A random attractor for the stochastic quasi-geostrophic dynamical system on unbounded domains

Nonlinear Analysis 90 (2013) 96–112

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A random attractor for the stochastic quasi-geostrophic dynamical system on unbounded domains Hong Lu a,b , Shujuan Lü a,b,∗ , Jie Xin c , Daiwen Huang d a

School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing, 100191, PR China

b

LMIB, Beijing University of Aeronautics and Astronautics, Beijing, 100191, PR China

c

School of Mathematics and Information, Ludong University, Yantai City, Shandong Province, 264025, PR China

d

Institute of Applied Physics and Computational Mathematics, Beijing, 100088, PR China

article

abstract

info

Article history: Received 5 November 2012 Accepted 24 May 2013 Communicated by Enzo Mitidieri MSC: 35B40 35B41 37L55 Keywords: Stochastic quasi-geostrophic equation Asymptotic compactness Random attractor Pullback attractor

In this paper, we prove the existence of a random attractor for the stochastic twodimensional quasi-geostrophic equation on an unbounded domain, which models a class of large-scale geophysical flows. It is sufficient for a closed absorbing set and the asymptotic compactness of the stochastic system to guarantee the existence of a random attractor for a continuous random dynamical system. Hence, the uniform estimates of solutions including the estimates on the tails of solutions are derived first. Second, the D -pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we study the asymptotic behavior of solutions of the stochastic quasi-geostrophic (Q-G) equation on an unbounded two-dimensional domain. Let E = D × R, where D is a bounded domain in R. Consider the following Q-G equation on E:

 d(△ψ − F ψ + β0 y) +

   ∂ψ ∂ ∂ψ ∂ 1 r 2 △ ψ − △ψ dt − Φ dw, − (△ψ − F ψ + β0 y)dt = ∂x ∂y ∂y ∂x Re 2

(1.1)

where (x, y) ∈ E , t > 0, F is the Froude constant (F ≈ O(1)), Re is the Reynolds constant (Re ≥ 102 ), β0 is a positive constant (β0 ≈ O(10−1 )), r is the Ekman dissipative constant (r ≈ O(1)). Φ = Φ (x, y) is a given function defined on E, and w which will be specified later is a two-sided real-valued Wiener process on a probability space. The two-dimensional quasi-geostrophic equations are derived from the primitive equation of the ocean or the atmosphere under the assumption that the velocity and pressure fields are independent of the vertical coordinate [1]. From the physical point of view, the two-dimensional quasi-geostrophic equations can be regarded as an approximation of the rotating shallow-water equations by a conventional asymptotic expansion for small Rossby number [1]. Attractors for the

∗

Corresponding author. Tel.: +86 13439159382. E-mail addresses: [email protected] (H. Lu), [email protected] (S. Lü), [email protected] (J. Xin), [email protected] (D. Huang).

0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.05.020

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

97

3D baroclinic quasi-geostrophic equations of large scale atmosphere in [2] and the existence of the global solutions for the dissipative quasi-geostrophic equations in the Besov space in [3] have been proved. The random attractors for dissipative quasi-geostrophic equations under stochastic forcing on a bounded domain in [4] have been proved. The random attractors of stochastic quasi-geostrophic equations on an unbounded domain are considered in this paper. The concept of pullback random attractor, which is an extension of the attractor theory of deterministic systems in [5–9], was introduced in [10,11]. In the case of bounded domains, the existence of random attractors for stochastic partial differential equations has been investigated by many authors, referring to [10–14] and the references therein. However, this problem is not well understood in the case of unbounded domains. Recently, the existence of random attractors was established in [15–18] when the domain is unbounded. The asymptotic behavior of stochastic Navier–Stokes equations on an unbounded domain was studied in [19]. The random attractors for the stochastic FitzHugh–Nagumo system defined on an unbounded domain were investigated in [16], and for the Benjamin–Bona–Mahony equation in [20]. Combining the classical method with the characteristic of stochastic Q-G equation, we obtain the existence of a pullback random attractor in this paper. As we know that it is sufficient for the asymptotic compactness and existence of a bounded absorbing set to guarantee the existence of a random attractor for a continuous random dynamical system. However, Sobolev embeddings are not compact on an unbounded domain. In this paper, we will employ a tail-estimates approach for the weakly dissipative stochastic Q-G equation and prove the existence of compact random attractors on an unbounded domain, and it is based on that the solutions of the equation are uniformly small when space and time variables are sufficiently large. The highlights of this paper are the uniform estimates of solutions. Thanks to the complexity of nonlinear term G(u), it is necessary to handle specially and to estimate precisely. Furthermore, using the theory of elliptic equations, we simplify the original equation. And applying the stochastic analysis and stationary process, we convert the stochastic equation (3.1) with a random term into a deterministic one with a random parameter. The sections of this paper are arranged as follows. In Section 2, we recall the pullback random attractors theory for the random dynamical system. In Section 3, we define a continuous random dynamical system for the stochastic Q-G equation on E. In Section 4, we derive the uniform estimates of solutions (including the estimates on the tails of solutions). In Section 5, we establish the asymptotic compactness of the random dynamical system and prove the existence of a pullback random attractor. Throughout this paper, we make the following notations. We denote by ∥ · ∥ the norm of H = L2 (E ) with usual inner product (·, ·), denote by ∥ · ∥p the norm of Lp (E ) for all 1 ≤ p ≤ ∞(∥ · ∥2 = ∥ · ∥) and denote by ∥ · ∥H k the norm of a usual Sobolev space H k (E ) for all 1 ≤ k < ∞. The letters C and Cj (j = 0, 1, 2, . . .) are generic positive constants which may change their values from line to line or even in the same line. In addition, we will frequently use the embedding inequality

∥u∥∞ 6 β1 ∥u∥H 1 (E ) , 0

∀u ∈ H01 (E ),

(1.2)

and the Poincaré inequality

∥∇ u∥2 ≥ λ∥u∥2 , ∀u ∈ H01 (E ), (1.3) where β1 and λ are positive constants. In addition, the following Gagliardo–Nirenberg inequality [21] is frequently used too. Theorem 1.1. Suppose that 1 ≤ q, r ≤ +∞ (j, m are integer), and 0 ≤ j < m. Assume j/m ≤ θ ≤ 1 (θ < 1, when m − j − N /r is a nonnegative integer), and p satisfies 1 p

=

j N

+θ



1 r

−

m N



1

+ (1 − θ ) . q

Then there exists a constant C (q, r , j, m, θ , N ) such that

∥Dj u∥Lp (E ) ≤ C ∥Dm u∥θLr (E ) ∥u∥1Lq−θ (E ) ,

∀ u ∈ Cc∞ (E ).

