ϕ‖∞ 󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩󵄩󵄩󵄩|x|∇D2jk uη 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 + (4 + 4η)󵄩󵄩󵄩|x|∇D2jk uη 󵄩󵄩󵄩󵄩󵄩󵄩D2jk uη 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩2 ≤ η󵄩󵄩󵄩|x|∇D2jk uη 󵄩󵄩󵄩 + α󵄩󵄩󵄩|x|D2jk uη 󵄩󵄩󵄩 + C(η)(󵄩󵄩󵄩|x|D2jk uη fη 󵄩󵄩󵄩 + ‖∇ϕ‖2H 2 󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 󵄩 󵄩2 + ‖∇ϕ‖2H 2 󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 + ‖uη ‖2H 2 ) 󵄨 󵄨2 󵄩 󵄩2 ≤ η ∫ |x|2 󵄨󵄨󵄨∇D2jk uη 󵄨󵄨󵄨 dx + C(η) + C(η)󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 . Hence, we get d 󵄨 󵄨2 󵄨 󵄨2 ∫ |x|2 󵄨󵄨󵄨D2jk uη 󵄨󵄨󵄨 dx + α ∫ |x|2 󵄨󵄨󵄨D2jk uη 󵄨󵄨󵄨 dx + η ∫ |x|2 |∇D2jk uη |2 dx dt 󵄩 󵄩2 ≤ C(η) + C(η)󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 . From equation (1.13.46), we have 1 d 󵄩󵄩 󵄩2 ∫ |x|2 |uη |2 dx. 󵄩󵄩|x|∇uη 󵄩󵄩󵄩 ≤ C(η) − η dt By Gronwall inequality, we obtain (note that uη (0, x) = 0) 󵄨 󵄨2 ∫ |x|2 󵄨󵄨󵄨D2jk uη 󵄨󵄨󵄨 dx

t

󵄩 󵄩2 ≤ ∫ e−2α(t−s) (C(η) + C(η)󵄩󵄩󵄩|x|∇uη 󵄩󵄩󵄩 (s))ds 0

t

≤ C(η) ∫ e−2α(t−s) (C(η) − 0

t

≤ C(η) − C(η) ∫ e−2α(t−s) 0

d 󵄨 󵄨2 ∫ |x|2 󵄨󵄨󵄨uη (s)󵄨󵄨󵄨 dx)ds ds

d 󵄩󵄩 2 󵄩2 󵄩|x| uη (s)󵄩󵄩󵄩 ds ds 󵄩

(1.13.49)

1.13 The dissipation Klein–Gordon–Schrödinger equations on R3

≤ C(η) − C(η)e

| 179

t

󵄩2 −2α(t−s) 󵄩 󵄩2 󵄨󵄨s=t ds 󵄩󵄩|x| uη 󵄩󵄩󵄩 󵄨󵄨󵄨 + C(η) ∫ 2α󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩 e 󵄨s=0

−2α(t−s) 󵄩 󵄩

2

0

󵄩2 󵄩 ≤ C(η) − C(η)󵄩󵄩󵄩|x|2 uη 󵄩󵄩󵄩 + C(η)C1 (η)(1 − e−2αt ) ≤ C2 (η). Integrating by parts, we have ∫ |x|2 |∇uη |2 dx = − ∫ 2x∇uη uη dx − ∫ |x|2 Δuη uη dx 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ 2󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩‖∇uη ‖ + 󵄩󵄩󵄩|x|Δuη 󵄩󵄩󵄩󵄩󵄩󵄩|x|uη 󵄩󵄩󵄩 ≤ C(η). Now we establish an inequality for vη . Taking the inner product of (1.13.33) and 2|x|2 vηt , we get d (∫ |x|2 |νηt |2 dx + ∫ |x|2 |νη |2 dx + ∫ |x|2 |∇νη |2 dx) + β ∫ |x|2 |vηt |2 dx dt = ∫(|ψ|2 + g)χL(η) |x|2 vηt dx − 4 ∫ x∇vη vηt dx

(1.13.50)

Taking the inner product of (1.13.33) and |x|2 vη , we get d 1 d󵄩 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩2 ∫ |x|2 vηt vη dx − 󵄩󵄩󵄩|x|vηt 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 + 2 ∫ x∇vη vη dx + β 󵄩󵄩󵄩|x|νη 󵄩󵄩󵄩 dt 2 dt = ∫(|ψ|2 + g)χL(η) |x|2 vη dx.

(1.13.51)

Then by equations (1.13.50) and (1.13.51), we deduce 1 d 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 (󵄩󵄩|x|vηt 󵄩󵄩󵄩 + (1 + δβ)󵄩󵄩󵄩|x|vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 + δ ∫ xvη vηt dx) dt 2 󵄩 󵄩2 󵄩 󵄩2 + (β − δ)‖|x|vηt ‖2 + δ󵄩󵄩󵄩|x|vη 󵄩󵄩󵄩 + δ󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 = ∫(|ψ|2 + g)χL(η) |x|2 νηt dx − 4 ∫ x∇vη vηt dx − 2δ ∫ x∇vη vη dx + δ ∫(|ψ|2 + g)χL(η) |x|2 vη dx.

(1.13.52)

If δ ≤ 21 β, (ψ, ϕ, θ) ∈ L∞ (R+ ; Y), and, by Lemma 1.13.5, ‖ϕη ‖H 2 ≤ C, thus we have vη = ϕ − ϕη ∈ L∞ (R+ ; H 2 ). Hence, the right-hand side of (1.13.52) 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩|x|χL(η) (|ψ|2 + g)󵄩󵄩󵄩|x|vηt 󵄩󵄩󵄩 + 4‖∇vη ‖󵄩󵄩󵄩|x|vηt 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 + 2δ󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩‖vη ‖ + 󵄩󵄩󵄩|x|χL(η) (|ψ|2 + g)󵄩󵄩󵄩|x|vη 󵄩󵄩󵄩 1 󵄩 󵄩2 1 󵄩 󵄩2 1 󵄩 󵄩2 ≤ (β − δ)󵄩󵄩󵄩|x|vηt 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 + δ󵄩󵄩󵄩|x|vη 󵄩󵄩󵄩 2 2 2 󵄩 󵄩2 + C(η)(󵄩󵄩󵄩|x|χL(η) (|ψ|2 + g)󵄩󵄩󵄩 + ‖∇vη ‖2 + ‖vη ‖2 ),

180 | 1 Attractor and its dimension estimation 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨󵄨 2 2 󵄨󵄨δ ∫ xvη vηt dx 󵄨󵄨󵄨 ≤ δ‖xvηt ‖‖vη ‖ ≤ ‖xvηt ‖ + C‖vη ‖ . 󵄨󵄨 󵄨󵄨 2 Inserting these inequalities into equation (1.13.41), and using Gronwall inequality, we get 󵄩2 󵄩2 󵄩 󵄩 󵄩2 󵄩󵄩 󵄩󵄩|x|vηt 󵄩󵄩󵄩 + (1 + δβ)󵄩󵄩󵄩|x|vη 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vη 󵄩󵄩󵄩 ≤ C3 (η). Differentiating (1.13.33) with respect to xk , k = 1, 2, 3, multiplying by 2|x|2 vηxk t and integrating over R3 by parts, we also obtain 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩|x|vηxk t 󵄩󵄩󵄩 + (1 + δβ)󵄩󵄩󵄩|x|vηxk 󵄩󵄩󵄩 + 󵄩󵄩󵄩|x|∇vηxk 󵄩󵄩󵄩 ≤ C4 (η), finishing the proof of the lemma. Now we prove the existence of an attractor. Theorem 1.13.3. Let f , g ∈ H 2 (R3 ), and let S(t) be the semigroup generated by problem (1.13.6)–(1.13.9). Then there exists a set A ⊂ X which satisfies (1) S(t)A = A , ∀t ≥ 0; (2) limt→∞ distX (S(t)B, A ) = limt→∞ supy∈A distX (S(t)y, A ) = 0, where B ⊂ Y is a bounded set; (3) A is compact in X. That is to say, A is a global attractor in X, it attracts all the bounded set of Y with respect to the topology of X. To prove the theorem, we need the following compact embedding lemma. Lemma 1.13.7. Let s > s1 be an integer. The embedding from H s (Rn ) ∩ H s1 (Rn , (1 + |x|2 )dx) to H s1 (Rn ) is compact. Proof. Let B ⊂ H s ∩ H s1 ((1 + |x|2 )dx) be a bounded set. We only need to prove that B has a finite ε-net (for any ε > 0). Firstly, since 󵄨 󵄨2 ∫ |x|2 ∑ 󵄨󵄨󵄨Dl u󵄨󵄨󵄨 dx ≤ C,

Rn

l≤s1

∀u ∈ B,

there exists A > 0 such that 1 C ε2 󵄨 󵄨2 󵄨 󵄨2 ∫ ∑ 󵄨󵄨󵄨Dl u󵄨󵄨󵄨 dx ≤ 2 ∫ |x|2 ∑ 󵄨󵄨󵄨Dl u󵄨󵄨󵄨 dx ≤ 2 ≤ . 2 A A l≤s l≤s

|x|>A

1

|x|>A

1

Let Ω = {x | |x| < A}. Then the embedding H s (Ω) 󳨅→ H s1 (Ω) is compact. Hence B|Ω = {u | u = v|Ω , v ∈ B} ⊂ H s (Ω) is relatively compact in H s1 (Ω), and it has finite √ε2 -net

1.14 Two-dimensional unbounded region derivative Ginzburg–Landau equation

| 181

{B(ũk , √ε2 ), k = 1, 2, . . . , m}, ũk ∈ B|Ω , ũk = uk |Ω , uk ∈ B. We need to prove that {B(uk , √ε2 )} ̃ k such that ̃ = u|Ω , there exists u is the ε-net of B in H s1 (Ω). In fact, for any u ∈ B, u ̃k − u ̃ ‖H s1 (Ω) < ‖u

ε . √2

Hence ̃−u ̃ k ‖2H s (Ω) + ‖u ̃−u ̃ k ‖2H n (Ω) ≤ ‖u − uk ‖2H s (Ω) = ‖u

ε2 󵄨 󵄨2 + ∫ ∑ 󵄨󵄨󵄨Dl u󵄨󵄨󵄨 dx < ε2 , 2 l≤s |x|>A

1

and the lemma is proved. Proof of Theorem 1.13.3. From Lemmas 1.13.6 and 1.13.7 we know that the operator S2η (t) defined by (1.13.32)–(1.13.34) is compact from Y to X. For any bounded set B ⊂ Y, we have α(S2η (t)B) = 0,

∀t ≥ 0.

From Lemma 1.13.5 we know, for ∀ε > 0, that there exist η and t0 > 0 such that 󵄩󵄩 󵄩 󵄩󵄩S1η (ψ0 , ϕ0 , θ0 )󵄩󵄩󵄩 ≤ ε, ∀t ≥ t0 , and (ψ0 , ϕ0 , θ0 ) ∈ B, B ⊂ Y is a bounded set. That is, for η > 0, α(S1η (t)B) ≤ 2ε,

t ≥ t0 .

Then we have α(S(t)B) ≤ α(S1η (t)B) + α(S2η (t)B) = α(S1η (t)B) ≤ 2ε,

t ≥ t0 .

Hence lim α(S(t)B) = 0

t→∞

and S(t) is asymptotically smooth. By the theory in [137], Theorem 1.13.3 is proved.

1.14 Two-dimensional unbounded region derivative Ginzburg–Landau equation As mentioned before, Guo and Wang [114] considered the following two-dimensional Ginzburg–Landau equation with derivative term (DGL): ut = ρu + (1 + iv)Δu − (1 + iu)|u|2σ u + αλ1 ⋅ ∇(|u|2 u) + β(λ2 ⋅ ∇u)|u|2 ,

(1.14.1)

182 | 1 Attractor and its dimension estimation where ρ > 0, α, β, v, u are real constants and λ1 , λ2 are real constant vectors. The authors proved that problem (1.14.1) with periodic initial value possesses a global attractor of bounded dimension, where they supposed that there exists a positive δ0 such that the following inequality is valid: 1 √1 +

μ−vδ2 ( 1+δ2 )2

−1

≥ σ ≥ 3.

(1.14.2)

In 1997, Gao and Duan [71] considered the Cauchy problem of the following twodimensional derivative GL equation: ut = α0 u + α1 Δu + α2 |u|2 ux + α3 |u|3 uy + α4 u2 ux + α5 u2 uy − α6 |u|2σ u, where α0 > 0, αj = aj + ibj , 1 ≤ j ≤ 6, a1 > 0, a6 > 0 and σ > 0. They proved the

existence of a global solution in H 2 . Assume that b1 b6 > 0, σ ≥ 1+ 2 10 ; and, when b6 = 0 or b1 b6 < 0, then there exists a positive number δ > 0 such that √

1 √1 +

)2

(b1 δ−b6 (1+δ)(a1 δ+a6 )

−1

≥σ≥

1 + √10 . 2

(1.14.3)

In 1997, Guo and Li [163] proved the existence of a global solution of the above problem in an unbounded domain, and improved condition (1.14.3). Now we consider the Cauchy problem for the following two-dimensional derivative GL equation: ut = γu + (1 + iv)Δu − (1 + iμ)|u|2σ u 󵄨 󵄨 + λ1 ⋅ ∇(|u|2 u) + (λ ⋅ ∇u)󵄨󵄨󵄨u2 󵄨󵄨󵄨, t > 0, x ∈ R2 , u(0, x) = u0 (x), x ∈ R2 ,

(1.14.4) (1.14.5)

where γ > 0, v, u are real constants and λ1 , λ2 are complex constant vectors. Let σ, v, u satisfy the following condition (A): σ>2

and −1 − vμ
0 is an appropriate weight function, which possesses the property:

󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨∇ρ(x)󵄨󵄨󵄨, 󵄨󵄨󵄨Δρ(x)󵄨󵄨󵄨 ≤ ρ0 ρ(x), such as ρ =

1 , cosh |x|

and

∫ ρ(x)dx = ρ0 < +∞,

(1.14.6)

ρ = e−|x| , and so on. Let Ty ρ(x) = ρ(x − y) denote the translation of 1

the weight function. The weighted Lp with norm is ‖μ‖p,p = (∫ ρ|μ|p dx) p , 1 < p < ∞. The uniform local norm is ‖u‖p,lu = sup ‖u‖p,Ty p y∈R2

Let Lpp denote the all weighted Lp norm function space, ‖u‖p,p < ∞; Lplu denotes all uniform local space of u, ‖u‖p,lu < +∞

and ‖Ty u − u‖p,lu → 0

when y → 0.

Both (Lpρ , ‖ ⋅ ‖p,ρ ) and (Lplu , ‖ ⋅ ‖p,lu ) are Banach spaces. We also define the weighted 1

Sobolev space Wρm,p , ‖u‖Wρm,p = (∑k≤m ‖Dk u‖pp,ρ ) p ; the uniform local Sobolev space

Wlum,p = completion of function space Cb∞ with respect to the norm ‖u‖W m,p supy∈R2 ‖u‖W m,p . In particular, Ty p

Hpm

=

Wρm,2 ,

m Hlu

=

Wlum,2 .

lu

=

Using the semigroup theory and contraction mapping principle, we can get the following existence theorem: 1 Theorem 1.14.2. Let u0 ∈ Hlu . Then there exists a unique solution of the Cauchy problem (1.14.4)–(1.14.5), 1 2 u(t) ∈ C([0, T∗ ), Hlu ) ∩ C((0, T∗ ), Hlu ).

If T∗ < +∞, then 󵄩 󵄩 lim 󵄩󵄩u(t)󵄩󵄩󵄩H 1 = +∞.

t→T∗ 󵄩

lu

In order to get the existence of a global solution and global attractor in an unbounded domain, we must give a uniform a priori estimate on the weighted space. Lemma 1.14.1. Suppose condition (A) is valid and 2≤p
0 such that ‖u‖pp,ρ ≤ C, whenever ‖u0 ‖p,ρ ≤ R.

t ≥ t0 (R),

184 | 1 Attractor and its dimension estimation Proof. By direct calculation, it follows that 1 d ‖u‖pp,ρ = Re ∫ ρ|u|p−2 uut dx p dt = Re ∫ ρ|u|p−2 u(γu + (1 + iv)Δu − (1 + iμ)|u|2σ u + (λ1 ⋅ ∇)(|u|2 u) + (λ2 ⋅ ∇u)|u|2 )dx

= γ‖u‖pp,ρ − ‖u‖p+2σ p+2σ,ρ + I1 + I2 ,

(1.14.8)

where I1 = Re ∫(1 + iv)ρ|u|p−2 uΔudx, I2 = ∫ ρ|u|p−2 u((λ1 ⋅ ∇)(|u|2 u) − (λ2 ⋅ Δu)|u|2 )dx. Integration by parts yields I1 = − Re ∫(1 + iv)∇ρ|u|p−2 uΔudx − Re ∫(1 + iv)ρ∇(|u|p−2 u)Δudx = − Re ∫(1 + iv)∇ρ|u|p−2 uΔudx −

1 Re ∫ ρ|u|p−4 (p|u|2 |∇u|2 + (1 + iv)(p − 2)u2 (∇u)2 )dx 2

= − Re ∫(1 + iv)∇ρ|u|p−2 uΔudx −

2 u𝜕 u 1 ∫ ρ|u|p−4 ∑(u𝜕j u, u𝜕j u)M(v, p) ( j ) dx, u𝜕j u 4 j=1

(1.14.9)

where tr

p ∗

M(v, p) = M(v, p) = (

(1 − iv)(p − 2) ). p

(1.14.10)

When equation (1.14.7) is satisfied, the smallest eigenvalue of M(v, p) is λM (v, p) = p − |p − 2|√1 + v2 > 0. Hence M(v, p) is positive, and we have 1 λ (v, p) ∫ ρ|u|p−2 |∇u|2 dx 4 M 2ρ2v 1 ≤ ∫ ρ|u|p dx − λM (ν, p) ∫ ρ|u|p−2 |∇u|2 dx, λM (v, p) 8

I1 ≤ ρv ∫ ρ|u|p−1 |∇u|dx −

where ρv = ρ0 √1 + v2 . By Cauchy inequality, we get |I2 | ≤ (3|λ1 | + |λ2 |) ∫ ρ|u|p+1 |∇u|dx ≤

2(3|λ1 | + |λ2 |)2 1 λM (v, p) ∫ ρ|u|p−2 |∇u|2 dx + ∫ ρ|u|p+4 dx. 8 λM (v, p)

(1.14.11)

1.14 Two-dimensional unbounded region derivative Ginzburg–Landau equation

| 185

Since σ > 2, by Hölder inequality we have 2(3|λ1 | + |λ2 |)2 ∫ ρ|u|p+1 dx λM (v, p)

1−2σ

2(3|λ1 | + |λ2 |)2 (∫ ρ|u|p dx) λM (v, p) 1 ≤ ‖u‖p+2σ + C1 ‖u‖pp,ρ , 2 p+2σ,ρ ≤

where C1 =

σ−2 σ ( ) σ 4

2(σ−2) σ2

2

2σ

(∫ ρ|u|p+2σ dx)

2σ

1 |−|λ2 |) ( 2(3|λ ) σ−2 . Then we get λ (ν,p) M

C C 1 d 1 ‖u‖pp,ρ ≤ C2 ‖u‖pp,ρ − ‖u‖p+2σ ≤ − 2 ‖u‖pp,ρ + 3 , p dt 2 p+2σ,p p p where C2 = γ +

2ρ2 λM (v,p)

2 (p+1) + C1 and C3 = σρ0 ( 2Cp+2σ )

‖u‖pp,ρ ≤ ‖u0 ‖pp,ρ e−C2 t +

p+2σ 2σ

. Gronwall inequality gives

C3 , C2

t ≥ 0,

proving the lemma. Remark 1.14.1. If σ = 2, we can suppose 6|λ1 | + 2|λ2 | < λM (v, p), then Lemma 1.14.1 is also valid. Lemma 1.14.2. Under condition (A), there exists a constant C, independent of R, and constant t1 (R) such that 󵄩󵄩 󵄩2 󵄩󵄩∇u(t)󵄩󵄩󵄩2,ρ ≤ C,

t ≥ t1 (R),

whenever ‖u0 ‖Hρ1 ≤ R. Proof. By equation (1.14.4), and integrating by parts, we obtain 1 d ∫ ρ|∇u|2 dx = Re ∫ ρ∇u∇ut dx 2 dt = Re ∫ ρ∇u∇(γu + (1 + iv)Δu − (1 + iμ)|u|2σ u + (λ1 ⋅ ∇)(|u|2 u) + (λ2 ⋅ ∇u)|u|2 )dx = γ‖∇u‖22,ρ − ‖Δu‖22,ρ − Re ∫(1 + iv)∇ρ∇uΔudx + Re ∫(1 + iμ)ρ|u|2σ uΔudx + Re ∫(1 + iμ)∇ρ∇u|u|2σ udx − Re ∫ ρΔu((λ1 ⋅ ∇)(|u|2 u) + (λ2 ⋅ ∇u)|u|2 )dx − Re ∫ ∇ρ∇u((λ1 ⋅ ∇)(|u|2 u) + (λ2 ⋅ ∇u)|u|2 )dx

186 | 1 Attractor and its dimension estimation 7

= γ‖∇u‖22,ρ − ‖Δu‖22,ρ + ∑ Ik , k=3

where

(1.14.12)

󵄨󵄨 󵄨󵄨 󵄨 󵄨 |I3 | = 󵄨󵄨󵄨− Re ∫(1 + iv)∇ρ∇uΔudx 󵄨󵄨󵄨 ≤ ρv ∫ ρ|∇u||Δu|dx, 󵄨󵄨 󵄨󵄨 I4 = Re ∫ ρ(1 − iμ)|u|2σ uΔudx,

󵄨󵄨 󵄨󵄨 󵄨 󵄨 |I5 | = 󵄨󵄨󵄨− Re ∫(1 + iμ)∇ρ∇u|u|2σ udx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ ρu ∫ ρ|u|2σ+1 |∇u|dx,

󵄨󵄨 󵄨󵄨 󵄨 󵄨 |I6 | = 󵄨󵄨󵄨− Re ∫ ∇ρ∇u((λ1 ⋅ ∇)(|u|2 u) + (λ2 ⋅ ∇u)|u|2 )dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (3|λ1 | + |λ2 |)ρ0 ∫ ρ|u|2 |∇u||∇u|2 dx,

󵄨󵄨 󵄨󵄨 󵄨 󵄨 |I7 | = 󵄨󵄨󵄨− Re ∫ ρΔu((λ ⋅ ∇)(|u|2 u) + (λ2 ⋅ ∇u)|u|2 dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (3|λ1 | + |λ2 |) ∫ ρ|u|2 |∇u||∇u||Δu|dx.

Let δ > 0 (choosing properly) and define δ 1 |u|2σ+2 )dx. Vδ (u(t)) = ∫ ρ( |∇u|2 + 2 2σ + 2

(1.14.13)

From equations (1.14.12) and (1.14.18), for p = 2σ + 2, we get d 2 4σ+2 V (u(t)) = γ(‖∇u‖22,ρ − δ‖u‖2σ+2 2σ+2,ρ ) − (‖Δu‖2,ρ + δ‖u‖4σ+2,ρ ) dt δ + (δI1 + I4 ) + I3 + I5 + I6 + δI2 + I7 2 4σ+2 ≤ γ(‖∇u‖22,ρ + δ‖u‖2σ+2 2σ+2,ρ ) − (‖Δu‖2,ρ + δ‖u‖4σ+2,ρ )

+

|u|2σ u 1 Re ∫ ρ(|u|2σ u, Δu) ⋅ N0 ⋅ ( ) dx Δu 2

+ ρv ∫ ρ|∇u||Δu|dx + ρμ ∫ ρ|u|2σ+1 |∇u|dx + (3|λ1 | + |λ2 |)ρ0 ∫ ρ|u|2 |∇u|2 dx + δ(3|λ1 | + |λ2 |) ∫ ρ|u|2σ+3 |∇u|dx + (3|λ1 | + |λ2 |) ∫ ρ|u|2 |∇u||Δu|dx,

(1.14.14)

). Arguing similarly as with equations (1.14.9)–(1.14.10), where N0 = N0T = ( ∗0 1+δ−i(δv−μ) 0 for any α such that |α|
0 is unknown), and adding to equation (1.14.14), we arrive at d 2σ+2 V (u(t)) ≤ γ(‖∇u‖22 + δ‖u‖2σ+2 ) − (1 − κ)(‖Δu‖22 + δ‖u‖4σ+2 4σ+2 ) dt δ − ηλM (α, 2σ + 2) ∫ ρ|u|2σ |∇u|2 dx +

|u|2σ u 1 ) dx Re ∫ ρ(|u|2σ u, Δu) ⋅ N ⋅ ( Δu 2

+ (ρμ + ηρα ) ∫ ρ|u|2σ+1 |∇u|dx + ρv ∫ ρ|∇u||Δu|dx + (3|λ1 | + |λ2 |)ρ0 ∫ ρ|u|2 |∇u|2 dx + δ(3|λ1 | + |λ2 |) ∫ ρ|u|2σ+3 |∇u|dx + (3|λ1 | + |λ2 |) ∫ ρ|u|2 |∇u||Δu|dx,

(1.14.16)

where 0 ≤ κ < 1 is undetermined and −2δκ ∗

N = N tr = (

1 + δ − η − i(δv − μ − αη) ). −2κ

For this matrix N, we have Proposition 1.14.1. When σ, ν and μ satisfy condition (A), we can choose proper δ, η √ which are positive, κ ∈ (0, 1) and |α| < 2σ+1 such that N is non-positive. Hence σ Re ∫ ρ(|u|2σ u, Δu) ⋅ N ⋅ (

|u|2σ u ) dx ≤ 0. Δu

188 | 1 Attractor and its dimension estimation For the last five integrals of equation (1.14.16), we perform the following estima-

tion:

(ρμ + ηρα ) ∫ ρ|u|2σ+1 |∇u|dx ≤

2(ρμ + ηρα )2 1 (1 − κ) ∫ ρ|u|4σ−2 dx + ∫ ρ|∇u|2 dx, 8 (1 − κ)

2ρ2μ 1 2 ρν ∫ ρ|∇u||Δu|dx ≤ (1 − κ) ∫ ρ|Δu| dx + ∫ ρ|∇u|2 dx, 8 (1 − κ) 2

1

1

(3|λ1 | + |λ2 |)ρ0 ∫ ρ|u|2 |∇u|2 dx = (3|λ1 | + |λ2 |)ρ0 ∫ ρ σ |u|2 |∇u| σ ⋅ ρ1− σ |∇u|2−2/σ dx 2σ

2

1 σ

1− σ1

2

≤ (3|λ1 | + |λ2 |)ρ0 (∫ ρ|u| |∇u| dx) (∫ ρ|∇u| dx) 1 ≤ ηλM (α, 2σ + 2) ∫ ρ|u|2σ |∇u|2 dx 2 σ

1

(σ − 1)[(3|λ1 | + |λ2 |)ρ0 ] σ−1 2 σ + ( ) ∫ ρ|∇u|2 dx, σ σ 1 4 2 2 ∫ ρ|u| |∇u|2 dx, ∫ ρ|u| |∇u||Δu|dx ≤ ε1 ∫ ρ|Δu| dx + 4ε1 1 ∫ ρ|u|2σ+3 |∇u|dx ≤ ε2 ∫ ρ|u|4σ+2 |∇u|2 dx + ∫ ρ|u|4 |∇u|2 dx, 4ε2 where ε1 > 0, ε2 > 0 are arbitrary. For σ > 2, by Young inequality, we have 4

∫ ρ|u|4 |∇u|2 dx = ∫ ρ|u|4 |∇u| σ |∇u|2(1−2/σ) dx 2

≤ ε3 ∫ ρ|u|2σ |∇u|2 dx +

σ − 2 2 σ−2 ( ) ∫ ρ|∇u|2 dx, σ σε3

∀ε3 > 0.

Choosing ε1 , ε2 and ε3 small enough so that ε1 (3|λ1 | + |λ2 |) ≤ 2(1 − κ), 1 ε1 δ(3|λ1 | + |λ2 |) ≤ (1 − κ), 2 1 δ 1 ε3 ( + )(3|λ1 | + |λ2 |) ≤ ηλM (α, 2σ + 2), 4ε1 4ε2 2 we obtain d V (u(t)) ≤ (γ + C4 )‖∇u‖22,ρ + δγ‖u‖2σ+2 2σ+2,ρ dt δ 1 − (1 − κ)(‖Δu‖22 + δ‖u‖2σ+2 4σ+2,ρ ), 2

(1.14.17)

1.14 Two-dimensional unbounded region derivative Ginzburg–Landau equation

| 189

where C4 is the sum of integral coefficients of ∫ ρ|∇u|2 dx in the above inequalities. Note that there exists C5 = C5 (γ, σ, κ) > 0 such that 1 2σ+2 δγ‖u‖2σ+2 2σ+2,ρ − δ(1 − κ)‖u‖4σ+2,ρ 2 2δγ 2γ |u|2σ+2 )dx = − ‖u‖2σ+2 + δC5 ρ0 . ≤ δ ∫ ρ(C5 − 2σ + 2 2σ + 2 2σ+2,ρ Using integration by parts and Cauchy inequality, we arrive at ‖∇u‖2,ρ = ∫ ρ∇u∇udx = − ∫ ∇ρ∇uudx − ∫ ρΔuudx ≤ ρ0 ‖u‖2,ρ ‖∇u‖2,ρ + ‖u‖2,ρ ‖Δu‖2,ρ

ρ2 (2γ + C4 ) 1−κ 1 ‖Δu‖22,ρ + ( 0 + )‖u‖22,ρ . ≤ ‖∇u‖22,ρ + 2 4(2γ + C4 ) 2 1−κ

By Lemma 1.14.1, ‖u‖2,ρ ≤ C0 , and 1 (γ + C4 )‖∇u‖2,ρ ≤ C6 + (1 − κ)‖Δu‖22,ρ − γ‖∇u‖22,ρ , 2 where C6 = (ρ20 (2γ + C4 ) +

2(2γ+C4 )2 )C0 , 1−κ

hence we have

d V (u(t)) ≤ −2γVδ (u(t)) + δC5 ρ0 + C6 . dt δ

(1.14.18)

Using Gronwall inequality, we obtain Vδ (u(t)) ≤ V(u0 )e−2γt +

δC5 ρ0 + C6 , 2γ

t ≥ 0,

(1.14.19)

proving the lemma. Corollary 1.14.1. Under the assumptions of Lemma 1.14.2, we have 󵄩󵄩 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩H 1 ≤ C, lu

t ≥ t1 (R),

whenever ‖u0 ‖H 1 ≤ R, with C independent of R. lu

Remark 1.14.2. When σ = 2 and 3|λ1 | + |λ2 | is small enough, Lemma 1.14.2 and Corollary 1.14.1 are valid. Based on local existence theorem and above a priori estimates, we have Theorem 1.14.3 (Global existence). Suppose condition (A) is valid. Then for any u0 ∈ 1 Hlμ , the problem (1.14.4)–(1.14.5) has a unique solution 1 2 u(t) ∈ C([0, +∞); Hlu ) ∩ C((0, +∞); Hlu ).

190 | 1 Attractor and its dimension estimation 1 The semigroup S(t) generated by the DGL equation is continuous in Hlu (t > 0) and there exist constants L1 and t1 (R) > 0 such that

󵄩 󵄩󵄩 󵄩󵄩u(t)󵄩󵄩󵄩H 1 ≤ L1 , lu

t ≥ t1 (R),

1 whenever ‖u0 ‖H 1 ≤ R. This time, B(0, L1 ) is an absorbing set in Hlu . For any q > 2, there lu exist constants L2 > 0 and t∗ (R) > 0 such that

󵄩󵄩 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩W l,q ≤ L2 , lu

t ≥ t∗ (R).

2 Proof. We first prove the regularity of a solution in Hlu . Since M1 = supt≥0 ‖u(t)‖H 1 < lu ∞, when 1 < p < 2,

󵄩󵄩 󵄩 2σ+1 + M13 ) ≜ M2 , 󵄩󵄩F(u(t))󵄩󵄩󵄩Lp ≤ C(p)(M1 + M1 lu

t ≥ 0,

and then equation (1.14.4) can be written as ut = Bp u + F(u), Bp = Ap − (R1 + 1),

F(u) = (γ + R1 + 1)u + (1 + iμ)|u|2σ u

+ (λ1 ⋅ ∇)(|u|2 u) + (λ2 ⋅ u)|u|2 ,

Ap u = (1 + iv)Δu. Hence, for q > 2, by the interpolation inequality (setting p = 1−

1 2q

∈ (0, 1)) and

󵄩󵄩 Bq t 󵄩󵄩 −( 1 + 1 − 1 ) −ωt 󵄩󵄩e u󵄩󵄩W s+l,q ≤ Mt p 2 q e ‖u‖W s,p , lu lu

2q 1+q

∈ (1, 2), θ = 21 − q1 + p1 =

∀u ∈ Wlus,p , 1 < p ≤ q, t > 0,

we arrive at θ 󵄩󵄩 Bq (t−s) 󵄩 󵄩 󵄩 1−θ 󵄩 󵄩 F(u(s))󵄩󵄩󵄩W l,q ≤ (󵄩󵄩󵄩eBq (t−s) F(u(s))󵄩󵄩󵄩Lp ) (󵄩󵄩󵄩eBq (t−s) F(u(s))󵄩󵄩󵄩W 2,p ) 󵄩󵄩e lu lu lu 󵄩 󵄩 ≤ M(t − s)−θ e−ω(t−s) 󵄩󵄩󵄩F(u(s))󵄩󵄩󵄩Lp lu

≤ MM2 (t − s)−θ e−ω(t−s) ,

t > s ≥ 0.

Through u(t) = e

Bq t

t

u(0) + ∫ eBq (t−s) F(u(s))ds, 0

(1.14.20)

1.14 Two-dimensional unbounded region derivative Ginzburg–Landau equation

| 191

we arrive at t

󵄩 󵄩 B (t−s) 󵄩 󵄩 󵄩󵄩 − 1 −ωt 󵄩 F(u(s))󵄩󵄩󵄩W 1,q ds, 󵄩󵄩u(t)󵄩󵄩󵄩W l,q ≤ Mt 2 e 󵄩󵄩󵄩u(0)󵄩󵄩󵄩Lq + ∫ 󵄩󵄩󵄩e q lu lu lu 0

t

1

CMt − 2 e−ωt ‖u0 ‖H 1 + ∫ MM2 (t − s)−θ e−ω(t−s) ds lu

0

1

≤ MM1 t − 2 e−ωt +

1 MM2 Γ( 2q )

ω

,

t > 0.

In particular, we have ‖u(t)‖W 1,4 ≤ M3 (t0 ), t ≥ t0 , for fixed t0 > 0. lu It follows that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2σ+1 2 󵄩󵄩F(u(t))󵄩󵄩󵄩H 1 ≤ C 󵄩󵄩󵄩u(t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩u(t)󵄩󵄩󵄩W 1,4 + ‖u(t)‖W 1,4 ‖u(t)‖H 2 lu lu lu lu lu 󵄩 󵄩󵄩 2σ+1 󵄩 ≤ C(M3 + 1) (1 + 󵄩󵄩u(t)󵄩󵄩H 2 ) lu 󵄩 󵄩 ≤ M4 (1 + 󵄩󵄩󵄩u(t)󵄩󵄩󵄩H 2 ), t ≥ t0 , lu which yields t

󵄩󵄩 󵄩 󵄩 B (t−t ) 󵄩 󵄩 B (t−s) 󵄩 F(u(t))󵄩󵄩󵄩H 2 ds 󵄩󵄩u(t)󵄩󵄩󵄩H 2 ≤ 󵄩󵄩󵄩e 2 0 u(t0 )󵄩󵄩󵄩H 2 + ∫ 󵄩󵄩󵄩e 2 lu lu lu t0

󵄩 󵄩 ≤ M(t − t0 ) e−ω(t−t0 ) 󵄩󵄩󵄩u(t0 )󵄩󵄩󵄩H 1 − 21

t

lu

1 󵄩 󵄩 + ∫ M(t − s)− 2 e−ω(t−s) 󵄩󵄩󵄩F(u(t))󵄩󵄩󵄩H 1 ds lu

t0

1 󵄩 󵄩 ≤ M(t − t0 )− 2 e−ω(t−t0 ) 󵄩󵄩󵄩u(t0 )󵄩󵄩󵄩H 1

t

lu

1 󵄩 󵄩 + ∫ MM4 (t − s)− 2 e−ω(t−s) (1 + 󵄩󵄩󵄩u(s)󵄩󵄩󵄩H 2 )ds, lu

t0

t > t0 .

From this, by Gronwall inequality, we get 󵄩󵄩 󵄩 −1 󵄩󵄩u(s)󵄩󵄩󵄩H 2 ≤ M5 (t − t0 ) 2 , lu

t0 < t ≤ T.

2 Since t0 > 0 is arbitrary, u(t) ∈ Hlu , and it is continuous for t > 0. Secondly, we prove that there exists t∗ > 0 such that u(t) is uniformly bounded in Wlu1,q , where ‖u0 ‖H 2 ≤ R. Form this we will know that B(0, L2 ) (that is, the ball with lu

radius L2 in Wlu1,q ) is an absorbing set in Hρ1 . Since supt≥t1 (R) ‖u(t)‖H 1 ≤ L1 , when 1 < p < 2, lu

󵄩󵄩 󵄩 2σ+1 3 󵄩󵄩F(u(t))󵄩󵄩󵄩Lp ≤ C(p)(L1 + L1 + L1 ) ≜ C1 (p), lu

t ≥ t1 (R),

192 | 1 Attractor and its dimension estimation where C1 (p) is independent of R, similarly to the proof above. For q > 2, by interpo2q 1 , θ = 21 − q1 + p1 = 1 − 2q ) and equation (1.14.20), we lation formula (setting p = 1+q have θ 󵄩 󵄩 1−θ 󵄩 󵄩 󵄩 󵄩󵄩 Bq (t−s) F(u(s))󵄩󵄩󵄩W 1,q ≤ (󵄩󵄩󵄩eBq (t−s) F(u(s))󵄩󵄩󵄩Lp ) (󵄩󵄩󵄩eBq (t−s) F(u(s))󵄩󵄩󵄩W 2,p ) 󵄩󵄩e lu lu lu 󵄩 󵄩 ≤ M(t − s)−θ e−ω(t−s) 󵄩󵄩󵄩F(u(s))󵄩󵄩󵄩Lp lu

≤ MC1 (p)t −θ e−ωt ,

t > s ≥ t1 .

Since u(t) = e

Bq (t−t1 )

t

u(t1 ) + ∫ eBq (t−s) F(u(s))ds, t1

we arrive at t

󵄩󵄩 󵄩 − 1 −ω(t−t1 ) 󵄩 󵄩󵄩u(t1 )󵄩󵄩󵄩 q + ∫ MC1 (p)(t − s)−θ e−ω(t−s) ds 󵄩󵄩u(t)󵄩󵄩󵄩W 1,q ≤ M(t − t1 ) 2 e 󵄩 󵄩Llu lu t1

1 MC1 (p)Γ( 2q ) 1 󵄩 󵄩 ≤ M(t − t1 )− 2 e−ω(t−t1 ) 󵄩󵄩󵄩u(t1 )󵄩󵄩󵄩H 1 + , lu ω

Taking t∗ = t1 + 1 +

1 ω

t > t1 .

lg(ML1 ), we obtain

1 ) MC1 (p)Γ( 2q 󵄩󵄩 󵄩 , 󵄩󵄩u(t)󵄩󵄩󵄩W 1,q ≤ L2 = 1 + lu ω

t ≥ t∗ (R).

Employing the theory of Hale [137] and Temam [197], together with the compact 2 embedding from Hlu to Hρ1 , Theorem 1.14.1 can be deduced. The global attractor A can be represented as the ω-limit set which is generated by the semigroup S(t) for DGL equation (1.14.4) with initial value (1.14.5). That is, A = ω(B(0, L1 )) = ⋂ ⋃ S(t)B(0, L1 ), s≥0 t≥s

where the closure is taken with respect to Hρ1 topology, and this set possesses the following properties: (1) A is translation invariant; (2) A is rotation invariant; (3) If σ is an integer and ρ is smooth, |Dm ρ(x)| ≤ ρm ρ(x), ∀m ≥ 1, then ∞

m

A ⊂ ⋂ Hlu . m=1

Properties (1) and (2) can be obtained from the translation invariance and rotation invariance of DGL equation (1.14.4). Property (3)can be deduced by induction through a similar argument as the proof of Theorem 1.14.1.

1.15 The relation between attractor and turbulence

| 193

1.15 The relation between attractor and turbulence With the in-depth research on dynamical system properties of Navier–Stokes equations, these results can reveal certain analogies with the turbulence theory of Kolmogorov. Now we indicate them briefly as follows:

1.15.1 The algebraic decay of characteristic norm Suppose that the solution u of Navier–Stokes equation can be expanded into a series: ∞

̂ j (t)ωj (x) u(x, t) = ∑ u

(1.15.1)

j=1

where {ωj (x)} is an orthogonal complete set of functions having eigenvalues {λj }, −Δωj + grad rj = λj ωj , { div ωj = 0,

(1.15.2)

ωj satisfies the boundary conditions as for u, and rj corresponds to the pressure term. Let | ⋅ |0 denote the L2 -norm of a vector function ϕ in Ω, 1

2 󵄨 󵄨2 |ϕ|0 = {∫ 󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dx} .

Ω

Use ‖ ⋅ ‖ to denote the L2 -norm of the gradient of ϕ: 1

2 󵄨 󵄨2 ‖ϕ‖ = |grad ϕ|0 = {∫ 󵄨󵄨󵄨grad ϕ(x)󵄨󵄨󵄨 dx} .

Ω

Use R0 to denote an upper of the norm in L2 for u with initial value u(x, 0), 󵄨󵄨 󵄨 󵄨󵄨u(⋅, 0)󵄨󵄨󵄨0 ≤ R0 .

(1.15.3)

Foias and coauthors got the following results in [68]: Theorem 1.15.1. Suppose that the space dimension is 2. Then there exists a constant k1 , which only depends on μ, |f |0 , Ω, and constant t1 , depending on the above parameters and R0 , such that 1

2 λ λ 󵄨󵄨 ̂ 󵄨 󵄨󵄨um (t)󵄨󵄨󵄨 ≤ k1 ( 1 )(1 + ln( 1 )) , λ λ

m

m

∀m, ∀t ≥ t1 ,

(1.15.4)

194 | 1 Attractor and its dimension estimation 2 where ν is viscosity coefficient and f is external force. Here λm = km , km is wave number. ̂ m (t) = u ̂ (km , t). From equation (1.15.4) we get Denote u 1

2 κ k 󵄨 󵄨󵄨 ̂ 󵄨󵄨u(km , t)󵄨󵄨󵄨 ≤ ( 21 )(1 + ln( 1 )) . k k

m

m

The decay rate of energy spectrum E is κ 󵄨̂ 󵄨2 E(k, t) ≈ k 󵄨󵄨󵄨u (k, t)󵄨󵄨󵄨 ≤ ( 3 ) ln k, k

t ≥ t1 .

Up to the factor ln k, it is consistent with the two-dimensional Kraichnan decay rate (see [152]). Similar results are valid in dimension 3. Suppose that no singularity occurs, and ‖u(⋅, t)‖ stays uniformly bounded 󵄩 󵄩 M1 = sup 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩 < +∞. t≥0

(1.15.5)

Similar to equation (1.15.4), we have 2

3 󵄨󵄨 ̂ 󵄨 󸀠 λ 󵄨󵄨um (t)󵄨󵄨󵄨 ≤ κ1 ( 1 ) , λ

m

∀m, ∀t ≥ t1󸀠 .

Hence x 󵄨󵄨 ̂ 󵄨 󵄨󵄨u(km , t)󵄨󵄨󵄨 ≤ 4 . km3 Energy spectrum is κ 󵄨̂ 󵄨2 E(k, t) ≈ k 2 󵄨󵄨󵄨u (k, t)󵄨󵄨󵄨 ≤ 2 , k3

t ≥ t1 .

If Grashof or Reynolds number is large enough, then it is consistent with Kolmogorov decay rate. 1.15.2 The Fourier coefficients of exponential decay Suppose that equation (1.15.1) can be expanded into standard Fourier series, ̂ j (t)eijx , u(x, t) = ∑ u j∈Zn

̂ 0 = 0, and where j = {j1 , j2 } or {j1 , j2 , j3 }. Suppose that ∫Ω u(x, t)dx = 0, hence u ̂j = u ̂j . u

(1.15.6)

1.15 The relation between attractor and turbulence

| 195

Due to incompressibility condition div u = 0, ̂ j = 0, j⋅u

(1.15.7)

∀j.

Assume that f (x) and pressure p also have Fourier series expansions, p(x, t) = ∑ p̂ j (t)eij⋅x .

f (x) = ∑ fĵ eij⋅x , j∈Zn

j∈Zn

Then Navier–Stokes equation is equivalent to the algebraic differential equations ̂j du dt

̂j + i + 4π 2 νu

∑

k,l∈Zn ,k+l=j

̂ j = fj , ̂ k )u ̂ l + ijp (l ⋅ u

∀j ≠ 0.

From equation (1.15.7) we get ̂j du dt

̂ j + i ∑ (j ⋅ u ̂ k )u ̂ j−k + ijp̂ k = fj , + 4π 2 νu k∈Zn

∀j ≠ 0.

(1.15.8)

Taking the inner product of (1.15.8) and j, and using equation (1.15.7), we obtain the ̂ j as expression of p ̂ j = −i p

j0 ⋅ f̂j |j|2

̂ k )(j ⋅ u ̂ j−k ). − ∑ (j ⋅ u k∈Zn

̂ j into equation (1.15.8), we have Substituting p ̂j du dt

̂ j + i ∑ (j ⋅ u ̂ k ){j ⋅ u ̂ j−k − + 4π 2 νu k∈Zn

j j j j ̂ )} = ̂fj − ( ⋅ ̂fj ). ( ⋅u |j| |j| j−k |j| |j|

(1.15.9)

If |j| is large, ̂fj disappears, so that ̂f = 0, j

|j| > j0 .

Suppose ̂fj is exponentially decaying in |j|, that is, there exist σ1 , σ2 > 0 such that |̂fj | ≤ σ1 e−σ2 |j| ,

∀j.

Foias et al. [64] proved the following: Theorem 1.15.2. Suppose that the space dimension is 2. Then there exist constants k2 , k3 , which only depend on ν, |f |0 , σ1 , σ2 , and t2 depending on ν and κ0 such that k 󵄨󵄨 ̂ 󵄨󵄨 −k |j| 󵄨󵄨uj (t)󵄨󵄨 ≤ ( 2 )e 3 , |j|

∀j ≠ 0, ∀t ≥ t2 .

(1.15.10)

196 | 1 Attractor and its dimension estimation When the space dimension is 3, as long as smoothness is kept, similar results are also valid. These results are consistent with the Kolmogorov turbulence. Equation (1.15.10) is completely consistent with the dissipation expression of Kolmogorov turbulence. The results are more detailed than Kolmogorov’s given in 1941, because they concretely apply to the decision flow and allow obtaining the obvious exponential decay rate. Meanwhile, their precision is inferior to the results of Kolmogorov, because the constants k2 , k3 are too large. 1.15.3 The dimension of the attractor Navier–Stokes equation can be written in functional form as du + νAu + B(u, u) = f , dt

(1.15.11)

where A is Stokes operator, B(⋅, ⋅) is a nonlinear term, f is external force, u = u(t) (= u(⋅, t)). Considering the solution u(t) of the linear operator for equation (1.15.11), Φ → νAΦ + B(u(t), Φ) + B(Φ, u(t)), we need to estimate the time average of the trace of the finite dimensional projection operator for any possible orbit u(t). For the proper function family Φj , j = 1, 2, . . . , m 1

(these functions are assumed to satisfy Φj ∈ D(A 2 ), and are orthogonal in L2 (Ω)m , 1

D(A 2 ) = V, ϕ ∈ L2 , ‖ϕ‖ = |∇ϕ|0 < +∞). We consider the following sum: m

∑⟨(νAΦj + B(u(t), Φj ) + B(Φj , u(t)), Φ)⟩ j=1

m

= ∑{ν‖Φj ‖2 + B(Φj , u(t), Φj )}. j=1

Let qm (t) be a lower bound of the above sum, where for any orbit and any Φj on the attractor, let T

1 qm = lim inf ∫ qm (t)dt. T→∞ T 0

We can prove that if m is large enough, qm > 0, and then the dimension of the attractor 1 is m. Suppose that the standard length (i. e., for the diameter or |Ω| n ) of field Ω is l0 = k0−1 . Kolmogorov dissipation length ld = kd−1 is defined as 1

Δ3 4 ld = ( ) , ε

1.15 The relation between attractor and turbulence

| 197

where ε is the energy when |∇u(x, t)|2 > average dissipation rate. The average rate of the attractor A is T

ε = ν lim sup{ sup t→∞

u(0)∈A

1 󵄨2 󵄨 ∫ sup 󵄨󵄨∇u(τ, x)󵄨󵄨󵄨 dτ}. t x∈Ω 󵄨 0

Constantin et al. [32] gave the following result: An upper bound of Hausdorff and fractal dimensions of attractor A is C(

n

kd ) , k0

n = 3,

(1.15.12)

where C is an absolute constant. The following remark is compelling. Let |∇v| denote an upper bound of |∇v(x)|, x ∈ Ω, v ∈ A . Obviously, ε ≤ ε = ν(|∇v|)2 ,

1

1

kd ≤ kd = (

ε 4 |∇v| 2 ) . ) =( 3 v v

By equation (1.15.12) we deduce that |∇v| ) Cl0n (

dim A ≤

1 n 2

v

,

n = 3.

(1.15.13)

1

v 2 The length of ( |∇v| ) is called the smallest scale by Henshow et al. (1991) and Bartuccelli et al. (1990) [10]. Constantin, Foias, Temam and coauthors have proved in [32] that

l2 ( 0 ) ≤ G. ld Hence dim A ≤ CG,

(1.15.14)

where G is Grashof number 1

1

G=

2 L 2 n−3 󵄨 󵄨2 {∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx} , 2 ν

Ω

1

1

with ν denoting dynamic viscosity, L the diameter of Ω or |Ω| n , and Re = G 2 . This result improves those of Babin and Vishik (1983) [7], Ladyzhenskaya (1982, 1985), Foias and Temam (1983) [61].

198 | 1 Attractor and its dimension estimation For the periodic case, the result has been further improved. Constantin et al. [87] have proved 1

2

3 l l dim A ≤ C( 0 ) (1 + ln( 0 )) , lη lη

(1.15.15)

where Kraichnan dissipation length lη replaces with the Kolmogorov dissipative length ld , 1

ν 6 lη = ( 3 ) , η and η is the space and time average of ν|curl curl u(x, t)2 | on the attractor A , that is, t

η = ν lim sup sup ∫ t→∞

Meanwhile, this proves

l0 lη

u(0)∈A

0

1 󵄨 󵄨2 ∫ 󵄨󵄨󵄨Δu(x, s)󵄨󵄨󵄨 dxds. |Ω| Ω

1

≤ G 3 . Hence by equation (1.15.15) one obtains 2

1

dim A ≤ CG 3 (1 + ln G) 3 .

(1.15.16)

A simplified proof of equations (1.15.15) and (1.15.16) appeared in Doering and Gillon (1991) and Ghidagila, Temam (1990) [77].

1.15.4 The best dimensional estimate of attractor Babin and Vishik [36] considered the situation of a spacial periodic flow in a narrow and long area Ω = (0, 2πL ) × (0, 2πL). Firstly, Ghidaglia and Temam (1990) proved: α 1

2

3 G3 g )(ln(1 + 1 )) , dim A ≤ C(1 + α α4

α ≠ 1,

(1.15.17)

where C is an absolute constant. When α → 0, we have the estimate dim A ≤ C(1 +

G

1

α2

).

(1.15.18)

Babin and Vishik have given an estimate of a lower bound of dim A . Suppose that the pressure f (x1 , x2 ) possesses the form f (x1 , x2 ) = (g(x2 ), 0),

(1.15.19)

1.15 The relation between attractor and turbulence

| 199

2πL

∫ g(x2 )dx2 = 0, 0

and Grashof number G is G=

G

1

α2

(1.15.20)

, 1

1 2

2πL

(2π) 2 ( ∫ g(y)dy) . G= ν2 0

It is easy to see that Navier–Stokes equation possesses a stationary solution us , ps , ps = 0, us = (U(x2 ), 0), −νU 󸀠󸀠 (x2 ) = g(x2 ), x2

2πL

0

0

x 1 U(x2 ) = − ∫(x2 ⋅ s)g(s)ds − 2 ∫ sg(s)ds + const. ν 2πLν Choosing the constant such that 2πL

∫ U(x2 )dx2 = 0 0

for small α, the region Ω becomes slender and the stationary solution becomes unstable. For the time being, we know that the global attractor contains A steady solution us and its unstable manifold of Uuns . Thus dim A ≥ dim Uuns . By using the properties of Orr–Somerfeld equation, one shows that there exists at least κ unstable norm, where the parameter κ depends on G, and α dim A ≥

κ(G) . α

(1.15.21)

Ghidaglia and Temam [76] generalized these results in dimension 3. Let Ω = (0, 2πL ) × (0, 2πL) × (0, 2πL ), 0 < β ≤ α, and α β f (x1 , x2 , x3 ) = (g(x2 ), 0, 0). When α is small enough, the stationary solution us belongs to unstable manifold and so dim A ≥ dim Uuns ≥

κ(G) . αβ

200 | 1 Attractor and its dimension estimation On the other hand, we can estimate dim U ≤ C

(1 + Re3 ) , αβ

where 1 2

t

L 1 󵄨2 󵄨 Re = sup lim sup( ∫ sup 󵄨󵄨󵄨u(x, s)󵄨󵄨󵄨 ds) < ∞. 2 u t→∞ t x∈Ω 0

Hence dim U ≃

κ . αβ

This is the best dimensional estimate of the invariant set U .

2 Inertial manifold The concept of inertial manifold was first put forward by Foias, Sell and Temam in 1985. The inertial manifold is a finite-dimensional manifold which is at least Lipschitz continuous, is invariant in the phase space, approaches the trajectory exponentially, and contains the global attractor [35]. In [65], the authors studied the general initial value problems of nonlinear evolution equations du + Au = f (u), dt u(0) = u0 , where the nonlinear semigroup S(t) defined on a Hilbert space is a self-adjoint operator. In 1988, Chow and Lu [28] took investigated a general equation with the bounded nonlinear term f ∈ C 1 in a Banach space, but the index attracting to manifold was not proved to be on the phase space of bounded subsets uniformly. Mallet-Paret and Sell [171] introduced the space average principle in 1988; for the case, when the spectral gap condition is not completely satisfied, they proved the existence of inertial manifolds for diffusion equations. In 1988, Constantin et al. [34] tried to portray the spectral gap condition by the concept of spectral barrier in a Hilbert space. In 1990, Bernal [17] considered the situation in a Banach space, but the proof was more complex, and the spectral gap condition was demanded more critically. In 1991, Demengel, Ghidaglia [47] first demonstrated the existence of an inertial manifold in a Hilbert space when A is self-adjoint and f is unbounded. In 1993, Debussche and Temam [43] gave another proof when f was essentially unbounded. In 1990, Debussche [40] by using Sacker’s equation gave another proof of the existence of the manifold for a general Banach space with f ∈ C 1 . In 1991, Fabes, Luskin and Sell [54] used the elliptic regularization method to construct the inertial manifold. For a Hilbert space, when A is a self-adjoint operator, the proof of existence of inertial manifolds could be found in [42] and [191]. For the construction of an inertial manifold in a Hilbert space, the strong extrusion and cone conditions, we refer to [188]. Conway, Hoff and Smoller [37] in 1978, Mane [172] in 1981, and Mora [177] in 1983 studied the reaction diffusion equations and parabolic equations which yielded the proof of existence of inertial manifolds. There were lots of works which developed general theory to be applicable to some specific equations. Especially, for the estimation of the minimum dimension of inertial manifolds; for instance, for the KS equation, see Foias et al. [69] and Temam and Wang [199]; for Cahn–Hilliard equation, nonlocal Burgers equation and some reaction diffusion equations, see Constantin et al. [34]; for compressible gas dynamics model, see Nicolaenko [180]; for one dimensional reaction–diffusion equations (including explicit structure of inertial manifolds), see Jolly [145]; for Swift–Hohenberg https://doi.org/10.1515/9783110549652-002

202 | 2 Inertial manifold convection model, see Jaboada [196]; for Ginzburg–Landau equation, see Demengel and Ghidaglia [47]; for phase flow equation, see Bates and Zheng [11], and so on. Many existence results depend on the spectral gap condition, which is very restrictive. For instance, for the Navier–Stokes equations this condition is not satisfied. In 1992, Kwak [158] pointed out that the partial periodic boundary conditions for 2D Navier–Stokes equation, and the ratio of the periods in the two directions being a rational number, could overcome this difficulty. His key idea is converting the original equation into a set of reaction diffusion equations by a nonlinear transformation of dependent variable. And then these equations satisfy the spectral gap condition and possess the asymptotic properties of the original NS equation. Of course, the inertial manifold of reaction diffusion equations is not proved cross-sectionally in manifolds. Therefore, whether any trajectory of NS equation is attracted to the finite-dimensional manifold exponentially is still an open question. New progress on inertial manifolds is mainly visible when discussing: (1) Generalized properties of equations, the accuracy of the spectral gap condition, the completely asymptotic behavior of inertial manifolds; (2) The existence of continuous invariant foliation, the growth features of a complete trajectory, u(t) = O(e−σt ), σ > 0, t → −∞; (3) C 1 regularity and normal hyperbolicity; (4) C m,α regularity. For these aspects we can refer to Rasa and Temam [189]. In 1995, Guo [26, 89] proved the existence of an inertial manifold of generalized Kuramoto–Sivashinsky type; in 1996, Gao and Guo [91] proved the existence of a finite-dimensional inertial manifold of the one-dimensional generalized Ginzburg–Landau equation. For other related work, see [66, 146, 173, 174, 198, 41, 63, 52, 44, 47, 205].

2.1 The inertial manifold for a class of nonlinear evolution equations In a Hilbert space H, let an inner product (⋅, ⋅) be given. The nonlinear evolution equation has the form du + Au + R(u) = 0, dt

(2.1.1)

R(u) = B(u, u) + Cu − f ,

(2.1.2)

where

A is a linear unbounded self-adjoint operator on H, and D(A) is dense in H. Let A be positive, that is, (Av, v) > 0,

∀v ∈ D(A), ν ≠ 0.

2.1 The inertial manifold for a class of nonlinear evolution equations | 203

Assume A−1 is compact and the mapping u → Au is an isomorphism from D(A) to H; As means the sth power of A for s ∈ R. The space V2s = D(As ) is a Hilbert space, which is endowed with the inner product (u, v)2s = (As u, As v),

∀u, v ∈ D(As ),

1

u ∈ Vs , and |u|s = (u, u)s2 . Because A−1 is compact and self-adjoint, there exists an orthogonal basis {ωj } of H which consists of the eigenvectors of A, Aωj = λj ωj ,

(2.1.3)

and the eigenvalues satisfy 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ,

λj → +∞,

j → ∞.

(2.1.4)

From equations (2.1.3)–(2.1.4) we readily get 1 1 󵄨󵄨 21 󵄨󵄨 󵄨󵄨A u󵄨󵄨 ≥ λ12 |u|, u ∈ D(A 2 ), 1 󵄨 p 󵄨 󵄨󵄨 p+ 21 󵄨󵄨 p+ 1 󵄨󵄨A u󵄨󵄨 ≥ λ12 󵄨󵄨󵄨A u󵄨󵄨󵄨, ∀u ∈ D(A 2 ), ∀p.

(2.1.5) (2.1.6)

Let PN be the orthogonal projection of H in the subspace spanned by {ω1 , ω2 , . . . , ωN }, N = 1, 2, . . . , QN = I − PN . The nonlinear terms R(u), B(u, u) in (2.1.2) are bilinear oper1 ators, D(A) × D(A) → H, C is a linear operator from D(A) to H, and f ∈ D(A 2 ). Furthermore, set (B(u, v), v) = 0,

∀u, v ∈ D(A),

(2.1.7)

1 1 󵄨󵄨 󵄨 󵄨 1 󵄨1 󵄨 1 󵄨1 󵄨󵄨B(u, v)󵄨󵄨󵄨 ≤ C1 |u| 2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 2 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨 2 |Av| 2 , 1 󵄨 1 󵄨1 |Cu| ≤ C2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 2 |Av| 2 ,

∀u, v ∈ D(A),

∀u ∈ D(A),

(2.1.8) (2.1.9)

where C1 , C2 and the following Ci , i = 3, 4, are positive constants. For operators B and C, the following continuity properties are assumed: 󵄨 󵄨󵄨 21 󵄨󵄨A B(u, v)󵄨󵄨󵄨 ≤ C3 |Au||Av|, ∀u, v ∈ D(A), 󵄨󵄨 21 󵄨󵄨 󵄨󵄨A Cu󵄨󵄨 ≤ C4 |Au|, ∀u ∈ D(A).

(2.1.10) (2.1.11)

Finally, if A + C is positive, then 󵄨 1 󵄨2 ((A + C)u, u) ≥ α󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ,

∀u ∈ D(A), α > 0.

(2.1.12)

Consider the initial value problem of equation (2.1.1), i. e., equation (2.1.1) with the initial condition u(0) = u0 ∈ H.

(2.1.13)

204 | 2 Inertial manifold Assume that problem (2.1.1), (2.1.13) has a unique solution S(t)u0 , ∀t ∈ R+ , S(t)u0 ∈ D(A), ∀t ∈ R+ . The mapping S(t) has the property of an ordinary semigroup. Now we establish an a priori estimate for equation (2.1.1). To proceed, we need the following inequality and lemma: for β > 0, p1 + q1 = 1, 1 < p, q < +∞, we have p β−q 󵄨 β 󵄨 Σ|xi yi | = Σ|βxi |󵄨󵄨󵄨β−1 yi 󵄨󵄨󵄨 ≤ (Σ|xi |p ) + (Σ|yi |q ). p q

(2.1.14)

Lemma 2.1.1. Assume that g(t), h(t), y(t) are three positive integrable functions, t0 ≤ t < ∞, which satisfy dy ≤ gy + h, dt

∀t ≥ t0 ,

(2.1.15)

and t−1

t+1

∫ g(s)ds ≤ α1 , ∫ h(s)ds ≤ α2 , t

t−1

∫ y(s)ds ≤ α3 ,

t

∀t ≥ t0 ,

(2.1.16)

t

where α1 , α2 , α3 are positive constants. Then we have y(1 + t) ≤ (α3 + α2 ) exp(α1 ),

∀t ≥ t0 .

(2.1.17)

Taking the inner product of (2.1.1) and u, and using equations (2.1.7) and (2.1.12), we get 1 1 d 2 −1 󵄨 1 󵄨2 󵄨 󵄨 󵄨 1 󵄨 α 󵄨 1 󵄨2 |u| + α󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(f , u)󵄨󵄨󵄨 ≤ λ1 2 |f |󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + |f |2 , 2 dt 2 2λ1 α which yields d 2 d 1 󵄨 1 󵄨2 |u| + αλ1 |u|2 ≤ |u|2 + α󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ |f |2 . dt dt αλ1

(2.1.18)

Taking the inner product of equation (2.1.1) and Δu, from equations (2.1.8), (2.1.9) and (2.1.14), we have 3 3 1 󵄨󵄨 󵄨 1 󵄨 󵄨 󵄨 1 󵄨1 󵄨󵄨B(u, u) + (u, Au)󵄨󵄨󵄨 ≤ C|u| 2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨|Au| 2 + C2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 2 |Au| 2 1 󵄨 1 󵄨4 󵄨 1 󵄨2 ≤ 54(C14 |u|2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + C24 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ) + |Au|2 . 4

Similarly, we have |(f , Au)| ≤ |f ||Au| ≤ |f |2 + 41 |Au|2 . From this we get 1 d 󵄨󵄨 21 󵄨󵄨2 d 󵄨 1 󵄨2 󵄨 1 󵄨2 2 󵄨󵄨A u󵄨󵄨 + λ1 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + |Au| 2 dt dt 󵄨 1 󵄨4 󵄨 1 󵄨2 ≤ C6 |u|2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + C7 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + 2|f |2 .

(2.1.19)

2.1 The inertial manifold for a class of nonlinear evolution equations | 205

Taking the inner product of equation (2.1.1) and A2 u, we have 1 d 󵄨 󵄨 󵄨 󵄨 3 󵄨2 󵄨 |Au|2 + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨B(u, u) + (Cu, A2 u)󵄨󵄨󵄨 + 󵄨󵄨󵄨(f , A2 u)󵄨󵄨󵄨 2 dt 3 3 󵄨 1 󵄨 󵄨 1 󵄨 ≤ 󵄨󵄨󵄨(A 2 (B(u, u) + Cu), A 2 u)󵄨󵄨󵄨 + 󵄨󵄨󵄨(A 2 f , A 2 u)󵄨󵄨󵄨 󵄨 3 󵄨 󵄨 1 󵄨󵄨 3 󵄨 󵄨 3 󵄨 ≤ C3 |Au|2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + C4 |Au|󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + 󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 1 1 3 󵄨 1 󵄨2 1 󵄨 3 󵄨2 ≤ C8 |Au|4 + C9 |Au|2 + 󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨 + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 , 2 2 2 2 which yields d d 󵄨 3 󵄨2 |Au|2 + λ1 |Au|2 ≤ |Au|2 + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 dt dt

󵄨2 󵄨 ≤ C8 |Au|4 + C9 |Au|2 + 3󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨 . 1

(2.1.20)

Applying equation (2.1.15) to (2.1.18), we have u(t) = S(t)u0 , 󵄨󵄨 󵄨2 󵄨 󵄨2 2 󵄨󵄨u(t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 exp(−αλ1 t) + ρ0 (1 − exp(−αλ1 t)), 1 |f |. 2λ1

where ρ0 =

(2.1.21)

Hence |u(t)| is uniformly bounded in t, and we have 󵄨 󵄨2 lim sup 󵄨󵄨󵄨u(t)󵄨󵄨󵄨 ≤ ρ20 .

t→∞

t+1

From equation (2.1.18), we have that ∫t (2.1.19), we get

(2.1.22)

1

|A 2 u|2 ds is uniformly bounded. By equation

1 d 21 2 󵄨 1 󵄨4 |A u| ≤ C10 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + (C7 − λ1 )|A 2 u|2 + 2|f |2 , dt

where C10 = C6 b20 , |u(t)|2 ≤ b20 , t ≥ 0. Then by Lemma 2.1.1, setting 󵄨 1 󵄨2 󵄨 1 󵄨2 g = C10 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 , y = 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 , 󵄨 1 󵄨2 h = (C7 − λ1 )󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + 2|f |2 , 1

we get that |A 2 u|2 is uniformly bounded in H. By equation (2.1.19), we know that t+1 ∫t |Au(s)|2 ds is uniformly bounded. Moreover, by equation (2.1.20), we get |Au(t)|2 t+1

and ∫t

3

|A 2 u(s)|2 ds are uniformly bounded in t. Hence 󵄨 1 󵄨2 lim sup 󵄨󵄨󵄨A 2 u(t)󵄨󵄨󵄨 ≤ ρ21 , t→+∞ 󵄨 󵄨2 lim sup 󵄨󵄨󵄨Au(t)󵄨󵄨󵄨 ≤ ρ22 . t→+∞

(2.1.23) (2.1.24)

206 | 2 Inertial manifold From equations (2.1.22), (2.1.23) and (2.1.24) we know that any solution of equation (2.1.1) enters into the following balls after some time(t ≥ t0 > 0), B0 = {x ∈ H, |x| ≤ 2ρ0 }, 1 󵄨 1 󵄨 B1 = {x ∈ D(A 2 ), 󵄨󵄨󵄨A 2 x 󵄨󵄨󵄨 ≤ 2ρ1 }, B2 = {x ∈ D(A), |Ax| ≤ 2ρ2 }, respectively, where the ω-limit set of B2 given by A = ω(B2 ) = ⋂ Cl(⋂ S(t)B2 ) s≥0

t≥s

is the global attractor of equation (2.1.1); the closure Cl is taken on the set H, and A ⊆ B2 ∩ B1 ∩ B0 .

We consider the inertial manifold of a truncated equation of (2.1.1). Let θ(s) be a smooth function from R+ to [0, 1]: θ(s) = 1, 0 ≤ s ≤ 1; θ(s) = 0, s ≥ 2, |θ󸀠 (s)| ≤ 2, s ≥ 0. Fix ρ = 2ρ2 and define s θρ (s) = θ( ), ρ

s ≥ 0.

Then the inertial manifold equation of (2.1.1) is du + Au + θρ (|Au|)R(u) = 0, dt

(2.1.25)

for which one can prove directly the existence and uniqueness of solution to (2.1.25) with initial value u(0) = u0 ∈ H. Obviously, when |Au| ≤ ρ, θs (|Au|) = 1, equation (2.1.24) and equation (2.1.1) are consistent. When |Au| ≥ 2ρ, θρ (|Au|) = 0, taking the inner product of equation (2.1.24) and A2 u, we get 1 d 1 d 󵄨 3 󵄨2 |Au|2 + λ1 |Au|2 ≤ |Au|2 + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ 0. 2 dt 2 dt Thus the trajectory u(t) will exponentially converge to the ball with radius ρ3 ≥ 2ρ in D(A). In addition, R(u) is locally Lipschitz continuous in u. And F(u) = θρ (|u|)R(u) is globally Lipschitz continuous. That is, there exists K, such that 󵄨󵄨 󵄨 󵄨󵄨F(u) − F(v)󵄨󵄨󵄨 ≤ K|u − v|,

∀u, v ∈ H.

(2.1.26)

Definition 2.1.1. The inertial manifold for a semigroup {S(t)}t≥0 is a finite-dimensional smooth manifold μ ∈ H (at least Lipschitz), which satisfies the following properties: (1) μ is invariant, that is, S(t)μ ⊂ μ;

(2.1.27)

2.1 The inertial manifold for a class of nonlinear evolution equations | 207

(2) μ attracts all the solutions of equation (2.1.25) exponentially. That is, there exist constants k1 > 0, k2 > 0 such that for u0 ∈ H, we have dist(S(t)u0 , μ) ≤ k1 e−k2 t ,

∀t ≥ 0;

(2.1.28)

(3) The attractor A belongs to μ. Now we will construct the inertial manifold, and in doing so we will prove its existence. Suppose PN is an N-dimensional orthogonal projection of H, QN = I − PN . And denote P = PN , Q = QN . Set u(t) to be the solution of equation (2.1.25). Let p(t) = Pu, q(t) = Qu. Then p(t), q(t) belong to PH and QH and satisfy dp + Ap + PF(u) = 0, dt dq + Aq + QF(u) = 0, dt

(2.1.29) (2.1.30)

where F(u) = θρ (|Au|)R(u), and u = p + q. We are looking for the inertial manifold μ, which can be constructed as the graph of the Lipschitz function Φ : PD(A) → QD(A). That is, μ = graph Φ. The function Φ is an operator, which is obtained by the fixed point of a function Fb,l , which is a function Φ : PD(A) → QD(A), satisfying 󵄨󵄨 󵄨 󵄨󵄨AΦ(p)󵄨󵄨󵄨 ≤ b, ∀p ∈ PD(A), 󵄨󵄨 󵄨 󵄨󵄨AΦ(p1 ) − AΦ(p2 )󵄨󵄨󵄨 ≤ l|Ap1 − Ap2 |, ∀p1 , p2 ∈ PD(A), supp Φ ⊂ {p ∈ PD(A) | |Ap| ≤ 4ρ},

(2.1.31) (2.1.32)

where b > 0, l > 0. When p = p(t), q = Φ(p(t)) satisfy equations (2.1.29)–(2.1.30), u = p(t)+Φ(p(t)) is the solution of (2.1.25). Suppose that Φ is given by Fb,l , p0 ∈ PD(A), then the solution p = p(t; p0 , Φ) for equation dp + Ap + PF(p + Φ(p)) = 0, dt

p(0) = p0 ,

(2.1.33)

exists and is unique. This is because σ → θρ R(σ + Φ(σ)) is Lipschitz continuous. Since p = p(t; p0 , Φ), ∀t ∈ R, similar to equation (2.1.30), we have dq + Aq + QF(p + Φ(p)) = 0, dt

p(0) = p0 .

(2.1.34)

Since QF(p + Φ(p)) is bounded, R → H, a solution of equation (2.1.34) exists and is unique; it is a bounded solution q(t) when t → ∞. From this, we get 0

q(0) = − ∫ eτAQ QF(p + Φ(p))dτ, −∞

(2.1.35)

208 | 2 Inertial manifold where p = p(τ) = p(τ, Φ, p0 ). This q(0) depends on Φ ∈ Fb,l and p ∈ PD(A); q(0) = q(0, p0 , Φ). Function p0 ∈ PD(A) → q(0; p0 , Φ) ∈ QD(A) maps PD(A) into QD(A), and is denoted by F Φ. Hence 0

F Φ(p0 ) = − ∫ e

τAQ

QF(u)dτ,

(2.1.36)

−∞

where u = u(τ) = p(τ, Φ, p0 ) + Φ(p(τ; Φ, p0 )). Since q(0) = Φ(p0 ), we seek conditions on N, b and l such that (1) F maps Fb,l into itself; (2) F is dense in Fb,l . Now we consider the function Φ : PD(A) → QD(A), P = PN , Q = QN = I − PN . Then Φ ∈ Fb,l , i. e., equations (2.1.31)–(2.1.33) are satisfied. With the distance 󵄨 󵄨 ‖Φ − Ψ‖ = sup 󵄨󵄨󵄨AΦ(p) − AΨ(p)󵄨󵄨󵄨, p∈D(A)

(2.1.37)

Fb,l is a complete metric space. For Φ ∈ Fb,l , the mapping F is defined on PD(A) as 0

F Φ(p0 ) = − ∫ e

τAQ

QF(u)dτ,

p0 ∈ PD(A),

(2.1.38)

−∞

where u(τ) = p(τ; Φ, p0 ) + Φ(p(τ; Φ, p0 )), p(τ; Φ, p0 ) is the solution of equation (2.1.29) such that p(0; Φ, p0 ) = p0 . In the following we study the properties of operator F . Lemma 2.1.2. Let α > 0, and τ < 0. The operator (AQ)α eτAQ is linear and continuous on QH, its norm on F (that is, |(AQ)α eτAQ |0p ) is bounded by K3 |τ|−α ,

α λN+1 eτλN+1 ,

−1 when − αλN+1 ≤ τ < 0;

−1 when − ∞ < τ ≤ −αλN−1 .

Proof. Let v = ∑∞ j=N+1 bj ωj be an element of QH. Then ∞

󵄨󵄨 α τAQ 󵄨2 α τλ 2 󵄨󵄨(AQ) e v󵄨󵄨󵄨 = ∑ (λj e j )bj j=N+1

≤ sup (λα eτλ )Σb2j λ≥λN+1

2

= sup (λα eτλ ) |v|2 . λ≥λN+1

(2.1.39)

2.1 The inertial manifold for a class of nonlinear evolution equations | 209

Hence 󵄨󵄨 α τAQ 󵄨 α τλ 󵄨󵄨(AQ) e 󵄨󵄨󵄨0p ≤ sup λ e . λ≥λN−1

Through elementary calculation, this shows that sup (λα eτAλ ) = {

λ≥λN+1

|τ|−α (αe−1 )α , α λN+1 eτλN+1 ,

−1 when − αλN+1 ≤ τ < 0, −1 when τ ≤ −αλN+1 .

Hence we get the equation (2.1.39), where K3 = K3 (α) = (αe−1 )α . As the direct corollary of equation (2.1.39), we have 0

󵄨 󵄨 −α−1 , ∫ 󵄨󵄨󵄨(AQ)α eτAQ 󵄨󵄨󵄨0p dτ ≤ (1 − α)−1 eα λN+1

0 < α < 1.

(2.1.40)

−∞

From equations (2.1.10) and (2.1.11) we get 1 󵄨󵄨 󵄨 󵄨 1 󵄨 󵄨 1 󵄨 󵄨 1 󵄨 󵄨󵄨(AQ) 2 R(u)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨A 2 B(u, u)󵄨󵄨󵄨 + 󵄨󵄨󵄨A 2 Cu󵄨󵄨󵄨 + 󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨 󵄨 1 󵄨 ≤ C3 |Au|2 − C4 |Au| + 󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨.

When |Au| > 2ρ and θρ (|Au|) = 0, we have 1 󵄨󵄨 󵄨 󵄨󵄨(AQ) 2 F(u)󵄨󵄨󵄨 ≤ K4 ,

(2.1.41)

1 2

where K4 = 4C3 ρ2 + 2C4 ρ + |A f | and F(u) is determined by equation (2.1.30). Lemma 2.1.3. Let p0 ∈ PD(A), F Φ(p0 ) ∈ QD(A). Then − 21 󵄨󵄨 󵄨 , 󵄨󵄨AF Φ(p0 )󵄨󵄨󵄨 ≤ K5 λN+1 1 5 − 󵄨 󵄨󵄨 4 4 , 󵄨󵄨A F Φ(p0 )󵄨󵄨󵄨 ≤ K6 λN+1

where K5 , K6 are absolute constants, independent of p0 , Φ. Proof. Since QeτAQ = eτAQ , it is easy to see that F Φ(p0 ) ∈ QD(A) and 0

󵄨 󵄨 󵄨󵄨 󵄨 τAQ 󵄨󵄨AF Φ(p0 )󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨AQe F(u)󵄨󵄨󵄨dτ −∞ 0

1 1 󵄨 󵄨 󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨(AQ) 2 eτAQ 󵄨󵄨󵄨0p 󵄨󵄨󵄨(AQ) 2 F(u)󵄨󵄨󵄨dτ,

−∞ 0

5 󵄨󵄨 45 󵄨 󵄨 󵄨 τAQ 󵄨󵄨A F Φ(p0 )󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨(AQ) 4 e F(u)󵄨󵄨󵄨dτ

−∞ 0

3 1 󵄨 󵄨 󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨(AQ) 4 eτAQ 󵄨󵄨󵄨0p 󵄨󵄨󵄨(AQ) 2 F(u)󵄨󵄨󵄨dτ.

−∞

(2.1.42) (2.1.43)

210 | 2 Inertial manifold Inequalities (2.1.43)–(2.1.42) can be obtained from equations (2.1.40)–(2.1.41). Now we take −1

2 b = K5 λN+1 .

(2.1.44)

Then for Φ ∈ Fb,l , similar to equation (2.1.31), F Φ satisfies 󵄨 󵄨󵄨 󵄨󵄨AF Φ(p0 )󵄨󵄨󵄨 ≤ b,

∀p0 ∈ PD(A). 5

(2.1.45) 1

By equation (2.1.43), F Φ is a bounded set in D(A 4 ). Since A− 4 is compact, the range of F Φ is a compact subset of QD(A), which does not depend on Φ. Now we consider the properties of the support and continuity properties of F Φ. Lemma 2.1.4. For every Φ ∈ Fb,l , the support of F Φ is included in {p ∈ PD(A); |Ap| ≤ 4ρ}. Proof. Set u = p + Φ(p). If |Ap| > 2ρ, then 󵄨 󵄨2 1 |Au| = (|Ap|2 + 󵄨󵄨󵄨AΦ(p)󵄨󵄨󵄨 ) 2 ≥ |Ap| > 2ρ. Hence, θρ (|Au|) = 0. Now if we set |Ap0 | > 4ρ, then in some interval of t, |Ap(t)| > 2ρ. Then equation (2.1.28) becomes dp + Ap = 0, dt from which we infer that 1 d 1 d|Ap|2 󵄨󵄨 32 󵄨󵄨2 |Ap|2 + λ1 |Ap|2 ≤ + 󵄨󵄨A p󵄨󵄨 = 0. 2 dt 2 dt Hence, for τ > 0, we have 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2ρ < 󵄨󵄨󵄨Ap(0)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Ap(τ)󵄨󵄨󵄨 exp(λ1 τ) ≤ 󵄨󵄨󵄨Ap(τ)󵄨󵄨󵄨. Thus θρ (|Au(τ)|) = 0, ∀τ < 0. By virtue of equation (2.1.38), we have F Φ(p0 ) = 0,

∀Φ ∈ Fb,l .

To proceed, we verify the Lipschitz property of nonlinear F(u). Lemma 2.1.5. If p1 , p2 ∈ PD(A), Φ1 , Φ2 ∈ Fb,l , ui = pi + Φi (pi ), then 1 󵄨󵄨 21 󵄨 󵄨󵄨A F(u1 ) − A 2 F(u2 )󵄨󵄨󵄨 ≤ K7 [(1 + l)|Ap1 − Ap2 | + ‖Φ1 − Φ2 ‖],

where constant K7 does not depend on pi or Φi , i = 1, 2.

(2.1.46)

2.1 The inertial manifold for a class of nonlinear evolution equations | 211

Proof. First, noting equations (2.1.10) and (2.1.10), we deduce 1 󵄨 󵄨󵄨 21 󵄨󵄨A R(u1 ) − A 2 R(u2 )󵄨󵄨󵄨 󵄨 1 󵄨 󵄨 1 󵄨 ≤ 󵄨󵄨󵄨A 2 [B(u1 , u1 ) − B(u1 , u2 ) + B(u1 , u2 ) − B(u2 , u2 )]󵄨󵄨󵄨 + 󵄨󵄨󵄨A 2 C(u1 − u2 )󵄨󵄨󵄨 ≤ C3 (|Au1 | + |Au2 |)|Au1 − Au2 | + C4 |Au1 − Au2 |

and 󵄨󵄨 21 󵄨 2 2 󵄨 1 󵄨 󵄨󵄨A R(u1 )󵄨󵄨󵄨 ≤ C3 |Au1 | + C4 |Au1 | + 󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨. Define G as follows: 1

1

G = A 2 F(u1 ) − A 2 F(u2 ) 1

1

= θρ (|Au1 |)A 2 R(u1 ) − θρ (|Au2 |)A 2 R(u2 ). We discuss three different cases: (1) 2ρ ≤ |Au1 |, 2ρ ≤ |Au2 |; (2) |Au1 | < 2ρ ≤ |Au2 | or |Au2 | < 2ρ ≤ |Au1 |; (3) |Au1 | ≤ 2ρ, |Au2 | ≤ 2ρ. For case (1), when |Au| ≥ 2ρ, θρ (|Au|) = 0 and |θ󸀠 | ≤ 2ρ−1 , we get G = 0. In case (2), we have 1 󵄨 󵄨 |G| = 󵄨󵄨󵄨θρ (|Au|)A 2 R(u)󵄨󵄨󵄨 1 1 󵄨 󵄨 = 󵄨󵄨󵄨θρ (|Au1 |)A 2 R(u1 ) − θρ (|Au2 |)A 2 R(u1 )󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 1 ≤ 󵄨󵄨󵄨θρ (|Au1 |) − θρ (|Au2 |)󵄨󵄨󵄨󵄨󵄨󵄨A 2 R(u1 )󵄨󵄨󵄨

󵄨 1 󵄨 ≤ 2ρ−1 ‖Au1 ‖ − ‖Au2 ‖(C3 |Au1 |2 + C4 |Au2 |2 + 󵄨󵄨󵄨A 2 f 󵄨󵄨󵄨) + [C3 (|Au1 | + |Au2 |) + C4 ]|Au1 − Au2 |.

Hence 1 󵄨 󵄨󵄨 21 󵄨󵄨A F(u1 ) − A 2 F(u2 )󵄨󵄨󵄨 ≤ K7 |Au1 − Au2 |,

(2.1.47)

1

where K7 = 2ρ−1 (C3 4ρ2 + C4 2ρ + |A 2 f |) + C3 4ρ + C4 . Since u1 − u2 = p1 − p2 + (Φ1 (p1 ) − Φ1 (p2 ))+(Φ1 (p2 )−Φ2 (p2 )), we have |Au1 −Au2 | ≤ (1+l)|Ap1 −Ap2 |+‖Φ1 −Φ2 ‖. Combining with equation (2.1.47), we get equation (2.1.46). Now we prove that under appropriate assumptions, the map F is Lipschitz from

Fb,l to Fb,l , and this is a strict restriction.

First, we set Φ to be fixed, take p01 , p02 ∈ PD(A), and let p = p1 (t), p = p2 (t) be the solutions of equation (2.1.28) which satisfy the respective initial conditions pi (0) = p0i , i = 1, 2.

212 | 2 Inertial manifold Let Δ = p1 − p2 , then Δ satisfies the equation dΔ + AΔ + PF(u1 ) − PF(u2 ) = 0, dt

(2.1.48)

where ui = pi + Φ(pi ), i = 1, 2. Taking the inner product of (2.1.48) with A2 Δ, we get 1 3 1 d 󵄨 3 󵄨2 |AΔ|2 + 󵄨󵄨󵄨A 2 Δ󵄨󵄨󵄨 = −(A 2 P(F(u1 ) − F(u2 )), A 2 Δ). 2 dt

(2.1.49)

By equation (2.1.46), we get 1 d 󵄨 3 󵄨2 󵄨 3 󵄨 |AΔ|2 + 󵄨󵄨󵄨A 2 Δ󵄨󵄨󵄨 ≤ K7 (1 + l)|AΔ|󵄨󵄨󵄨A 2 Δ 󵄨󵄨󵄨. 2 dt 3

3

d Hence |AΔ| dt |AΔ| ≥ −|A 2 Δ|2 − K7 (1 + l)|AΔ||A 2 Δ|. Since Δ ∈ PD(A), we have 1 󵄨󵄨 32 󵄨󵄨 󵄨󵄨A Δ󵄨󵄨 ≤ λN2 |AΔ|,

which yields |AΔ|

1 d |AΔ| ≥ −λN |AΔ|2 − K7 (1 + l)λN2 |AΔ|2 , dt

or equivalently, 1 d |AΔ| + (λN + K7 (1 + l)λN2 )|AΔ| ≥ 0. dt

(2.1.50)

From equation (2.1.50) we get 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨AΔ(τ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨AΔ(0)󵄨󵄨󵄨 exp(−τ(λN + K7 (1 + l)λN2 )), τ ≤ 0.

(2.1.51)

1

Lemma 2.1.6. Assume that γN = λN+1 − λN − K7 (1 + l)λN2 > 0. Then for Φ ∈ Fb,l and p01 , p02 ∈ PD(A), we have 󵄨󵄨 󵄨 󵄨󵄨AF Φ(p01 ) − AF Φ(p02 )󵄨󵄨󵄨 ≤ L|Ap01 − Ap02 |, where −1

1

2 L = K7 (1 + l)λN+1 [1 + (1 − γN αN )−1 ]e− 2 exp(

γN =

λN , λN+1

which implies that Φ ∈ Fb,l .

−1

2 αN = 1 + K7 (1 + l)λN+1 ,

γN αN ), 2

(2.1.52)

2.1 The inertial manifold for a class of nonlinear evolution equations | 213

Proof. By equations (2.1.38) and (2.1.46), we have 0

󵄨 󵄨 󵄨 󵄨󵄨 τAQ 󵄨󵄨AF Φ(p01 ) − AF Φ(p02 )󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨AQe (F(u1 ) − F(u2 ))󵄨󵄨󵄨dτ −∞

0

1 󵄨 󵄨 󵄨 󵄨 ≤ K7 (1 + l) ∫ 󵄨󵄨󵄨(AQ) 2 eτAQ 󵄨󵄨󵄨0p 󵄨󵄨󵄨AΔ(τ)󵄨󵄨󵄨dτ,

−∞

where Δ = p1 − p2 . By Lemma 2.1.2 and equation (2.1.51), we have 0

1 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨(AQ) 2 eτAQ 󵄨󵄨󵄨0p 󵄨󵄨󵄨AΔ(τ)󵄨󵄨󵄨dτ

−∞

−1 − 21 λN+1

−1

−1

2 exp[τ(λN+1 − λN − K7 (1 + l)λN 2 )]dτ ∫ λN+1

≤

−∞

+

0

1 1 1 ∫ K3 ( )|τ|− 2 exp[−τ(λN + K7 (1 + l)λN2 )]dτ|Ap01 − Ap02 |. 2

−1 − 21 λN+1

Through elementary calculations, we show that the right-hand expression is bounded by −1

1

2 e− 2 [1 + (1 − γN αN )−1 ] exp( λN+1

γN αN )|Ap01 − Ap02 |, 2

which proves equation (2.1.52). By equations (2.1.44), (2.1.52) and Lemma 2.1.4, we get F Φ ∈ Fb,l .

Up to now, we have proved that F : Fb,l → Fb,l . Now we prove that the map F is Lipschitz. To this end, we consider two functions Φ1 and Φ2 with the same initial value. Let pi = p(t; Φi , p0 ),

ui = pi + Φi (pi ),

where i = 1, 2.

We estimate |AF Φ1 (p0 ) − AF Φ2 (p0 )|. Using a similar method as before, we get 1 d |AΔ| + λN αN |AΔ| ≥ −K7 λN2 ‖Φ1 − Φ2 ‖, dt

(2.1.53)

1

2 where Δ = p1 − p2 , αN = (1 + K7 (1 + l)λN+1 ). Since Δ(0) = 0, from equation (2.1.53) we get 1

2 󵄨󵄨 󵄨 K7 λN ‖Φ1 − Φ2 ‖ (exp(−αN λN τ)) − 1 󵄨󵄨AΔ(τ)󵄨󵄨󵄨 ≤ α λ 1 2

N N

≤ K7 λN ‖Φ1 − Φ2 ‖ exp(−αN λN τ),

τ ≤ 0.

(2.1.54)

214 | 2 Inertial manifold As in Lemma 2.1.6, from equations (2.1.38),(2.1.46) and (2.1.54), we get 󵄨 󵄨󵄨 󵄨󵄨AF Φ1 (p0 ) − AF Φ2 (p0 )󵄨󵄨󵄨 0

󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨AQeτAQ (F(u1 ) − F(u2 ))󵄨󵄨󵄨dτ −∞

0

1 󵄨 󵄨 ≤ K7 ∫ 󵄨󵄨󵄨(AQ) 2 eτAQ 󵄨󵄨󵄨0p [(1 + l)|AΔ| + ‖Φ1 − Φ2 ‖]dτ

−∞

0

1

− 󵄨 󵄨 ≤ K7 ‖Φ1 − Φ2 ‖ ∫ 󵄨󵄨󵄨(AQ) 2 eτAQ 󵄨󵄨󵄨0p (1 + K7 λN 2 (1 − l)e−λN αN τ )dτ. 1

(2.1.55)

−∞

From Lemma 2.1.2, the integral of the right-hand side of (2.1.55) is bounded by 0

1 −1 1 1 2 2e− 2 λN+1 + ∫ K3 ( )|τ|− 2 exp(−λN αN τ)dτ 2

−α

+ K7 (1 +

−1 l)λN 2 (

−α

−1

2 exp[τ(λN+1 − λN αN )]dτ) ∫ λN+1

−∞

≤ 2e

− 21

− 21 λN+1 −1

1

−1

−1

1

2 + K7 (1 + l)λN 2 [λN+1 e− 2 (1 + (1 − γN αN )−1 ) exp(

−1

γN αN )] 2

2 ≤ 2e− 2 λN+1 + λN 2 L.

Then, we have 󵄨󵄨 󵄨 󸀠 󵄨󵄨AF Φ1 (p0 ) − AF Φ2 (p0 )󵄨󵄨󵄨 ≤ L ‖Φ1 − Φ2 ‖, 1

−1

∀p0 ∈ PD(A),

(2.1.56)

−1

2 + λN 2 L). where L󸀠 = K7 (2e− 2 λN+1 As mentioned above, we seek conditions to ensure that the mapping F is from Fb,l to itself, as well as a strict contractive compression on Fb,l . This requires us to find sufficient conditions (for λN and λN−1 ) to guarantee

L ≤ l;

L󸀠 < 1. −1

Firstly, we note that γN = λN+1 − λN − kl (1 + l)λN 2 > 0 is equivalent to 1 − γN αN > 0,

(2.1.57)

1 > γN αN > 0.

(2.1.58)

or Then from equation (2.1.57) we deduce that −1

2 L ≤ K7 (1 + l)λN+1 [1 + (1 − γN αN )−1 ].

2.1 The inertial manifold for a class of nonlinear evolution equations | 215

In order to have L ≤ l, we need to select N appropriately so that the following two inequalities are valid: 1 l 2 ≤ , K7 (1 + l)λN+1 2 l − 21 −1 K7 (1 + l)λN+1 (1 − γN αN ) ≤ . 2

(2.1.59) (2.1.60)

Inequality (2.1.59) can be rewritten as 1

(2.1.61)

2 , K10 ≤ λN+1

where K10 = 2K7 (1+l)L−1 . Now we select N, such that equation (2.1.61) holds. Inequality (2.1.60) can be written as −1

2 K10 λN+1 ≤ 1 − γN αN ,

(2.1.62)

which is equivalent to −1

−1

1

2 2 K10 λN+1 − 1 + γN + K7 (1 + l)λN+1 γN2 ≤ 0,

where γN =

λN . λN+1

(2.1.63)

Let 1

−1

−1 2 γN2 + K10 λN+1 = (γN λN+1 )

− 21

−1

2 + K10 γN+1 ≤ 1,

(2.1.64)

apply to equation (2.1.64) twice, to obtain −1

1

−1

−1

1

2 2 2 K10 λN+1 − 1 + γN + K10 λN+1 γN2 ≤ K10 λN+1 − 1 + γN2 ≤ 0.

(2.1.65)

Since l ≤ 81 , we have K7 (1 + l) ≤ K10 . By equation (2.1.64), we deduce equation (2.1.63). Moreover, we obtain that (2.1.62) holds. Therefore, in order to ensure the map F takes Fb,l into itself, we need to suppose γN > 0 or 1 − γN αN > 0. This assumption is assured by equation (2.1.62). Sufficient conditions for which the mapping F takes Fb,l into itself are (2.1.59) and (2.1.62), this is assured by equations (2.1.61), (2.1.64). It is easy to see that these two inequalities are corollaries of 1

1

2 K10 ≤ λN+1 − λN2 .

(2.1.66)

In order to let the mapping F be a contraction on Fb,l , we must have L󸀠 < 1. Set L ≤ 21 , then we can deduce 󸀠

1

2 K11 ≤ λN+1 ,

(2.1.67)

216 | 2 Inertial manifold 1

where K11 = 2K7 (2e− 2 +L). Hence, under conditions in (2.1.66) and (2.1.67), the mapping F takes Fb,l into itself and is a contraction. Therefore, there exists a fixed point of F . Now we prove M = graph(Φ) is fixed under the action of S(t). That is, S(t)M ⊂ M,

∀t ≥ 0,

(2.1.68)

and prove that it attracts all the orbits and approaches M exponentially. We firstly prove the invariance of M. In fact, from the expression 0

Φ(p0 ) = − ∫ eτAQ QF(u(τ, p0 ))dτ,

(2.1.69)

−∞

where u(τ, p0 ) = p(τ, Φ, p0 ) + Φ(p(τ), Φ, p0 ), inserting p(t) = p(t; Φ, p0 ) with p0 into equation (2.1.69), and noting that p(τ, Φ, p(t; Φ, p0 )) = p(τ + t; Φ, p0 ), we deduce that 0

Φ(p(t)) = − ∫ eτAQ QF(u(τ, p(t)))dτ −∞ t

= − ∫ e−(t−τ)AQ QF(u(τ), p0 )dτ,

∀t ∈ R.

(2.1.70)

−∞

Differentiating with respect to t, it is readily seen that (p(t), q(t)) is the solution of problem (2.1.29)–(2.1.30), and u(t) = p(t) + q(t) is the solution of problem (2.1.25), where q(t) = Φ(p(t)). This means that S(t)M ⊆ M, ∀t ≥ 0. In order to prove that the set M attracts all solutions of (2.1.25) exponentially, we first describe excursion properties of equation (2.1.20): Excursion properties For every T > 0, γ > 0, r > 0, there exist constants K2 , K3 (they depend on T, γ, r and constant C1 = C4 , but not on S(t) or N), such that for every N ≥ 1, one of the following inequalities is valid:

or

󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨QN (S(t)u0 − S(t)v0 )󵄨󵄨󵄨 ≤ γ 󵄨󵄨󵄨PN (S(t)u0 − S(t)v0 )󵄨󵄨󵄨,

(2.1.71)

󵄨󵄨 󵄨 󵄨󵄨S(t)u0 − S(t)v0 󵄨󵄨󵄨 ≤ K2 exp(−K3 αλN+1 t)|u0 − v0 |.

(2.1.72)

The results are used for all t, satisfying t0 ≤ t ≤ 2t0 , where t0 = ( 2K1 ) lg 2 and K1 are 1 constants. When |Au0 | ≤ r, |Au0 | ≤ r, we have 󵄨󵄨 󵄨 󵄨󵄨S(t)u0 − S(t)v0 󵄨󵄨󵄨 ≤ exp(K1 t)|u0 − v0 |,

∀t ≥ t0 ,

2.1 The inertial manifold for a class of nonlinear evolution equations | 217

γ = 81 , N ≥ N0 , where N0 satisfies λN0 +1 ≥ (2K3 αt0 )−1 lg(2K2 ).

(2.1.73)

Now equations (2.1.71)–(2.1.72) become 󵄨 1󵄨 󵄨 󵄨󵄨 󵄨󵄨QN (S(t)u0 − S(t)v0 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨PN (S(t)u0 − S(t)v0 )󵄨󵄨󵄨, 8 󵄨 1 󵄨󵄨 󵄨󵄨S(t)u0 − S(t)v0 󵄨󵄨󵄨 ≤ |u0 − v0 |, 2

(2.1.74) (2.1.75)

where u0 , v0 ∈ D(A), |Au0 | ≤ r, |Av0 | ≤ r, t0 ≤ t ≤ 2t0 . For a fixed r = 4ρ + b, the orbit of equations (2.1.24)–(2.1.25) will enter into the ball with origin 0 and radius 4ρ = 8ρ2 in D(A). Let |Au0 | ≤ 4ρ,

󵄨 󵄨󵄨 󵄨󵄨AS(t)u0 󵄨󵄨󵄨 ≤ 4ρ,

∀t ≥ 0.

We first prove that for any t1 , t0 ≤ t1 ≤ 2t0 , we have dist(S(t1 )u0 , μ) ≤

1 dist(u0 , M), 2

where dist(Φ, M) = inf{|ϕ − v| : v ∈ M}. To achieve this, we select v0 such that |u0 − ν0 | = dist(u0 , M). Then v0 = Pv0 + Φ(P(v0 )). We request |APv0 | ≤ 4ρ. Otherwise, if |APv0 | > 4ρ ≥ |APu0 |, then Φ(Pv0 ) = 0, v0 = Pv0 . In addition, there exists β, 0 < β ≤ 1, such that |APvβ | = 4ρ, where vβ = βPu0 + (1 − β)v0 ∈ PD(A). Then we have Φ(vβ ) = 0, hence vβ ∈ u, and |vβ − u0 |2 = |vβ − Pu0 |2 + |Qu0 |2 󵄨 󵄨2 = 󵄨󵄨󵄨(1 − β)(ν0 − Pu0 )󵄨󵄨󵄨 + |Qu0 |2 < |v0 − Pu0 |2 + |Qu0 |2 = |v0 − u0 |2 . This is contradiction to |v0 − u0 | = dist(v0 , M). Since |AΦ(Pv0 )| ≤ b, we have 󵄨 󵄨 |Av0 | ≤ |APv0 | + 󵄨󵄨󵄨AΦ(Pv0 )󵄨󵄨󵄨 ≤ 4ρ + b = r. Secondly, for S(t)u0 and S(t)v0 , using the excursion property (2.1.74)–(2.1.75), if equation (2.1.75) is established, then 󵄨 󵄨 dist(S(t1 )u0 , μ) ≤ 󵄨󵄨󵄨S(t1 )u0 − S(t1 )v0 󵄨󵄨󵄨 1 1 ≤ |u0 − v0 | ≤ dist(u0 , μ). 2 2 If (2.1.74) is established, then 󵄨 󵄨 dist(S(t1 )u0 , μ) ≤ 󵄨󵄨󵄨S(t1 )u0 − (P(S(t1 )u0 ) + Φ(PS(t1 )u0 ))󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨QS(t1 )u0 − QS(t1 )v0 󵄨󵄨󵄨 + 󵄨󵄨󵄨Φ(PS(t1 )v0 ) − Φ(PS(t1 )u0 )󵄨󵄨󵄨 λ 1 󵄨 󵄨 ≤ ( + l N )󵄨󵄨󵄨PS(t1 )v0 − PS(t1 )u0 󵄨󵄨󵄨. 8 λN+1

218 | 2 Inertial manifold From equations (2.1.74)–(2.1.32), and −1 |q| ≤ λN+1 |Aq|,

q ∈ QD(A),

p ∈ PD(A),

|Ap| ≤ λN |p|, we have dist(S(t1 )u0 , M) ≤

1 󵄨󵄨 1 󵄨 1 󵄨S(t )v − S(t1 )u0 󵄨󵄨󵄨 ≤ |v0 − u0 | ≤ dist(u0 , M). 4󵄨 1 0 2 2

Then, for any t ≥ t0 , t = nt1 , t0 ≤ t1 ≤ 2t0 , we have n

1 dist(S(t)u0 , μ) ≤ ( ) dist(u0 , M) 2 t = exp(− ln 2) dist(u0 , M) t1 ≤ exp(−

t ln 2) dist(u0 , M), 2t0

(2.1.76)

which implies that the exponential decay rate is 2t1 ln 2. 0 We prove the global attractor A ⊂ M. In fact, if u ∈ A , then the solution S(t)u is defined for all t. By equations (2.1.23)–(2.1.24), we have dist(S(t)u, M) ≤ 2ρ2 ,

∀t ∈ R.

Letting v = S(−t)u, where t ≥ t0 , from equation (2.1.76), we have dist(u, M) = dist(S(t)v, M) ≤ exp(−

t ) ⋅ 2ρ2 . t0

From this we get distu∈A (u, M) = 0. Hence A ⊂ M. All in all, we get the following theorem: Theorem 2.1.1. Assume that equations (2.1.1)–(2.1.11) are satisfied, and 0 < l < 81 . Let N0 be determined by equation (2.1.73). Then if there are constants K10 , K11 (depending on l and the initial value) such that N ≥ N0 ,

1

2 λN+1 ≥ K11 ,

1

1

2 λN+1 − λN2 ≥ K10 ,

then there exists b > 0 such that (i) The mapping F maps Fb,l into itself; (ii) Fb,l has one fixed point; (ii) M = graph(Φ) is the inertial manifold of (2.1.25); (iv) M includes the global attractor of (2.1.1).

(2.1.77)

2.1 The inertial manifold for a class of nonlinear evolution equations | 219

Theorem 2.1.2. Assume that equations (2.1.1) and (2.1.25) are given in H, where the nonlinear term F(u) = θρ (|Au|)R(u) satisfies equation (2.1.26). Suppose l is given, 0 < l < 81 . Assume there exists ρ0 such that for every solution of equation (2.1.1) inequality (2.1.21) is satisfied. If there exist constants N0 , K12 , K13 (they depend on l and the initial value) such that N ≥ N0 ,

λN+1 ≥ K12 ,

λN+1 − λN ≥ K13 ,

(2.1.78)

then the conclusion of Theorem 2.1.1 remains true. Proof. The nonlinear term F(u) = θρ (|Au|)R(u) satisfies the Lipschitz condition 󵄨󵄨 󵄨 󵄨󵄨F(u) − F(v)󵄨󵄨󵄨 ≤ K|u − v|,

∀u, v ∈ H,

where the parameter is taken as ρ = 2ρ0 . The space Fb,l is comprises functions Φ : PH → QH, satisfying: 󵄨󵄨 󵄨 󵄨󵄨Φ(p)󵄨󵄨󵄨 ≤ b, ∀p ∈ PD(A), 󵄨󵄨 󵄨 󵄨󵄨Φ(p1 ) − Φ(p2 )󵄨󵄨󵄨 ≤ l|p1 − p2 |, ∀p1 , p2 ∈ D(A), supp Φ ⊆ {p ∈ PD(A)‖p‖ ≤ 4ρ}.

The operator F is defined as 0

F Φ(p0 ) = − ∫ e

τAQ

QF(u)dτ,

−∞

where u = u(τ) = p(τ; Φ, p0 ) + Φ(p(τ; Φ, p0 )). The inequality (2.1.41) becomes 󵄨󵄨 󵄨 󸀠 󵄨󵄨F(u)󵄨󵄨󵄨 ≤ K4 , where K4󸀠 depends on R(u), θ and ρ. Lemma 2.1.2 and inequality (2.1.40) is both valid when α = 0. Equation (2.1.42) becomes 󵄨󵄨 󵄨 󸀠 −1 󵄨󵄨F Φ(p0 )󵄨󵄨󵄨 ≤ K5 λN+1 , −1 where K5󸀠 = K 󸀠 . Hence, we can take b = K5󸀠 λN+1 . Lemma 2.1.4 allows changing the norm |Av| into norm |v|, then inequality (2.1.46) turns into

󵄨󵄨 󵄨 󸀠 󵄨󵄨F(u1 ) − F(u2 )󵄨󵄨󵄨 ≤ K7 [(1 + l)|p1 − p2 | + ‖Φ1 − Φ2 ‖],

(2.1.79)

where ‖Φ‖ = sup{Φ(p)|; p ∈ PD(A)}. Let Δ = p1 − p2 , then equation (2.1.48) stays the same. Taking the inner product of equation (2.1.48) and Δ, we obtain 1 d 2 󵄨󵄨 21 󵄨󵄨2 |Δ| + 󵄨󵄨A Δ󵄨󵄨 = −(P(F(u1 ) − F(u2 )), Δ). 2 dt

220 | 2 Inertial manifold From equation (2.1.79), we get 󵄨 󵄨󵄨 1 d 󵄨 󵄨 1 󵄨2 󵄨󵄨 󵄨󵄨 |Δ|2 + 󵄨󵄨󵄨A 2 Δ󵄨󵄨󵄨 󵄨󵄨󵄨 ≤ K7󸀠 (1 + l)󵄨󵄨󵄨Δ|2 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 dt from which we deduce |Δ|

d 󵄨 1 󵄨2 |Δ| ≥ −󵄨󵄨󵄨A 2 Δ󵄨󵄨󵄨 − K7󸀠 (1 + l)|Δ|2 dt ≥ −|λN Δ|2 − K7󸀠 (1 + l)|Δ|2 .

Then we have 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󵄨󵄨Δ(τ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Δ(0)󵄨󵄨󵄨 exp(−τ(λN + K7 (1 + l))),

τ < 0,

which can be replaced with inequality (2.1.51). In Lemma 2.1.6 the assumptions for γN can be replaced with 󵄨󵄨 󵄨 󵄨󵄨F Φ(p01 ) − F Φ(p02 )󵄨󵄨󵄨 ≤ L|p01 − p02 |, where L = K7󸀠 (1 + l)λN−1 . In fact, 󵄨󵄨 󵄨 󵄨󵄨F Φ(p01 ) − F Φ(p02 )󵄨󵄨󵄨 0

󵄨 󵄨 󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨eτAQ 󵄨󵄨󵄨0p 󵄨󵄨󵄨F(u1 ) − F(u2 )󵄨󵄨󵄨dτ −∞ ≤ K7󸀠 (1

+ l)|p01 − p02 |

0

⋅ ∫ exp(τ(λN+1 − λN − K7󸀠 (1 + l))dτ −∞

≤ K7󸀠 (1 + l)(λN+1 − λN − K7󸀠 (1 + l))) |p01 − p02 |. −1

Similarly, we have 󵄨󵄨 󵄨 󸀠 󵄨󵄨F Φ1 (p0 ) − F Φ2 (p0 )󵄨󵄨󵄨 ≤ L ‖Φ1 − Φ2 ‖, −1 where L󸀠 = K7󸀠 λN+1 + K7 (λN + K7󸀠 (1 + l))−1 L. Set L ≤ l, L󸀠 ≤ 21 . Then, when l < 81 and

K12 ≤ λN+1 ,

K13 ≤ λN+1 − λN ,

(2.1.80)

the mapping F maps Fb,l into Fb,l , and has a fixed point. The lemma is proved. Now we consider the Galerkin approximation equation of (2.1.25), duM + AuM + PM F(uM ) = 0, dt

(2.1.81)

where F(u) = θρ (|Au|)R(u), uM is defined on PM D(A). Regarding the Galerkin approximation equation (2.1.81), we have the following theorem:

2.2 Inertial manifold and normal hyperbolicity property | 221

Theorem 2.1.3. Assume that the assumptions of Theorem 2.1.1 hold, l > 0, and N satisfies the condition of Theorem 2.1.1. then for every M > N, equation (2.1.81) possesses an inertial manifold MM . It is made of Lipschitz graph of function ΦM , where ΦM : PM D(A) → QPM D(A) ⊂ QD(A), the Lipschitz constant L of ΦM is the same as that of Φ : PD(A) → QD(A) in Theorem 2.1.1. Finally, −1

1

4 4 λM+1 , ‖ΦM − Φ‖ ≤ 2K6 λN+1

where

󵄨 󵄨 ‖ΦM − Φ‖ = sup 󵄨󵄨󵄨AΦM (p) − AΦ(p)󵄨󵄨󵄨. p∈PD(A)

2.2 Inertial manifold and normal hyperbolicity property We consider the following abstract evolution equation in a Banach space E : du + Au = f (u), dt

u(0) = u0 ,

(2.2.1)

where f is globally Lipschitz continuous, mapping a Banach space E into another Banach space F, and we assume E ⊂ F ⊂ E , the inclusion mappings are continuous, E in F and F in E are both are dense. Suppose there exists M1 > 0, which is the Lipschitz constant of f , such that 󵄨󵄨 󵄨 󵄨󵄨f (u) − f (v)󵄨󵄨󵄨F ≤ M1 |u − v|E ,

u, v ∈ E.

(2.2.2)

Suppose that −A generates a strongly continuous semigroup {e−tA }t≥0 on ε, e−tA ⊂ E , ∀t > 0, and there exist two sequences {λn }n∈N , {Λ n }n∈N , 0 < λn < Λ n , ∀n ∈ N. Assume that there exists a sequence of finite-dimensional projections {Pn }n∈N , such that Qn = I − Pn , and the following exponential dichotomy was established: Pn E is invariant under the action of {e−tA }t≥0 , which can be extended on Pn E to a strongly continuous semigroup {e−tA Pn }t∈R such that 󵄩󵄩 −tA 󵄩󵄩 ≤ K1 e−λnt , 󵄩e Pn 󵄩 {󵄩󵄩 −tA 󵄩󵄩L (E) α −λnt 󵄩󵄩e Pn 󵄩󵄩 , 󵄩 󵄩L (F,E) ≤ K1 λn e

∀t ≤ 0, ∀t ≤ 0,

(2.2.3)

and Qn E is positively invariant with respect to e−tA , ∀t ≥ 0. We have 󵄩󵄩 −tA 󵄩󵄩 ≤ K2 e−Λ n t , 󵄩e Qn 󵄩 {󵄩󵄩 −tA 󵄩󵄩L (E) −α α −Λ n t 󵄩󵄩e Qn 󵄩󵄩 , 󵄩 󵄩L (F,E) ≤ K2 (t + Λ n )e where K1 , K2 ≥ 1, 0 ≤ α < 1.

∀t ≥ 0,

∀t > 0,

(2.2.4)

222 | 2 Inertial manifold Assume equation (2.2.1) determines a continuous flow {S(t)}t≥0 , S(t)u0 = u(t) in E, where u(t) is the solution for the following integral equation (2.2.1): t

u(t) = e

−tA

u0 + ∫ e−(t−s)A f (u(s))ds,

∀t > 0.

0

Setting (t, u0 ) → S(t)u0 to be continuous, [0, ∞) × E → E, and e γ dγ,

+∞ −τ −α

∫0

γα = {

0,

0 < α < 1, α = 0,

we can get the following result: Theorem 2.2.1. Under the above assumptions, if for some n ∈ N , the following spectral gap condition is satisfied: Λ n − λn > 3M1 K1 K3 λnα + 3M1 K1 K2 (1 + γα )Λαn ,

(2.2.5)

then the semiflow {S(t)}t≥0 generated by equation (2.2.1) possesses the inertial manifold M = graph Φ, where Φ : Pn E → Qn E is Lipschitz continuous and has Lipschitz constant < 1. Manifold M has the positive and negative constants. Furthermore, M is asymptotically complete, that is, for any u0 ∈ E, there exists v0 ∈ E such that 󵄨 󵄨󵄨 −ηt 󵄨󵄨S(t)u0 − S(t)v0 󵄨󵄨󵄨E ≤ Kη (|u0 |E )e ,

∀t ≥ 0,

where η is an arbitrary number satisfying the following inequality: 0 < η < Λ n − 2M1 K2 (1 + γα )Λαn . Here Kη depends on η and |u0 |E . Remark 2.2.1. Theorem 2.2.1 only includes the main results. Actually, under the same assumptions as in Theorem 2.2.1, we can get more results, namely, the inertial manifold is the union of all complete orbits, the orbits are confined by e−σt , t → −∞. For some σ > 0, the existence of a continuous foliation E = ⋃v0 ∈M Nv0 in the space E also can be obtained, where every leaf is a Lipschitz function from Qn E to Pn E, and Nv0 can be expressed as 󵄨 󵄨 Nv0 = {u0 ∈ E, 󵄨󵄨󵄨S(t)u0 − S(t)v0 󵄨󵄨󵄨E = o(e−ηt ), t → +∞},

η > 0.

We can also get a continuous contraction mapping π : E → M, there is π −1 πu0 = Nu0 such that S(t)πu0 is the unique orbit in M, which possesses |S(t)u0 − S(t)πu0 |E = O (e−ηt ), t → ∞, η > 0. This gives the asymptotic completeness of M.

2.2 Inertial manifold and normal hyperbolicity property | 223

Usually, the nonlinear term f in (2.2.1) is Lipschitz on a bounded set. Set 󵄨 󵄨󵄨 󵄨󵄨f (u) − f (v)󵄨󵄨󵄨F ≤ d1 (r)|u − v|E ,

󵄨 󵄨󵄨 󵄨󵄨f (u)󵄨󵄨󵄨F ≤ d0 (r),

∀u, v ∈ B(E)(r) = {u ∈ E; |u|E ≤ r}. Now we can get for {S(t)}t≥0 in E that there exists an absorbing set B , that is, B ⊂ E is bounded, and for any bounded set B ⊂ E, there exists t0 (B) > 0, such that S(t)B ⊂ B , ∀t ≥ t0 . Take a function θ ∈ C 1 ([0, +∞)), θ(s) = 1, s ∈ [0, 1], θ(s) = 0, 4s ∈ [2, −∞), 󸀠 |θ (s)| ≤ 2, ∀s ≥ 0. Using the truncation function fθ (μ) instead of f , we have fθ (u) = θ(

|u|2E )f (u), ρ2

∀u ∈ E,

where ρ > 0, such that B ⊂ BE (ρ); here B means the closure in E of B . We consider the truncation function in the equation du + Au = fθ (u), dt

u(0) = u0 ,

which determines the continuous semiflow {Sθ (t)}t≥0 in E, and possesses the same initial value as equation (2.2.1). Theorem 2.2.2. Under the above assumptions, if for some n ∈ N , the following spectral gap condition is met: Λ n − λn > 3M1 K1 K2 λnα + 3M1 K1 K2 (1 + γα )Λαn , d (√2ρ)

where M1 := d1 (√2ρ) + 4√2 0 ρ , then equation (2.2.1) has inertial manifold M, which is the graph of a Lipschitz function Φ : O ⊂ Pn E → Qn E, where O is an open subset of Pn E. Furthermore, M is asymptotically complete, that is, for any u0 ∈ E, there exists v0 ∈ E such that 󵄨󵄨 󵄨 −ηt 󵄨󵄨S(t + t0 )u0 + S(t)v0 󵄨󵄨󵄨E ≤ e ,

∀t ≥ 0,

where η is an arbitrary number satisfying the following inequality: 0 < η < Λ n − 2M1 K2 (1 + γα )Λαn , where t0 depends on η and |u0 |E . Proof of Theorem 2.2.1. For simplicity, fix n ∈ N , suppose equation (2.2.5) is valid, and denote P = Pn , Q = Qn , λ = λn , Λ = Λ n , 0 < α < 1. When α = 0, the proof is almost the same. Take γα = 0. Introduce the space 󵄨 σt 󵄨 Fσ = {φ ∈ C((−∞, 0], E) : ‖φ‖Fσ = sup(e 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨E ) < +∞}, t≤0

224 | 2 Inertial manifold which is a Banach space with the norm ‖ ⋅ ‖Fσ . For φ ∈ Fσ , y ∈ PE, we consider the formal mapping 0

J (φ, y)(t) = e

−tA

Py − ∫ e−(t−s)A Pf (φ(s))ds t

t

+ ∫ e−(t−s)A Qf (φ(s))ds,

t ≤ 0.

(2.2.6)

−∞

The construction of an invariant manifold is based on the fact that for appropriate σ, a function φ ∈ Fσ is the solution of equation (2.2.1) if and only if φ is the fixed point of J , which requires proving that for an appropriate σ, the mapping J : Fσ × PE → Fσ defined by equation (2.2.6) is a strict contraction in Fσ and uniform for PE. Therefore, there exists a mapping φ : PE → Fσ such that J (φ(y0 ), y0 ) = φ(y0 ), ∀y0 ∈ PE, φ(y0 ) is a solution of equation (2.2.1). We define the mapping: Φ : PE → QE 0

Φ(y0 ) = Qφ(y0 )(0) = ∫ esA Qf (φ(y0 )(s))ds. −∞

Hence φ(y0 )(0) = y0 + Φ(y0 ). Let M = graph Φ, and prove M is Lipschitz, invariant and possesses asymptotic completeness property. Thus M is our sought inertial manifold. Lemma 2.2.1. When λ < σ < Λ, we have J : Fσ × PE → Fσ . Proof. Choose φ ∈ Fσ , y ∈ PE. Since f is globally Lipschitz, there exists a constant M0 > 0 such that 󵄨󵄨 󵄨 󵄨󵄨f (u)󵄨󵄨󵄨F ≤ M0 + M1 |u|E ,

∀u ∈ E.

It follows that t

󵄨󵄨 󵄨 󵄨 −(t−s)A 󵄨 Qf (φ(s))󵄨󵄨󵄨E ds 󵄨󵄨QJ (φ, y)󵄨󵄨󵄨E ≤ ∫ 󵄨󵄨󵄨e −∞

t

󵄨 󵄨 ≤ K2 ∫ (e−Λ(t−s) ((t − s)−α + Λα )(M0 + M1 )󵄨󵄨󵄨φ(s)󵄨󵄨󵄨E )ds. −∞

Hence, for λ < σ < Λ, we have σt 󵄨󵄨

t

󵄨 e 󵄨󵄨QJ (φ, y)(t)󵄨󵄨󵄨E ≤ M0 K2 Λα e(σ−Λ)t ∫ eΛs ds −∞

(2.2.7)

2.2 Inertial manifold and normal hyperbolicity property | 225

+ M1 K2 e

σt

t

∫ e−Λ(t−s) (t − s)−α ds −∞

t

σ

+ M1 K2 Λ ‖φ‖Fσ ∫ e(σ−Λ)(t−s) ds t

−∞

+ M1 K2 ‖φ‖Fσ ∫ e(σ−Λ)(t−s) (t − s)−α ds −∞

γ M0 K2 σt M1 K2 Λα M1 K2 γα M K e + ‖φ‖Fσ , ‖φ‖Fσ + ≤ 01−α2 eσt + α 1−α Λ−σ Λ Λ (Λ − σ)1−σ where we used t

∫ e−r(t−s) (t − s)−α ds = −∞

γα , γ 1−α

t

1 ∫ e−r(t−s) ds = , r

∀r > 0.

(2.2.8)

−∞

Hence, QJ (φ, y)(t) ∈ QE, ∀t ≤ 0, and M K (1 + γα )Λα 󵄨 󵄨 sup(eσt 󵄨󵄨󵄨QJ (φ, y)(t)󵄨󵄨󵄨E ) ≤ M0 K2 (1 + γα ) + 1 2 ‖φ‖Fσ . Λ−σ t≤0

(2.2.9)

Similarly, 0

󵄨󵄨 󵄨 󵄨 −tA 󵄨 󵄨 −Λ(t−s) 󵄨 Pf (φ(s))󵄨󵄨󵄨E ds 󵄨󵄨P J (φ, y)(t)󵄨󵄨󵄨E ≤ 󵄨󵄨󵄨e Py󵄨󵄨󵄨E + ∫ 󵄨󵄨󵄨e t

0

󵄨 󵄨 ≤ K1 e−λt |y|E + K1 λα ∫ e−λ(t−s) (M0 + M1 󵄨󵄨󵄨φ(s)󵄨󵄨󵄨E )ds. t

Hence 󵄨 󵄨 eσt 󵄨󵄨󵄨P J (φ, y)(t)󵄨󵄨󵄨E ≤ K1 e

(σ−λ)t

α (σ−λ)t

|y|E + M0 K1 λ e

0

λs

α

0

∫ e ds + M1 K1 λ ‖φ‖Fσ ∫ e(σ−λ)(t−s) ds t

M0 K1 (σ−λ)t M1 K1 λα e + ‖φ‖Fσ . ≤ K1 e(σ−λ)t |y|E + 1−α σ−λ λ

t

Thus, P J (φ, y)(t) ∈ PE, ∀t ≤ 0, and M0 K1 M1 K1 λα 󵄨 󵄨 sup(eσt 󵄨󵄨󵄨P J (φ, y)(t)󵄨󵄨󵄨E ) ≤ K1 |y|E + 1−α + ‖φ‖Fσ . σ−λ λ t≤0

(2.2.10)

226 | 2 Inertial manifold Hence, J (φ, y)(t) ∈ E, ∀t ≤ 0. The continuity of the mapping t → J (φ, φ) from (−∞, 0] to E is also proved. From equations (2.2.9)–(2.2.10), we know J (φ, y) ∈ Fσ , and M0 K1 M K 󵄩 󵄩󵄩 + (1 + γα ) 01−α2 󵄩󵄩J (φ, y)(t)󵄩󵄩󵄩Fσ ≤ K1 |y|E + 1−α λ Λ M1 K2 λα M1 K2 (1 + γα )Λα + }‖φ‖Fσ . +{ σ−λ Λ−σ This proves that J is a well-defined mapping, Fσ × PE → Fσ . Lemma 2.2.2. For any σ, such that λ < σ < Λ, we have 󵄩󵄩 󵄩 󵄩󵄩J (φ1 , y) − J (φ2 , y)󵄩󵄩󵄩Fσ ≤ θσ ‖φ1 − φ2 ‖Fσ ,

∀φ1 , φ2 ∈ Fσ , y ∈ PE,

where θσ =

M1 K1 λα M1 K2 (1 + γα )Λα + . σ−λ Λ−σ

Proof. Taking φ1 , φ2 ∈ Fσ , y ∈ PE, for t ≤ 0, we have 󵄨󵄨 󵄨 󵄨󵄨QJ (φ1 , y)(t) − QJ (φ2 , y)(t)󵄨󵄨󵄨E t

󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨e−(t−s)A Q[f (φ1 (s)) − f (φ2 (s))]󵄨󵄨󵄨E ds −∞

t

󵄨 󵄨 ≤ K2 ∫ ((t − s)−α + Λα )e−Λ(t−s) 󵄨󵄨󵄨f (φ1 (s)) − f (φ2 (s))󵄨󵄨󵄨F ds −∞

t

󵄨 󵄨 ≤ M1 K2 ∫ ((t − s)−α + Λα )e−Λ(t−s) 󵄨󵄨󵄨φ1 (s) − φ2 (s)󵄨󵄨󵄨E ds −∞

t

≤ M1 K2 ‖φ1 − φ2 ‖Fσ ∫ ((t − s)−α + Λα )e−Λ(t−s) e−σs ds. −∞

Hence, using equations (2.2.7) and (2.2.8), we get 󵄨 󵄨 eσt 󵄨󵄨󵄨QJ (φ1 , y)(t) − QJ (φ2 , y)(t)󵄨󵄨󵄨E t

≤ M1 K2 ‖φ1 − φ2 ‖Fσ ∫ e(σ−Λ)(t−s) (t − s)−α ds −∞

t

+ M1 K2 Λα ‖φ1 − φ2 ‖Fσ ∫ e(σ−Λ)(t−s) ds ≤

−∞

M K Λα M1 K2 γ 2‖φ1 − φ2 ‖Fσ + 1 2 ‖φ1 − φ2 ‖Fσ 1−α (Λ − σ) (Λ − σ) α

2.2 Inertial manifold and normal hyperbolicity property | 227

≤

M1 K2 (1 + γα ) ‖φ1 − φ2 ‖Fσ . (Λ − σ)

(2.2.11)

Similarly we have 󵄨 󵄨󵄨 󵄨󵄨P J (φ1 , y)(t) − P J (φ2 , y)(t)󵄨󵄨󵄨E 0

󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨e−(t−s)A P[f (φ1 (s)) − f (φ2 (s))]󵄨󵄨󵄨E ds t

0

󵄨 󵄨 ≤ M1 K1 λ ∫ e−λ(t−s) 󵄨󵄨󵄨φ1 (s) − φ2 (s)󵄨󵄨󵄨E ds. α

t

Hence 󵄨 󵄨 eσt 󵄨󵄨󵄨P J (φ1 , y)(t) − P J (φ2 , y)(t)󵄨󵄨󵄨E 0

≤ M1 K1 ‖φ1 − φ2 ‖Fσ ∫ e(σ−Λ)(t−s) ds t

M K Λα ≤ 1 1 ‖φ1 − φ2 ‖Fσ . σ−λ

(2.2.12)

Then by equations (2.2.11) and (2.2.12) we get 󵄩󵄩 󵄩 󵄩󵄩J (φ1 , y) − J (φ2 , y)󵄩󵄩󵄩Fσ ≤ θσ ‖φ1 − φ2 ‖Fσ , where θσ =

M1 K1 λα M1 K2 (1 + γα )Λα + . σ−λ Λ−σ

Now by the spectral gap condition (2.2.5), we have λ + 2M1 K1 λα < λ + 3M1 K1 K2 λα

< Λ − 3M1 K1 K2 (1 + γα )Λα

< Λ − 2M1 K2 (1 + γα )Λα . Thus, we can select σ which satisfies

λ − 2M1 K1 λα < σ < Λ − 2M1 K2 (1 + γα )Λα ,

(2.2.13)

such that M1 K1 λα 1 < , σ−λ 2

M1 K2 (1 + γα )Λα 1 < . Λ−σ 2

Therefore, θσ < 1. That is, J is a strict contraction in Jσ , uniform for PE, where σ satisfies equation (2.2.13), which allows us to define the mapping, PE → QE.

228 | 2 Inertial manifold Lemma 2.2.3. Assume that for any σ satisfying equation (2.2.13), θσ < 1, J is a strict contraction in Jσ , for PE being uniform. Then there exists a mapping φ : PE → Fσ such that J (φ(y0 ), y0 ) = φ(y0 ), ∀φ(y0 ) ∈ PE. Hence we can define mapping, PE → QE, by 0

Φ(y0 ) = Qφ(y0 )(0) = ∫ esA Qf (φ(y0 )(s))ds.

(2.2.14)

−∞

Proposition 2.2.1. Consider the manifold M = graph Φ, where Φ is defined by equation (2.2.14). Then M is invariant under the equation (2.2.1), that is, S(t)M = M, ∀t ≥ 0, of finite dimension, Lipschitz continuous, and has Lipschitz constant less than 1; M can be characterized as M = {u0 ∈ E, u0 belongs to a complete orbit {u(t; u0 )}t∈R ,

the solution of equation (2.2.1) satisfies |u(t; u0 )|E = O(e−σt ), t → −∞}, (2.2.15)

where σ is an arbitrary number satisfying equation (2.2.13). Furthermore, for any y0 ∈ PE, Φ satisfies 0

Φ(y0 ) = ∫ esA Qf (y(s) + Φ(y(s)))ds,

(2.2.16)

−∞

where y = y(t), t ≤ 0, is the solution of dy + Ay = Pf (y + Φ(y)), dt

y(0) = y0 .

(2.2.17)

Meanwhile, Φ(y0 ) = z(0), where z = z(t) ∈ QE is the solution of dz + Az = Qf (y + Φ(y)), dt

t ≤ 0,

(2.2.18)

|z(t)|E = O(e−σt ), t → −∞, where σ satisfies equation (2.2.13), y = y(t) ∈ PE, t ≤ 0 is the solution of equation (2.2.17). Proof. To see that M is characterized by equation (2.2.15), we fix σ to satisfy equation (2.2.13), and use Mσ to denote the right-hand side of equation (2.2.15). By definition, u0 ∈ Mσ if and only if there exists a continuous function (−∞, 0] ∋ t → u(t; u0 ) ∈ E, u(0; u0 ) = u0 , u(⋅, u0 ) ∈ Fσ is the solution of equation (2.2.1) and t

u(t; u0 ) = e−(t−τ)A u(τ; u0 ) + ∫ e−(t−τ)A f (u(s; u0 ))ds,

(2.2.19)

τ

∀τ ≤ t ≤ 0. Then we have 󵄨󵄨 −(t−τ)A 󵄨 󵄨 󵄨 Qu(τ; u0 )󵄨󵄨󵄨E ≤ K2 󵄨󵄨󵄨e−Λ(t−τ) u(τ; u0 )󵄨󵄨󵄨E 󵄨󵄨e 󵄩 󵄩 ≤ K2 e−Λt 󵄩󵄩󵄩u(⋅; u0 )󵄩󵄩󵄩F e(Λ−σ)τ → 0, σ

τ → −∞.

(2.2.20)

2.2 Inertial manifold and normal hyperbolicity property | 229

Acting with Q on the integral equation (2.2.19), and setting τ → −∞, we deduce t

Qu(t; u0 ) = ∫ e−(t−s) Qf (u(s; u0 ))ds, ∀t ≤ 0.

(2.2.21)

−∞

On the other hand, acting with P on equation (2.2.19), setting t = 0, we get t

τA

Pu0 = e Pu(τ; u0 ) + ∫ esA Pf (u(s; u0 ))ds.

(2.2.22)

−∞

Since {etA P}t∈R generates a group, multiplying equation (2.2.22) by e−τA P, we get 0

Pu(τ; u0 ) = e

−τA

Pu0 − ∫ e−(τ−s)A Pf (u(s; u0 ))ds,

τ ≤ 0.

(2.2.23)

τ

Therefore, based on equations (2.2.21) and (2.2.23), by (2.2.19) we deduce 0

u(t; u0 ) = e

−τA

Pu0 − ∫ e−(τ−s)A Pf (u(s; u0 ))ds t

t

+ ∫ e−(τ−s)A Qf (u(s; u0 ))ds,

∀t ≤ 0.

(2.2.24)

−∞

This means that u(0; u0 ) ∈ Fσ is the fixed point of J (⋅, Pu0 ). By Lemma 2.2.3, we have u(⋅, u0 ) = φ(Pu0 ). Hence u0 = φ(Pu0 )(0) = Pu0 + Φ(Pu0 ) ∈ graph Φ = M, which proves Mσ ⊂ M. On the contrary, if u0 ∈ graph Φ, then we have u0 = Pu0 + Φ(Pu0 ) = φ(Pu0 )(0). Since φ(Pu0 ) is the fixed point of J (⋅, Pu0 ), we get φ(Pu0 )(t) − e−(t−τ)A φ(Pu0 )(τ)

0

= e−tA Pu0 − e−(t−s)A−τA Pu0 − ∫ e−(t−s)A Pf (φ(Pu0 ))(s)ds 0

τ

+ e−(t−τ)A ∫ e−(τ−s)A Pf (φ(Pu0 ))(s)ds t

+ ∫ e −∞

τ

−(t−s)A

τ

Qf (φ(Pu0 )(s))ds − e

−(t−τ)A

∫ e−(τ−s)A Qf (φ(Pu0 )(s))ds −∞

230 | 2 Inertial manifold t

t

= ∫e τ

−(t−s)A

Pf (φ(Pu0 )(s))ds + ∫ e−(t−s)A Qf (φ(Pu0 )(s))ds,

τ ≤ t ≤ 0.

τ

Hence t

φ(Pu0 )(t) = e

−(t−τ)A

φ(Pu0 )(τ) + ∫ e−(t−s)A f (φ(Pu0 )(s))ds, τ

which means φ(Pu0 ) is the solution of equation (2.2.1), u(Pu0 ) ∈ Fσ is the fixed point of J (⋅, Pu0 ). Hence, u0 ∈ Mσ , M ⊂ Mσ , thus M = Mσ . It can be seen from the characterizing equation (2.2.15) that M is invariant. Hence equations (2.2.16)–(2.2.18) are valid. For the Lipschitz continuity of Φ, taking y1 , y2 ∈ PE, and noting that σ satisfies equation (2.2.13), we get 0

󵄨󵄨 󵄨 󵄨 τA 󵄨 󵄨󵄨Φ(y1 ) − Φ(y2 )󵄨󵄨󵄨E ≤ ∫ 󵄨󵄨󵄨e Q[f (φ(y1 )(s)) − f (φ(y2 )(s))]󵄨󵄨󵄨E ds −∞

0

󵄨 󵄨 ≤ M1 K2 ∫ eΛs (|s|−α + Λα )󵄨󵄨󵄨φ(y1 )(s) − φ(y2 )(s)󵄨󵄨󵄨E ds −∞

0

󵄩 󵄩 ≤ M1 K2 󵄩󵄩󵄩φ(y1 ) − φ(y2 )󵄩󵄩󵄩F ∫ (|s|−α + Λα )e(Λ−σ)s ds σ

−∞

M K (1 + γα )Λα 󵄩󵄩 󵄩 ≤ 1 2 󵄩󵄩φ(y1 ) − φ(y2 )󵄩󵄩󵄩Fσ , Λ−σ

(2.2.25)

where 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩φ(y1 ) − φ(y2 )󵄩󵄩󵄩Fσ = 󵄩󵄩󵄩J (φ(y1 ), y1 ) − J (φ(y2 ), y2 )󵄩󵄩󵄩Fσ 󵄩 󵄩 ≤ 󵄩󵄩󵄩J (φ(y1 ), y1 ) − J (φ(y2 ), y1 )󵄩󵄩󵄩F σ 󵄩󵄩 󵄩󵄩 + 󵄩󵄩J (φ(y2 ), y1 ) − J (φ(y2 ), y2 )󵄩󵄩F σ 󵄩 󵄩 󵄩 󵄩 ≤ θσ 󵄩󵄩󵄩φ(y1 ) − φ(y2 )󵄩󵄩󵄩F + 󵄩󵄩󵄩e−tA P(y1 − y2 )󵄩󵄩󵄩F σ σ 󵄩󵄩 󵄩󵄩 ≤ θσ 󵄩󵄩φ(y1 ) − φ(y2 )󵄩󵄩F + K1 |y1 − y2 |E σ

(2.2.26)

and θσ < 1. Hence by equation (2.2.26), we obtain K1 󵄩󵄩 󵄩 |y − y2 |E . 󵄩󵄩φ(y1 ) − φ(y2 )󵄩󵄩󵄩Fσ ≤ 1 − θσ 1

(2.2.27)

Inserting equation (2.2.27) into equation (2.2.25), we get M K K (1 + γα )Λα 󵄩󵄩 󵄩 |y − y2 |E . 󵄩󵄩Φ(y1 ) − Φ(y2 )󵄩󵄩󵄩E ≤ 1 1 2 (1 − θσ )(Λ − α) 1

(2.2.28)

2.2 Inertial manifold and normal hyperbolicity property | 231

By the spectral gap condition (2.2.25), we select σ = Λ − 3M1 K1 K2 (1 + γα )Λα , where σ satisfies equation (2.2.13). For the σ, we have M1 K1 Λ α M1 K2 (1 + γα ) + σ−λ Λ−α M1 K2 (1 + γα )Λα M1 K1 Λ α + . = α Λ − λ − 3M1 K1 K2 (1 + γα )Λ 3M1 K1 K2 (1 + γα )Λα

θσ =

Since Λ − λ − 3M1 K1 K2 (1 + γα )Λα > 3M1 K1 K2 λα , we obtain θσ
λ, we have 󵄨 󵄨 󵄨 󵄨󵄨 −(t−τ)A P[S(τ)v0 − S(τ)u0 ]󵄨󵄨󵄨E ≤ K1 e−λ(t−τ) 󵄨󵄨󵄨ψ(t)󵄨󵄨󵄨E = O(e(λ−σ)t ), 󵄨󵄨e

τ → +∞.

Hence, if we let τ → +∞ in equation (2.2.31), using equation (2.2.30), we get t

ψ(t) = e

−tA

(z0 − Qu0 ) + ∫ e−(t−s)A Q[f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds 0

∞

− ∫ e−(t−s)A P[f (S(s)v0 + ψ(s)) − f (S(s)u0 )]ds.

(2.2.32)

t

On the contrary, let the function ψ ∈ C([0, +∞), E) be such that ψ(t) = O(e−σt ), t → +∞. And for z0 ∈ QE, assume u0 ∈ E satisfies equation (2.2.32). Then y0 = Pu0 + Pψ(0), v0 = y0 + z0 . Set v(t) = ψ(t) + S(t)u0 ,

t ≥ 0.

First, we note that v(0) = ψ(0) + u0 = Pψ(0) + z0 − Qu0 + u0

= y0 − Pu0 + z0 − Qu0 + u0 = y0 + z0 = v0 .

Second, y0 = Pu0 + Pψ(0) +∞

= Pu0 − ∫ esA P[f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds. 0

Thus, equation (2.2.32) can be rewritten as ψ(t) = e−tA (z0 − Qu0 ) + e−tA (y0 − Pu0 ) t

+ ∫ e−(t−s)A Q[f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds 0

t

+ ∫ e−(t−s)A P[f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds 0

=e

−tA

t

(v0 − u0 ) + ∫ e−(t−s)A [f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds. 0

Then t

v(t) = ψ(t) + S(t)u0 = e

−tA

v0 + ∫ e−(t−s)A f (v(s))ds 0

(2.2.33)

2.2 Inertial manifold and normal hyperbolicity property | 233

and v(t) is the solution having v(0) = v0 = y0 + z0 and satisfying equation (2.2.1), v(t) = S(t)v0 . Finally, we have 󵄨 󵄨 󵄨 󵄨󵄨 −σt 󵄨󵄨S(t)v0 − S(t)u0 󵄨󵄨󵄨E = 󵄨󵄨󵄨ψ(t)󵄨󵄨󵄨E = O(e ),

t → +∞.

Hence, we have the following lemma: Lemma 2.2.4. Let ψ ∈ C([0, +∞), E), |ψ(t)|E = O(e−σt ), t → +∞, where σ > λ. Then ψ satisfies equation (2.2.32), u0 ∈ E, z0 ∈ QE, if and only if ψ possesses the form ψ(t) = S(t)v0 − S(t)u0 ,

∀t ≥ 0, v0 ∈ E.

Furthermore, v0 , u0 , z0 and ψ have the following relation: v0 = Pu0 + Pψ(0) + z0 . The above lemma enlightens us to make a similar definition of Fσ . Define the space 󵄨 σt 󵄨 Gσ = {ψ ∈ C([0, +∞), E) : ‖ψ‖Gσ = sup(e 󵄨󵄨󵄨ψ(t)󵄨󵄨󵄨E ) < +∞}, t≥0

which is a Banach space with norm ‖ ⋅ ‖Gσ . For u0 ∈ E, z0 ∈ QE, ψ ∈ Gσ , t ≥ 0. We define a formal mapping t

W(ψ, u0 , z0 )(t) = e

−tA

(z0 − Qu0 ) + ∫ e−(t−s)A Q[f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds 0

+∞

− ∫ e−(t−s)A P[f (S(s)u0 + ψ(s)) − f (S(s)u0 )]ds.

(2.2.34)

t

Note that for u0 ∈ E, z0 ∈ QE, ψ satisfies (2.2.32) if and only if ψ = W(ψ, u0 , z0 ). Lemma 2.2.5. Let σ satisfy (2.2.32). Equation (2.2.34) defines a mapping W : Gσ × E × QE → Gσ which satisfies 󵄩󵄩 󵄩 󵄩󵄩W(ψ, u0 , z0 )󵄩󵄩󵄩Gσ ≤ K2 |z0 − Qu0 |E + θσ ‖ψ‖Gσ , 󵄩󵄩 󵄩 󵄩󵄩W(ψ1 , u0 , z0 ) − W(ψ2 , u0 , z0 )󵄩󵄩󵄩Gσ ≤ θσ ‖ψ1 − ψ2 ‖Gσ ,

(2.2.35) (2.2.36)

where u0 ∈ E, z0 ∈ QE, ψ, ψ1 , ψ2 ∈ Gσ ; θσ is defined in Lemma 2.2.2, in particular, θσ < 1. Proof. Fixing σ, which satisfies (2.2.13), taking u0 ∈ E, z0 ∈ QE, ψ ∈ Gσ , we have t

󵄨󵄨 󵄨 󵄨 −Λt −α α −Λ(t−s) 󵄨󵄨 󵄨󵄨W(ψ, u0 , z0 )(t)󵄨󵄨󵄨E ≤ K2 e |z0 − Qu0 |E + M1 K2 ∫((t − s) + Λ )e 󵄨󵄨ψ(s)󵄨󵄨󵄨E ds 0

+∞

󵄨 󵄨 + M1 K1 λα ∫ e−λ(t−s) 󵄨󵄨󵄨ψ(s)󵄨󵄨󵄨E ds. t

234 | 2 Inertial manifold Hence, 󵄨 󵄨 eσt 󵄨󵄨󵄨W(ψ, u0 , z0 )(t)󵄨󵄨󵄨E ≤ K2 e(σ−Λ)t |z0 − Qu0 |E t

+ M1 K2 ‖ψ‖Gσ ∫((t − s)−α + Λα )e−(Λ−σ)(t−s) ds 0

+∞

+ M1 K1 λα ‖ψ‖Gσ ∫ e(σ−Λ)(t−s) ds t

≤ K2 |z0 − Qu0 |E + {

M1 K2 (1 + γα )Λα M1 K1 λα + }‖ψ‖Gσ , Λ−α σ−λ

and then we have 󵄩󵄩 󵄩 󵄩󵄩W(ψ, u0 , z0 )(t)󵄩󵄩󵄩G ≤ K2 |z0 − Qu0 |E + θσ ‖ψ‖Gσ . σ

This proves equation (2.2.35). Now for ψ1 , ψ2 ∈ Gσ , 󵄨󵄨 󵄨 󵄨󵄨W(ψ1 , u0 , z0 )(t) − W(ψ2 , u0 , z0 )(t)󵄨󵄨󵄨E t

≤ ∫ K2 ((t − s)−α + Λα )e−Λ(t−s) M1 |ψ1 (s) − ψ2 (s)|E ds 0 +∞

+ ∫ K1 λα e−λ(t−s) M1 |ψ1 (s) − ψ2 (s)|E ds. t

Hence 󵄨 󵄨 eσt 󵄨󵄨󵄨W(ψ1 , u0 , z0 )(t) − W(ψ2 , u0 , z0 )(t)󵄨󵄨󵄨E t

≤ M1 K2 ‖ψ1 − ψ2 ‖Gσ ∫((t − s)−α + Λα )e(σ−Λ)(t−s) ds 0

+∞

+ M1 K1 λα ‖ψ1 − ψ2 ‖Gσ ∫ e(σ−λ)(t−s) ds, t

and then we have 󵄩󵄩 󵄩 󵄩󵄩W(ψ1 , u0 , z0 )(t) − W(ψ2 , u0 , z0 )(t)󵄩󵄩󵄩Gσ ≤ θσ ‖ψ1 − ψ2 ‖Gσ . This proves equation (2.2.36). Since θσ < 1, we have proved the lemma. Definition 2.2.1. Using Lemma 2.2.5, we define the mapping W : Gσ × E × QE → Gσ , which is a strict contraction in Fσ , uniformly in E × QE, hence there exists a mapping ψ : E × QE → Gσ , such that W(ψ(u0 , z0 ), u0 , z0 ) = ψ(u0 , z0 ), ∀u0 ∈ E, ∀z0 ∈ QE. We can define a mapping Ψu0 : QE → PE, for any u0 ∈ E, by Ψu0 (z0 ) = Pu0 + Pψ(u0 , z0 )(0),

z0 ∈ QE.

(2.2.37)

2.2 Inertial manifold and normal hyperbolicity property | 235

Defining Nu0 = graph Ψu0 , ∀u0 ∈ E, we have Proposition 2.2.2. For any u0 ∈ E, we have 󵄨 󵄨󵄨 󸀠 󵄨󵄨Ψu0 (z1 ) − Ψu0 (z2 )󵄨󵄨󵄨E ≤ l |z1 − z2 |E ,

∀z1 , z2 ∈ QE,

(2.2.38)

where 0 < l󸀠 < 1. Furthermore, Ψu0 satisfies +∞

Ψu0 (z0 ) = Pu0 − ∫ esA P[f (S(s)(Ψu0 (z0 ) + z0 )) − f (S(s)(u0 ))]ds,

(2.2.39)

0

and Nu0 can be characterized as 󵄨 󵄨 Nu0 = {v0 ∈ E; 󵄨󵄨󵄨S(t)v0 − S(t)u0 󵄨󵄨󵄨E = O(e−σt ), t → +∞},

(2.2.40)

where σ is an arbitrary number, which satisfies equation (2.2.13). Proof. Fix u0 ∈ E, and consider z1 , z2 ∈ QE, ψi = ψ(u0 , zi ), i = 1, 2. By definition, Ψu0 (z1 ) − Ψu0 (z2 ) = Pu0 + Pψ(u0 , z1 )(0) − Pu0 − Pψ(u0 , z0 )(0) + PW(ψ1 , u0 , z1 )(0) − PW(ψ2 , u0 , z2 )(0) +∞

= − ∫ esA P[f (S(s)u0 + ψ1 (s)) − f (S(s)u0 + ψ2 (s))]ds. 0

Since σ satisfies equation (2.2.13), we have +∞

󵄨 󵄨󵄨 󵄨 α −λs 󵄨 󵄨󵄨Ψu0 (z1 ) − Ψu0 (z2 )󵄨󵄨󵄨E ≤ M1 K1 λ ∫ e 󵄨󵄨󵄨ψ1 (s) − ψ2 (s)󵄨󵄨󵄨E ds 0

α

+∞

≤ M1 K1 λ ‖ψ1 − ψ2 ‖Gσ ∫ e(σ−λ)s ds =

α

0

M1 K1 λ ‖ψ1 − ψ2 ‖Gσ . σ−λ

Since ψi is the fixed point of W(⋅, u0 , zi ), i = 1, 2, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ψ1 − ψ2 󵄩󵄩󵄩Gσ = 󵄩󵄩󵄩W(ψ1 , u0 , z1 ) − W(ψ2 , u0 , z2 )󵄩󵄩󵄩Gσ 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩W(ψ1 , u0 , z1 ) − W(ψ2 , u0 , z1 )󵄩󵄩󵄩G + 󵄩󵄩󵄩W(ψ2 , u0 , z1 ) − W(ψ2 , u0 , z2 )󵄩󵄩󵄩G σ σ 󵄩 󵄩 ≤ θσ ‖ψ1 − ψ2 ‖Gσ + 󵄩󵄩󵄩e−tA Q(z1 − z2 )󵄩󵄩󵄩G σ ≤ θσ ‖ψ1 − ψ2 ‖Gσ + K2 |z1 − z2 |E .

Since 0 < θσ < 1, σ satisfies equation (2.2.13), and then we get ‖ψ1 − ψ2 ‖Gσ ≤

K2 |z − z2 |E . 1 − θσ 1

236 | 2 Inertial manifold Therefore 󵄨 󵄨󵄨 󸀠 󵄨󵄨Ψu0 (z1 ) − Ψu0 (z2 )󵄨󵄨󵄨E ≤ l |z1 − z2 |E , where l󸀠 ≤

M1 K1 K2 λα . (1 − θσ )(σ − λ)

By the spectral gap condition (2.2.5), we can take σ = λ +3M1 K1 K2 λα to satisfy equation (2.2.13). Then by Proposition 2.2.1, θσ < 32 , and 1 < 3. 1 − θσ Thus l󸀠
ε,

(2.2.46)

where ε > 0, and {ujk }j∈N is a subsequence. By Lemma 2.2.6, {πujk } is bounded in M. Since M is locally compact (its dimension is finite), a subsequence of {πujk } still denoted by πujk converges to ν0 ∈ M, k → ∞: πujk → v0 ,

k → +∞.

By Proposition 2.2.3, it follows that 󵄨󵄨 󵄨 −σt 󵄨󵄨S(t)ujk − S(t)πujk 󵄨󵄨󵄨E ≤ Ce ,

∀t ≥ 0, ∀k ∈ N,

(2.2.47)

where σ satisfies equation (2.2.13), C is independent of k, t. For each fixed point t ≥ 0, in equation (2.2.47) by letting k → +∞, we obtain 󵄨󵄨 󵄨 −σt 󵄨󵄨S(t)u0 − S(t)v0 󵄨󵄨󵄨E ≤ Ce ,

∀t ≥ 0.

Hence, by the characterization of Nu0 , we get v0 ∈ Nu0 . But v0 ∈ M, by the definition, πu0 = u0 . This contradicts equation (2.2.46), hence we know πuj → πu0 , and π is continuous. Lemma 2.2.7. The mapping (u0 , z0 ) → Ψu0 (z0 )

(2.2.48)

is continuous, E × QE → PE, and the mapping (u0 , z0 , t) → ψ(u0 , z0 )(t) is continuous, E × QE × [0, ∞) → E.

(2.2.49)

240 | 2 Inertial manifold Proof. Fix u0 ∈ E, z0 ∈ QE, and select any sequences {uj }j∈N and {zj }j∈N such that uj → u0 ,

zj → z0 ,

j → +∞.

From equation (2.2.35), if ψ(uj , zj ) is the fixed point of W(⋅, uj , zj ) we know K2 󵄩 󵄩󵄩 |z − Quj |E . 󵄩󵄩ψ(uj , zj )󵄩󵄩󵄩Gσ ≤ 1 − θσ j Hence 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨ψuj (zj )󵄨󵄨󵄨E = 󵄨󵄨󵄨Puj + Pψ(uj , zj )(0)󵄨󵄨󵄨E 󵄩 󵄩 ≤ K1 |uj |E + K1 󵄩󵄩󵄩ψ(uj , zj )󵄩󵄩󵄩G σ K1 K2 |z − Quj |E , ≤ K1 |uj |E + 1 − θσ j which means {Ψuj (zj )} is bounded in PE, for the PE’s dimension is finite. If {Ψuj (zj )} does not converge to Ψu0 (z0 ), then as in the previous proof, we can select subsequences {ujk }k∈N and {zjk }k∈N such that 󵄨󵄨 󵄨 󵄨󵄨Ψujk (zjk ) − Ψu0 (z0 )󵄨󵄨󵄨E ≥ ε, ε > 0, Ψujk (zjk ) → ỹ0 , k → +∞, ỹ0 ∈ PE.

(2.2.50)

Noting that zj + Ψuj (zj ) ∈ Nuj , and it is bounded, by Propositions 2.2.2 and 2.2.3, it follows that 󵄨 󵄨󵄨 −σt 󵄨󵄨S(t)(zjk + Ψujk (zjk )) − S(t)πujk 󵄨󵄨󵄨E ≤ Ce ,

∀t ≥ 0,

where σ satisfies equation (2.2.13), C is independent of k, t. Letting k → ∞, we obtain 󵄨󵄨 󵄨 −σt 󵄨󵄨S(t)(z0 + ỹ0 ) − S(t)πu0 󵄨󵄨󵄨E ≤ Ce ,

∀t ≥ 0.

Thus z0 + ỹ0 ∈ Nu0 = graph Ψu0 . Then we have ỹ0 = Ψu0 (z0 ). Since Ψujk (zjk ) → Ψu0 (z0 ),

k → +∞,

we obtain a contradiction to equation (2.2.50). Hence Ψuj (zj ) → πu0 (z0 ), and equation (2.2.48) is proved. For the equation (2.2.49), by Lemma 2.2.4 we have ψ(u0 , z0 )(t) = S(t)(Ψu0 (z0 ) + z0 ) − S(t)u0 . Hence, from equation (2.2.48) and (t, u0 ) → S(t)u0 , the continuity of the mapping follows by equation (2.2.49).

2.2 Inertial manifold and normal hyperbolicity property | 241

Finally, we have Proposition 2.2.5. A continuous foliation structure of E is given by E = ⋃u0 ∈M Nu0 ; furthermore, the foliation along the semiflow possesses the “translation” property: S(t)Nu0 = NS(t)u0 , t ≥ 0, ∀u0 ∈ E. S(t) ⋅ π = π ⋅ S(t), ∀t ≥ 0. Proof. Firstly, we note that E = ⋃v0 ∈M Nu0 , by Propositions 2.2.2 and 2.2.3, u0 ∈ Nπu0 , ∀u0 ∈ E. Let h : PE × QE → E be defined as h(y, z) = Ψy+Φ(y) (Φ(y) + z) + Φ(y) + z. From equation (2.2.48), we know h is continuous. Furthermore, h(y, QE) = graph Ψy+Φ(y) = foliation across y + Φ(y) = foliation through Ψy+Φ(y) (Φ(y)) + Φ(y) = foliation through h(y, 0), where we use y = Ψy+Φ(y) (Φ(y)) (Lemma 2.2.6). In a similar way, we have h(y, 0) = Ψy+Φ(y) (Φ(y)) + Φ(y) = y + Φ(y), whence it follows that h(PE, 0) = graph Φ = M and then E = ⋃u0 ∈M Nu0 is a continuous leaf structure. From the characterization of Nu0 , it is easy to get S(t)Nu0 = NS(t)u0 , the same characterization and the definition of π give S(t) ⋅ π = π ⋅ S(t). Proof of Theorem 2.2.2. We know that 󵄨󵄨 󵄨 󵄨󵄨fθ (u) − fθ (v)󵄨󵄨󵄨F ≤ M1 |u − v|E ,

∀u, v ∈ E,

(2.2.51)

where M1 = d1 (√2ρ) +

4√2d0 (√2ρ) . ρ

Based on equation (2.2.51) and the spectral gap condition, using Theorem 2.2.2 for truncation equation, we know that {Sθ (t)}t≥0 has an inertial manifold Mθ = graph Φθ , where Φθσ : PE → QE has Lipschitz constant less than 1. Now we start from Mθ to get the inertial manifold of equation (2.2.1). Denote B0 = BE (ρ) and let B ⊂ B0 be an absorbing set of the original equation. Since B is absorbing, there exists t0 = t0 (B0 ) such that S(t)B0 ⊂ B ,

∀t ≥ t0 .

Let B1 = ⋃ S(t)B0 . t≥t0

(2.2.52)

242 | 2 Inertial manifold then B1 ⊂ B , B1 ⊂ B0 , S(t)B1 ⊂ B , ∀t ≥ 0. Hence S(t)|B1 = Sθ (t)|B1 , and B1 is an absorbing set of the original equation. Let M1 = Mθ ∩ B1 . Then S(t)M1 = Sθ (t)M1 = Sθ (t)Mθ ∩ Sθ (t)B1 ⊂ Mθ ∩ B1 = M1 ,

∀t ≥ 0.

(2.2.53)

Thus, M1 is positive invariant for the original and truncated equations. Since B1 ⊂ B0 , we can select ε, 0 < ε < 1, such that vε (B1 ) ⊂ B0 , where vε (B1 ) = {ν ∈ E, |v − u|E < ε, u ∈ B1 }. Let Mε = Mθ ∩ νε (B1 ). Then Mε is an open neighborhood of M1 in Mθ . Now we consider the inertial form {Sθ (t)|Mθ }t≥0 in Mθ . Since Mθ is Lipschitz continuous, of finite dimension and invariant, we get that Sθ (⋅) is continuous, [0, t0 ] × Mθ → Mθ . Hence Sθ (t)|Mθ is a homeomorphism of Mθ , t ≥ 0. Since Sθ (t)M1 ⊂ M1 , t ≥ 0, Mε is a domain of Mθ (t) in M1 , and from the continuity of Sθ (⋅) as a map [0, t0 ] × Mθ → Mθ , we deduce that there exists Mθ in a domain Mδ of M1 such that Sθ (t)Mδ ⊂ Mε ,

∀t ∈ [0, t0 ].