2. Preliminaries In this section, we first review some basic concepts related to random attractors for stochastic dynamical systems. The reader may refer to [10,13,22] for more details. Let (X , ∥ · ∥X ) be a separable Hilbert space with Borel σ -algebra B (x), and (Ω , F , P) be a probability space. Definition 2.1. (Ω , F , P, (θt )t ∈R ) is called a measurable dynamical systems, if θ : R × Ω → Ω is (B (R) × F , F )measurable, θ0 = I, θt +s = θt ◦ θs for all t , s ∈ R, and θt A = A for all t ∈ R and A ∈ F . Definition 2.2. A stochastic process φ(t , ω) is called a continuous random dynamical system (RDS) over (Ω , F , P, (θt )t ∈R ) if φ is (B (R+ ) × F × B (X ), B (X ))-measurable, and for all ω ∈ Ω

• the mapping φ : R+ × Ω × X → X is continuous; • φ(0, ω) = I on X ; • φ(t + s, ω, χ ) = φ(t , θs ω, φ(s, ω, χ )) for all t , s ≥ 0 and χ ∈ X (cocycle property).

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H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

Definition 2.3. A random bounded set {B(ω)}ω∈Ω ⊆ X is called tempered with respect to (θt )t ∈R if for P-a.e. ω ∈ Ω and all ϵ>0 lim e−ϵ t d(B(θ−t ω)) = 0

t →∞

where d(B) = supχ∈B ∥χ∥X . Consider a continuous random dynamical system φ(t , w) over (Ω , F , P, (θt )t ∈R ) and let D be the collection of all tempered random set of X .

˜ = {D˜ (ω) ⊆ X : ω ∈ Ω } with D˜ (ω) ⊆ D(ω) for Definition 2.4. D is called inclusion-closed if D = {D(ω)}ω∈Ω ∈ D and D ˜ ∈ D. all ω ∈ Ω imply that D Definition 2.5. Let D be a collection of random subsets of X and {K (ω)}ω∈Ω ∈ D . Then {K (ω)}ω∈Ω is called an absorbing set of φ in D if for all B ∈ D and P-a.e. ω ∈ Ω there exist tB (ω) > 0 such that

φ(t , θ−t ω, B(θ−t ω)) ⊆ K (ω),

t ≥ tB (ω).

Definition 2.6. Let D be a collection of random subsets of X . Then φ is said to be D -pullback asymptotically compact in X if for P-a.e. ω ∈ Ω , {φ(tn , θ−tn ω, χn )}∞ n=1 has a convergent subsequence in X whenever tn → ∞, and χn ∈ B(θ−tn ω) with {B(ω)}ω∈Ω ∈ D . Definition 2.7. Let D be a collection of random subsets of X and {A(ω)}ω∈Ω ∈ D . Then {A(ω)}ω∈Ω is called a D -random attractor (or D -pullback attractor) for φ if the following conditions are satisfied, for P-a.e. ω ∈ Ω ,

• A(ω) is compact, and ω → d(χ , A(ω)) is measurable for every χ ∈ X ; • {A(ω)}ω∈Ω is strictly invariant, i.e., φ(t , ω, A(ω)) = A(θt ω), ∀t ≥ 0 and for a.e. ω ∈ Ω ; • {A(ω)}ω∈Ω attracts all sets in D , i.e., for all B ∈ D and a.e. ω ∈ Ω we have lim d(φ(t , θ−t ω, B(θ−t ω)), A(ω)) = 0, t →∞

where d is the Hausdorff semi-metric given by d(Y , Z ) = supy∈Y infz ∈Z ∥y − z ∥X , for any Y , Z ⊆ X . According to [11], we can infer the following result. Proposition 2.1. Let D be an inclusion-closed collection of random subsets of X and φ a continuous RDS on X over (Ω , F , P, (θt )t ∈R ). Suppose that {K (ω)}ω∈Ω ∈ D is a closed absorbing set of φ and φ is D -pullback asymptotically compact in X . Then φ has a unique D -random attractor which is given by {A(ω)}ω∈Ω  φ(t , θ−t ω, K (θ−t ω)). A(ω) = κ≥0 t ≥κ

In this paper, we will prove that the stochastic Q-G equation has a D -random attractor in L2 (E ). 3. Stochastic quasi-geostrophic equations In this section, we discuss the existence of a continuous random dynamical system for the stochastic Q-G equation defined on unbounded domain. Let D be a bounded domain in R and E = D × R. Consider the stochastic Q-G equation defined on E:

 d(△ψ − F ψ + β0 y) +

   ∂ψ ∂ ∂ψ ∂ 1 r 2 △ ψ − △ψ dt − Φ dw, − (△ψ − F ψ + β0 y)dt = ∂x ∂y ∂y ∂x Re 2

(3.1)

with the boundary condition

ψ(x, y, t ) = △ψ(x, y, t ) = 0,

(x, y) ∈ ∂ E .

(3.2)

Next, in order to be convenient for the research, we simplify the Eq. (3.1). For any u ∈ L2 (E ), by the following elliptic equation with Dirichlet boundary condition:



F ψ − △ψ = u, ψ|∂ E = 0,

(3.3)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

99

we obtain ψ = (FI − △)−1 u = P (u). Applying the elliptic regular theory, we know that P : L2 (E ) → H01 (E ) ∩ H 2 (E ). Therefore the Eq. (3.1) can be rewritten as du −

1 Re

△udt = G(u)dt + Φ dw,

F F ∂u ∂u where ψ = P (u), G(u) = −J (ψ, u)+β0 ψx +( Re − 2r )u − F ( Re − 2r )ψ , J the is Jacobian operator, and J (ψ, u) = ∂ψ − ∂ψ . ∂x ∂y ∂y ∂x In this paper, we will consider the following problem

du −

1 Re

△udt = G(u)dt + Φ dw,

(x, y) ∈ E , t > 0,

(3.4)

with the boundary condition u|∂ E = 0,

t > 0,

(3.5)

and the initial condition u(x, y, 0) = u0 (x, y),

(x, y) ∈ E .

(3.6)

In the sequel, we consider the probability space (Ω , F , P) where

Ω = {ω ∈ C (R, R) : ω(0) = 0}, F is the Borel σ -algebra induced by the compact-open topology of Ω , and P the corresponding Wiener measure on (Ω , F ). Defined the time shift by

θt ω(·) = ω(· + t ) − ω(t ),

ω ∈ Ω , t ∈ R.