(2.2.54)

Without loss of generality, we can set Mδ to have the form Mδ = Mθ ∩ vδ (B1 ), 0 < δ < ε. By equations (2.2.52) and (2.2.54), we get Sθ (t)Mδ ⊂ Mε ,

S(t)|Mδ = Sθ (t)|Mδ ,

∀t ≥ 0.

Denote M = ⋃ S(t)Mδ . t≥0

We show that M is the required inertial manifold. In fact, by definition S(t)M ⊂ M, ∀t ≥ 0. Now we know that Sθ (t)|Mθ is a homeomorphism of Mθ , t ≥ 0. Thus it maps an open set of Mθ into an open set of Mθ . Since S(t)Mδ (= Sθ (t)Mδ ) is open in Mθ , t ≥ 0, the set M itself is open in Mθ and I + Φθ : PE → Mθ is continuous. The set O = (I − Φθ )−1 (M) is open in PE, M = graph Φ, where Φ is a Lipschitz function given by the following map: Φ = Φθ |O : O ⊂ PE → QE, and has Lipschitz constant less than 1. From the asymptotic completeness property of M, if B ⊂ E is bounded, because B1 for the original equations is absorbing, then there exists t1 ≥ 0, t1 = t1 (B ), such that S(t)B ⊂ B1 , ∀t ≥ t1 .

2.2 Inertial manifold and normal hyperbolicity property | 243

̃ 0 = S(t)u0 , ũ0 ∈ B1 . By the asymptotic completeness of trunGiven u0 ∈ B , set u cated equation Mθ , there exists ũ0 ⊂ Mθ , such that 󵄨 󵄨󵄨 −ηt 󵄨󵄨Sθ (t)ũ0 − Sθ (t)ṽ0 󵄨󵄨󵄨E ≤ Kη e ,

∀t ≥ 0,

(2.2.55)

where 0 < η < Λ − 2M1 K2 (1 + γα )λα and Kη only depends on η, since ũ0 ∈ B1 . Taking t2 = t2 (η) such that Kη e−ηt2 < δ

(2.2.56)

and setting v0 = Sθ (t2 )ṽ0 , we then deduce that v0 ∈ M. Furthermore, from equations (2.2.55)–(2.2.56), we get 󵄨󵄨 󵄨 󵄨󵄨S(t + t1 + t2 )u0 − S(t)v0 󵄨󵄨󵄨E 󵄨 󵄨 = 󵄨󵄨󵄨S(t + t2 )S(t1 )u0 − Sθ (t)v0 󵄨󵄨󵄨E 󵄨 󵄨 = 󵄨󵄨󵄨S(t + t2 )ũ0 − Sθ (t)Sθ (t2 )ṽ0 󵄨󵄨󵄨E 󵄨 󵄨 = 󵄨󵄨󵄨Sθ (t + t2 )ũ0 − Sθ (t + t2 )ṽ0 󵄨󵄨󵄨E ≤ Kη e−η(t+t2 ) = Kη e−ηt2 e−ηt ≤ δe−ηt .

Hence, setting t3 = t1 + t2 , t3 = t3 (B , η), and δ < ε < 1, we obtain 󵄨󵄨 󵄨 −ηt 󵄨󵄨S(t + t2 )u0 − S(t)v0 󵄨󵄨󵄨E ≤ e ,

∀t ≥ 0,

which proves the asymptotic completeness of M. Finally, if the original equation has a global absorbing attractor A , then from the asymptotic completeness and invariance property, S(t)A = A , t ≥ 0, we get distE (A , M) = distE (S(t)A , M) = O(e−ηt ),

t → +∞, η > 0.

Hence A ⊂ M. But since A ⊂ B1 , M ⊃ Mθ ∩ νδ (B1 ), thus A ⊂ M. This finishes the proof. In the following, we consider the normal and normal hyperbolicity property of inertial manifold. Theorem 2.2.3. If f ∈ C 1 (E, F), then Theorem 2.2.1 defines inertial manifold M = graph Φ ∈ C 1 , where Φ satisfies Sack equation DΦ(y)(−Ay + Pn f (y + Φ(y)) + AΦ(y)) = Qn f (y + Φ(y)),

(2.2.57)

where y is defined in the domain of Φ. If we consider that it satisfies the assumption of Theorem 2.2.2, fη ∈ C 1 (E, F), then the same conclusion is valid.

244 | 2 Inertial manifold Proof. Because the inertial manifold in Theorem 2.2.2 is seen as a restriction of the inertial manifold for the truncated equation, this fully proves the result when f is globally Lipschitz continuous. Assume that all the conditions of Theorem 2.2.1 are satisfied and f ∈ C 1 (E, F). The rest of the proof can be divided into three steps. For convenience, set φ(y)(t) = φ(y, t), y ∈ E, t ≤ 0, where φ : PE → Fσ is given by Lemma 2.2.3. Step 1: The selection of differential. In Lemma 2.2.3, we have seen that Φ(y) = Pφ(y, 0) and 0

φ(y, t) = e

−tA

Py − ∫ e−(t−s)A Pf (φ(y, s))ds

t

t

+ ∫ e−(t−s)A Qf (φ(y, s))ds.

(2.2.58)

−∞

We consider the differential of Φ. First, we seek the differential of φ. Since DΦ(y) = Q𝜕y φ(y, 0), by formally differentiating equation (2.2.58) with respect to y, we see that 𝜕y φ(y) is a fixed point of L1 (⋅, y), where L1 is given by t

0

L1 (Δ, y)(t) = e

−tA

P − ∫ e−(t−s)A PDf (φ(y, s))Δ(s)ds + ∫ e−(t−s)A QDf (φ(y, s))Δ(s)ds. t

−∞

Similar as for the map L , we must verify that the mapping L1 is completely determined, which is strictly contracting in some appropriate space of Δ and is consistent with y. When σ satisfies equation (2.2.13), we consider the following space: 󵄩 σt 󵄩 F1,σ = {Δ ∈ C((−∞, 0], L (PE, E)); ‖Δ‖F1,σ = sup(e 󵄩󵄩󵄩Δ(t)󵄩󵄩󵄩L (PE,F) ) < +∞}, t≤0

which is a Banach space with norm ‖ ⋅ ‖F1,σ . By equation (2.2.2), we have 󵄩󵄩 󵄩 󵄩󵄩Df (u)󵄩󵄩󵄩L (E,F) ≤ M1 ,

∀u ∈ E.

(2.2.59)

From this, we clearly see that L1 is entirely determined as a function, F1,σ ×PE → F1,σ , which is Lipschitz continuous in Δ and has Lipschitz constant θσ . When σ satisfies equation (2.2.13), θσ < 1. From this we deduce that there exists a function Δ : PE → F1,σ such that L1 (Δ(y), y) = Δ(y),

∀y ∈ PE.

For convenience, set Δ(y)(t) = Δ(y, t); Δ is related with the choice of differential φ.

2.2 Inertial manifold and normal hyperbolicity property | 245

Step 2: Δ is continuous. For fixed y0 ∈ PE, we consider y ∈ PE which approaches y0 . Similar as equation (2.2.26), we get 1 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩L (Δ(y0 ), y) − L1 (Δ(y0 ), y0 )󵄩󵄩󵄩F1,σ . 󵄩󵄩Δ(y) − Δ(y0 )󵄩󵄩󵄩F1,σ ≤ 1 − θσ 󵄩 1 Hence, in order to prove the continuity of Δ, we only need to prove that 󵄩 󵄩󵄩 󵄩󵄩L1 (Δ(y0 ), y) − L1 (Δ(y0 ), y0 )󵄩󵄩󵄩F1,σ → 0,

y → y0 .

Taking μ < σ, it satisfies equation (2.2.13). Then by step 1, we know Δ(y0 ) ∈ F1,μ . Since 󵄨󵄨 󵄨 󵄨󵄨L1 (Δ(y0 ), y)(t) − L1 (Δ(y0 ), y0 )(t)󵄨󵄨󵄨E t

󵄩 󵄩 ≤ 󵄩󵄩󵄩Δ(y0 )󵄩󵄩󵄩F K2 ∫ e−Λ(t−s) ((t − s)−α + Λα )N(s, y)e−μs ds 1,u −∞

0

󵄩 󵄩 + 󵄩󵄩󵄩Δ(y0 )󵄩󵄩󵄩F K1 λα ∫ e−λ(t−s) N(s, y)e−μs ds, 1,u

t

where 󵄩 󵄩 N(s, y) = 󵄩󵄩󵄩Df (φ(y0 , s)) − Df (φ(y, s))󵄩󵄩󵄩J (E,F) , we get 󵄩󵄩 󵄩 󵄩 󵄩 α ̃ 󵄩󵄩L1 (Δ(y0 ), y) − L1 (Δ(y0 ), y0 )󵄩󵄩󵄩F1,σ ≤ 󵄩󵄩󵄩Δ(y0 )󵄩󵄩󵄩F1,u (K1 λ + K2 )N(y), where t

0

−∞

t

̃ N(y) = sup[e(σ−Λ)t ∫ ((t − s)−α + Λα )e(Λ−μ)s N(s, y)ds + e(σ−λ)s ∫ e(λ−μ)t N(s, y)ds]. t≤0

̃ In order to prove that, when y → y0 , N(y) → 0, we use proof by contradiction. Assume ̃ N(yj ) > ε, where ε > 0, for a sequence {yj }j∈N in PE which satisfies |yj − y0 |E → 0, j → +∞. Then there exists a non-positive sequence {tj }j∈N such that tj

e

(σ−Λ)tj

0

∫ ((tj − s)−α + Λα )e(λ−u)t N(s, yj )ds + e(σ−λ)tj ∫ e(λ−u)s N(s, yj )ds ≥ ε, tj

−∞

∀j.

(2.2.60) But by equation (2.2.59), N = N(s, y) is consistently bounded in 2M1 , hence 󵄨󵄨 󵄨 󵄨󵄨left-hand side of equation (2.2.60)󵄨󵄨󵄨 tj

≤ 2M1 e

(σ−Λ)tj

0

∫ ((tj − s)−α + Λα )e(Λ−μ)s ds + 2M1 e(σ−Λ)tj ∫ e(λ−μ)s ds −∞

tj

246 | 2 Inertial manifold

≤ 2M1 e(σ−μ)tj [

(1 + γα )Λα 1 + ]. Λ−μ μ−λ

Therefore, based on equation(2.2.60), tj must have a lower bound. Let −∞ < T ≤ tj ≤ 0, ∀j, T ≤ 0, Then we have 󵄨 󵄨󵄨 󵄨󵄨left-hand side of equation(2.2.60)󵄨󵄨󵄨 tj

≤e

(σ−Λ)tj

∫ ((tj − s) e

−α (Λ−u)(tj −s)

α (σ−Λ)T

N(s, yj )ds + Λ e

−∞

0

∫ e(Λ−μ)s N(s, yj )ds −∞

0

+ e(σ−Λ)tj ∫ e(λ−μ)s N(s, yj )ds. T

Then, by changing variables in the first integral on the right-hand side, 󵄨󵄨 󵄨 󵄨󵄨left-hand side of equation (2.2.60)󵄨󵄨󵄨 +∞

≤ ∫ s e

−α −(Λ−μ)s

0

α (σ−Λ)T

N(tj − syj )ds + Λ e

0

0

∫ e(Λ−μ)s N(s, yj )ds −∞

+ ∫ e(λ−μ)s N(s, yj )ds.

(2.2.61)

T

But from equation (2.2.27), we have 󵄨󵄨 󵄨 󵄩 󵄩 −σ(t −s) 󵄨󵄨φ(y0 , tj − s) − φ(y, tj − s)󵄨󵄨󵄨E ≤ 󵄩󵄩󵄩φ(y0 ) − φ(y)󵄩󵄩󵄩Jσ e j K1 −σ(T−s) e |yj − y0 |E → 0, ≤ 1 − θσ

j → ∞.

Thus N(tj −s, yj ) → 0, j → +∞. For a fixed s ≥ 0, N(s, yj ) → 0. By applying Lebesgue dominated convergence theorem on the right-hand side of equation (2.2.61), we obtain that |left-hand side of equation (2.2.60)| → 0, j → +∞. This contracts (2.2.60). Therẽ fore, when |y − y0 |E → 0, we have N(y) → 0, hence Δ = Δ(y) is a continuous function from PE to F1,σ . Step 3: 𝜕y φ(y) = Δ(y). Considering y, h ∈ PE, we have φ(y + h, t) − φ(y, t) − Δ(y, t)h t

= ∫ e−(t−s)A Q[f (φ(y + h, s)) − f (φ(y, s)) − Df (φ(y, s))Δ(y, s)h]ds −∞

0

+ ∫ e−(t−s)A P[f (φ(y + h, s)) − f (φ(y, s)) t

+ Df (φ(y, s))Δ(y, s)h]ds.

(2.2.62)

2.2 Inertial manifold and normal hyperbolicity property | 247

Let |φ(y + h, t) − φ(y, t) − Δ(y, t)h|E , ∀y, h ∈ PE, ∀t ≤ 0, |h|E |f (u + ω) − f (u) − Df (u)ω|F , ∀u, ω ∈ E, r(u, ω) = |ω|E

ρ(y, h, t) =

R(y, h, t) = r(φ(y, t), φ(y + h, t) − φ(y, t)),

∀y, h ∈ PE, ∀t ≤ 0.

Hence, adding and subtracting Df (φ(y, s)(φ(y+h, s)−(φ(y, s)))) in the brackets of equation (2.2.62), we can estimate ρ = ρ(y, h, t) by t

ρ(y, h, t) ≤ K2 ∫ e−Λ(t−s) ((t − s)−α + Λα )R(y, h, s) −∞

0

+ K1 λα ∫ e−λ(t−s) R(y, h, s) t

|φ(y + h, t) − φ(y, t)|E ds |h|E

|φ(y + h, t) − φ(y, t)|E ds |h|E

t

+ M1 K2 ∫ e−Λ(t−s) ((t − s)−α + Λα )ρ(y, h, s)ds −∞ 0

+ M1 K1 ∫ e−λ(t−s) ρ(y, h, s)ds. t

Let ρ̃(y, h) = sup(eσt ρ(y, h, t)) =

‖φ(y + h, ⋅) − φ(y, ⋅) − Δ(y, ⋅)h‖Fσ

t≤0

|h|E

∀y, h ∈ PE.

,

Thus, by the above inequalities we have t

0

−∞

t

̃ h) + ρ̃(y, h){ ∫ e(σ−λ)(t−s) ((t − s)−α + Λα )ds + M1 K1 λα ∫ e(σ−λ)(t−s) ds} ρ̃(y, h) ≤ R(y, ̃ h) + θσ ρ̃(y, h), ≤ R(y, where t

̃ h) = sup{K2 eσt ∫ ((t − s)−α + Λα )e−Λ(t−s) R(y, h, s) |φ(y + h, s) − φ(y, s)|E ds R(y, |h|E t≤0 −∞

0

+ K1 λα eσt ∫ e−λ(t−s) R(y, h, s) t

|φ(y + h, t) − φ(y, t)|E ds}. |h|E

Then, since θσ < 1, we obtain ρ̃(y, h) ≤

1 ̃ R(y, h). 1 − θσ

248 | 2 Inertial manifold ̃ = N(y), ̃ ̃ h) → 0, when |h|E → 0. Here we use the As in step 2 for N we can prove R(y, following inequality: K1 −μs |φ(y + h, s) − φ(y, s)|E ≤ e , |h|E 1 − θμ

μ satisfies equation (2.2.13), μ < σ.

Then ρ̃(y, h) → 0, when |h|E → 0. This proves 𝜕y φ(y) = Δ(y). Step 4: Φ ∈ C 1 (PE, QE). From steps 2 and 3, φ(y) = 𝜕y φ(y, 0), and we have |Φ(y + h) − Φ(y) − DΦ(y)|E |QΦ(y + h, 0) − QΦ(y, 0) − Q𝜕y Φ(y, 0)h|E = |h|E |h|E = ρ(y, h, 0)

≤ ρ̃(y, h) → 0,

h → 0,

∀y ∈ PE,

where ρ̃(y, h, t) and ρ̃(y, h) have been defined in step 3. Proposition 2.2.6. If f ∈ C 1 (E, F), then each leaf Nu0 ∈ C 1 . More specifically, for all u0 ∈ E, we have Ψu0 (z0 ) ∈ C 1 (QE, PE), (u0 , z0 ) → DΨu0 (z0 ) is continuous, E × QE → L (QE, PE). Proof. Similar to the proof of Theorem 2.2.3 on the regularity of Φ (omitted). Since Φ and Ψu0 ∈ C 1 , we consider the tangent bundles Tu0 M = {η + DΦ(Pu0 )η; η ∈ PE},

Tu0 Nu0 = {ξ − DΦu0 (Qu0 )ξ ; ξ ∈ QE},

∀u0 ∈ M.

The next lemma states that E = Tu0 M ⊕ Tu0 Nu0 . Hence Tu0 Nu0 is the normal vector bundle of M at u0 . Lemma 2.2.8. For any u0 ∈ M, there is the decomposition E = Tu0 M ⊕ Tu0 Nu0 , and the decomposition is continuous in u0 . Proof. For a fixed u0 ∈ M and given μ ∈ E, the decomposition μ = η + ξ is unique, η ∈ Tu0 M, ξ ∈ Tu0 Nu0 . This is equivalent to η = η + DΦ(Pu0 )η + DΨu0 (Qu0 )ξ + ξ ,

(2.2.63)

where η ∈ PE, ξ ∈ QE. But equation (2.2.36) is equivalent to Pμ = η + DΨu0 (Qu0 )ξ + ξ ,

Qu = ξ + DΦ(Pu0 )η,

that is, {

η = Pμ − DΨu0 (Qu0 )Qμ + DΨu0 (Qu0 )DΦ(Pu0 )η, ξ = Qμ − DΦ(Pu0 )Pμ + DΦ(Pu0 )DΨu0 (Qu0 ).

(2.2.64)

2.2 Inertial manifold and normal hyperbolicity property | 249

Since Φ and Ψu0 have Lipschitz constant less than 1, their differentials DΦ(Pu0 ) and DΨu0 (Qu0 ) have norm less than 1, so I − DΨu0 (Qu0 )DΦu0 (Pu0 ) and I − DΦu0 (Pu0 ) × DΨu0 (Qu0 ) are invertible operators in PE and QE, respectively. Then, equation (2.2.64) is equivalent to η = (I − DΨu0 (Qu0 )DΦ(Pu0 )) (P − DΨu0 (Qu0 )Q)μ,

{

−1

ξ = (I − DΦ(Pu0 )DΨu0 (Qu0 )) (Q − DΦ(Pu0 )P)μ. −1

Hence, it can be written uniquely as μ = P(u0 )μ + Q(u0 )μ, where P(u0 ) is the projection of E to Tu0 Nu0 along Tu0 M, given as P(u0 ) = (I + DΦ(Pu0 ))(I − DΨu0 (Qu0 )DΦ(Pu0 )) (P − DΨu0 (Qu0 )Q), −1

and Q(u0 ) is the projection of E to Tu0 Nu0 along Tu0 M, given as Q(u0 ) = (I + DΨu0 (u0 ))(I − DΦ(u0 )DΨu0 (Qu0 )) (Q − DΦ(Pu0 )P). −1

By the regularity of Φ and Proposition 2.2.6, P(u0 ) and Q(u0 ) are continuous in u0 , so the decomposition E = Tu0 M ⊕ Tu0 Nu0 is continuous. Now we define the tangent and normal bundles TM = {(u, μ) ∈ E × E; u ∈ M, μ ∈ Tu M},

NM = {(u, μ) ∈ E × E; u ∈ M, μ ∈ Nu M}, where Nu M = Tu Nu . We consider equation (2.2.1) and its first variation dμ du + Au = f (u), + Aμ = f (u)μ, dt dt u(0) = u0 , μ(0) = μ0 .

(2.2.65)

We need to prove M is normal hyperbolic, namely, the tangent bundle TM and normal bundle NM are invariant under equation (2.2.65), and the exponential dichotomy of equation (2.2.65) applies to these bundles. More precisely, we have the following results: Lemma 2.2.9. The tangent bundle is invariant under equation (2.2.65), and 󵄨󵄨 󵄨 2 −(λ+2M1 K1 λα )t , 󵄨󵄨μ(t)󵄨󵄨󵄨E ≤ 2K1 |μ0 |E e

∀t ≤ 0, (u0 , μ0 ) ∈ TM.

250 | 2 Inertial manifold Proof. The invariance of TM follows from the invariance of M. In fact, set (u0 , μ0 ) ∈ TM. Then u0 = y0 + Φ(y0 ),

μ0 = η0 + DΦ(y0 )η0 ,

y0 , η0 ∈ PE.

Let y = y(t, y0 ), t ∈ R be a global solution with the following inertial form: dy + Ay = Pf (y + Φ(y)), dt

y(0, y0 ) = y0 ,

(2.2.66)

and let η = η(t) be the first variational global solution, that is, dη + Aη = PDf (y + Φ(y))(η + DΦ(y)η), dt

η(0) = η0 .

(2.2.67)

Since M = graph Φ is invariant under equation (2.2.1), Φ ∈ C 1 . We deduce that Φ(y) satisfies dΦ(y) + AΦ(y) = Qf (y + Φ(y)), dt Φ(y(0, y0 )) = Φ(y0 ).

(2.2.68)

Differentiating equation (2.2.68) with respect to y0 and acting on η0 , we arrive at dDΦ(y)η + ADΦ(y)η = QDf (y + Φ(y))(η − DΦ(y)η), dt

∀t ∈ R.

(2.2.69)

Adding equations (2.2.69) and (2.2.67), we get d(η + DΦ(y)η) + A(η + DΦ(y)η) = Df (y + Φ(y))(η + DΦ(y)η). dt Thus, μ(t) = η(t) + DΦ(y(t; y0 ))η(t) is the solution for first variational equation (2.2.1), ∀t ∈ R. And it belongs to the tangent space to Φ in y(t; y0 ) ∈ Φ(y(t; y0 )) ∈ graph Φ = M. Hence, if we set u(t) = y(t; y0 ) + Φ(y(t; y0 )), then we find that (u, μ) is the solution of equation (2.2.65) with (u(0), μ(0)) = (u0 , μ0 ). It belongs to TM, ∀t ∈ R. This proves that TM is invariant under equation (2.2.65). Regarding the estimation of μ(t), from equation (2.2.67) we note that 0

η(t) = e−tA η0 − ∫ e−(t−s)A PDf (y(s) + Φ(y(s))(η(s) + DΦ(y(s))η(s))ds. t

Thus for t ≤ 0, we arrive at 0

󵄨󵄨 󵄨 󵄨 −λt α −λ(t−s) 󵄨󵄨 󵄨󵄨η(t)󵄨󵄨󵄨E ≤ K1 e |η0 |E + M1 K1 (1 + l)λ ∫ e 󵄨󵄨η(s)󵄨󵄨󵄨E ds, t

2.2 Inertial manifold and normal hyperbolicity property | 251

where l < 1 is the Lipschitz constant of Φ. Then 0

󵄨 󵄨 󵄨 e 󵄨󵄨η(t)󵄨󵄨󵄨E ≤ K1 |η0 |E + 2M1 K1 λα ∫ eλs 󵄨󵄨󵄨η(s)󵄨󵄨󵄨E ds, λt 󵄨󵄨

t

and, by Gronwall lemma, we obtain 󵄨 󵄨󵄨 −(λ+2M1 K1 λα )t , 󵄨󵄨η(t)󵄨󵄨󵄨E ≤ K1 |η0 |E e

∀t ≤ 0.

Therefore 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨μ(t)󵄨󵄨󵄨E = 󵄨󵄨󵄨η(t) + DΦ(y(t))η(t)󵄨󵄨󵄨E

≤ (1 + l)K1 |y0 |E e−(λ+2M1 K1 λ

α

)t

α

≤ 2K1 |Pμ0 |E e−(λ+2M1 K1 λ )t . Since ‖P‖L (E) ≤ K1 , we finally get 󵄨󵄨 󵄨 2 −(λ+2M1 K1 λα )t , 󵄨󵄨μ(t)󵄨󵄨󵄨E ≤ 2K1 |μ0 |E e

∀t ≤ 0.

Lemma 2.2.10. The normal bundle NM is invariant under equation (2.2.65), and α 󵄨󵄨 󵄨 21+l |μ | e−(Λ−2M1 K2 (1+γα )Λ )t , 󵄨󵄨μ(t)󵄨󵄨󵄨E ≤ 2K2 1 − l󸀠 0 E

󸀠

∀t ≥ 0,

where (u0 , μ0 ) ∈ NM, l󸀠 < 1. Proof. The invariance of NM comes from the invariance of M and translation invariance of the fiber. In fact, set (u0 , μ0 ) ∈ NM. For each v0 ∈ Nu0 = graph Ψu0 , it can be written as v0 = Ψu0 (z0 ) + z0 ,

z0 ∈ QE.

Noticing that S(t)Nu0 = NS(t)u0 = graph ΨS(t)u0 , we obtain S(t)u0 = ΨS(t)u0 (z(t)) + z(t),

z(t) ∈ QE, t ≥ 0,

that is, z(t) = QS(t)v0 ,

ΨS(t)u0 (z(t)) = PS(t)v0 .

(2.2.70)

252 | 2 Inertial manifold Hence, for t ≥ 0, we have t

z(t) = e

−tA

z0 + ∫ e−(t−s)A Qf (S(s)(Ψu0 z0 ) + z0 )ds 0

t

= e−tA z0 + ∫ e−(t−s)A Qf (Ψu0 (z0 ) + z0 )ds

(2.2.71)

0

and t

ΨS(t)u0 (z(t)) = e

−tA

Ψu0 (z0 ) + ∫ e−(t−s)A Pf (ΨS(t)u0 (z(s)) + z(s))ds.

(2.2.72)

0

Let ξ = ξ (t), t ≥ 0 be the solution for the first variational equation (2.2.71), that is, t

ξ (t) = e−tA ξ0 + ∫ e−(t−s)A QDf (ΨS(t)u0 (z(s)) + z(s))(DΨS(t)u0 (z(s))ξ (s) + ξ (s))ds, (2.2.73) 0

where we take ξ0 = Qu0 , μ0 = DΨu0 (Pu0 )ξ0 + ξ0 . Now differentiating equation (2.2.72) with respect to z0 , and acting on ξ0 , we get DΨS(t)u0 (z(t))ξ (t) =e

−tA

t

DΨu0 (z0 )ξ0 + ∫ e−(t−s)A PDf (ΦS(t)u0 (z(s)) + z(s))(DΨS(s)u0 (z(s))ξ (s) + ξ (s))ds. 0

(2.2.74)

Thus, from equations (2.2.73) and (2.2.74) we obtain that μ(t) = DΨS(t)u0 (z(t))ξ (t) + ξ (t) ∈ TS(t)v0 NS(t)u0 is the solution for the first variational equation (2.2.1): μ󸀠 (t) + Aμ = PDf (S(t)v0 )μ,

μ(0) = DΨu0 (z0 )ξ0 + ξ0 .

If we set z0 = Pu0 , then we have v0 = u0 , μ(0) = μ0 . Hence μ(t) ∈ TS(t)u0 NS(t)u0 = NS(t)u0 M,

∀t ≥ 0.

Therefore (u(t), μ(t)) ∈ NM, ∀t ≥ 0, where u(t) = S(t)u0 . This means that NM is positive invariant under equation (2.2.65). For the estimation of ξ (t), from equation (2.2.73) when t ≥ 0 we have t

󵄨󵄨 󵄨 󵄨 󵄨 −Λt 󵄨 󸀠 −α α −Λ(t−s) 󵄨󵄨 󵄨󵄨ξ (t)󵄨󵄨󵄨E ≤ K2 e 󵄨󵄨󵄨ξ (0)󵄨󵄨󵄨E + M1 K2 (1 + l ) ∫((t − s) + Λ )e 󵄨󵄨S(s)󵄨󵄨󵄨E ds, 0

2.2 Inertial manifold and normal hyperbolicity property | 253

where l󸀠 < 1 is a bound on the Lipschitz constant for ΨS(t)u0 , and b = 2M1 K2 (1 + γα )Λα . Then t

󵄨 󵄨 󵄨 󵄨 e(Λ−b)t 󵄨󵄨󵄨ξ (t)󵄨󵄨󵄨E ≤ K2 e−bt |ξ0 |E + M1 K2 (1 + l󸀠 ) ∫((t − s)−α + Λα )e−b(t−s) e(Λ−b)s) 󵄨󵄨󵄨ξ (s)󵄨󵄨󵄨E ds. 0

Let 󵄨 󵄨 G(t) = max e(Λ−b)s) 󵄨󵄨󵄨ξ (s)󵄨󵄨󵄨E . 0≤s≤t

Hence, for 0 ≤ t ≤ T, t

󵄨 󵄨 e(Λ−b)t 󵄨󵄨󵄨ξ (t)󵄨󵄨󵄨E ≤ K2 |ξ0 |E + M1 K2 (1 + l󸀠 )G(t) ∫((t − s)−α + Λα )e−b(t−s) ds 0

M K (1 + l󸀠 )(1 + γα ) ≤ K2 |ξ0 |E + 1 2 G(t). b Taking the maximum of the above equation over t ∈ [0, T] and inserting it into the expression of b, we obtain G(T) ≤ K2 |ξ0 |E + which yields G(t) ≤

2K2 |ξ | , 1−l󸀠 0 E

1 + l󸀠 G(T), 2

∀t ≥ 0. Then

α 2K2 󵄨󵄨 󵄨 |ξ0 |E e−(Λ−2M1 K2 (1+γα )Λ )t , 󵄨󵄨ξ (t)󵄨󵄨󵄨E ≤ 󸀠 1−l 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨μ(t)󵄨󵄨E = 󵄨󵄨DΨS(t)u0 (z(t))ξ (t) + ξ (t)󵄨󵄨E 󵄨 󵄨 ≤ (1 + l󸀠 )󵄨󵄨󵄨ξ (t)󵄨󵄨󵄨E

≤ 2K2

∀t ≥ 0,

α 1 + l󸀠 |ξ0 |E e−(Λ−2M1 K2 (1+γα )Λ )t , 󸀠 1−l

∀t ≥ 0.

Since |ξ0 |E = |Qμ0 |E ≤ K2 |μ0 |E , we finally obtain α 󵄨󵄨 󵄨 21+l |μ | e−(Λ−2M1 K2 (1+γα )Λ )t , 󵄨󵄨μ(t)󵄨󵄨󵄨E ≤ 2K2 1 − l󸀠 0 E

󸀠

∀t ≥ 0.

Theorem 2.2.4. Under the assumptions of Theorem 2.2.1, if f ∈ C 1 (E, F), then the inertial manifold is normal hyperbolic.

254 | 2 Inertial manifold Proof. From Lemmas 2.2.9, 2.2.10 and 2.2.11, together with the spectral gap condition, we arrive at λ + 2M1 K1 λα < Λ − 2M1 K2 (1 + γα )Λα . Hence the contraction along the normal direction of M is stronger than along the tangential. The theorem is proved. Now we consider the high-order regularity of inertial manifold. Theorem 2.2.5. Suppose the assumptions of Theorem 2.2.1 are satisfied, and f ∈ C j,v (E, F), j = 1, 2, . . . , k; 0 ≤ v ≤ 1, k ∈ N. Then Dj f is uniformly bounded and possesses global Hölder continuity with the index v. Furthermore, if the following strong spectral gap condition is valid: Λ n − 2M1 K2 (1 + γα )Λαn ≥ (k + ν)(λn + 2M1 K1 λnα ),

(2.2.75)

then the inertial manifold M = graph Φ is satisfied Φ ∈ C j,ν (Pn E, Qn E), j = 1, 2, . . . , k. In order to prove Theorem 2.2.5, we introduce the space: s

Fk,σ = {Δ ∈ C((−∞, 0], Mk (PE, E)); ‖Δ‖Fk,σ ≡ sup e t≤0

kσt 󵄩 󵄩

󵄩󵄩 󵄩󵄩Δ(t)󵄩󵄩Mks (PE,E) < +∞},

which is a Banach space with norm ‖⋅‖Fk,σ ; Mks (PE, E) means the space of all the continuous symmetric k-linear mappings PE → E, with the norm ‖ ⋅ ‖(PE,E) . For convenience, set λ = λn , Λ = Λ n , P = Pn , Q = Qn , and assume condition (2.2.75). Lemma 2.2.11. Under the assumptions of Theorem 2.2.5, there exist 𝜕yj φ, j = 1, . . . , k, and 𝜕yj φ ∈ Fj,σ , where φ(y) is given by Lemma 2.2.2, and σ satisfies Λ − 2M1 K2 (1 + γα )Λα > kσ > σ > λ + 2M1 K1 λα .

(2.2.76)

Proof. We prove by induction on k. When k = 1, the claim has been proved in Theorem 2.2.3. Now suppose the claim is valid when j ranges from 1 to k − 1, k ≥ 2. Next we prove that the theorem is true when j = k. The proof is divided into three steps. Step 1: Preparations. Since φ(y) is the fixed point of L (⋅, y), differentiating equation φ(y) = L (φ(y), y) with respect to y, we know that 𝜕yk φ(y) is the fixed point of the formal mapping Lk (⋅, y), 0

Lk (Δ, y)(t) = − ∫ e t

−(t−s)A

P[G(y, s) + Df (φ(y, s))Δ(s)]ds

t

+ ∫ e−(t−s)A Q[G(y, s) + Df (φ(y, s))Δ(s)]ds, −∞

2.2 Inertial manifold and normal hyperbolicity property | 255

where Δ ∈ Fk,σ , G = G(y, s) involves derivatives of φ of order from the first to (k − 1)th. From the induction hypothesis, they exist. Now we set 𝜕yj φ(y) ∈ Fj,σ , j = 1, . . . , k, and since the derivative of f is bounded, we have 󵄩 󵄩󵄩 −kσt , 󵄩󵄩G(y, t)󵄩󵄩󵄩M s (PE,E) ≤ mσ e k

∀t ≤ 0,

(2.2.77)

where σ satisfies equation (2.2.77), Lk is a well-defined mapping, Fk,σ × PE → Fk,σ , which satisfies 󵄩󵄩 󵄩 󵄩󵄩Lk (Δ 1 , y) − Lk (Δ 2 , y)󵄩󵄩󵄩Fk,σ ≤ θkσ ‖Δ 1 − Δ 2 ‖Fk,σ , where y ∈ PE, Δ 1 , Δ 2 ∈ Fk,σ , θkσ =

M1 K2 λα M1 K2 (1 + λα )Λα + , kσ − λ Λ − kσ

and σ satisfies equation (2.2.76). Then θkσ < 1. Hence, Lk is a strictly contracting mapping in Fk,σ , uniform in PE. Then there exists a function Δ : PE → Fk,σ , which satisfies Lk (Δ(y), y) = Δ(y), ∀y ∈ PE. Step 2: The continuity of Δ. Fix y0 ∈ PE, and consider y ∈ PE. Let σ satisfy equation (2.2.76), and assume μ satisfies equation (2.2.76), too, μ < σ. Similar to equation (2.2.26), we get 1 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩L (Δ(y0 ), y) − Lk (Δ(y0 ), y0 )󵄩󵄩󵄩Fk,σ , 󵄩󵄩Δ(y) − Δ(y0 )󵄩󵄩󵄩Fk,σ ≤ 1 − θk,σ 󵄩 k

(2.2.78)

Δ ∈ Fk,σ , which yields 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Lk (Δ(y0 ), y) − Lk (Δ(y0 ), y0 )󵄩󵄩󵄩Fk,σ ≤ M(y) + 󵄩󵄩󵄩Δ(y0 )󵄩󵄩󵄩Fk,σ N(y),

(2.2.79)

where α kσt

M(y) = sup{K1 λ e t≤0

t

0

󵄩 󵄩 ∫ e−λ(t−s) 󵄩󵄩󵄩G(y, s) − G(y0 , s)󵄩󵄩󵄩M s (PE,F) ds t

k

󵄩 󵄩 + K2 ekσt ∫ ((t − s)−α + Λα )e−Λ(t−s) 󵄩󵄩󵄩G(y, s) − G(y0 , s)󵄩󵄩󵄩M s (PE,F) ds}, k

−∞ α kσt

N(y) = sup{K1 λ e t≤0

t

0

󵄩 󵄩 ∫ e−λ(t−s) 󵄩󵄩󵄩Df (φ(y, s)) − Df (φ(y0 , s))󵄩󵄩󵄩L (E,F) e−kμs ds t

󵄩 󵄩 + K2 ekσt ∫ e−Λ(t−s) 󵄩󵄩󵄩Df (φ(y, s)) − Df (φ(y0 , s))󵄩󵄩󵄩L (E,F) e−kμs ds}. −∞

As in the proof of Theorem 2.2.3, when y → y0 , M(y), N(y) → 0. Then from equations (2.2.78)–(2.2.79), we deduce that when y → y0 , ‖Δ(y) − Δ(y0 )‖Fk,σ → 0. This proves the continuity of Δ.

256 | 2 Inertial manifold Step 3: 𝜕yk φ(y) = Δ(y). For y, h ∈ PE, set ρ(y, h, t) =

1 [φ(y + h, t) − φ(y, t) − 𝜕y φ(y, t)h − ⋅ ⋅ ⋅ |h|kE −

1 1 𝜕yk−1 φ(y)(h, . . . , h) − [Δ(y)(h, . . . , h)|E ]]. (k − 1)! k!

By induction, we prove that μ and σ satisfy equation (2.2.76), μ < σ. We have e

kσt

α kσt

ρ(y, h, t) ≤ K1 λ e

0

∫ e−λ(t−s) R(y, h, s)e−kμs ds t

t

+ K2 ekσt ∫ ((t − s)−α + Λα )e−Λ(t−s) R(y, h, s)e−kμσ ds −∞

+ θkσ sup(ekσs ρ(y, h, s)), s≤0

where R(y, h, s) is uniformly bounded and, when |h|E → 0, R(y, h, s) → 0. For each point s and t ≤ 0, taking an upper bound, we have sup(ekσt ρ(y, h, t)) ≤ t≤0

1 ̃ R(y, h), 1 − θkσ

̃ h, s) → 0. where by Lebesgue dominated convergence theorem, when |h|E → 0, R(y, k k This gives the existence of 𝜕y φ(y) and 𝜕y φ(y) = Δ(y) ∈ Fk,σ . Proof of Theorem 2.2.5. By the spectral gap condition (2.2.75), we can select σ which satisfies equation (2.2.76), 𝜕yj φ ∈ Fj,σ , j = 1, 2, . . . , k. Since Φ(y) = Qφ(y, 0), then Φ ∈

C k (PE, QE). We show Hölder continuity of DΦ. Taking y1 , y2 ∈ PE, we have 󵄩󵄩 󵄩 󵄩󵄩𝜕y φ(y1 , t) − 𝜕y φ(y2 , t)󵄩󵄩󵄩L (E,F) 0

󵄩 󵄩 󵄩 󵄩 ≤ K1 λα ∫ e−λ(t−s) 󵄩󵄩󵄩Df (φ(y1 , s)) − Df (φ(y2 , s))󵄩󵄩󵄩L (E,F) 󵄩󵄩󵄩𝜕y φ(y1 , s)󵄩󵄩󵄩L (E,F) ds t

0

󵄩 󵄩 󵄩 󵄩 + K1 λ ∫ e−λ(t−s) 󵄩󵄩󵄩Df (φ(y2 , s))󵄩󵄩󵄩L (E,F) 󵄩󵄩󵄩𝜕y φ(y1 , s) − 𝜕y φ(y2 , t)󵄩󵄩󵄩L (E,F) ds α

t

t

+ K1 ∫ ((t − s)−α Λα)e−Λ(t−s) −∞

󵄩 󵄩 󵄩 󵄩 × 󵄩󵄩󵄩Df (φ(y1 , s)) − Df (φ(y2 , s))󵄩󵄩󵄩L (E,F) 󵄩󵄩󵄩𝜕y φ(y1 , s)󵄩󵄩󵄩L (E,F) ds

2.2 Inertial manifold and normal hyperbolicity property | 257 t

+ K2 ∫ ((t − s)−α Λα)e−Λ(t−s) −∞

󵄩 󵄩 󵄩 󵄩 × 󵄩󵄩󵄩Df (φ(y2 , s))󵄩󵄩󵄩L (E,F) 󵄩󵄩󵄩𝜕y φ(y1 , s) − 𝜕y φ(y2 , s)󵄩󵄩󵄩L (E,F) ds 0

󵄩v 󵄩 󵄩 󵄩 ≤ K1 λα 󵄩󵄩󵄩𝜕y φ(y1 )󵄩󵄩󵄩B ∫ e−λ(t−s) M2 󵄩󵄩󵄩φ(y1 , s) − φ(y2 , s)󵄩󵄩󵄩E e−μs ds 1,μ t

t

󵄨 󵄨v 󵄩 󵄩 + K2 󵄩󵄩󵄩𝜕y φ(y1 )󵄩󵄩󵄩B ∫ ((t − s)−α Λα)e−Λ(t−s) M2 󵄨󵄨󵄨φ(y1 , s) − φ(y2 , s)󵄨󵄨󵄨E e−μs ds 1,μ + θσ e

−∞

−σt 󵄩 󵄩

󵄩 󵄩󵄩𝜕y φ(y1 ) − 𝜕y φ(y2 )󵄩󵄩󵄩B1,σ ,

where σ, μ satisfy equation (2.2.13), (1 + v)μ < σ, and M2 is the Hölder constant of M2 . If we replace μ with σ in equation (2.2.27), we get K1v 󵄩󵄩 M2 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝜕y φ(y1 ) − 𝜕y φ(y2 )󵄩󵄩󵄩B1,σ ≤ 󵄩𝜕 φ(y1 )󵄩󵄩󵄩B1,μ 1 − θσ (1 − θσ )v 󵄩 y |y1 − y2 |νE {

K1 λα K1 (1 + λα )Λα + }. σ−λ Λ−σ

Since 𝜕y φ(y1 ) = L1 (𝜕y φ(y1 ), y1 ), we have K1 󵄩󵄩 󵄩 . 󵄩󵄩𝜕y φ(y1 )󵄩󵄩󵄩B1,σ ≤ (1 − θμ ) Therefore 󵄩󵄩 󵄩 v 󵄩󵄩𝜕y φ(y1 , s) − 𝜕y φ(y2 , s)󵄩󵄩󵄩B1,σ ≤ C|y1 − y2 |E , where C=

K11+v M2 K1 γ α K1 (1 + λα )Λα { + }. 1 − θσ (1 − θσ )1+v σ − λ Λ−σ

Thus for DΦ(y) = Q𝜕y φ(y, 0), it follows that 󵄩󵄩 󵄩 v 󵄩󵄩DΦ(y1 ) − DΦ(y2 )󵄩󵄩󵄩L (PE,QE) ≤ C|y1 − y2 |E , which proves the Hölder continuity of DΦ. The Hölder continuity of Dj Φ, j = 1, . . . , k, can be proved by induction. As a consequence, we get Theorem 2.2.6. Suppose the assumptions of Theorem 2.2.5 are met; in particular, the spectral gap condition (2.2.75) is satisfied. Then the leaf Nv0 = graph Ψv0 is such that Ψν0 ∈ C jv (QE, PE), j = 1, 2, . . . , k, ∀v0 ∈ E. Proof. Similar to the proof of Theorem 2.2.5.

258 | 2 Inertial manifold

2.3 The finite-dimensional inertial form for the one-dimensional generalized Ginzburg–Landau equation In [97], the authors consider the following equation: 𝜕t u + νux = χu + (γr + iγi )uxx − (βr + iβi )|u|2 u − (δr + iδi )|u|4 u − (λr + iλi )|u|2 ux − (μr + iμi )u2 ux ,

x ∈ R, t > 0,

(2.3.1)

which was put forward in [46]. As shown in [209], when γr > 0, ν > 0, βi = 1, βr < 0, χ < 0, δr = λr = μr = μi = 0, and when the initial value of u is large enough, the spacial periodic solution of the equation will blow up. But the corresponding physical motion did not provide evidence of the blow-up phenomenon [79]. So the above equation was proposed in [46] to describe the objective reality more accurately. Therefore studies of the well-posedness of the solution of this equation and its long time behavior are necessary. In [97], the authors got the following conclusion: Theorem 2.3.1. When χ > 0, δr > 0, γr > 0, 4δr γr > (λi − μi )2 , a spacial periodic 1 solution for equation (2.3.1) exists and is unique in Hper [0, L]. And for the semigroup S(t) determined by the solution (namely, S(t)u(t) = S(t)u0 , t ≥ 0, where u0 is the initial value) 1 there exists a finite-dimensional attractor in Hper [0, L]. In the following, we will prove that the condition 4δr γr > (λi − μi )2 cannot be improved. If 4δr γr ≤ (λi − ui )2 , we consider a spacial periodic solution u = Reikx . Substituting it into equation (2.3.1), taking the real part and solving for R2 , we get R2 =

1 2 (−(βr − (λi − μi )k)) ± √(βr − (λi − ui )k) − (4δr γr k 2 − 4δr χ) 2δr 2

=

β β 4δ χ k (−( r − (λi − μi ))) ± √( r − (λi − μi )) − 4δr γr + 2r . 2δr k k k

We can see that, when k → ±∞, R2 → +∞. This illustrates that the solution blows up. Thus 4δr γr > (λi − μi ) cannot be improved. Because the semigroup S(t) of equation (2.3.1) has a finite-dimensional attractor, in order to effectively consider the long time behavior of solutions, we expect to use finite-dimensional ordinary differential equations to describe the attractor. Therefore, we must take into account the inertial manifold; for the definitions of attractor and inertial manifold, refer to [197]. An important condition for the existence of inertial α α manifolds is the spectral gap condition, λm+1 − λm > M02 1+l (λm+1 + λm ), here {λm }∞ m=1 is l 2πm 2 a sequence of eigenvalues for −γr 𝜕xx (λm = γr ( L ) , L is the length of spacial period) with periodic boundary conditions, l ≤ 81 is given, M0 is a constant of estimation. The

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 259

spectral gap conditions are met if γr (

2

1 + l 2π 21 2π ) (2m + 1) > M0 γ (2m + 1), L l L r

or 1 + l − 21 2π γ L≤ . l r M0 From the above equation we can see that, when L is small enough or γr is large enough, this is indeed the case. By the conclusions in [197], now we get the existence of an inertial manifold of equation (2.3.1); here we do not discuss this in detail. The purpose of this section is to try to improve the results. Consider a simplified form of equation (2.3.1): ut = χμ + γuxx − (βr + iβi )|u|2 u − (δr + iδi )|u|4 u − (λr + iλi )|(u|2 u)x ,

(2.3.2)

x ∈ R, t > 0, γ, δr > 0. Obviously, equation (2.3.3) is a special case of equation (2.3.2). Equation (2.3.2) is a generalized form for the following Ginzburg–Landau equation studied in [53]: ut = uxx + (1 − |u|2 )u, where u is a complex function. Here using a nonlinear term linearized method, the long time dynamic behavior of equation (2.3.2) can be described by a group of finite-dimensional ordinary differential equations. Firstly, we introduce some notations: H = L2per [0, L] = {u ∈ L2 [0, 1], u(x) = u(x + L)}, its inner product and norm are respectively defined by L

(u, v) = ∫ uvdx, 0

|u|20 = (u, u),

u, v ∈ H,

n Hper [0, L] = {u : u ∈ H, ux ∈ H, . . . , ux...x ∈ H},

n ≥ 1.

Suppose A = −(μ + γ𝜕xx ), for γ > 0 and appropriate μ. Then the characterization problem with periodic boundary condition, −(μ + γ𝜕xx )g = λg, has no zero eigenvalue. Hence A is a linear self-adjoint unbounded positive operator. It follows that we can define the power of A as Aα , α ∈ [0, 1], V2α = D(Aα ) (domain of Aα ). By [141], 1 1 2 V1 = D(A 2 ) = Hper [0, L], V2 = D(A) = Hper [0, L]; the norms of V1 and V2 can be defined as | ⋅ |1 and | ⋅ |2 . Since equation (2.3.2) is a special case of equation (2.3.1), then γ, δr > 0 and, whenever 4δr γ > λi2 , the global solution of equation (2.3.2) exists uniquely in V1 , and there exists a finite-dimensional attractor AGGL in V1 . Further, we have

260 | 2 Inertial manifold Proposition 2.3.1. If u ∈ AGGL , and if there exist ρ0 , ρ1 , ρ2 > 0 such that |u|0 ≤

ρ0 , 2

|u|1 ≤

ρ1 , 2

|u|2 ≤

ρ2 , 2

2 then we have AGGL ⊂ Hper [0, L]. ρ

ρ

Proof. By [97] we know that, if u ∈ AGGL , then it is obvious that |u|0 ≤ 20 , |u|1 ≤ 21 . We ρ only need to prove |u|2 ≤ 22 . By [97] we know, when t > 0, u(t) is smooth enough, then taking the inner product of (2.3.2) and uxxxx and then the real component, we obtain L

1 d |u |2 + γ|uxxx |20 = χ|uxx |20 − Re((βr + iβi ) ∫ |u|2 uuxxxx dx) 2 dt xx 0 0

L

− Re((δr + iδi ) ∫ |u|4 uuxxxx dx) 0

L

− Re((λr + iλi ) ∫(|u|2 u)x uxxxx dx).

(2.3.3)

0

By [97] we know that, there exists T > 0 such that when t ≥ T we have |u|0 ≤ ρ0 ,

|u|1 ≤ ρ1 .

In the following estimation, we use the embedding inequality |u|L∞ ≤ C|u|1 and Gagliardo–Nirenberg inequality. The constant C in the estimate is a positive number only depending on the coefficients ρ0 , ρ1 in equation (2.3.2). Here we do not distinguish them. For t ≥ T, we have 1 d |u |2 + γ|uxxx |20 2 dt xx 0 ≤

|χ||uxx |20

+

√βr2

+

󵄨󵄨

L

󵄨󵄨

󵄨 󵄨 βi2 󵄨󵄨󵄨 ∫(|u|2 u)x uxxx dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 0

+

√δr2

󵄨󵄨 L 󵄨󵄨 󵄨 󵄨 = √λr2 + λi2 󵄨󵄨󵄨 ∫(|u|2 u)xx uxxx dx 󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨

+

󵄨󵄨

L

󵄨󵄨

󵄨 󵄨 δi2 󵄨󵄨󵄨 ∫(|u|4 u)x uxxx dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 0 (2.3.4)

0

Obviously, we only need to deal with the last term: L 󵄨󵄨 L 󵄨󵄨 γ 1 2 󵄨󵄨 󵄨󵄨 2 2 2 2 󵄨󵄨 ∫(u| u)xx uxxx dx 󵄨󵄨 ≤ |uxxx |0 + ∫(6|u||ux | + 3|u| |uxx |) dx 󵄨󵄨 4 󵄨󵄨 γ 0

0

L

γ ≤ |uxxx |20 + C(ρ0 , ρ1 ) ∫ |ux |4 dx + C(ρ0 , ρ1 )|uxx |20 . 4 0

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation |

261

L

For ∫0 |ux |4 dx, we use Gagliardo–Nirenberg inequality, and equation (2.3.4) becomes d |u |2 ≤ C + C|uxx |20 . dt xx 0 By Lemma 2.4 in [97], we have t+1

󵄨2 󵄨 ∫ 󵄨󵄨󵄨uxx (⋅, τ)󵄨󵄨󵄨0 dτ ≤ C(ρ0 , ρ1 ),

t ≥ T.

t

Using the uniform Gronwall inequality, we obtain that |uxx |20 ≤ ρ2 , t ≥ T + 1, ρ only depends on the coefficients ρ0 , ρ1 of the equation. Hence |u|2 ≤

ρ2 , 2

u ∈ AGGL ,

and Proposition 2.2.3 has been proved. This generalized the results of [97]. In order to keep the dissipation of equations after transformation, we need to alter equation (2.3.2) appropriately, namely ut = γuxx − (λr + iλi )(|u|2 u)x − φρ ⋅ g(u) − (1 − φρ )(δr + 9γ + 9

λr2 + λi2 ) × |u|4 u, γ

(2.3.5)

where g(u) = (βr + iβi )|u|2 u + (δr + iδi )|u|4 u − χu, φρ = φ(

|u|20 + 2|ux |20 + |u|6L6 ρ2

),

0 < ρ ≤ ∞,

φ : R+ → [0, 1] is a smooth monotonic function such that φ(s) = 1, 0 ≤ s ≤ 1, φ(s) = 0, s ≥ 2|φ󸀠 (s)| ≤ 2. We notice that φρ is independent of the spacial variables. If ρ = ∞, then φρ = 1, and equation (2.2.5) is (2.2.2). Given the initial value u0 ∈ V1 , similar to the discussion in [97], we have Proposition 2.3.2. Given the initial value u0 ∈ V1 , γ, δr > 0, 4δr γ > λi2 , equation (2.2.5) n has a unique solution u ∈ C([0, ∞); V1 ) ∩ L2 (0, T; V2 ) ∩ C(0, +∞; Hper [0, L]) (n ≥ 2 is arbitrary). Furthermore, there exist constants r0 , r1 , r2 (independent of u0 and ρ) such that lim |u|0 ≤ r0 ,

t→∞

lim |u|1 ≤ r1 ,

t→∞

lim |u|2 ≤ r2 .

t→∞

Thus it can be seen that equation (2.2.5) possesses the global attractor Aρ (its finite dimension is shown similarly as in [97]), and when ρ2 ≥ 4r 2 = 4(r02 + 2r12 + r32 ) (here r3 is a bound of |u|3L6 ), Aρ = AGGL . The proof of Proposition 2.3.2 is similar to the proof of existence of an absorbing set in [97] and the proof of Proposition 2.3.2; here we omit it.

262 | 2 Inertial manifold By Proposition 2.3.2, we can introduce the following function transform: J(u) = (u, ux , |u|2 u) = (u, v, ω). Then u, v, ω respectively satisfy the following equations (we are only using equation (2.3.5) and calculating directly): u = γuxx − (λr + iλi )ωx + η1 , { { t {vt = γvxx − (λr + iλi )ωxx + η2 , { {ωt = γωxx + η3 ,

(2.3.6)

where η1 = −φρ g(u) − (1 − φρ )(9γ + δr + 9

λr2 + λi2 )|u|4 u, γ

η2 = −φρ h(u, v) − (1 − φρ )(9γ + δr + 9

λr2 + λi2 )(3|u|4 v + 2|u|2 u2 v), γ

h(u, v) = (βr + iβi )(2|u|2 v + u2 v) + (δr + iδi )(2|u|4 v + 2|u|2 u2 v), η3 = −(4γ|v|2 u2 + 2γv2 u) + 2|u|2 η1 + u2 η1

− 2(λr + iλi )(2|u|4 v + |u|2 u2 v) − (λr − iλi )(2|u|2 v + |u|4 v).

For equations (2.3.6), coupled with some additional terms below (and to keep the dissipative property), ut = γuxx − (λr + iλi )ωx + η1 + ξ1 , { { { { v = γvxx − (λr + iλi )ωxx + η2 − k1 (v − ux ) + ξ2 , { { t { {ω = γω − (4γ|ν|2 u + 2γv2 u) t xx { 2 { + 2|u| (η1 + ξ1 ) + u2 (η1 + ξ 1 ) { { { { { 1 { − (k2 − 16(λr2 + λi2 ))(1 + |u|2 )(ω − f (u)), { γ

(2.3.7)

where f (u) = |u|2 u; k1 , k2 are unknown constants; ξ1 , ξ2 can be taken as ξ1 = −2γ|u|2 (ω − f (u)) + γu2 (ω − f (u)), ξ2 = 2γuv(ω − f (u)) + 2γuv(ω − f (u)) + 2γuv(ω − f (u)). Note that if v = ux , ω = f (u), the additional terms disappear. Setting J(u) = (u, ux , f (u)), u ∈ V1 , these additional terms will lead to the solutions of (2.3.7), which are exponentially converging to J(u). So far we have seen that if u(t) is the solution of equation (2.3.5) with initial value u0 ∈ V1 , then J(u(t)) is the solution of equation (2.3.7). By the uniqueness of solution of equation (2.3.7), the reversed conclusion is also valid. The uniqueness and existence of solution of equation (2.3.7) will be given in the following. Hence, the solutions of equations (2.3.5) and (2.3.7) have the same dynamics in the set J(V1 ).

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 263

Proposition 2.3.3. If u(t) is the solution of equation (2.3.5) with the initial value u0 ∈ V1 , then J(u(t)) is the solution of equation (2.3.7). Conversely, if (u, v, ω) is the solution of equation (2.3.7) with the initial value J(u0 ), u0 ∈ V1 , then u(t) is the solution of equation (2.3.5). In the following we study the uniqueness and existence of solution for equation (2.3.7). Letting U = (u, v, ω)t (t denotes the transpose), in the space D(A) × D(A) × D(A) we define the operator A by Au + (λr + iλi )ωx AU = (Av + k1 ux + (λr + iλi )ωxx ) , Aω where A is defined as before. Similarly, define F̃ = (F1 , F2 , F3 )t where F1 = η1 + ξ1 + μu, F2 = η2 + ξ2 − k1 v + μv, F3 is the right-hand side of the third equation in (2.3.7) with the term μω added. Then equation (2.3.7) can be written as d U = −AU + F(U). dt

(2.3.8)

It is obvious that the operator A is not self-adjoint, but we can prove A is a fan-shaped operator, that is, −A generates an analytic semigroup in H = H × H × H. Lemma 2.3.1. The operator A is a fan-shaped operator in H . Proof. First, we note that A is a fan-shaped operator in H, hence there exist 0 < θ < and M ≥ 1 such that

π 2

ρ(A) ⊃ Σ = {λ | θ < |arg λ| ≤ π, λ ≠ 0}, M 󵄩󵄩 −1 󵄩 , 󵄩󵄩(λ − A) 󵄩󵄩󵄩op ≤ |λ|

λ ∈ Σ.

(2.3.9)

Define (A − λ)U = F where λ ∈ Σ, F = (f1 , f2 , f3 )t ∈ H . Then the equation for ω can be written as Aω − λω = f3 .

(2.3.10)

M |f | . |λ| 3 0

(2.3.11)

From equation (2.3.9) we have |ω|0 ≤

264 | 2 Inertial manifold Taking the inner product of equation (2.3.10) and ω, and then taking its real component, and using equation (2.3.11), we get M2 + M 2 󵄨󵄨 21 󵄨󵄨2 2 |f3 |0 . 󵄨󵄨A ω󵄨󵄨0 ≤ |λ||ω|0 + |f3 |0 |ω|0 ≤ |λ|

(2.3.12)

From equation (2.3.10), we get |ωxx |0 ≤ (|λ| + |u|)|ω|0 + |f3 |0 .

(2.3.13)

The equation about u is Au + (λr + iλi )ωx − λu = f1 . Using equation (2.3.9) and (2.3.12), we arrive at M (|f | + √λr2 + λi2 |ωx |0 ) |λ| 1 0

|u|0 ≤

1

M M2 + M 2 2 ≤ (|f1 |0 + ( ) √λr + λi2 |f3 |0 ) |λ| |λ| M ≤ 1 (|f1 |0 + |f3 |0 ), M1 ≥ 1, |λ| 1

(2.3.14)

1

where we used |ux |0 ≤ |A 2 u|0 , u ∈ D(A 2 ). Similarly, we have 󵄨󵄨 21 󵄨󵄨2 2 󵄨󵄨A u󵄨󵄨0 ≤ λ|u|0 + √λr2 + λi2 |ωx |0 |u|0 + |f1 |0 |u|0 1 1 ≤ (|λ| + 1)|u|20 + (λr2 + λi2 )|ωx |20 + |f1 |20 . 2 2

(2.3.15)

Finally, we consider the equation for v. From equation (2.3.9) we have |v|0 ≤

M (k |u | + √λr2 + λi2 |ωxx |0 + |f2 |0 ). |λ| 1 x 0

By equations (2.3.12), (2.3.13) and (2.3.15), we have |v|0 ≤

M2 (|f | + |f2 |0 + |f3 |0 ), |λ| 1 0

M2 ≥ 1.

(2.3.16)

From equations (2.3.11), (2.3.14) and (2.3.16), we obtain that for λ ∈ Σ, (A − λI)−1 exists, and M 󵄩󵄩 −1 󵄩 󵄩󵄩(−λI + A) 󵄩󵄩󵄩op ≤ 3 , |λ|

M3 ≥ 1.

This proves that A is a fan-shaped operator in H . By [185, Th. 5.2], we get that −A generates an analytic semigroup in H .

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation |

265

Since −A is a sectoral operator, we can define the fractional power of A, and it is 1 easy to see that F̃ : D(A 2 ) = V1 × V1 × V1 → H is locally Lipschitz continuous. Hence if 1 U(0) = U0 ∈ D(A 2 ), then there exists a unique strong solution of equation (2.3.7) such that 1

U ∈ C([0, T); D(A 2 )),

0 < T ≤ ∞.

In the following, we will prove equations (2.3.7) possess the global attractor A . In fact, A is the image of Aρ under embedding J, that is, J(Aρ ) = A . If U is a smooth enough solution of equation (2.3.7), then the equation of u is ut = γuxx − (λr + iλi )ωx + η1 + ξ1 .

(2.3.17)

Differentiating equation (2.3.17) with respect to x, we get (ux )t = γuxxx − (λr + iλi )ωxx + η1x + ξ1x . Then subtracting the above equation from that of v in (2.3.7), we obtain (v − ux )t = γ(v − uxx )x + η2 − η1x + ξ2 − ξ1x − k1 (v − ux ).

(2.3.18)

̃ = f (|u|2 u), it is easy to verify that ω ̃ satisfies Letting ω ̃t = γω ̃xx − (4γ|ux |2 u + 2γu2x u) ω

+ 2|u|2 (η1 + ξ1 ) + u2 (η1 + ξ 1 )

− 2|u|2 (λr + iλi )ωx − u2 (λr − iλi )ωx , and then we have ̃)t = γ(ω − ω ̃)xx − 4γu(|v|2 − |ux |2 ) − 2γu(v2 − u2 ) (ω − ω − 2(λr + iλi )|u|2 (2|u|2 v + u2 v − ωx )

− (λr − iλi )u2 (2|u|2 v + u2 v − ωx )

1 ̃). − (k2 − 16(λr2 + λi2 )(1 + )|u|4 )(ω − ω γ

(2.3.19)

Firstly, we estimate a uniform bound of ̃|20 . |u|20 + |ux |20 + |v − ux |20 + |ω − ω Then by Minkowski inequality, we get a uniform bound of |U|K . That is, we need to prove the following proposition: Proposition 2.3.4. Let k1 > 0, k2 > γ +

λr2 +λi2 , 2α

γ > 2√λr2 + λi2 (α is a constant). If U is the 1

solution of equation (2.3.7) with the initial value U0 ∈ D(A 2 ), then, when ρ4 ≥ √2ρ, there exists lim |U|2K ≤ ρ24 .

t→∞

266 | 2 Inertial manifold Proof. Let U = (u, v, ω)t be the smooth solution of equation (2.3.7). Then taking the inner product of equation (2.3.17) and u, and then taking its real part, we obtain L

1 d 2 |u| = −γ|ux |20 − Re((λr + iλi ) ∫ ωx udx) + Re(η1 + ξ1 , u). 2 dt 0 0

Since L

− Re((λr + iλi ) ∫ ωx udx) 0

L

= Re((λr + iλi ) ∫ ωux dx) 0

L

L

= Re((λr + iλi ) ∫(ω − f (u))ux dx) + Re((λr + iλi ) ∫ f (u)ux dx) 0

0

L

̃, ux )) − λi Im ∫ |u|2 uux dx, = Re((λr + iλi )(ω − ω Re(η1 + ξ1 , u)

0 L

= − Re(φρ ∫((βr + iβi )|u|4 + (δr + iδi )|u|6 − χ|u|2 )dx) 0

L

− Re((1 − φρ )(δr + 9γ + 9

λr2 + λi2 ) ∫ |u|6 dx) + Re(ξ1 , u) γ

L

L

L

0

0

0

0

L

≤ −δr ∫ |u|6 dx + |βr | ∫ |u|4 dx + |χ| ∫ |u|2 dx − Re((1 − φρ )9γ ∫ |u|6 dx) + Re(ξ1 , u). 0

According to the choice of ξ1 , we have L

̃|dx. Re(ξ1 , u) ≤ 3γ ∫ |u|3 |ω − ω 0

Hence

1 d 2 ̃, ux )) |u| ≤ −γ|ux |20 − Re((λr + iλi )(ω − ω 2 dt 0 L

L

L

0

6

L

L

4

L

+ |λi | ∫ |u| |ux |dx − δr ∫ |u| dx + |βr | ∫ |u| dx + |χ| ∫ |u|2 dx 0

3

0

̃|dx. − (1 − φρ )9γ ∫ |u|6 dx + 3γ ∫ |u|3 |ω − ω 0

0

0

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 267

By virtue of L

L

0

0

̃|dx ≤ 9γ ∫ |u|6 dx + 3γ ∫ |u|3 |ω − ω

γ ̃|20 |ω − ω 4

and the assumption 4δr γ > λi2 , we can choose 2α = 2γ − b2 ,

β = 2δr − a2 ,

|a ⋅ b| = |λ|.

The rest of estimation follows [97], and we have L

1 d 2 |u| ≤ −2α|ux |20 − |u|20 + P + 18φρ γ ∫ |u|6 dx 2 dt 0 0

γ ̃|20 − 2 Re((λr + iλi )(ω − ω ̃, ux )) + |ω − ω 2 ≤

−α|ux |20

−

|u|20

L

+ P + 18φρ γ ∫ |u|6 dx

2 2 γ λ + λi ̃|20 , +( + r )|ω − ω 2 α

0

(2.3.20)

(|β |+1)2

where P = 81 ( r β + 2|χ| + 1)2 . Taking the inner product of equation (2.3.17) and uxx , and then taking its real part, we obtain L

1 d 2 |u| = −γ|uxx |20 + Re((λr + iλi ) ∫ ωx uxx dx) − Re(η1 + ξ1 , uxx ). 2 dt 0 0

By virtue of L

L

Re((λr + iλi ) ∫ ωx uxx dx) = Re((λr + iλi ) ∫(ωx − ω)̃ x uxx dx) 0

0

L

+ Re((λr + iλi ) ∫(|u|2 u)x uxx dx) 0

γ ≤ |uxx |20 + 2

λr2

L

+ λi2 󵄨󵄨 9 2 ̃)x 󵄨󵄨󵄨󵄨0 + (λr2 + λi2 ) ∫ |u|4 |ux |2 dx, 󵄨(ω − ω γ 󵄨 γ

Re(η1 + ξ1 , uxx ) = −φρ Re(g(u), uxx ) + Re(ξ1 , uxx ) − (1 − φρ )(9γ + δr + 9

0

λr + iλi ) Re(|u|4 u, uxx ), γ

268 | 2 Inertial manifold L

L

Re(|u| u, uxx ) = 3 ∫ |u| |ux | dx + 2 ∫ |u|2 u2 u2x dx 4

L

4

2

0

0

≥ ∫ |u|4 |ux |2 dx, 0

L

̃|dx Re(ξ1 , uxx ) ≤ 3γ ∫ |u|2 |ω − ω 0

L

̃|2 dx + ≤ 9γ ∫ |u|4 |ω − ω 0

γ |u |2 . 4 xx 0

Together with the above calculation, we obtain L

γ 18 1 d |u |2 ≤ − |uxx |20 + φρ ⋅ (λr2 + λi2 ) ∫ |u|4 |ux |2 dx 2 dt x 0 2 γ 0

2(λr2 + λi2 ) 󵄨󵄨 󵄨2 − 2φρ Re(g(u), uxx ) + 󵄨󵄨(ωx − ω)x 󵄨󵄨󵄨0 γ L

̃|2 dx. + 9γ ∫ |u|4 |ω − ω

(2.3.21)

0

Taking the inner product of equation (2.3.18) and v − ux , and then taking its real part, we obtain 1 d 󵄨 󵄨2 |v − ux |20 = −γ 󵄨󵄨󵄨(v − ux )x 󵄨󵄨󵄨0 + Re(η2 − η1x , v − ux ) 2 dt + Re(ξ2 − ξ1x , v − ux ) − k1 |v − ux |20 . Let g1 (u) = (βr + iβi )(2|u|2 ux + u2 ux )

+ (δr + iδi )(3|u|4 ux + 2|u|2 u2 ux ) − χux .

It follows that Re(η2 − η1x , v − ux ) = φρ Re((g1 (u) − h(u, v), v − ux )) − (1 − φρ )(9γ + δr + 9 L

× Re ∫ (3|u|4 |v − ux |2 + 2|u|2 u2 (v̄ − ū x )2 )dx 0

λr2 + λi2 ) γ

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 269

≤ φρ Re((g1 (u) − h(u, v), v − ux )) L

λ2 + λi2 ) ∫ |u|4 |v − ux |2 dx, − (1 − φρ )(9γ + δr + 9 r γ 0

Re(ξ2 − ξ1x , v − ux ) L

̃)(v − ux ) + 2u(v + ux )(ω − ω ̃)(v − ux ) = γ[Re ∫(2u(v + ux )(ω − ω 0

̃)(v − ux ))dx] + 2u(v + ux )(ω − ω L

̃)x + u2 (ω − ω ̃)x )(v − ux )dx], + γ Re[∫(2|u|2 (ω − ω 0

̃)(v − u)x + 2u(v + ux )(ω − ω ̃)(v − ux )] Re[2u(v + ux )(ω − ω ̃)(|v|2 − |ux |2 − vux + vux ) = 4 Re(Re(u(ω − ω

̃))(|v|2 − |ux |2 ). = 4 Re(u(ω − ω Therefore, we have L

̃)(|v|2 − |ux |2 )dx Re(ξ2 − ξ1x , v − ux ) ≤ 4γ Re ∫ u(ω − ω 0

L

L

2

2

̃)(ν − ux ) dx + 3γ ∫ |u|2 (ω − ω ̃)x ‖v − ux ‖dx + 2γ Re ∫ u(ω − ω 0

0

L

γ󵄨 9 2 ̃)x 󵄨󵄨󵄨󵄨0 + γ ∫ |u|4 |v − ux |2 dx, ≤ I + 󵄨󵄨󵄨(ω − ω 2 2 0

where L

L

0

0

̃)(|v|2 − |ux |2 )dx + 2γ Re ∫ u(ω − ω ̃)(v2 − u2x )dx. I = 4γ Re ∫ u(ω − ω Thus, we arrive at d 2 󵄨 󵄨2 󵄨 ̃)x 󵄨󵄨󵄨󵄨0 |v − ux |20 ≤ −2γ 󵄨󵄨󵄨(v − ux )x 󵄨󵄨󵄨0 − 2k1 |v − ux |20 + 2I + γ 󵄨󵄨󵄨(ω − ω dt L

+ φρ (9γ ∫ |u|4 |v − ux |2 dx − (h(u, v) − g1 (u), v − ux )). 0

(2.3.22)

270 | 2 Inertial manifold ̃, and taking its real part, we obtain Taking inner product of equation (2.3.19) and ω − ω 1 d 2 ̃|20 ̃|20 = −γ 󵄨󵄨󵄨󵄨(ω − ω ̃)x 󵄨󵄨󵄨󵄨0 − k2 |ω − ω |ω − ω 2 dt L

̃))dx) − Re ( ∫(4γu(|v|2 − |ux |2 ) − 2γu(v2 − u2x )(ω − ω 0

L

̃))dx) − Re ( ∫(2|u|2 (λr + iλi )(2|u|2 v + u2 v − ωx )(ω − ω 0

L

̃)dx) − Re ( ∫ u2 (λr − iλi )(2|u|2 v + u2 v − ωx )(ω − ω 0

− (9γ +

16(λr2

+

λi2 )(1

L

1 ̃|2 dx. + )) ∫ |u|4 |ω − ω γ 0

2

̃x For the terms 2|u|2 v + u2 ν̄ − ωx and 2|u|2 v + u v − ωx , we handle them by inserting ω 2 ̃x (ω ̃ = f (u) = |u| u). We omit the specific steps. Finally, we arrive at and ω 1 d 2 2 ̃|20 ≤ −γ 󵄨󵄨󵄨󵄨(ω − ω ̃)x 󵄨󵄨󵄨󵄨0 − k2 󵄨󵄨󵄨󵄨(ω − ω ̃)󵄨󵄨󵄨󵄨0 − I |ω − ω 2 dt L

̃|dx + 12√λr2 + λi2 ∫ |u|4 |v − ux ||ω − ω 0

L

̃|󵄨󵄨󵄨󵄨(ω − ω ̃)x 󵄨󵄨󵄨󵄨dx + 4√λr2 + λi2 ∫ |u|2 |ω − ω 0

− (9γ +

16(λr2

+

λi2 )(1

L

1 ̃|2 dx + )) ∫ |u|4 |ω − ω γ 0

3 󵄨 2 ̃)x 󵄨󵄨󵄨󵄨0 − k2 |ω − ω ̃|20 − I ≤ − γ 󵄨󵄨󵄨(ω − ω 4 L

L

̃|2 dx. + 9 ∫ |u|4 |v − ux |2 dx − 9γ ∫ |u|4 |ω − ω 0

0

Adding equation (2.3.20) to (2.3.23), we get d ̃|20 ) (|u|20 + |ux |20 + |v − ux |20 + |ω − ω dt 2 2 γ λ + λi ̃|20 ≤ −|u|20 + P − α|ux |20 + ( + r − 2k2 )|ω − ω 2 α L

L

λ2 + λi2 γ + 18φρ γ ∫ |u| dx − |uxx |20 + 18φρ r ∫ |u|4 |ux |2 dx 2 γ 0

6

0

(2.3.23)

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 271

L

− 2φρ Re(g(u), uxx ) + 18γφρ ∫ |u|4 |v − ux |2 dx +(

λr2

+ γ

λi2

0

γ 󵄨 2 2 ̃)x 󵄨󵄨󵄨󵄨0 − 2k1 |v − ux |20 − 2γ 󵄨󵄨󵄨󵄨(v − ux )x 󵄨󵄨󵄨󵄨0 . − )󵄨󵄨󵄨(ω − ω 2

(2.3.24)

For a given ρ > 0, without loss of generality, suppose |u|20 + |ux |20 + |v|20 + |ω|20 ≥ 2ρ2 , otherwise Proposition 2.3.4 is proved. At this time, φρ = 0, where φρ = φ(

|u|20 +|ux |20 +|v|20 +|ω|20 ). ρ2

λr2 +λi2 , γ > 2√λr2 + λi2 , from equation (2.3.24) we deduce 2α 2 2 2 2 ̃|0 exponentially decays, and is uniformly bounded. that |u|0 + |ux |0 + |v − ux |0 + |ω − ω

Hence, when k1 > 0, k2 > γ + Also

̃|20 + |ω ̃|20 , |u|20 + |ux |20 + |v|20 + |ω|20 ≤ |u|20 + 2|ux |20 + |v − ux |20 + |ω − ω L

̃| = ∫ |u|6 dx. |ω 0

By using Sobolev embedding theorem, L

3

∫ |u|6 dx ≤ C(|ux |20 + |u|20 ) 0

and there exists ρ4 ≥ 2ρ such that lim {|u|20 + |ux |20 + |v|20 + |ω|20 } ≤ ρ24 .

t→∞

Thus Proposition 2.3.4 has been proved. As corollaries of Proposition 2.3.4, we get that the solution of equation (2.3.7) exponentially converges to the set J(V1 ). Corollary 2.3.1. Under the assumptions of Proposition 2.3.4, if U(t) is the solution of equation (2.3.7), then there exist T1 , K1 > 0 such that, when t ≥ T1 , we have d ̃|20 ) ≤ (K1 − 2k1 )|v − ux |20 − k2 |ω − ω ̃|20 . (|v − ux |20 + |ω − ω dt Proof. Adding equation (2.3.22) to (2.3.23), we get d 3 2 2 ̃|20 ) ≤ −2γ 󵄨󵄨󵄨󵄨(v − ux )x 󵄨󵄨󵄨󵄨0 − 2k1 |v − ux |20 − γ 󵄨󵄨󵄨󵄨(ω − ω ̃)x 󵄨󵄨󵄨󵄨0 (|v − ux |20 + |ω − ω dt 2 − 2k2 |ω −

̃|20 ω

L

+ 18γφρ ∫ |u|4 |v − ux |2 dx. 0

272 | 2 Inertial manifold By virtue of L

∫ |u|4 |v − ux |2 dx ≤ |u|4L∞ |v − ux |20 , 0

1 due to Proposition 2.3.4 and inclusion Hper (I) 󳨅→ L∞ (I), we know that there exists T1 > 0 such that

|u|L∞ ≤ C1 ,

t ≥ T1 .

Hence k1 = 18γC14 , and Corollary 2.3.1 has been proved. In the following, we will prove that the solution of equation (2.3.7) is uniformly 1 1 bounded in D(A 2 ) and has the global attractor in D(A 2 ). Proposition 2.3.5. When k1 >

K1 , k2 2 1 2

>γ+

λr2 +λi2 , 2α

γ > 2√λr2 + λi2 , the solution of equation 1

(2.3.7) is uniformly bounded in D(A ) and has the global attractor A in D(A 2 ).

Proof. Taking the inner product of equation (2.3.7) and v, u, ω, respectively, then taking its real parts, and using Hölder inequality and Proposition 2.3.4, we obtain 1 d 2 |u| ≤ −γ|ux |20 + √λr2 + λi2 |ωx |0 |u|0 + C1 , 2 dt 0 1 d 2 |v| ≤ −γ|vx |20 + √λr2 + λi2 |ωx |0 |v|0 + C2 , 2 dt 0 1 d |ω|2 ≤ −γ|ωx |20 + C3 (|v|2L4 + 1), 2 dt 0 where Ci , i = 1, 2, 3, only depend on the coefficients of the equation and the uniform bound of Proposition 2.3.4. By Sobolev embedding theorem and Young inequality, we have |v|2L4 ≤ C|v|0 |v|1 ≤ C1 (|v|20 + |v|0 |vx |0 ) ≤ C4 (1 + |vx |0 ),

(2.3.25)

which yields λ2 + λi2 γ 1 d (|u|20 + |ν|20 + |ω|20 ) ≤ −(γ|ux |20 + |vx |20 + (γ − r )|ωx |20 ) + C5 . 2 dt 4 γ

(2.3.26)

Taking the inner product of the three equation of equations (2.3.7) and −uxx , −vxx , −ωxx , respectively, then taking its real parts, and using Hölder inequality, we obtain 1 d |u |2 ≤ −γ|uxx |20 + √λr2 + λi2 |ωx |0 |uxx |0 + C6 |uxx |0 , 2 dt x 0

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 273

1 d |v |2 ≤ −γ|vxx |20 + √λr2 + λi2 |ωxx |0 |vxx |0 + C7 |vxx |0 , 2 dt x 0 1 d |ω |2 ≤ −γ|ωxx |20 + C8 (|v|2L4 + 1)|ωxx |0 . 2 dt x 0 Using Young inequality and equation (2.3.25), we get 1 d (|u |2 + |vx |20 + |ωx |20 ) 2 dt x 0 λ2 + λi2 λ2 + λi2 1 )|ωxx |20 + r |ωx |20 + C9 |vx |20 + C10 . ≤ − (γ − r 2 γ 2γ

(2.3.27)

From equations (2.3.26)–(2.3.27), assumption γ > 2√λr2 + λi2 and uniform Gronwall in1

equality, we get that U is uniformly bounded in D(A 2 ). As for the existence of the global attractor, which can be established as in [197, Th. 1.1] with operator A being sectoral 1 operator (it generates the semigroup which is compact in D(A 2 ) when t > 0). Similarly as C1 , C2 , C3 , the above Ci , i = 4, . . . , 10, only depend on the coefficients of the equation and the uniform bound of Proposition 2.3.4. Using Corollary 2.3.1, we can prove the following lemma: Lemma 2.3.2. Assume that k1 > into the set J(V1 ).

K1 , 2

k2 > γ +

λr2 +λi2 , 2α

γ > 2√λr2 + λi2 . Then A is included

Proof. First we note that there exists a constant ρ5 such that 󵄨 󵄨2 |v − ux |20 + 󵄨󵄨󵄨ω − f (u)󵄨󵄨󵄨0 ≤ ρ25 ,

∀(u, v, ω) ∈ A .

Now set U = (U0 , t) = (u(t), v(t), ω(t))t to be the solution of equation (2.3.7) with the initial value U0 ∈ A . By Corollary 2.3.1, we get 2 󵄨󵄨 󵄨2 󵄨 ̃(t)󵄨󵄨󵄨󵄨0 󵄨󵄨v(t) − ux (t)󵄨󵄨󵄨0 + 󵄨󵄨󵄨ω(t) − ω 2 󵄨 󵄨2 󵄨 ̃(T1 )󵄨󵄨󵄨󵄨0 ) ≤ e−μt (󵄨󵄨󵄨v(T1 ) − ux (T1 )󵄨󵄨󵄨0 + 󵄨󵄨󵄨ω(T1 ) − ω

≤ ρ25 e−μt

(due to the invariance of A ).

(2.3.28)

For t ≥ T1 , here μ = min(2k1 − K1 , 2k2 ). Furthermore, by the invariance of A , for any U ∗ ∈ A and t ≥ T1 , there exists U0 such that U(U0 , t) = U ∗ . Hence by equation (2.3.28), we obtain dist(U ∗ , J(V1 )) ≤ ρ25 e−μt ,

t ≥ T1 ,

dist(A , J(V1 )) = sup dist(U, J(V1 )) ≤ ρ25 e−μt , U∈A

dist(A , J(V1 )) = 0.

t ≥ T1 ,

274 | 2 Inertial manifold Next we only need to prove that J(V1 ) is closed in H , which can be deduced from A ⊂ J(V1 ). In fact, if (un , (un )x , f (un )) ∈ H converges to (u, v, ω) ∈ H , then ν is the

weak derivative of u, v ∈ H, and it follows that

󵄨 󵄨 󵄨 󵄨 |v|0 ≤ 󵄨󵄨󵄨v − (un )x 󵄨󵄨󵄨0 + 󵄨󵄨󵄨(un )x 󵄨󵄨󵄨0 < ∞, and 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨ω − f (u)󵄨󵄨󵄨0 ≤ 󵄨󵄨󵄨ω − f (un )󵄨󵄨󵄨0 + 󵄨󵄨󵄨f (un ) − f (u)󵄨󵄨󵄨0 , L

󵄨󵄨 󵄨 6 6 2 󵄨󵄨f (un ) − f (u)󵄨󵄨󵄨0 = (∫(|u| − |u| ) dx)

1 2

0

≤ (|u|2L∞ + |un |L∞ |u|L∞ + |un |2L∞ ) × (|un |3L∞ + |u|3L∞ )|un − u0 |0 .

Since |un |L∞ is uniformly bounded, then so is |u|L∞ . Hence ω = f (u). The lemma has been proved. In the following, we give the main result: Theorem 2.3.2. Assume that k1 >

K1 , k2 2

>γ+

λr2 +λi2 , 2α

γ > 2√λr2 + λi2 . Then A = J(Aρ ).

Proof. It is obvious that J(Aρ ) ⊂ A , and A is the invariant set in J(V1 ), but since Aρ is the global attractor of equation (2.3.5), J(Aρ ) is the maximal invariant set in J(V1 ). Hence, A ⊂ J(Aρ ), and it follows that A = J(Aρ ). Next, we consider the existence of inertial manifold and inertial form. According to the previous discussion, we know that equation (2.3.7) keeps the long time dynamic behavior of equation (2.3.2). In particular, by virtue of Propositions 2.3.4 and 2.3.5, there exists a constant ρ6 > 0 such that 󵄨󵄨 21 󵄨󵄨 󵄨󵄨A U 󵄨󵄨0 ≤ ρ6 ,

∀U ∈ A .

In order to prove the existence of an inertial manifold for equation (2.3.7), we truncate ̃ That is, considering the nonlinear term F. dU + AU = F(U) dt 1

(2.3.29)

2

|A 2 U| ̃ we can see that equations (2.3.29) and (2.3.7) for the where F(U) = φ( ρ 0 )F(U), 6 global attractor A have the same long time behavior. For simplicity, here we just illustrate the conclusion.

Proposition 2.3.6. Under the above assumptions, equation (2.3.29) has the inertial 1 manifold μ = graph Φ, where Φ : P H → QH ∩ D(A 2 ) is a Lipschitz mapping; P is the

2.3 The finite-dimensional inertial form for the 1-dimensional generalized GL equation

| 275

projection onto the subspace spanned by the first N +1 eigenvectors of A in H ; Q = I −P; and A is the differential operator having the following form: A A = (−k1 𝜕x 0

0 A 0

(λr + iλi )𝜕x −(λr + iλi )𝜕xx ) . A

Similar to the discussion of Lemma 2.3.1, A has discrete spectrum. By the classical functional analysis, P H and QH are invariant under the action of A. Because A is not a self-adjoint operator, we can prove Proposition 2.3.6 using the results of [191]. At this time, the spectrum of A is {( 2πn )2 }n≥1 , so the gap condition is satisfied. L Corollary 2.3.2. The essential long time behavior dynamics for the semigroup of equation (2.3.2) can be described by the following ordinary differential equation: d PU = −APU + PF(PU + Φ(PU)). dt

(2.3.30)

From the exponential attractivity property of inertial manifold, we have Corollary 2.3.3. If u(t) is a solution of equation (2.3.2), then there exists a solution PU(t) of ordinary differential equation (2.3.30) which satisfies (C, α are positive constants) 󵄨󵄨 󵄨 −αt 󵄨󵄨u(t) − (P1 (t) + Φ1 (PU))󵄨󵄨󵄨 ≤ Ce ,

∀t > 0,

where PU(t) = (P1 (t), P2 (t), P3 (t)),

Φ = (Φ1 , Φ2 , Φ3 ).

In the following, we give the corresponding conclusion of Kwak. Define S = U ∩ J(V1 ). We denote Sμ as the first component of S. Proposition 2.3.7. The set Su is invariant under the action of solution semigroup of equation (2.3.2) and it attracts all the orbits of equation (2.3.2). We cannot prove that Su is a finite-dimensional manifold, but can prove U u (the first component of U ) is a finite-dimensional manifold which attracts all bounded set 1 in D(A 2 ), and it has no invariance under the action of the semigroup. Finally, we provide a lower dimensional ordinary differential equation than equation (2.3.30) to describe the dynamics of Su . Theorem 2.3.3. The essential long time dynamics of the solution semigroup of equation (2.3.2) can be completely described by the following ordinary differential equation: d PU + APU = PF(J(u)) dt where PU = (P1 , P1x , P3 ), u(t) = P1 (t) − Φ1 (P1 (t), P1x (t), P3 (t)).

276 | 2 Inertial manifold

2.4 The existence of inertial manifolds for the generalized KS equation Consider the existence of inertial manifold and its dimension estimation of the periodic initial value problem for the following generalized Kuramoto–Sivashinsky equation (GKS) [59]: ut + αuxx + γuxxx + f (u)x + φ(u)xx = g(u) + h(x), u|t=0 = u0 (x),

(x, t) ∈ R × R+ ,

x ∈ R,

u(x + D, t) = u(x − D, t),

(2.4.1) (2.4.2)

∀x ∈ R, t ∈ R+ , D > 0, (2.4.3)

where α ≥ 0, γ > 0. Obviously, when α = 0, f (u) = 0, equation (2.4.1) is the well-known generalized Cahn–Hilliard equation. For the problem (2.4.1)–(2.4.3) we will give a time t a priori estimate and will then prove the invariant cone property (ICP) and strong squeezing property (SSP) of the GKS equation, giving the existence of inertial manifolds for the prepared GKS equation (PGKS). Finally, we will prove the existence of an inertial manifold for the periodic initial value problem of GKS equation. Lemma 2.4.1. If the following conditions are satisfied: (1) φ󸀠 (u) ≤ φ0 , φ0 > 0; α+φ +1 (2) g(0) = 0, g 󸀠 (u) ≤ g0 , g0 < − 20 , γ > 21 (α + φ0 ); (3) h(x) ∈ L2 (Ω), Ω = (−D, D); (4) u0 (x) ∈ L2 (Ω), then for any smooth solution of problem (2.4.1)–(2.4.3), we have the following estimate: 2

α+φ +1 α+φ +1 ‖h(x)‖L2 (Ω) 󵄩󵄩 󵄩2 2(g0 + 20 )t 2(g + 0 )t 󵄩 󵄩󵄩u0 (x))󵄩󵄩󵄩2 ) 󵄩󵄩u(⋅, t)󵄩󵄩󵄩L2 (Ω) ≤ e 0 2 󵄩 󵄩L2 (Ω) + |2g + α + φ + 1| (1 − e 0 0

0 ≤ t < ∞.

Furthermore, we have 1

1 󵄩 󵄩 󵄩2 󵄩2 lim 󵄩󵄩u(⋅, t)󵄩󵄩󵄩L (Ω) + lim ∫ 󵄩󵄩󵄩uxx (⋅, s)󵄩󵄩󵄩L (Ω) ds ≤ E0 , 2 2 t→∞ 󵄩 t→∞ t

(2.4.4)

0

where the constant E0 depends on ‖u0 (x)‖2L2 (Ω) and ‖h0 (x)‖L2 (Ω) . Proof. Taking the inner product of equation (2.4.1) and u, we obtain (u, ut + αux + γuxxxx + f (u)x + φ(u)xx − g(u) − h(x)) = 0, where (f (u)x , u) = −(f (u), ux ) = 0,

(2.4.5)

2.4 The existence of inertial manifolds for the generalized KS equation

|

277

(φ(u)xx , u) = −(φ󸀠 (u)ux , ux ) ≥ −φ0 ‖ux ‖2L2 , (u, g(u)) ≤ g0 ‖u‖2L2 ,

1 (u, h(x)) ≤ (‖u‖2L2 + ‖h‖2L2 ). 2

For the periodic function u(⋅, t), we have

1 ‖ux ‖2L2 ≤ ‖u‖L2 ‖uxx ‖L2 ≤ (‖u‖2L2 + ‖uxx ‖2L2 ). 2 Therefore, from (2.4.5) we have α + φ0 α + φ0 + 1 1󵄩 1 d 󵄩2 ‖u‖2L2 + (γ − )‖uxx ‖2L2 ≤ (g0 + )‖u‖2L2 + 󵄩󵄩󵄩h(x)󵄩󵄩󵄩L . 2 2 dt 2 2 2 Since γ −

α+φ0 2

(2.4.6)

> 0, using α + φ0 + 1 󵄩󵄩 d 󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩2 )󵄩󵄩u(⋅, t)󵄩󵄩󵄩L + 󵄩󵄩󵄩h(x)󵄩󵄩󵄩L , 󵄩󵄩u(⋅, t)󵄩󵄩󵄩L2 ≤ 2(g0 + 2 2 dt 2

it is then easy to get 2

‖h(x)‖L2 (Ω) 󵄩󵄩 󵄩2 (2g0 +α+φ0 +1)t (2g +α+φ0 +1)t 󵄩 󵄩󵄩u0 (x)󵄩󵄩󵄩2 + ) 󵄩󵄩u(⋅, t)󵄩󵄩󵄩L2 ≤ e 0 󵄩 󵄩L2 |2g + α + φ + 1| (1 − e 0 0 Integrating equation (2.4.6) with respect to time, we get the estimation t

1 󵄩󵄩 󵄩2 ∫ 󵄩󵄩uxx (⋅, s)󵄩󵄩󵄩L ds ≤ E0 . 2 t→∞ t lim

0

Lemma 2.4.2. Suppose the conditions of Lemma 2.4.1 are met, and let (1) |f (u)| ≤ A|u|p , 1 ≤ p < 7, A > 0, 󵄨󵄨 󸀠 󵄨󵄨 q 󵄨󵄨φ (u)󵄨󵄨 ≤ B|u| ,

0 ≤ q < 4, B > 0;

(2) u0 (x) ∈ H 1 (Ω), h(x) ∈ L2 (Ω). Then we have the following estimate of a solution for problem (2.4.1)–(2.4.3): 󵄩󵄩 󵄩2 󵄩2 g t󵄩 󵄩󵄩ux (⋅, t)󵄩󵄩󵄩L2 (Ω) ≤ e 0 󵄩󵄩󵄩u0x (x)󵄩󵄩󵄩L2 (Ω) 2 󵄨 max 󵄨󵄨C (D, p, + |g0 | t∈[0,∞) 󵄨 5

󵄩󵄩 󵄩 󵄩 󵄩 󵄨 󵄩󵄩u(⋅, t)󵄩󵄩󵄩L2 , 󵄩󵄩󵄩h(x)󵄩󵄩󵄩L2 )󵄨󵄨󵄨,

∀t ≥ 0,

(2.4.7)

where the function C5 (⋅, ⋅, ξ , η) is continuous and increasing in the variables ξ , η. Furthermore, we have t

1 󵄩 󵄩 󵄩2 󵄩2 lim 󵄩󵄩ux (⋅, t)󵄩󵄩󵄩L (Ω) + lim ∫ 󵄩󵄩󵄩uxxx (⋅, s)󵄩󵄩󵄩L ds ≤ E1 , 2 2 t→∞ 󵄩 t→∞ t 0

where the constant E1 depends on ‖u0 (x)‖H 1 (Ω) , ‖h(x)‖L2 (Ω) and E0 .

(2.4.8)

278 | 2 Inertial manifold Proof. Taking the inner product of equation (2.4.1) and uxx , this gives (uxx , ut + αuxx + γuxxxx + f (u)x + φ(u)xx − g(u) − h(u)) = 0,

(2.4.9)

where 󵄩 󵄨 󵄩 󵄨 󵄨 󵄨󵄨 󵄨󵄨(f (u)x , uxx )󵄨󵄨󵄨 = 󵄨󵄨󵄨(f (u), uxxx )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩f (u)󵄩󵄩󵄩L2 ‖uxxx ‖L2 γ 3󵄩 󵄩2 ≤ ‖uxxx ‖2L2 + 󵄩󵄩󵄩f (u)󵄩󵄩󵄩L . 2 6 2γ

(2.4.10)

Using the assumptions of the lemma and Sobolev estimate for a smooth periodic function [88], we have 5 1 1 󵄩󵄩 󵄩 ‖u‖L2 + √2‖u‖L6 ‖uxx ‖L6 , 󵄩󵄩u(x)󵄩󵄩󵄩L∞ (Ω) ≤ 2 2 √D 1 󵄩󵄩 󵄩󵄩 󵄩󵄩u(x)󵄩󵄩L∞ (Ω) ≤ (2D) 2 ‖uxx ‖L2 , 1 󵄩󵄩 j+1 󵄩󵄩 21 󵄩󵄩󵄩Dj u󵄩󵄩󵄩 󵄩󵄩󵄩Dj−1 u󵄩󵄩󵄩 2 ≤ x x 󵄩 󵄩L∞ (Ω) 󵄩 󵄩L∞ (Ω) 󵄩󵄩Dx 󵄩󵄩L∞ (Ω) , j = 1, 2, . . . .

Then we use the generalized Young inequality ab ≤ εp

ap bq + ε−q , p q

1 1 + = 1, p q

p, q > 1, ε, a, b > 0,

which yields 3 3 󵄨󵄨 3 󵄨2 2p 2p−2 2 󵄨f (u)󵄨󵄨󵄨L2 ≤ A2 ‖u‖L2p ≤ A2 ‖u‖L2 ‖u‖L2 2γ 󵄨 2γ 2γ 2(5p+1) γ ≤ ‖uxxx ‖2L2 + c1 ‖u‖2p + c2 ‖u‖L 7−p , L 2 2 6

(2.4.11)

where c1 =

3A2 (p − 1) , γDp−1

5+p

(7p−13)

p−1

12

− 5+p

c2 = (7 − p) ⋅ 2 7−p (p − 1) 7−p ⋅ 3 7−p ⋅ A 7−p ⋅ γ 7−p 3󵄩 󸀠 󵄨󵄨 󵄨 󵄨 󸀠 󵄩2 󵄨 γ 2 󵄨󵄨(φ(u)xx , uxx 󵄨󵄨󵄨 = 󵄨󵄨󵄨(φ (u)ux , uxxx )󵄨󵄨󵄨 ≤ ‖uxxx ‖ + 󵄩󵄩󵄩φ (u)ux 󵄩󵄩󵄩L2 , 6 2γ

3 󵄩󵄩 󸀠 3 󸀠 󵄩2 2 2 󵄩φ (u)ux 󵄩󵄩󵄩L2 ≤ ‖φ (u)‖L∞ ⋅ ‖ux ‖L2 2γ 󵄩 2γ 2(5p+4) γ 4−q + c ‖u‖ , ≤ ‖uxxx ‖2L2 + c3 ‖u‖3p−2 4 L2 L2 6 where 3

4

c3 = 4√3γ −2 q 2 B3 D− 3 q , 7p+2

6

c4 = 2 4−q 3 4−q γ

− q+8 4−q

12

6

2+q

B 7−p q 4−q (q + 2) 4−q ,

(2.4.12)

2.4 The existence of inertial manifolds for the generalized KS equation

|

−(g(u), uxx ) = (g 󸀠 (u)ux , ux ) ≤ g0 ‖ux ‖2L2 , 󵄩 󵄨 󵄩 󵄨󵄨 󵄨󵄨(h(x), uxx )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩h(x)󵄩󵄩󵄩L2 ‖uxx ‖L2 2 1 󵄩 󵄩 ≤ 󵄩󵄩󵄩h(x)󵄩󵄩󵄩L ‖u‖L2 ‖uxxx ‖L3 2 2 2 − q2

γ 2 γ ≤ ‖uxxx ‖2L2 + ( ) 6 3 2

279

(2.4.13)

1 󵄩3 󵄩 ‖u‖L2 󵄩󵄩󵄩h(x)󵄩󵄩󵄩L2 . 2 2

(2.4.14)

Using Young inequality, 2

4

α‖uxx ‖2L2 ≤ α‖uxx ‖L3 ‖uxxx ‖L3 2

≤

2

3

γ α γ ‖u ‖2 + ( ) ‖u‖2L2 , 6 xxx L2 3 4 −2

(2.4.15)

and then from equations (2.4.9)–(2.4.15) we deduce γ 1 d ‖u (⋅, t)‖2L2 + ‖uxxx ‖2L2 ≤ g0 ‖ux ‖2L2 + 2c5 (p, D, q, ‖u‖L2 , ‖h‖L2 ), 2 dt x 6

(2.4.16)

where c5 (p, q, D, ‖u‖L2 , ‖h‖L2 )

2(5p+4)

2(5p+1)

7−p + c3 ‖u‖3q+2 = c1 ‖u‖2p + c4 ‖u‖L 4−p L L + c2 ‖u‖L 2

2 γ + ( ) 3 2

− q2

2

1 2

‖u‖L ‖h‖L + 2

2

2

3 2

2

3

α γ ( ) ‖u‖2L2 . 3 4 −2

From equation (2.4.16) we get 2 󵄨 󵄨 󵄩󵄩 󵄩2 󵄩2 2g t 󵄩 max 󵄨󵄨C (p, D, q, ‖u‖L2 , ‖h‖L2 )󵄨󵄨󵄨. 󵄩󵄩ux (⋅, t)󵄩󵄩󵄩L2 (Ω) ≤ e 0 󵄩󵄩󵄩u0x (x)󵄩󵄩󵄩L2 (Ω) + |g0 | t∈[0,∞) 󵄨 5 By Lemma 2.4.1 and equation (2.4.16), we get 1

1 󵄩 󵄩2 󵄩 󵄩2 lim 󵄩󵄩ux (⋅, t)󵄩󵄩󵄩L (Ω) + lim ∫ 󵄩󵄩󵄩uxxx (⋅, s)󵄩󵄩󵄩L (Ω) ds ≤ E1 . 2 2 t→∞ 󵄩 t→∞ t

(2.4.17)

0

Lemma 2.4.3. Assume that the conditions of Lemma 2.4.2 are met, and set u0 (x) ∈ H 2 (Ω). Then for a smooth solution of problem (2.4.1)–(2.4.3), we have 3 1 󵄩󵄩 󵄩2 󵄩2 2g t 󵄩 (1 − e2g0 t )( ‖h‖2L2 (Ω) + C6 ), 󵄩󵄩uxx (⋅, t)󵄩󵄩󵄩L2 (Ω) ≤ e 0 󵄩󵄩󵄩u0xx (x)󵄩󵄩󵄩L2 (Ω) + |g0 | γ

(2.4.18)

where the continuous function C6 depends on ‖u(⋅, t)‖H 1 (Ω) . Further, we have t

1 󵄩 󵄩 󵄩2 󵄩2 lim 󵄩󵄩uxx (⋅, t)󵄩󵄩󵄩L (Ω) + lim ∫ 󵄩󵄩󵄩uxxxx (⋅, s)󵄩󵄩󵄩L (Ω) ds ≤ E2 , 2 2 t→∞ 󵄩 t→∞ t 0

where the constant E2 depends on ‖u0xx (x)‖L2 , ‖h(x)‖L2 and E1 .

(2.4.19)

280 | 2 Inertial manifold Proof. Similar to the proof of Lemmas 2.4.1 and 2.4.2. Lemma 2.4.4. Suppose the conditions of Lemma 2.4.3 are met, and set u0 (x) ∈ H 3 (Ω). Then for a smooth solution of problem (2.4.1)–(2.4.3), we have 1 󵄩 󵄩2 󵄩󵄩 󵄨 󵄨 2g t 󵄩 max 󵄨󵄨C (‖u‖H 2 (Ω) , ‖h‖H 1 (Ω) )󵄨󵄨󵄨. 󵄩󵄩uxxx (⋅, t)󵄩󵄩󵄩L2 (Ω) ≤ e 0 󵄩󵄩󵄩u0 (x)󵄩󵄩󵄩H 3 (Ω) + |g0 | 0≤t 0,

(2.4.25)

2.4 The existence of inertial manifolds for the generalized KS equation

|

281

Definition 2.4.2 (Squeezing property). If there exists a constant β > 0 such that 󵄩 󵄩󵄩 −βt 󵄩󵄩(S(t)u − S(t)v󵄩󵄩󵄩H ≤ e ‖u − v‖H ,

∀t ≥ 0,

(2.4.26)

where (u, v) ∈ B×B, (S(t)u, S(t)v) ∈ ̸ CN (γ), then we call it the squeezing property, which means B × B \ CN (γ) is exponentially squeezed. Definition 2.4.3 (Strong squeezing property). If ICP and the squeezing property hold simultaneously, we call it the strong squeezing property, SSP for short. For the GKS equation (2.4.1), setting the operator A = DA = {u ∈ V :

d4 u dx4

be defined on subspace

d3 u d4 u ∈ L2 (−D, D), ∈ V}, 4 dx dx3

A is a self-adjoint operator in L2 (Ω), which satisfies AWk = λk Wk , k = 1, 2, . . . , where λk = ( 2πk )4 . It is easy to verify that D V =D

1

A4

1

A2 u =

󵄩 1 󵄩 󵄩 1 󵄩 ‖u‖1 = 󵄩󵄩󵄩A 4 u󵄩󵄩󵄩L = 󵄩󵄩󵄩A 4 u󵄩󵄩󵄩,

,

2

2

du , dx 2

u∈D

1

A2

,

u ∈ V,

󵄩 3 󵄩 ‖u‖3 = 󵄩󵄩󵄩A 4 u󵄩󵄩󵄩,

u∈D

3

A4

.

Then GKS equation (2.4.1) can be rewritten in the operator form: 1 du + γAu − αA 2 u + f (u)x + φ(u)xx = g(u) + h(x). dt

(2.4.27)

Lemma 2.4.5. Let u01 , u02 ∈ B and assume u0i (x), i = 1, 2, and h(x) are odd periodic functions, u1 (t) = S(t)u01 , u2 (t) = S(t)u02 . Then we have the following estimate 󵄩󵄩 󵄩 󵄩 0 0 󵄩 2Kt 󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩u1 − u2 󵄩󵄩󵄩e ,

∀t ≥ 0,

(2.4.28)

where S(t)u0i means the semigroup generated by equation (2.4.27) with the initial value u0i (x), i = 1, 2, B = ⋃ S(t)B2E , t≥0

B2E = {u ∈ D

3

A4

, ‖u‖3 ≤ 2(E0 + E1 + E2 + E3 )},

(2.4.29)

and the constants are determined by equations (2.4.4), (2.4.8), (2.4.19), and (2.4.21), as well as

282 | 2 Inertial manifold 1 3 3α2 2γ { { K = sup (‖R2x ‖L∞ (Ω) + 2‖R1 ‖L∞(Ω) + ‖R3 ‖2L∞(Ω) ) + − λ1 , { { 2 0≤t 0, ∀t ∈ D. Hence 󵄩󵄩 󵄩 󵄩󵄩p(t) + φ(p(t))󵄩󵄩󵄩1 > 2E, and then we have F(p(t) + φ(p(t))) = 0,

‖p + φ(p)‖1 ) = 0. θ( E

Φ(p(t) + Φ(p(t))) = 0,

Equation (2.4.59) boils down to 1

1

A 2 p(τ) = e−γAPN τ A 4 p0 . The above equation is well-defined at least for small |τ|, for τ < 0 we have 󵄩 󵄩󵄩 󵄩 󵄩 1 󵄩 γλ |τ| 󵄩 1 γλ |τ| 󵄩󵄩p(τ)󵄩󵄩󵄩1 = 󵄩󵄩󵄩A 4 p(τ)󵄩󵄩󵄩 ≥ e 1 󵄩󵄩󵄩A 4 p0 󵄩󵄩󵄩 = e 1 ‖p0 ‖1 . Thus

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩p(τ) + φ(p(τ))󵄩󵄩󵄩1 ≥ 󵄩󵄩󵄩p(τ)󵄩󵄩󵄩1 > 4E,

which yields that F0 φ(p0 ) = 0.

The lemma has been proved.

∀τ ≤ 0,

2.4 The existence of inertial manifolds for the generalized KS equation

|

293

Lemma 2.4.12 ([69]). For α > 0 and τ < 0, the operator (AQ)α eτAQ is linear and continuous in QH. Furthermore, its norm ‖(AQ)α eτAQ ‖0p is bounded in L (QH) by k2 (α)|τ|−α ,

−1 when −αλN+1 ≤ τ < 0,

α λN+1 eτλN+1 ,

If α < 1, then

when −∞ < τ ≤

(2.4.69)

−1 −αλN+1 .

(2.4.70)

󵄩 󵄩 α−1 , ∫ 󵄩󵄩󵄩(AQ)α eτAQ 󵄩󵄩󵄩L (QH) dτ ≤ k3 (α)λN+1

(2.4.71)

0

−∞

where λN+1 is the smallest eigenvalue of A|QH , k2 (α), k3 (α) are all some known constants, depending on α. 1

4 , b, l ∈ (0, 1]. Then for N + 1 ≥ N0󸀠 (E), we have Lemma 2.4.13. Let φ ∈ Fb,l

󵄩󵄩 󵄩 󵄩 1 󵄩 󵄩󵄩F0 (φ)(p)󵄩󵄩󵄩1 = 󵄩󵄩󵄩A 4 F0 φ(p)󵄩󵄩󵄩 ≤ b.

(2.4.72)

Proof. From the expression (2.4.64) of F0 φ(p), we get 0

1 1 󵄩󵄩 41 󵄩 󵄩 τγAQN 󵄩 󵄩󵄩󵄩󵄩󵄩A 4 φ(p)󵄩󵄩󵄩dτ 󵄩󵄩A F0 φ(p)󵄩󵄩󵄩 ≤ α ∫ 󵄩󵄩󵄩(AQN ) 2 e 󵄩󵄩 󵄩

−∞ 0

1 󵄩 󵄩󵄩 󵄩 + ∫ 󵄩󵄩󵄩(AQN ) 4 eτγAQN 󵄩󵄩󵄩󵄩󵄩󵄩QN B1 (u, u)󵄩󵄩󵄩dτ

−∞ 0

1 󵄩󵄩 1 󵄩 󵄩 + ∫ 󵄩󵄩󵄩(AQN ) 2 eτγAQN 󵄩󵄩󵄩󵄩󵄩󵄩A 4 Φ(u)󵄩󵄩󵄩dτ,

−∞

where

‖u‖1 )(−g(u) − h(x)) E 󵄩󵄩 󵄩 󵄩 󸀠 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩QN B1 (u, v)󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩f (u)󵄩󵄩󵄩L∞ ‖u‖1 + 󵄩󵄩󵄩g(u)󵄩󵄩󵄩 + 󵄩󵄩󵄩h(x)󵄩󵄩󵄩2 ≤ C1 (E) 󵄩 󵄩 󸀠 󵄩 󵄩󵄩 41 󵄩󵄩A Φ(u)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩φ (u)󵄩󵄩󵄩L∞ ‖u‖1 ≤ C2 (E). B1 (u, u) = f (u)x + θ(

Then by Lemma 2.4.12 and equation (2.4.73) we get 󵄩 󵄩 1 󵄩 󵄩󵄩 󵄩󵄩F0 φ(p)󵄩󵄩󵄩1 = 󵄩󵄩󵄩A 4 F0 φ(p)󵄩󵄩󵄩 1 21 1 − 43 1 − 21 + C1 (E)k3 ( )λN+1 + C2 (E)k3 ( )λN+1 ≤ abk3 ( )λN+1 2 4 2 3 1 1 −2 1 −4 = k3 ( )λN+1 (ab + C2 (E)) + C1 (E)k3 ( )λN+1 2 4 −1

−3

4 2 ≤ C3 (E)[λN+1 + λN+1 ],

(2.4.73)

294 | 2 Inertial manifold where λN+1 =

(N+1)4 ̃4 . D

̃0 (E), we have Hence for N + 1 ≥ N 󵄩󵄩 󵄩 󵄩󵄩F0 φ(p)󵄩󵄩󵄩1 ≤ b.

Suppose there are two groups of differential inequality: dy + ay + bz ≥ 0, dt dz − cy + dz ≤ 0, dt

(2.4.74) t ∈ I ⊂ R.

(2.4.75)

Consider the set Cγ given by Cγ = {(y, e) ∈ R+ × R+ | e ≥ γy },

γ > 0.

(2.4.76)

Lemma 2.4.14. Let a, b, c, d, γ > 0 and d − γ −1 c > 0,

d − a − γb − γ −1 c > 0.

(2.4.77)

Then for the orbit (y(t), z(t)) of (2.4.74)–(2.4.75), when (y(t0 ), z(t0 )) ∈ Cγ we have z(t) ≤ z(t0 ) exp{−(d − γ −1 C)(t − t0 )},

∀t ≥ t0 , (y(t), z(t)) ∈ Cγ .

(2.4.78)

Theorem 2.4.2. For any b, l ∈ (0, 1], there exists N1 = N1 (γ, ̃ D) such that when N + 1 > N1 we have 1

1

4 4 F0 Fb,l ⊂ Fb,l .

(2.4.79)

Proof. By Lemmas 2.4.11 and 2.4.14, we only need to prove 󵄩󵄩 41 󵄩 󵄩 1 󵄩 󵄩󵄩A (F0 φ(p01 ) − F0 φ(p02 ))󵄩󵄩󵄩 ≤ l󵄩󵄩󵄩A 4 (p01 − p02 )󵄩󵄩󵄩, 1

l > 0, ∀p01 , p02 ∈ PD(A 4 ).

(2.4.80)

Now we consider the functional operator F0 φ with two different initial values p01 , p02 . Let Uj = pj + φ(pj ), j = 1, 2, where pj = pj (τ; p0j ; φ) is the solution of the equation ṗ j + γApj + θ(

‖Uj ‖1 E

1

)(1 − αA 2 pj ) + pN B(Uj , Uj ) = 0,

(2.4.81)

which has the initial value pj (0; p0j , φ) = p0j .

(2.4.82)

F0 φ(p0j ) = qj (0),

(2.4.83)

Then

2.4 The existence of inertial manifolds for the generalized KS equation

|

295

where qj (τ) is the solution of equation q̇ j + γAQN qj + θ(

‖Uj ‖1 E

1

)(−αA 2 QN Uj ) + QN B(Uj , Uj ) = 0.

(2.4.84)

Let δ(τ) = p1 (τ; p01 ; φ) − p2 (τ; p02 ; φ), { Δ(τ) = q1 (τ) − q2 (τ).

(2.4.85)

Then δ(τ), Δ(τ) satisfy the equations δ̇ + γAδ + PN D(U1 , U2 ) = 0, Δ̇ + γAΔ + QN D(U1 , U2 ) = 0,

δ(0) = p01 − p02 ,

(2.4.86) (2.4.87)

where 1 ‖U1 ‖1 )(−αA 2 U1 ) + B(U1 , U1 ) E 1 ‖U ‖ − θ( 2 1 )(−αA 2 U2 ) − B(U2 , U2 ). E ‖U ‖ B(U, U) = F(U)x + Φ(U)xx + θ( 1 1 )(−g(U) − h(x)) E = B1 (U, U) + Φ(U)xx .

D(U1 , U2 ) = θ(

(2.4.88)

(2.4.89)

For simplicity, introduce the shorthand θ1 = θ(

‖U1 ‖1 ), E

θ2 = θ(

‖U2 ‖1 ). E

Then D(U1 , U2 ) can be written as 1 1 1 1 D(U1 , U2 ) = (θ1 + θ2 )(−αA 2 (U1 − U2 )) + (θ1 − θ2 )(−αA 2 (U1 + U2 )) 2 2 + (F 󸀠 (U1 ) − F 󸀠 (U2 ))U1x + F 󸀠 (U2 )(U1x − U2x ) 1 + (Φ(U1 ) − Φ(U2 ))xx − (θ1 + θ2 )(g(U1 ) − g(U2 )) 2 1 − (θ1 − θ2 )(g(U1 ) + g(U2 )) 2 = D1 (U1 , U2 ) + (Φ(U1 ) − Φ(U2 ))xx .

(2.4.90)

1

Taking the inner product of equation (2.4.86) and A 2 δ, we get 1 1 d 󵄩󵄩 41 󵄩󵄩2 󵄩 3 󵄩2 󵄩A δ󵄩󵄩 + γ 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 + (PN D(U1 , U2 ), A 2 δ) = 0, 2 dt 󵄩

(2.4.91)

296 | 2 Inertial manifold where 1

1

(D(U1 , U2 ), A 2 δ) = (D1 (U1 , U2 ) + (Φ(U1 ) − Φ(U2 ))xx , A 2 δ) 1

3

= (D1 , A 2 δ) − ((Φ(U1 ) − Φ(U2 ))x , A 4 δ) 1

3

= (D1 , A 2 δ) − ((Φ󸀠 (U1 ) − Φ󸀠 (U2 ))U1x + Φ󸀠 (U2 )(U1x − U2x ), A 4 δ). Therefore 1 󵄨 α 󵄩 3 󵄩󵄩 1 󵄩 󵄨󵄨 󵄨󵄨(D(U1 , U2 ), A 2 δ)󵄨󵄨󵄨 ≤ (θ1 + θ2 )󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 4 (U1 − U2 )󵄩󵄩󵄩 2 α 󵄩 1 󵄩󵄩 3 󵄩 + (θ1 − θ2 )󵄩󵄩󵄩A 4 (U1 + U2 )󵄩󵄩󵄩󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 2 󵄩 󵄩 󵄩 1 󵄩 + 󵄩󵄩󵄩F 󸀠󸀠 (U2 )󵄩󵄩󵄩L ‖U1x ‖‖U1 − U2 ‖L∞ 󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 ∞ 󵄩 󵄩 󵄩 1 󵄩 + 󵄩󵄩󵄩F 󸀠 (U2 )󵄩󵄩󵄩L ‖U1 − U2 ‖1 󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 ∞ 1 󵄩 󵄩 󵄩 1 󵄩 + (θ1 + θ2 )󵄩󵄩󵄩g 󸀠 (U)󵄩󵄩󵄩L ‖U1 − U2 ‖󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 ∞ 2 1 󵄩 󵄩󵄩 1 󵄩 + |θ1 − θ2 |󵄩󵄩󵄩g(U1 ) + g(U2 )󵄩󵄩󵄩󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 2 󵄩󵄩 󸀠󸀠 󵄩󵄩 󵄩 3 󵄩 + 󵄩󵄩Φ (U)󵄩󵄩L ‖U1x ‖‖U1 − U2 ‖L∞ 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 ∞ 󵄩 󵄩 󵄩 3 󵄩 + 󵄩󵄩󵄩Φ󸀠 (U2 )󵄩󵄩󵄩L ‖U1 − U2 ‖1 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩, ∞

where 󵄩 󵄩 ‖U1 − U2 ‖1 ≤ ‖p1 − p2 ‖1 + 󵄩󵄩󵄩φ(p1 ) − φ(p2 )󵄩󵄩󵄩1 󵄩 1 󵄩 󵄩 1 󵄩 ≤ (1 + l)󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 ≤ 2󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩, 󵄨󵄨󵄨 ‖U ‖ ‖U ‖ 󵄨󵄨󵄨 |θ1 − θ2 | = 󵄨󵄨󵄨θ( 1 1 ) − θ( 2 1 )󵄨󵄨󵄨 E E 󵄨󵄨 󵄨󵄨 2 ≤ ‖U1 − U2 ‖1 E 4󵄩 1 󵄩 ≤ 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩. E Then 1 󵄨󵄨 󵄨 󵄩 3 󵄩󵄩 1 󵄩 󵄩 1 󵄩󵄩 3 󵄩 󵄨󵄨(D(U1 , U2 ), A 2 δ󵄨󵄨󵄨 ≤ 2α󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 + 2α‖U1 + U2 ‖1 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 󵄩 󵄩 󵄩 1 󵄩󵄩 1 󵄩 + C 󵄩󵄩󵄩F 󸀠󸀠 (U)󵄩󵄩󵄩L ‖U1x ‖󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 ∞ 󵄩 󵄩 󵄩 1 󵄩󵄩 1 󵄩 + 2󵄩󵄩󵄩F 󸀠 (U2 )󵄩󵄩󵄩L 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 ∞ 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩 41 󵄩󵄩󵄩󵄩 21 󵄩󵄩 + 2󵄩󵄩g (U)󵄩󵄩L 󵄩󵄩A δ󵄩󵄩󵄩󵄩A δ󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 1 󵄩 + 2󵄩󵄩g(U1 ) + g(U2 )󵄩󵄩󵄩󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 2 δ󵄩󵄩󵄩 󵄩 󵄩 󵄩 1 󵄩󵄩 3 󵄩 + 2󵄩󵄩󵄩Φ󸀠󸀠 (U)󵄩󵄩󵄩L ‖U1x ‖󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 ∞

2.4 The existence of inertial manifolds for the generalized KS equation

|

297

󵄩 󵄩 1 󵄩󵄩 3 󵄩 󵄩 + 2󵄩󵄩󵄩Φ󸀠 (U2 )󵄩󵄩󵄩L 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 ∞ γ 󵄩󵄩 43 󵄩󵄩2 󵄩 3 󵄩 󵄩 1 󵄩󵄩 1 󵄩 ≤ 󵄩󵄩A δ󵄩󵄩 + C1 (󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 + 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩)󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩. 2 Thus by equation (2.4.91) we get 󵄨 󵄨󵄨 1 d 󵄩 3 󵄩2 󵄨󵄨 󵄨󵄨 󵄩󵄩 41 󵄩󵄩2 󵄨󵄨 󵄩󵄩A δ󵄩󵄩 + γ 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 dt 󵄩󵄩 43 󵄩󵄩 󵄩󵄩 41 󵄩󵄩 󵄩󵄩 41 󵄩󵄩 ≤ C1 (󵄩󵄩A δ󵄩󵄩 + 󵄩󵄩A δ󵄩󵄩)󵄩󵄩A δ󵄩󵄩 +

γ 󵄩󵄩 43 󵄩󵄩2 󵄩A δ󵄩󵄩 . 2󵄩

(2.4.92)

In a similar way, we get 1 d 󵄩󵄩 41 󵄩󵄩2 󵄩A Δ󵄩󵄩 + 2 dt 󵄩

γ 󵄩󵄩 43 󵄩󵄩2 󸀠 󵄩 3 󵄩 󵄩 1 󵄩 󵄩󵄩A Δ󵄩󵄩 ≤ C1 (󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 + 󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩)‖U1 − U2 ‖1 2 󵄩 3 󵄩 󵄩 1 󵄩󵄩 1 󵄩 ≤ C2 (󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 + 󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩)󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩.

(2.4.93)

1

1

Let y = A 4 δ, z = A 4 Δ. Then from equations (2.4.92)–(2.4.93) we get 1 2 d ‖y‖2 ≥ − λN ‖y‖2 − C1 (λN2 ‖y‖ + ‖y‖)‖y‖ dt 3γ 3 d ‖z‖ 2 󵄩 󵄩 󵄩 3 󵄩 ‖z‖2 + γ 󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 ≤ 2C2 (󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 + ‖z‖) . dt γ1

By virtue of 󵄩 3 󵄩2 γ 󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 ≥ γλN+1 ‖z‖2 ,

2

2C2 󵄩󵄩 43 󵄩󵄩 󵄩 3 󵄩2 1 2C 2 󵄩A Δ󵄩󵄩‖z‖ ≤ γ/2󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 + ( 2 ) ‖z‖ , γ1 󵄩 2γ γ1 we thus have 1 3γ d ‖y‖ + ( λN + C1 λN2 + 1)‖y‖ ≥ 0, dt 4 γ d ‖z‖ + ( λN+1 − C3 (γ, γ1 ))‖z‖ ≤ 0, dt 4

where C3 (γ, γ1 ) =

C2 2C + 2. γ1 γγ12

By Lemma 2.4.14, where d = γ/4λN+1 − C3 (γ, γ1 ) > 0, C = 0,

b = 0,

N ≥ N0 , 1

a = 3γ/4λN + C1 λN2 + 1,

(2.4.94) (2.4.95)

298 | 2 Inertial manifold d − a − γ1 b − γ1−1 C =

1 3γ γ λN+1 − λN − C1 λN2 − 1 − C3 > 0, 4 4

N ≥ N0 ,

we arrive at γ 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩z(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩z(t0 )󵄩󵄩󵄩 exp{−( λN+1 − C3 )(t − t0 )} 4 γ 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩z(0)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩z(t0 )󵄩󵄩󵄩 exp{−( λN+1 − C3 )t0 }. 4

(2.4.96)

In the above equation, letting t0 → −∞, by Lemma 2.4.10, since 󵄩 󵄩 󵄩 󵄩 1 sup{󵄩󵄩󵄩A 4 Δ(t)󵄩󵄩󵄩 = 󵄩󵄩󵄩z(t)󵄩󵄩󵄩 : −∞ < t ≤ 0} < ∞, we have 󵄩󵄩 󵄩 󵄩󵄩z(0)󵄩󵄩󵄩 = 0. But ‖z(0)‖ > l‖y(0)‖(≠ 0) is not possible, so it follows that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩z(0)󵄩󵄩󵄩 ≤ l󵄩󵄩󵄩y(0)󵄩󵄩󵄩, namely 󵄩󵄩 41 󵄩 󵄩 1 󵄩 󵄩󵄩A (F0 φ(p01 ) − F0 φ(p02 ))󵄩󵄩󵄩 ≤ l󵄩󵄩󵄩A 4 (p01 − p02 )󵄩󵄩󵄩.

(2.4.97)

If N is large enough, N ≥ max(N0 , N2󸀠 ) = N1 (γ, γ1 , ̃ D). Theorem 2.4.2 is proved. Now we prove the contractivity of functional operator F0 . Theorem 2.4.3. For any b, l ∈ (0, 1], there exists N2 = N2 (γ, γ1 , b, l, ̃ D) such that when N + 1 ≥ N2 we have 1 d(F0 φ1 , F0 φ2 ) ≤ d(φ1 , φ2 ), 2

1

4 . ∀φ1 , φ2 ∈ Fb,l

(2.4.98)

1

4 Proof. Let φ1 , φ2 ∈ Fb,l and set 1 󵄩 󵄩 d0 = d(φ1 , φ2 ) = max{󵄩󵄩󵄩φ1 (p) − φ2 (p)󵄩󵄩󵄩1 , p ∈ PD(A 4 )}.

(2.4.99)

In the proof of Theorem 2.4.2, set Uj = pj + φj (pj ),

j = 1, 2,

(2.4.100)

where pj = pj (τ; p0 ; φj ) is the solution of equation (2.4.81), namely p󸀠j + γApj + θ(

‖Uj ‖1 E

1

)(−αA 2 pj ) + PN B(Uj , Uj ) = 0,

(2.4.101)

2.4 The existence of inertial manifolds for the generalized KS equation

|

299

satisfying the same initial value condition pj (0; p0 ; φj ) = p0 .

(2.4.102)

Let qj (τ) be the bounded solution of equation (2.4.84) in (−∞, 0], that is, qj󸀠 + γAQN qj + θ(

‖Uj ‖1 E

1

)(−αA 2 QN Uj ) + QN B(Uj , Uj ) = 0.

(2.4.103)

Let δ(τ) = p1 (τ; p0 , φ1 ) − p2 (τ; p0 , φ2 ),

(2.4.104)

Δ(τ) = q1 (τ) − q2 (τ).

Then by equations (2.4.101)–(2.4.102), we have δ̇ + γAδ + PN D(U1 , U2 ) = 0, δ(0) = 0, Δ̇ + γAΔ + QN D(U1 , U2 ) = 0.

(2.4.105) (2.4.106)

Similar to inequality (2.4.92) and equation (2.4.93), we have 󵄨󵄨 1 d 󵄨 󵄨󵄨 󵄩󵄩 41 󵄩󵄩2 γ 󵄩󵄩 43 󵄩󵄩2 󵄨󵄨󵄨 󵄩 3 󵄩 󵄩 1 󵄩 󵄨󵄨 󵄩󵄩A δ󵄩󵄩 + 󵄩󵄩A δ󵄩󵄩 󵄨󵄨 ≤ C1 (󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩 + 󵄩󵄩󵄩A 4 δ󵄩󵄩󵄩)‖U1 − U2 ‖1 , 󵄨󵄨 2 dt 󵄨󵄨 2 1 d 󵄩󵄩 41 󵄩󵄩2 γ 󵄩󵄩 43 󵄩󵄩2 󵄩 3 󵄩 󵄩 1 󵄩 󵄩󵄩A δ󵄩󵄩 + 󵄩󵄩A δ󵄩󵄩 ≤ C2 (󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 + 󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩)‖U1 − U2 ‖1 . 2 dt 2

(2.4.107) (2.4.108)

Since 󵄩 󵄩 ‖U1 − U2 ‖1 ≤ ‖p1 − p2 ‖1 + 󵄩󵄩󵄩φ1 (p1 ) − φ1 (p2 )󵄩󵄩󵄩1 󵄩 󵄩 󵄩 󵄩 ≤ ‖p1 − p2 ‖1 + 󵄩󵄩󵄩φ1 (p1 ) − φ1 (p2 )󵄩󵄩󵄩1 + 󵄩󵄩󵄩φ1 (p1 ) − φ1 (p2 )󵄩󵄩󵄩1 󵄩 1 󵄩 ≤ 2󵄩󵄩󵄩A 4 Δ󵄩󵄩󵄩 + d0 , 1

1

setting y = A 4 δ, z = A 4 Δ, we then get 󵄨󵄨 1 d γ 󵄩 1 󵄩2 󵄨󵄨󵄨 󵄨󵄨 󵄩 1 󵄩 ‖y‖2 + 󵄩󵄩󵄩A 2 y󵄩󵄩󵄩 󵄨󵄨󵄨 ≤ C1 (󵄩󵄩󵄩A 2 y󵄩󵄩󵄩 + ‖y‖)(2‖y‖ + d0 ), 󵄨󵄨 󵄨󵄨 2 dt 󵄨󵄨 2 γ 󵄩 1 󵄩2 d 󵄩 1 󵄩 ‖z‖2 + 󵄩󵄩󵄩A 2 z 󵄩󵄩󵄩 ≤ C2 (󵄩󵄩󵄩A 2 z 󵄩󵄩󵄩 + ‖z‖)(2‖y‖ + d0 ) dt 2 C 󵄩 1 󵄩 ≤ 2 (󵄩󵄩󵄩A 2 z 󵄩󵄩󵄩 + ‖z‖)(2‖z‖ + γ1 d0 ), γ1

(2.4.109)

for ‖z‖ > γ1 ‖y‖,

(2.4.110)

where 󵄩󵄩 󵄩 󵄩󵄩y(0)󵄩󵄩󵄩 = 0,

󵄩 󵄩 󵄩 󵄩 1 sup{󵄩󵄩󵄩A 4 Δ(t)󵄩󵄩󵄩 = 󵄩󵄩󵄩z(t)󵄩󵄩󵄩 : −∞ < t ≤ 0} < ∞.

(2.4.111)

300 | 2 Inertial manifold Consider the open subset J = {t ∈ (−∞, 0] : ‖z‖ > 21 d0 } on (−∞, 0]. If 0 ∈ J, set J0 = (−T, 0] to the part of J including 0. On J0 , by inequalities (2.4.110)–(2.4.111) we get 󵄨󵄨 1 d γ 󵄩 1 󵄩2 󵄨󵄨󵄨 2 󵄩 1 󵄩 󵄨󵄨 ‖y‖2 + 󵄩󵄩󵄩A 2 y󵄩󵄩󵄩 󵄨󵄨󵄨 ≤ 2C1 (󵄩󵄩󵄩A 2 y󵄩󵄩󵄩 + ‖y‖) (‖y‖ + ‖z‖) 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 dt 2 γ 󵄩 1 󵄩2 2C 󵄩 1 󵄩 1 d ‖z‖2 + 󵄩󵄩󵄩A 2 z 󵄩󵄩󵄩 ≤ 2 (󵄩󵄩󵄩A 2 z 󵄩󵄩󵄩 + ‖z‖)(‖z‖ + γ1 ‖z‖) 2 dt 2 γ1 2C2 󵄩 1 󵄩 = (γ + 1)[󵄩󵄩󵄩A 2 z 󵄩󵄩󵄩‖z‖ + ‖z‖2 ] γ1 1 ≤

γ 󵄩󵄩 21 󵄩󵄩2 2C1 (γ1 + 1) 2C1 (γ1 + 1) ( + 1)‖z‖2 , 󵄩A z 󵄩󵄩 + 4󵄩 γ1 γγ1

hence we have 1 γ d ‖y‖ + λN ‖y‖ + 2C1 (λN2 + 1)(‖y‖ + ‖z‖) ≥ 0, dt 2 2C (γ + 1) 2C1 (γ1 + 1) γ d ( + 1)‖z‖ ≤ 0. ‖z‖ + λN+1 ‖z‖ − 1 1 dt 4 γ1 γγ1

(2.4.112) (2.4.113)

Since ‖z(0)‖ ≥ 21 d0 , ‖y(0)‖ = 0, and 2C (γ + 1) 2C1 (γ1 + 1) γ λ − 1 1 ( + 1) > 0, 4 N+1 γ1 γγ1 1 γ 2 c = 0, α = λN + 2C1 (λN+1 + 1) > 0, 2

d=

N + 1 ≥ N20 ,

1

b = 2C1 (λN2 + 1) > 0, 1 γ γ 2 + 1) d − a − γ1 b − γ1−1 c = λN+1 − λN − 2C1 (λN+1 4 2 1 2C (γ + 1) 2C1 (γ1 + 1) 2 − 2C1 γ1 (λN+1 + 1) − 1 1 ( − 1) γ1 γγ1 1 γ γ = λN+1 − λN − 2C1 (γ1 + 1)λN2 − 2C1 (γ1 + 1) 4 2 2C1 (γ1 + 1) 2C1 (γ1 + 1) − ( + 1) > 0, N ≥ N21 . γ1 γγ1 By Lemma 2.4.14, when N + 1 ≥ max(N20 , N21 ) = N2 , we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩z(0)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩z(t)󵄩󵄩󵄩 exp(dt),

t ∈ J0 .

Since sup{‖z(t)‖ : −∞ ≤ t ≤ 0} < ∞, if T = ∞ then we have 1 󵄩󵄩 󵄩 󵄩󵄩z(0)󵄩󵄩󵄩 = 0 ≤ d0 . 2 Otherwise by equation (2.4.114) and for −T ∈ ̸ J, we have 󵄩󵄩 󵄩 󵄩 󵄩 1 󵄩󵄩z(0)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩z(−T)󵄩󵄩󵄩 ≤ d0 . 2

(2.4.114)

2.4 The existence of inertial manifolds for the generalized KS equation

| 301

Thus we prove 1 󵄩 1 󵄩 󵄩 󵄩󵄩 󵄩󵄩F0 φ1 (p0 ) − F0 φ2 (p0 )󵄩󵄩󵄩 = 󵄩󵄩󵄩z(0)󵄩󵄩󵄩 ≤ d0 = d(φ1 , φ2 ). 2 2 1

Since p0 ∈ PD(A 2 ) is arbitrary, then we have 1 d(F0 φ1 , F0 φ2 ) ≤ d(φ1 , φ2 ), 2

N + 1 ≥ N2 .

Now we reveal the structure of the inertial manifold M of PGKS equation, namely: 1 (1) M = graph(Φ), where Φ is a Lipschitz continuous mapping from PD(A 4 ) to 1 QD(A 4 ); 1 1 (2) Φ has compact support, supp Φ ⊂ {p ∈ PD(A 4 ) : ‖A 4 p‖ < 2E}; (3) Sp (t)M ⊂ M, t ≥ 0; 1

(4) There exists a constant λ > 0, such that for any u0 ∈ D(A 4 ), there exists μ0 > 0 (u0 is established uniformly for bounded set) such that dist(Sp (t)u0 , M) ≤ μ0 exp(−λt),

t ≥ 0;

(5) The attractor Ap of PGKS equation is in M. 1

Since Φ ∈ Ab,l4 is known, claims (1) and (2) are true. Now we prove that (3), (4) and (5) are also true. Theorem 2.4.4. Let b, l ∈ (0, 1] and N + 1 > N2 (γ, γ1 , b, l, ̃ D). 1

4 Then F0 is the unique fixed point Φ ∈ Fb,l such that

1 ̃) ≤ d(Φ, φ ̃), d(Φ, F0 φ 2

1

4 ∀φ ∈ Fb,l ,

(2.4.115)

and the Lipschitz manifold 1

M = graph(Φ) = {p + φ(p) : p ∈ PD(A 4 )}

(2.4.116)

is invariant for {Sp (t)}t≥0 . Proof. The first part of the claim follows from Theorem 2.4.2. Since Φ is the fixed point of F0 , namely F0 Φ = Φ, by Theorem 2.4.3 we have 1 d(Φ, F0 Φ) = d(F0 ϕ, F0 φ) ≤ d(ϕ, φ). 2 1

4 By the definition of F0 and from F0 Φ = Φ, Φ ∈ Fb,l , we have

Sp (t)M ⊂ M.

302 | 2 Inertial manifold Theorem 2.4.5. For any u ∈ B2E and τ ≥ 0, we have 󵄩 󵄩 1 dist(Sp (τ)u, M) = inf{󵄩󵄩󵄩A 4 (Sp (τ)u − v)󵄩󵄩󵄩, v ∈ M} ≤ 4E exp{−dτ},

(2.4.117)

where M is defined in (2.4.116) and d=

γ (N + 1)4 2C1 (γ1 + 1) 2C1 (γ1 + 1) − ( + 1) > 0. 4 ̃ γ1 γγ1 D4

Proof. For u ∈ B2E and τ > 0, let p(t) = p(t; pN Sp (t)u; Φ) be the solution of equation 1 ‖U‖1 dp + γAp + θ( )(−αA 2 p) + PN B(U, U) = 0, dt E

(2.4.118)

where U = p + Φ(p). Define v = p(−τ) + Φ(p(−τ)), and set 1

̃ = A 4 P [S (t)u − S (t)v], δ(t) N p p 1

̃ = A 4 QN [Sp (t)u − Sp (t)v], Δ(t)

t ∈ [0, τ].

Then as in the proof of Theorem 2.4.2, we have 󵄨󵄨 1 d 󵄨 1 1 󵄨󵄨 ̃ 2 + γ 󵄩󵄩󵄩A 2 δ̃󵄩󵄩󵄩2 󵄨󵄨󵄨󵄨 ≤ C (󵄩󵄩󵄩A 2 δ̃󵄩󵄩󵄩 + ‖δ‖)‖ ̃ δ‖, ̃ ‖δ‖ 󵄨󵄨 1 󵄩 󵄩 󵄩 󵄩 󵄨󵄨 󵄨󵄨 2 dt 2 󵄨 1 d ̃ 2 γ 󵄩󵄩 21 ̃ 󵄩󵄩2 󵄩 1 ̃ 󵄩󵄩 ̃ ̃ δ‖. ‖Δ‖ + 󵄩󵄩A Δ󵄩󵄩 ≤ C2 (󵄩󵄩󵄩A 2 Δ 󵄩󵄩 + ‖Δ‖)‖ 2 dt 2 ̃ ̃ Since |Δ(0)| ≥ |δ(0)| = 0, by Lemma 2.4.14 we then have 󵄩󵄩 󵄩 󵄩󵄩Sp (τ)u − {PN Sp (τ)u + Φ[PN Sp (t)u]}󵄩󵄩󵄩1 󵄩 ̃ 󵄩󵄩 󵄩󵄩 ̃ 󵄩󵄩 −dτ = 󵄩󵄩󵄩Δ(τ) 󵄩󵄩 ≤ 󵄩󵄩Δ(0)󵄩󵄩e . If ‖p(−τ)‖1 ≥ 2E, then QN v = 0 and 󵄩 󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩 41 󵄩󵄩Δ(0)󵄩󵄩 = 󵄩󵄩A QN [Sp (t)u − Sp (0)v]󵄩󵄩󵄩 󵄩 1 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩A 4 QN u󵄩󵄩󵄩1 = 󵄩󵄩󵄩QN u󵄩󵄩󵄩1 ≤ 2E. If ‖p(−τ)‖1 < 2E, then 󵄩 󵄩 󵄩 󵄩 ‖QN v‖1 = 󵄩󵄩󵄩Φ[p(−τ)]󵄩󵄩󵄩1 = 󵄩󵄩󵄩Φ[p(−τ)] − Φ(2E)󵄩󵄩󵄩1 ≤ l(2E − ‖p(−τ)‖1 ) ≤ 2E. ̃ Hence ‖Δ(0)‖ ≤ ‖QN (u − v)‖1 ≤ 4E, which yields 󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩Δ(0)󵄩󵄩 ≤ max(2E, 4E) = 4E. The theorem has been proved.

(2.4.119)

2.4 The existence of inertial manifolds for the generalized KS equation

| 303

Corollary 2.4.1. The attractor Ap of PGKS equation is in the set M. Proof. Since Ap = Sp (t)Ap ⊂ S(t)B2E , ∀t ≥ 0, where B2E is an absorbing set, for u ∈ Ap , by inequality (2.4.119) we have dist(u, M) ≤ 4Ee−dt ,

d > 0.

When t → ∞, we can get that Ap is included in the set M. By the above results, we conclude that ̃ γ1 , b, l, ̃ Theorem 2.4.6. For any N ≥ N(γ, D), there exists an inertial manifold M of PGKS equation, which is the graph of the fixed point of equation (2.4.64) defined by the operator 1

4 F0 on Fb,l .

Now we will consider the relations between the inertial manifold of PGKS equation and that of GKS equation (2.4.1). Moreover, we will consider the Galerkin approximation of inertial manifold M for PGKS equation. Lemma 2.4.15. The attractor A and absorbing set of GKS equation (2.4.1) given by B 3 E = {u ∈ H 1 , ‖u‖1 ≤ 2

3 (E + E1 )} 2 0

have the following relation: A = ⋂ S(t)B 3 E .

(2.4.120)

2

t≥0

Proof. By the definition of the global attractor A , we know A = ⋂ ⋃ S(t)B 3 E

(2.4.121)

2

t≥0 t≥τ

where the closure is taken in L2 . Denoting A 󸀠 = ⋂t≥0 S(t)B 3 E , from equation (2.4.121) 2

we know A 󸀠 ⊂ A . On the other hand, since B 3 E is absorbing, there exists t1 > 0 such 2 that S(t)B 3 E ⊂ B 3 E , 2

2

t ≥ t1 .

Hence A ⊂ ⋃ S(t)B 3 E ⊂ S(τ − t1 )B 3 E , t≥τ

2

2

and it follows that A ⊂ A 󸀠 , thus we have A =A . 󸀠

τ ≥ t1 ,

304 | 2 Inertial manifold Theorem 2.4.7. Let M be the inertial manifold of the PGKS equation, which satisfies: 1 (1) M = graph(Φ), where Φ is the Lipschitz continuous mapping, PN D(A 4 ) → 1 QN D(A 4 ); (2) Φ has compact support, when Φ is outside of B2E , Φ ≥ 0. Then in an open neighborhood of M, M2 is an inertial manifold of GKS equation (2.4.1). Proof. Let B 1 E = {u ∈ H 1 : ‖u‖1 ≤ 21 E}. Then there exists t0 ≥ 0 such that 2

B1 = ⋃ S(t)B 1 E ⊂ BE {u ∈ H 1 : ‖u‖1 ≤ E}. t≥t0

2

Noting that S(t)B1 ⊂ B1 , Sp (t)B1 = S(t)B, ∀t ≥ 0, and setting M1 = M ∩ B1 , we then have S(t)M1 = Sp (t)M1 ⊂ Sp (t)M ∩ Sp (t)B1

= Sp (t)M ∩ S(t)B1 ⊂ M ∩ B1 = M1 ,

∀t ≥ 0.

Let M1 = 𝜕M + M2 , that is, M2 is the set of inner points of M1 in M. Because Sp (t)M is a homeomorphism on M, then S(t)M2 is also in the interior of Sp (t)M1 = Sp (t)M1 ⊂ M1 . Therefore M2 satisfies properties (1) and (2), namely: 1 1 (1) M2 = graph(Φ), where Φ is a Lipschitz continuous mapping, PN D(A 4 ) → QN D(A 4 ); (2) S(t)M2 ⊂ M2 , t ≥ 0. As in [182], we can show that the attractor A of GKS equation ⊂ M2 . And there exists dist(S(t)u, M2 ) ≤ μ1 e−δt ,

μ1 > 0, δ > 0, t ≥ 0, u ∈ B,

where B is any bounded set in B1 . Then M2 is an inertial manifold of GKS equation. Now consider the mth order approximation of PGKS equation: Acting with Pm on PGKS equation, we get 1 ‖u(m) ‖1 d (m) u + γAu(m) + θ( )(−αA 2 u(m) ) + Pm B(u(m) , u(m) ) = 0, dt E 1

u(m) ∈ Pm D(A 4 ).

(2.4.122)

̃ the Galerkin approximate equation (2.4.122) has m-dimensional When m ≥ N ≥ N, inertial manifold M (m) , and we have the following theorem:

2.4 The existence of inertial manifolds for the generalized KS equation

| 305

̃ then the Galerkin approximate equation (2.4.122) has Theorem 2.4.8. Let m ≥ N ≥ N, 1 (m) inertial manifold M = {p + Φm (p), p ∈ PN D(A 4 )}, where Φm is a fixed point of operator Pm F , 1

1

4 4 → Pm Fb,l , Pm F0 : Pm Fb,l

1

1

4 4 Pm Fb,l = {Pm φ : φ ∈ Pm Fb,l }.

Suppose m is large enough, and we can set up an approximate estimation of Φm . ̃ is fixed. Then we have Theorem 2.4.9. Suppose m ≥ N d(Φ, Φm ) ≤

C(̃ D) (m + 1)3

(2.4.123)

where 󵄩 󵄩 d(φ1 , φ2 ) = max{󵄩󵄩󵄩φ1 (p) − φ2 (p)󵄩󵄩󵄩1 , p ∈ PN D(A 4 )}, 1

and N is the dimension of inertial manifolds graph Φ and graph Φm . ̃ and set Proof. Let m > N ≥ N, U (m) = p(m) + Φm (p(m) ),

Pm U (m) = U (m) ,

(2.4.124)

where p(m) (τ, p0 ; Φm ) is the solution of equation 1 ‖U (m) ‖1 d (m) u + γAu(m) + θ( )(−αA 2 p(m) ) + PN B(U (m) , U (m) ) = 0, (2.4.125) dt E

p(m) (0; p0 ; Φm ) = p0 ,

(2.4.126)

which yields Φm (p0 ) = Pm F0 Φm (p0 ) 0

= − ∫ eτγAQN ⋅ θ( −∞

1 ‖U (m) ‖1 )QN [−αA 2 U (m) E

+ PN B(U (m) , U (m) )]dτ.

(2.4.127)

Now considering the definition of F0 Φm , equations (2.4.58)–(2.4.59) are consistent 1 1 with equations (2.4.124)–(2.4.125). Since PN D(A 2 ) ⊂ Pm D(A 2 ), Pm U (m) = U (m) , then 0

F0 Φm (p0 ) = − ∫ e −∞

τγAQN

⋅ θ(

1 ‖U (m) ‖1 )QN [−αA 2 U (m) + PN B(U (m) , U (m) )]dτ. E

(2.4.128)

306 | 2 Inertial manifold From equations (2.4.127)–(2.4.128), we arrive at 󵄩 󵄩󵄩 󵄩󵄩Φm (p0 ) − F0 Φm (p0 )󵄩󵄩󵄩1 󵄩 󵄩 = 󵄩󵄩󵄩Pm F0 Φm (p0 ) − F0 Φm (p0 )󵄩󵄩󵄩1

󵄨󵄨 0 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 τγAQN (m) (m) 󵄨 4 = 󵄨󵄨 ∫ (AQN ) e ⋅ QN (1 − Pm )B(U , U )dτ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −∞ 󵄨 0

1

≤ ∫ (AQm ) 4 eτγAQm ⋅ Qm B(U (m) , U (m) )dτ −∞ −γ

≤

Ce 4

3

(λm+1 ) 4

.

3 The approximate inertial manifold In the previous discussions on the global attractor and inertial manifold, we saw that the global attractor may not be smooth. Although the inertial manifold is smooth, we must solve the integral equations on an infinite interval to seek them, which brings much trouble in the calculation. Therefore it is natural that we would like to use the approximate and smooth manifold which can be obtained more easily to approximate the global attractor and inertial manifold. This chapter describes recent developments [213, 214] on approximate inertial manifold. In the following, we would like to use the two-dimensional Navier–Stokes equations as an example to illustrate the variety of approximate inertial manifolds and their error estimation.

3.1 Two-dimensional Navier–Stokes equation Let Navier–Stokes equation have the following form [31, 60, 61]: du + νAμ + B(u, u) = f , dt u(0) = u0 ,

(3.1.1) (3.1.2)

where Au = −PΔu, ∀u ∈ D(A), B(u, v) = P[(u ⋅ ∇)ω], ∀u, ω ∈ D(A), here P means the orthogonal projection from (L2 (Ω))2 to H, while H means the closure of V with respect to (L2 (Ω))2 . When u|𝜕Ω = 0, V = {v ∈ {C0 (Ω)}2 , div v = 0}, here A is a linear unbounded self-adjoint operator, A−1 is compact, D(A) is dense in H. Hence, H has the orthogonal basis {ωj }∞ j=1 , Aωj = λj ωj , j = 1, 2, . . . , which are the eigenvectors of operator A, and 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ , eigenvalues λm satisfy C0 λ1 m ≤ λm ≤ C1 λ1 m,

m = 1, 2, . . . ,

(3.1.3)

where C0 , C1 are some known constants. In the following C0 , C1 , . . . represent positive constants. It is easy to verify that B(u, v) satisfies 1 1 1 1 1 󵄨󵄨 󵄨 󵄨󵄨(B(u, v), ω)󵄨󵄨󵄨 ≤ C2 |u| 2 ‖u‖ 2 ‖v‖ 2 |ω| 2 ‖ω‖ 2 , ∀u, v, ω ∈ V, 󵄨󵄨 󵄨 󵄨󵄨(B(u, v), ω)󵄨󵄨󵄨 ≤ C3 ‖u‖L∞ (Ω) ‖v‖|ω|, ∀u ∈ D(A), ∀v ∈ V, ∀ω ∈ H,

(3.1.4) (3.1.5)

where u ∈ H, |u|2 = ∫Ω |u(x)|2 dx; u ∈ V, ‖u‖2 = ∫Ω |∇v(x)|2 dx. By Brezis–Gallouet inequality, we have ‖u‖L∞ (Ω) ≤ C4 ‖u‖[1 + lg[ https://doi.org/10.1515/9783110549652-003

|Au| 1

λ12 ‖u‖

1 2

]] ,

∀u ∈ D(A).

(3.1.6)

308 | 3 The approximate inertial manifold From equations (3.1.5)–(3.1.6), we have 1

2 |Au| 󵄨 󵄨󵄨 ]] , 󵄨󵄨(B(u, v), ω)󵄨󵄨󵄨 ≤ C5 ‖v‖|ω|‖u‖[1 + 2 lg[ 1 λ12 ‖u‖

∀u ∈ D(A), ∀v ∈ V, ∀ω ∈ H,

(3.1.7)

1

2 |Aω| 󵄨 󵄨󵄨 ]] , 󵄨󵄨(B(u, v), ω)󵄨󵄨󵄨 ≤ C6 ‖v‖|u|‖ω‖[1 + 2 lg[ 1 λ12 ‖ω‖

∀u ∈ H, ∀v ∈ V, ∀ω ∈ D(A). (3.1.8)

Furthermore, B(u, v) also satisfies the following functional equation (∗): (B(u, v), ω) = −(B(u, ω), v),

∀u ∈ H, ∀v, ω ∈ D(A).

For the solution u(t) of problem (3.1.1)–(3.1.2), we have proved that there exists t0 , which depends on u0 , ν, |f |, and λ1 , such that 󵄨󵄨 󵄨 󵄨󵄨u(t)󵄨󵄨󵄨 ≤ M0 ,

󵄩󵄩 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ M1 ,

∀t ≥ t0 ,

(3.1.9)

where the constants M0 , M1 depend on ν, |f |, and λ1 . Now we consider the approximate inertial manifold of problem (3.1.1)–(3.1.2). Let Pm be the orthogonal projection of H onto Hm = span{ω1 , . . . , ωm }, Qm = I − Pm . Let p = Pm u, q = Qm u. Then equation (3.1.1) is equivalent to dp + νAp + Pm B(p + q, p + q) = Pm f , dt dq + νAq + Qm B(p + q, p + q) = Qm f . dt

(3.1.10) (3.1.11)

The inertial manifold of equation (3.1.1) is the subset of M ⊂ H, possessing the following properties: (i)

M is a finite-dimensional Lipschitz manifold.

(3.1.12)

(ii)

The convection M is a positive invariant set. That is, if u0 ∈ M, then the solution of equation (3.1.1)–(3.1.2) u(t) ∈ M, ∀t > 0.

(iii)

(3.1.13)

M attracts all orbits exponentially. For any solution u(t) of equations (3.1.1)–(3.1.2) it follows that dist(u(t), M) → 0, exponentially as t → ∞. From this we deduce that the global attractorA ⊂ M. (3.1.14)

If we demand that M is the graph of a Lipschitz function Φ : Hm → Qm H, then the invariance condition (3.1.13) is equivalent to q(t) = Φ(p(t)) being valid for any solution p(t), q(t) of problem (3.1.10)–(3.1.11) with q(0) = Φ(p(0)). Therefore, if the function Φ exists, then equations (3.1.10)–(3.1.11) for the inertial manifold M are equivalent to the following system of differential equations (called the inertia form): dp + νAp + Pm B(p + Φ(p), p + Φ(p)) = Pm f , dt

p ∈ Hm .

(3.1.15)

3.1 Two-dimensional Navier–Stokes equation

| 309

In order to use the smooth manifold to approximate the global attractor, we introduce the approximate inertial manifold. Obviously, in equation (3.1.15), if Φ ≜ 0, then we can obtain the ordinary Galerkin approximation [140]: dum + νAum + Pm B(um , um ) = Pm f , dt

um ∈ Hm .

(3.1.16)

Now we introduce a finite-dimensional analytic manifold, μ0 = graph(Φ0 ), Φ0 (p) = (νA)−1 [Qm f − Qm B(p, p)],

p ∈ Hm ,

(3.1.17)

which can be a better approximation of the global attractor. Theorem 3.1.1. Let m be large enough such that λm+1 ≥ (

2

2C2 M ). ν

(3.1.18)

Then for any solution u(t) = p(t) + q(t) of equation (3.1.10)–(3.1.11), which satisfies 1 󵄨󵄨 󵄨 −1 󵄨󵄨q(t)󵄨󵄨󵄨 ≤ K0 λm+1 L 2 ,

(3.1.19)

1 − 21 󵄨󵄨 󵄨 L2 , 󵄨󵄨q(t)󵄨󵄨󵄨 ≤ K1 λm+1 1 󵄨󵄨 󵄨 󵄨󵄨Aq(t)󵄨󵄨󵄨 ≤ K2 L 2 , 1 󵄨󵄨󵄨 dq 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ K 󸀠 λ−1 L 2 , 0 m+1 󵄨󵄨 dt 󵄨󵄨 󵄨 󵄨 󵄩󵄩 󵄩 −1 󵄩󵄩q(t) − Φm (p(t))󵄩󵄩󵄩 ≤ K1 λm+1 L, ∀t ≥ T∗ ,

(3.1.20) (3.1.21) (3.1.22) (3.1.23)

where T∗ > 0 only depends on ν, λ1 , |f | and R0 , |u(0)| ≤ R0 , L = (lg

λm ) + 1, λ1

and where K0 , K0󸀠 , K1 , K2 are positive constants depending on ν, λ1 and |f |. Let B = {p ∈ Hm | ‖p‖ ≤ 2M1 }, B − = {q ∈ Qm V | ‖p‖ ≤ 2M1 }, where M1 satisfies equation (3.1.13). When m is large enough, there exists a mapping Φs : B → Qm V which satisfies Φs (p) = (νA)−1 [Qm f − Qm B(p + Φs (p), p + Φs (p))],

∀p ∈ B .

(3.1.24)

Let M s = graph Φs , which is a C-analytic manifold, and contains all the stationary solutions of equation (3.1.1). Now to prove the existence of Φs , its upper bound is given.

310 | 3 The approximate inertial manifold Theorem 3.1.2. Let m be large enough such that λm+1 ≥ max{4r22 ,

r22 }. 4M12

(3.1.25)

Then there exists a unique mapping Φs : B → Qm V, which satisfies equation (3.1.24), and − 21 󵄩󵄩 s 󵄩󵄩 r1 , 󵄩󵄩Φ (p)󵄩󵄩 ≤ λm+1

(3.1.26)

where −1

1

2 r1 = ν−1 C5 8M12 L 2 + ν−1 C2 8M12 + ν−1 λm+1 |f |, 1

r2 = [ν−1 C5 2M1 L 2 + ν−1 C2 6M1 ], λ L = 1 + lg m . λ1 Proof. Suppose p ∈ B is fixed, define Tp : B ⊥ → Qm V such that Tp (q) = (νA)−1 [Qm f − Qm B(p + q, p + q)]. To prove that Tp has a unique fixed point, first we need to prove Tp : B ⊥ → B ⊥ . Let q ∈ B ⊥ , ω ∈ H, |ω| = 1, then 󵄨󵄨 21 󵄨 󵄨 −1 󵄨 −1 −1 󵄨󵄨(A Tp (q), ω)󵄨󵄨󵄨 ≤ ν [󵄨󵄨󵄨(B(p + q, p + q), A 2 Qm ω)󵄨󵄨󵄨 + |A Qm f ||ω|] 1 󵄨 󵄨 ≤ ν−1 [󵄨󵄨󵄨(B(p, p + q), A− 2 Qm ω)󵄨󵄨󵄨 1 󵄨 󵄨 −1 |f |. + 󵄨󵄨󵄨(B(q, p + q), A− 2 Qm ω)󵄨󵄨󵄨] + ν−1 λm+1 By equations (3.1.7) and (3.1.4) we get 1

|Ap| 2 󵄨 󵄨 󵄨󵄨 21 󵄨 −1 −1 ) 󵄨󵄨(A Tp (q), ω)󵄨󵄨󵄨 ≤ ν C5 ‖p + q‖󵄨󵄨󵄨A 2 Qm ω󵄨󵄨󵄨‖p‖(1 + lg 1 ‖p‖λ12

1 1 1 󵄨 1 󵄨1 + ν−1 C2 |q| 2 ‖q‖ 2 ‖p + q‖󵄨󵄨󵄨A− 2 Qm ω󵄨󵄨󵄨 2 |ω| 2 + (νλm+1 )−1 |f |

−1

1

2 ≤ ν−1 C5 8M12 λm+1 (1 + lg λm /λ1 ) 2

−1

2 + ν−1 C2 λm+1 8M12 + (νλm+1 )−1 |f |,

which yields − 21 󵄩󵄩 󵄩 r. 󵄩󵄩Tp (q)󵄩󵄩󵄩 ≤ λm+1

From equation (3.1.25), we obtain ‖Tp (q)‖ ≤ 2M1 .