Then (Ω , F , (θt )t ∈R ) is a metric dynamical system. In this paper, we need to convert the stochastic equation (3.1) with a random term into a deterministic one with a random parameter. To this end, we consider the stationary process

η(θt ω) = −α



0

eατ (θt ω)(τ )dτ ,

t ∈ R,

−∞

which satisfies the following one-dimensional equation dη + αηdt = dw(t ),

(3.7)

where α is a positive constant. ˜ ⊆ Ω of full P measure such that y(θt ω) is continuous in t for every It is known that there exists a θt -invariant set Ω ˜ , and the random variable |η(ω)| is tempered (see [11,12,14]). Therefore, it follows from Proposition 4.3.3 in [12] that ω∈Ω there exists a tempered function r (ω) > 0 such that

|η(ω)|2 ≤ r (ω),

(3.8)

where r (ω) satisfies, for P-a.e. ω ∈ Ω , α

r (θt ω) ≤ e 2 |t | r (ω),

t ∈ R.

(3.9)

Then it follows from (3.8) and (3.9) that, for P-a.e. ω ∈ Ω , α

|η(θt ω)|2 ≤ e 2 |t | r (ω),

t ∈ R.

(3.10)

Put z (θt ω) = (I − △)−1 Φ η(θt ω) where △ is the Laplacian with domain H01 (E ) ∩ H 2 (E ). By (3.7) we find that dz = d(△z ) − α(z − △z )dt + Φ dw. Let v(t , ω) = u(t , ω) − z (θt ω), where u(t , ω) satisfies (3.4)–(3.6). Then for v(t , ω) we infer that

vt −

1 Re

 △v = G(v + z (θt ω)) + α z (θt ω) +

1 Re

 − α △z (θt ω),

(3.11)

with the boundary condition

v|∂ E = 0,

(3.12)

and the initial condition

v(0, ω) = v0 (ω).

(3.13)

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H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

Referring to the Galerkin method, it can be proved that, for P-a.e. ω ∈ Ω and for all v0 ∈ L2 (E ), problem (3.11)–(3.13) has a unique solution v(·, ω, v0 ) ∈ C ([0, ∞), L2 (E )) with v(0, ω, v0 ) = v0 . Then the solution v(t , ω, v0 ) is continuous with respect to v0 in L2 (E ) for all t > 0. Throughout this paper, we always write u(t , ω, u0 ) = v(t , ω, v0 ) + z (θt ω),

with v0 = u0 − z (ω).

(3.14)

Then u is a solution of problem (3.11)–(3.13) in some sense. We now define a mapping Ψ : R × Ω × L (E ) → L (E ) by +

Ψ (t , ω, u0 ) = u(t , ω, u0 ),

2

∀ (t , ω, u0 ) ∈ R+ × Ω × L2 (E ).

2

(3.15)

Note that Ψ satisfies conditions in Definition 2.2. Therefore, Ψ is a continuous random dynamical system associated with the stochastic Q-G equation on E. In that follows, we will establish uniform estimates for Ψ and prove that Ψ has a D -random attractor in L2 (E ), where D is a collection of random subsets of L2 (E ) given by

D = B : B = {B(ω)}ω∈Ω , B(ω) ⊆ L2 (E ) and e−δ t d(B(θ−t ω)) → 0 as t → −∞ ,





(3.16)

where δ is the positive constant in (4.17) and d(B(θ−t ω)) =

sup

u∈B(θ−t ω)

∥u∥L2 (E ) .

Notice that D contains all tempered random sets, especially all bounded deterministic subsets of L2 (E ). 4. Uniform estimates of solutions To prove the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system, in this section, we will derive uniform estimates on the solutions of the stochastic Q-G equation defined on E when t → ∞, which include the uniform estimates on the tails of solutions as both (x, y) and t approach infinity. From now on, we always suppose that D is the collection of random subsets of L2 (E ) given by (3.16). Firstly, we derive the following uniform estimates on v in L2 (E ).    Throughout this paper, ‘‘dxdy’’ is omitted in E fdxdy. So we write E fdxdy as E f . Lemma 4.1. Suppose that Φ ∈ L2 (E ). Let B = {B(ω)}ω∈Ω ∈ D and v0 (ω) ∈ B(ω). Then for P-a.e. ω ∈ Ω , there is T = T (B, ω) > 0 such that for all t ≥ T ,

∥v(t , θ−t ω, v0 (θ−t ω))∥ ≤ r1 (ω), where r1 (ω) is a positive random function satisfying e−δ t r1 (θ−t ω) → 0 as t → ∞.

(4.1)

Proof. Taking the inner product of (3.11) with v in L2 (E ), we obtain 1 d 2 dt

2

∥v∥ +

1 Re





2

G(v + z (θt ω))v +

∥∇v∥ =

 α z (θt ω) +



1 Re

E



− α △z (θt ω), v ,

(4.2)

where



G(v + z (θt ω))v = E



−J (P (v + z (θt ω)), v + z (θt ω)) v +



E

β0 (P (v + z (θt ω)))x v E

 

F

E

Re

+

−

r 2

 (v + z (θt ω))v −



 F E

F Re

−

r

 P (v + z (θt ω))v.

2

(4.3)

Now we estimate every term on the right-hand side of (4.3). Note that



J (P (v), v)v = 0, E



J (P (z (θt ω)), v)v = 0. E

Then, for the first term on the right-hand side of (4.3), we have

 E

        −J (P (v + z (θt ω)), v + z (θt ω)) v ≤  J (P (z (θt ω)), z (θt ω))v  +  J (P (v), z (θt ω))v  . E

E

(4.4)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

101

By integration by parts and Hölder inequality, the first term on the right-hand side of (4.4) satisfies

         J (P (z (θt ω)), z (θt ω))v  =  ∂ P (z (θt ω)) ∂ z (θt ω) v − ∂ P (z (θt ω)) ∂ z (θt ω) v      ∂x ∂y ∂y ∂x E E     ∂ P (z (θt ω)) ∂v   ∂ P (z (θt ω)) ∂v   ≤ z (θt ω) z (θt ω) + ∂x ∂y   E ∂y ∂x  E         ∂ P (z (θt ω))   ∂v   ∂ P (z (θt ω))   ∂v    +    ≤ ∥z (θt ω)∥∞     ∂y     ∂x  ∂x ∂y ≤ 2∥z (θt ω)∥∞ ∥∇ P (z (θt ω))∥ ∥∇v∥.

(4.5)

By (3.3), we deduce that P (z (θt ω)) = ψ = (FI − △) z (θt ω). Then F ψ − △ψ = z (θt ω) holds. Taking the inner product of above equality with ψ in L2 (E ), then applying integration by parts and Young inequality, we infer that −1

F ∥ψ∥2 + ∥∇ψ∥2 = (z (θt ω), ψ) ≤

F 2

∥ψ∥2 +

1 2F

∥z (θt ω)∥2 .

Then we have

∥∇ P (z (θt ω))∥ = ∥∇ψ∥ ≤ ∥z (θt ω)∥.

(4.6)

Substituting (4.6) into (4.5), we obtain that

     J (P (z (θt ω)), z (θt ω))v  ≤  

2∥z (θt ω)∥∞ ∥z (θt ω)∥ ∥∇v∥

≤

4Re∥z (θt ω)∥2∞ ∥z (θt ω)∥2 +

≤

4Reβ12 ∥z (θt ω)∥2H 1 ∥z (θt ω)∥2 +

E

1

≤ 4 Reβ12 C0 |η(θt ω)|4 +

4Re

1 4Re

∥∇v∥2 1

4Re

∥∇v∥2

∥∇v∥2 .

(4.7)

Similarly to inequality (4.7), for the second term on the right-hand side of (4.4), we have

     J (P (v), z (θt ω))v  ≤ 2∥z (θt ω)∥∞ ∥∇ P (v)∥ ∥∇v∥   E

≤ 2β1 ∥z (θt ω)∥H 1 ∥v∥ ∥∇v∥ ≤ 4Reβ12 ∥Φ ∥2 |η(θt ω)|2 ∥v∥2 +

1 4Re

∥∇v∥2 ,

(4.8)

Inserting (4.7) and (4.8) into (4.4), we obtain that the first term on the right-hand side of (4.3) is dominated by



1

−J (P (v + z (θt ω)), v + z (θt ω)) v ≤

2Re

E

∥∇v∥2 + 4Reβ12 ∥Φ ∥2 |η(θt ω)|2 ∥v∥2 + 4Reβ12 C0 |η(θt ω)|4 .

(4.9)

Next, we estimate the three terms left on the right-hand side of (4.3). The three terms left on the right-hand side of (4.3) can be rewritten as



β0 (P (v + z (θt ω)))x v + E

= β0



(P (z (θt ω)))x v +

+

β0



E

Re F

E

Re

 (P (v))x v +

F

 

E



 

F Re

E

−

r

−

− r

 (v + z (θt ω))v −

2 r

2

 z (θt ω)v − F



F E



2

 v −F 2

E

F Re

F Re

−

r

−



2

According to the definition of the operator P, we have

β0





(P (v))x v + E

F Re

−

r 2



 v −F 2

E





F Re

−

r 2



v P (v) E

r 2

F Re



−

r 2

 P (v + z (θt ω))v

v P (z (θt ω)) E

 v P (v) . E

(4.10)

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H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

β0

=



F

β0

=

F

v(vx + △ψx ) +





(F ψ − △ψ)△ψx −



F Re

−

r



2



F

F

2

∥△ψ∥ +

2

Re

r

−

 v − 2



r 2

F Re

E

−

Re

E



r

−

Re

E

F

=F



−

r



2

v(v + △ψ) E

(F ψ − △ψ)△ψ E

 ∥△ψ∥2 .

2

By the assumption of the parameter β0 , F , Re, r, we obtain that

β0





(P (v))x v +

F Re

E

−

r





2

v −F 2

F Re

E

−

r



2

v P (v) ≤ 0.

(4.11)

E

So the three terms left on the right-hand side of (4.3) are dominated by



β0 (P (v + z (θt ω)))x v +

 

F

E

Re

E

−

r 2

 (v + z (θt ω))v −



 F E

F Re

−

r



2

P (v + z (θt ω))v

                F r r F    − − v P (z (θt ω)) z (θt ω)v  + F ≤ β0  (P (z (θt ω)))x v  +      Re 2 Re 2 E E E        F F r r   ≤ β0 ∥∇ P (z (θt ω))∥ ∥v∥ +  −  ∥z (θt ω)∥ ∥v∥ + F −  ∥z (θt ω)∥ ∥v∥  Re 2  Re 2 ≤

3λ 16Re

∥v∥2 + C1 |η(θt ω)|2 .

(4.12)

Substituting (4.9) and (4.12) into (4.3), we obtain that



G(v + z (θt ω))v ≤ E

1 2Re

  3λ ∥∇v∥2 + 4Reβ12 ∥Φ ∥2 |η(θt ω)|2 ∥v∥2 + ∥v∥2 + C2 (|η(θt ω)|2 + |η(θt ω)|4 ). (4.13) 16Re

By using the Young inequality, the second term on the right-hand side of (4.2) is bounded by

      1 λ   − α △z (θt ω), v  ≤ ∥v∥2 + C3 |η(θt ω)|2 .  α z (θt ω) +   16Re Re

(4.14)

Substituting (4.13) and (4.14) into (4.2), we obtain that d dt

∥v∥2 +

1 Re

  λ ∥∇v∥2 ≤ 8Reβ12 ∥Φ ∥2 |η(θt ω)|2 ∥v∥2 + ∥v∥2 + C4 (|η(θt ω)|2 + |η(θt ω)|4 ).

(4.15)

2Re

By Poincaré inequality, (4.15) can be rewritten as d dt

∥v∥2 +



λ 2Re

 − 8Reβ12 ∥Φ ∥2 |η(θt ω)|2 ∥v∥2 ≤ C4 (|η(θt ω)|2 + |η(θt ω)|4 ).

(4.16)

Denote by

δ=

1 2Re

λ and β = 8Reβ12 ∥Φ ∥2 .

(4.17)

t

2 Multiplying (4.16) by e 0 (δ−β|η(θτ ω)| )dτ and then integrating over (0, s) with s ≥ 0, we infer that

∥v(s, ω, v0 (ω))∥2 ≤ e−δs+β

s 0

|η(θτ ω)|2 dτ

∥v0 (ω)∥2 + C4

s



(|η(θσ ω)|2 + |η(θσ ω)|4 )eδ(σ −s)+β

0

Now we replace ω by θ−t ω with t ≥ 0 in (4.18) to deduce that, for any t ≥ 0 and s ≥ 0, s

∥v(s, θ−t ω, v0 (θ−t ω))∥2 ≤ e−δs+β 0 |y(θτ −t ω)| dτ ∥v0 (θ−t ω)∥2  s s 2 + C4 (|η(θσ −t ω)|2 + |η(θσ −t ω)|4 )eδ(σ −s)+β σ |η(θτ −t ω)| dτ dσ 0

2

s

2 σ |η(θτ ω)| dτ d

σ.