(3.1.27)

3.1 Two-dimensional Navier–Stokes equation

|

311

Next we prove that Tp is contracting. We observe that 𝜕 T (q)η = (νA)−1 Qm [B(p + q, η) + B(η, p + q)], 𝜕q p

∀η ∈ Qm V,

set ω ∈ H, |ω| = 1, and then arrive at 󵄨󵄨 1 𝜕 󵄨󵄨 1 󵄨 󵄨󵄨 2 󵄨 󵄨 Tp (q)η, ω)󵄨󵄨󵄨 ≤ ν−1 󵄨󵄨󵄨(B(p, η), A− 2 Qm ω)󵄨󵄨󵄨 󵄨󵄨(A 󵄨󵄨 󵄨 𝜕q 󵄨 1 1 󵄨 󵄨 󵄨 󵄨 + ν−1 󵄨󵄨󵄨(B(q, η), A− 2 Qm ω)󵄨󵄨󵄨 + ν−1 󵄨󵄨󵄨(B(η, p + q), A− 2 Qm ω)󵄨󵄨󵄨 1 1 󵄨 󵄨 1 ≤ ν−1 C5 ‖η‖󵄨󵄨󵄨A− 2 Qm ω󵄨󵄨󵄨‖p‖(1 + lg |Ap|/λ12 ‖p‖) 2 1 1 1 󵄨 1 󵄨1 + ν−1 C2 |q| 2 ‖q‖ 2 ‖η‖󵄨󵄨󵄨A− 2 Qm ω󵄨󵄨󵄨 2 |ω| 2 1 1 1 󵄨 1 󵄨1 + ν−1 C2 |η| 2 ‖η‖ 2 ‖p + q‖󵄨󵄨󵄨A− 2 Qm ω󵄨󵄨󵄨 2 |ω| 2 1

2 λ − 21 ≤ [ν C5 2M1 (1 + lg( m )) + ν−1 C2 6M1 ]λm+1 ‖η‖, λ1

−1

which yields 󵄩󵄩 󵄩󵄩 𝜕 − 21 󵄩 󵄩󵄩 . ≤ r3 λm+1 󵄩󵄩 Tp (q)󵄩󵄩󵄩 󵄩󵄩L (Qm V) 󵄩󵄩 𝜕q

(3.1.28)

In virtue of equation (3.1.25), from (3.1.28) we deduce 󵄩󵄩 𝜕 󵄩󵄩 1 󵄩󵄩 󵄩 ≤ , 󵄩󵄩 Tp (q)󵄩󵄩󵄩 󵄩󵄩 𝜕q 󵄩󵄩L (Qm V) 2 completing the proof. From the contraction mapping principle, we deduce that there exists a unique q(p) ∈ B ⊥ such that q(p) = Tp (q). Let Φs (p) = q(p). From equation (3.1.27), we deduce that (3.1.26) is valid and M s = graph Φs is a C-analytic manifold. Every orbit u(t) = p(t) + q(t) is located in a small neighborhood of manifold M, and the global attractor is also included in this small neighborhood. Theorem 3.1.3. Suppose that m is large enough, so that equation (3.1.25) is true, then for every solution u(t) = p(t) + q(t) of equations (3.1.10)–(3.1.11), we have 󸀠 󵄩󵄩 󵄩 2K − 32 21 s L , 󵄩󵄩q(t) − Φ (p(t))󵄩󵄩󵄩 ≤ 0 λm+1 ν

∀t ≥ T∗ ,

where T∗ and K0󸀠 are the constants from Theorem 3.1.1. Proof. Let Δ(t) = q(t) − Φs (p(t)). By equations (3.1.11) and (3.1.17), we have νAΔ + Qm [B(Δ, p + Φs (p(t)) + B(p + q, Δ)] +

dq = 0. dt

(3.1.29)

312 | 3 The approximate inertial manifold Taking the inner product of the equation above and Δ on H, by (∗) we have 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 dq 󵄨 ν‖Δ‖2 ≤ 󵄨󵄨󵄨B(Δ, p + Φs (p(t)), Δ)󵄨󵄨󵄨 + 󵄨󵄨󵄨( , Δ)󵄨󵄨󵄨. 󵄨󵄨 dt 󵄨󵄨 From equation (3.1.4) we have 󵄨 󵄨 󵄩 󵄨󵄨 dq 󵄨󵄨 󵄩 ν‖Δ‖2 ≤ C2 |Δ|‖Δ‖󵄩󵄩󵄩p + Φs (p)󵄩󵄩󵄩 + 󵄨󵄨󵄨 󵄨󵄨󵄨|Δ|, 󵄨󵄨 dt 󵄨󵄨

(3.1.30)

when t > T∗ , ‖p(t)‖ ≤ M1 . By Theorem 3.1.2, we have Φs (p(t)) ≤ 2M. Substituting equation (3.1.22) into equation (3.1.30), we get −1

−1

1

2 2 ν‖Δ‖2 ≤ C2 λm+1 ‖Δ‖2 (M1 + 2M1 ) + K0󸀠 λm+1 L 2 ‖Δ‖.

From equation (3.1.25), we get ‖Δ‖ ≤

2K0󸀠 − 32 1 λ L2 , ν m+1

proving the theorem. Now we consider another approximation Φs , which can be successively approximated and explicitly solved for. We have the following theorem: Theorem 3.1.4. Assume that m is large enough, so that equation (3.1.25) is true. As in Theorem 3.1.2, we define Tp : B ⊥ → B ⊥ by Tp (q) = (νA)−1 [Qm f − Qm B(p + q, p + q)],

∀q ∈ B ⊥ .

Denote Φs (p) = Tp (0), { 0s Φn+1 (p) = Tp (Φsn (p)),

∀p ∈ B , ∀p ∈ B ; n = 0, 1, 2, . . . .

(3.1.31)

Then 1 − 21 n+1 − 21 −1 󵄨󵄨 s 󵄨 s ) λm+1 ν [|f | + 4C5 M12 L 2 ], 󵄨󵄨Φ (p) − Φn (p)󵄨󵄨󵄨 ≤ (2r2 λm+1

(3.1.32)

where r2 is defined in Theorem 3.1.2. Proof. Firstly, we note that Φs0 (p) ≜ Φ0 (p), ∀p ∈ B . By Theorem 3.1.2 and equations (3.1.28)–(3.1.35), it is easy to get − 21 n+1 󵄩 s 󵄨󵄨 s 󵄨 󵄩 s ) 󵄩󵄩󵄩Φ0 (p)󵄩󵄩󵄩. 󵄨󵄨Φ (p) − Φn (p)󵄨󵄨󵄨 ≤ 2(r2 λm+1

(3.1.33)

Hence, we only need to estimate ‖Φs0 (p)‖. From equation (3.1.31), we have Φs0 (p) = Φ0 (p) = (νA)−1 [Qm f − Qm B(p, q)],

(3.1.34)

3.1 Two-dimensional Navier–Stokes equation

|

313

which yields 1

|Ap| 2 󵄨󵄨 s 󵄨󵄨 −1 −1 2 ) , 󵄨󵄨AΦ0 (p)󵄨󵄨 ≤ ν |f | + ν C5 ‖p‖ (1 + lg 1 ‖p‖λ12 1 󵄨󵄨󵄨AΦs (p)󵄨󵄨󵄨 ≤ ν−1 |f | + ν−1 C5 4M 2 L 2 . 1 󵄨 0 󵄨 Hence 1 − 21 −1 󵄩󵄩 s 󵄩󵄩 ν [|f | + 4C5 M12 L 2 ]. 󵄩󵄩Φ0 (p)󵄩󵄩 ≤ λm+1

(3.1.35)

Combining with equations (3.1.33)–(3.1.35), we deduce that equation (3.1.32) holds. Corollary 3.1.1. Assume that m is large enough, so that equation (3.1.25) is true. Then for every solution u(t) = p(t) + q(t) of equations (3.1.10)–(3.1.11), we have 1 − 32 2K0 1 − 21 n+1 − 21 −1 󵄩󵄩 󵄩 s L 2 + 2(r2 λm+1 ) λm+1 ν [|f | + 4C5 M12 L 2 ], 󵄩󵄩q(t) − Φn (p(t))󵄩󵄩󵄩 ≤ λm+1 ν

∀t ≥ T∗ , n = 0, 1, 2, . . . ,

(3.1.36)

where Φsn is determined by equation (3.1.31); T∗ , L and K0󸀠 are determined in Theorem 3.1.1; r2 is determined in Theorem 3.1.2. Proof. This is a corollary of Theorems 3.1.3 and 3.1.4. In order to estimate the finite-dimensional approximation error of the Galerkin method, we need to prove the following lemma: Lemma 3.1.1. Assume that m is large enough, such that equation (3.1.25) is valid, then for any positive integer k ≥ m + 1, we have − 21 󵄩󵄩 󵄩 s , 󵄩󵄩Qk Φ (p)󵄩󵄩󵄩 ≤ K1 λk+1

where K1 =

16C2 M12 (1 ν

+ lg

λk 21 ) λ1

+ 1.

Proof. From equation (3.1.24), we have νAQk Φs (p) + Qk B(p + Φs (p), p + Φs (p)) = Qk f . Taking the inner product of the equation above and Φs (p) on H, we have 󵄩 󵄩2 󵄨 󵄨 ν󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨(B(p + Φs (p), p + Φs (p)), Qk Φs (p))󵄨󵄨󵄨 󵄨 󵄨 + |f |󵄨󵄨󵄨Qk Φs (p)󵄨󵄨󵄨, 󵄩 󵄩2 󵄨 󵄨 ν󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨(B(p + Pk Φs (p), p + Φs (p)), Qk Φs (p))󵄨󵄨󵄨 󵄨 󵄨 󵄩 󵄩 + 󵄨󵄨󵄨(B(Qk Φs (p), p + Φs (p)), Qk Φs (p))󵄨󵄨󵄨 + |f |󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩.

(3.1.37)

314 | 3 The approximate inertial manifold From equations (3.1.4) and (3.1.7), we have 1

λ 2 󵄨 󵄩2 󵄨 󵄩 󵄩2 󵄩 ν󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩 ≤ C5 󵄩󵄩󵄩p + Φs (p)󵄩󵄩󵄩 + 󵄨󵄨󵄨Qk Φs (p)󵄨󵄨󵄨(1 + lg k ) λ1 󵄩 󵄩 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄨󵄨󵄩󵄩 󵄨󵄨 s s s + C2 󵄨󵄨Qk Φ (p)󵄨󵄨󵄩󵄩Qk Φ (p)󵄩󵄩p + Φ (p)󵄩󵄩󵄩󵄩 + |f |󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩. Through equation (3.1.26) and by definition of B , we have 1

λ 2 − 21 󵄩 󵄩 ν󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩 ≤ C5 8M12 λk+1 (1 + lg k ) λ1 − 21 󵄩 − 21 󵄩 + C2 √8M1 λk+1 󵄩󵄩󵄩Qk Φs (p)󵄩󵄩󵄩 + λk+1 |f |. From equation (3.1.25), we can deduce that equation (3.1.37) is valid. Let k ≥ m + 1, where m be large enough and satisfies equation (3.1.25). Consider the ordinary kth order Galerkin approximation: duk + νAuk + Pk B(uk , uk ) = Pk f , dt

uk ∈ Hk .

(3.1.38)

By using Theorem 3.1.2, it is easy to prove that equation (3.1.38) possesses a unique solution Φs,k : B → Pk Qm V, which satisfies νAΦs,k (p) + Pk Qm B(p + Φs,k (p), p + Φs,k (p)) = Pk Qm f ,

∀p ∈ B .

(3.1.39)

We note that Φs,k possesses all the stationary solutions of equation (3.1.38). Lemma 3.1.2. Assume that m is large enough, so that equation (3.1.25) is true. Then for any positive integer k ≥ m + 1, we have − 21 󵄩󵄩 s 󵄩 s,k , 󵄩󵄩Φ (p) − Φ (p)󵄩󵄩󵄩 ≤ K3 λm+1

(3.1.40)

∀p ∈ B ,

where 1

(2C2 + C3 ) λ 2 − 21 K3 = [1 + 2√8M1 λm+1 (1 + lg k ) ]K1 ν λ1 and K1 is defined in Lemma 3.1.1. Proof. For p ∈ B , set u = p + Φs (p), v = p + Φs,k (p), Δ = Pk (u − v), η = Qk (u − v), u − v = Δ + η. From equation (3.1.24) and (3.1.39), we have νAΔ + Pk Qm [B(u − v, u) + B(v, u − v)] = 0,

νAΔ + Pk Qm [B(Δ + η, u) + B(v, Δ + η)] = 0. Taking the inner product of above equation and Δ on H, and using equation (∗), we get 󵄨 󵄨 󵄨 󵄨 ν‖Δ‖2 ≤ 󵄨󵄨󵄨(B(Δ + η, u), Δ)󵄨󵄨󵄨 + 󵄨󵄨󵄨(B(v, η), Δ)󵄨󵄨󵄨.

3.1 Two-dimensional Navier–Stokes equation

|

315

From equation (∗), we get 󵄨 󵄨 󵄨 󵄨 ν‖Δ‖2 ≤ 󵄨󵄨󵄨(B(Δ, u), Δ)󵄨󵄨󵄨 + 󵄨󵄨󵄨(B(η, u), Δ)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨(B(Pk v, Δ), η)󵄨󵄨󵄨 + 󵄨󵄨󵄨(B(Qk v, Δ), η)󵄨󵄨󵄨. Using equations (3.1.4), (3.1.8) and (3.1.7), we get ν‖Δ‖2 ≤ C2 |Δ|‖Δ‖‖u‖ 1

+ C5 |η|‖u‖‖Δ‖(1 + lg

λk 2 ) λ1 1

λ 2 + C5 ‖v‖‖Δ‖|η|(1 + lg k ) λ1 1

1

1

1

+ C2 |Qk v| 2 ‖Qk v‖ 2 ‖Δ‖|η| 2 ‖η‖ 2 .

(3.1.41)

By equation (3.1.26) we deduce that ‖u‖ ≤ √8M1 . Similarly, we have ‖v‖ ≤ √8M1 . From equation (3.1.41), we deduce −1

2 ν‖Δ‖ ≤ C2 λm+1 ‖Δ‖√8M1

+

− 21 C2 2√8M1 ‖η‖λm+1 (1

1

λ 2 + lg k ) λ1 1 2

λ −2 + C2 √8M1 ‖η‖λm+1 (1 + lg k ) . λ1 1

From equation (3.1.25), we get 1

1 2 2(2C5 + C2 ) √8M1 ‖η‖λ− 2 (1 + lg λk ) . ‖Δ‖ ≤ m+1 ν λ1

(3.1.42)

Finally, by equations (3.1.37) and (3.1.42), we get equation (3.1.40). Theorem 3.1.5. Assume that m is large enough, so that equation (3.1.25) is true. Then for any positive integer k ≥ m + 1, for any p ∈ B , we define Tp : B ⊥ → B ⊥ , like in Theorem 3.1.2, Tp (q) = (νA)−1 [Qm f − Qm B(p + q, p + q)],

∀q ∈ B .

Let Φs,k 0 (p) = Pk Tp (0),

Φs,k m+1 (p)

=

∀p ∈ B ,

Pk Tp (Φs,k 0 (p)),

∀p ∈ B ; n = 0, 1, 2, . . .

(3.1.43)

Then 1 − 21 n+1 − 21 −1 󵄩󵄩 s,k 󵄩 s,k ) λm+1 ν [|f | + 4C5 M12 L 2 ]. 󵄩󵄩Φ (p) − Φn (p)󵄩󵄩󵄩 ≤ 2(r2 λk+1

(3.1.44)

316 | 3 The approximate inertial manifold Furthermore, for a solution u(t) = p(t) + q(t) of equations (3.1.10)–(3.1.11), we have 󸀠 󵄩󵄩 󵄩 2K − 32 21 s,k L 󵄩󵄩q(t) − Φn (p(t))󵄩󵄩󵄩 ≤ 0 λm+1 ν −1

n+1 − 21 −1 λm+1 ν [|f |

2 + (2r2 λm+1 )

−1

2 + K2 λk+1 ,

1

+ 4C5 M12 L 2 ]

∀t ≥ T∗ , n = 0, 1, 2, . . . ,

(3.1.45)

where T∗ , L and K0󸀠 are determined in Theorem 3.1.1, r2 is determined in Theorem 3.1.2, and K3 is determined in Lemma 3.1.2. Proof. To get (3.1.44), we repeat the proof of Theorem 3.1.4, and only replace Φs with Φs,k . The estimate (3.1.45) is a direct consequence of equations (3.1.29), (3.1.40) and (3.1.44).

3.2 The Gevrey regularity of solutions Consider Navier–Stokes equations with spatial periodic boundary conditions 𝜕u − νΔu + (u ⋅ ∇)u + ∇p = f , 𝜕t ∇ ⋅ u = 0,

(3.2.1) (3.2.2)

where u = u(x, t), p = p(x, t) are unknown functions, u = {u1 , u2 } in dimension 2; u = {u1 , u2 , u3 } in dimension 3; x ∈ Rn , n = 2 or 3. The external force f is given, while ν > 0 is dynamic viscosity. Let f , u, p along any spatial direction have period 2π. Denote Ω as the cube of period (0, 2π)n and assume that u, p are periodic on Ω.

(3.2.3)

For simplicity, suppose that the average of u over Ω is zero, namely ∫ u(x, t)dx = 0,

∀t ∈ R+ .

(3.2.4)

Ω

Usually problem (3.2.1)–(3.2.4) can be written in an abstract form du + νAu + B(u) = f . dt

(3.2.5)

In the Hilbert space H, operator A (corresponding to the Stokes operator with spatial periodic boundary conditions) is a linear self-adjoint unbounded positive operator, D(A) ⊂ H. We expand u into Fourier series with respect to x, u = ∑ uj eijx , j∈Z n

uj ∈ C n ,

u−j = uj ,

u0 = 0,

(3.2.6)

3.2 The Gevrey regularity of solutions | 317

j ⋅ uj = 0,

(3.2.7)

∀j, 2 |u|2 < ∞. ∑ |uj |2 = n (2π) n j∈Z

(3.2.8)

Then equation (3.2.7) is equivalent to equation (3.2.2). The condition u0 = 0 is implied by equation (3.2.4). Consider the domain for a positive power of operator of A, D(Aα ), α > 0, that is, the set of functions u which satisfy problem (3.2.6)–(3.2.8) and 󵄨 󵄨2 (2π)n ∑ |j|2α |uj |2 = 󵄨󵄨󵄨Aα u󵄨󵄨󵄨 < ∞.

(3.2.9)

j∈Z n

s

For the given τ, s > 0, we consider the Gevrey category D(eτA ), which is the set of solutions u of equations (3.2.6)–(3.2.8) satisfying 2s 󵄨 s 󵄨2 (2π)n ∑ e2τ|j| |uj |2 = 󵄨󵄨󵄨eτA u󵄨󵄨󵄨 < ∞.

j∈Zn

(3.2.10)

Finally, in equation (3.2.5), B(u) = B(u, u), where B(u, v) is defined as n

(B(u, v), ω) = b(u, v, ω) = ∑ ∫ uj j,k=1 Ω

𝜕vk ω dx. 𝜕xj k

We supplement equation (3.2.5) with the initial condition u(0) = u0 .

(3.2.11)

Then we consider the Gevrey regularity for the solution of the initial value problem (3.2.5) and (3.2.11). 1 2

Lemma 3.2.1. Assume that u, v, ω are given in D(eτA ), τ > 0. Then B(u, v) belongs to 1 2

D(eτA ), and for the two- and three-dimensional spaces, we have 1

1

󵄨󵄨 τA 2 󵄨 τA 2 󵄨󵄨(e B(u, v), e Aω)󵄨󵄨󵄨 1 1 1 1 󵄨 2 1 󵄨 1 󵄨 2 󵄨 1 󵄩 2 1 󵄩󵄨 2 󵄨 ≤ C1 󵄨󵄨󵄨eτA A 2 u󵄨󵄨󵄨 2 󵄨󵄨󵄨eτA Au󵄨󵄨󵄨 2 󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩󵄨󵄨󵄨eτA Aω󵄨󵄨󵄨,

(3.2.12)

where C1 > 0 is an appropriate constant. Proof. Let u = ∑ uj eijx , j∈Zn

u∗ = ∑ u∗j eijx , j∈Zn

u∗ = eτ|j| uj .

Using similar symbols for v, ω, we have (B(u, v), ω) = (2π)n i ∑ (uj ⋅ k)(vk ⋅ ωl ), j−k=l

(3.2.13)

318 | 3 The approximate inertial manifold where j, k, l ∈ Zn . It follows that 1 2

1 2

(eτA B(u, v), eτA Aω) = (2π)n i ∑ (uj ⋅ k)(vk ⋅ ωl )|l|2 e2τ|l| j+k=l

= (2π) i ∑ (u∗j ⋅ k)(vk∗ ⋅ ω∗l )|l|2 eτ(|l|−|j|−|k|) . n

(3.2.14)

j+k=l

Noting that |l| − |j| − |k| = |j + k| − |j| − |k| ≤ 0, we have 1

1

2 2 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 (eτA B(u, v), eτA Aω) ≤ (2π)n ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨|k|󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨|l|2 .

(3.2.15)

j+k=l

The right-hand side of equation (3.2.15) equals the integral ∫ ξ (x)Φ(x)θ(x)dx, Ω

where

󵄨 󵄨 ξ (x) = ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨eijx , j∈Zn

󵄨 󵄨 Φ(x) = ∑ |k|󵄨󵄨󵄨νk∗ 󵄨󵄨󵄨eikx , j∈Zn

󵄨 󵄨 θ(x) = ∑ |l|2 󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨eilx . j∈Zn

This integral can be estimated by (B(u∗ , v∗ ), Aω∗ ); an estimate involving B is standard in the NS equation. So we get equation (3.2.12). 1 2

1

Theorem 3.2.1. Assume that u0 is given in D(A 2 ), and f is given in D(eσ1 A ), σ1 > 0. Then 1 there exists T∗ , which depends on the initial value u0 and |A 2 u0 |, such that the following claims are valid: (1) For the two- and three-dimensional spaces, equations (3.2.5) and (3.2.11) on (0, T∗ ) 1 possess a unique regular solution (it is continuous from [0, T∗ ] to D(A 2 )), such that 1 2

1

t → eΨ(t)A A 2 u(t) is analytic in (0, T∗ ), Ψ(t) = min(t, σ1 , T∗ ); (2) If solutions of equations (3.2.5) and (3.2.11) for any t > 0 exist and stay bounded 1 in D(A 2 ), then any such u is analytic in the interval (T∗ , ∞), its value belongs to 1

1 2

D(A 2 eσA ), where σ > 0, and T∗ is defined as before.

Proof. The main idea is to establish an a priori estimate of a solution. The rest involves using the Galerkin method to establish the approximate solution. Suppose that we take the complex framework, t = ζ ∈ C, φ(t) = min(t, σ1 ); (i) Real case. At the time τ, taking the inner product of equation (3.2.5) and Au(τ) 1 2

with respect to D(eφ(τ)A ) gives 1

1

1

2 2 2 󵄨 󵄨2 (eφ(τ)A u󸀠 (τ), Aeφ(τ)A u(τ)) + ν󵄨󵄨󵄨eφ(τ)A Au(τ)󵄨󵄨󵄨 1 2

1 2

1 2

1 2

= (eφ(τ)A f , eφ(τ)A Au(τ)) − (eφ(τ)A B(u(τ)), eφ(τ)A Au(τ)),

(3.2.16)

3.2 The Gevrey regularity of solutions | 319 1 2

where (⋅, ⋅)2 and | ⋅ |2 mean the inner product and norm in D(eτA ), respectively, but 1 2

1

((⋅, ⋅)) and ‖ ⋅ ‖2 mean the inner product and norm in D(eτA A 2 ), respectively. Then 1 2

1 2

(eφ(t)A u󸀠 (t), eφ(t)A Au(t)) 1

1 2

1 2

1 2

1

= (A 2 (eφ(t)A u(t)) − φ󸀠 (t)Aeφ(t)A u(t), eφ(t)A A 2 u(t)) 󸀠

1 1 1 1 d 󵄨󵄨 21 φ(t)A 21 2 2 󵄨2 u(t)󵄨󵄨󵄨 − φ󸀠 (t)(Aeφ(t)A u(t), eφ(t)A A 2 u(t)) 󵄨󵄨A e 2 dt 1 1 d 󵄩󵄩 󵄩2 󸀠 = 󵄩u(t)󵄩󵄩󵄩φ(t) − φ (t)(Au(t), A 2 u(t))φ(t) 2 dt 󵄩

=

1 d 󵄩󵄩 󵄩2 󵄨 󵄨 󵄩 󵄩 󵄩u(t)󵄩󵄩󵄩φ(t) − 󵄨󵄨󵄨Au(t)󵄨󵄨󵄨φ(t) 󵄩󵄩󵄩u(t)󵄩󵄩󵄩φ(t) 2 dt 󵄩 ν󵄨 1󵄩 1 d 󵄩󵄩 󵄩2 󵄨2 󵄩2 ≥ 󵄩u(t)󵄩󵄩󵄩φ(t) − 󵄨󵄨󵄨Au(t)󵄨󵄨󵄨φ(t) − 󵄩󵄩󵄩u(t)󵄩󵄩󵄩φ(t) . 2 dt 󵄩 4 ν ≥

(3.2.17)

The right-hand side of equation (3.2.16) can be written as (f , Au)φ − (B(u), Au)φ .

(3.2.18)

By Lemma 3.2.1, (3.2.12) and Schwarz inequality, we get the bound of equation (3.2.18), namely 3

3

|f |φ |Au|φ + C1 ‖u‖φ2 |Au|φ2 ≤

C ν 2 |Au|2φ + |f |2φ + 32 ‖u‖6φ , 4 ν ν

where C1 , C2 and Ci , Ci󸀠 are constants. Then by equation (3.2.16) we have 2C d 4 2 ‖u‖2φ + ν|Au|2φ ≤ |f |2φ + ‖u‖2φ + 32 |u|6φ dt ν ν ν 3C2 2 4 2 ≤ |f |φ + C3 + 3 |u|φ . ν ν

(3.2.19)

Finally, we get y󸀠 ≤ K1 y3 , 󵄩 󵄩2 y(t) = 1 + 󵄩󵄩󵄩u(t)󵄩󵄩󵄩φ ,

1

(3C2 ) 3 4 . K1 = |f |2σ1 + C3 + ν ν

(3.2.20)

Hence 1

1 2 󵄨 󵄨2 󵄨2 󵄨 1 y(t) = 1 + 󵄨󵄨󵄨eφ(t)A A 2 u(t)󵄨󵄨󵄨 ≤ 2y(0) = 2 + 2󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨 , 2 󵄨 󵄨 1 󵄨2 −2 󵄨 1 (1 + 󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨 ) , t ≤ T1 (󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨) = K1 1 2

(3.2.21) (3.2.22)

1

and then u(t) belongs to D(eφ(t)A A 2 ), so equation (3.2.21) is established when t ∈ 1 (0, T1 ), u(0) ∈ D(A 2 ). In particular, 1

󵄨2 󵄨 1 󵄨2 󵄨󵄨 φ(T1 )A 2 21 A u(T1 )󵄨󵄨󵄨 ≤ 2 + 2󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨 . 󵄨󵄨e

(3.2.23)

320 | 3 The approximate inertial manifold If we know that 󵄨 󵄨󵄨 21 󵄨󵄨A u(t)󵄨󵄨󵄨 ≤ M,

∀t ≥ 0,

(3.2.24)

then equation (3.2.23) for two-dimensional space is always true, while in dimension 3 must give some assumptions. Next we repeat the above principle: for any t0 > 0, 1

󵄨󵄨 σ2 A 2 21 󵄨2 A u(t)󵄨󵄨󵄨 ≤ 2 + 2M12 , 󵄨󵄨e

∀t ≥ T2 ,

(3.2.25)

where σ2 = φ(T2 ) = min(T2 , σ1 ), and we have T2 = T2 (M1 ) =

2 −2 (1 + M12 ) . K1

(3.2.26)

(ii) Complex case. In order to get the analyticity of time t, we consider equation (3.2.5) with complex time ζ ∈ C. Then u is a complex function and H is the complexification of the original space H. The inner product, norm and operators A, B can be also extended accordingly. Equation (3.2.5) can then be written as du + νAu + Bu = f , dζ

(3.2.27)

where ζ = seiθ , s > 0, cos θ > 0, hence Re ζ > 0. Taking the inner product of equation 1 2

(3.2.27) and Au(seiθ ) with respect to D(eφ(s cos θ)A ), multiplying this equation by e−iθ , and taking its real part, we get Re e−iθ (eφ(s cos θ)A 1

= Re (A 2

1 2

du iθ φ(s cos θ)A 21 (se ), e Au(seiθ )) dζ

d φ(s cos θ)A 21 (e u(seiθ )) ds

1 2

1

− (e−iθ φ󸀠 (s cos θ) cos θAu(seiθ ), eφ(s cos θ)A A 2 u(seiθ ))) 1 d 󵄨󵄨 21 iθ 󵄨2 󵄨A u(se )󵄨󵄨󵄨φ(s cos θ) 2 ds 󵄨 󵄨 󵄨 󵄨 󵄨 1 − cos2 θφ󸀠 (s cos θ)󵄨󵄨󵄨Au(seiθ )󵄨󵄨󵄨φ(s cos θ) 󵄨󵄨󵄨A 2 u(seiθ )󵄨󵄨󵄨φ(s cos θ) 1 d 󵄨󵄨 21 ν cos θ 󵄨󵄨 iθ 󵄨2 iθ 󵄨2 ≥ 󵄨󵄨A u(se )󵄨󵄨󵄨φ(s cos θ) − 󵄨󵄨Au(se )󵄨󵄨󵄨φ(s cos θ) 2 ds 4 1 󵄩󵄩 󵄩2 iθ 󵄩 − 󵄩u(se )󵄩󵄩φ(s cos θ) ν cos θ 󵄩 =

1 2

1 2

≥ Re e−iθ (eφ(s cos θ)A Au(seiθ ), eφ(s cos θ)A Au(seiθ )) = cos θ|Au(seiθ )|2φ(s cos θ) .

3.2 The Gevrey regularity of solutions | 321

Replacing φ with φ(s cos θ) (s can be ignored if necessary), we get 1 d 3 1 ‖u‖2φ + ν cos θ|Au|2φ − ‖u‖2φ 2 ds 4 ν cos θ ≤ Re e−iθ (f , Au)φ − Re e−iθ (B(u), Au)φ .

(3.2.28)

The right-hand side of inequality (3.2.28) can be estimated as follows: 3

3

|f |φ |Au|φ + C1 ‖u‖φ2 ‖Au‖φ2 ≤

C4 ν cos θ 2 ‖Au‖2φ + ‖f ‖2 + ‖u‖6φ . 4 ν cos θ φ (ν cos θ)3

Similar to equation (3.2.19), we get 2C4 d 4 2 ‖u‖2 + ν cos θ|Au|2φ ≤ ‖f ‖2 + ‖u‖2φ + ‖u‖6φ ds φ ν cos θ φ ν cos θ (ν cos θ)3 4C4 4 ≤ ‖f ‖2 + C5 + ‖u‖6φ , ν cos θ φ (ν cos θ)3

(3.2.29)

where u = u(seiθ ), φ = φ(s cos θ). If we restrict cos θ ≥ √2/2, then we obtain equations whose form is similar to equation (3.2.20), i. e., dy ≤ K2 y3 , ds 1 1 2 󵄨 󵄨2 y(s) = 1 + 󵄨󵄨󵄨eφ(s cos θ)A A 2 u(seiθ )󵄨󵄨󵄨 , K2 =

25 C 8 2 |f |σ1 + C5 + 3 4 , ν ν

(3.2.30)

then also y(s) ≤ 2y(0). That is, 1

󵄨󵄨 φ(s cos θ)A 2 21 󵄨2 󵄨2 󵄨 1 A u(seiθ )󵄨󵄨󵄨 ≤ 2 + 2󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨 , 󵄨󵄨e

(3.2.31)

where 2 󵄨 1 󵄨 󵄨2 −2 󵄨 1 [1 + 󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨 ] , 0 ≤ s ≤ T3 (󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨) = K2 √2 ≤ cos θ ≤ 1. 2

(3.2.32) (3.2.33)

1

This implies that, if u(0) ∈ D(A 2 ), then the complex angle region which is determined 1 2

1

by u(seiθ ) in equations (3.2.32)–(3.2.33) belongs to D(eφ(s cos θ)A A 2 ). In particular, 1

󵄨󵄨 σ3 A 2 21 󵄨2 󵄨 1 󵄨2 A u(T3 eiθ )󵄨󵄨󵄨 ≤ 2 + 2󵄨󵄨󵄨A 2 u(0)󵄨󵄨󵄨 . 󵄨󵄨e From this we have 1 ≥ cos θ ≥ √2/2, σ3 = φ(T3 , √2/2) = min(T √2/2, σ1 ).

(3.2.34)

322 | 3 The approximate inertial manifold If equation (3.2.24) is valid, then we can repeat the above process, such that for any t0 > 0, 1

󵄨2 󵄨󵄨 σ4 A 2 21 A u(seiθ + t0 )󵄨󵄨󵄨 ≤ 2 + 2M12 , 󵄨󵄨e

(3.2.35)

where 0 ≤ s ≤ T4 , √2/2 ≤ cos θ ≤ 1, T4 = T4 (M1 ) =

2 −2 (1 + M12 ) , K2

(3.2.36)

σ4 = φ(T4 , √2/2) = min(T4 √2/2, σ1 ), in particular the estimate (3.2.25) holds for φ = seiθ + t0 . On the domain Δ(M1 ), 󵄨󵄨 󵄨 √2 T (M ) 󵄨󵄨Im(s)󵄨󵄨󵄨 ≤ 2 4 1

Re(s) ≥ T4 (M1 ),

(3.2.37)

are established. Here T∗ = T4 (M1 ),

σ = min((√2/2)T∗ , σ1 ).

Thus the theorem has been proved. 1 2

Remark 3.2.1. (i) If f depends on t, and f is analytic with respect to time t in D(eσ1 A ), then the analyticity domain of u is the intersection of Δ(M1 ) and the analyticity domain of f . (ii) In the proof of Lemma 3.2.1, for the left-side estimate of equation (3.2.12), we can use another estimate for B. For instance, for two-dimensional space, from 󵄨󵄨 󵄨 1 ∗ 󵄨󵄨 1 ∗ 󵄨󵄨 ∗ 󵄨 ∗ ∗ ∗ 󵄨 󵄨󵄨(B(u , v ), ω )󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨A 2 u 󵄨󵄨󵄨󵄨󵄨󵄨A 2 v 󵄨󵄨󵄨󵄨󵄨󵄨Aω 󵄨󵄨󵄨(1 + lg

|Au∗ |2 1

4π 2 |A 2 u∗ |2

)

1 2

we have 1

1

󵄨󵄨 τA 2 󵄨 τA 2 󵄨󵄨(e B(u, v), e ω)󵄨󵄨󵄨 ≤

1

󵄨 2 C1󸀠 󵄨󵄨󵄨eτA

1

1

󵄨󵄨 2 1 󵄨󵄨 2 󵄨 A u󵄨󵄨󵄨󵄨󵄨󵄨eτA A 2 v󵄨󵄨󵄨󵄨󵄨󵄨eτA Aω󵄨󵄨󵄨(1 + lg 1 2

1 2

|eτA Au|2 1

1

4π 2 |eτA 2 A 2 u|2

1 2

) .

(3.2.38)

Replacing equation (3.2.38) with equation (3.2.12) could increase the size of T4 (M1 ) and area (3.2.37). If which replace with equation (3.2.39), we get dy ≤ C7 (ν cos θ)−1 y2 lg y. ds

(3.2.39)

Hence, if equation (3.2.24) is satisfied, we can get equation (3.2.36) for any t0 ≥ 0. For any s, θ satisfying 0 ≤ s ≤ T5 (M1 ), √2/2 ≤ cos θ ≤ 1, T5 (M1 ) =

C8

( ν1 )(|f |2σ1 + M12 ) − lg((1/ν2 )(|f |2σ1 + M12 ))

.

(3.2.40)

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 323

The analyticity domain of u contains Δ󸀠 (M), similar to Δ(M1 ); T5 is replaced 1 1 with T4 . The analyticity domain is in D(A 2 eσ5 A 2 ), σ5 = min((T5 √2/2), σ1 ). 1 2

(iii) Let u∗ (t) = eφ(t)A u(t). Then from equation (3.2.5) we get the following equation for u∗ : 1 du∗ + νAu∗ − φ󸀠 A 2 u + e−φ B(u∗ ) = f ∗ . dt

(3.2.41)

The related results in this aspect can refer to the reference [121, 122].

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations Suppose that H is an infinite-dimensional Hilbert space, which possesses inner product (⋅, ⋅) and norm | ⋅ |, A is a given unbounded, positive self-adjoint linear operator, D(A) ⊂ H, A−1 is compact in H, a nonlinear operator R : D(A) → H on D(A) is analytic in a finite-dimensional subspace, which can be extend in accordance with the complexity of the subspace. Because A−1 is compact in H, there exists a basis of functions {ωj } of H, which are the eigenvectors of A. Namely, there exists an orthogonal basis in H of eigenfunctions of A with corresponding set of eigenvalues λj ∈ R+ , Aωj = λj ωj , |ωj | = 1,

0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ .

When j → ∞, λj → +∞. Likewise we can define powers of A, namely As , s ∈ R. Then As 1

maps D(As ) to H. In particular, for V = D(A 2 ), its norm and inner product are denoted by ‖⋅‖ and ((⋅)), respectively, and |⋅|1 = (‖⋅‖2 +|⋅|2 )2 . For the vector u, Au = (Au1 , . . . , Auk ). Consider the following abstract initial value problem: u󸀠 (t) + DAu(t) + R(u(t)) = f (t), u(0) = u0 ,

t ∈ R+ , t = 0,

(3.3.1) (3.3.2)

where u is a vector function R+ → D(A), f : R+ → H is analytic, D = (dij )k×k is a real positive-definite matrix. Use Hc , Vc and D(A)c to denote the complexification of H, V and D(A), respectively. For example, Hc is the subset of H ⊕ H which consists of the basic elements h1 + ih2 , h1 , h2 ∈ R, A(h1 + ih2 ) = Ah1 + iAh2 , (h1 + ih2 , g1 + ig2 ) = (h1 , g1 ) + (h2 , g2 ) + i[(h2 , g1 ) − (h1 , g2 )],

324 | 3 The approximate inertial manifold A−1 is compact in Hc . In fact, using the orthogonal basis {ωj } of space H, it is easy to construct the Hc ’s orthogonal basis {ωj } which is formed from the eigenvectors of A. Suppose that 1 ≤ γ < ∞, K > 0, and the function C ∈ C((0, ∞); R+ ) is such that for any ε > 0, R satisfies 2γ 󵄨 󵄨󵄨 2 󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 ≤ ε|Au| + C(ε)|u|1 + K,

∀u ∈ D(A)c .

(3.3.3)

When vm → v from X to D(A) converge in the weak topology, R(vm ) → R(v)

weakly with respect to L2 (X, H).

(3.3.4)

Suppose there exists M > 0 such that f satisfies 󵄩󵄩 󵄩 󵄩󵄩f (t)󵄩󵄩󵄩∗ ≤ M,

∀t ∈ R+ .

(3.3.5)

Theorem 3.3.1. Suppose u is a solution of equations (3.3.1)–(3.3.2). Under the above assumptions (3.3.3), (3.3.4) and (3.3.5), there exists θ0 , |θ0 | ≤ π4 , and function T1 ∈ C(R+ , R− ) such that if |u(0)|1 is finite, then u takes values in D(A)c and can be extended analytically to 󵄨 󵄨 Δ(󵄨󵄨󵄨u(0)󵄨󵄨󵄨1 ) = {z = seiθ | |θ| ≤ θ0 , 0 ≤ s ≤ T1 (|u0 |1 )}.

(3.3.6)

Furthermore, if |u(t)|1 is constrained by B for t ∈ (a, b), then the above set can be extended to Δ = ⋃ t + Δ(B). t∈(a,b)

For any compact set K included in the area of analyticity, the following inequalities hold: 󵄨󵄨 dk u 󵄨󵄨 1 1 󵄨 󵄨 sup 󵄨󵄨󵄨 k (z)󵄨󵄨󵄨 ≤ 2 2 (2/d)k (k!)(1 + |u0 |21 ) 2 , 󵄨󵄨1 z∈K 󵄨󵄨 dz 󵄨󵄨 󵄨 sup 󵄨󵄨Au(z)󵄨󵄨󵄨 ≤ T2 (K) < ∞, z∈K

d = dist(K, 𝜕Δ),

󵄨󵄨 󵄨󵄨 dk u −k 󵄨 󵄨 sup 󵄨󵄨󵄨A( k (z))󵄨󵄨󵄨 ≤ 2k (k!)[d(K, 𝜕Δ(u0 ))] T2 (K 󸀠 ), 󵄨 󵄨 dz 󵄨 z∈K 󵄨

(3.3.7) (3.3.8) (3.3.9)

where K 󸀠 ≜ {z ∈ Δ(u0 ) | d(z, 𝜕Δ(u0 )) ≥ 21 d(K, 𝜕Δ(u0 ))}. Proof. Considering the Galerkin approximation of complex time, for the complex differential equations on Hm ≜ {Cω1 + ⋅ ⋅ ⋅ + Cωm }. Let Pm be the projection of Hm . We seek solutions m

um (z) = ∑ gi (z)ωi , i=1

gi : C → C

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 325

𝜕um + DAum + R(um ) − f (t), v) = 0, 𝜕z um (0) = Pm u0 .

∀v ∈ Hm ,

(

(3.3.10) (3.3.11)

Because on Hm the operator A possesses a very simple form, namely Aum (z) = ∑ λi gi (z)ωi , equations (3.3.10)–(3.3.11) boil down to m ordinary differential equations. Based on the well-known Cauchy–Kovalevskaya theorem, we obtain a unique analytic solution in a complex neighborhood of the origin. Moreover, um can also be used in the Galerkin approximation of the real time problem (3.3.1)–(3.3.2) (restriction on the real axis). Now we make an a priori estimate. Setting v = Aum + um in equation (3.3.10) and ignoring the subscript m, we have (

𝜕u 𝜕u , Au) + ( , u) + (DAu, Au) + (DAu, u) 𝜕z 𝜕z + (R(u), Au + u) − (f (t), Au + u) = 0.

iθ

− π2

π . 2

(3.3.12)

iθ

Let z = se , ≤θ≤ Multiply equation (3.3.12) by e and take its real part. Since D is a real positive matrix, there exists α0 ≥ 0, such that Re(Dz, z) ≥ α0 |z|2 , ∀z ∈ Cn . Then we get 1 d (‖u‖2 + |u|2 ) + α0 cos θ(|Au|2 + ‖u‖2 ) 2 ds ≤ sin θ‖D‖∗ (|Au|2 + ‖u‖2 ) 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 + 󵄨󵄨󵄨(f (t), Au + u)󵄨󵄨󵄨. Using condition (3.3.3), taking ε = (α0 cos θ)/8, and combining with condition (3.3.5), we get 1 d (‖u‖2 + |u|2 ) + α0 cos θ(|Au|2 + ‖u‖2 ) 2 ds ≤ |sin θ|‖D‖∗ (|Au|2 + ‖u‖2 ) + (α0 /8) cos θ|Au|2 2γ

+ K + C(θ)|u|1 + M(|Au| + |u|).

(3.3.13)

Further we restrict θ so that α0 cos θ ≥ ‖D‖∗ |sin θ|, for example, we can select π ). 4

|θ| ≤ min(arctan(α0 /4‖D‖∗ ) ⋅

Hence, using Young inequality in the last term of equation (3.3.13), we get 1 d 2 |u| + α0 cos θ(|Au|2 + ‖u‖2 ) 2 ds 1 ≤ (α0 /4) cos θ[|Au|2 + ‖u‖2 ]

2γ

+ (α0 /8) cos θ|Au|2 + K + C(θ)|u|1

+ 4M 2 /(α0 cos θ) + (α0 /8) cos θ|Au|2 + M|u|2 ,

(3.3.14)

326 | 3 The approximate inertial manifold which can be reduced to d 2 |u| + α0 cos θ(|Au|2 + ‖u‖2 ) ds 1 2γ ≤ C(θ)|u|1 + 4M 2 /(α0 cos θ) + K.

(3.3.15)

If θ satisfies equation (3.3.14), then the coefficient in the bound of equation (3.3.15) is independent of θ. Thus d 󵄨󵄨 󵄨 iθ 󵄨2 2 2 iθ 󵄨2γ 󵄨u(se )󵄨󵄨󵄨1 + C1 (|Au| + ‖u‖ ) ≤ C2 + C3 󵄨󵄨󵄨u(se )󵄨󵄨󵄨1 . ds 󵄨

(3.3.16)

Ignoring C1 (|Au|2 + ‖u‖2 ), and denoting y(s) = 1 + |u(seiθ )|21 , we get the differential inequality y󸀠 (s) ≤ C4 yγ (s),

s ≥ 0,

where C4 = max{C2 , C3 }. Integrating the above inequality, we obtain 1/(1−γ)

0 < y(s) ≤ (y(0)1−γ − (γ − 1)C4 s) 1−γ

0 ≤ s < y(0)

,

−1

[(γ − 1)C4 ] .

This means that there exists a constant T1 = y(0)1−γ (1 − ( 21 )γ−1 )/((γ − 1)C4 ), which depends on |u0 |1 , but not on m, such that 1 1 󵄨󵄨 󵄨 2 iθ 󵄨 2󵄨 󵄨󵄨um (se )󵄨󵄨󵄨1 ≤ 2(1 + 󵄨󵄨󵄨um (0)1 󵄨󵄨󵄨) 2 ≤ 2(1 + |u0 |1 ) 2 , ∀θ satisfies equation (3.3.14), 0 ≤ s ≤ T1 (|u0 |1 ),

(3.3.17)

where we have recovered the subscript m. So from the existence of solutions for ordinary differential equations, we can deduce that um can be extended to an analytic solution of equation (3.3.16) in the set Δ(|u(0)|1 ) = {z = seiθ | 0 < s < T1 (|u0 |1 ), θ satisfies the equation (3.3.14)}, shown in Figure 3.1. Namely, we have 1 󵄨 󵄨 sup 󵄨󵄨󵄨um (z)󵄨󵄨󵄨1 ≤ 2(1 + |u0 |21 ) 2 ,

z ∈ Δ(|u0 |1 ).

Furthermore, by Cauchy formula we have dk um = (k!)/(2πi) dz k

∫

um (η)(η − z)−(k+1) dη,

|z−η|= d2

where d = d(z, 𝜕Δ(u0 )). Hence 󵄨󵄨 dk u 󵄨󵄨 󵄨󵄨 󵄨 󵄨 k m 󵄨󵄨 󵄨󵄨 󵄨 ≤ (2/d) (k!) sup 󵄨󵄨󵄨um (z)󵄨󵄨󵄨1 . 󵄨󵄨 dz k 󵄨󵄨󵄨1 z∈Δ(|u0 |1 )

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 327

Figure 3.1

In particular, for any K which is a compact subset of Δ(|u0 |1 ), we have 󵄨󵄨 dk u 󵄨󵄨 1 1 󵄨 󵄨󵄨 k 2 m sup 󵄨󵄨󵄨 (z) 󵄨󵄨 ≤ 2 2 (2/d) (k!)(1 + |u0 |1 ) 2 , k 󵄨 󵄨 󵄨1 z∈K 󵄨 dz

(3.3.18)

where d = d(K, 𝜕Δ(|u0 |1 )). In order to get a bound of um in D(A), for any compact subset K of Δ(|u0 |1 ), and any z = seiθ ∈ K, we have 󵄨󵄨 d 󵄨 󵄨󵄨 du 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 iθ 󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨um (se )󵄨󵄨󵄨1 󵄨󵄨󵄨 = 󵄨󵄨󵄨2( m , um )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 dz 󵄨󵄨 ds 󵄨󵄨 du 󵄨󵄨 󵄨 󵄨 ≤ 2󵄨󵄨󵄨 m 󵄨󵄨󵄨 |um |1 ≤ 4(2/d)(1 + |u0 |21 ), 󵄨󵄨 dz 󵄨󵄨1

(3.3.19)

where d = d(K, 𝜕Δ(|u0 |1 )). Inserting this into equation (3.3.15), and ignoring ‖um ‖2 , we get γ

C1 |Aum |2 ≤ C2 + C3 (2(1 + |u0 |21 ) ) + 4(2/d)(1 + |u0 |21 ).

(3.3.20)

Hence, for any compact subset K ⊂ Δ(|u0 |1 ), um in L∞ (K, D(A)) is uniformly bounded (is independent of m). 󵄨 󵄨 sup 󵄨󵄨󵄨Aum (z)󵄨󵄨󵄨 ≤ T2 < ∞, z∈K

T2 = T2 (K).

By Cauchy formula, we get, for any compact subset K, that when d = d(K, 𝜕Δ(|u0 |1 )) > d > 0, A(dk um /dz k )(z) = (k!)/(2πi)

∫ |z−η|= d2

Aum (η)/(η − z)k+1 dη,

∀z ∈ K.

328 | 3 The approximate inertial manifold Letting K 󸀠 be the compact subset {z ∈ Δ(u0 ) | d(z, 𝜕Δ(|u0 |1 )) ≥ 21 d(K, 𝜕Δ(|u0 |1 ))} ⊃ K, we have −k 󵄨 󵄨 󵄨 󵄨󵄨 k k k 󵄨󵄨A(d um /dz )(z)󵄨󵄨󵄨 ≤ 2 (k!)[d(z, 𝜕Δ(u0 ))] sup 󵄨󵄨󵄨Aum (z)󵄨󵄨󵄨, 󸀠 z∈K

−k 󵄨 󵄨 sup 󵄨󵄨󵄨A(dk um /dz k )(z)󵄨󵄨󵄨 ≤ 2k (k!)[d(z, 𝜕Δ(u0 ))] T2 (K 󸀠 ). z∈K

(3.3.21)

Now we take the limit m → ∞. The functions um : C → D(A) on Δ(|u0 |1 ) are uniformly bounded and analytic. On the basis of classical Montel theorem, for the sequence {um }, we can select a subsequence which converges uniformly to function a u∗ with respect to the D(A) norm on any compact set Δ(u0 ), which is D(A) analytic in Δ(u0 ). Furthermore, because um |0,T corresponds to the real Galerkin approximation, um |0,T → u is a solution of equation (3.3.31)–(3.3.32). Thus u∗ is just the analytic continuation of u on Δ(u0 ). So u∗ is the unique analytical continuation of u on Δ(u0 ), u∗ = u. Therefore any subsequence of {um } converges to u, and in fact the whole sequence converges to u. Then we have 󵄨 󵄨 sup 󵄨󵄨󵄨Au(z)󵄨󵄨󵄨 ≤ T2 (K), z∈K

∀ compact set K ⊂ Δ(u0 ).

Similarly, by equations (3.3.18)–(3.3.21) we deduce that dk um dk u → dz k dz k is consistently convergent in Δ(u0 ) with respect to D(A). Since u satisfies equations (3.3.38)–(3.3.39), from the results on the uniform convergence, it is readily seen that u is the solution of equations (3.3.31)–(3.3.32). If |u|1 ≤ R, t ∈ (α, β), then we can repeat the principle at t = 0 to get the same at any point t ∈ (α, β), and we have that u : C → D(A)c is a D(A)-valued analytic function on the open set 󵄨 󵄨 ⋃ [t + Δ(󵄨󵄨󵄨u(t)󵄨󵄨󵄨1 )] ⊃ ⋃ [t + Δ(R)].

t≥0

t∈(α,β)

Remark 3.3.1. Under the assumptions of Theorem 3.3.1, for 0 < t ≤ ( 43 )T1 (|u0 |1 ), we have 󵄨󵄨 dk u 󵄨󵄨 1 󵄨󵄨 󵄨 2 k −k 1 󵄨󵄨 k (t)󵄨󵄨󵄨 ≤ 2 k!C (2 2 )(1 + |u0 |1 ) 2 , 󵄨󵄨 dz 󵄨󵄨1 󵄨󵄨 dk u 󵄨󵄨 1 k −k 2 γ 2 2 󵄨󵄨󵄨A 󵄨󵄨 󵄨󵄨 dz k (t)󵄨󵄨󵄨 ≤ 2 k!C (C2 + C3 (2(1 + |u0 |1 ) ) + 4(4/C)(1 + |u0 |1 ) C1 ), 󵄨 󵄨 1

(3.3.22) (3.3.23)

where C = (tm /(1 + m)2 ) 2 is the distance from t to 𝜕Δ, m = α0 /4‖D‖∗ , C1 , C2 and γ are constants. In the following, we list some specific applications of Theorem 3.3.1.

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 329

Example: The reaction diffusion equation Let Ω ⊂ Rn be a bounded open set. The reaction diffusion equation is written as 𝜕u − DΔu + g(u) = 0, 𝜕t u(0) = u0 ,

(3.3.24) (3.3.25)

where u = (u1 , u2 , . . . , uk ) is the vector function defined on Ω × R+ , D is a positive diagonal matrix, the diagonal elements of which are d1 , d2 , . . . , dk . The function g : Rk → Rk has its components of rth order (r ≥ 2) polynomial form: α

α

gi (x) = ∑ Cαi x1 1 ⋅ ⋅ ⋅ xk k .

(3.3.26)

|α|≤r

Supplement one of the following boundary conditions: u(x, t) = 0,

x ∈ 𝜕Ω, t > 0,

(3.3.27)

n

u(⋅, t) is Ω periodic, t ≥ 0, Ω = (0, L) , 𝜕u (x, t) = 0, ∀x ∈ 𝜕Ω, ∀t ≥ 0. 𝜕n

(3.3.28) (3.3.29)

Corresponding to the above boundary conditions, we have H = L2 (Ω),

L2 (Ω),

L2 (Ω),

V = H01 (Ω),

1 Hper (Ω),

H 1 (Ω),

D(A) = H 2 (Ω) ∩ H01 (Ω),

2 Hper (Ω),

{ν ∈ H 2 (Ω) |

𝜕v = 0}. 𝜕n

Write equation (3.3.24) in the form of equation (3.3.31), so that we have A = −Δ, R(u) = g(u), u ∈ D(A), and consider that R(u) satisfies condition (3.3.3) and inequality α 󵄨2 󵄨󵄨 󵄨 󵄨 i α 󵄨󵄨R(u)󵄨󵄨󵄨 = ∫ ∑ 󵄨󵄨󵄨(Cα x1 1 ⋅ ⋅ ⋅ uk k )󵄨󵄨󵄨 dx 1≤i≤k

α 󵄨2 α 󵄨 ≤ C ∫ ∑ ∑ 󵄨󵄨󵄨(Cαi x1 1 ⋅ ⋅ ⋅ uk k )󵄨󵄨󵄨 dx ≤ C ∫ ∑ ∑ Cαi ‖u‖2|α dx. 1≤i≤k |α|≤γ

1≤i≤k |α|≤γ

We can find constants C1 and C2 , which are independent of i, such that 󵄨 󵄨 ∑ 󵄨󵄨󵄨Cαi 󵄨󵄨󵄨‖u‖2|α ≤ C1 |u|2γ + C2 .

|α|≤γ

Hence γ 󵄨󵄨 󵄨 󵄨󵄨R(u)󵄨󵄨󵄨 ≤ C(|u|L2γ ) + C|Ω|, 󵄨 󵄨󵄨 α 󵄨󵄨 󵄨 󵄨󵄨 󵄨 i α 󵄨󵄨R(u), Au + u󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ ∑ ∑ (Cα u1 1 ⋅ ⋅ ⋅ uk k )(−Δui + ui )dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 1≤i≤k |α|≤γ

330 | 3 The approximate inertial manifold 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨 ∑ {∫ ∑ 󵄨󵄨󵄨Cαi 󵄨󵄨󵄨|u||α| |Δui |dx + ∫ ∑ 󵄨󵄨󵄨Cαi 󵄨󵄨󵄨|u||α|+1 dx}󵄨󵄨󵄨 1≤i≤k

|α|≤γ

|α|≤γ

≤ ∑ { ∫(C1 |u|γ |Δui | + C2 |Δui |)dx + ∫(C3 |u|γ+1 + C4 )dx} 1≤i≤k

γ

γ+1

1

≤ C1 (|u|L2γ ) |Au| + C2 |Ω| 2 |Au| + C3 (|u|Lγ+1 )

+ C4 |Ω|.

(3.3.30)

When n = 1 or n = 2, we have |u|Lp ≤ Cp |u|H 1 ,

p ≥ 1.

Thus, using Young inequality again, we obtain 1 γ+1 γ 󵄨󵄨 󵄨 󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 ≤ C1 (C2γ |u|1 ) |Au| + C2 |Ω| 2 |Au| + C3 (Cγ+1 |u|1 ) + C4 |Ω| 1 2γ γ+1 ≤ ε|Au|2 + C( )(|u|1 ) + |Ω| + C(|u|1 ) + C1 |Ω|. ε

Therefore the condition of equation (3.3.3) holds. At this point γ is an arbitrary positive integer. When n ≥ 3, we have the following lemma: Lemma 3.3.1. There exist σ and τ which satisfy 1 > τ, σ ≥ 0, rσ < 1, (r + 1)τ < 2 such that r

rσ/2

(|u|L2r ) ≤ C(r, n, Ω)(|Au|2 + |u|2 ) r+1

(|u|Lr+1 )

1 ≤ r < (n + 2)/(n − 2), 2

(|u|1 )

2 (r+1)r/2

≤ C(r, n, Ω)(|Au| + |u| )

1 ≤ r < (n + 6)/(n − 2).

r(1−σ)

(|u|1 )

,

(r+1)(1−r)

(3.3.31) , (3.3.32)

Proof. First, we have the embedding theorem: H σ+1 is embedded into L2r , when 1 1 ≥ − (1 + σ)/n, 2r 2

2r ≤ 2n/(n − 2(1 + σ)).

(3.3.33) 1

By using H 1+σ which can be obtained by interpolation of H 1 and H 2 , and (|Au|2 + |u|2 ) 2 which is equivalent to the H 2 -norm: σ

1+σ

|u|H 1+σ ≤ C(|u|H 2 ) (|u|H 1 ) r

2 rσ/2

(|u|H 1+σ ) ≤ C(|Au|2 + |u| )

,

(|u|H 1 )

as long as rσ < 1, we have 1 2r < 2n/(n − 2)(1 + ), r r < (n + 2)/(n − 2).

r(1−σ)

,

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 331

Equation (3.3.31) has been proved. For equation (3.3.32), by Sobolev embedding theorem, H 1+τ embeds into Lr+1 , when 1 − (1 + τ)/n, 2 r + 1 ≤ 2n/(n − 2(1 + τ)).

1/(r + 1) ≥

(3.3.34)

Considering H 1+τ , which can be obtained by interpolating H 1 and H 2 , we arrive at 1+r

(|u|H 1+τ )

≤ C(|Au|2 + |u|2 )

(r+1)τ/2

(r+1)(1−τ)

(|u|H 1 )

.

Taking (r + 1)τ/2 < 1, that is, (r + 1)τ < 2, and substituting into equation (3.3.34), we have r + 1 < 2n/(n − 2(1 + τ/(r + 1))), r < (n + 6)/(n − 2), which yields equation (3.3.32). By use of Lemma 3.3.1, we get 1 r(1−s) 󵄨󵄨 󵄨 2 2 rs/2 |Au| + C2 |Ω| 2 |Au| 󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 ≤ C(|Au| + |u| ) (|u|1 )

+ C(|Au|2 + |u|2 )

(r+1)τ/2

(|u|1 )

(r+1)(1−τ)

+ C4 |Ω|

1 r(1−s) 󵄨󵄨 󵄨 C rs+1 + |u|rs )(|u|1 ) + C2 |Ω| 2 |Au| 󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 ≤ (|Au| 2

+ C(|Au|(r+1)τ + |u|(r+1)τ )(|u|1 )

(r+1)(1−τ)

+ C4 |Ω|.

From this we can obtain Theorem 3.3.2. In dimension 1 or 2, there exist θ0 , |θ0 | ≤ π4 , and function T1 ∈ C(R+ , R+ ), which depends on g and spacial dimension, such that if for any t, |u(t)|1 is finite, then u has D(A)c analytic extension to the complex domain 󵄨 󵄨 Δ = t + Δ(󵄨󵄨󵄨u(t)󵄨󵄨󵄨1 ), and equations (3.3.37), (3.3.38) and (3.3.39) are valid. While in dimension 3, the same holds as long as the degree of polynomial of g(u) is 1, 2, 3 or 4. For space dimension 4 or 5, we restrict r to 1 or 2. Example: Ginzburg–Landau equation Let Ω ⊂ Rn be an bounded open set, n = 1, 2, 3. The Ginzburg–Landau equation is 𝜕u − (ν + iα)Δu + (k + iβ)|u|2 u − γu = 0, 𝜕t u(0) = u0 ,

(3.3.35) (3.3.36)

332 | 3 The approximate inertial manifold where u(x, t) is a complex function, (x, t) ∈ Ω × R+ , the parameters ν, α, k, β, γ are real constants, and ν > 0, k > 0. Suppose that one of the following boundary conditions holds: u(x, t) = 0,

∀x ∈ 𝜕Ω, t ≥ 0,

(3.3.37)

n

u(⋅, t) is Ω periodic, ∀t ≥ 0, Ω = (0, L) , 𝜕u (x, t) = 0, ∀x ∈ 𝜕Ω, t ≥ 0. 𝜕n

(3.3.38) (3.3.39)

Write equation (3.3.35) in the form of equation (3.3.31), u = u1 + iu2 , A = −Δ, v D=( α

−α ), ν

R1 (u) = (u21 + u22 )(ku1 − βu2 ) − γu1 ,

R2 (u) = (u21 + u22 )(ku2 − βu1 ) − γu2 .

Now we check condition (3.3.3) which involves estimating 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨 󵄨 2 󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(k + iβ)󵄨󵄨󵄨󵄨󵄨󵄨 ∫ ∇(|u| u + (|γ| + 1)u) ⋅ ∇udx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 2 ≤ C|u|L4 |∇u|L4 + (|γ| + 1)‖u‖ . 3

3

When n = 1, 2, 3, H 4 can be continuously embedded into L4 , and H 4 can be constructed by the interpolation of L2 and H 1 . We have 3

1

|u|L4 ≤ C|u| 4 (|u|1 ) 4 .

(3.3.40)

1

By the equivalence of the (|Au|2 + |u|2 ) 2 and H 2 norms, we get 3

1

|∇u|L4 ≤ C|u| 4 (|Au|2 + |u|2 ) 8 .

(3.3.41)

From equations (3.3.40) and (3.3.41), we get 3 3 1 1 󵄨󵄨 󵄨 2 2 2 󵄨󵄨(R(u), Au + u)󵄨󵄨󵄨 ≤ C|u| 2 (|u|1 ) 2 ‖u‖ 2 (|u| + |Au| ) 4 + (|γ| + 1)‖u‖

6

≤ ε|Au|2 + ε|u|2 + (ε)−3 C|u|2 ‖u‖2 (|u|1 ) + (|γ| + 1)‖u‖2 10

≤ ε|Au|2 + C(ε)−3 (|u|1 ) + K, where C = max((4ε)−1 C, |γ| + 1). Then condition (3.3.3) is satisfied, where γ = 5. Hence the conclusion of Theorem 3.3.2 is true. Then the two real components of the solution of Ginzburg–Landau equation are D(A)-analytic functions. Since n = 1, 2, there exists an absorbing set in V. We have Theorem 3.3.3. In dimension 1 or 2, there exist d, τ > 0 such that the real and imaginary components of the solution for the Ginzburg–Landau equation with the initial–boundary value problem (3.3.35) can be analytically continued on the set 󵄨 󵄨 Δ d = {z ∈ C | Re(z) > τ, 󵄨󵄨󵄨Im(z)󵄨󵄨󵄨 < d},

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 333

and the equations (3.3.37)–(3.3.39) are valid in Δ d (particularly for t > τ). For spacial dimension 3, when |u(t)|1 is finite, we can continue the real and imaginary parts of the solution of this problem analytically to Δ(|u(t)|1 ). To introduce the following Gevrey regularity classes, suppose we are given a (2p)th order linear self-adjoint unbounded elliptic operator A, which possesses the form A = ∑ aα D2α , |α|=p

where α = (α1 , . . . , αn ), |α| = α1 + ⋅ ⋅ ⋅ + αn . Let Ω = [0, 2π]n be the set in the periodic ̂ which is a homogeneous (2p)th boundary condition. The Fourier transform of A is A, order positive polynomial: ̂ ) ≤ C|ξ |2p . C −1 |ξ |2p ≤ A(ξ

(3.3.42)

The eigenvectors of A are the exponential functions {eijx }j∈Zn , the corresponding eigen̂ j∈Zn , values are {λj = A(j)} u = ∑ uj eijx , |u|2 ≜ (

n

uj = (u1j , . . . , unj ) ∈ Cn ,

1 ) ∑ |uj |2 < ∞. 2π

(3.3.43)

For simplicity, set ∫ u(x, t)dx = 0,

∀t ≥ 0, for all the solutions u.

(3.3.44)

Ω

Equation (3.3.44) is equivalent to condition u0 = 0. On D(A), |Au|2 = (

n

1 ) ∑ |λj |2 |uj |2 . 2π

̂ ) = |ξ |2p and Defining the differential operator B = (−1)p Δp , we have B(ξ |Bu|2 = (

n

1 ) ∑ |j|4p |uj |2 . 2π

Hence, in equation (3.3.42), if we set ξ = j, then on D(A) we have C −1 |Bu|2 ≤ |Au|2 ≤ C|Bu|2 .

(3.3.45) 1

Now for τ > 0, we define the Gevrey category, D(exp(2B 2p )), as the set of functions having the form (3.3.43)–(3.3.44) such that n

1 1 󵄨 󵄨2 |u|2τ = 󵄨󵄨󵄨exp(τB 2p )u󵄨󵄨󵄨 = ( ) ∑ e2τ|j| |uj |2 < ∞. 2π

(3.3.46)

334 | 3 The approximate inertial manifold Now we consider an abstract initial value problem (3.3.31)–(3.3.32) with periodic boundary conditions. And nonlinear term R is a polynomial of {Dα }|α|≤d . Since Fourier transform preserves distances in L2 , there exists a function F, which only depends on Fourier coefficient uj and multiindex αj of R, such that (R(u), Au) = F(uj , jαj ). We require that γ, K and C are as in equation (3.3.33), and for arbitrary ε > 0, F satisfies 󵄨󵄨 󵄨 αj 󵄨 󵄨 2 2γ 󵄨󵄨F(uj , 󵄨󵄨󵄨j 󵄨󵄨󵄨)󵄨󵄨󵄨 ≤ ε|Au| + C(ε)‖u‖ + K.

(3.3.47)

But for condition (3.3.34), there is a replacement: there are constants M, σ > 0 such that f satisfies 󵄨󵄨 󵄨 󵄨󵄨f (t)󵄨󵄨󵄨σ ≤ M,

∀t ∈ R+ .

(3.3.48)

Theorem 3.3.4. Suppose that a differential operator A satisfies the previous conditions, the initial value is u0 ∈ V, f satisfies equation (3.3.48), and R(u) satisfies equations (3.3.47) and (3.3.34). There exist T∗ , which only depends on ‖u0 ‖, and region Δ ⊇ (0, T∗ ) in the complex plane such that the following statements are valid: (i) There exists a unique regular solution u for the problem (3.3.31) and equation (3.3.32) 1 1 such that the mapping t → [A 2 exp(ϕ(t)B 2p )]u takes values on H and is analytic in (0, T∗ ) ⊂ Δ, ϕ(t) = min(t, σ, T∗ ). (ii) If a solution of equation (3.3.31)–(3.3.32) exists and is uniformly bounded in V 1

1

(t ∈ R+ ), then u takes values on D(A 2 exp(σB 2p )) and is analytic for t ∈ (0, ∞), here Δ is included in (0, ∞). In order to prove Theorem 3.3.4, we first need the following lemma: 1

1 2p

Lemma 3.3.2. Let u ∈ D(A 2 eτB ), τ > 0. If R satisfies equation (3.3.47), then we have 󵄨󵄨 󵄨 󵄨 1 󵄨2γ 2 󵄨󵄨(R(u), Au)τ 󵄨󵄨󵄨 ≤ ε(|Au|τ ) + C(ε)󵄨󵄨󵄨A 2 u󵄨󵄨󵄨τ + K, where γ, k, C are as in equation (3.3.33). Proof. Let u = ∑ uj eijx , 1

u∗ = ∑ u∗j eijx = exp(τB 2p u), where u∗ = eτ|j| uj ,

j ∈ Zn .

∀ε > 0,

(3.3.49)

3.3 Time analyticity of solution for a class of dissipative nonlinear evolution equations | 335

Let d

R(u) = ∑ ∏ Dαk uik , k=1

where α1 , . . . , αd are arbitrary multiindices; i1 , . . . , id are integers between 1 and n; u = (u1 , . . . , un ). Then we have (R(u), Au)τ = (R(u), exp(2τB1+(2p) )Au)τ , d

j

j α

(R(u), Au)τ = ∫(∏ ∑ ujk jk k eijk x )( ∑ uj0 λj0 e2τ|j0 | e−ij0 x )dx. Ω

k=1 jk

k

∈Zn

j0

∈Zn

0

This integral is zero, due to the property of the exponential function, unless −j0 + j1 + j2 + ⋅ ⋅ ⋅ + jk = 0. In this case, the integral value is (2π)n . So (R(u), Au)τ = (2π)n ∑ 󵄨󵄨 󵄨 n 󵄨󵄨(R(u), Au)τ 󵄨󵄨󵄨 ≤ (2π) ∑

i

i

i α

α

∑

uj1 ⋅ ⋅ ⋅ ujd ujd j1 d ⋅ ⋅ ⋅ jdd λj0 e2τ|j0 | ,

∑

󵄨󵄨 ∗i0 󵄨󵄨 󵄨󵄨 ∗id 󵄨󵄨󵄨󵄨 α1 󵄨󵄨 󵄨󵄨 αd 󵄨󵄨 τ(|j |−|j |−⋅⋅⋅−|jd |) , 󵄨󵄨uj 󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨uj 󵄨󵄨󵄨󵄨j1 󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨jd 󵄨󵄨λj0 e 0 1 0 d

j1 +⋅⋅⋅+jd =j0 j1 +⋅⋅⋅+jd =j0

d

d

d

but |j0 | = |j1 + ⋅ ⋅ ⋅ + jk | ≤ |j1 | + ⋅ ⋅ ⋅ + |jk |, yielding exp(τ(|j0 | − |j1 | − ⋅ ⋅ ⋅ − |jd |)) ≤ 1. Therefore we get 󵄨󵄨 󵄨 n 󵄨󵄨(R(u), Au)τ 󵄨󵄨󵄨 ≤ (2π)

∑

j1 +⋅⋅⋅+jd =j0

󵄨 ∗i 󵄨 󵄨 ∗i 󵄨 󵄨 α 󵄨 ∑ 󵄨󵄨󵄨uj 0 󵄨󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨󵄨uj d 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨jdd 󵄨󵄨󵄨λj0 , 0

d

1

where u∗ = exp(τB 2p )u. Then we have 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨2 󵄨 1 󵄨2γ (R(u), Au)τ ≤ 󵄨󵄨󵄨F(󵄨󵄨󵄨u∗j 󵄨󵄨󵄨, 󵄨󵄨󵄨jαj 󵄨󵄨󵄨)󵄨󵄨󵄨 ≤ ε󵄨󵄨󵄨Au∗ 󵄨󵄨󵄨 + C(ε)󵄨󵄨󵄨A 2 u∗ 󵄨󵄨󵄨 + R, completing the proof of the lemma. Now we use Lemma 3.3.2 to prove Theorem 3.3.4. As in the framework of Theorem 3.3.1, the key point is to get an a priori estimate. Firstly, complexify equation 1

(3.3.31), and set ϕ(t) = min(t, σ), z = seiθ , s > 0, cos θ > 0, Es = exp(ϕ(s cos θ)B 2p ), θ ∈ (− π2 , π2 ). Taking the inner product of equation (3.3.31) and Au(seiθ ) in D(Es ), multiplying by eiθ , and taking its real part, we get Re eiθ [(Es

𝜕u iθ (se ), Es Au(seiθ )) + (Es DAu, Es Au) + (Es R(u), Es Au)] 𝜕t

= Re eiθ (Es f , Es Au).

(3.3.50)

336 | 3 The approximate inertial manifold Using the relation

d ds

d = e−iθ dz , we get

𝜕u iθ (se ), Es Au(seiθ ))} 𝜕t 1 d 1 = Re(A 2 (Es u(seiθ )) − ϕ󸀠 (s cos θ)(cos θ)Es Au(seiθ ), Es A 2 u(seiθ )) ds 1 d 󵄨󵄨 21 󵄨 1 󵄨 󸀠 iθ 󵄨2 = 󵄨󵄨A u(se )󵄨󵄨󵄨ϕ(s cos θ) − (cos θ)ϕ (s cos θ)|Au|ϕ(s cos θ) 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨ϕ(s cos θ) 2 ds 1 d 󵄨󵄨 21 iθ 󵄨2 2 ≥ 󵄨A u(se )󵄨󵄨󵄨ϕ(s cos θ) − α0 cos θ/4|Au|ϕ(s cos θ) 2 ds 󵄨 󵄨 1 󵄨2 − (cos θ/α0 )󵄨󵄨󵄨A 2 u󵄨󵄨󵄨ϕ(s cos θ) . (3.3.51)

Re eiθ {(Es

As estimated before, we have Re eiθ (Es DAu, Es Au) ≥ cos θα0 |Au|2ϕ(s cos θ) − |sin θ|‖D‖∗ |Au|2ϕ(s cos θ) . Restricting θ to satisfy equation (3.3.51), we get 3 Re eiθ (Es DAu, Es Au) ≥ ( ) cos θα0 |Au|2ϕ(s cos θ) . 4

(3.3.52)

Substituting equations (3.3.51) and (3.3.52) into (3.3.50), and replacing ϕ(s cos θ) with ϕ, we get 1 d 󵄨 󵄨 󵄨 󵄨 ‖u‖2 + (α0 cos θ)/2|Au|2ϕ ≤ 󵄨󵄨󵄨(f , Au)ϕ 󵄨󵄨󵄨 − 󵄨󵄨󵄨(R(u), Au)ϕ 󵄨󵄨󵄨. 2 ds ϕ

(3.3.53)

Dealing with the term |(f , Au)ϕ | as in Lemma 3.3.1, we get the inequality d 󵄩󵄩 󵄨 󵄨 iθ 󵄩2 iθ 󵄨2 iθ 󵄨2σ 󵄩󵄩u(se )󵄩󵄩󵄩ϕ + C1 󵄨󵄨󵄨Au(se )󵄨󵄨󵄨ϕ ≤ C2 + C3 󵄨󵄨󵄨u(se )󵄨󵄨󵄨ϕ , ds

(3.3.54)

where constants C2 , C3 only depend on the initial value, and not on θ. Letting 1 󵄨2 󵄨 y(s) = 1 + 󵄨󵄨󵄨Es A 2 u(seiθ )󵄨󵄨󵄨 ,

󸀠

(3.3.55)

γ

y (s) ≤ C4 y (s),

(3.3.56)

we get 1 󵄨󵄨 󵄨 1 󵄨2 iθ 󵄨2 󵄨󵄨Es A 2 u(se )󵄨󵄨󵄨 ≤ 2 + 2󵄨󵄨󵄨A 2 u0 󵄨󵄨󵄨 .

(3.3.57)

The region Δ(‖u0 ‖) is given by γ−1

1 0 ≤ s ≤ T1 (‖u0 ‖) = y(0)1−γ (1 − ( ) 2 π |θ| ≤ min(arctan(α0 /4‖D‖∗ ), ). 4

)/((γ − 1)C4 ), (3.3.58)

3.4 Two-dimensional Ginzburg–Landau equation

| 337

1

1

So, if u0 ∈ D(A 2 ), then u(seiθ ) ∈ D(Es A 2 ) in the angular region Δ(‖u0 ‖). If ‖u‖ ≤ M, t ∈ R+ , then Δ(‖u0 ‖) can be extended to Δ = ⋃t>0 (t + Δ(M)). Similarly, for any compact set K ⊂ Δ, we have the following inequality; 󵄩󵄩 dk u 󵄩󵄩 1 1 󵄩 󵄩 sup 󵄩󵄩󵄩 k (seiθ )󵄩󵄩󵄩 ≤ 2 2 (2/d)k (k!)(1 + ‖u0 ‖2 ) 2 , 󵄩󵄩ϕ 󵄩 dz z∈K 󵄩 󵄨󵄨 iθ 󵄨󵄨 sup 󵄨󵄨Au(se )󵄨󵄨ϕ ≤ T2 (K) < ∞,

d = dist(K, 𝜕Δ),

z∈K k

󵄩󵄩 d u 󵄩󵄩 −k 󵄩 󵄩 sup 󵄩󵄩󵄩A( k (seiθ ))󵄩󵄩󵄩 ≤ 2k (k!)[d(K, 𝜕Δ(u0 ))] T2 (K 󸀠 ), 󵄩 󵄩 dz 󵄩ϕ z∈K 󵄩

(3.3.59) (3.3.60) (3.3.61)

where 1 K 󸀠 ≜ {z ∈ Δ(u0 ) | d(z, 𝜕Δ(u0 )) ≥ d(K, 𝜕Δ(u0 ))}. 2 Remark 3.3.2. If solution u is uniformly bounded by M (∀t > 0) in V, then from equation (3.3.57) we have 1 󵄨󵄨 󵄨2 2 󵄨󵄨Et A 2 u(t)󵄨󵄨󵄨 ≤ 2 + 2M .