(4.18)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

 s−t

2 −t |η(θτ ω)| dτ

= e−δs+β  + C4

s−t

103

∥v0 (θ−t ω)∥2

(|η(θσ ω)|2 + |η(θσ ω)|4 )eδ(σ −s+t )+β

 s−t σ

|η(θτ ω)|2 dτ

dσ .

(4.19)

−t

By (4.19) we obtain that, for any t ≥ 0, 0

∥v(t , θ−t ω, v0 (θ−t ω))∥2 ≤ e−δt +β −t |η(θτ ω)| dτ ∥v0 (θ−t ω)∥2  0 0 2 (|η(θσ ω)|2 + |η(θσ ω)|4 )eδσ +β σ |η(θτ ω)| dτ dσ . + C4 2

(4.20)

−t α

Note that |η(θt ω)|2 ≤ e 2 t r (ω), t ≥ 0. Then we deduce that

β



0

|η(θτ ω)| dτ ≤ β 2



0

−t

−t

2

α

e 2 τ r (ω)dτ ≤ β r (ω)

α

α

1 − e− 2 t



≤

2β

r (ω).

α

(4.21)

By (4.20) and (4.21), we obtain that, for any t ≥ T0 (ω), 2β

∥v(t , θ−t ω, v0 (θ−t ω))∥2 ≤ e−δt + α

r (ω)

0



∥v0 (θ−t ω)∥2 + C4

(|η(θσ ω)|2 + |η(θσ ω)|4 )eδσ +β

0

2 σ |η(θτ ω)| dτ d

σ.

(4.22)

−t

Note that |η(θσ ω)| is tempered, and hence by (3.10), the integrand of the second term on the right-hand side of (4.22) is convergent to zero exponentially as σ → −∞. This shows that the following integral is convergent: r0 (w) = C4



0

(|η(θσ ω)|2 + |η(θσ ω)|4 )eδσ +β

0

2 σ |η(θτ ω)| dτ d

σ.

(4.23)

−∞

It follows from (4.22) and (4.23) that, for all t ≥ T0 (ω), 2β

∥v(t , θ−t ω, v0 (θ−t ω))∥2 ≤ e−δt + α

r (ω)

∥v0 (θ−t ω)∥2 + r0 (w).

(4.24)

In addition, by assumption B = {B(ω)}ω∈Ω ∈ D and hence we have e−δ t ∥v0 (θ−t ω)∥2 → 0

as t → ∞,

from which and (4.24) we find that there is T = T (B, ω) > 0 such that for all t ≥ T ,

∥v(t , θ−t ω, v0 (θ−t ω))∥2 ≤ 2r0 (ω). √ Let r1 (ω) = 2r0 (ω). Then we obtain that, for all t ≥ T , ∥v(t , θ−t ω, v0 (θ−t ω))∥ ≤ r1 (w).

(4.25)

In what follows, we prove r1 (ω) satisfies (4.1). Replacing ω by θ−t ω in (4.23) we obtain that r0 (θ−t ω) = C4



0

(|η(θσ −t ω)|2 + |η(θσ −t ω)|4 )eδσ +β

−∞  −t

= C4 −∞  −t

≤ C4

(|η(θσ ω)|2 + |η(θσ ω)|4 )eδ(σ +t )+β 3

0

2 σ |η(θτ −t ω)| dτ d

σ

 −t

|η(θτ ω)|2 dτ

σ

(|η(θσ ω)|2 + |η(θσ ω)|4 )e 2 δ(σ +t )+β

 −t σ

dσ

|η(θτ ω)|2 dτ

dσ

−∞

≤ C4 e

3 δt 2



0

3

(|η(θσ ω)|2 + |η(θσ ω)|4 )e 2 δσ +β

0

2 σ |η(θτ ω)| dτ d

σ.

(4.26)

−∞

The integrand converges to zero exponentially by (4.21) as σ → −∞, hence the last integral of the above formula is indeed convergent. Then we deduce that e−σ t r1 (θ−t ω) = e−σ t

≤





2r0 (θ−t ω)

2C4 e

− 41 δ t

0



(|η(θσ ω)| + |η(θσ ω)| )e 2

−∞

→ 0 as t → ∞, which along with (4.25) completes the proof.



4

 3 δσ +β σ0 2

|η(θτ ω)|2 dτ

dσ

 12

104

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

Lemma 4.2. Suppose that Φ ∈ H01 (E ). Let B = {B(ω)}ω∈Ω ∈ D and v0 (ω) ∈ B(ω). Then for P-a.e. ω ∈ Ω , there is T1 = T1 (B, ω) > 0 such that for all t ≥ T1 ,

  ∥∇v(t , θ−t ω, v0 (θ−t ω))∥2 ≤ C 1 + r0 (ω) + r (ω) + r02 (ω) + r 2 (ω) + r05 (ω) + r 5 (ω) .

(4.27)

Proof. Taking the inner product of (3.11) with △v in L2 (E ), we deduce that 1 d

∥∇v∥2 +

2 dt

1 Re

∥△v∥2 = −





G(v + z (θt ω))△v −

 α z (θt ω) +



1 Re

E



− α △z (θt ω), △v ,

(4.28)

where





G(v + z (θt ω))△v =

− E

J (P (v + z (θt ω)), v + z (θt ω))△v −

 

F

E

Re

E

− β0



(P (v + z (θt ω)))x △v +

E



 F E

F Re

−

r 2

−

r 2

 (v + z (θt ω))△v

 P (v + z (θt ω))△v.

(4.29)

Next, we estimate every term on the right-hand side of (4.29). Firstly, we consider ∥P (v + z (θt ω))∥ and ∥△P (v + z (θt ω))∥. By (3.3), we have P (v + z (θt ω)) = ψ = (FI − △)−1 (v + z (θt ω)). Then F ψ − △ψ = v + z (θt ω).

(4.30)

Taking the inner product of above equality with ψ in L2 (E ), then applying integration by parts and Young inequality, we obtain that F ∥ψ∥2 + ∥∇ψ∥2 = (v + z (θt ω), ψ) ≤

F 2

∥ψ∥2 +

1 2F

∥v + z (θt ω)∥2 .

Then we have

∥P (v + z (θt ω))∥ = ∥ψ∥ ≤

1 F

∥v + z (θt ω)∥,

(4.31)

∥∇ P (v + z (θt ω))∥ = ∥∇ψ∥ ≤ ∥v + z (θt ω)∥.

(4.32)

Taking the inner product of (4.30) with −△ψ in L2 (E ), then applying integration by parts and Young inequality, we obtain that 1

F ∥∇ψ∥2 + ∥△ψ∥2 = −(v + z (θt ω), △ψ) ≤

2

1

∥△ψ∥2 + ∥v + z (θt ω)∥2 . 2

Then we have

∥△P (v + z (θt ω))∥ = ∥△ψ∥ ≤ ∥v + z (θt ω)∥.

(4.33)

By applying Hölder inequality, Gagliardo–Nirenberg inequality, Young inequality, (4.31) and (4.33), the first term on the right-hand side of (4.29) is bounded by

     J (P (v + z (θt ω)), v + z (θt ω))△v    E    ∂ P (v + z (θt ω)) ∂(v + z (θt ω)) ∂ P (v + z (θt ω)) ∂(v + z (θt ω))  =  △v − △v  ∂x ∂y ∂y ∂x E ≤ 2∥△v∥ ∥∇ P (v + z (θt ω))∥4 ∥∇(v + z (θt ω))∥4 3

3

1

≤ C5 ∥△v∥ ∥△P (v + z (θt ω))∥ 4 ∥△(v + z (θt ω))∥ 4 ∥v + z (θt ω)∥ 4 5

3

≤ C6 ∥△v∥ ∥v + z (θt ω)∥ 4 ∥△(v + z (θt ω))∥ 4 ≤ ≤

1 8Re 1 4Re

5

3

∥△v∥2 + C7 ∥v + z (θt ω)∥ 2 ∥△(v + z (θt ω))∥ 2 ∥△v∥2 + C8 (∥△z (θt ω)∥2 + ∥v + z (θt ω)∥10 ).

(4.34)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

105

Similarly, the left terms on the right-hand side of (4.29) are bounded by

          r F r F   − (v + z (θt ω)) + F − P (v + z (θt ω)) △v  −β0 (P (v + z (θt ω)))x −   E  Re 2 Re 2        F r r  F  − ≤ β0 ∥∇ P (v + z (θt ω))∥ ∥△v∥ +  −  ∥v + z (θt ω)∥ ∥△v∥ + F  ∥∇ P (v + z (θt ω))∥ ∥∇v∥  Re 2  Re 2         F r r  F  − ≤ β0 +  −  ∥v + z (θt ω)∥ ∥△v∥ + F  ∥v + z (θt ω)∥ ∥∇v∥  Re 2  Re 2 1

≤

4Re

∥△v∥2 +

1 8Re

∥∇v∥2 + C9 ∥v + z (θt ω)∥2 .

(4.35)

Substituting (4.34) and (4.35) into (4.29), we infer that



G(v + z (θt ω))△v ≤

− E

1 2Re

∥△v∥2 +

1 8Re

∥∇v∥2 + C9 ∥v + z (θt ω)∥2

+ C8 (∥△z (θt ω)∥2 + ∥v + z (θt ω)∥10 ).

(4.36)

For the second term on the right-hand side of (4.28), we deduce that



 α z (θt ω) +



1 Re



− α △z (θt ω), △v

≤

1 4Re

∥△v∥2 + C10 (∥z (θt ω)∥2 + ∥△z (θt ω)∥2 ).

(4.37)

Substituting (4.36) and (4.37) into (4.28), we infer that d dt

∥∇v∥2 +

1 2Re

∥△v∥2 ≤

1 4Re

∥∇v∥2 + C11 (1 + ∥△z (θt ω)∥2 + ∥v + z (θt ω)∥10 + ∥z (θt ω)∥2 ).

(4.38)

Applying integration by parts and Young inequality, we deduce that 1

∥∇v∥2 ≤ ∥△v∥ ∥v∥ ≤ ∥△v∥2 + ∥v∥2 ,

(4.39)

4

namely, 1

− ∥v∥2 + ∥∇v∥2 ≤ ∥△v∥2 .

(4.40)

4

Combining (4.40) with (4.38), we obtain that d dt

∥∇v∥2 +

1 4Re

∥∇v∥2 ≤

1 8Re

∥v∥2 + C12 (1 + ∥△z (θt ω)∥2 + ∥v + z (θt ω)∥10 + ∥z (θt ω)∥2 )

≤ C13 (∥v∥2 + ∥v∥10 ) + C14 (1 + |η(θt ω)|2 + |η(θt ω)|10 ).

(4.41)

In view of (4.41), it is deduced that d dt

∥∇v∥2 ≤

1 8Re

∥v∥2 + C12 (1 + ∥△z (θt ω)∥2 + ∥v + z (θt ω)∥10 + ∥z (θt ω)∥2 )

≤ C13 (∥v∥2 + ∥v∥10 ) + C14 (1 + |η(θt ω)|2 + |η(θt ω)|10 ).

(4.42)

Then there exists a positive constant T1 (B), take t ≥ T1 (B) and s ∈ (t , t + 1). Integrating (4.42) over (s, t + 1), we infer that

∥∇v(t + 1, ω, v0 (ω))∥2 ≤ C14

t +1



(1 + |η(θτ ω)|2 + |η(θτ ω)|10 )dτ + ∥∇v(s, ω, v0 (ω))∥2 s t +1



(∥v(τ , ω, v0 (ω))∥2 + ∥v(τ , ω, v0 (ω))∥10 )dτ

+ C13 s

≤ ∥∇v(s, ω, v0 (ω))∥2 + C14

t +1



(1 + |η(θτ ω)|2 + |η(θτ ω)|10 )dτ t

t +1



(∥v(τ , ω, v0 (ω))∥2 + ∥v(τ , ω, v0 (ω))∥10 )dτ .

+ C13 t

(4.43)

106

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

Now integrating the above with respect to s over (t , t + 1), we infer that

∥∇v(t + 1, ω, v0 (ω))∥2 ≤

t +1



∥∇v(s, ω, v0 (ω))∥2 ds + C14

t +1



(1 + |η(θτ ω)|2 + |η(θτ ω)|10 )dτ t

t t +1



(∥v(τ , ω, v0 (ω))∥2 + ∥v(τ , ω, v0 (ω))∥10 )dτ .

+ C13

(4.44)

t

Replacing ω by θ−t −1 ω, we obtain that

∥∇v(t + 1, θ−t −1 ω, v0 (θ−t −1 ω))∥2 ≤

t +1



∥∇v(s, θ−t −1 ω, v0 (θ−t −1 ω))∥2 ds t t +1



(1 + |η(θτ −t −1 ω)|2 + |η(θτ −t −1 ω)|10 )dτ

+ C14 t t +1



(∥v(τ , θ−t −1 ω, v0 (θ−t −1 ω))∥2

+ C13 t

+ ∥v(τ , θ−t −1 ω, v0 (θ−t −1 ω))∥10 )dτ .