(3.3.62)

In particular, by the definition of Et we have 󵄨󵄨 󵄨2 2 −1 −2|j|ϕ(t) . 󵄨󵄨uj (t)󵄨󵄨󵄨 ≤ (2 + 2M )λj e

(3.3.63)

This suggests that the Fourier coefficients decay exponentially. Remark 3.3.3. Suppose that α is a given multiindex. Then n

1 󵄨󵄨 α 󵄨󵄨 󵄨 α 󵄨2 2 󵄨󵄨D u󵄨󵄨 ≤ ( ) ∑ 󵄨󵄨󵄨j 󵄨󵄨󵄨 |uj | , 2π |u|2τ = (

n

j∈Z n

1 ) ∑ e2τ|j| |uj |2 . 2π j

There exists a constant M(α, τ) > 0 such that 󵄨󵄨 α 󵄨󵄨 2|j| 󵄨󵄨j 󵄨󵄨 ≤ e ,

j ∈ Zn , |j| ≥ M.

From this we can see, if |u|τ < ∞ (for some τ > 0), then |Dα u| < ∞. Hence the Gevrey function category is included in C ∞ (Ω).

3.4 Two-dimensional Ginzburg–Landau equation In 1996, Guo and Wang considered the time analyticity, Gevrey regularity and approximate inertial manifold for the two-dimensional Ginzburg–Landau equation [25, 117].

338 | 3 The approximate inertial manifold Suppose we are given the following Ginzburg–Landau equation: 𝜕u = ρu + (1 + iv)Δu − (1 + iu)|u|2σ u + αλ1 ⋅ ∇(|u|2 u) + β(λ2 ⋅ ∇u)|u|2 , 𝜕t (x, t) ∈ Ω × R+ ,

(3.4.1)

u(x, 0) = u0 (x),

x ∈ Ω,

(3.4.2)

Ω = (0, L1 ) × (0, L2 ),

(3.4.3)

u is Ω periodic,

where u is an unknown complex function, σ ∈ N, ρ > 0, u, α, β are constants, λ1 , λ2 are real vectors. In [114], we have proved that if u0 ∈ H 2 (Ω), and there exists δ > 0 such that 2 0,

(3.4.5)

and there exists a constant K, which depends the parameters (σ, ρ, ν, μ, α, β, λ1 , λ2 , δ, Ω), such that 󵄩󵄩 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩H 1 ≤ K1 ,

∀t ≥ t1 ,

(3.4.6)

where t1 depends on the parameters (σ, ρ, v, μ, α, β, λ1 , λ2 , δ, Ω) and R, whenever ‖u0 ‖H 1 ≤ R. Let u(t) = u1 (t) + iu2 (t), u1 (t) and u2 (t) are real functions. Then taking the real and imaginary parts of equation (3.4.1), it follows that 𝜕u1 = ρu1 + Δu1 − νΔu2 − |u|2σ (u1 − μu2 ) 𝜕t + αλ1 cot ∇(|u|2 u1 ) + β(λ2 ⋅ ∇u1 )|u|2 , 𝜕u2 = ρu2 + Δu2 + νΔu1 − |u|2σ (u2 + μu1 ) 𝜕t + αλ1 ⋅ ∇(|u|2 u2 ) + β(λ2 ⋅ ∇u2 )|u|2 .

(3.4.7)

(3.4.8)

For simplicity, use u(t) to denote the vector (u1 (t), u2 (t)). Then the formulas (3.4.7)– (3.4.8) can be written as 𝜕u = ρu + DΔu − D1 |u|2σ u + αλ1 ⋅ ∇(|u|2 u) + β(λ2 ⋅ ∇u)|u|2 , 𝜕t where 1 ν

D=(

−ν ), 1

1 μ

D1 = (

−μ ). 1

3.4 Two-dimensional Ginzburg–Landau equation

| 339

Then we have du(t) + DAu(t) + R(u(t), u(t), u(t)) = 0, dt

(3.4.9)

where A = −Δ is an unbounded self-adjoint operator, D(A) = {u ∈ H 2 (Ω) × H 2 (Ω) : u satisfies equation (3.4.3)}, and R(u, v, ω) = −ρω + D1 (u ⋅ v)σ ω − αλ1 ⋅ ∇((u ⋅ v)ω) − β(λ2 ⋅ ∇ω)(u ⋅ v).

(3.4.10)

The operator R : D(A) × D(A) × D(A) → H = H × H, and for R(u, v, ω) we can establish the following estimate. Lemma 3.4.1. Suppose that u, v, ω ∈ D(A). Then R(u, v, ω) ∈ H , and 󵄩󵄩 󵄩 σ σ 󵄩󵄩R(u, v, ω)󵄩󵄩󵄩 ≤ ρ‖ω‖ + C‖u‖H 1 ‖v‖H 1 ‖ω‖H 1 1

1

+ C‖ω‖H2 1 ‖Aω‖ 2 ‖u‖H 1 ‖v‖H 1

+ C‖ω‖H 1 ‖u‖H 1 ‖v‖H 1 .

Hereafter, we use C and Ci , i = 1, 2, . . . , to denote any constants which only depend on the parameters (σ, ρ, ν, μ, α, β, λ1 , λ2 , δ, Ω). Proof. From equation (3.4.10) we get 󵄩󵄩 󵄩 󵄩 σ 󵄩 󵄩󵄩R(u, v, ω)󵄩󵄩󵄩 ≤ ρ‖ω‖ + 󵄩󵄩󵄩D1 (u, v) ω󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩αλ1 ⋅ ∇((u, v)ω)󵄩󵄩󵄩 + 󵄩󵄩󵄩β(λ2 ⋅ ∇ω)(u ⋅ v)󵄩󵄩󵄩, 󵄩󵄩 2σ 2σ 2 σ 󵄩 󵄩󵄩D1 (u, v) ω󵄩󵄩󵄩 = √1 + μ2 (∫ |u| |v| |ω| dx)

(3.4.11)

1 2

Ω

≤ √1 + μ2 ‖u‖σ8σ ‖v‖σ8σ ‖ω‖4 ≤ C1 ‖u‖σH 1 ‖v‖σH 1 ‖ω‖H 1 , 󵄩󵄩 󵄩 2 2 2 󵄩󵄩β(λ2 ⋅ ∇ω)(u ⋅ v)󵄩󵄩󵄩 ≤ |βλ2 |(∫ |∇ω| |u| |v| dx)

(3.4.12)

1 2

Ω 1

1

≤ βλ2 ‖∇ω‖4 ‖u‖8 ‖v‖8 ≤ C‖∇ω‖ 2 ‖∇ω‖H2 1 ‖u‖H 1 ‖v‖H 1 1

1

≤ C‖ω‖H2 1 ‖ω‖H2 2 ‖u‖H 1 ‖v‖H 1 1

1

≤ C‖ω‖H2 1 (‖ω‖ + ‖Aω‖) 2 ‖u‖H 1 ‖v‖H 1 1

1

1

≤ C‖ω‖H2 1 (‖ω‖ 2 + ‖Aω‖ 2 )‖u‖H 1 ‖v‖H 1 1

1

≤ C2 ‖ω‖H2 1 ‖Aω‖ 2 ‖u‖H 1 ‖v‖H 1 + C3 ‖ω‖H 1 ‖u‖H 1 ‖v‖H 1 .

(3.4.13)

340 | 3 The approximate inertial manifold Since αλ1 ⋅ ∇((u ⋅ v)ω) = α(λ1 ⋅ ∇ω)(u ⋅ v) + α(λ1 ⋅ ∇u)(v ⋅ ω) + α(λ1 ⋅ ∇v)(u ⋅ ω), similar to equation (3.4.13), we get 1 1 󵄩 󵄩󵄩 󵄩󵄩αλ1 ⋅ ∇((u ⋅ v)ω)󵄩󵄩󵄩 ≤ C4 ‖ω‖H2 1 ‖Aω‖ 2 ‖u‖H 1 ‖v‖H 1 1

1

1

1

+ C5 ‖u‖H2 1 ‖Au‖ 2 ‖v‖H 1 ‖ω‖H 1 + C6 ‖v‖H2 1 ‖Av‖ 2 ‖u‖H 1 ‖ω‖H 1

+ C7 ‖ω‖H 1 ‖u‖H 1 ‖v‖H 1 .

(3.4.14)

Lemma 3.4.1 can be deduced from equations (3.4.11)–(3.4.14). As a corollary of Lemma 3.4.1, we have 󵄩󵄩 󵄩 2σ+1 󵄩󵄩R(u, u, u)󵄩󵄩󵄩 ≤ ρ‖u‖H 1 + C1 ‖u‖H 1 5

1

+ C8 ‖u‖H2 1 ‖Au‖ 2 + C9 ‖u‖3H 1 .

(3.4.15)

From this we get 󵄨󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄨󵄨(R(u, u, u), Au + u)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩R(u)󵄩󵄩󵄩‖Au‖ + 󵄩󵄩󵄩R(u)󵄩󵄩󵄩‖u‖ 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩R(u)󵄩󵄩󵄩‖Au‖ + 󵄩󵄩󵄩R(u)󵄩󵄩󵄩‖u‖H 1

5

3

2 2 ≤ ρ‖u‖H 1 ‖Au‖ + C1 ‖u‖2σ+1 H 1 ‖Au‖ + C8 ‖u‖H 1 ‖Au‖

+ C9 ‖u‖3H 1 ‖Au‖ + ρ‖u‖2H 1 + C1 ‖u‖2σ+2 H1 7

1

+ C8 ‖u‖H2 1 ‖Au‖ 2 + C9 ‖u‖4H 1

≤ ε‖Au‖2 + C10 ‖u‖4σ+2 H 1 + C11 ,

∀ε > 0.

(3.4.16)

From equations (3.4.16) and (3.4.6), we get Theorem 3.4.1. Suppose that equation (3.4.4) holds and u ∈ H 2 (Ω). Then there exist θ0 and T0 such that each component of the solution for the problem (3.4.1)–(3.4.2) possesses D(A)-valued analytic continuation to the following complex region: Δ 1 = {t + seiθ : t ≥ t1 , |θ| ≤ θ0 , 0 ≤ s ≤ T0 }, where t1 is determined in equation (3.4.6); θ0 and T0 depend on the initial value and |θ0 | ≤ π4 . Moreover, there exists a constant K, depending on the initial value, such that 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩 󵄩󵄩u(z)󵄩󵄩󵄩, 󵄩󵄩󵄩A 2 u(z)󵄩󵄩󵄩, 󵄩󵄩󵄩Au(z)󵄩󵄩󵄩 ≤ K,

∀z ∈ Δ 2 ,

(3.4.17)

where Δ 2 = {z : Re z ≥ a, |Im z| ≤ b}, with a, b constants, which depend on initial value and R, whenever |u0 |H 1 ≤ R. Proof. Noting equations (3.4.16) and (3.4.6), we deduce that the conditions of Theorem 1.1 in [186] are satisfied, and from it we get the implication of Theorem 3.4.1.

3.4 Two-dimensional Ginzburg–Landau equation

| 341

By Theorem 3.4.1 and Cauchy formula, we get Proposition 3.4.1. Suppose that equation (3.4.3) is true and u0 ∈ H 2 (Ω). Then we have 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 1 d 󵄩󵄩 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 u(t)󵄩󵄩󵄩, 󵄩󵄩󵄩A 2 u(t)󵄩󵄩󵄩, 󵄩󵄩󵄩A u(t)󵄩󵄩󵄩 ≤ K2 , 󵄩󵄩 󵄩󵄩 dt 󵄩󵄩 󵄩󵄩 dt 󵄩󵄩 󵄩󵄩 dt

∀t ≥ t2 ,

where constant K2 depends on the initial data, t2 > t1 only depends on the initial value and R, whenever ‖u0 ‖H 1 ≤ R. In the following, we construct an approximate inertial manifold for problem (3.4.1)–(3.4.3). First of all, we know that the eigenvectors of A = −Δ form an orthogonal basis {ωj }∞ j=1 in H such that Aωj = λj ωj ,

0 = λ1 < λ2 ≤ ⋅ ⋅ ⋅ ≤ λj → ∞, j → ∞.

Given m, let P = Pm be the orthogonal projection of H to the subspace spanned by {ω1 , . . . , ωm }. Letting Q = Qm = I − Pm and acting with Pm and Qm on equation (3.4.9), we have dp + DAp + Pm R(p + q, p + q, p + q) = 0, dt dq + DAq + Qm R(p + q, p + q, p + q) = 0, dt

(3.4.18) (3.4.19)

where p = Pm u, q = Qm u, and we have 󵄩󵄩 γ 󵄩󵄩 γ γ 󵄩󵄩A p󵄩󵄩 ≤ λm ‖p‖, γ > 0, p ∈ Pm D(A ), γ 󵄩󵄩 γ 󵄩󵄩 γ 󵄩󵄩A q󵄩󵄩 ≤ λm+1 ‖q‖, γ > 0, q ∈ Qm D(A ), 󵄩 󵄩 󵄩󵄩 21 󵄩󵄩 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 1 󵄩󵄩A u󵄩󵄩 = 󵄩󵄩 󵄩󵄩, u ∈ H (Ω), 󵄩󵄩 𝜕x 󵄩󵄩 ‖Pm u‖ ≤ ‖u‖, ‖Qm u‖ ≤ ‖u‖, ∀u ∈ H.

(3.4.20) (3.4.21) (3.4.22) (3.4.23)

By equation (3.4.7) and Proposition 3.4.1, we obtain 󵄩󵄩 󵄩 󵄩󵄩Au(t)󵄩󵄩󵄩 ≤ C,

󵄩󵄩 󵄩󵄩 d 󵄩 󵄩󵄩 󵄩󵄩 u(t)󵄩󵄩󵄩 ≤ C, 󵄩󵄩 󵄩󵄩 dt

∀t ≥ t∗ .

(3.4.24)

For this C and t∗ , as in Proposition 3.4.1, equations (3.4.21), (3.4.23) and (3.4.24) imply 󵄩󵄩 󵄩 −1 󵄩󵄩q(t)󵄩󵄩󵄩 ≤ Cλm+1 , 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩 −1 󵄩󵄩 q(t)󵄩󵄩󵄩 ≤ Cλm+1 , 󵄩󵄩 dt 󵄩󵄩

− 21 󵄩 󵄩󵄩 21 󵄩󵄩A q(t)󵄩󵄩󵄩 ≤ Cm+1

∀t > t∗ .

(3.4.25)

Now we construct an approximate inertial manifold of problem (3.4.1)–(3.4.3). To achieve this, we define the mapping Φ : Pm H → Qm H such that ∀p ∈ Pm H, Φ(p) = Ψ is given by the following equation: DAΨ + Qm R(p, p, p) = 0.

(3.4.26)

342 | 3 The approximate inertial manifold Letting Σ = graph(Φ), we prove that Σ is an approximate inertial manifold. Then we get Theorem 3.4.2. Let equation (3.4.4) be valid, and u0 ∈ H 2 (Ω). Then there exists constant K, depending on the initial value, such that −3

2 , distH (u(t), Σ) ≤ Kλm+1

t ≥ t∗ ,

(3.4.27)

where u(t) is the solution of problem (3.4.1)–(3.4.3), t∗ depends on the initial value and R∗ , whenever ‖u0 ‖H 1 ≤ R. Proof. From equations (3.4.19) and (3.4.26), we get dq + Qm R(u) − Qm R(p), dt R(u) − R(p) = −ρq + D1 |u|2σ u − D1 |p|2σ p

D(AΨ − Aq) =

− αλ1 ⋅ ∇(|u|2 u) + αλ1 ⋅ ∇(|p|2 p)

− β(λ2 ⋅ ∇u)|u|2 + β(λ2 ⋅ ∇p)|p|2 ,

(3.4.28)

(3.4.29)

where in every term of estimate (3.4.29), f (s) = s2σ , and ξ is between |u| and |p|. Moreover, 󵄩󵄩 󵄩 󵄩 2σ 2σ 󵄩 2σ 2σ 󵄩 󵄩 2σ 󵄩󵄩D1 |u| u − D1 |p| p󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩D1 (u − |p| )u󵄩󵄩󵄩 + 󵄩󵄩󵄩D1 |p| (u − p)󵄩󵄩󵄩 󵄩 󵄩 ≤ √1 + μ2 ‖u‖∞ 󵄩󵄩󵄩|u|2σ − |p|2σ 󵄩󵄩󵄩 + √1 + μ2 ‖p‖2σ ∞ ‖q‖ 󵄩 󵄩 ≤ √1 + μ2 ‖u‖∞ 󵄩󵄩󵄩f 󸀠 (ξ )(|u| − |p|)󵄩󵄩󵄩 + √1 + μ2 ‖p‖2σ ∞ ‖q‖ 󵄩 󵄩 ≤ √1 + μ2 ‖p‖∞ 󵄩󵄩󵄩f 󸀠 (ξ )󵄩󵄩󵄩∞ ‖q‖ + √1 + μ2 ‖p‖2σ ∞ ‖q‖.

(3.4.30)

Equation (3.4.17) and the following inequalities C1 ‖u‖H 2 (Ω) ≤ ‖u‖ + ‖Δu‖ ≤ C2 ‖u‖H 2 (Ω) ,

∀u ∈ H 2 (Ω),

yield ‖u‖H 2 (Ω) ≤ C,

∀t ≥ t∗ ,

(3.4.31)

where t∗ depends the initial value and R, whenever ‖u0 ‖H 1 ≤ R. From this, we deduce 1

1

‖u‖∞ ≤ C‖u‖ 2 ‖u‖H2 2 (Ω) ≤ C,

∀t ≥ t∗ .

(3.4.32)

Similarly, we have 1

1

‖p‖∞ ≤ C2 ‖p‖ 2 ‖p‖H2 2 (Ω) 1

1

1

1

≤ C2 ‖p‖ 2 (‖p‖ + ‖Δp‖) 2 ≤ C2 ‖u‖ 2 (‖u‖ + ‖Au‖) 2 ≤ C3 .

(3.4.33)

3.4 Two-dimensional Ginzburg–Landau equation

| 343

From equations (3.4.32) and (3.4.33) we get ‖ξ ‖∞ ≤ ‖p‖∞ + ‖u‖∞ ≤ C,

∀t ≥ t∗ .

(3.4.34)

By virtue of equations (3.4.40), (3.4.32)–(3.4.34), we get 󵄩󵄩 2σ 2σ 󵄩 󵄩󵄩D1 |u| u − D2 |p| p󵄩󵄩󵄩 ≤ C5 ‖q‖, 󵄩󵄩 2 2󵄩 󵄩󵄩β(λ2 ⋅ ∇u)|u| − β(λ2 ⋅ ∇p)|p| 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩β(λ2 ⋅ (∇u − ∇p))|u|2 󵄩󵄩󵄩 + 󵄩󵄩󵄩β(λ2 ⋅ ∇p)(|u|2 − |p|2 )󵄩󵄩󵄩 󵄩 󵄩 ≤ |βλ2 |‖u‖2∞ ‖∇q‖ + 󵄩󵄩󵄩β(λ2 ⋅ ∇p)(u + p)(u − p)󵄩󵄩󵄩 ≤ |βλ2 |‖u‖2∞ ‖∇q‖ + |βλ2 |‖u + p‖∞ (∫(∇p)2 |q|2 ) ≤ C6 ‖∇q‖ + C7 ‖∇p‖4 ‖q‖4 1

(3.4.35)

1 2

Ω 1

≤ C6 ‖∇q‖ + C8 ‖∇p‖ 2 ‖∇p‖H2 1 ‖q‖H 1

1 󵄩 1 󵄩 󵄩 1 󵄩 󵄩 1 󵄩1 󵄩 1 󵄩 ≤ C6 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩 + C8 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 2 (󵄩󵄩󵄩A 2 p󵄩󵄩󵄩 + ‖Ap‖) 2 (‖q‖ + 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩)

1 󵄩 1 󵄩1 󵄩 1 󵄩 󵄩 1 󵄩 󵄩 1 󵄩 ≤ C6 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩 + C8 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 2 (󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 + ‖Au|) 2 (‖q‖ + 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩) 󵄩 1 󵄩 ≤ C9 ‖q‖ + C10 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩, αλ1 ⋅ ∇(|u|2 u) = α(λ1 ⋅ ∇u)|u|2 + 2α(λ1 ⋅ ∇u)uu.

(3.4.36) (3.4.37)

By using equation (3.4.37), similar to equation (3.4.36), we get 󵄩󵄩 2 2 󵄩 󵄩󵄩αλ1 ⋅ ∇(|u| u) − αλ1 ⋅ ∇(|p| p)󵄩󵄩󵄩 󵄩 1 󵄩 ≤ C11 ‖q‖ + C12 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩.

(3.4.38)

From equations (3.4.29), (3.4.35), (3.4.36) and (3.4.38), we know that there exists a constant C such that 󵄩󵄩 󵄩 󵄩 1 󵄩 󵄩󵄩R(u) − R(p)󵄩󵄩󵄩 ≤ C‖q‖ + C 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩.

(3.4.39)

By equations (3.4.28) and (3.4.39) we get 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 dq 󵄩󵄩 󵄩 󵄩 󵄩󵄩D(AΨ − Aq)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 󵄩󵄩󵄩 + 󵄩󵄩󵄩R(u) − R(p)󵄩󵄩󵄩 󵄩󵄩 dt 󵄩󵄩 −1

−1 −1

−1

−1 2 2 2 ≤ Cλ2 2 λm+1 + Cλm+1 . ≤ Cλm+1 + Cλm+1

(3.4.40)

From the formula 󵄩󵄩 󵄩 󵄩󵄩D(AΨ − Aq)󵄩󵄩󵄩 = √1 + ν2 ‖AΨ − Aq‖ ≥ √1 + ν2 λm+1 ‖Ψ − q‖,

(3.4.41)

344 | 3 The approximate inertial manifold together with equations (3.4.40) and (3.4.41), we get −3

2 ‖Ψ − q‖ ≤ Cλm+1 ,

∀t ≥ t∗ ,

(3.4.42)

where t∗ is taken as in equation (3.4.31). Then we know 󵄩 󵄩 dH (u(t), Σ) ≤ 󵄩󵄩󵄩u(t) − (p(t) + Φ(p(t)))󵄩󵄩󵄩 − 32 󵄩 󵄩 = 󵄩󵄩󵄩Ψ(t) − q(t)󵄩󵄩󵄩 ≤ Cλm+1 ,

and the theorem has been proved. Now we consider the Gevrey regularity of the solution for problem (3.4.1)–(3.4.3), which can be used to improve the convergence rate of the approximate inertial manifold. Set Ω = (0, 2π)2 , and ∫ u(x, t)dx = 0,

∀t > 0.

(3.4.43)

Ω 1 2

Lemma 3.4.2. Let u, v, ω ∈ D(eτA A), τ > 0. Then for R(u, v, ω) we have the following estimate: 1

1

1

1

2 2 󵄩 2 󵄩󵄩 2 󵄩 (eτA R(u, v, ω), eτA Ay) ≤ ρ󵄩󵄩󵄩eτA ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩

1

1

1

1

󵄩 2 1 󵄩σ 󵄩 2 1 󵄩σ 󵄩 2 1 󵄩󵄩 2 󵄩 + C 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩 1

1

1

󵄩 2 1 󵄩󵄩 2 1 󵄩󵄩 2 1 󵄩 1 + C 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩 2 1

1

󵄩 2 1 󵄩1 󵄩 2 󵄩 × 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩 1

1

1

󵄩 2 1 󵄩1 󵄩 2 󵄩1 󵄩 2 1 󵄩 + C 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA Au󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩 1

1

󵄩 2 1 󵄩󵄩 2 󵄩 × 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩 1

1

1

󵄩 2 1 󵄩󵄩 2 1 󵄩󵄩 2 1 󵄩 1 + C 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩 2 1

1

󵄩 2 1 󵄩󵄩 2 󵄩 × 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩. Proof. Letting 1 2

u = ∑ uj eijx ,

u∗ = eτA u = ∑ u∗j eijx ,

v = ∑ vj eijx ,

v∗ = eτA v = ∑ vj∗ eijx ,

j∈Z2

j∈Z2

ω = ∑ ωj eijx , j∈Z2

j∈Z2

1 2

j∈Z2

1 2

ω∗ = eτA ω = ∑ ω∗j eijx , j∈Z2

u∗ = eτ|j| uj , v∗ = eτ|j| vj , ω∗ = eτ|j| ωj ,

(3.4.44) (3.4.45) (3.4.46)

3.4 Two-dimensional Ginzburg–Landau equation 1 2

y = ∑ yj eijx ,

y∗ = eτA y = ∑ yj∗ eijx , j∈Z2

j∈Z2

y∗ = eτ|j| yj ,

| 345

(3.4.47)

we have (R(u, v, ω), y) = −ρ(ω, y) + ((u, v)σ D1 ω, y) − (αλ1 ⋅ ∇(u, v)ω, y)

− (β(λ2 ⋅ ∇ω)(u, v), y).

(3.4.48)

All the terms of estimate (3.4.48) are as follows: −ρ(ω, y) = −ρ ∫ ∑ ωl eilx ⋅ ∑ ys e−isx dx = 4π 2 ρ ∑ ωl ys , s

Ω l

l=s

(3.4.49)

((u, v)σ D1 ω, y) = ∫(∑ uj1 eij1 x ⋅ ∑ vk1 eik1 x ) ⋅ ⋅ ⋅ (∑ ujσ eijσ x ⋅ ∑ νkσ eikσ x ) j1

Ω

jσ

k1

kσ

× (D1 ∑ ωl eilx ⋅ ∑ ys e−isx )dx = 4π

s

l

2

(uj1 ⋅ vk1 ) ⋅ ⋅ ⋅ (ujσ ⋅ vkσ )(D1 ωl ⋅ ys ), (3.4.50)

∑

j1 +k1 +⋅⋅⋅+jσ +kσ +l=s

−(β(λ2 ⋅ ∇ω)(u, v), y) = −β ∫(u ⋅ v)(λ2 ⋅ ∇ω)ydx Ω

= −β ∫(∑ uj eijx ⋅ ∑ vk eikx ) j

Ω

k

× (∑(λ2 ⋅ il)ωl eilx ⋅ ∑ ys e−isx )dx s

l 2

= −4π β ∑ (uj ⋅ vk )(λ2 ⋅ il)(ωl ⋅ ys ), j+k+l=s

−α(λ1 ⋅ ∇((u, v)ω), y) = −α((λ1 ⋅ ∇ω)(u ⋅ v), y)

− α((λ1 ⋅ ∇u)ω, y) − α(((λ1 ⋅ ∇v)u)ω, y).

(3.4.51)

(3.4.52)

Similar to equation (3.4.51) we get −α((λ1 ⋅ ∇ω)(u, v), y) = −4π 2 α ∑ (uj ⋅ vk )(λ1 ⋅ il)(ωl ⋅ ys ), j+k+l=s

(3.4.53)

−α(((λ1 ⋅ ∇u)v)ω, y) = −4π 2 α ∑ (λ1 ⋅ ij)(uj ⋅ vk )(ul ⋅ ys ) j+k+l=s

= −α ∫(∑(λ1 ⋅ ij)uj eijx ⋅ ∑ vk eikx ) j

Ω

k

× (∑ ωl eilx ∑ ys e−isx )dx. l

s

(3.4.54)

346 | 3 The approximate inertial manifold Similarly, we arrive at − α(((λ1 ⋅ ∇v)u)ω, y)

= −4π 2 α ∑ (λ1 ⋅ ik)(uj ⋅ vk )(ωl ⋅ ys ). j+k+l=s

(3.4.55)

By equations (3.4.48)–(3.4.55) we get (R(u, v, ω), y) = −4π 2 ρ ∑ ωl ⋅ ys + 4π

2

l=s

(uj1 ⋅ vk1 ) ⋅ (ujσ ⋅ ⋅ ⋅ vkσ )(D1 ωl ⋅ ys )

∑

j1 +k1 +⋅⋅⋅+jσ +kσ +l=s

− 4π 2 α ∑ (uj ⋅ vk )(λ1 ⋅ il)(ωl ⋅ ys ) j+k+l=s

2

− 4π α ∑ (λ1 ⋅ ij)(uj ⋅ vk )(ωl ⋅ ys ) j+k+l=s

2

− 4π α ∑ (λ1 ⋅ ik)(uj ⋅ vk )(ωl ⋅ ys ) j+k+l=s

2

− 4π β ∑ (uj ⋅ vk )(λ2 ⋅ il)(ωl ⋅ ys ), j+k+l=s

1 2

1 2

(3.4.56)

1 2

(eτA R(u, v, ω), eτA Ay) = (R(u, v, ω), e2τA Ay)

= −4π 2 ρ ∑(ω∗l , y∗s )|s|2 + 4π 2

l=s

∑

(u∗j1 ⋅ vk∗1 ) ⋅ (u∗jσ ⋅ ⋅ ⋅ vk∗σ )

j1 +k1 +⋅⋅⋅+jσ +kσ +l=s

× (D1 ω∗l ⋅ y∗s )|s|2 eτ(|s|−|j1 |−|k1 |−⋅⋅⋅−|jσ |−|kσ |−|l|)

− 4π 2 α ∑ (u∗j ⋅ vk∗ )(λ1 ⋅ il)(ω∗l ⋅ y∗s )|s|2 eτ(|s|−|j|−|k|−|l|) j+k+l=s

2

− 4π α ∑ (λ1 ⋅ ij)(u∗j ⋅ vk∗ )(ω∗l ⋅ y∗s )|s|2 eτ(|s|−|j|−|k|−|l|) j+k+l=s

2

− 4π α ∑ (λ1 ⋅ ik)(u∗j ⋅ vk∗ )(ω∗l ⋅ y∗s )|s|2 eτ(|s|−|j|−|k|−|l|) j+k+l=s

2

− 4π β ∑ (u∗j ⋅ vk∗ )(λ2 ⋅ il)(ω∗l ⋅ y∗s )|s|2 eτ(|s|−|j|−|k|−|l|) . j+k+l=s

(3.4.57)

Note that |s| = |j1 + k1 + ⋅ ⋅ ⋅ + jσ + kσ + l| ≤ |j1 | + |k1 | + ⋅ ⋅ ⋅ + |jσ | + |kσ | + |l|, hence eτ(|s|−|j1 |−|k1 |−⋅⋅⋅−|jσ |−|kσ |−|l|) ≤ 1.

(3.4.58)

3.4 Two-dimensional Ginzburg–Landau equation

| 347

Similarly, for s = j + k + l, we have eτ(|s|−|j|−|k|−|l|) ≤ 1.

(3.4.59)

Then from equations (3.4.57)–(3.4.59) we have 1

1

󵄨󵄨 τA 2 󵄨 τA 2 󵄨󵄨(e R(u, v, ω), e Ay)󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 ≤ 4π 2 ρ ∑ 󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 l=s

+ 4π

2√

1 + μ2

∑

j1 +k1 +⋅⋅⋅+jσ +kσ +l=s

󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨 󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨 2 󵄨󵄨uj1 󵄨󵄨󵄨󵄨vk1 󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨ujσ 󵄨󵄨󵄨󵄨vkσ 󵄨󵄨󵄨󵄨ωl 󵄨󵄨󵄨󵄨ys 󵄨󵄨|s|

󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 + 4π α|λ1 | ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨|l|󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 2

j+k+l=s

󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 + 4π α|λ1 | ∑ |j|󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 2

j+k+l=s

󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 + 4π α|λ1 | ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨|k|󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 2

j+k+l=s

󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 + 4π β|λ2 | ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨|l|󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 . 2

j+k+l=s

(3.4.60)

Obviously, 󵄨 󵄨󵄨 󵄨 4π 2 ρ ∑ 󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 = ρ ∫ ξ (x)θ(x)dx, l=s

(3.4.61)

Ω

where 󵄨 󵄨 ξ (x) = ∑ 󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨eilx ,

󵄨 󵄨 θ(x) = ∑ |s|2 󵄨󵄨󵄨y∗s 󵄨󵄨󵄨eisx .

l

Hence

(3.4.62)

󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 4π 2 ρ ∑ 󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 ≤ ρ󵄨󵄨󵄨 ∫ ξ (x)θ(x)dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 l=s Ω

1

1

󵄩 2 󵄩󵄩 2 󵄩 ≤ ρ‖ξ ‖‖θ‖ = ρ󵄩󵄩󵄩eτA ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩, 󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨 󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨 2 4π 2 √1 + μ2 ∑ 󵄨󵄨uj1 󵄨󵄨󵄨󵄨vj1 󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨ujσ 󵄨󵄨󵄨󵄨vkσ 󵄨󵄨󵄨󵄨ωl 󵄨󵄨󵄨󵄨ys 󵄨󵄨|s|

(3.4.63)

j1 +k1 +⋅⋅⋅+jσ +kσ +l=s

= √1 + μ2 ∫ φj1 (x)Ψk1 (x) ⋅ φjσ (x)Ψkσ (x)ξ (x)θ(x)dx,

(3.4.64)

Ω

where ξ (x), θ(x) are shown in equation (3.4.62) and 󵄨 󵄨 φj1 (x) = 󵄨󵄨󵄨u∗j1 󵄨󵄨󵄨eij1 x , 󵄨 󵄨 φjσ (x) = 󵄨󵄨󵄨u∗jσ 󵄨󵄨󵄨eijσ x ,

.. .

󵄨 󵄨 Ψk1 (x) = 󵄨󵄨󵄨vk∗1 󵄨󵄨󵄨eik1 x , 󵄨 󵄨 Ψkσ (x) = 󵄨󵄨󵄨vk∗σ 󵄨󵄨󵄨eikσ x .

(3.4.65)

348 | 3 The approximate inertial manifold From equation (3.4.64) we get 4π 2 √1 + μ2

∑

j1 +k1 +⋅⋅⋅+jσ +kσ +l=s

󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨 󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨󵄨󵄨 ∗ 󵄨󵄨 2 󵄨󵄨uj1 󵄨󵄨󵄨󵄨vk1 󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨ujσ 󵄨󵄨󵄨󵄨vkσ 󵄨󵄨󵄨󵄨ωl 󵄨󵄨󵄨󵄨ys 󵄨󵄨|s|

󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ √1 + μ2 󵄨󵄨󵄨 ∫ φj1 (x)Ψk1 (x) ⋅ φjσ (x)Ψkσ (x)ξ (x)θ(x)dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

≤ √1 + μ2 ‖φj1 ‖4σ+2 ‖Ψk1 ‖4σ+2 ⋅ ⋅ ⋅ ‖φjσ ‖4σ+2 ‖Ψkσ ‖4σ+2 ‖ξ ‖4σ+2 ‖θ‖ ≤ √1 + μ2 ‖φj1 ‖H 1 ‖Ψk1 ‖H 1 ⋅ ⋅ ⋅ ‖φjσ ‖H 1 ‖Ψkσ ‖H 1 ‖ξ ‖H 1 ‖θ‖

󵄩 󵄩 1 󵄩󵄩 1 󵄩󵄩 1 󵄩 󵄩 1 󵄩󵄩 1 ≤ C1 󵄩󵄩󵄩A 2 φj1 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψk1 󵄩󵄩󵄩 ⋅ ⋅ ⋅ 󵄩󵄩󵄩A 2 φjσ 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψkσ 󵄩󵄩󵄩󵄩󵄩󵄩A 2 ξ 󵄩󵄩󵄩‖θ‖ 1

1

1

1

󵄩 2 1 󵄩σ 󵄩 2 1 󵄩σ 󵄩 2 1 󵄩󵄩 2 󵄩 ≤ C2 󵄩󵄩󵄩eτA A 2 y󵄩󵄩󵄩 󵄩󵄩󵄩eτA A 2 ν󵄩󵄩󵄩 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩.

(3.4.66)

By virtue of 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 4π 2 α|λ1 | ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨|l|󵄨󵄨󵄨ω∗k 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 j+k+l=s

= α|λ1 | ∫ φ(x)Ψ(x)η(x)θ(x)dx,

(3.4.67)

Ω

where θ(x) is defined by equation (3.4.62), and since 󵄨 󵄨 φ(x) = 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨eijx ,

󵄨 󵄨 Ψ(x) = 󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨eikx ,

󵄨 󵄨 η(x) = |l|󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨eilx ,

(3.4.68)

by equation (3.4.67) we obtain 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 4π 2 α|λ1 | ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨|l|󵄨󵄨󵄨ω∗k 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 j+k+l=s

󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ |α||λ1 |󵄨󵄨󵄨 ∫ φ(x)Ψ(x)η(x)θ(x)dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω

≤ |α||λ1 |‖φ‖8 ‖Ψ‖8 ‖η‖4 ‖θ‖ 1

1

≤ C3 ‖φ‖H 1 ‖Ψ‖H 1 ‖η‖ 2 ‖η‖H2 1 ‖θ‖

󵄩 1 󵄩󵄩 1 󵄩 󵄩 1 󵄩 1 ≤ C4 󵄩󵄩󵄩A 2 φ󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩‖η‖󵄩󵄩󵄩A 2 η󵄩󵄩󵄩 2 ‖θ‖ 1

1

1

󵄩 2 1 󵄩󵄩 2 1 󵄩󵄩 2 1 󵄩 1 ≤ C5 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩 2 1

1

󵄩 2 󵄩1 󵄩 2 󵄩 × 󵄩󵄩󵄩eτA Aω󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩.

(3.4.69)

Similar to equation (3.4.69), we have 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 4π 2 α|λ1 | ∑ |j|󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 j+k+l=s 1

1

1

1

1

󵄩 2 1 󵄩 1 󵄩 2 󵄩 1 󵄩 2 1 󵄩󵄩 2 1 󵄩󵄩 2 1 󵄩 ≤ C6 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA Au󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 y󵄩󵄩󵄩,

(3.4.70)

3.4 Two-dimensional Ginzburg–Landau equation

| 349

󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 4π 2 α|λ1 | ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨|k|󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2 j+k+l=s 1

1

1

1

1

1

1

󵄩 2 1 󵄩󵄩 2 1 󵄩 1 󵄩 2 1 󵄩 1 󵄩 2 1 󵄩󵄩 2 󵄩 ≤ C7 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩, 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 4π 2 β ∑ 󵄨󵄨󵄨u∗j 󵄨󵄨󵄨󵄨󵄨󵄨vk∗ 󵄨󵄨󵄨|l|󵄨󵄨󵄨ω∗l 󵄨󵄨󵄨󵄨󵄨󵄨y∗s 󵄨󵄨󵄨|s|2

(3.4.71)

j+k+l=s

1

1

1

󵄩 2 1 󵄩󵄩 2 1 󵄩󵄩 2 1 󵄩 1 󵄩 2 1 󵄩 1 󵄩 2 󵄩 ≤ C8 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 v󵄩󵄩󵄩󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA A 2 ω󵄩󵄩󵄩 2 󵄩󵄩󵄩eτA Ay󵄩󵄩󵄩.

(3.4.72)

From equations (3.4.60), (3.4.63), (3.4.66)–(3.4.69), we deduce the conclusion of the lemma. From Lemma 3.4.2, we get 1

1

󵄨󵄨 τA 2 󵄨 τA 2 󵄨󵄨(e R(u, u, u), e Au)󵄨󵄨󵄨 1 1 1 1 1 1 󵄩 2 1 󵄩2σ+1 󵄩󵄩 τA 2 󵄩󵄩 󵄩 τA 2 1 󵄩 5 󵄩 τA 2 󵄩 3 󵄩 2 󵄩󵄩 2 󵄩 ≤ ρ󵄩󵄩󵄩eτA u󵄩󵄩󵄩󵄩󵄩󵄩eτA Au󵄩󵄩󵄩 + C 󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 󵄩󵄩e Au󵄩󵄩 + C 󵄩󵄩󵄩e A 2 u󵄩󵄩󵄩 2 󵄩󵄩󵄩e Au󵄩󵄩󵄩 2 1 1 1 1 󵄩 2 󵄩2 󵄩 2 󵄩2 󵄩 2 1 󵄩4σ+2 󵄩 2 1 󵄩10 ≤ ε󵄩󵄩󵄩eτA Au󵄩󵄩󵄩 + C1 (ε)󵄩󵄩󵄩eτA u󵄩󵄩󵄩 + C2 (ε)󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 + C3 (ε)󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 1 1 1 󵄩 2 󵄩2 󵄩 2 󵄩4σ+2 󵄩 2 󵄩4σ+2 ≤ ε󵄩󵄩󵄩eτA Au󵄩󵄩󵄩 + C4 (ε)󵄩󵄩󵄩eτA u󵄩󵄩󵄩 + C5 (ε)󵄩󵄩󵄩eτA u󵄩󵄩󵄩 + C6 1 1 1 󵄩 2 󵄩4σ+2 󵄩 2 󵄩2 ≤ ε󵄩󵄩󵄩eτA Au󵄩󵄩󵄩 + C7 (ε)󵄩󵄩󵄩eτA A 2 u󵄩󵄩󵄩 + C6 . (3.4.73) Theorem 3.4.3. Suppose that conditions (3.4.4) is satisfied, u0 ∈ H 2 (Ω). Then there exists a constant k that depends on the initial value, such that each component of the solu1 1 tion for the problem (3.4.1)–(3.4.3) belongs to D(A 2 exp(kA 2 )), the analytic continuation in the complex domain is given as follows: Δ = {t + seiθ : t ≥ t∗ , |θ| ≤ θ0 , 0 ≤ s ≤ T0 }

(3.4.74)

and 1

󵄩 󵄩󵄩 kA 2 21 󵄩󵄩e A u(Z)󵄩󵄩󵄩 ≤ K,

Z ∈ Δ,

(3.4.75)

where θ0 , T0 and K depend on the initial value, |θ0 | ≤ π/4, t∗ depends on the initial value and R, whenever ‖u0 ‖H 1 ≤ R. Proof. From equations (3.4.6) and (3.4.73), we know Theorem 3.1 of [186] is valid. Hence the theorem has been proved. From Theorem 3.4.3 and Cauchy formula, we get 󵄩󵄩 1 1 d 󵄩󵄩 󵄩󵄩 kA 2 2 󵄩 u(t)󵄩󵄩󵄩 ≤ K1 , 󵄩󵄩e A 󵄩󵄩 󵄩󵄩 dt

∀t ≥ t1 .

(3.4.76)

Through equations (3.4.75) and (3.4.76) we get, when t is large enough, that 1

󵄩󵄩 kA 2 21 󵄩 󵄩󵄩e A q(t)󵄩󵄩󵄩 ≤ K,

󵄩󵄩 1 1 d 󵄩󵄩 󵄩󵄩 kA 2 2 󵄩 q(t)󵄩󵄩󵄩 ≤ K1 , 󵄩󵄩e A 󵄩󵄩 󵄩󵄩 dt

(3.4.77)

350 | 3 The approximate inertial manifold where q(t) = Qm u(t). Hence when t ≥ t∗ , 1

1

− 21 −kλ 2 󵄩 󵄩 󵄩 󵄩󵄩 21 −kλ 2 e m+1 , 󵄩󵄩A q(t)󵄩󵄩󵄩 ≤ Ke m+1 , 󵄩󵄩󵄩q(t)󵄩󵄩󵄩 ≤ Kλm+1 1 󵄩󵄩 dq(t) 󵄩󵄩 − 21 −kλ 2 󵄩󵄩 󵄩󵄩 e m+1 . 󵄩󵄩 󵄩󵄩 ≤ K1 λm+1 󵄩󵄩 dt 󵄩󵄩

(3.4.78)

Using equation (3.4.78) instead of equation (3.4.25), similar to Theorem 3.4.2, we have Theorem 3.4.4. Suppose that condition (3.4.4) holds, u0 ∈ H 2 (Ω). Then there exists a constant E, depending on the initial value, such that 1 2

−1 −kλm+1 distH (Σ, u(t)) ≤ Eλm+1 e ,

t ≥ t∗ ,

(3.4.79)

where u(t) is the solution of problem (3.4.1)–(3.4.3), Σ is its approximate inertial manifold, t∗ depends on the initial value and R, whenever ‖u0 ‖H 1 ≤ R.

3.5 Bernard convection equation Two-dimensional Newton–Boussinesq equation can be used to describe the famous Bernard convection: R 𝜕t ξ + u𝜕x ξ + v𝜕y ξ = Δξ − α 𝜕x θ, { { { Pγ { { { ΔΨ = ξ , u = 𝜕y Ψ, v = −𝜕x Ψ, { { { { { {𝜕t θ + u𝜕x θ + ν𝜕y θ = 1 Δθ, Pγ { where (u, v) ≜ u is the velocity vector; θ is the temperature; Ψ is the stream function; ξ is vorticity; Pγ > 0 is the Prandtl constant; and Rα > 0 is Rayleigh number. The above equation can be rewritten as R 𝜕 𝜕 ΔΨ + J(Ψ, ΔΨ) = Δ2 Ψ − α θ , 𝜕t Pγ 𝜕x 𝜕θ 1 + J(Ψ, θ) = Δθ, 𝜕t Pγ

(3.5.1) (3.5.2)

where J(u, v) = uy vx − ux vy . The above equation can be endowed with the initial value Ψ(x, y, 0) = Ψ0 (x, y),

θ(x, y, 0) = θ0 (x, y)

(3.5.3)

3.5 Bernard convection equation

| 351

and periodic boundary conditions Ψ(x + 2D, y, t) = Ψ(x, y, t),

Ψ(x, y + 2D, t) = Ψ(x, y, t),

θ(x + 2D, y, t) = θ(x, y, t),

θ(x, y + 2D, t) = θ(x, y, t).

(3.5.4)

In 1987, Foias et al. [49] proved the existence and the finiteness of dimension of the global attractor. In [86] Guo proved the convergence of the spectral method and the existence and uniqueness of a global smooth solution. The nonlinear Galerkin method was proposed and its convergence was proved by Guo [90] in 1995. Guo and Wang [120] in 1996 proved the existence of approximate inertial manifolds. In order to simplify the problem using functional formulation, we let Au = −Δu. The inner product on H = L2 (Ω) is (⋅), its norm is ‖ ⋅ ‖, D(A) = {u ∈ H 2 (Ω); u satisfies the equation (3.5.4)}. Let Ω = (0, 2D)×(0, 2D). Then equations (3.5.1)–(3.5.2) can be written as R d AΨ + J(Ψ, AΨ) + A2 Ψ − α B(θ) = 0, dt Pγ

(3.5.5)

d 1 θ + J(Ψ, θ) + A(θ) = 0, dt Pγ

(3.5.6)

where B(θ) = 𝜕θ is a linear operator. From [90] we know that when (Ψ0 , θ0 ) ∈ H 2 × H 1 , 𝜕x the problem (3.5.1)–(3.5.4) has a unique solution (Ψ, θ), Ψ ∈ L∞ (R+ ; H 2 (Ω)), ∞

+

1

θ ∈ L (R ; H (Ω)),

ΔΨ ∈ L2 (0, T; H 1 (Ω)), 2

∀T > 0,

Δθ ∈ L (0, T; H).

One of the properties of equation (3.5.2) is that the average of the solution of the equations is conserved (t > 0): m(θ(t)) =

1 ∬ θ(x, y, t)dxdy |Ω| Ω

1 = ∬ θ0 (x, y, t)dxdy = m(θ0 ). |Ω| Ω

Hence, there exists an unbounded absorbing set in the whole space H. We introduce the subset of H: Hα = {θ ∈ H : |m(θ)| ≤ α},

α is a fixed number,

Lemma 3.5.1 (Uniform Gronwall lemma, [197]). Suppose g, h, y are three positive locally integrable functions on [t0 , ∞), and y󸀠 is also locally integrable on the interval [t0 , ∞). Let g, h, y satisfy the inequalities dy ≤ gy + h, dt

t ≥ t0 ,

352 | 3 The approximate inertial manifold t+r

t+r

∫ g(s)ds ≤ a1 ,

∫ h(s)ds ≤ a2 ,

t

t

t+r

∫ y(s)ds ≤ a3 ,

t ≥ t0 .

t

where r, a1 , a2 , a3 are positive constants. Then we have y(t + r) ≤ (

a3 + a2 ) exp(a1 ), r

t ≥ t0 .

Lemma 3.5.2. Let Ψ0 ∈ H 4 , θ0 ∈ Hα ∩ H 2 . Then for the solution (Ψ, θ) of problem (3.5.2)–(3.5.4), we have the following estimates: 󵄩 󵄩 󵄩 3 󵄩 󵄩 1 󵄩 󵄩󵄩 d 󵄩󵄩 ‖AΨ‖, 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩, ‖θ‖, 󵄩󵄩󵄩A 2 θ󵄩󵄩󵄩, 󵄩󵄩󵄩 θ󵄩󵄩󵄩 ≤ M0 , 󵄩󵄩 dt 󵄩󵄩

∀t ≥ t∗ ,

where M0 depends on the parameters (α, Ω, Pγ , Rα ); t∗ depends on (α, Ω, Pγ , Rα ) and R, whenever ‖Ψ0 ‖H 4 ≤ R and ‖θ0 ‖H 2 ≤ R. Proof. Similarly as in Lemma 2.3 in [90], it is easy to see that there exists a constant C, which only depends on the initial value, such that t+1

󵄩 3 󵄩2 󵄩 1 󵄩 ‖AΨ‖, ‖θ‖, 󵄩󵄩󵄩A 2 θ󵄩󵄩󵄩, ∫ 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 dt ≤ C,

∀t ≥ t0 ,

(3.5.7)

t

where t0 depends on the initial value and R, whenever ‖Ψ0 ‖H 1 ≤ R, ‖θ0 ‖ ≤ R; C here means the parameter depends only on the constants. Taking the inner product of equation (3.5.5) with A2 Ψ, we obtain 1 d 󵄩󵄩 32 󵄩󵄩2 󵄩 2 󵄩2 R 2 2 󵄩A Ψ󵄩󵄩 + (J(Ψ, AΨ), A Ψ) + 󵄩󵄩󵄩A Ψ󵄩󵄩󵄩 − α (B(θ), A Ψ) = 0, 2 dt 󵄩 Pγ 󵄨󵄨 󵄨 󵄨 󵄩 2 󵄩󵄨 2 󵄨󵄨(J(Ψ, AΨ), A Ψ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨J(Ψ, AΨ)󵄩󵄩󵄩A Ψ󵄩󵄩󵄩󵄨󵄨󵄨 󵄩 1󵄩 󵄩2 󵄩 3 󵄩4 󵄩 3 󵄩2 󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 󵄩󵄩󵄩A2 Ψ󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩A2 Ψ󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 , 4 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩󵄩󵄩 2 󵄩󵄩 󵄩󵄩 21 󵄩󵄩󵄩󵄩 2 󵄩󵄩 2 󵄨󵄨(B(θ), A Ψ)󵄨󵄨 ≤ 󵄩󵄩B(θ)󵄩󵄩󵄩󵄩A Ψ󵄩󵄩 ≤ 󵄩󵄩A θ󵄩󵄩󵄩󵄩A Ψ󵄩󵄩 󵄩2 󵄩 󵄩 1󵄩 ≤ C 󵄩󵄩󵄩A2 Ψ󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩A2 Ψ󵄩󵄩󵄩 + C. 4 From equations (3.5.8)–(3.5.10) we get d 󵄩󵄩 32 󵄩󵄩2 󵄩󵄩 2 󵄩󵄩2 󵄩 3 󵄩4 󵄩A Ψ󵄩󵄩 + 󵄩󵄩A Ψ󵄩󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 + C, dt 󵄩 d 󵄩󵄩 32 󵄩󵄩2 󵄩 3 󵄩4 󵄩A Ψ󵄩󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 + C. dt 󵄩

(3.5.8)

(3.5.9)

(3.5.10)

3.5 Bernard convection equation

| 353

In order to use the claim of the uniform Gronwall lemma, we set 󵄩 3 󵄩2 y = 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 ,

󵄩 3 󵄩2 g = C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 ,

h = C.

Then from equation (3.5.7), we know that conditions of Lemma 3.5.1 are satisfied, hence we have 󵄩󵄩 32 󵄩󵄩2 󵄩󵄩A Ψ󵄩󵄩 ≤ C,

∀t ≥ t0 + 1,

(3.5.11)

where t0 is like in equation (3.5.7). Differentiating equations (3.5.5)–(3.5.6) with t, we obtain R d AΨ + J(Ψt , AΨ) + J(Ψ, AΨt ) + A2 Ψt − α B(θt ) = 0, dt t Pγ

d 1 θ + J(Ψt , θ) + J(Ψ, θt ) + A(θt ) = 0. dt t Pγ

Since Ψ0 ∈ H 4 ,

θ0 ∈ H 2 ,

using equations (3.5.5)–(3.5.6), we get AΨt (0) ∈ H,

θt (0) ∈ H.

Repeating the proof in [90], we get 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩A Ψ󵄩󵄩󵄩 ≤ C, 󵄩󵄩 dt 󵄩󵄩

󵄩󵄩 d 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 θ󵄩󵄩󵄩 ≤ C, 󵄩󵄩 dt 󵄩󵄩

∀t ≥ t0󸀠 ,

where t0󸀠 depends on the initial value and R, whenever ‖Ψ0 ‖H 4 ≤ R, ‖θ0 ‖H 2 ≤ R. Let t∗ = max{t0󸀠 , t0 + 1}. When t ≥ t∗ , from the above inequalities (3.5.7) and (3.5.11), we get the claim of the lemma. Let {ωj (x, y)}, j = 1, 2, . . . , be the periodic eigenvectors of A, which satisfy Aωj = λj ωj ,

j = 1, 2, . . . ,

λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ,

λj → +∞,

j → ∞.

Fixing a positive integer m, set P = Pm to be the projection from H to the subspace span{ω1 , ω2 , . . . , ωm }. Let Q = Qm = I − Pm . Then acting with P, Q on the formulas (3.5.5)–(3.5.6), we get the coupling equations for Ψ1 = PΨ, Ψ2 = QΨ, θ1 = Pθ and θ2 = Qθ as follows: R d { AΨ + Pm J(Ψ, AΨ) + A2 Ψ1 − α Pm B(θt ) = 0, { { { dt 1 Pγ { R d { { { AΨ2 + Qm J(Ψ, AΨ) + A2 Ψ2 − α Qm B(θt ) = 0, dt Pγ {

(3.5.12)

354 | 3 The approximate inertial manifold d 1 { θ1 + Pm J(Ψ, θ) + Aθ1 = 0, { { { dt Pγ { d 1 { { { θ2 + Qm J(Ψ, θ) + Aθ2 = 0. dt P γ {

(3.5.13)

Hereafter, we usually use the following inequality: 󵄩 3 󵄩󵄩 1 󵄩 󵄨 󵄨󵄨 󵄨󵄨J(u, v)󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩󵄩󵄩󵄩A 2 v󵄩󵄩󵄩,

u ∈ H 3, v ∈ H 1.

(3.5.14)

The proof of equation (3.5.14) is obvious, since 1

2 󵄨󵄨 󵄨 2 󵄨󵄨J(u, v)󵄨󵄨󵄨 = (∬(uy vx − ux vy ) dxdy)

Ω

≤ (‖uy ‖∞ + ‖ux ‖∞ )‖∇v‖ 1

1

1

1

≤ C(‖uy ‖ 2 ‖Δuy ‖ 2 + ‖ux ‖ 2 ‖Δux ‖ 2 )‖∇v‖ 1

1

≤ C‖∇u‖ 2 ‖∇Δu‖ 2 ‖∇v‖ 󵄩 1 󵄩1 󵄩 3 󵄩1 󵄩 1 󵄩 󵄩 3 󵄩󵄩 1 󵄩 ≤ C 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 2 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 2 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩󵄩󵄩󵄩A 2 v󵄩󵄩󵄩, thus we get equation (3.5.14). Now we give the long-time estimate for Ψ2 , θ2 and their derivatives. Lemma 3.5.3. Let Ψ0 ∈ H 4 , θ0 ∈ H0 ∩ H 2 . Then there exists a constant M, which depends on parameter t∗ and R, such that whenever ‖Ψ0 ‖H 4 ≤ R, ‖θ0 ‖H 2 ≤ R, − 21 󵄩 󵄩󵄩 32 , 󵄩󵄩A Ψ2 (t)󵄩󵄩󵄩 ≤ Mλm+1 − 21 󵄩 󵄩󵄩 32 , 󵄩󵄩A θ2 (t)󵄩󵄩󵄩 ≤ Mλm+1

󵄩󵄩 󵄩󵄩 d 󵄩 󵄩󵄩 −1 󵄩󵄩 AΨ2 󵄩󵄩󵄩 ≤ Mλm+1 , 󵄩󵄩 󵄩󵄩 dt 󵄩󵄩󵄩 d 󵄩󵄩󵄩 󵄩󵄩 θ2 󵄩󵄩 ≤ Mλ−1 . m+1 󵄩󵄩󵄩 dt 󵄩󵄩󵄩

Proof. Taking the inner product of equation (3.5.12) with A2 Ψ2 over H, we obtain 1 d 󵄨󵄨 32 󵄨󵄨2 󵄨 2 󵄨2 R 2 2 󵄨A Ψ2 󵄨󵄨 + (J(Ψ, AΨ), A Ψ2 ) + 󵄨󵄨󵄨A Ψ2 󵄨󵄨󵄨 − α (B(θ) − A Ψ2 ) = 0. 2 dt 󵄨 Pγ From equation (3.5.14), we have 󵄨󵄨 󵄨 󵄩 󵄩󵄩 2 󵄩 2 󵄨󵄨(J(Ψ, AΨ), A Ψ2 )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩J(Ψ, AΨ)󵄩󵄩󵄩󵄩󵄩󵄩A Ψ2 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 3 󵄩2 󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 󵄩󵄩󵄩A2 Ψ2 󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩A2 Ψ2 󵄩󵄩󵄩, 󵄩󵄩 2 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 21 󵄩󵄩󵄩󵄩 2 󵄩󵄩 2 󵄨󵄨(B(θ), A Ψ2 )󵄨󵄨 ≤ 󵄩󵄩A θ󵄩󵄩󵄩󵄩A Ψ2 󵄩󵄩 ≤ C 󵄩󵄩A Ψ2 󵄩󵄩. Hence by equation (3.5.15) we have 1 d 󵄨󵄨 32 󵄨󵄨2 󵄩󵄩 2 󵄩󵄩2 󵄩 2 󵄩 󵄨A Ψ2 󵄨󵄨 + 󵄩󵄩A Ψ2 󵄩󵄩 ≤ C 󵄩󵄩󵄩A Ψ2 󵄩󵄩󵄩 ≤ 2 dt 󵄨

1 󵄩󵄩 2 󵄩󵄩2 󵄩A Ψ2 󵄩󵄩 + C. 2󵄩

(3.5.15)

3.5 Bernard convection equation

| 355

3

Since ‖A2 Ψ2 ‖2 ≥ λm+1 ‖A 2 Ψ2 ‖2 , we get 1 d 󵄩󵄩 32 󵄩󵄩2 󵄩 3 󵄩2 󵄩󵄩A Ψ2 󵄩󵄩 + λm+1 󵄩󵄩󵄩A 2 Ψ2 󵄩󵄩󵄩 ≤ C. 2 dt Through Gronwall Lemma we know that 󵄩󵄩 32 󵄩2 󵄩 3 󵄩2 −λ (t−t ) −1 󵄩󵄩A Ψ2 (t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩A 2 Ψ2 (t∗ )󵄩󵄩󵄩 e m+1 ∗ + Cλm+1 ,

t ≥ t∗ ,

where t∗ is similar to that in Lemma 3.5.2. Since 󵄩󵄩 32 󵄩 󵄩 3 󵄩 󵄩󵄩A Ψ2 (t∗ )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩A 2 Ψ(t∗ )󵄩󵄩󵄩 ≤ M, we have 󵄩󵄩 32 󵄩2 2 −λ (t−t ) −1 −1 󵄩󵄩A Ψ2 (t)󵄩󵄩󵄩 ≤ M e m+1 ∗ + Cλm+1 ≤ Cλm+1 ,

∀t ≥ t∗ ,

(3.5.16)

M2λ

where t∗󸀠 = supm max{t∗ , t∗ + λ 1 lg Cm+1 }. m+1 Taking the inner product of equation (3.5.13) with Aθ2 over H, we get 1 1 d 󵄩󵄩 21 󵄩󵄩2 2 󵄩󵄩A θ2 󵄩󵄩 + (Qm J(Ψ, θ), Aθ2 ) + ‖Aθ2 ‖ = 0. 2 dt Pγ Thus we have 1 d 󵄩󵄩 21 󵄩󵄩2 1 󵄨 󵄨 2 󵄩󵄩A θ2 󵄩󵄩 + ‖Aθ2 ‖ ≤ 󵄨󵄨󵄨J(Ψ, θ)󵄨󵄨󵄨|Aθ2 | 2 dt Pγ

1 󵄩 3 󵄩󵄩 1 󵄩 ‖Aθ2 ‖2 + C, ≤ C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩󵄩󵄩󵄩A 2 θ󵄩󵄩󵄩‖Aθ2 ‖ ≤ C‖Aθ2 ‖ ≤ 2Pγ

which yields d 󵄩󵄩 21 󵄩󵄩2 1 2 󵄩A θ2 󵄩󵄩 + ‖Aθ2 ‖ ≤ C, dt 󵄩 Pγ

d 󵄩󵄩 21 󵄩󵄩2 λm+1 󵄩󵄩 21 󵄩󵄩2 󵄩A θ2 󵄩󵄩 + 󵄩A θ2 󵄩󵄩 ≤ C. dt 󵄩 Pγ 󵄩 By Gronwall lemma, we get 󵄩󵄩 21 󵄩2 󵄩 1 󵄩2 − 1 λm+1 (t−t∗ ) −1 + Cλm+1 󵄩󵄩A θ2 (t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩A 2 θ2 (t∗ )󵄩󵄩󵄩 e Pγ ≤ M2e

− P1 λm+1 (t−t∗ ) γ

−1 + Cλm+1 ,

∀t ≥ t∗ ,

where t∗ is similar to that in Lemma 3.5.2. This proves 󵄩󵄩 21 󵄩2 −1 󵄩󵄩A θ2 (t)󵄩󵄩󵄩 ≤ Cλm+1 , where t∗󸀠 = supm max{t∗ , t∗ +

Pγ λm+1

lg

M 2 λm+1 }. C

∀t ≥ t∗󸀠 ,

(3.5.17)

356 | 3 The approximate inertial manifold Differentiating equations (3.5.12) and (3.5.13) with respect to t and applying the above method, when t is large enough, we have 󵄩󵄩2 󵄩󵄩 d 󵄩 󵄩󵄩 −1 󵄩󵄩 AΨ2 (t)󵄩󵄩󵄩 ≤ Mλm+1 , 󵄩󵄩 󵄩󵄩 dt

󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩 −1 󵄩󵄩 θ2 (t)󵄩󵄩󵄩 ≤ Mλm+1 . 󵄩󵄩 dt 󵄩󵄩

(3.5.18)

d d Of course, at this time J not only involves Ψ, θ, but also includes dt Ψ, dt θ. So we also need to use Lemma 3.5.2. From equations (3.5.16)–(3.5.18) we get the claim of the lemma.

Since Ψ2 , θ2 ∈ Qm H, by Lemma 3.5.3 we get −1 ‖AΨ2 ‖ ≤ Mλm+1 ,

−1 ‖θ2 ‖ ≤ Mλm+1 ,

∀t ≥ t∗ .

We now construct two-dimensional Newton–Boussinesq equations explicitly approximating inertial manifold. In [90], the authors introduced the following nonlinear Galerkin method, and proved its convergence. The idea was to look for an approximate solution (Ψ1 , θ1 ) ∈ Pm H × Pm H, satisfying R d { AΨ + PJ(Ψ1 , AΨ1 ) + PJ(φ1 , AΨ1 ) + PJ(Ψ1 , AΨ1 ) + A2 Ψ1 − α PB(θ1 ) = 0, { { dt 1 Pγ (3.5.19) { R { 2 α {A φ1 + QJ(Ψ1 , AΨ1 ) − QB(θ1 ) = 0, Pγ { d 1 { θ1 + PJ(Ψ1 , θ1 ) + PJ(φ1 , θ1 ) + PJ(Ψ1 , φ2 ) + Aθ1 = 0, { { { dt P { γ { 1 (3.5.20) { Aφ + αJ(Ψ , θ ) = 0, { 2 1 1 { P { γ { { {Ψ1 (0, x, y) = PΨ0 (x, y), θ1 (0, x, y) = Pθ0 (x, y), where φ1 , φ2 ∈ Qm H. We notice that a nonlinear mapping F is as in the nonlinear Galerkin method. In the following, we prove that Σ1 = graph(F) is an approximate inertial manifold. The mapping F : Pm H × Pm H → Qm H × Qm H is such that F(Ψ1 , θ1 ) = (φ1 , φ2 ) is valid for any (Ψ1 , θ1 ) ∈ Pm H × Pm H. Here φ1 , φ2 are determined by equations (3.5.19)–(3.5.20). Theorem 3.5.1. There exists a constant M, which only depends on parameters, such that −3

2 distH 2 ×H ((Ψ, θ), Σ1 ) ≤ Mλm+1 ,

∀t ≥ t∗

where t∗ depends on the parameters and R, whenever ‖Ψ0 ‖H 4 ≤ R, ‖θ0 ‖H 2 ≤ R, and (Ψ, θ) is the solution of equations (3.5.1)–(3.5.4). Proof. Taking the difference of equations (3.5.19) and (3.5.12), we get A2 (φ1 − Ψ2 ) + QJ(Ψ1 , AΨ1 ) − QJ(Ψ, AΨ) −

Rα R d QB(θ1 ) + α QB(θ) − AΨ2 = 0. Pγ Pγ dt

3.5 Bernard convection equation

| 357

By bilinear property of J(u, v), we have A2 (φ1 − Ψ2 ) + QJ(−Ψ2 , AΨ1 ) + QJ(Ψ1 , −AΨ2 ) +

Rα d QB(θ2 ) − AΨ2 = 0, (3.5.21) Pγ dt

− 21 󵄩 3 󵄩󵄩 3 󵄩 󵄩 3 󵄩󵄩 3 󵄩 󵄨 󵄨󵄨 , 󵄨󵄨J(−Ψ2 , AΨ1 )󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩A 2 Ψ2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ1 󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩 ≤ Cλm+1 1 3 3 −2 󵄩 󵄨 󵄨󵄨 󵄩󵄩 󵄩 , 󵄨󵄨J(Ψ1 , −AΨ2 )󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ2 󵄩󵄩󵄩 ≤ Cλm+1 1 1 −2 󵄩 󵄨 󵄩 󵄨󵄨 . 󵄨󵄨B(θ2 )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩A 2 θ2 󵄩󵄩󵄩 ≤ Cλm+1

(3.5.22) (3.5.23) (3.5.24)

By equations (3.5.21)–(3.5.24) and Lemma 3.5.1, we get − 21 󵄩󵄩 2 2 󵄩 . 󵄩󵄩A φ1 − A Ψ2 󵄩󵄩󵄩 ≤ Cλm+1

Hence we get −3

(3.5.25)

2 . ‖Aφ1 − AΨ2 ‖ ≤ Cλm+1

Taking the difference of equations (3.5.20) and (3.5.13), it follows that 1 d (Aφ2 − Aθ2 ) + QJ(Ψ1 , θ1 ) − QJ(Ψ1 , θ) − θ2 = 0, Pγ dt that is, d 1 (Aφ2 − Aθ2 ) + QJ(−Ψ2 , θ1 ) + QJ(Ψ, −θ2 ) − θ2 = 0. Pγ dt Then we get 󵄩 󵄩 󵄩 3 󵄩󵄩 1 󵄩 󵄩 3 󵄩󵄩 1 󵄩 󵄩󵄩 d 󵄩󵄩 ‖Aφ2 − Aθ2 ‖ ≤ C 󵄩󵄩󵄩A 2 Ψ2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 θ1 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩󵄩󵄩󵄩A 2 θ2 󵄩󵄩󵄩 + 󵄩󵄩󵄩 θ2 󵄩󵄩󵄩 󵄩󵄩 dt 󵄩󵄩 −1

−1

−1 2 2 + Cλm+1 ≤ Cλm+1 , ≤ Cλm+1

which implies −3

2 ‖φ2 − θ2 ‖ ≤ Cλm+1 .

(3.5.26)

For any solution (Ψ(t), θ(t)) = (Ψ1 + Ψ2 , θ1 + θ2 ) of problem (3.5.1)–(3.5.4), we have 󵄩 󵄩 󵄩 󵄩 distH 2 ×H ((Ψ, θ), Σ1 ) ≤ 󵄩󵄩󵄩Ψ − (Ψ1 + φ1 )󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩θ − (θ1 + φ2 )󵄩󵄩󵄩 ≤ ‖Ψ2 − φ1 ‖H 2 + ‖θ2 − φ2 ‖ −3

2 , ≤ C‖AΨ2 − Aφ1 ‖ + ‖θ2 − φ2 ‖ ≤ Cλm+1

∀t ≥ t∗ ,

where t∗ is chosen as in Lemmas 3.5.2 and 3.5.3. Theorem 3.5.1 has been proved.

358 | 3 The approximate inertial manifold Now consider another implicit approximate inertial manifold. The manifold is determined by the contraction mapping, and provides a high-order approximation for the global attractor on Σ1 . Introduce the balls: 󵄩 3 󵄩 Bm = {Ψ1 ∈ Pm H : 󵄩󵄩󵄩A 2 Ψ1 󵄩󵄩󵄩 ≤ 2M0 }, 󵄩 1 󵄩 Om = {θ1 ∈ Pm H : 󵄩󵄩󵄩A 2 θ1 󵄩󵄩󵄩 ≤ 2M0 }, 󵄩󵄩 32 󵄩󵄩 B⊥ m = {g ∈ Qm H : 󵄩 󵄩A g 󵄩󵄩 ≤ 2M0 }, 󵄩󵄩 21 󵄩󵄩 ⊥ Om = {h ∈ Qm H : 󵄩󵄩A h󵄩󵄩 ≤ 2M0 }, where M0 is a constant as in Lemma 3.5.2. ⊥ Define the mapping G : Bm × Om → B⊥ m × Om such that G(Ψ1 , θ1 ) = (g, h) for any (Ψ1 , θ1 ) ∈ Bm × Om and where (g, h) is determined by the following equations: A2 g + QJ(Ψ1 + g, AΨ1 + Ag) −

Rα QB(θ1 − h) = 0, Pγ

1 Ah + QJ(Ψ1 + g, θ1 + h) = 0. Pγ

(3.5.27) (3.5.28)

Firstly we prove that (g, h) is uniquely determined by equations (3.5.27)–(3.5.28). Lemma 3.5.4. There exists an integer m0 , which only depends on parameters, such that ⊥ for m ≥ m0 , equations (3.5.27)–(3.5.28) possess a unique solution (g, h) ∈ B⊥ m × Om , ∀(Ψ1 , θ1 ) ∈ Bm × Om . ̃ : Proof. Use the fixed point principle, by setting (Ψ1 , θ1 ) ∈ Bm × Om and defining G ⊥ ⊥ ⊥ ⊥ ̃ , h ), which are Bm × Om → Qm H × Qm H as follows: for (g1 , h1 ) ∈ Bm × Bm , (g, h) = G(g 1 1 determined by the following equations: A2 g + QJ(Ψ1 + g1 , AΨ1 + Ag1 ) −

Rα QB(θ1 + h1 ) = 0, Pγ

1 Ah + QJ(Ψ1 + g1 , θ1 + h1 ) = 0. Pγ

(3.5.29) (3.5.30)

̃ has a fixed point, namely the solution of equations (3.5.27)–(3.5.28). InObviously, G deed, ̃ maps B⊥ × O⊥ into itself. By equation (3.5.29), we (1) When m is large enough, G m m get 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 󵄩 R 󵄩 󵄩 󵄩󵄩A g 󵄩󵄩 ≤ 󵄩󵄩J(Ψ1 + g1 , AΨ1 + Ag1 )󵄩󵄩󵄩 + α 󵄩󵄩󵄩B(θ1 + h1 )󵄩󵄩󵄩 Pγ

3 1 󵄩 3 󵄩2 R 󵄩 1 󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ1 + A 2 g1 󵄩󵄩󵄩 + α 󵄩󵄩󵄩A 2 θ1 + A 2 h1 󵄩󵄩󵄩 ≤ C. P

γ

3.5 Bernard convection equation

| 359

From the above, we get − 21 󵄩󵄩 32 󵄩󵄩 . 󵄩󵄩A g 󵄩󵄩 ≤ Cλm+1

Since λm+1 → ∞, there exists k1 , which depends on parameters, such that, when m ≥ k1 , g ∈ B⊥ m. Through equation (3.5.28) we get 1 󵄩 󵄩 ‖Ah‖ ≤ 󵄩󵄩󵄩J(Ψ1 + g1 , θ1 + h1 )󵄩󵄩󵄩 Pγ 3 1 󵄩 3 󵄩󵄩 1 󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ1 + A 2 g1 󵄩󵄩󵄩󵄩󵄩󵄩A 2 θ1 + A 2 h1 󵄩󵄩󵄩 ⊥ ≤ C(Ψ1 , θ1 ) ∈ Bm × Qm , (g1 , h1 ) ∈ B⊥ m × Om . Hence if − 21 󵄩󵄩 21 󵄩󵄩 , 󵄩󵄩A h󵄩󵄩 ≤ Cλm+1

then there exists k2 , which depends on parameters, such that, when m ≥ k2 , h ∈ O⊥ m. ̃ is contracting. (2) G ⊥ Let (g1 , h1 ), (g2 , h2 ) ∈ B⊥ m × Om . By equation (3.5.29) we get A2 g(g1 , h1 ) − A2 g(g2 , h2 ) + QJ(Ψ1 + g1 , AΨ1 + Ag1 ) − QJ(Ψ1 + g2 , AΨ1 + Ag2 ) − +

Rα QB(θ1 + h2 ) = 0, Pγ

Rα QB(θ1 + h1 ) Pγ

thus we have A2 g(g1 , h1 ) − A2 g(g2 , h2 ) + QJ(g1 − g2 , AΨ1 + Ag1 ) + QJ(Ψ1 + g2 , Ag1 − Ag2 ) −

Rα QB(h1 − h2 ) = 0. Pγ

Then also 󵄩 󵄩󵄩 2 2 󵄩󵄩A g(g1 , h1 ) − A g(g2 , h2 )󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 R 󵄩 󵄩 ≤ 󵄩󵄩󵄩QJ(g1 − g2 , AΨ1 + Ag1 )󵄩󵄩󵄩 + 󵄩󵄩󵄩QJ(Ψ1 + g2 , Ag1 − Ag2 )󵄩󵄩󵄩 + α 󵄩󵄩󵄩QB(h1 − h2 )󵄩󵄩󵄩 Pγ 3 3 3 3 󵄩󵄩 3 󵄩 󵄩󵄩 3 󵄩 󵄩 󵄩 3 󵄩 3 󵄩 1 ≤ C 󵄩󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ1 + A 2 g1 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ1 + A 2 g2 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 (h1 − h2 )󵄩󵄩󵄩 3 1 󵄩 3 󵄩 󵄩 󵄩 1 ≤ C 󵄩󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 h1 − A 2 h2 󵄩󵄩󵄩, which yields 3 󵄩 󵄩󵄩 32 󵄩󵄩A g(g1 , h1 ) − A 2 g(g2 , h2 )󵄩󵄩󵄩 1 3 1 1 − 21 󵄩 3 󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩 + Cλ− 2 󵄩󵄩󵄩A 2 h1 − A 2 h2 󵄩󵄩󵄩. ≤ Cλm+1 󵄩 󵄩 󵄩 m+1 󵄩

(3.5.31)

360 | 3 The approximate inertial manifold Through equation (3.5.30), we get 1 1 Ah(g1 , h1 ) − Ah(g2 , h2 ) + QJ(Ψ1 + g1 , θ1 + h1 ) Pγ Pγ − QJ(Ψ2 + g2 , θ2 + h2 ) = 0, that is, 1 1 Ah(g1 , h1 ) − Ah(g2 , h2 ) + QJ(g1 − g2 , θ1 + h1 ) Pγ Pγ + QJ(Ψ1 + g2 , h1 − h2 ) = 0. Therefore 1 󵄩󵄩 󵄩 󵄩Ah(g1 , h1 ) − Ah(g2 , h2 )󵄩󵄩󵄩 Pγ 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩QJ(g1 − g2 , θ1 + h1 )󵄩󵄩󵄩 + 󵄩󵄩󵄩QJ(Ψ1 + g2 , h1 − h2 )󵄩󵄩󵄩 3 1 3 1 󵄩󵄩 1 󵄩 󵄩󵄩 1 󵄩 󵄩 3 󵄩 3 ≤ C 󵄩󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 θ1 + A 2 h1 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 Ψ1 + A 2 g2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 h1 − A 2 h2 󵄩󵄩󵄩 3 1 󵄩 󵄩 󵄩 3 󵄩 1 ≤ C 󵄩󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 h1 − A 2 h2 󵄩󵄩󵄩, which suggests 1 󵄩 󵄩󵄩 21 󵄩󵄩A h(g1 , h1 ) − A 2 h(g2 , h2 )󵄩󵄩󵄩 3 1 − 21 󵄩 3 − 21 󵄩 1 󵄩 󵄩󵄩A 2 h1 − A 2 h2 󵄩󵄩󵄩. ≤ Cλm+1 󵄩󵄩󵄩A 2 g1 − A 2 g2 󵄩󵄩󵄩 + Cλm+1 󵄩 󵄩

(3.5.32)

Since λm+1 → ∞, by equations (3.5.31)–(3.5.32) we know that there exists m0 ≥ k0 such ̃ is contracting. that whenever m ≥ m0 , G ̃ has a unique fixed point in B⊥ ×Q⊥ . Lemma 3.5.4 By the preceding we know that G m m has been proved. Let Σ2 = graph(G). Then Σ2 is an approximate inertial manifold. Indeed, we have Theorem 3.5.2. There exists m0 ≥ k0 such that for any m ≥ m0 , there exists a constant M, for which we have −2 distH 2 ×H ((Ψ(t), θ(t)), Σ2 ) ≤ Mλm+1 ,

∀t ≥ t∗ ,

where (Ψ(t), θ(t)) is any solution of equations (3.5.1)–(3.5.4), M only depends on parameters, t∗ depends on parameters and R, whenever ‖Ψ0 ‖H 4 ≤ R, ‖θ0 ‖H 2 ≤ R. Proof. From equations (3.5.27)–(3.5.32) we get A2 g − A2 Ψ2 + QJ(Ψ1 + g, AΨ1 + Ag) − QJ(Ψ1 , AΨ) −

Rα R d QB(θ1 + h) + α QB(θ) − AΨ2 = 0, Pγ Pγ dt

3.5 Bernard convection equation

|

361

that is, A2 g − A2 Ψ2 + QJ(g − Ψ2 , AΨ1 + Ag) + QJ(Ψ, Ag − AΨ2 ) −

Rα d QB(h − θ2 ) − AΨ2 = 0. Pγ dt

Then we arrive at 󵄩 󵄩 󵄩󵄩 2 2 󵄩 󵄩󵄩A g − A Ψ2 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩J(g − Ψ2 , AΨ1 + Ag)󵄩󵄩󵄩

󵄩󵄩 󵄩 󵄩 󵄩 R 󵄩 󵄩 󵄩󵄩 d 󵄩 + 󵄩󵄩󵄩J(Ψ, Ag − AΨ2 )󵄩󵄩󵄩 + α 󵄩󵄩󵄩B(h − θ2 )󵄩󵄩󵄩 + 󵄩󵄩󵄩 AΨ2 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 dt Pγ 3 3 3 3 󵄩󵄩 󵄩 󵄩 ≤ C 󵄩󵄩󵄩A 2 g − A 2 Ψ2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 Ψ1 + A 2 g 󵄩󵄩󵄩 󵄩󵄩 󵄩 3 1 1 󵄩 󵄩 3 󵄩󵄩 3 󵄩 󵄩 󵄩󵄩 d + C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩󵄩󵄩󵄩A 2 g − A 2 Ψ2 󵄩󵄩󵄩 + C‖A 2 h − A 2 θ2 󵄩󵄩󵄩 + 󵄩󵄩󵄩 AΨ2 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 dt 1 3 1 󵄩 󵄩 󵄩 󵄩󵄩 32 −1 ≤ C 󵄩󵄩A g − A 2 Ψ2 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 h − A 2 θ2 󵄩󵄩󵄩 + Cλm+1 1 − 21 󵄩 2 󵄩 1 󵄩 2 󵄩 −1 ≤ Cλm+1 󵄩󵄩󵄩A g − A Ψ2 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 h − A 2 θ2 󵄩󵄩󵄩 + Cλm+1 .

Thus there exists m0 ≥ k0 such that for all m ≥ m0 , we have 1 󵄩󵄩 2 󵄩 1 󵄩 2 󵄩 −1 󵄩󵄩A g − A Ψ2 󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩A 2 h − A 2 θ2 󵄩󵄩󵄩 + Cλm+1

−1

−1 2 ≤ Cλm+1 ‖Ah − Aθ2 ‖ + Cλm+1 .

(3.5.33)

Taking the difference of equations (3.5.28) and (3.5.13), it follows that d 1 (Ah − Aθ2 ) + QJ(Ψ1 + g, θ1 + h) − QJ(Ψ1 , θ) − θ2 = 0, Pγ dt that is, 1 d (Ah − Aθ2 ) + QJ(g − Ψ2 , θ1 + h) + QJ(Ψ, h − θ2 ) − θ2 = 0, Pγ dt which implies 3 1 󵄩 3 󵄩󵄩 1 󵄩 ‖Ah − Aθ2 ‖ ≤ C 󵄩󵄩󵄩A 2 g − A 2 Ψ2 󵄩󵄩󵄩󵄩󵄩󵄩A 2 θ1 + A 2 h󵄩󵄩󵄩 1 󵄩 󵄩 3 󵄩󵄩 1 −1 + C 󵄩󵄩󵄩A 2 Ψ󵄩󵄩󵄩󵄩󵄩󵄩A 2 h − A 2 θ2 󵄩󵄩󵄩 + Cλm+1 3 1 󵄩 3 󵄩 󵄩 󵄩 1 −1 ≤ C 󵄩󵄩󵄩A 2 g − A 2 Ψ2 󵄩󵄩󵄩 + C 󵄩󵄩󵄩A 2 h − A 2 θ2 󵄩󵄩󵄩 + Cλm+1 − 21 󵄩 2 − 21 2 󵄩 −1 ‖Ah − Aθ2 ‖ + Cλm+1 . ≤ Cλm+1 󵄩󵄩󵄩A g − A Ψ2 󵄩󵄩󵄩 + Cλm+1

(3.5.34)

By virtue of (3.5.33)–(3.5.34), we deduce that, when m ≥ m0 , 1 − 21 󵄩 2 󵄩󵄩 2 2 󵄩 󵄩󵄩A g − A2 Ψ2 󵄩󵄩󵄩 + Cλ− 2 ‖Ah − Aθ2 ‖ + Cλ−1 . 󵄩󵄩A g − A Ψ2 󵄩󵄩󵄩 + ‖Ah − Aθ2 ‖ ≤ Cλm+1 m+1 󵄩 󵄩 m+1

362 | 3 The approximate inertial manifold Since λm+1 → ∞, we know that there exists k0 ≥ m0 , such that for all m ≥ k0 we have 󵄩󵄩 2 −1 2 󵄩 󵄩󵄩A g − A Ψ2 󵄩󵄩󵄩 + ‖Ah − Aθ2 ‖ ≤ Cλm+1 . Hence, we have −2 ‖Ag − AΨ2 ‖ ≤ Cλm+1 , −2 ‖h − θ2 ‖ ≤ Cλm+1 .

Then for any solution (Ψ(t), θ(t)) of problem (3.5.1)–(3.5.4), we have 󵄩 󵄩 󵄩 distH 2 ×H ((Ψ, θ), Σ2 ) ≤ 󵄩󵄩󵄩Ψ(t) − (Ψ1 + g)󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩θ(t) − (θ1 + h)‖ ≤ ‖Ψ2 − g‖H 2 + ‖θ2 − h‖ −2 ≤ C‖AΨ2 − Ag‖ + ‖θ2 − h‖ ≤ Cλm+1 .

Theorem 3.5.2 has been proved. Finally, we introduce the nonlinear Galerkin method to approximate the inertial manifold Σ2 . To seek for the approximate solution Ψm , θm ∈ Pm H, we need the following equations: R d { AΨm + PJ(Ψm , AΨm ) + PJ(g, AΨm ) + PJ(Ψm , Ag) + A2 Ψm − α PB(θm ) = 0, { { { dt Pγ { { dθm 1 { + PJ(Ψm , θm ) + PJ(g, θm ) + PJ(Ψm , h) + Aθm = 0, { { dt Pγ { { { Ψ (0, x, y) = P Ψ (x, y), θ (0, x, y) = P θ m m 0 m m 0 (x, y), {

(3.5.35)

where (g, h) ∈ Qm H are determined by equations (3.5.27)–(3.5.28). To achieve this, (Ψ1 , θ1 ) is replaced with (Ψm , θm ). By the method in [90], we can prove the convergence of the approximate solution of the above form. Theorem 3.5.3. Let Ψ0 (x, y) ∈ H 2 , and assume θ0 (x, y) ∈ H 1 is a periodic function of x, y. Then the approximate solutions (Ψm , θm ), determined by equation (3.5.35), when m → ∞ converge to the generalized solution (Ψ, θ) of problem (3.5.1)–(3.5.4).

3.6 Long wave–short wave (LS) equation In 1977, Djordjevic et al. [48] studied two-dimensional gravity wave packet movement and first put forward the interaction of long and short waves. The general theory of interaction between long and short waves was studied by Grimshaw [82] and Denney [13]. In 1987, Guo [85] studied the existence and uniqueness of a smooth solution for the generalized LS equation. In 1991, Guo [87] studied the initial and periodic initial value problem for the generalized LS equations. 1994, Tsutsumi and Hatano [203] proved the existence and uniqueness of a solution in H 1 . In 1996, Guo and Miao [107]

3.6 Long wave–short wave (LS) equation

| 363

considered an even more general LS equation and improved the results of [203] for the LS equation; [203] also contained an open problem. Guo and Chen [94] proved the orbital stability of a solitary wave for the LS equation. In 1996, Guo and Wang [118] studied the long time behavior for a nonlinear dissipative LS equation. In 1998, Guo and Wang [123] proved the existenc of attrators. Consider the following generalized LS equations with the dissipative term [12]: iut + uxx − nu + iαu + βg(|u|2 )u + h1 (x) = 0, nt +

|u|2x

(3.6.1)

2

(3.6.2)

x ∈ Ω(−D, D), D > 0,

(3.6.3)

+ δn + γf (|u| )u + h2 (x) = 0,

with the initial value u|t=0 = u0 (x),

n|t=0 = n0 (x),

and periodic boundary condition u(x − D, t) = u(x + D, t),

n(x − D, t) = n(x + D, t),

∀x ∈ R, t ≥ 0,

(3.6.4)

where u = (u1 (x, t), . . . , uN (x, t)) is an unknown complex vector; n(x, t) is an unknown real function; g(s) and f (s), 0 ≤ s < ∞, are known real functions; h1 (x) and h2 (x) denote vector replication function and numerical function, respectively; α, β, γ and δ are real constants, α > 0. In order to construct inertial manifolds of problem (3.6.1)–(3.6.4), we prove the existence and uniqueness of a global smooth solution for problem (3.6.1)–(3.6.4). Lemma 3.6.1. Let u0 (x), h1 (x) and h2 (x) ∈ L2 (Ω). Then for any solution (u(t), n(t)) of problem (3.6.1)–(3.6.4), we have 󵄩 󵄩 sup 󵄩󵄩󵄩u(t)󵄩󵄩󵄩 ≤ M1 ,

0≤t≤T

∀T > 0,

where M1 = M1 (α, β, γ, δ, f , g, h1 , h2 , T) is a positive constant. Proof. Taking the inner product of equation (3.6.1) and u over H, we get (iut + uxx − nu + iαu + βg(|u|2 )u + h1 (x), u) = 0.

(3.6.5)

Taking the imaginary part of equation (3.6.5), we obtain 1 d 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩u(t)󵄩󵄩󵄩 + α󵄩󵄩󵄩u(t)󵄩󵄩󵄩 + Im(h, u) = 0. 2 dt 󵄩 By Gronwall lemma, we get the claim of Lemma 3.6.1. Lemma 3.6.2. Suppose (1) ρg(s) ≤ B1 s2−σ + C1 , s > 0, B1 > 0, C1 > 0, σ > 0; 3 (2) |f (s)| ≤ B2 s 2 + C2 , s > 0, B2 > 0, C2 > 0;

(3.6.6)

364 | 3 The approximate inertial manifold (3) h1 (x), h2 (x) ∈ L2 (Ω); (4) u0 (x) ∈ H 1 (Ω), n0 (x) ∈ L2 (Ω). Then we have 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩ux (t)󵄩󵄩󵄩 + sup 󵄩󵄩󵄩n(t)󵄩󵄩󵄩 ≤ M2 , 0≤t≤T

0≤t≤T

∀T > 0,

where M2 = M2 (T, ‖u0 ‖H 1 , ‖n0 ‖) is a positive constant. Proof. Taking the inner product of equation (3.6.1) and ut over H, we get (iut + uxx − nu + iαu + βg(|u|2 )u + h1 (x), ut ) = 0.

(3.6.7)

Taking the imaginary part of equation (3.6.7), we obtain −

1 d 1 d ‖ux ‖2 − ∫ n |u|2 dx + Re(iαu, ut ) 2 dt 2 dt 1 2 d + β ∫ g(|u| ) |u|2 dx + Re(h1 , ut ) = 0. 2 dt

By virtue of ∫n

d 2 d |u| dx = ∫ n|u|2 dx − ∫ |u|2 nt dx dt dt d = ∫ n|u|2 dx + δ ∫ n|u|2 dx + γ ∫ f (|u|2 )|u|2 dx + ∫ h2 |u|2 dx, dt

and letting

s

G(s) = ∫ g(τ)dτ, 0

we get the following result: −

1 d 1 d d 1 ‖ux ‖2 − ∫ n|u|2 dx − δ ∫ n|u|2 dx 2 dt 2 dt 2 dt 1 1 2 2 2 − γ ∫ f (|u| )|u| dx − ∫ h2 |u| dx 2 2 1 d d 2 Re(h1 , u) = 0. + β ∫ G(|u| )dx + Re(iαu, ut ) + 2 dt dt

Integrating equation (3.6.8) over t ∈ (0, T), we get t

‖ux ‖2 + ∫ n|u|2 dx + δ ∫ ∫ n|u|2 dx t

0 2

t

+ γ ∫ ∫ f (|u| )|u| dx + ∫ ∫ h2 |u|2 dx − β ∫ G(|u|2 )dx 0

2

0

(3.6.8)

3.6 Long wave–short wave (LS) equation

| 365

t

− 2 ∫ Re(iαu, ut )dt − 2 Re(h1 , u) 0

󵄨2 󵄨 󵄩2 󵄩 = 󵄩󵄩󵄩ux (0)󵄩󵄩󵄩 + ∫ n(0)󵄨󵄨󵄨u(0)󵄨󵄨󵄨 dx 󵄨2 󵄨 − β ∫ G(󵄨󵄨󵄨u(0)󵄨󵄨󵄨 )dx − 2 Re(h1 , u(0)).

(3.6.9)

Now we estimate each term of equation (3.6.9). For all ρ > 0 we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 4 2 2 󵄨󵄨 ∫ n|u| dx󵄨󵄨󵄨 ≤ ρ‖n‖ + C‖u‖4 ≤ ρ‖n‖ + ρ‖ux ‖ + M. 󵄨󵄨 󵄨󵄨

(3.6.10)

Similarly we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 2 󵄨󵄨δ ∫ n|u| dx󵄨󵄨󵄨 ≤ ρ‖n‖ + ρ‖ux ‖ + M, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 5 2 󵄨󵄨γ ∫ f (|u| )|u| dx󵄨󵄨󵄨 ≤ |γ| ∫ B2 |u| dx + C2 |γ| ∫ |u| dx 󵄨󵄨 󵄨󵄨 3

≤ M‖u‖H2 + M ≤ M‖ux ‖2 + M, 1

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 󵄨󵄨 ∫ h2 |u| dx󵄨󵄨󵄨 ≤ ρ‖ux ‖ + M. 󵄨󵄨 󵄨󵄨

(3.6.11)

(3.6.12) (3.6.13)

Using assumption (1), we have βG(s) ≤

1 B s3−σ + C1 s, 3 1

∀s > 0,

and then 1 β ∫ G(|u|2 )dx ≤ B1 ∫ |u|6−2σ dx + C1 ∫ |u|2 dx 3 ≤ M‖ux ‖2−σ + M ≤ M‖ux ‖2 + M, 󵄨󵄨 󵄨 󵄨󵄨2 Re(h1 , u)󵄨󵄨󵄨 ≤ 2‖h1 ‖‖u‖ ≤ M. Taking the real part of (3.6.5), we obtain − Re(iu, ut ) = ‖ux ‖2 + ∫ n|u|2 dx − β ∫ g(|u|2 )|u|2 dx − Re(u, h1 ). By virtue of assumption (1), we have β ∫ g(|u|2 )|u|2 dx ≤ B1 ∫ |u|6−2σ dx + C1 ∫ |u|2 dx ≤ ‖ux ‖2 + M, which yields − Re(iu, ut ) ≥ ∫ n|u|2 dx − M,

(3.6.14) (3.6.15)

366 | 3 The approximate inertial manifold that is, − 2α Re(iu, ut ) ≥ 2α ∫ n|u|2 dx − M.

(3.6.16)

From equations (3.6.9)–(3.6.16), we get t

‖ux ‖2 ≤ ρ‖n‖2 + 2ρ‖ux ‖2 + (2α + 1)ρ ∫ ‖n‖2 dt 0

t

+ (2αρ + 2ρ + M) ∫ ‖ux ‖2 dt + Mt + M.

(3.6.17)

0

Taking the inner product of equation (3.6.2) and n over H, we obtain 1 d ‖n‖2 + ∫ n|u|2x dx + δ‖n‖2 + γ ∫ f (|u|2 )ndx + ∫ h2 ndx = 0. 2 dt

(3.6.18)

Since ∫ n|u|2x dx = ∫ nux udx + ∫ nuux dx = i ∫(ut ux − ut ux )dx + 2 Re ∫ iαuux dx + 2 Re ∫ h1 ux dx,

(3.6.19)

d ∫(iuux − iux u)dx = i ∫(ut ux + uuxt − uxt u − ux ut )dx dt = i ∫(ut ux − ut ux + ut ux − ut ux )dx = 2i ∫(ut ux − uux )dx,

(3.6.20)

from equations (3.6.18)–(3.6.20), we get 1 d 1 d ‖n‖2 + ∫(iuux − iux u)dx + 2 Re ∫ iαuux dx 2 dt 2 dt + 2 Re ∫ h1 ux dx + δ‖n‖2 + γ ∫ f (|u|2 )ndx + ∫ h2 ndx = 0.

(3.6.21)

Integrating equation (3.6.21) over t ∈ (0, t), we get t

2

‖n‖ + ∫ iuux dx − ∫ iux udx + 4 Re ∫ ∫ iαuux dx t

t

0

0

0

t

+ 4 Re ∫ ∫ h1 ux dx + 2δ ∫ ‖n‖2 dx + 2γ ∫ ∫ f (|u|2 )ndxdt t

0

+ 2 ∫ ∫ h2 ndxdt 0

󵄩 󵄩2 = 󵄩󵄩󵄩n(0)󵄩󵄩󵄩 + i ∫ u(0)ux (0)dx − i ∫ ux (0)u(0)dx.

(3.6.22)

3.6 Long wave–short wave (LS) equation

| 367

Since 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 󵄨󵄨i ∫(uux − ux u)dx󵄨󵄨󵄨 ≤ ρ‖ux ‖ + M, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 󵄨󵄨4 Re ∫ iαuux dx󵄨󵄨󵄨 ≤ 4α ∫ |u||ux | dx ≤ ρ‖ux ‖ + M, 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨4 Re ∫ h1 ux dx󵄨󵄨󵄨 ≤ 4 ∫ |h1 ||ux |2 dx ≤ ρ‖ux ‖2 + M, 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 3 󵄨󵄨2γ ∫ f (|u| )ndx󵄨󵄨󵄨 ≤ C ∫ |u| |n|dx + C ∫ |n|dx 󵄨󵄨 󵄨󵄨 ≤ ‖n‖2 + M‖ux ‖2 + M,

(3.6.23) (3.6.24) (3.6.25)

(3.6.26)

together with equations (3.6.22)–(3.6.26), we get t

‖n‖2 ≤ ρ‖ux ‖2 + (2ρ + M) ∫ ‖ux ‖2 dt 0

t

+ 2(1 + |δ|) ∫ ‖n‖2 dt + Mt + M.

(3.6.27)

0

From equations (3.6.17)–(3.6.27), we get ‖ux ‖2 + ‖n‖2 ≤ ρ‖n‖2 + 3ρ‖ux ‖2 + (2αρ + 4ρ + M) ∫ ‖ux ‖2 dt t

+ (2αρ + ρ + 2 + 2|δ|) ∫ ‖n‖2 dt + Mt + M. 0

Choosing ρ = 41 , from the above equation we get t

‖ux ‖2 + ‖n‖2 ≤ M ∫(‖ux ‖2 + ‖n‖2 )dt + M,

∀0 ≤ t ≤ T.

0

By Gronwall lemma, we get ‖ux ‖2 + ‖n‖2 ≤ M,

∀0 ≤ t ≤ T.

Lemma 3.6.2 has been proved. By virtue of Lemmas 3.6.1 and 3.6.2, we get sup ‖u‖H 1 ≤ M,

0≤t≤T

sup ‖u‖∞ ≤ M.

0≤t≤T

(3.6.28)

368 | 3 The approximate inertial manifold Lemma 3.6.3. Let conditions of Lemma 3.6.2 be valid, and g ∈ C 1 [0, +∞), u0 ∈ H 2 . Then we have sup ‖nt ‖ + sup ‖ut ‖ ≤ M3 ,

0≤t≤T

0≤t≤T

∀T > 0,

where M3 = M3 (T, ‖u0 ‖H 2 , ‖n0 ‖) is a positive constant. Proof. From equation (3.6.2) we get 󵄩󵄩󵄩 󵄩󵄩󵄩 d 󵄩 󵄩 ‖nt ‖ ≤ 󵄩󵄩󵄩 (uu)󵄩󵄩󵄩 + |δ|‖n‖ + 󵄩󵄩󵄩γf (|u|2 )󵄩󵄩󵄩 + ‖h2 ‖ 󵄩󵄩 󵄩󵄩 dt 󵄩 󵄩 ≤ ‖ux u + uux ‖ + |δ|‖n‖ + 󵄩󵄩󵄩γf (|u|2 )󵄩󵄩󵄩 + ‖h2 ‖.

(3.6.29)

In virtue of inequality (3.6.28) we get ‖nt ‖ ≤ M. Differentiating (3.6.1) with respect to t, we get i

d u + uxxt − nt u − nut + iαut + βg 󸀠 (|u|2 )|u|2 ut dt t + βg 󸀠 (|u|2 )|u|2 ut + βg(|u|2 )ut = 0.

(3.6.30)

Taking the inner product of equation (3.6.30) and ut over H, and taking its imaginary part, we obtain 1 d ‖u ‖2 + α‖ut ‖2 − Im ∫ nt uut dx + Im ∫ βg 󸀠 (|u|2 )u2 u2t dx = 0. 2 dt t From the above equation, we get sup ‖ut ‖ ≤ M.

0≤t≤T

Lemma 3.6.4. Let the conditions of Lemma 3.6.3 be satisfied and consider f ∈ C 1 [0, +∞), n0 , h2 ∈ H 1 . Then we have sup ‖nx ‖ ≤ M4 ,

0≤t≤T

where M4 = M4 (T, ‖u0 ‖H 2 , ‖n0 ‖H 2 ) is a positive constant. Proof. Differentiating equation (3.6.2) with respect to x, we get nxt + uxx u + 2ux ux + uuxx + δnx + γf 󸀠 (|u|2 )|u|2x + h󸀠2 = 0. Taking the inner product of equation (3.6.31) and nx over H, we obtain 1 d ‖n ‖2 + ∫ uxx unx dx + 2 ∫ ux ux nx dx 2 dt x + ∫ uuxx nx dx + δ‖nx ‖2 + γ ∫ f 󸀠 (|u|2 )ux unx dx + γ ∫ f 󸀠 (|u|2 )ux unx dx + ∫ h󸀠2 nx dx = 0.

(3.6.31)

3.6 Long wave–short wave (LS) equation

| 369

After a detailed calculation, we get d ‖n ‖2 ≤ M‖nx ‖2 + M. dt x By Gronwall lemma, we get the claim of the lemma. Lemma 3.6.5. Let the conditions of Lemma 3.6.4 be satisfied and consider g ∈ C 2 [0, +∞), h1 , n0 ∈ H 1 , u0 ∈ H 3 . Then we have sup ‖ntx ‖ + sup ‖utx ‖ ≤ M5 ,

0≤t≤T

0≤t≤T

where M5 = M5 (T, ‖u0 ‖H 3 , ‖n0 ‖H 1 ) is a positive constant. Proof. From equation (3.6.31), we get 󵄩 󵄩 󵄩 󵄩 ‖nxt ‖ ≤ ‖uxx u‖ + 2‖ux ux ‖ + ‖uuxx ‖ + ‖δnx ‖ + 󵄩󵄩󵄩γf 󸀠 (|u|2 )|u|2x 󵄩󵄩󵄩 + 󵄩󵄩󵄩h󸀠2 󵄩󵄩󵄩 ≤ M.

(3.6.32)

Differentiating equation (3.6.1) with respect to x and t, we get iuttx + uxxxt − nxt u − nx ut − nt ux − nuxt + iαuxt

+ βg 󸀠󸀠 (|u|2 )|u|2t |u|2x u + βg 󸀠 (|u|2 )|u|2tx u + βg 󸀠 (|u|2 )|u|2x ut + βg 󸀠 (|u|2 )|u|2t ux + βg(|u|2 )uxt = 0.

Taking the inner product of above equation and uxt over H, and then taking its imaginary part, we obtain 1 d ‖u ‖2 + α‖utx ‖2 ≤ ∫ |nxt unxt |dx + ∫ |nx ut uxt |dx + ∫ |nt ux uxt |dx 2 dt tx 󵄨 󵄨 + ∫ 󵄨󵄨󵄨βg 󸀠󸀠 (|u|2 )uux (ux ut u2 + ux ut |u|2 + ut ut |u|2 + ux ut u2 )󵄨󵄨󵄨dx 󵄨 󵄨 + ∫ 󵄨󵄨󵄨βg 󸀠 (|u|2 )uuxt (utx u + ux ut + ux ut + uutx )󵄨󵄨󵄨dx 󵄨 󵄨 + ∫ 󵄨󵄨󵄨βg 󸀠 (|u|2 )(ux u + uux )ut utx 󵄨󵄨󵄨dx 󵄨 󵄨 + ∫ 󵄨󵄨󵄨βg 󸀠 (|u|2 )(ut u + uut )ux uxt 󵄨󵄨󵄨dx ≤ M‖uxt ‖2 + M, and then we have d ‖u ‖2 ≤ M‖uxt ‖2 + M. dt xt

(3.6.33)

Lemma 3.6.5 is proved. Lemma 3.6.6. Let the conditions of Lemma 3.6.4 be satisfied and consider u0 ∈ H 4 , n0 ∈ H 2 . Then we have sup ‖ntt ‖ + sup ‖utt ‖ ≤ M6 ,

0≤t≤T

0≤t≤T

where M6 = M6 (T, ‖u0 ‖H 4 , ‖n0 ‖H 2 ) is a positive constant.

370 | 3 The approximate inertial manifold Proof. Differentiating (3.6.2) with respect to t, we get 󵄩 󵄩 ‖ntt ‖ ≤ ‖uxt u‖ + ‖ux ut ‖ + ‖ut ux ‖ + ‖uuxt ‖ + ‖δnt ‖ + 󵄩󵄩󵄩γf 󸀠 (|u|2 )(ut u + uut )󵄩󵄩󵄩 󵄩 󵄩 ≤ 2‖u‖∞ ‖uxt ‖ + 2‖ux ‖∞ ‖ut ‖ + ‖δnt ‖ + 2󵄩󵄩󵄩γf 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖ut ‖ ≤ M.

(3.6.34)

Differentiating equation (3.6.2) with respect to x twice, we get iuttt + uxxtt − ntt u − nt ut − nutt + iαutt

+ βg 󸀠󸀠 (|u|2 )|u|2 |u|2t u + βg 󸀠 (|u|2 )|u|2tt u + βg 󸀠 (|u|2 )|u|2t ut

+ βg(|u|2 )utt = 0.

Taking the inner product of the above equation and utt over H, and taking its imaginary part, we obtain 1 d ‖u ‖2 + α‖utt ‖2 − Im ∫ ntt uutt dx − 2 Im ∫ nt ut utt dx 2 dt tt + Im ∫ βg 󸀠󸀠 (|u|2 )uutt (ut u + uut )2 dx + Im ∫ βg 󸀠 (|u|2 )uutt |u|2tt dx + 2 Im ∫ βg 󸀠 (|u|2 )ut utt (ut u + uut )dx = 0, which yields 1 d 1 ‖u ‖2 + α‖utt ‖2 ≤ ‖u‖∞ (‖ntt ‖2 + ‖utt ‖2 ) + ‖ut ‖∞ (‖nt ‖2 + ‖utt ‖2 ) 2 dt tt 2 󵄩 󵄩 + 2󵄩󵄩󵄩βg 󸀠󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖2∞ ‖ut ‖2∞ (‖u‖2 + ‖utt ‖2 ) 󵄩 󵄩 + 󵄩󵄩󵄩βg 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖2∞ ‖utt ‖2 󵄩 󵄩 + 3󵄩󵄩󵄩βg 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖ut ‖2∞ (‖u‖2 + ‖utt ‖2 ) ≤ M‖utt ‖2 + M.

By Gronwall inequality, we have sup ‖utt ‖ ≤ M,

0≤t≤T

∀T > 0.

Lemma 3.6.7. Let the conditions of Lemma 3.6.6 be satisfied and assume (1) g(s) ∈ C 2k−2 , f (s) ∈ C 2k−3 , 0 ≤ s < ∞, k ≥ 2; (2) h1 ∈ H 2k−2 , h2 ∈ H 2k−3 ; (3) u0 ∈ H 2k , n0 ∈ H 2k−2 . Then we have 󵄩 󵄩󵄩 + 󵄩󵄩󵄩Dk−1 Dx u󵄩󵄩󵄩 + 󵄩󵄩󵄩Dk u󵄩󵄩󵄩 + 󵄩󵄩󵄩Dk n󵄩󵄩󵄩) ≤ M7 , sup (󵄩󵄩󵄩Dk−1 t Dx n󵄩 󵄩 󵄩 t 󵄩 󵄩 t 󵄩 󵄩 t 󵄩

0≤t≤T

where M7 = M7 (T, ‖u0 ‖H 2k , ‖n0 ‖H 2k−2 ) is a positive constant.

(3.6.35)

3.6 Long wave–short wave (LS) equation

| 371

Proof. We can prove the claim by induction. The proof is similar to that of Lemma 9 in [87]. From Galerkin method and the technique used in [87], together with the a priori estimate of the above lemma, we obtain Theorem 3.6.1. Assume (1) g(s) ∈ C 2k−2 [0, ∞), βg(s) ≤ B1 s2−σ + C1 , s > 0, B1 > 0, C1 > 0, σ > 0; 3 (2) f (s) ∈ C 2k−3 [0, ∞), |f (s)| ≤ B2 s 2 + C2 , s > 0, B2 > 0, C2 > 0; (3) h1 ∈ H 2k−2 , h2 ∈ H 2k−3 ; (4) u0 ∈ H 2k , n0 ∈ H 2k−2 , k ≥ 2. Then there exists a unique global smooth solution (u(x, t), n(x, t)) of problem (3.6.1)–(3.6.4), u(x, t) ∈ L∞ ((0, T); H 2k (Ω)),

n(x, t) ∈ L∞ ((0, T); H 2k−2 (Ω)),

j

Dt u ∈ L∞ ((0, T); H 2k−2j (Ω)),

D3t n ∈ L∞ ((0, T); H 2k−2j (Ω)).

Similarly, by Lemmas 3.6.1–3.6.4, we get an important case of approximate inertial manifolds. Theorem 3.6.2. Suppose that the conditions of Lemma 3.6.2 are satisfied and u0 ∈ H 2 (Ω), n0 ∈ H 1 (Ω). Then there exists a global unique solution (u(x, t), n(x, t)) of problem (3.6.1)–(3.6.4), u(x, t) ∈ L∞ ((0, T); H 2 (Ω)),

n(x, t) ∈ L∞ ((0, T); H 1 (Ω)).

We note that, if f satisfies 3 󵄨󵄨 󵄨 󵄨󵄨f (s)󵄨󵄨󵄨 ≤ B2 s 2 ,

s > 0, B2 > 0,

then all the a priori estimates are consistent, all are independent of Ω = (−D, D). Applying again the tools from [87], and in the problem (3.6.1)–(3.6.4) letting D → ∞, we get Theorem 3.6.3. Let the conditions of Lemma 3.6.1 be satisfied. Then there exists a unique global smooth solution (u(x, t), n(x, t)) such that u(x, t) ∈ L∞ ((0, T); H 2k (R)),

n(x, t) ∈ L∞ ((0, T); H 2k−2 (R)).

For simplicity, we consider the simplified LS equations as follows:

nt +

iut − uxx − nu + iαu + h1 (x) = 0, |u|2x

2

+ δn + γf (|u| ) + h2 (x) = 0,

(3.6.36) (3.6.37)

372 | 3 The approximate inertial manifold with initial value u|t=0 = u0 (x),

n|t=0 = n0 (x),

x ∈ Ω = (−D, D), D > 0,

(3.6.38)

and periodic boundary condition u(x − D, t) = u(x + D, t),

n(x − D, t) = n(x + D, t),

∀x ∈ R, t ≥ 0.

(3.6.39)

In order to construct an approximate inertial manifold of problem (3.6.36)–(3.6.39), we set (H1 ) α > 0, σ > 0; 3 (H2 ) f (s) ∈ C 2 [0, ∞), |f (s)| ≤ B2 s 2 −ν + C1 , B1 > 0, C1 > 0, ν > 0; (H3 ) h1 , h2 ∈ H 1 (Ω); (H4 ) u0 ∈ H 3 (Ω), n0 ∈ H 1 (Ω). Note that if conditions (H1 )–(H4 ) are satisfied, then by Theorem 3.6.2 we know that there exists a unique global solution (u(x, t), n(x, t)) of problem (3.6.36)–(3.6.39), and the following lemma can be proved: Lemma 3.6.8. Suppose that (H1 )–(H2 ) are satisfied. Then we have ‖nx ‖ + ‖ntx ‖ ≤ C,

∀t ≥ t1 ,

where C1 depends only on the initial value; t1 depends on the initial value and R, whenever ‖(u0 , n0 )‖H 2 ×H 1 ≤ R. Lemma 3.6.9. Under assumptions (H1 )–(H2 ), we have ‖uxx ‖ + ‖uxxx ‖ ≤ C2 ,

∀t ≥ t2 ,

where C2 depends only on the initial value; t2 depends on the initial value and R, whenever ‖(u0 , n0 )‖H 3 ×H 1 ≤ R. In order to construct an approximate inertial manifold of problem (3.6.36)–(3.6.39), we write problem (3.6.36)–(3.6.37) in the form of abstract differential equations: du − Au − B(u, n) + iαu + h1 = 0, dt dn + H(u) + δn + γf (|u|2 ) + h2 = 0, dt i

(3.6.40) (3.6.41)

where B(u, n) = nu is a bilinear operator, H 1 × H → H, H(u) = |u|2x is a nonlinear operator, H 1 → H, A = −𝜕xx is an unbounded self-adjoint operator, and D(A) = {u ∈ H 2 : u(x + 2D) = u(x), n(x + 2D) = n(x)}.

3.6 Long wave–short wave (LS) equation

| 373

Then there exists an orthogonal basis {ωj }∞ j=1 composed of eigenvectors of A such that Aωj = λj ωj

0 ≤ λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λj → +∞,

j → ∞.

Also ∀m, let P = Pm be the projection of H onto the subspace spanned by {ω1 , . . . , ωm }. Set Q = Qm = I − Pm . Acting with Pm and Qm on equations (3.6.40) and (3.6.41), respectively, we have dy { { {i dt − Ay − Pm B(u, n) + iαy + Pm h1 = 0, { { {i dz − Az − Q B(u, n) + iαz + Q h = 0, m m 1 { dt

(3.6.42)

dp 2 { { { dt + Pm H(u) + δp + γPm f (|u| ) + Pm h2 = 0, { { { dq + Q H(u) + δq + γQ f (|u|2 ) + Q h = 0, m m m 2 { dt

(3.6.43)

where y = Pm u,

z = Qm u,

p = Pm n,

q = Qm n.

By Lemmas 3.6.8 and 3.6.9, we get 󵄩󵄩 21 󵄩󵄩 󵄩󵄩 21 󵄩󵄩 󵄩󵄩 21 󵄩󵄩 󵄩󵄩 21 󵄩󵄩 󵄩󵄩A z 󵄩󵄩, 󵄩󵄩A zt 󵄩󵄩, 󵄩󵄩A q󵄩󵄩, 󵄩󵄩A qt 󵄩󵄩 ≤ C, ‖z‖, ‖zt ‖, ‖q‖, ‖qt ‖ ≤

− 21 Cλm+1 ,

∀t ≥ t∗ , ∀t ≥ t∗

Define the mapping Φ : Pm H×Pm H → Qm H×Qm H such that for any (y, p) ∈ Pm H×Pm H, Φ(y, p) = (Ψ1 , Ψ2 ), which satisfies −AΨ1 − Qm B(y, p) + Qm h1 = 0, 2

δΨ2 + Qm H(y + Ψ1 ) + γQm f (|y| ) + Qm h2 = 0.

(3.6.44) (3.6.45)

Let Σ1 = graph(Φ). We will prove that it is an approximate inertial manifold of problem (3.6.36)–(3.6.39). Theorem 3.6.4. Suppose that (H1 )–(H4 ) are satisfied. Then there exists a constant K, which depends on the initial value, such that −1

2 distH 2 ×H ((u(t), n(t)), Σ1 ) ≤ Kλm+1 ,

∀t ≥ t∗ ,

where (u(t), n(t)) is a global solution of problem (3.6.36)–(3.6.39); t∗ only depends on the initial value and R, whenever ‖(u0 , n0 )‖H 3 ×H 1 ≤ R.

374 | 3 The approximate inertial manifold Proof. Taking the difference of equation (3.6.44) and (3.6.42), it follows that AΨ1 − Az = Qm B(u, n) − Qm B(y, p) − iαz − i

dz dt

= Qm B(u − y, n) + Qm B(y, n − p) − iαz − i dz , dt 󵄩󵄩 dz 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ‖AΨ1 − Az‖ ≤ 󵄩󵄩󵄩B(z, n)󵄩󵄩󵄩 + 󵄩󵄩󵄩B(z, q)󵄩󵄩󵄩 + α‖z‖ + 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 dt 󵄩󵄩 󵄩󵄩 dz 󵄩󵄩 󵄩 󵄩 ≤ ‖n‖∞ ‖z‖ + ‖y‖∞ ‖q‖ + α‖z‖ + 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 dt 󵄩󵄩

dz dt

= Qm B(z, n) + Qm B(y, q) − iαz − i

−1

2 ≤ Cλm+1 .

(3.6.46)

Taking away (3.6.45) from (3.6.43), we arrive at δΨ2 − δq = Qm H(n) − Qm H(y + Ψ1 ) + γQm f (|u|2 ) − γQm f (|y|2 ) +

dq . dt

(3.6.47)

Since H(u) − H(u − Ψ1 ) = |u|2x − |y + Ψ|2x

d d Ψ )u + (yx + Ψ1 )(u − y − Ψ1 ) dt 1 dt d d + u(ux − yx − Ψ1 ) + (u − y − Ψ1 )(yx + Ψ1 ), dt dt

= (ux − yx −

we have 1 󵄩 󵄩󵄩 󵄩 󵄩 1 󵄩󵄩H(u) − H(u + Ψ1 )󵄩󵄩󵄩 ≤ 2‖u‖∞ 󵄩󵄩󵄩A 2 Ψ1 − A 2 z 󵄩󵄩󵄩 1 󵄩 1 󵄩 + 2‖Ψ1 − z‖∞ 󵄩󵄩󵄩A 2 y + A 2 Ψ1 󵄩󵄩󵄩.

(3.6.48)

By equation (3.6.44) we have 󵄩 󵄩 ‖AΨ1 ‖ ≤ 󵄩󵄩󵄩B(y, p)󵄩󵄩󵄩 + ‖h1 ‖ ≤ ‖y‖∞ ‖p‖ + ‖h1 ‖ ≤ C‖y‖H 1 ‖n‖ + ‖h1 ‖ ≤ C,

and then 󵄩󵄩 21 󵄩󵄩 󵄩󵄩A Ψ1 󵄩󵄩 ≤ C.

(3.6.49)

From equations (3.6.48)–(3.6.49), we obtain 1 󵄩󵄩 󵄩 󵄩 1 󵄩 󵄩 1 󵄩 󵄩 1 󵄩 󵄩󵄩H(u) − H(y + Ψ1 )󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩A 2 Ψ1 − A 2 z 󵄩󵄩󵄩 + C(󵄩󵄩󵄩A 2 y󵄩󵄩󵄩 + 󵄩󵄩󵄩A 2 Ψ1 󵄩󵄩󵄩)‖Ψ1 − z‖H 1 1

1

1

2 ≤ C‖A 2 Ψ − A 2 z‖ ≤ Cλm+1 ,

(3.6.50)

3.7 One-dimensional ferromagnetic chain equation

󵄩 󸀠 󵄩󵄩 2 2 󵄩 2 2 󵄩 󵄩󵄩f (|u| ) − f (|y| )󵄩󵄩󵄩 = C 󵄩󵄩󵄩f (ξ )(|u| − |y| )󵄩󵄩󵄩 󵄩 󵄩 ≤ C 󵄩󵄩󵄩2(u + y)󵄩󵄩󵄩‖z‖ ≤ C(‖u‖∞ + ‖y‖∞ )‖z‖ −1 −1 ≤ C‖z‖ ≤ Cλm+1 ‖Az‖ ≤ Cλm+1 .

| 375

(3.6.51)

From equation (3.6.47), inequality (3.6.50) and equation (3.6.50) we have 󵄩󵄩 dq 󵄩󵄩 − 21 󵄩 󵄩 −1 ‖Ψ2 − q‖ ≤ Cλm+1 + 󵄩󵄩󵄩 󵄩󵄩󵄩 ≤ Cλm+1 . 󵄩󵄩 dt 󵄩󵄩

(3.6.52)

Therefore distH 2 ×H ((u(t), n(t)), Σ1 ) 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩u(t) − (y(t) + Ψ1 )󵄩󵄩󵄩H 2 + 󵄩󵄩󵄩n(t) − (p(t) + Ψ2 )󵄩󵄩󵄩 1

2 ≤ ‖AΨ1 − Az‖ + ‖Ψ2 − q‖ ≤ Cλm+1 .

Thus Theorem 3.6.4 has been proved.

3.7 One-dimensional ferromagnetic chain equation Consider the following one-dimensional ferromagnetic chain equation: 𝜕t u = −αu × (u × uxx ) + β(u × uxx ),

(x, t) ∈ Ω × R+

(3.7.1)

with initial value 󵄨󵄨 󵄨 󵄨󵄨u0 (x)󵄨󵄨󵄨 = 1,

u(x, 0) = u0 (x),

x ∈ Ω = (−D, D), D > 0,

(3.7.2)

and periodic boundary condition u(x − D, t) = u(x + D, t),

∀x ∈ R, t ≥ 0,

(3.7.3)

where × denotes the vector cross-product in R3 ; u = (u1 , u2 , u3 ) : R × R+ → R3 is a rotating vector field; α, β are constants; α > 0 is Gilbert damping constant. In 1995, Guo and Wang [115] studied the existence of an approximate inertial manifold of problem (3.7.1)–(3.7.3). From [217], we know that if u0 ∈ H 2 (Ω), |u0 (x)| = 1, then problem (3.7.1)–(3.7.3) possesses a unique global smooth solution u(x, t) such that u(x, t) ∈ L∞ (R+ , H 2 (Ω)). In order to construct an approximate inertial manifold, we need to establish a uniform in t estimate.

376 | 3 The approximate inertial manifold Lemma 3.7.1. Let |u0 (x)| = 1, x ∈ R. Then for any smooth solution of problem (3.7.1)–(3.7.3), we have 󵄨2 󵄨󵄨 + 󵄨󵄨u(x, t)󵄨󵄨󵄨 = 1, (x, t) ∈ R × R , 󵄩󵄩2 󵄩󵄩2 󵄩󵄩 󵄩󵄩 + 󵄩󵄩ux (t)󵄩󵄩 ≤ 󵄩󵄩ux (0)󵄩󵄩 , t ∈ R .

(3.7.4) (3.7.5)

Proof. Taking the dot product of equation (3.7.1) with u, we get 𝜕 󵄨󵄨 󵄨2 󵄨u(x, t)󵄨󵄨󵄨 = 0, 𝜕t 󵄨

(x, t) ∈ R × R+ ,

and thus equation (3.7.4) is established. Taking the inner product of equation (3.7.1) and uxx , we get −

1 d ‖u ‖2 = −α ∫(u × (u × uxx )) ⋅ uxx dx 2 dt x Ω

= −α ∫(uxx × u) ⋅ (u × uxx )dx = α ∫ |u × uxx |2 dx. Ω

Hence

Ω

d ‖u ‖2 ≤ 0, dt x

which yields equation (3.7.5). Introduce the subset Hρ of H as 󵄨 󵄨 Hρ = {u ∈ H : 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 = 1, ‖ux ‖ ≤ ρ}. Lemma 3.7.1 claims that Hρ is a subspace of H. Hereafter we assume u0 ∈ Hρ . Lemma 3.7.2. Suppose that the conditions of Lemma 3.7.1 are satisfied. Then smooth solution of problem (3.7.1)–(3.7.3) if and only if u is a smooth solution following problem: 𝜕u = αuxx + βu × uxx + α|ux |2 u, 𝜕t 󵄨 󵄨 u(x, 0) = u0 (x), 󵄨󵄨󵄨u0 (x)󵄨󵄨󵄨 = 1, x ∈ Ω, u(x − D, 0) = u0 (x − D, t),

+

x ∈ R, t ∈ R .

u is a of the (3.7.6) (3.7.7) (3.7.8)

Proof. See [217, Lemma 2]. Based on Lemma 3.7.2, hereafter we merely need to study the problem (3.7.6)–(3.7.8). Lemma 3.7.3. Let u0 ∈ H 2 ∩ Hρ . Then for any solution of problem (3.7.6)–(3.7.8) we have 󵄩󵄩 󵄩 󵄩󵄩uxx (t)󵄩󵄩󵄩 ≤ C1 ,

t+1

∫ ‖uxxx ‖2 dt ≤ C1 ,

∀t ≥ t1 ,

t

where C1 is a constant which depends on parameters (α, β, ρ, Ω); t1 depends on parameters (α, β, ρ, Ω) and R, whenever ‖u0 ‖H 2 ≤ R.

3.7 One-dimensional ferromagnetic chain equation

| 377

Proof. For simplicity, hereafter C denotes as arbitrary constant that merely depends on parameters (α, β, ρ, Ω). From |u(x, t)| = 1, we know that if |ux | ≠ 0, then u, ux and u×ux form an orthogonal basis of R3 . Let uxx = α1 u + α2 ux + α3 u × ux . By a simple calculation, we get α1 = −|ux |2 ,

α2 =

ux ⋅ uxx , |ux |2

α3 =

(u × ux ) ⋅ uxx . |ux |2

Differentiating equation (3.7.6) with respect to x twice, and taking the inner product with uxx , we get 1 d ‖u ‖2 = ∫ uxx ⋅ (αuxx + βu × uxx + α|ux |2 u)xx dx 2 dt xx Ω

= −α‖uxxx ‖2 − β ∫(ux × uxx ) ⋅ uxxx dx Ω

− α ∫ uxxx ⋅ (|ux |2 ux + 2u(ux ⋅ uxx ))dx.

(3.7.9)

Ω

Differentiating equation (3.7.4) with respect to x thrice, we obtain 3 u ⋅ uxxx = − (|ux |2 )x . 2

(3.7.10)

Using equation (3.7.10), all terms on the right-hand side of estimate (3.7.9) can be written as follows: ∫ uxxx ⋅ (|ux |2 ux )dx = − ∫ |ux |2 |uxx |2 dx − 2 ∫ |ux ⋅ uxx |2 dx, Ω

Ω

(3.7.11)

Ω

∫ uxxx ⋅ (u(ux ⋅ uxx ))dx = ∫(uxxx ⋅ u)(ux ⋅ uxx )dx Ω

Ω

= −3 ∫ |ux ⋅ uxx |2 dx, Ω

∫(ux × uxx ) ⋅ uxxx dx = ∫[ux × (−|ux |2 u + Ω

Ω

(3.7.12) (u × ux ) ⋅ uxx (u × ux ))] ⋅ uxxx dx |ux |2

= ∫ |ux |2 (u × ux ) ⋅ uxxx dx Ω

+∫ Ω

(u × ux ) ⋅ uxx (ux × (u × ux )) ⋅ uxxx dx |ux |2

= ∫ |ux |2 (u × ux ) ⋅ uxxx dx − Ω

3 ∫(|ux |2 )x (u × ux ) ⋅ uxx dx 2 Ω

3 = ∫ |ux |2 (u × ux ) ⋅ uxxx dx + ∫ |ux |2 (u × ux ) ⋅ uxx dx 2 Ω

Ω

378 | 3 The approximate inertial manifold 3 ∫ |ux |2 (u × ux ) ⋅ uxxx dx 2

+

Ω

5 = ∫ |ux |2 (u × ux ) ⋅ uxxx dx. 2

(3.7.13)

Ω

By equations (3.7.9), (3.7.11)–(3.7.13), we obtain 1 d ‖u ‖2 + α‖uxxx ‖2 = α ∫ |ux |2 |uxx |2 dx 2 dt xx Ω

5 + 8α ∫ |ux ⋅ uxx |2 dx − β ∫ |ux |2 (u × ux ) × uxxx dx. 2

(3.7.14)

Ω

Ω

In addition, 1 d ∫ |ux |4 dx = ∫ |ux |2 ux ⋅ (αuxx + βu × uxx + α|ux |2 u)x dx 4 dt Ω

Ω

= ∫ |ux |2 ux ⋅ (αuxxx + βux × uxx + βu × uxxx Ω

+ α|ux |2 ux + 2αu(ux ⋅ uxx ))dx

= −α ∫ |ux |2 |uxx |2 dx − 2α ∫ |ux ⋅ uxx |2 dx Ω

Ω 6

+ α ∫ |ux | dx − β ∫ |ux |2 (ux × ux ) ⋅ uxxx dx, Ω

Ω

that is, we have − β ∫ |ux |2 (u × ux ) ⋅ uxxx dx Ω

=

1 d ∫ |ux |4 dx + α ∫ |ux |2 |uxx |2 dx + 2α ∫ |ux ⋅ uxx |2 dx − α ∫ |ux |6 dx. 4 dt Ω

Ω

(3.7.15)

Ω

Ω

Substituting equation (3.7.15) into equation (3.7.14), we get d 5 d ‖u ‖2 − ∫ |ux |4 dx + 2α‖uxx ‖2 + 5α ∫ |ux |6 dx dt xx 4 dt Ω

2

Ω

2

2

= 7α ∫ |ux | |uxx | dx + 26α ∫ |ux ⋅ uxx | dx. Ω

Ω

By virtue of 7α ∫ |ux |2 |uxx |2 dx + 26α ∫ |ux ⋅ uxx |2 dx Ω

Ω

≤ 33α ∫ |ux |2 |uxx |2 dx Ω

(3.7.16)

3.7 One-dimensional ferromagnetic chain equation

5

| 379

7

≤ 5α ∫ |ux |6 dx + C ∫ |uxx |3 dx ≤ 5α ∫ |ux |6 dx + C‖ux ‖ 4 ‖uxxx ‖ 4 Ω

Ω

α ≤ 5α ∫ |ux | dx + ‖uxxx ‖2 + C, 2

Ω

6

Ω

and equation (3.7.16), we deduce d 5 d 3 ‖u ‖2 − ∫ |ux |4 dx + α‖uxxx ‖2 ≤ C. dt xx 4 dt 2

(3.7.17)

Ω

Since ux is periodic, ∫Ω uxx (x)dx = 0. Using Poincaré inequality, we get α ‖u ‖2 ≥ K‖uxx ‖2 . 2 xxx

(3.7.18)

Hence from equation (3.7.17) we get 5 5 d (‖uxx ‖2 − ∫ |ux |4 dx) + K(‖uxx ‖2 − ∫ |ux |4 dx) + α‖uxxx ‖2 ≤ C, dt 4 4 Ω

(3.7.19)

Ω

that is, 5 5 d (‖uxx ‖2 − ∫ |ux |4 dx) + K(‖uxx ‖2 − ∫ |ux |4 dx) ≤ C. dt 4 4 Ω

Ω

By Gronwall lemma, we get 󵄩󵄩 󵄩2 5 4 󵄩󵄩uxx (t)󵄩󵄩󵄩 − ∫ |ux | dx 4 Ω

C 󵄩 󵄨4 󵄩2 5 󵄨 ≤ (󵄩󵄩󵄩uxx (0)󵄩󵄩󵄩 − ∫ 󵄨󵄨󵄨ux (0)󵄨󵄨󵄨 dx)e−Kt + 4 K Ω

󵄩 󵄩2 ≤ 󵄩󵄩󵄩uxx (0)󵄩󵄩󵄩 e−Kt + C ≤ R2 e−Kt + C, where t∗ =

1 K

t ≥ 0 ≤ 2C, t ≥ t∗ ,

(3.7.20)

2

ln RC . Based on the interpolation inequality 3 1 5 ‖ux ‖4 ≤ C|ux | 4 ‖uxx ‖ 4 ‖ux ‖24 4

1 ≤ C‖ux ‖3 ‖uxx ‖ ≤ C‖uxx ‖ ≤ ‖uxx ‖2 + C, 2

(3.7.21)

from equations (3.7.20)–(3.7.21) we get 󵄩󵄩 󵄩 󵄩󵄩uxx (t)󵄩󵄩󵄩 ≤ C,

∀t ≥ t∗ .

(3.7.22)

380 | 3 The approximate inertial manifold Integrating equations (3.7.19)–(3.7.22) with respect to t over (t, t + 1), we get t+1

∫ ‖uxxx ‖2 dt ≤ C,

∀t ≥ t∗ .

t

Thus Lemma 3.7.3 is proved. Lemma 3.7.4. Let u0 ∈ H k+1 ∩ Hρ , k ≥ 1. Then we have 󵄩󵄩 k+1 󵄩2 󵄩󵄩Dx u(t)󵄩󵄩󵄩 ≤ Ck ,

t ≥ tk ,

󵄩 󵄩󵄩2 ∫ 󵄩󵄩󵄩Dk+2 x u(t)󵄩 󵄩 dt ≤ Ck ,

t ≥ tk ,

t+1 t

where constant Ck only depends on parameter k; tk depend on parameter k and R; whenever ‖u0 ‖H 2 ≤ R. Proof. We use induction to prove the lemma: (1) When k = 1, the claim of Lemma 3.7.4 boils down to that of Lemma 3.7.3, which has been proved. (2) Suppose the lemma is true up to k − 1. Now we prove it is true also for k, k ≥ 2. Differentiating (k + 1) times formula (3.7.6) with respect to x, and taking the inner product with Dk+1 x , we get 1 d 󵄩󵄩 k+1 󵄩󵄩2 󵄩 k+2 󵄩2 󵄩D u󵄩 + α󵄩󵄩󵄩Dx u󵄩󵄩󵄩 2 dt 󵄩 x 󵄩 k+1

i = β ∑ Ck+1 u × Dix uxx ) ⋅ Dk+1 ∫(Dk+1−i x x udx i=0

Ω

k+1

i + α ∑ Ck+1 u ⋅ Dk+1 ∫ Dix |u0 |2 (Dk+1−i x x u)dx. i=0

(3.7.23)

Ω

Through simple calculation we get k+1

i β ∑ Ck+1 u × Dix uxx ) ⋅ Dk+1 ∫(Dk+1−i x x udx i=0

Ω

󵄩 󵄩󵄩󵄩󵄩 k+2 󵄩󵄩 󵄩󵄩 k+1 󵄩󵄩2 ≤ C 󵄩󵄩󵄩Dk+1 x u󵄩 󵄩󵄩󵄩Dx u󵄩󵄩 + C 󵄩󵄩Dx u󵄩󵄩 α 2 u‖2 + C‖Dk+1 ≤ ‖Dk+2 x u‖ , 4 x

(3.7.24)

k+1

i u ⋅ Dk+1 α ∑ Ck+1 ∫ Dix |u0 |2 (Dk+1−i x x u)dx i=0

Ω

󵄩 󵄩󵄩2 󵄩󵄩 k+1 󵄩󵄩2 󵄩󵄩 k 󵄩󵄩2 󵄩󵄩 k+1 󵄩󵄩󵄩󵄩 k+2 󵄩󵄩 ≤ 󵄩󵄩󵄩Dk+1 x u󵄩 󵄩 + C 󵄩󵄩Dx u󵄩󵄩 󵄩󵄩Dx u󵄩󵄩 + C 󵄩󵄩Dx u󵄩󵄩󵄩󵄩Dx u󵄩󵄩 α󵄩 󵄩2 󵄩 󵄩󵄩2 ≤ 󵄩󵄩󵄩Dk+2 u󵄩󵄩 + C 󵄩󵄩󵄩Dk+1 x u󵄩 󵄩. 4 x 󵄩

(3.7.25)

3.7 One-dimensional ferromagnetic chain equation

| 381

From equations (3.7.23)–(3.7.25), we get d 󵄩󵄩 k+1 󵄩󵄩2 󵄩 k+1 󵄩2 α 󵄩 k+2 󵄩2 󵄩 k+2 󵄩2 󵄩D u󵄩 + α󵄩󵄩󵄩Dx u󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩Dx u󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Dx u󵄩󵄩󵄩 + C, dt 󵄩 x 󵄩 2

(3.7.26)

d 󵄩󵄩 k+1 󵄩󵄩2 α 󵄩󵄩 k+1 󵄩󵄩2 󵄩D u󵄩 + 󵄩D u󵄩 ≤ C, dt 󵄩 x 󵄩 2K 󵄩 x 󵄩

(3.7.27)

which yields

where 󵄩󵄩 k+1 󵄩󵄩 󵄩 k+2 󵄩 󵄩󵄩Dx u󵄩󵄩 ≤ K 󵄩󵄩󵄩Dx u󵄩󵄩󵄩

(∫ Dk+1 x udx = 0).

Then we get 󵄩󵄩 k+1 󵄩󵄩2 󵄩󵄩Dx u󵄩󵄩 ≤ C,

t ≥ t∗ ,

(3.7.28)

󵄩 󵄩󵄩 ∫ 󵄩󵄩󵄩Dk+1 x u󵄩 󵄩dt ≤ C,

t ≥ t∗ .

(3.7.29)

t+1 t

Thus Lemma 3.7.4 has been proved. Lemma 3.7.5. Let u0 ∈ H k+2 ∩ Hρ , k ≥ 0. Then there exists a constant Ck , depending only on parameter k, such that 󵄩󵄩 k 󵄩󵄩 󵄩󵄩Dx ut 󵄩󵄩 ≤ Ck ,

t ≥ tk ,

where tk depends on parameter k and R, whenever ‖u0 ‖H 2 ≤ R. Proof. Differentiating k times the equation with respect to x and applying Lemma 3.7.4, we get the claim. Now to construct an approximate inertial manifold of the ferromagnetic chain equation for the initial value problem (3.7.6)–(3.7.8), write (3.7.6)–(3.7.8) in an abstract form: du + αAu + B(u, u) + R(u) = 0, dt

(3.7.30)

where A = −𝜕xx is an unbounded self-adjoint operator, D(A) = {u ∈ H 2 : u satisfies equation (3.7.8)}, B(u, v) = −βu × vxx is a bilinear operator, R(u) = −α|ux |2 is a nonlinear operator, D(A) → H. Let {ωj }∞ j=1 be the orthogonal basis of H, which is composed of eigenvectors of A: Aωj = λj ωj ,

0 = λ1 < λ2 ≤ ⋅ ⋅ ⋅ ≤ λj → +∞,

j → ∞.

For a given m, let P = Pm : H → span{ω1 , ω2 , . . . , ωm } be the projection, Q = Qm = I−Pm . Then acting with Pm and Qm on equation (3.7.30), we get

382 | 3 The approximate inertial manifold dp { { { dt + αAp + Pm B(p + q, p + q) + Pm R(p + q) = 0, { { { dq + αAq + Q B(p + q, p + q) + Q R(p + q) = 0, m m { dt

(3.7.31)

where p = Pm u, q = Qm u. Using Lemmas 3.7.1, 3.7.4 and 3.7.5, we have 󵄩 3 󵄩 ‖u‖, 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 ≤ C,

t ≥ t∗ ,

󵄩󵄩 k 󵄩󵄩 󵄩󵄩 k+1 󵄩󵄩 󵄩󵄩 󵄩󵄩A u󵄩󵄩, 󵄩󵄩A u󵄩󵄩, 󵄩󵄩A

k+ 32

󵄩 󵄩 󵄩 󵄩󵄩 𝜕u 󵄩󵄩 u󵄩󵄩󵄩, 󵄩󵄩󵄩Ak u󵄩󵄩󵄩 ≤ C, 󵄩󵄩 𝜕t 󵄩󵄩

(3.7.32) t ≥ t∗ ,

(3.7.33)

where constants Ck , tk are defined as before. Note that 󵄩 󵄩 󵄩󵄩 21 󵄩󵄩 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 1 󵄩󵄩A u󵄩󵄩 = 󵄩󵄩 󵄩󵄩, u ∈ H , 󵄩󵄩 𝜕x 󵄩󵄩 ‖Pm u‖ ≤ ‖u‖, ‖Qm u‖ ≤ ‖u‖, u ∈ H, 󵄩󵄩 α 󵄩󵄩 α α 󵄩󵄩A u󵄩󵄩 ≥ λm+1 ‖u‖, α > 0, u ∈ Qm D(A ).

(3.7.34)

By equations (3.7.33)–(3.7.34) we deduce that 󵄩 󵄩 󵄩 3 󵄩 󵄩󵄩 d 󵄩󵄩 −k , ‖q‖, ‖Aq‖, 󵄩󵄩󵄩A 2 q󵄩󵄩󵄩, 󵄩󵄩󵄩 q󵄩󵄩󵄩 ≤ Ck λm+1 󵄩󵄩 dt 󵄩󵄩

t ≥ tk .

(3.7.35)

Define the mapping Φ : Pm H → Qm H such that p ∈ Pm H, Φ(p) = Ψ is determined by αAΨ + Qm B(p, p) + Qm R(p) = 0.

(3.7.36)

Letting Σ = graph(Φ), we can prove that Σ is an approximate inertial manifold of problem (3.7.6)–(3.7.8). Theorem 3.7.1. Let u0 ∈ H 2k+2 ∩ Hρ . Then for any positive integer k, there exists a constant Ck , depending on parameter k, such that −k distH 2 (u(t), Σ) ≤ Ck λm+1 ,

t ≥ tk ,

where u(t) is a solution of problem (3.7.6)–(3.7.8); tk depends on parameter k and R; whenever ‖u0 ‖H 2 ≤ R. Proof. Taking the difference of equation (3.7.36) and (3.7.31), it follows that dq + Qm B(p + q, p + q) − Qm B(p, q) + Qm R(p + q) − Qm R(p) dt dq = + Qm B(p + q, p + q) dt + Qm B(p, q) + Qm R(p + q) − Qm R(p).

αAΨ − αAq =

(3.7.37)

3.8 Nonlinear Schrödinger equation

Since

󵄩 󵄩󵄩 󵄩󵄩B(q, p + q) − B(p, q)󵄩󵄩󵄩 ≤ β‖q × Au‖ + β‖p × Aq‖ ≤ β‖Au‖∞ ‖q‖ + β‖p‖‖Aq‖∞ 󵄩 3 󵄩 󵄩 3 󵄩 −k ≤ β󵄩󵄩󵄩A 2 u󵄩󵄩󵄩‖q‖ + β‖p‖󵄩󵄩󵄩A 2 q󵄩󵄩󵄩 ≤ Ck λm+1 , 󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 2 󵄩 󵄩󵄩R(p + q) − R(p)󵄩󵄩 = α󵄩󵄩|ux | u − |px | p󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ α󵄩󵄩󵄩(|ux |2 − |px |2 )u󵄩󵄩󵄩 + α󵄩󵄩󵄩|px |2 (u − p)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ α󵄩󵄩󵄩(|ux | + |px |)(|ux | − |px |)u󵄩󵄩󵄩 + α󵄩󵄩󵄩|px |2 q󵄩󵄩󵄩

≤ α(|ux ‖∞ + ‖px ‖∞ )‖qx ‖∞ ‖u‖ + α‖px ‖2∞ ‖q‖ −k ≤ 2α‖Au‖‖Aq‖‖u‖ + α‖Au‖‖q‖ ≤ Ck λm+1 ,

| 383

(3.7.38)

(3.7.39)

by virtue of equations (3.7.37)–(3.7.39), we deduce −k ‖AΨ − Aq‖ ≤ Ck λm+1 ,

Therefore

t ≥ tk .

󵄩 󵄩 distH 2 (u(t), Σ) ≤ 󵄩󵄩󵄩u(t) − (p(t) + Φ(p(t)))󵄩󵄩󵄩H 2 󵄩 󵄩 ≤ 󵄩󵄩󵄩Φ(p(t)) − q(t)󵄩󵄩󵄩H 2 −k ≤ ‖AΨ − Aq‖ ≤ Ck λm+1 ,

t ≥ tk .

Finally, Theorem 3.7.1 has been proved.

3.8 Nonlinear Schrödinger equation In 1988, Ghidaglia [74] studied the existence of a global attractor and its dimension estimate for a class of nonlinear Schrödinger equations with damping. In 1996, Guo and Wang constructed two kinds of approximate inertial manifold [119] for such equations. Consider the following nonlinear Schrödinger equation [20, 21, 126, 195]: i

𝜕u 𝜕2 u + + g(|u|2 )u + iαu + h = 0, 𝜕t 𝜕x2

(3.8.1)

with initial value condition u(x, 0) = u0 (x),

x ∈ Ω = (−D, D)

(3.8.2)

and one of the following boundary conditions: (Dirichlet) (Neumann) (periodic boundary)

u(−D, t) = u(D, t) = 0, t ∈ R; 𝜕u 𝜕u (−D, t) = (D, t) = 0, t ∈ R; 𝜕x 𝜕x u(x + 2D, t) = u(x, t), x ∈ R, t ∈ R+ ,

(3.8.3)

384 | 3 The approximate inertial manifold where u is an unknown complex-valued function; α > 0 is constant. Also h(x) ∈ L2 (Ω), while g(s), 0 ≤ s < ∞, is a real-valued smooth function which satisfies lim

s→+∞

G+ (s) = 0, s3

(3.8.4)

and ∃ω > 0 such that lim

s→+∞

f (s) − ωG( s) s3

≤ 0,

(3.8.5)

s

where f (s) = sg(s), G(s) = ∫0 g(τ)dτ, G+ (s) = max{G(s), 0}. In [74] we have proved that under condition (3.8.4)–(3.8.5), for any u0 ∈ H 1 (Ω), problem (3.8.1)–(3.8.3) has a unique global solution u(x, t) such that u(x, t) ∈ L∞ (R+ ; H 1 (Ω)), and if u0 (x) ∈ H 2 (Ω) then u(x, t) satisfies ‖u‖, ‖ux ‖, ‖uxx ‖, ‖ut ‖, ‖u‖∞ , ‖ux ‖∞ ≤ K,

0 ≤ t ≤ T,

(3.8.6)

t ≥ t∗ ,

(3.8.7)

and ‖u‖, ‖ux ‖, ‖uxx ‖, ‖ut ‖, ‖u‖∞ , ‖ux ‖∞ ≤ C,

where K is a constant, which depends on parameters (α, Ω, g) and T, R; ‖u0 ‖H 2 ≤ R; t∗ depends on parameters (α, Ω, g) and R; here and hereafter C denotes arbitrary constant only depending on parameters (α, Ω, g). In order to construct an approximate inertial manifold, we need to provide a higher order a priori estimate. Lemma 3.8.1. Suppose the conditions (3.8.4)–(3.8.5) are satisfied and u0 ∈ H 3 (Ω), h(x) ∈ H 1 (Ω). Then for any solution u(x, t) of problem (3.8.1)–(3.8.3) we have 󵄩󵄩 󵄩 󵄩󵄩uxxx (⋅, t)󵄩󵄩󵄩 ≤ C,

t ≥ t∗ ,

where the constant C only depends on parameters (α, Ω, g), t∗ depends on parameters (α, Ω, g) and R, whenever ‖u0 ‖H 2 ≤ R. Proof. Differentiating equation (3.8.1) with respect to x, we get iutx + uxxx + iαux + g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux + h󸀠 (x) = 0. Taking the inner product of equation (3.8.8) and uxxxt + αuxxx , we get (iutx + uxxx + iαux + g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux + h󸀠 (x), uxxxt + αuxxx ) = 0.

(3.8.8)

3.8 Nonlinear Schrödinger equation

| 385

Taking the real part of the above equation, we arrive at 1 d ‖u ‖2 + α‖uxxx ‖2 + Re(h󸀠 , uxxxt + αuxxx ) 2 dt xxx + Re(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux , uxxxt + αuxxx ) = 0.

(3.8.9)

Since 󵄨󵄨 󸀠 󵄨 2 2 2 󵄨󵄨(g (|u| )|u|x u + g(|u| )ux , αuxxx )󵄨󵄨󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ C 󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ 󵄩󵄩󵄩(ux u + uux )󵄩󵄩󵄩∞ ‖uxxx ‖ + C 󵄩󵄩󵄩g(|u|2 )󵄩󵄩󵄩∞ ‖ux ‖∞ ‖uxxx ‖ ≤ C‖uxxx ‖,

d Re(h󸀠 , uxxxt + αuxxx ) = Re ∫ h󸀠 uxxx dx + Re(h󸀠 , αuxxx ) dt 󵄨󵄨 󵄨 󵄩 󸀠󵄩 󸀠 󵄨󵄨Re(h , αuxxx )󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩h 󵄩󵄩󵄩‖uxxx ‖ ≤ C‖uxxx ‖,

(3.8.10) (3.8.11) (3.8.12)

and due to equations (3.8.11)–(3.8.12), we have Re(h󸀠 , uxxxt + αuxxx ) ≥

d Re ∫ h󸀠 uxxx dx − C‖uxxx ‖. dt

(3.8.13)

Then by equations (3.8.9), (3.8.10) and (3.8.13) we have 1 d d ‖u ‖2 + α‖uxxx ‖2 + Re ∫ h󸀠 uxxx dx 2 dt xxx dt + Re(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux , uxxxt ) ≤ C‖uxxx ‖.

(3.8.14)

Obviously, Re(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux , uxxxt ) d = Re ∫(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux )uxxx dx dt d − Re ∫ (g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux )uxxx dx dt d = Re ∫(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux )uxxx dx dt − Re ∫ g 󸀠󸀠 (|u|2 )|u|2t |u|2x u ⋅ uxxx dx − Re ∫ g 󸀠 (|u|2 )|u|2xt u ⋅ uxxx dx − Re ∫ g 󸀠 (|u|2 )|u|2x ut ⋅ uxxx dx − Re ∫ g 󸀠 (|u|2 )|u|2t ux ⋅ uxxx dx − Re ∫ g(|u|2 )uxt ⋅ uxxx dx.

(3.8.15)

386 | 3 The approximate inertial manifold From equations (3.8.14)–(3.8.15), we have d 1 d ‖uxxx ‖2 + α‖uxxx ‖2 + Re ∫ h󸀠 ⋅ uxxx dx 2 dt dt d 󸀠 2 + Re ∫(g (|u| )|u|2x u + g(|u|2 )ux ) ⋅ uxxx dx dt ≤ C‖uxxx ‖ + Re ∫ g 󸀠󸀠 (|u|2 )|u|2t |u|2x u ⋅ uxxx dx + Re ∫ g 󸀠 (|u|2 )|u|2xt u ⋅ uxxx dx + Re ∫ g 󸀠 (|u|2 )|u|2x ut ⋅ uxxx dx + Re ∫ g 󸀠 (|u|2 )|u|2t ux ⋅ uxxx dx + Re ∫ g(|u|2 )uxt ⋅ uxxx dx.

(3.8.16)

Now we estimate each term on the right-hand side of equation (3.8.16): 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠󸀠 2 2 2 󵄨󵄨 ∫ g (|u| )|u|t |u|x u ⋅ uxxx dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨 ∫ g 󸀠󸀠 (|u|2 )(ut u + uut )(ux u + uux )u ⋅ uxxx dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩󵄩 󸀠󸀠 2 󵄩󵄩 3 ≤ 4󵄩󵄩g (|u| )󵄩󵄩∞ ‖u‖∞ ‖ux ‖∞ ∫ |ut uxxx |dx ≤ C‖ut ‖‖uxxx ‖ ≤ C‖uxxx ‖, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠 2 2 󵄨󵄨 ∫ g (|u| )|u|x ut ⋅ uxxx dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨 ∫ g 󸀠 (|u|2 )(ux u + uux )ut uxxx dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩 󵄩 ≤ 2󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖ux ‖∞ ∫ |ut uxxx |dx

(3.8.17)

≤ C‖ut ‖‖uxxx ‖ ≤ C‖uxxx ‖, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠 2 2 󵄨󵄨 ∫ g (|u| )|u|t ux ⋅ uxxx dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩 󵄩 ≤ 2󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖ux ‖∞ ∫ |ut uxxx |dx

(3.8.18)

≤ C‖uxxx ‖,

(3.8.19)

2

Re ∫ g(|u| )uxt uxxx dx = −Re ∫ g 󸀠 (|u|2 )|u|2x uxt uxx dx − 1 ∫ g 󸀠 (|u|2 )|u|2t |uxx |dx 2 1 d ≤− ∫ g(|u|2 )|uxx |2 dx 2 dt +

1 d ∫ g(|u|2 )|uxx |2 dx 2 dt

3.8 Nonlinear Schrödinger equation

| 387

󵄩 󵄩 + 2󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖ux ‖∞ ∫ |uxt uxx |dx 󵄩 󵄩 + 󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖uxx ‖∞ ∫ |ut uxxx |dx ≤−

1 d ∫ g(|u|2 )|uxx |2 dx + C‖uxxx ‖ + C, 2 dt

(3.8.20)

Re ∫ g 󸀠 (|u|2 )u|u|2xt uuxxx dx = − Re ∫ g 󸀠󸀠 (|u|2 )|u|2x |u|2xt uuxx dx − Re ∫ g 󸀠 (|u|2 )|u|2xxt uuxx dx − Re ∫ g 󸀠 (|u|2 )|u|2xt ux uxx dx, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠󸀠 2 2 2 󵄨󵄨 ∫ g (|u| )|u|x |u|xt uuxx dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨 ∫ g 󸀠󸀠 (|u|2 )(ux u + uux )(uxt u + ux ut + ut ux + uuxt )uuxx dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ C‖uxt ‖‖uxx ‖ + C‖ut ‖‖uxx ‖ ≤ C‖uxxx ‖ + C.

(3.8.21)

(3.8.22)

Similarly, we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠 2 2 󵄨󵄨 ∫ g (|u| )|u|xt ux uxx dx󵄨󵄨󵄨 ≤ C‖uxxx ‖ + C. 󵄨󵄨 󵄨󵄨

(3.8.23)

Since − Re ∫ g 󸀠 (|u|2 )|u|2xxt uuxxx dx = − Re ∫ g 󸀠 (|u|2 )uuxx {uxxt u + uxx ut + 2uxt ux + 2ux uxt + ut uxx + uuxxt }dx,

(3.8.24)

we can bound each term on the right-hand side of equation (3.8.24): 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠 2 󵄨󵄨 ∫ g (|u| )uuxx (uxx ut + 2uxt ux + 2ux uxt + ut uxx )dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩 󵄩 ≤ 2󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖uxx ‖∞ ∫ |ut uxx |dx 󵄩 󵄩 + 4󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖ux ‖∞ ∫ |uxx uxt |dx ≤ C‖ux ‖H 1 ‖ut ‖‖uxx ‖ + C‖uxt ‖‖uxx ‖ ≤ C‖uxxx ‖ + C, 󸀠

2

2

− Re ∫ g (|u| )|u| uxx uxxt dx 1 d 1 ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx + ∫ g 󸀠󸀠 (|u|2 )|u|2t |u|2 |uxx |2 dx 2 dt 2 1 + ∫ g 󸀠 (|u|2 )|u|2t |uxx |2 dx 2 1 d 󵄩 󵄩 ≤− ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx + 󵄩󵄩󵄩g 󸀠󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖3∞ ‖uxx ‖∞ ∫ |ut uxx |dx 2 dt

=−

(3.8.25)

388 | 3 The approximate inertial manifold 󵄩 󵄩 + 󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ‖uxx ‖∞ ∫ |ut uxx |dx 1 d ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx + C‖uxx ‖H 1 ‖ut ‖‖uxx ‖ 2 dt 1 d ≤− ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx + C‖uxxx ‖ + C, 2 dt ≤−

(3.8.26)

− Re ∫ g 󸀠 (|u|2 )u2 uxx uxxt dx =−

d Re ∫ g 󸀠 (|u|2 )u2 u2xx dx + Re ∫ g 󸀠󸀠 (|u|2 )|u|2t u2 u2xx dx dt

+ Re ∫ g 󸀠 (|u|2 )2uut u2xx dx + Re ∫ g 󸀠 (|u|2 )u2 uxxt uxx dx. Then − Re ∫ g 󸀠 (|u|2 )u2 uxx uxxt dx 1 d Re ∫ g 󸀠 (|u|2 )u2 u2xx dx 2 dt 1 + Re ∫ g 󸀠󸀠 (|u|2 )|u|2t u2 u2xx dx + Re ∫ g 󸀠 (|u|2 )uut uxx dx 2 1 d ≤− Re ∫ g 󸀠 (|u|2 )u2 u2xx dx 2 dt 󵄩 󵄩 + 󵄩󵄩󵄩g 󸀠󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖3∞ ‖uxx ‖∞ ∫ |ut uxx |dx =−

󵄩 󵄩 + 󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖∞ ∫ |ut uxx |dx 1 d Re ∫ g 󸀠 (|u|2 )u2 |uxx |2 dx + C‖uxx ‖H 1 ‖ut ‖‖uxx ‖ 2 dt 1 d Re ∫ g 󸀠 (|u|2 )u2 u2xx dx + C‖uxxx ‖ + C. ≤− 2 dt ≤−

(3.8.27)

From equations (3.8.21)–(3.8.27) we get Re ∫ g 󸀠 (|u|2 )|u|2xt uu2xxx dx ≤−

1 d 1 d Re ∫ g 󸀠 (|u|2 )|u2 ||uxx |2 dx − Re ∫ g 󸀠 (|u|2 )u2 ū 2xx dx + C‖uxxx ‖ + C. 2 dt 2 dt (3.8.28)

Moreover, from equations (3.8.16)–(3.8.20) and (3.8.28) we get d d d ‖u ‖2 + 2 Re ∫ h󸀠 uxxx dx + ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx dt xxx dt dt d + 2 Re ∫(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux )uxxx dx dt d + Re ∫ g 󸀠 (|u|2 )u2 u2xx dx dt

3.8 Nonlinear Schrödinger equation

d ∫ g(|u|2 )|uxx |2 dx + 2α‖uxxx ‖2 dt α ≤ C‖uxxx ‖ + C ≤ ‖uxxx ‖2 + C. 2

| 389

+

(3.8.29)

Let E(t) = ‖uxxx ‖2 + 2 Re ∫ h󸀠 u2xxx dx + 2 Re ∫(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux )uxxx dx + ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx + Re ∫ g 󸀠 (|u|2 )u2 ū 2xx dx + ∫ g(|u|2 )|uxx |2 dx.