(4.45)

By (4.15), we deduce that

 t +1 1 ∥v(t + 1, ω, v0 (ω))∥2 − ∥v(t , ω, v0 (ω))∥2 + ∥∇v(s, ω, v0 (ω))∥2 ds Re t    t +1  λ 2 2 2 4 ≤ β|η(θτ ω)| + ∥v(τ , ω, v0 (ω))∥ + C4 (|η(θτ ω)| + |η(θτ ω)| ) dτ . 2Re

t

Replacing ω by θ−t −1 ω, we have 1 Re

t +1



∥∇v(s, θ−t −1 ω, v0 (θ−t −1 ω))∥2 ds t

≤ ∥v(t , θ−t −1 ω, v0 (θ−t −1 ω))∥2 + C4

t +1



(|η(θτ −t −1 ω)|2 + |η(θτ −t −1 ω)|4 )dτ t

t +1



 β|η(θτ −t −1 ω)| + 2

+ t

λ 2Re

 ∥v(τ , θ−t −1 ω, v0 (θ−t −1 ω))∥2 dτ .

(4.46)

Applying Lemma 4.1 and (4.46), we deduce that t +1



  ∥∇v(s, θ−t −1 ω, v0 (θ−t −1 ω))∥2 ds ≤ C r0 (ω) + r (ω) + r02 (ω) + r 2 (ω) .

(4.47)

t

By (4.45) and (4.47), we obtain that

  ∥∇v(t + 1, θ−t −1 ω, v0 (θ−t −1 ω))∥2 ≤ C 1 + r0 (ω) + r (ω) + r02 (ω) + r 2 (ω) + r05 (ω) + r 5 (ω) , which completes the proof.

(4.48)



To prove the asymptotic compactness of the random dynamical system, we will derive the uniform estimates on the tails of solutions when (x, y) and t approach to infinite. To this end, for every (x, y) ∈ E = D × R, where x ∈ D, and y ∈ R. Given k > 0, denote by Ek = {(x, y) ∈ E : |y| < k}, and E \ Ek the complement of Ek . Lemma 4.3. Suppose that Φ ∈ L2 (E ). Let B = {B(ω)}ω∈Ω ∈ D and v0 (ω) ∈ B(ω). Then for every ε > 0, and P-a.e. ω ∈ Ω , there exist T2 = T2 (B, ω, ε) > 0 and k0 = k0 (ω, ε) > 0 such that for all t ≥ T2 ,



|v(t , θ−t ω, v0 (θ−t ω))|2 ≤ ε. E \Ek

(4.49)

0

Proof. Take a smooth χ such that 0 ≤ χ ≤ 1 for all s ∈ R and

χ (s) =

1, 0,



if |s| < 1, if |s| > 2.

(4.50)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

107 y2

Then there exists a positive constant c such that |χ ′ (s)| + |χ ′′ (s)| ≤ c for all s ∈ R. Multiplying (3.11) by χ 2 ( k2 )v and then integrating with respect to x, y respectively on E, we obtain that



χ



2

y2



vvt −

k2

E

1



Re

χ

2



y2



k2

E



v△v =

χ



2





k2

E

+

y2

χ2



v G(v + z (θt ω))

y2

     1 − α △ z (θ ω) . v α z (θ ω) + t t 2

k

E

Re

(4.51)

Note the first term on the left-hand side of the above, and we have





χ2

y2



vvt =

k2

E

1 d 2 dt



χ2



y2



k2

E

|v|2 .

(4.52)

Applying integration by part, we deduce that

−



1 Re

χ2



y2



k2

E

v△v =



1 Re

χ2



y2



k2

E

|∇v|2 +



1 Re

E

  2  y v ∇v∇χ 2 2 .

(4.53)

k

Substituting (4.52) and (4.53) into (4.51), we obtain that 1 d



2 dt

χ2

y2



k2

E



|v|2 +

1 Re



χ2



y2



k2

E

|∇v|2 = −



1 Re



E

χ2

+

  2    2 y y v ∇v∇χ 2 2 + χ 2 2 v G(v + z (θt ω)) k



     y 1 v α z (θt ω) + − α △z (θt ω) . 2 k

E

k

E

2

Re

(4.54)

Next, we estimate every term on the right-hand side of (4.54). For the first term we infer that

   2   2      1    y 2 − ≤ 1 2χ χ ′ y  · 2|y| v ∇v∇χ |v| · |∇v| ·  Re   2 2 k Re E k  k2 E   2     1 2χ χ ′ y  · 2|y| ≤ |v| · |∇v| · √  2 Re k≤|y|≤ 2k k  k2  c ≤ √ |v| |∇v| k

≤

c k

k≤|y|≤ 2k

c

∥v∥ ∥∇v∥ ≤ (∥v∥2 + ∥∇v∥2 ).

(4.55)

k

For the second term on the right-hand side of (4.54) we obtain that





χ2

y2



k2

E

v G(v + z (θt ω)) = −



χ2





χ2



E

 −F



k2

E

+ β0

y2

F Re

−

v J (P (v + z (θt ω)), v + z (θt ω))     2 F r y2 y 2 v(P (v + z (θt ω)))x + − χ v(v + z (θt ω)) 2 2 k

r 2

Re



χ2 E



2

y

k2



2

v P (v + z (θt ω)).

E

k

(4.56)

Now, we estimate every term on the right-hand side of (4.56). Similarly to (4.34), the first term on the right-hand side of the above is bounded by





y2



v J (P (v + z (θt ω)), v + z (θt ω))   2  y2 ∂ P (v + z (θt ω)) ∂(v + z (θt ω)) y ∂ P (v + z (θt ω)) ∂(v + z (θt ω)) 2 2 = χ v − χ v 2 2 k ∂ y ∂ x k ∂x ∂y E E  2   2  2     ∂ P (v + z (θt ω)) 2 y ∂v ∂ P (v + z (θt ω)) y 2y y ∂v =− χ + 2χ χ ′ v + χ2 2 (v + z (θt ω)) 2 2 2 ∂ y k ∂ x ∂ x k k k ∂y E  c ≤ 2∥∇ P (v + z (θt ω))∥4 ∥v + z (θt ω)∥4 ∥∇v∥ + √ |∇ P (v + z (θt ω))| · |v| · |v + z (θt ω)| χ

−

2

E

k2





k

k≤|y|≤ 2k

≤ C (∥v + z (θt ω)∥ ∥v + z (θt ω)∥4 ∥∇v∥ + ∥∇ P (v + z (θt ω))∥4 ∥v∥4 ∥v + z (θt ω)∥)

108

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

  3 1 1 1 ≤ C ∥v + z (θt ω)∥ 2 ∥∇(v + z (θt ω))∥ 2 ∥∇v∥ + ∥v + z (θt ω)∥2 ∥∇v∥ 2 ∥v∥ 2 ≤ C (∥∇(v + z (θt ω))∥ + ∥∇v∥2 ∥v + z (θt ω)∥3 + ∥∇v∥ + ∥v + z (θt ω)∥4 ∥v∥) ≤ C (∥∇v∥ + ∥∇v∥4 + ∥∇ z (θt ω)∥ + ∥v∥2 + ∥v + z (θt ω)∥6 + ∥v + z (θt ω)∥8 ).