(3.8.30)

Then from equation (3.8.29) we get dE α + αE + ‖uxxx ‖2 ≤ 2 Re ∫ h󸀠 uxxx dx dt 2 + 2α Re ∫(g 󸀠 (|u|2 )|u|2x u + g(|u|2 )ux )uxxx dx + α ∫ g 󸀠 (|u|2 )|u|2 |uxxx |2 dx + α Re ∫ g 󸀠 (|u|2 )u2 u2xx dx + α ∫ g(|u|2 )|uxx |2 dx + C.

(3.8.31)

We also estimate each term on the right-hand side of formula (3.8.31): 󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄩 󵄩 2󵄨󵄨󵄨 Re ∫ h󸀠 uxxx dx󵄨󵄨󵄨 ≤ 2󵄩󵄩󵄩h󸀠 󵄩󵄩󵄩‖uxxx ‖ ≤ C‖uxxx ‖ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨2 Re ∫(g 󸀠 (|u|2 )|u|2 u + g(|u|2 )ux )uxxx dx󵄨󵄨󵄨 x 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ C‖ux ‖‖uxxx ‖ ≤ C‖uxxx ‖, 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ∫ g 󸀠 (|u|2 )|u|2 |uxx |2 dx 󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩 ‖u‖2 ‖uxx ‖2 ≤ C. 󵄨󵄨 󵄨󵄨 󵄩 󵄩∞ ∞ 󵄨 󵄨

(3.8.32)

(3.8.33) (3.8.34)

Similarly, we have 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󸀠 2 2 󵄨󵄨 Re ∫ g (|u| )u uxx dx󵄨󵄨󵄨 ≤ C, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2 2 󵄨󵄨 ∫ g(|u| )|uxx | dx󵄨󵄨󵄨 ≤ C. 󵄨󵄨 󵄨󵄨 By equations (3.8.31)–(3.8.36) we get α α dE + αE + ‖uxxx ‖2 ≤ C‖uxxx ‖ + C ≤ ‖uxxx ‖2 + C, dt 2 2

(3.8.35) (3.8.36)

390 | 3 The approximate inertial manifold thus dE + αE ≤ C. dt

(3.8.37)

From Gronwall lemma we have E(t) ≤ E(t∗ )e−α(t−t∗ ) +

C , α

t ≥ t∗ .

(3.8.38)

Similarly, replacing equation (3.8.6) with equation (3.8.7), we have dE + αE ≤ K, dt

0 ≤ t ≤ T,

(3.8.39)

where the constant K depends on parameters (α, Ω, g), T and R, whenever ‖u0 ‖H 3 ≤ R. Thus we get E(t∗ ) ≤ E(0)e−αt∗ +

C ≤ C(R), α

(3.8.40)

where C(R) depends on parameters (α, Ω, g), R and ‖u0 ‖H 3 . By equations (3.8.38) and (3.8.40) we get C(R))e−α(t−t∗ ) + Cα , E(t) ≤ { 2C α

t ≥ t∗ , t ≥ t∗ ,

(3.8.41)

where t∗ = max{t∗ , t∗ + α1 ln αC(R) }. C Based on equation (3.8.30) and equations (3.8.32)–(3.8.36), we get 1 ‖uxxx ‖2 ≤ C‖uxxx ‖ + C ≤ ‖uxxx ‖2 + C, t ≥ t∗󸀠 . 2 From this we get the claim. Lemma 3.8.2. Under the conditions of Lemma 3.8.1, we have the estimate ‖utx ‖ ≤ C,

t ≥ t∗ .

Proof. By equation (3.8.8) we get 󵄩 󵄩 ‖utx ‖ ≤ ‖uxxx ‖ + α‖ux ‖ + 2󵄩󵄩󵄩g 󸀠 (|u|2 )󵄩󵄩󵄩∞ ‖u‖2∞ ‖ux ‖ 󵄩 󵄩 + 󵄩󵄩󵄩g(|u|2 )󵄩󵄩󵄩∞ ‖ux ‖ + ‖h‖H 1 ≤ C. Thus Lemma 3.8.2 has been proved. Now we construct an approximate inertial manifold of problem (3.8.1)–(3.8.3). Write equation (3.8.1) in an abstract differential equation form: i

du − Au + g(|u|2 )u + iαu + h = 0, dt

(3.8.42)

3.8 Nonlinear Schrödinger equation

| 391

where A = −𝜕xx is an unbounded self-adjoint operator D(A) = {u ∈ H 2 : u satisfies boundary condition (3.8.3)}. Let {ωj }∞ j=1 be the orthogonal basis of H, which is composed of eigenvectors of A, that is, Aωj = λj ωj ,

0 ≤ λ1 < λ2 ≤ ⋅ ⋅ ⋅ ≤ λj → +∞, j → ∞.

For a given m, set P = Pm : H → span{ω1 , . . . , vm } to be the projection, Q = Qm = I − Pm . Acting with Pm and Qm on equation (3.6.42), we have dp 2 { { {i dt − Ap + Pm (g(|u| )u) + iαp + Pm h = 0, { { {i dq − Aq + Q (g(|u|2 )u) + iαq + Q h = 0, m m { dt

(3.8.43)

where p = Pm u, q = Qm u. Note that 󵄩 󵄩 󵄩󵄩 21 󵄩󵄩 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 { { A u = 󵄩 󵄩 󵄩󵄩 󵄩󵄩, { 󵄩 󵄩 { 󵄩󵄩 𝜕x 󵄩󵄩 { ‖P u‖ ≤ ‖u‖, ‖Qm u‖ ≤ ‖u‖, { { {󵄩󵄩 m α 󵄩 󵄩󵄩 ≥ λα ‖u‖, A u 󵄩 m+1 󵄩 {󵄩

u ∈ H 1, u ∈ H, α > 0, u ∈ Qm D(Aα ).

(3.8.44)

Then from equation (3.8.7) and Lemma 3.8.2 we deduce that 󵄩󵄩 21 󵄩󵄩 󵄩󵄩 21 󵄩󵄩 󵄩A q󵄩󵄩, 󵄩󵄩A qt 󵄩󵄩 ≤ C, {󵄩 − 21 ‖q‖, ‖qt ‖ ≤ λm+1 ,

t ≥ t∗ , t ≥ t∗ .

(3.8.45)

Now we define the mapping Φ : Pm H → Qm H such that for any p ∈ Pm H, Φ(p) = Ψ is given by the following equation: − AΨ + Qm g(|p|2 )p + iαΨ + Qm h = 0.

(3.8.46)

We first prove the existence of a unique solution Ψ ∈ Qm H of equation (3.8.46). Lemma 3.8.3. There is an integer m0 depending on α and Ω such that when m ≥ m0 , there exists a unique solution Ψ ∈ Qm H of equation (3.8.46) for any p ∈ Pm H. Proof. We will use the fixed point principle to prove the theorem. Let p ∈ Pm H be fixed and introduce the mapping G : Qm H → Qm H such that for any φ ∈ Qm H, G(φ) = Ψ is given by the following equation: − AΨ + Qm g(|p|2 )p + iαφ + Qm h = 0. Obviously, any fixed point of G is a solution of equation (3.8.46).

(3.8.47)

392 | 3 The approximate inertial manifold In the following we prove that there exists an integer m0 , which depends on α and Ω, such that when m ≥ m0 , G : Qm H → Qm H is a compact mapping. Let φ1 , φ2 ∈ Qm H, then by equation (3.8.47) we have AΨ1 − AΨ2 = iα(φ1 − φ2 ). Hence ‖AΨ1 − AΨ2 ‖ = α‖φ1 − φ2 ‖.

(3.8.48)

From equations (3.8.44) and (3.8.48), we have −1 ‖Ψ1 − Ψ2 ‖ = αλm+1 ‖φ1 − φ2 ‖.

Since λm+1 → ∞, we can find an m0 , merely depending on α and Ω, such that when m ≥ m0 , Ψ is compact. Hence G is the unique fixed point in Qm H. This finishes the proof of the lemma. Letting Σ1 = graph(Φ), we can prove that Σ1 is an approximate inertial manifold of problem (3.8.1)–(3.8.3). In fact, we have Theorem 3.8.1. Suppose conditions (3.8.4)–(3.8.5) hold and u0 ∈ H 3 , h ∈ H 1 . Then there exists a constant m0 , which depends on parameters (α, Ω, g), such that when m ≥ m0 , we have −1

2 distH 2 (u(t), Σ1 ) ≤ Kλm+1 ,

t ≥ t∗ ,

where the constant K depends on parameters (α, Ω, g); u(t) is the solution of problem (3.8.1)–(3.8.3); t∗ only depends on parameters (α, Ω, g) and R, whenever ‖u0 ‖H 2 ≤ R. Proof. Subtracting equation (3.8.46) from (3.8.43), we obtain AΨ − Aq = Qm g(|p|2 )p − Qm g(|u|2 )u + iα(Ψ − q) − iqt .

(3.8.49)

Furthermore, we have 󵄩󵄩 2 2 󵄩 󵄩󵄩g(|p| )p − g(|u| )u󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(g(|p|2 ) − g(|u|2 ))p󵄩󵄩󵄩 + 󵄩󵄩󵄩g(|u|2 )q󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩g 󸀠 (ξ )(|p|2 − |u|2 )p󵄩󵄩󵄩 + 󵄩󵄩󵄩g(|u|2 )q󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩g 󸀠 (ξ )(ppq − pqu)󵄩󵄩󵄩 + 󵄩󵄩󵄩g(|u|2 )q󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩g 󸀠 (ξ )󵄩󵄩󵄩∞ ‖p‖2∞ ‖q‖ + 󵄩󵄩󵄩g 󸀠 (ξ )󵄩󵄩󵄩∞ ‖p‖∞ ‖u‖∞ ‖q‖ + 󵄩󵄩󵄩g(|u|2 )󵄩󵄩󵄩∞ ‖q‖ −1

2 ≤ C‖q‖ ≤ Cλm+1 .

(3.8.50)

3.8 Nonlinear Schrödinger equation

| 393

From equations (3.8.49)–(3.8.50), we have −1

2 ‖AΨ − Aq‖ ≤ Cλm+1 + α‖Ψ − q‖ + ‖qt ‖

−1

−1 2 ≤ Cλm+1 + αλm+1 ‖AΨ − Aq‖.

Since λm+1 → ∞, we can find a positive integer m0 , merely depending on parameters (α, Ω, g), such that when m ≥ m0 , we have −1

2 , ‖AΨ − Aq‖ ≤ Cλm+1

t ≥ t∗ .

Hence 󵄩 󵄩 distH 2 (u(t), Σ1 ) ≤ 󵄩󵄩󵄩u(t) − (p(t) + Ψ(t))󵄩󵄩󵄩H 2

− 21 󵄩 󵄩 󵄩 󵄩 , ≤ 󵄩󵄩󵄩Ψ(t) − q(t)󵄩󵄩󵄩H 2 = 󵄩󵄩󵄩AΨ(t) − Aq󵄩󵄩󵄩 ≤ Cλm+1

t ≥ t∗ .

Theorem 3.8.1 has been proved. Note that, if u0 ∈ H 2 , by equation (3.8.7) we have ‖q‖, ‖qt ‖ ≤ C,

t ≥ t∗ .

(3.8.51)

From this, we get Theorem 3.8.2. Let the conditions (3.8.4)–(3.8.5) be satisfied and u0 ∈ H 2 , h ∈ L2 . Then there exists a constant m0 , which depends on parameters (α, Ω, g), such that when m ≥ m0 , we have −1

2 distH 1 (u(t), Σ1 ) ≤ Kλm+1 ,

t ≥ t∗ ,

where the constant K depends on parameters (α, Ω, g); t∗ only depends on parameters (α, Ω, g) and R, whenever ‖u0 ‖H 2 ≤ R; and u(t) is the solution of problem (3.8.1)–(3.8.3). In the following, we introduce a simpler explicit inertial manifold Σ1 = graph(Φ∗ ), in which Φ∗ is the mapping Pm H → Qm H such that for any p ∈ Pm H, Φ∗ (p) = φ is given by the following equation: −Aφ + Qm g(|p|2 )p + Qm h = 0. We can prove Theorem 3.8.3. Suppose conditions (3.8.4)–(3.8.5) are satisfied and u0 ∈ H 3 , h ∈ H 1 . Then there exists a constant K which depends only on parameters (α, Ω, g) such that −1

2 distH 2 (u(t), Σ2 ) ≤ Kλm+1 ,

t ≥ t∗ ,

where u(t) is the solution of problem (3.8.1)–(3.8.3); t∗ only depends on parameters (α, Ω, g) and R, whenever ‖u0 ‖H 3 ≤ R.

394 | 3 The approximate inertial manifold Theorem 3.8.4. Suppose that conditions (3.8.4)–(3.8.5) are satisfied and u0 ∈ H 2 , h ∈ L2 . Then there exists a constant K which depends only on parameters (α, Ω, g) such that −1

2 distH 1 (u(t), Σ2 ) ≤ Kλm+1 ,

t ≥ t∗ ,

where u(t) is the solution of problem (3.8.1)–(3.8.3); t∗ depends only on parameters (α, Ω, g) and R, whenever ‖u0 ‖H 2 ≤ R.

3.9 The convergence of approximate inertial manifolds Approximate inertial manifold plays a vital role in approximation calculation, but whether or not we can prove the existence of an inertial manifold by constructing approximate inertial manifolds is not clear. The problem had not been solved for a long time. In 1994, Debussche and Temam [44] constructed a sequence of approximate inertial manifolds and proved that it converges to the inertial manifold. Of course, the spectral gap conditions were still required. Consider a class of nonlinear evolution equations in a Banach space E of the form { du + Au = f (u), { dt {u(0) = u0 ,

(3.9.1)

where A is a linear density operator in E ; f ∈ C 1 is a nonlinear operator, E → F, where E and F are two Banach spaces, E ⊂ F ⊂ E, and the mapping is continuous. In E , E, F, their norms are denoted by | ⋅ |ε , | ⋅ |E , and | ⋅ |F , respectively. Assume that the function f is globally Lipschitz, E → F. Let M1 be a positive constant such that for any x, y ∈ E, 󵄨󵄨 󵄨 󵄨f (x) − f (y)󵄨󵄨󵄨F ≤ M1 |x − y|E , {󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨f (x)󵄨󵄨F ≤ M1 (1 + |x|E ). For the operator A, suppose the linear equation is { du + Au = 0, { dt {u(0) = u0 . In ε, we define a strong continuous semigroup (e−At )t≥0 such that e−At F ⊂ E,

∀t > 0.

(3.9.2)

3.9 The convergence of approximate inertial manifolds | 395

Suppose there exists an eigenvector sequence (Pn )n∈N of A and two series (λn )n∈N , (Λ n )n∈N such that Λ n ≥ λn ≥ 0,

∀n ≥ 0,

λn → ∞, n → ∞, Λn is bounded as n → ∞. λn

(3.9.3a) (3.9.3b) (3.9.3c)

Also assume that for Qn = I − Pn we have (∗) Pn E and Qn E is invariant under the action of e−At , ∀t ≥ 0; (∗∗) (e−At )t≥0 can be extended to a semigroup (e−At )t∈R in Pn E , where these projections can define the exponential dichotomy of (e−At )t≥0 : there exist K1 , K2 > 0, α ∈ (0, 1), which are independent of n, such that (H1 ) for t ≤ 0, 󵄨󵄨 −At 󵄨󵄨 −λ t 󵄨󵄨e Pn 󵄨󵄨L (E) ≤ K1 e n , 󵄨󵄨 −At 󵄨󵄨 α −λ t 󵄨󵄨e Pn 󵄨󵄨L (F,E) ≤ K1 λn e n ; (H2 ) for t > 0, 1 󵄨󵄨 −At 󵄨󵄨 α −Λ t 󵄨󵄨e Qn 󵄨󵄨L (F,E) ≤ K2 ( α + Λ n )e n , t 󵄨󵄨 −1 −At 󵄨󵄨 α−1 −Λ t 󵄨󵄨A e Qn 󵄨󵄨L (F,E) ≤ K2 Λ n e n . Finally, suppose that equation (3.9.1) defines a continuous semigroup (S(t))t≥0 in E. An approximate inertial manifold is constructed in the following. A sequence of approximate inertial manifolds is obtained by constructing the inertial manifold through the approximate Lyapunov–Perron method as the fixed point of a mapping defined by an integral equation. As is well known, the Lyapunov–Perron method consists in looking for Ψ as the fixed point of the mapping J given by 0

As

J Ψ(y0 ) = ∫ e Qn f (y(s)) + Ψ(y(s))ds,

(3.9.4)

−∞

where y satisfies { dy + Ay = Pn (y + Ψ(y)), { dt {y(0) = y0 .

(3.9.5)

Now we select the time step τ and positive integer N. Approximate formulas (3.9.4)– (3.9.5) as follows: yk+1 = Rτ yk + Sτ Pn f (yk + Ψ(yk ))

(3.9.6)

396 | 3 The approximate inertial manifold where k = 0, . . . , N − 1, Rτ and Sτ are linear operators, which satisfy (H3 )

|R P | ≤ eτλn , { τ n L (E) |Sτ Pn |L (F,E) ≤ K3 λnα−1 (eτλn − 1),

and where positive constant K3 does not depend on n. Define the mapping TNτ by N−1

TNτ Ψ(y) = A−1 (I − e−Aτ ) ∑ e−kAτ Qn f (yk + Φ(yk )) k=0

−1 −NAτ

+A e

Qn f (yN + Φ(yN )),

where (yk )k=0,1,...,N are calculated by equation (3.9.6). For any Ψ ∈ Fl,b , Fl,b = {Ψ : Pn E → Qn E | Lip Ψ < l, sup

y∈Pn E

|Ψ(y)|E ≤ b}, 1 + |y|E

(3.9.7)

yn ∈ Pn E, y is the solution of equation (3.9.5), ỹk = y(−kτ). Let (H4 )

ỹ = Rτ ỹk + Sτ Pn f (ỹk + Ψ(ỹk )) + εk , α { k+1 |εk |E ≤ α1 (λn )τ2 (1 + |y0 |E )eτ(k+1)(λn +K4 λn )

and suppose that the derivative with respect to y does not increase too fast, that is, (H5 )

󵄨󵄨 dy 󵄨󵄨 󵄨󵄨 󵄨󵄨 −(λ +K λα )t 󵄨󵄨 󵄨󵄨 ≤ α2 (λn )(1 + |y0 |E )e n 5 n , 󵄨󵄨 dt 󵄨󵄨E

t ∈ (−∞, 0],

where K4 , K5 do not depend on N, τ or n; α1 (λn ), α2 (λn ) depends on λn , but not on N, τ. For many partial differential equations with dissipation, the semigroup defined by equation (3.9.1) has a global attractor A . In general it is embedded in a Banach subspace of E. From the definition of the attractor, u0 ∈ A . The solution of equation (3.9.1) is defined on R and is still in A . As a result, u is uniformly bounded and its time derivative is uniformly bounded. For the real number field, the conditions (H4 ), (H5 ) can be replaced with (H4 )󸀠 (H5 )󸀠

ỹk+1 = Rτ ỹk + Sτ Pn f (u(−kτ)) + εk , { |εk |E ≤ τ2 β1 , ∀k ≤ N, 󵄨󵄨󵄨 dy 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ β2 , ∀t ≤ 0, 󵄨󵄨 dt 󵄨󵄨 󵄨 󵄨

respectively, where y = Pn u is the projection of solution of the equation (3.9.1) on Pn E in A ; ỹk = y(−kτ) = Pu(−kτ), β1 , β2 do not depend on N, τ or n. Now we construct approximate inertial manifolds. Take a strictly positive sequence (τn )n∈N , define a sequence (Φn )n∈N such that Φ = 0, { 0 ΦN+1 = TNτN (ΦN ),

N ≥ 0,

(3.9.8)

3.9 The convergence of approximate inertial manifolds | 397

where T0τ is defined as T0τ Φ(y0 ) = A−1 Qn f (y0 + Φ(y0 )). Then graph(Φ1 ) is an approximate inertial manifold. Letting M1 = graph(ΦN ), we now estimate yk and TNτ . Lemma 3.9.1. Let (H3 ) be satisfied. Then the following is true: (i) Assume Ψ ∈ Fl,b , y0 ∈ Pn E, and using equation (3.9.6) define (yk )k=0,1,...,N . Then we have α

|yk |E ≤ ekτ(λn +K3 M1 (1+b)λn ) (|y0 |E + 1),

∀k ∈ N,

and λn1−α ≥ K3 M1 (1 + b). (ii) Assume Ψi ∈ Fl,b , y0i ∈ Pn E, and by equation (3.9.6) define (yki )k=0,...,N , y0 = y0i , i = 1, 2. Then we have 󵄨󵄨 1 2󵄨 kτ(λ −K M (1+l)λnα ) 󵄨󵄨 1 2󵄨 󵄨󵄨yk − yk 󵄨󵄨󵄨 ≤ e n 3 1 󵄨󵄨y0 − y0 󵄨󵄨󵄨E α 󵄨 󵄨 + λnα kτekτ(λn +K6 λn ) K3 M1 󵄨󵄨󵄨Ψ1 − Ψ2 󵄨󵄨󵄨∞ (1 + |y0 |E ), where K5 = K3 M1 max(1 + l, 1 + b). Proof. From (H3 ) and equations (3.9.2)–(3.9.7), we get |yk+1 |E ≤ eτλn |yk |E + K3 λnα−1 (eλn − 1)(M1 (1 + b)(1 + |yk |E )) α

≤ eτ(λn +K3 M1 (1+b)λn ) |yk |E + K3 λnα−1 M1 (1 + b)(eλn − 1),

where the following basic inequalities were used: eτλn − 1 ≤ τλn eτλn ,

α

1 + τK1 M1 (1 + b)λnα ≤ eτK3 M1 (1+b)λn . By successive iteration, we get α

|yk |E ≤ ekτ(λn +K3 M1 (1+b)λn ) (|y0 |E + K3 M1 (1 + b)λnα−1 ). Hence (i) is proved. Take (yki )k=0,...,N , i = 1, 2, as in (ii) and denote yk = yk1 − yk2 . Then we have 󵄨 󵄨 |yk+1 |E ≤ eτλn |yk |E + K3 λnα−1 (eτλn − 1)(|yk |E + 󵄨󵄨󵄨Ψ1 (yk1 ) − Ψ2 (yk2 )󵄨󵄨󵄨E ) 󵄨 󵄨 󵄨 󵄨 ≤ eτλn |yk |E + K3 M1 λnα−1 (eτλn − 1)(1 + l)|yk |E + 󵄨󵄨󵄨Ψ1 − Ψ2 󵄨󵄨󵄨∞ (1 + 󵄨󵄨󵄨yk1 󵄨󵄨󵄨E ). Applying (i) to (yki )k=0,...,N , we have: α α 󵄨 󵄨 󵄨 󵄨 |yk+1 |E ≤ eτ(λn +K3 M1 (1+l)λn ) |yk |E + τK3 M1 λnα 󵄨󵄨󵄨Ψ1 − Ψ2 󵄨󵄨󵄨∞ (1 + 󵄨󵄨󵄨y01 󵄨󵄨󵄨E )e(k+1)τ(λn +K3 M1 (1+b)λn ) .

Again by the circle principle, we deduce (ii). This completes the proof.

398 | 3 The approximate inertial manifold Lemma 3.9.2. Let (H3 ) be satisfied. Assume Ψ ∈ Fl,b , y0 ∈ Pn E, let (ek )k=0,...,N be a sequence in Pn E and (yk )k=0,...,N be defined by equation (3.9.6). Consider the sequence (ỹk )k=0,...,N defined as follows: ỹk+1 = Rτ ỹk + Sτ Pn f (ỹk + Ψ(ỹk )) + ek , ỹ0 = y0 .

Then we have k−1

α

|ỹk − yk |E ≤ ∑ e(k−1−j)τ(λn +K3 M1 (1+l)λn ) |ej |E , j=0

k = 0, . . . , N.

̃l,b to replace Fl,b : Suppose ρ is bounded, M0 = supu∈E |f (u)|F , and use F 󵄨

󵄨

̃l,b = {Ψ : Pn E → Qn E | 󵄨󵄨Ψ(y)󵄨󵄨 ≤ b, ∀y ∈ Pn E, F 󵄨 󵄨E

󵄨󵄨 󵄨 󵄨󵄨Ψ(x) − Ψ(y)󵄨󵄨󵄨E ≤ l|x − y|E , ∀x, y ∈ Pn E},

(3.9.9)

which leads to a simpler calculation. Proposition 3.9.1. Assume that (H2 ) and (H3 ) hold. Then there exist two constants C1 and C2 such that if (N + 1)τ ≤

C1 , λnα

λn ≥ C2 ,

̃l,b → F ̃l,b , ∀b ≥ b0 . then there exist l, b0 such that FNτ : F ̃l,b , y0 ∈ Pn E, and define function y(s), s ∈ (−∞, 0] by Proof. Consider Ψ ∈ F {

ỹ(s) = yk , ỹ(s) = yk ,

s ∈ ((−k + 1)τ, −kτ], s ∈ (−∞, −Nτ),

k = 0, . . . , N − 1,

(3.9.10)

where (yk )k=0,...,N is calculated by formula (3.9.6). Then FNτ Ψ(y0 ) can be written as τ FN Ψ(y0 )

0

= ∫ eAs Qn f (y(s) + Ψ(y(s)))ds −∞ 0

=

eAs Qn f (y(s) + Ψ(y(s)))ds + A−1 e−(N+1)Aτ Qn f (yN + Ψ(yN )).

∫ −(N+1)τ

From the bound of (H2 ) and f , we get 0

1 󵄨󵄨 τ 󵄨 α Λ s 󵄨󵄨FN Ψ(y0 )󵄨󵄨󵄨 ≤ ∫ K2 M0 ( α + Λ n )e n ds |s| −∞

0

≤ K2 M0 ( ∫ −∞

ey dy + 1)Λα−1 n |y|α

≤ K2 M0 (Γ(−1 − α) + 1)Λα−1 n ≤ b.

3.9 The convergence of approximate inertial manifolds | 399

If b ≥ b0 = K2 M0 (Γ(−1 − α) + 1)C2α−1 ,

(3.9.11)

choosing y01 , y02 ∈ Pn E, by equation (3.9.6) defining (yki )k=0,...,N , y0 = y0i , and constructing yi as before i = 1, 2, then τ

1

τ

2

FN Ψ(y0 ) − FN Ψ(y0 ) 0

=

∫

eAs Qn f (y1 (s) + Ψ(y1 (s))) − f (y2 (s) + Ψ(y2 (s)))ds

−(N+1)τ

+ A−1 e−A(N+1)τ Qn (f (yN1 + Ψ(yN1 )) − f (yN2 + Ψ(yN2 ))). By (H2 ), through equations (3.9.9) and (3.9.2) we get 󵄨󵄨 τ τ 2 󵄨 1 󵄨󵄨FN Ψ(y0 ) − FN Ψ(y0 )󵄨󵄨󵄨E 0

≤ K2 M1 (1 + l)

∫ ( −(N+1)τ

1 󵄨 󵄨 + Λαn )󵄨󵄨󵄨y1 (s) − y2 (s)󵄨󵄨󵄨E ds |s|α

2󵄨 −Λαn (N+1)τ 󵄨󵄨 1 + K2 M1 (1 + l)Λα−1 󵄨󵄨yN − yN 󵄨󵄨󵄨E . n e

̃l,b ⊂ Fl,b , Using Lemma 3.9.1(ii), we deduce that for Ψ1 , Ψ2 ∈ F 󵄨󵄨 1 2󵄨 kτ(λ −K M (1+l)λnα ) 󵄨󵄨 1 2󵄨 󵄨󵄨yk − yk 󵄨󵄨󵄨E ≤ e n 3 1 󵄨󵄨y0 − y0 󵄨󵄨󵄨E 󵄨󵄨̃1 󵄨 2 −s(λ +K M (1+l)λnα ) 󵄨󵄨 1 2󵄨 󵄨󵄨y (s) − ỹ (s)󵄨󵄨󵄨E ≤ e n 3 1 󵄨󵄨y0 − y0 󵄨󵄨󵄨E . Hence, for s we have 󵄨󵄨 τ 1 τ 2 󵄨 󵄨󵄨FN Ψ(y0 ) − FN Ψ(y0 )󵄨󵄨󵄨E 󵄨 󵄨 ≤ K2 M1 (1 − l)󵄨󵄨󵄨y01 − y02 󵄨󵄨󵄨E

0

∫ ( −(N+1)τ

α 1 + Λαn )e(Λ n −λn −K3 M1 (1+l)λn )s ds |s|α

󵄨 󵄨 −(Λ n −λn −K3 M1 (1+l)λnα )(N+1)τ . + K2 M1 (1 + l)󵄨󵄨󵄨y01 − y02 󵄨󵄨󵄨E Λα−1 n e From equation (3.9.3) and inequality (N + 1)τ ≤ 0

∫ −(N+1)τ

C1 , λnα

we have

1 (Λ n −λn −K3 M1 (1+l)λnα )s e ds |s|α

≤ eK3 M1 C1 (1+l)

0

∫ −(N+1)τ

1 α(α−1) 1−α 1 ds ≤ eK3 M1 C1 (1+l) λ C1 |s|α 1−α n

400 | 3 The approximate inertial manifold 󵄨󵄨 τ 1 τ 2 󵄨 󵄨󵄨FN Ψ(y0 ) − FN Ψ(y0 )󵄨󵄨󵄨E 󵄨 󵄨 ≤ K2 M1 (1 + l)󵄨󵄨󵄨y01 − y02 󵄨󵄨󵄨E eK3 M1 (1+l) ×(

Λαn 1 α(α−1) 1−α 󵄨󵄨 1 2 󵄨󵄨 λn + Λα−1 C1 + n ) ≤ l󵄨󵄨y0 − y0 󵄨󵄨E . 1−α K3 M1 (1 + l)λnα

If (K2 M1 (

α

K2 Λ n 1 α(α−1) 1−α λ C1 + Λα−1 ( ) )eK3 M1 C1 (1+l) ≤ l, n )(1 + l) + 1−α n K3 λn

(3.9.12)

we can choose C2 and δ0 such that, when λn ≥ C2 , C1 ≤ δ0 , K2 M1 (

1 α(α−1) 1−α 1 K3 M1 C1 (1+l) λ C1 + Λα−1 ≤ . n )e 1−α n 2

Therefore, when 1 ( + C3󸀠 )eK3 M1 C1 l ≤ l, 2 C3󸀠 =

α

Λ K2 1 sup( k ) eK3 M1 δ0 + , K3 k λn 2

C1 = min(δ0 ,

ln 32

6K3 M1 C3󸀠

l = 6C3󸀠 ,

),

equation (3.9.12) is true. This proves the proposition. The result implies that if the sequence (τN )N∈N is chosen so that τN ≤

C1 , (N + 1)λnα

∀N,

(3.9.13)

̃l,b , where b, l are given in Proposition 3.9.1. then the sequence (ΦN )N∈N ⊂ F Now we give an estimate of thickness of an approximate manifold, which involves the attractor in a neighborhood of MN . Proposition 3.9.2. Let the assumptions of Proposition 3.9.1, and conditions (H4 )󸀠 , (H5 )󸀠 ̃l,b , we be satisfied. Then there exist three constants C3 , C4 , C5 such that for any Ψ ∈ F have 󵄨 󵄨 󵄨󵄨 󵄨󵄨 max 󵄨󵄨󵄨FNτ Ψ(y) − z 󵄨󵄨󵄨E ≤ C3 Λα−1 n max 󵄨󵄨Ψ(y) − z 󵄨󵄨E y+z∈A

u=y+z∈A

−1 + C4 (Λα−1 n β2 + Λ n β1 )τ + C5 (

((N + 1)τ)−α −Λ n (N+1)τ + Λα−1 . n )e Λn

̃l,b , u0 = y0 + z0 ∈ A . Let (yi )i=0,...,N be the sequence constructed Proof. Suppose Ψ ∈ F k ̄ be defined by (3.9.10), s ∈ (−∞, 0], y = Pn u, z = Qn u, by equation (3.9.6) and let y(s)

3.9 The convergence of approximate inertial manifolds | 401

ỹk = y(−kτ). Then 0

τ

As

FN Pn (y0 ) = ∫ e Qn f (ỹ(s) + Ψ(ỹ(s)))ds −∞ 0

z0 = ∫ eAs Qn f (y(s) + z(s))ds. −∞

Hence 0

󵄨󵄨 τ 󵄨 󵄨 As 󵄨 󵄨󵄨FN Pn (y0 ) − z0 󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨e Qn f (ỹ(s) + Ψ(ỹ(s))) − f (y(s) + z(s))󵄨󵄨󵄨E ds. −∞

Using condition (H2 ) and equations (3.9.2), (3.9.9) we get 󵄨 󵄨󵄨 τ 󵄨󵄨FN Pn (y0 ) − z0 󵄨󵄨󵄨 0

≤ K2 M1

∫ ( −(N+1)τ

s 1 󵄨 󵄨 󵄨 󵄨 + Λαn )eΛ n ((1 + l)󵄨󵄨󵄨y(s) − ỹ(s)󵄨󵄨󵄨E + 󵄨󵄨󵄨Ψ(y(s)) − z(s)󵄨󵄨󵄨E )ds |s|α

−(N+1)τ

+ 2M0 K2

∫ ( −∞

s 1 + Λαn )eΛ n ds. |s|α

(3.9.14)

Since (H4 )󸀠 is assumed, we have ỹk+1 = Rτ ỹk + Sτ Pn f (ỹk + z(−kτ)) + εk . By Lemma 3.9.2, we have k−1

α

|yk − ỹk | ≤ ∑ e(k−l−j)τ(λn +K3 M1 (1+l)λn ) j=0

󵄨 󵄨 × (|εj |E + 󵄨󵄨󵄨Sτ Pn (f (ỹj + z(−jτ)) − f (ỹj + Ψ(ỹj )))󵄨󵄨󵄨E ) 󵄨 󵄨 ≤ [τ2 β1 − K3 M1 λnα−1 (eτλn − 1) max 󵄨󵄨󵄨Ψ(y) − z 󵄨󵄨󵄨E ] y+z∈A α

α

× ekτ(λn +K3 M1 (1+l)λn ) /[eτ(λn +K3 M1 (1+l)λn ) − 1] ≤[

τβ1 󵄨 󵄨 + K3 M1 λnα−1 max 󵄨󵄨󵄨Ψ(y) − z 󵄨󵄨󵄨E ] y+z∈A λn + K3 M1 (1 + l)λnα α

× ekτ(λn +K3 M1 (1+l)λn ) .

Since s ∈ (−(k + 1)τ, −kτ), it follows from (H5 )󸀠 that 󵄨󵄨󵄨y(s) − y(s)󵄨󵄨󵄨 ≤ τβ2 + |yk − yk |E 󵄨 󵄨E τβ1 󵄨 󵄨 ≤ τβ2 + [ + K3 M1 λnα−1 max 󵄨󵄨󵄨Ψ(y) − z 󵄨󵄨󵄨E ] y+z∈A λn + K3 M1 (1 + l)λnα α

× e−skτ(λn +K3 M1 (1+l)λn ) .

402 | 3 The approximate inertial manifold Plugging the above equation into equation (3.9.14), we get 0

eu 󵄨󵄨 τ 󵄨 du + 1)Λα−1 󵄨󵄨FN Φ(y0 ) − z0 󵄨󵄨󵄨E ≤ K2 M1 β2 (1 + l)( ∫ n τ |u|α −∞

+ K2 M1 (1 + l)[ 0

×

∫ ( −(N+1)τ

τβ1 󵄨 󵄨 + K3 M1 λnα−1 max 󵄨󵄨󵄨Ψ(y) − z 󵄨󵄨󵄨E ] y+z∈A λn + K3 M1 (1 + l)λnα

α 1 + Λαn )e−K3 M1 (1+l)λn s ds |s|α

󵄨 󵄨 + K2 M1 (Γ(−1 − α) + 1)Λα−1 max 󵄨󵄨󵄨Ψ(y) − z 󵄨󵄨󵄨E n x+z∈A + 2K2 M0

((N + 1)τ)−α + Λαn −Λ n (N+1)τ e . Λn

Through the proof of Proposition 3.9.1, we get 0

K2 M1 (1 + l)

∫ ( −(N+1)τ

1 + Λαn )e−K3 M1 (1+l)s ds ≤ l. |s|α

Hence 󵄨󵄨 τ 󵄨 α−1 −1 󵄨󵄨FN Φ(y0 ) − z0 󵄨󵄨󵄨 ≤ C4 (Λ n β2 + Λ n β1 )τ 󵄨 󵄨 + C3 Λα−1 max 󵄨󵄨󵄨Ψ(y) − z 󵄨󵄨󵄨E n y+z∈A

+ 2K2 M0

((N + 1)τ)−α + Λαn −Λ n (N+1)τ , e Λn

where C3 , C4 are independent of N, τ and n. Thus, if τN satisfies equation (3.9.13), λn ≥ C2 , then we have 󵄨 󵄨 󵄨 󵄨 max 󵄨󵄨󵄨ΨN+1 (y) − z 󵄨󵄨󵄨E ≤ C3 Λα−1 max 󵄨󵄨󵄨ΨN (y) − z 󵄨󵄨󵄨E n y+z∈A

u=y+z∈A

−1 + C4 (Λα−1 n β2 + Λ n β1 )τN + C5

((N + 1)τN )−α + Λαn −Λ n (N+1)τN e . Λn

Proposition 3.9.2 has been proved. Theorem 3.9.1. Let (H2 ), (H3 ), (H4 )󸀠 , (H5 )󸀠 be satisfied. Assume the sequence (τN )N∈N satisfies C6 ≤ τN (N + 1)λnα ≤ C1 ,

∀N,

where C6 is an appropriate constant, and constant C1 is given in Proposition 3.9.2. Then the sequence (ΦN )N∈N defined by equation (3.9.8) satisfies:

3.9 The convergence of approximate inertial manifolds | 403

󵄨 󵄨 max 󵄨󵄨󵄨ΨN (y) − z 󵄨󵄨󵄨E

u=y+z∈A

N−1

N

j

1−α

α−1 −1 α−1 α−1 −C6 Λ n ≤ (C3 Λα−1 , n ) max |Qn u|E + C4 (Λ n β2 + Λ n β1 ) ∑ (C3 Λ n ) τN−1−j + 4C5 Λ n e u∈A

j=0

provided λn ≥ C7 , where C7 is another constant. This yields an estimate of distance from MN to A : dE (A , MN ) = sup inf |v − ω|E ≤

v∈A ω∈MN N (C3 Λα−1 n )

−1 max |Qn u|E + C4 (Λα−1 n β2 + Λ n β1 )

u=y+z∈A

N−1

j

1−α

α−1 −C6 Λ n × ∑ (C3 Λα−1 . n ) τN−1−j + 4C5 Λ n e j=0

(3.9.15)

The first and second terms on right-hand side of inequality (3.9.15) tend to zero when N → ∞. So the above distance decreases and comes close to a very small number 1−α

−C6 Λ n 4C5 Λα−1 . n e

Hence, if N is large enough, MN is an explicit inertial manifold of order 1−α

−C6 Λ n 8C5 Λα−1 . n e

Now we prove the convergence of FNτ when N → ∞. Proposition 3.9.3. Let (H2 ) and (H3 ) be satisfied. If there exists a constant C8 such that Λ n − λn ≥ C8 (Λαn + λnα ), then we have: for any N and τ > 0, FNτ maps Fl,b into itself, and FNτ is a strict compact mapping, its compression constant is less than 21 . Proof. Let Ψ ∈ Fl,b , y0 ∈ Pn E, and by equation (3.9.6) construct (yki )k=0,...,N . By equation (3.9.10) define y. From Lemma 3.9.1 we get that if C8 is large enough, C8 ≥ K3 M1 (1 + Λ l)(supk λ k )1−α , then k

󵄨 󵄨󵄨 −s(λ +K M (1+b)λnα ) (|y0 |E + 1), 󵄨󵄨y(s)󵄨󵄨󵄨E ≤ e n 3 1 Denote

0

τ

As

s ≤ 0.

FN Ψ(y0 ) = ∫ e Qn f (y(s) + Ψ(y(s)))dx. −∞

Using (H2 ) and equations (3.9.2), (3.9.7), we then get 0

󵄨󵄨 τ 󵄨 󵄨󵄨FN Ψ(y0 )󵄨󵄨󵄨E ≤ ∫ K2 M1 (1 + b)(1 + |y0 |E ) −∞

404 | 3 The approximate inertial manifold

×(

α 1 + Λαn )e(Λ n −λn −K3 M1 (1+b)λn )s ds α |s|

0

+ ∫ K2 M1 (1 + b)( −∞

1 + Λαn )eΛ n s ds |s|α

≤ K2 M1 (1 + b)(Γ(−1 − α) + 1)

Λαn + Λα−1 n )(1 + |y0 |E ) Λ n − λn − K3 M1 (1 + b)λnα ≤ b(1 + |y0 |E ), ×(

where C8 ≥ max(K3 M1 (1 + b),

2K2 M1 (1 + b )(Γ(−1 − α) + 1). b

Now we take y01 , y02 ∈ Pn E, and using equation (3.9.6) define (yki )k=0,...,N , y0 = y0i , and yi as before for i = 1, 2. From Lemma 3.9.1 we get 󵄨󵄨 1 󵄨 2 −s(λ +K λα ) 󵄨 1 2󵄨 󵄨󵄨y (s) − y (s)󵄨󵄨󵄨E ≤ e n 6 n 󵄨󵄨󵄨y0 − y0 󵄨󵄨󵄨E ,

s ≤ 0.

Hence from (H2 ) and equations (3.9.2) (3.9.3), we get 󵄨󵄨 τ 1 τ 2 󵄨 󵄨󵄨FN Ψ(y0 ) − FN Ψ(y0 )󵄨󵄨󵄨E 0

󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨eAs Qn f (y1 (s) + Ψ(y1 (s))) − f (y2 (s) + Ψ(y2 (s)))󵄨󵄨󵄨E ds −∞

󵄨 󵄨 ≤ K2 M1 (1 + l)󵄨󵄨󵄨y01 − y02 󵄨󵄨󵄨E (Γ(−1 − α) + 1) 󵄨 󵄨 ≤ l󵄨󵄨󵄨y01 − y02 󵄨󵄨󵄨E ,

Λαn Λ n − λn − K6 λnα

where C8 ≥ max(K6 , K2 M1l(1+l) ). Then we prove that FNτ is strictly compact. Taking Ψ1 , Ψ2 ∈ Fl,b , y0 ∈ Pn E, we use Pn = Ψ1 , Pn = Ψ2 and equation (3.9.6) to construct (yk1 )k=0,...,N , (yk2 )k=0,...,N , and y1 , y2 , respectively, as before. By Lemma 3.9.1, since y0 = y01 = y02 , we get 󵄨󵄨 1 󵄨 2 α −s(λ +K λα ) 󵄨󵄨y (s) − y (s)󵄨󵄨󵄨E ≤ K3 M1 λn |Ψ1 − Ψ2 |∞ (1 + |y0 |E )|s|e n 6 n . Writing τ 1 FN Ψ (y0 )

−

τ 2 FN Ψ (y0 )

0

≤ ∫ eAs Qn f (y1 (s) + Ψ(y1 (s))) −∞

− f (y2 (s) + Ψ(y2 (s)))ds,

3.9 The convergence of approximate inertial manifolds | 405

from (H2 ) and equations (3.9.2), (3.9.7) we get 󵄨󵄨 τ 1 󵄨 τ 2 󵄨󵄨FN Ψ (y0 ) − FN Ψ (y0 )󵄨󵄨󵄨E 0

≤ K2 M1 ∫ ( −∞

1 󵄨 󵄨 + Λαn )eΛ n s (1 + l)󵄨󵄨󵄨ȳ01 (s) − ȳ02 (s)󵄨󵄨󵄨E |s|α

󵄨 󵄨 + |Ψ1 − Ψ2 |∞ (1 + 󵄨󵄨󵄨y1 (s)󵄨󵄨󵄨E )ds ≤ K2 M1 |Ψ1 − Ψ2 |∞ [K3 M1 (1 + l)λnα (1 + |y0 |E ) 0

α

0

× ∫ (|s|1−α + Λαn |s|)e(Λ n −λn −K6 λn )s ds + ∫ ( −∞

−∞

1 + Λαn )eΛ n s ds |s|α

0

α

+ (1 + |y0 |E ) ∫ (|s|1−α + Λαn |s|) × e(Λ n −λn −K3 M4 (1+b)λn )s ds]. −∞

Then for y1 using Lemma 3.9.1, by direct calculation we get 󵄨󵄨 τ 1 󵄨 τ 2 󵄨󵄨FN Ψ (y0 ) − FN Ψ (y0 )󵄨󵄨󵄨E ≤ K2 M1 (1 + |y0 |E )|Ψ1 − Ψ2 |∞ × [Λα−1 n (Γ(−1 − α) + 1) + K3 M1 (1 + l) × ((1 − α)Γ(−1 − α) + 1) + 1 ≤ (1 + |y0 |E )|Ψ1 − Ψ2 |∞ , 2

Λαn λnα (Λ n − λn − K6 λnα )2

Λαn (Γ(−1 − α) + 1)] (Λ n − λn − K3 M1 (1 + b)λnα )

where we assume that C8 is greater than an appropriate constant which is independent of n. Thus Proposition 3.9.3 has been proved. Below, in order to compare the distance between FNτ ΨN and Φ or Ψ, we first state a theorem on the existence of inertial manifolds for equation (3.9.1) in [45]. Let 0

JΨ(y0 ) = ∫ eAs Qn f (y(s) + Pn (y(s)))ds. −∞

Theorem 3.9.2. If there exists a constant C0 , depending on f , K1 , K2 , l and b, such that Λ n − λn ≥ C0 (Λαn + λnα ), then the functional J maps Fl,b into itself, and J is a strictly compact mapping. Hence it has a fixed point, that is, J Φ = Φ, graph Φ is a C1 -inertial manifold of equation (3.9.1). Proposition 3.9.4. Under conditions of Proposition 3.9.3, Theorem 3.9.2, and assuming Λ −λ (H4 ) and (H5 ), there exist two constants C9 , C10 such that if Λαn +λnα ≥ C9 ≥ C8 , then for all n

n

406 | 3 The approximate inertial manifold Ψ ∈ Fl,b and all N, τ, we have 1 󵄨󵄨 τ 󵄨 󵄨󵄨FN Pn − Φ󵄨󵄨󵄨∞ ≤ |Ψ − Φ|∞ + ε(N, τ), 2 where ε(N, τ) = C10 ((α2 (λn ) +

α α1 (λn ) α (λ ) )τ + 2 αn e−Λ n Nτ ). λnα Λn

Proof. From Proposition 3.9.3, we have 1 󵄨󵄨 τ 󵄨 τ 󵄨 󵄨 󵄨󵄨FN Ψ − Φ󵄨󵄨󵄨∞ ≤ |Ψ − Φ|∞ + 󵄨󵄨󵄨FN Φ − Φ󵄨󵄨󵄨∞ . 2 We estimate |FNτ Φ − Φ|∞ successively. Suppose that y is the solution of the following equation: { dy + Ay = Pn f (y + Φ(y)), { dt {y(0) = y0 . Taking yk = y(−kτ), by equation (3.9.6) we construct (yki )k=0,...,N , Ψ = Φ, and then by (H4 ) and Lemma 3.9.2, with ek = εk , we have k−1

α

|yk − yk |E ≤ ∑ e(k−1−j)τ(λn +K3 M1 (1+l)λn ) |εj |E j=0

󸀠 α

≤ α1 (λn )τ2 (1 + |y0 |E )kekτ(λn +C4 λn ) ,

(3.9.16)

where C4󸀠 = max(K4 , K3 M1 (1 + l)). Using equation (3.9.10) we define y. Then by employing (H5 ) and equation (3.9.16), we arrive at 󵄨󵄨 󵄨 −s(λ +K λα ) 󵄨󵄨y(s) − y(s)󵄨󵄨󵄨E ≤ τα2 (λn )(1 + |y0 |E )e n 5 n

󸀠 α

+ τα1 (λn )(1 + |y0 |E )|s|e−s(λn +C4 λn ) ,

s ∈ ( − (k + 1)τ, −kτ].

(3.9.17)

Similarly, we can deduce that 󵄨󵄨 󵄨 −s(λ +K λα ) 󵄨󵄨y(s) − y(s)󵄨󵄨󵄨E ≤ |s + Nτ|α2 (λn )(1 + |y0 |E )e n 5 n 󸀠 α

+ τα1 (λn )(1 + |y0 |E )|s|e−s(λn +C4 λn ) ,

s ∈ (−∞, −Nτ]. Since τ FN Φ(y0 )

0

− Φ(y0 ) = ∫ eAs Qn f (y(s) + Φ(y(s))) − f (y(s) + Φ(y(s)))ds, −∞

(3.9.18)

3.9 The convergence of approximate inertial manifolds | 407

together with (H2 ) and equations (3.9.2), (3.9.7), we obtain 󵄨󵄨 τ 󵄨 󵄨󵄨FN Φ(y0 ) − Φ(y0 )󵄨󵄨󵄨E 0

≤ K2 M1 (1 + l) ∫ ( −∞

1 󵄨 󵄨 + Λαn )eΛ n s 󵄨󵄨󵄨y(s) − y(s)󵄨󵄨󵄨E sds |s|α

≤ K2 M1 (1 + l)(1 + |y0 |E ) 0

× [α2 (λn )τ ∫ ( −∞

α 1 + Λαn )eΛ n −λn −K5 λn sds |s|α

0

󸀠 α

+ α1 (λn ) ∫ (|s|α−1 + Λαn )e(Λ n −λn −C4 λn ) sds −∞ −Nτ

+ α2 (λn ) ∫ |s + Nτ|( −∞

≤ (1 + |y0 |E )C10 (α2 (λn ) + α1 (λn ) + α2 (λn )

α 1 + Λαn )eΛ n −λn −K5 λn sds] |s|α

Λαn τ Λ n − λn − K5 λnα

Λαn τ (Λ n − λn − C4󸀠 λnα )2

α Λαn e−(Λ n −λn −K5 λn )Nτ ) α 2 (Λ n − λn − K5 λn )

≤ C10 (1 + |y0 |E )(α2 (λn )τ +

α1 (λn ) α2 (λn ) −Λαn Nτ + e ), Λαn Λαn

where C9 ≥ max(K5 , 1, C8 , C4󸀠 ). The constant C10 can be calculated in detail, and does not depend on N, τ or n. Since, when τ → 0, Nτ → ∞, this leads to ε(N, τ) → 0. It is then easy to get Theorem 3.9.3. Under the assumptions of Proposition 3.9.4, defining the sequence (ΦN )N∈N by equation (3.9.8) which, it converges to Φ with respect to | ⋅ |∞ , where (τN )N∈N is satisfied that NτN → ∞ as N → ∞. Below we prove that the sequence (ΦN )N∈N converge to Φ under very few conditions in the C-topology. Introduce the set 󵄨

󵄨

Gl = {Δ : Pn E → L (Pn E, Qn E | sup 󵄨󵄨󵄨Δ(y)󵄨󵄨󵄨L (P E,Q E) ≤ l)}, n n

󵄨 󵄨 |Δ|∞ = sup 󵄨󵄨󵄨Δ(y)󵄨󵄨󵄨L (P E,Q E) . n n y∈PN E

y∈Pn E

408 | 3 The approximate inertial manifold For any Ψ ∈ Fl,b , define the mapping TΨ : Δ ∈ G → TΨ (Δ) by 0

TΨ (Δ)(y0 )η0 = ∫ eAs Qn Df (y(s) + Ψ(y(s)))(η(s) + Δ(y(s))η(s))ds,

(3.9.19)

−∞

where y is the solution of the following problem: { dy + Ay = Pn f (y + Ψ(y)), { dt {y(0) = y0 ,

(3.9.20)

{ dη + Aη = Pn f (y + Ψ(y))(η + Δ(y)η), { dt {η(0) = η0 .

(3.9.21)

and assume η satisfies

Under the assumption of Theorem 3.9.2, we can prove for Ψ ∈ Fl,b , that TΨ maps Fl into itself. Moreover, it is a strictly compact mapping, and this holds uniformly with respect to Ψ. Hence, we can prove Φ ∈ C 1 . Now by the construction method of FNτ we come up with an approximation of TΨ . τ For all N ∈ N, τ > 0, Ψ ∈ Fl,b , we define TN,Ψ in Fl as follows: N−1

τ TN,Ψ (Δ)(y0 )η0 = A−1 (I − e−τA ) ∑ e−kτA Qn Df (yk + Ψ(yk ))(ηk + Δ(yk )ηk ) k=0

+ A−1 e−NτA Qn Df (yN + Ψ(yN ))(ηk + Δ(yN )ηN ), where (yki )k=0,...,N and (ηk )k=0,...,N can be calculated by the following equations: yk+1 = Rτ yk + Sτ Pn f (yk + Ψ(yk )),

ηk+1 = Rτ ηk + Sτ Pn Df (yk + Ψ(yk ))(ηk + Δ(yk )ηk ).

(3.9.22) (3.9.23)

Suppose that (ΦN )N∈N is defined by equation (3.9.8). Then it is easy to get τ

N DΦN+1 = TN,Φ (DΦN ). N

τ Under some additional assumptions, we prove that TN,Φ is approaching TΨ in a certain sense, and thus DΦN is close to DΦ. Firstly, because of the technical reasons, we suppose f has finite support. This is valid in some application expamples. Because the function F can be truncated, suppose

(H6 )

|Rτ y|E ≥ |y|E , { {󵄨 −At 󵄨󵄨e y󵄨󵄨󵄨 ≥ |y|E , 󵄨E { {󵄨 {|y + z|E ≥ |y|E ,

τ > 0, t > 0, y ∈ Pn E, z ∈ Qn E.

(3.9.24)

3.9 The convergence of approximate inertial manifolds | 409

We say that equations (3.9.22)–(3.9.23) approximate (3.9.20)–(3.9.21) if for y(η) defined on the interval (−∞, 0], equal to yk (ηk ) on (−(k + 1)τ, −kτ], and equal to yN (ηN ) on (−∞, −Nτ], respectively, we have (H7 ) for all T > 0, y, η converge to the solution y, η of equation (3.9.20)–(3.9.21) in [−T, 0] consistently, where y0 , η0 ∈ Pn E, for any bounded set, τ → 0, Nτ → ∞. Theorem 3.9.4. Suppose the assumptions of Theorem 3.9.3 hold, (H6 ) and (H7 ) are satisfied, f has compact support, and Df is consistently continuous. Then the sequence (ΦN )N∈N converges to Φ in the C 1 -topology, when τN → 0, NτN → ∞. Proof. From Theorem 3.9.3, ΦN consistently converges to Φ in any bounded set of Pn E. Suppose that B is a ball in the support of f , and R1 is its radius. Choosing any y0 ∈ Pn E such that |y0 |E ≥ R1 , by (H6 ) we deduce that when y is the solution of equation (3.9.5) and (yki )k=0,...,N is defined by equation (3.9.6), we have 󵄨󵄨 󵄨 󵄨󵄨y(s)󵄨󵄨󵄨E ≥ R1 , |yk |E ≥ R1 ,

s ≤ 0, k = 0, . . . , N.

From equations (3.9.4) and (3.9.7), we deduce that Φ(y0 ) = ΦN (y0 ) = 0. Hence, for all N, ΦN has its support included in B and ΦN is uniformly convergent to Φ in Pn E. Denote τN τN DΦN+1 − DΦ = TN,Φ (DΦN ) − TN,Φ (DΦN ) N

N

τ

N + TN,Φ (DΦN ) − TΦ (DΦ). N

(3.9.25)

Using the method of Proposition 3.9.3, we get 1 τN 󵄨󵄨 τN 󵄨 (DΦ)󵄨󵄨󵄨∞ ≤ |DΦN − DΦ|∞ . 󵄨󵄨TN,ΦN (DΦN ) − TN,Φ N 2

(3.9.26)

Consider y0 , η0 ∈ Pn E, |η0 | = 1, and let (yki )k=0,...,N and (ηk )k=0,...,N , ((ykN )k=0,...,N and (ηNk )k=0,...,N ) be defined by equations (3.9.22)–(3.9.23). Letting Ψ = Φ, Δ = DΦ (Ψ = ΦN , Δ = DΦN ), define y, η (yN , ηN ) as in (H7 ). Then if y, η are the solutions of equations (3.9.20)–(3.9.21) with Ψ = Φ, Δ = DΦ, we have τ

N TN,Φ (DΦ)(y0 )(η0 ) − TΦ (DΦ)(y0 )(η0 ) N

0

= ∫ eAs Qn (Df (yN (s) + ΦN (yN (s))) −∞

410 | 3 The approximate inertial manifold × (ηN (s) + DΦ(yN (s))ηN (s)) − Df (y(s) + Φ(y(s)))(η(s) + DΦ(y(s))η(s))ds.

(3.9.27)

If |y0 |E ≥ R1 , then we can prove τ

N TΦ (DΦ)(y0 )(η0 ) = TN,Φ (DΦ)(y0 )(η0 ) = 0. N

There exists a constant

C5󸀠

such that the integral of equation (3.9.27) is less than

C5󸀠 (

α 1 + Λαn )|η0 |E e(Λ n −λn −K2 M1 (1+l)λn )s . |s|α

Hence, the integral for y0 , η0 is uniformly convergent. Since |η0 |E = 1, for ε > 0, there exists T, which is independent of y0 , η0 , such that −T

󵄨 ∫ 󵄨󵄨󵄨eAs Qn (Df (yN (s) + ΦN (yN (s)))

−∞

× (ηN (s) + DΦ(yN (s))ηN (s))

ε 󵄨 − Df (y(s) + Φ(y(s)))(η(s) + DΦ(y(s))η(s))󵄨󵄨󵄨E ds ≤ . 2 The remainder of the integral (3.9.27) is divided into the following sum of integrals: 0

I1 = ∫ eAs Qn (Df (yN (s) + ΦN (yN (s))) −T

× (ηN (s) + DΦ(yN (s))ηN (s)) − Df (y(s) + Φ(y(s)))(η(s) + DΦ(y(s))η(s))ds, 0

I2 = ∫ eAs Qn (Df (y(s) + ΦN (y(s))) −T

× (η(s) + DΦ(y(s))η(s)) − Df (y(s) + Φ(y(s)))(η(s) + DΦ(y(s))η(s))ds. In the proof of Proposition 3.9.3, we have shown that there exists a constant C6󸀠 (T, n) such that 󵄨󵄨 󵄨 N 󸀠 󵄨󵄨y(s) − y (s)󵄨󵄨󵄨E ≤ C6 (T, n)|ΦN − Φ|∞ (1 + |y0 |E ) ≤ C6󸀠 (T, n)(1 + R1 )|ΦN − Φ|∞ . (3.9.28) We can find a bounded set BT , which depends on n, R1 and T, such that y(s) and yN (s) are in BT , ∀s ∈ [−T, 0]. Therefore for any α > 0, if we choose αn such that 󵄨󵄨 󵄨 󵄨 󵄨 N N N 󵄨󵄨y(s) − y (s)󵄨󵄨󵄨E + 󵄨󵄨󵄨Φ(y(s)) − Φ (y (s))󵄨󵄨󵄨E ≤ αN , ∀s ∈ [−T, 0], |y0 | ≤ R1 ,

3.9 The convergence of approximate inertial manifolds | 411

we can prove that there exists C7󸀠 (T, n) such that 󵄨 󵄨󵄨 N 󸀠 󵄨󵄨η (s) − η(s)󵄨󵄨󵄨E ≤ C7 (T, n)(M1 (αN ) + M2 (αN )) s ∈ [−T, 0], |y0 |E ≤ R1 , |η0 |E = 1,

(3.9.29)

where 󵄨 󵄨 M1 (α) = sup 󵄨󵄨󵄨Df (x) − Df (y)󵄨󵄨󵄨L (E,F) , |x−y|≤α

M2 (α) =

sup

x,y∈BT , |x−y|E ≤α

󵄨󵄨 󵄨 󵄨󵄨DΦ(x) − DΦ(y)󵄨󵄨󵄨L (E) .

From equations (3.9.28)–(3.9.29), Df in Pn E and DΦ in BT are uniformly continuous, which implies that I1 uniformly converges to zero for given y0 , η0 , which satisfy |y0 |E ≤ R1 , |η0 |E = 1. Then by (H7 ), we know that I2 uniformly converges to zero for y0 , η0 , since |y0 |E ≤ R1 , |η0 |E = 1. τN Hence, JN,Φ (DΦ)(y0 )η0 − TΦ (DΦ)(y0 )η0 uniformly converges to zero for y0 , η0 N satisfying |y0 |E ≤ R1 , |η0 |E = 1. But since |y0 | ≤ R1 , it is zero. Hence 󵄨󵄨 τN 󵄨 󵄨󵄨TN,Φ (DΦN ) − TΦ (DΦ)󵄨󵄨󵄨∞ → 0,

N → ∞.

From this and equations (3.9.25)–(3.9.26) we get |DΦ − DΦN |∞ → 0,

N → ∞.

Thus the theorem has been proved. Now we give three examples to illustrate the established results. Example 1 Consider the classic parabolic semilinear equation with symmetric linear part, namely { du + Au = g(u), { dt {u(0) = u0 ,

(3.9.30)

where the operator A is a dense linear positive unbounded self-adjoint operator in a Hilbert space, which has norm | ⋅ |. This equation has a compact pre-solution set, which is composed of an orthogonal basis {ωj }j∈N of eigenvectors with the corresponding eigenvalues 0 < μ0 ≤ μ1 ≤ ⋅ ⋅ ⋅ ≤ μj → +∞. Space D(As ), s > 0 is defined as usual, with the norm | ⋅ |s = |As ⋅ |. Let g ∈ C 1 : D(Aα+γ ) → D(Aγ ), γ ≥ 0, α ∈ [0, 1). Take E = H,

F = D(Aγ ),

E = D(Aα+γ ).

412 | 3 The approximate inertial manifold Define Pn as the characteristic projection of A. Then Pn H and Qn H are invariant under the action of e−At (t ≥ 0). The semigroup (e−At )t≥0 can be extended to a group in Pn H, e−At D(Aγ ) ⊂ D(Aα+γ ), ∀t > 0. Assume that conditions (H1 ) and (H2 ) are easy to verify that they are satisfied, and just take K1 = K2 = 1,

λn = μn ,

Λ n = μn+1 .

Under suitable assumptions, equation (3.9.30) in D(Aα+γ ) defines a continuous semigroup (S(t))t≥0 . Generally speaking, the function g is not globally Lipschitz. In order to overcome this difficulty, we can set (S(t))t≥0 to be a bounded absorbing set in D(Aα+γ ). Take R ≥ 0 appropriately large, so that the ball with a radius of R in D(Aα+γ ) contains this absorbing set. The function f is defined as: f (u) = θ(

|u|2α−γ R2

)g(u),

where θ(x) ∈ C 1 is such that θ(x) = 1, { { θ(x) ≤ 1, { { {θ(x) = 0,

x ≤ 1, ∀x, x ≥ 2.

In general, we consider the truncated equation { du + Au = f (u), { dt {u(0) = u0 ,

(3.9.31)

Consider the setting of Proposition 3.9.3, and the gap condition is given by lim sup n→∞

μn+1 − μn = ∞. μαn+1 + μαn

(3.9.32)

When A is an elliptic operator on a bounded domain Ω ⊂ Rn , its eigenvalues possess the asymptotical property μn ∼ Cnp ,

(3.9.33)

which can justify the assumption of equation (3.9.32). Condition p(α − 1) > 1

(3.9.34)

is not implied by equation (3.9.32). For example, let A = −Δ, with periodic boundary conditions on [0, L1 ] × [0, L2 ] ⊂ R2 . When LL1 is a rational number and α = 0, equation 2 (3.9.32) is satisfied, but p(1 − α) = 1, and so inequality (3.9.34) does not hold.

3.9 The convergence of approximate inertial manifolds | 413

The approximation formula (3.9.6) for equation (3.9.5) can be discretized using different methods. First, consider the simple Euler scheme: yk − yk−1 + Ayk = Pn f (yk + Ψ(yk )), τ which gives R = I + τA, { τ Sτ = −τI.

(3.9.35)

We verify that (H3 ), (H4 ), (H5 ), (H4 )󸀠 , (H5 )󸀠 hold. First, (H3 ) is obvious if we take K3 = 1. And it is easy to verify (H5 ). By equation (3.9.5) we get 󵄨󵄨 dy 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (λn + M1 (1 + b)λnα )|y|α+1 + λnα M1 (1 + b). 󵄨󵄨 󵄨󵄨 󵄨󵄨 dt 󵄨󵄨α+γ Denote

t

y(t) = e−At y(0) + ∫ e−A(t−s) Pn f (y(s) + Ψ(y(s)))ds. 0

Then we get t

󵄨 󵄨󵄨 󵄨 󵄨 −λ (t−s) 󵄨󵄨 −λ t 󵄨 α 󵄨󵄨f (y(s) + Ψ(s))󵄨󵄨󵄨γ ds 󵄨󵄨y(t)󵄨󵄨󵄨α+γ ≤ e n 󵄨󵄨󵄨y(0)󵄨󵄨󵄨α+γ + λn ∫ e n ≤e

0

−λn t 󵄨󵄨

󵄨 α−1 󵄨󵄨y(0)󵄨󵄨󵄨α+γ + λn M1 (1 + b) t

󵄨 󵄨 + M1 (1 + b)λnα ∫ e−λn (t−s) 󵄨󵄨󵄨y(s)󵄨󵄨󵄨α+γ ds. 0

Using Gronwall lemma, we arrive at 󵄨󵄨 󵄨 󵄨 󵄨 α−1 −(λ +M (1+b)λnα )t . 󵄨󵄨y(t)󵄨󵄨󵄨α+γ ≤ 2(󵄨󵄨󵄨y(0)󵄨󵄨󵄨α+γ + λn M1 (1 + b))e n 1 Therefore,

󵄨󵄨 dy 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ 3(λn + M1 (1 + b)λnα ) 󵄨󵄨 󵄨󵄨 󵄨󵄨 dt 󵄨󵄨α+γ α 󵄨 󵄨 × (󵄨󵄨󵄨y(0)󵄨󵄨󵄨α+γ + λnα−1 M1 (1 + b))e−(λn +M1 (1+b)λn )t ,

and so (H5 ) is established, where α2 (λ) = 3(λn + M1 (1 + b)λnα ) max(1, M1 (1 + b) sup λnα−1 ), K5 = M1 (1 + b).

n

414 | 3 The approximate inertial manifold Now we prove (H4 ). Let y be the solution of equation (3.9.5). If we set ỹk = y(−kτ), then ỹk − ỹk+1 + Aỹk = Pn f (ỹk + Ψ(ỹk )) + εk󸀠 , τ where εk󸀠

1 = τ

−kτ

∫ ( −(k+1)τ

dy dy (s) − (−kτ))ds. dt dt

Hence, εk = τεk󸀠 . Using equation (3.9.5), 󵄨󵄨 dy 󵄨󵄨 dy 󵄨󵄨 󵄨 󵄨 󵄨 (−kτ)󵄨󵄨󵄨 ≤ 󵄨󵄨Ay(s) − Ay(−ks)󵄨󵄨󵄨α+γ 󵄨󵄨 (s) − 󵄨󵄨α+γ 󵄨 dt 󵄨󵄨 dt 󵄨 󵄨 + 󵄨󵄨󵄨Pn (f (y(s) + Ψ(y(s))) − f (ỹ(s)k + Ψ(ỹk )))󵄨󵄨󵄨α+γ . Since Ψ ∈ Fl,b and due to equation (3.9.2), we have 󵄨󵄨 󵄨󵄨 dy dy 󵄨 󵄨 󵄨 󵄨󵄨 (−kτ)󵄨󵄨󵄨 ≤ (λn + M1 (1 + l)λnα )󵄨󵄨󵄨y(s) − y(−kτ)󵄨󵄨󵄨α+γ 󵄨󵄨 (s) − 󵄨󵄨α+γ 󵄨󵄨 dt dt s

󵄨󵄨 dy 󵄨󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨 (θ)󵄨󵄨󵄨 dθ 󵄨󵄨 dt 󵄨󵄨α+γ

≤ τ(λn + M1 (1 +

l)λnα )

≤ τ(λn + M1 (1 +

l)λnα )(kτ

−kτ

α

+ s)α2 (λ)e−(λn +K5 λn )t .

From this we can deduce (H4 ), where α1 (λ) = (λn + M1 (1 + l)λnα )(kτ + s)α2 (λ), K4 = K5 .

In order to verify the (H4 )󸀠 , (H5 )󸀠 , each specific equation requires a special proof. First of all, it is necessary to prove the existence of an attractor and show that it is bounded in D(Aα+γ ). Secondly, the bounds of time derivatives can be obtained from the time analyticity of solution and the Cauchy inequality. In the end, the proof of (H6 ) is direct. Condition (H7 ) can be obtained from the classic, but cumbersome calculation. In summary, the inertial manifolds exist, and {ΦN }N∈N converges in the C 1 -topology, as long as equation (3.9.32) holds when n is sufficiently large. Another possible approximation of equation (3.9.5) is yk+1 = eAτ yk + A−1 (eAτ − 1)Pn f (yk + Ψ(yk )), where Rτ = eAτ ,

Sτ = A−1 (eAτ − I).

All assumptions can be verified similarly.

(3.9.36)

3.9 The convergence of approximate inertial manifolds | 415

Example 2 Consider the steady solution u of equation (3.9.31), namely Au = f (u). Set v = u − u, A v = Au − Df (u)v + γv,

h(v) = f (u + v) − f (u) − Df (u)v + γv, where the parameter γ is chosen such that A ≥ 0 (that is, (A v, v) > 0, ∀v ∈ D(A ) = D(A)). Then equation (3.9.31) can be changed into { dv + A v = h(v), { dt {v(0) = u0 − u.

(3.9.37)

If we set μj ∼ Cjp ,

E = H,

p > 0,

F = D(Aγ ) = D(A γ ),

E = D(Aα+γ ) = E(A α+γ ),

and if p(1 − α) > 1, then there exists the eigenvector sequence {Pnk }k∈N of A such that (H1 ), (H2 ) hold, where Pk = Pnk ,

λk = μnk + C 󸀠 μαnk ,

Λ k = μnk+1 − C 󸀠 μαnk+1 ,

where C 󸀠 is a constant, which depends on α and f . Function h is globally Lipschitz and satisfies equation (3.9.2). Since equation (3.9.31) implies the existence of an absorbing ball in D(Aα+γ ), this leads to a similar result for equation (3.9.37). Now we construct inertial manifolds of equation (3.9.37). Since Λ k − λk = μnk+1 − μnk − C 󸀠 (μαnk + μαnk+1 ), μnk ∼ npk ,

and condition p(1 − α) > 1 holds, this implies that Theorem 9.6 and spectral gap condition are satisfied. As an approximation of equation (3.9.5) one can consider yk − yk+1 + A yk = Pn f (yk + Ψ(yk )). τ For Rτ = I + τA , Sτ = −τI,

416 | 3 The approximate inertial manifold we need to change the norm on Pk D(Aα+γ ) to ensure (H3 ) is satisfied. For any y ∈ D(Aα+γ ), we define |A l y| , λkl l≥0 󵄩 󵄩 ‖y‖α+γ = 󵄩󵄩󵄩A α+γ y󵄩󵄩󵄩H , 󵄩 󵄩 ‖y‖γ = 󵄩󵄩󵄩A γ y󵄩󵄩󵄩H , ‖y‖H = sup

which can prove that there exists a C 󸀠󸀠 such that 󵄨󵄨 l 󵄨󵄨 󸀠󸀠 l 󵄨󵄨A y󵄨󵄨 ≤ C λk |y|,

∀l ≥ 0, k ≥ 0, y ∈ Pk D(Aα+γ ).

Introduce the norms ‖y + z‖H = ‖y‖h + |z|,

‖y + z‖α+γ = ‖y‖α+γ + |z|α+γ , ‖y + z‖γ = ‖y‖γ + |z|γ ,

which are equivalent to the previous respective norms. And their constants do not depend on k. It is easy to prove that (H3 ) is established for these norms. Conditions (H4 ), (H5 ), (H4󸀠 ), (H5󸀠 ) are checked as in the previous estimation, while (H6 ), (H7 ) are also easy to verify. An approximation of equation (3.9.5) can be considered as follows: yk+1 = eA τ yk + A −1 (eA τ − I)Pn f (yk + Ψ(yk )). At this time we can use different forms of the norm to meet the first part of (H3 ), the second part can be obtained from the following: A

−1

(e

Aτ

τ

− I)Pn = ∫ eA τ Pn ds. 0

Example 3 A class of equations with linear antisymmetric operators is the following: { du + A0 u + Cu + f (u) = 0, { dt { u(0) = u0 , where A0 satisfies the assumption of A discussed in Example 1, C is a linear bounded α+γ γ antisymmetric operator, D(As0 ) → H, s0 > 0; F is a C 1 function, D(A0 ) → D(A0 ), γ ≥ 0, α ∈ (0, 1]. Suppose that C and A0 commute. Take A = A0 + C,

α+γ

E = D(A0 ),

F = D(A0 ),

E = H,

(Pn )n∈N , (λn )n∈N , (Λ n )n∈N as in Example 1. Conditions (H1 ), (H2 ) are then easy to verify.

3.9 The convergence of approximate inertial manifolds | 417

The first example of such equation is the laser equation. In this case, H = L2 (Ω),

Ω ⊂ Rm is a bounded open set,

A0 u = −Δu + u, with Dirichlet or Neumann boundary condition, Cu = iα(A0 u − u),

F(u) = (1 + γ)u + (1 + iβ)f (|u|2 )u, s , δ > 0. Ginzburg–Landau equation is the second example. The nonlinear f (s) = 1+δs function of the GL equation is not globally Lipschitz. But it has an absorbing sphere. Proposition 3.9.3 is applicable for both equations, as long as the spectral gap condition is satisfied. When the space dimension is 1 or Ω = (0, L1 )×(0, L2 ) ⊂ R2 , and LL1 is a finite 2 number, then this gap condition holds. An approximation of equation (3.9.5) is

yk+1 = e(A0 +C)τ yk + (A0 + C)−1 (e(A0 +C)τ − 1)Pn f (yk + Ψ(yk )). Using the equality τ

(A0 + C)−1 (e(A0 +C)τ − I) = ∫ e(A0 +C)s ds, 0

condition (H3 ) is verified. The laser equation is not dissipative, and does not have attractors. Proposition 3.9.3 and Theorem 3.9.3 are applicable. For the GL equation, the existence of the attractor was proved. And the solution is time analytic. Therefore, (H4 )󸀠 and (H5 )󸀠 are established, while (H6 ) and (H7 ) are also easy to verify.

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Index ∗ convergent 17 α measure 172 absorbing set 19 antisymmetric operator 416 approximate inertial manifold 307, 351, 360, 373 approximate solution 318 Asymptotic completeness 238 asymptotic completeness property 242 asymptotical behavior 80 attractive set 1 attractor 1 Banach space 1 Bernard convection 350 bilinear property 357 blow up 258 Brezis–Gallouet inequality 307 C-analytic manifold 309, 311 C 1 -topology 409, 414 Cauchy formula 326 compact attractor 53, 74 compact embedding 180 compact global attractor 2 compact imbedding 19 complete continuous property 2 conneceted 2 continuous operator 2 contraction mapping principle 311 convex 2 Dirichlet 52, 57 elliptic operator 333 embedding theorem 28, 116 Euler scheme 413 existence 2 existence and uniqueness 363 fan-shaped operator 263 Fatou lemma 129 fixed point 229, 360, 391 foliation 241 fractal 1, 38, 53, 74, 107 fractal dimension 3, 21, 84

fractional power 265 Fréchet derivative 48, 98 Fréchet differentiable 4, 60 Gagliardo–Nirenberg inequality 55, 60, 260 Galekin method 45 Galerkin approximate equation 305 Galerkin approximation 220, 309, 314, 324 Galerkin method 313, 362 gauge 79, 86 generalized Young inequality 278 Gevrey regularity 316, 344 Gilbert damping 86 global attractor 1, 30, 78, 84 Gram determinant 34 Gronwall inequality 54, 57, 81, 117, 135, 178 harmonic mapping 146 Hausdorff 1, 38, 53, 74, 107 Hausdorff dimension 84 Hausdorff measure 3, 21 heteroclinic 163 high-order regularity 254 Hilbert transform 123 Hölder inequality 54, 78 inertial manifold 201 interpolation 190, 330 interpolation inequality 13, 40, 66 k-linear mappings 254 Kraichnan decay rate 194 Laplace operator 52 Laplace–Beltrami operator 145, 152, 160 Lebesgue dominated convergence theorem 246 linear density operator 394 linear mapping 102 Lipschitz condition 219 Lipschitz continuous 206 Lipschitz continuous mapping 301 Lipschitz function 207 Lipschitz manifold 308 lower bound 22, 50 Lyapunov energy functional 89 Lyapunov index 5, 38, 123, 145 Lyapunov–Perron method 395

430 | Index

manifold 1 maximal invariant set 274 maximum and minimum principles 122 nonlinear Galerkin method 356 normal bundles 249 normal hyperbolic 253 ω-limit set 206 ω limiting set 2 orthogonal basis 203, 307, 373, 377, 381 orthogonal projection 4, 280, 289, 341 orthonormal basis 50 orthonormal set 77 Poincaré inequality 94, 379 Prandtl constant 350 quasi-periodic orbit 1 R-linear inner product 121, 143 Rayleigh number 350 Riemannian manifold 145, 146 self-adjoint operator 307, 339 semigroup 1, 59 separable topological space 127

shrinkage 239 smooth monotonic function 261 Sobolev embedding 43, 83, 272, 331 Sobolev estimate 278 Sobolev inequality 52, 132, 152, 157 Sobolev interpolation 91 Sobolev–Lieb–Thirring inequality 61 spectral gap condition 241 squeezing property 281 stream function 350 strict compact mapping 403 strict contraction 234 strict contractive compression 214 tangent bundles 248 time-periodic orbit 1 uniqueness 18 upper bound 139, 309 variational problem 33 vorticity 350 weighted Sobolev space 183 weighted space 183 Young inequality 24, 64, 68, 272