(4.57)

The second term on the right-hand side of (4.56) is bounded by



β0

χ

2



y2



k2

E

v(P (v + z (θt ω)))x ≤ β0 ∥∇ P (v + z (θt ω))∥ ∥v∥ ≤ β0 ∥v + z (θt ω)∥ ∥v∥ ≤ C (∥z (θt ω)∥2 + ∥v∥2 ).

(4.58)

Similarly, for the two terms left on the right-hand side of (4.56) we deduce that



F

−

Re



r 2

χ

2

y2





k2

E

v(v + z (θt ω)) − F



F Re

−



r 2

χ

2



y2 k2

E



v P (v + z (θt ω)) ≤ C (∥v∥2 + ∥z (θt ω)∥2 ). (4.59)

By substituting (4.57), (4.58) and (4.59) into (4.56), it is deduced that



χ

2



y2



v G(v + z (θt ω)) ≤ C (∥v∥2 + ∥∇v∥ + ∥∇v∥4 + ∥z (θt ω)∥2

k2

E

+ ∥∇ z (θt ω)∥ + ∥v + z (θt ω)∥6 + ∥v + z (θt ω)∥8 ).

(4.60)

By using the Young inequality, the last term on the right-hand side of (4.54) is bounded by

   2     2     χ 2 y v α z (θt ω) + 1 − α △z (θt ω)  ≤ 1 λ χ 2 y |v|2 + C (∥z (θt ω)∥2 + ∥△z (θt ω)∥2 ). (4.61)  4Re  k2 Re k2 E

E

According to (4.55), (4.60) and (4.61), (4.54) can be rewritten as 1 d



χ



y2





1



y2



|v| + |∇v|2 χ Re E k2  2  1 y 2 ≤ λ χ |v|2 + C (∥v∥2 + ∥∇v∥ + ∥∇v∥2 + ∥∇v∥4 + ∥z (θt ω)∥2 2

2 dt

2

2

2

k2

E

4Re

E

k

+ ∥△z (θt ω)∥2 + ∥∇ z (θt ω)∥ + ∥v + z (θt ω)∥6 + ∥v + z (θt ω)∥8 ).

(4.62)

Note that

 2  2  2       2  2     ∇ χ y v  = v∇χ y + χ y ∇v      2 2 2 k k k E E   2 2     2 2    y   + 2 χ y  |∇v|2 ≤ 2 |v|2 ∇χ  2  2  k

E

 ≤2

√ k≤|y|≤ 2k

≤ ≤

c k2 c k2

k

E

  2 2  2   y y  |2y|2 2 |v|2 χ ′ 2  + 2 χ |∇v|2 4 2 k



|v|2 + 2

√ k≤|y|≤ 2k

χ2



E



2

k



χ

∥v∥ + 2

2



E

y2



k2

E

2

y

k2



k

|∇v|2

|∇v|2 .

(4.63)

Applying Poincaré inequality, we infer that



χ2 E



y2



k2

|v|2 ≤

 2     2  2    ∇ χ y v  ≤ c ∥v∥2 + 2 χ 2 y |∇v|2 ,  λ E k2 k2 λ λ E k2 1

namely,



χ2 E



y2 k2



|∇v|2 ≥

1 2

λ



χ2 E



y2 k2



|v|2 −

c 2k2

∥v∥2 .

(4.64)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

109

Combining (4.64) with (4.62), we obtain that



d dt

χ



2

y2



|v| +

k2

E

2

1 2Re

λ



χ

2



y2



k2

E

|v|2 ≤

C k2

(∥v∥2 + ∥∇v∥ + ∥∇v∥2 + ∥∇v∥4 + ∥z (θt ω)∥2

+ ∥△z (θt ω)∥2 + ∥∇ z (θt ω)∥ + ∥v + z (θt ω)∥6 + ∥v + z (θt ω)∥8 ).

(4.65)

Applying Gronwall inequality over (0, t ), we obtain that, for all t ≥ 0,



χ2



y2



k2

E

1

|v(t , ω, v0 (ω))|2 ≤ e− 2Re λt



χ2 E



y2



k2

|v0 (ω)|2 +

C k2



t

1

e 2Re λ(s−t ) (∥∇ z (θs ω)∥

0

+ ∥v(s, ω, v0 (ω))∥2 + ∥∇v(s, ω, v0 (ω))∥ + ∥z (θs ω)∥2 + ∥∇v(s, ω, v0 (ω))∥2 + ∥∇v(s, ω, v0 (ω))∥4 + ∥△z (θs ω)∥2 + ∥v(s, ω, v0 (ω)) + z (θs ω)∥6 + ∥v(s, ω, v0 (ω)) + z (θs ω)∥8 )ds.

(4.66)

Replacing ω by θ−t ω in the above, we find that, for all t ≥ 0,





y2



|v(t , θ−t ω, v0 (θ−t ω))|2  2  t  C y 1 1 e 2Re λ(s−t ) (∥z (θs−t ω)∥2 + ∥∇ z (θs−t ω)∥ ≤ e− 2Re λt χ 2 2 |v0 (θ−t ω)|2 + 2 χ2

k2

E

k

E

k

0

+ ∥v(s, θ−t ω, v0 (θ−t ω))∥2 + ∥∇v(s, θ−t ω, v0 (vω))∥ + ∥∇v(s, θ−t ω, v0 (θ−t ω))∥2 + ∥v(s, θ−t ω, v0 (θ−t ω)) + z (θs−t ω)∥8 + ∥△z (θs−t ω)∥2 + ∥∇v(s, θ−t ω, v0 (θ−t ω))∥4 + ∥v(s, θ−t ω, v0 (θ−t ω)) + z (θs−t ω)∥6 )ds.

(4.67)

In the follows, we estimate every term on the right-hand side of (4.67). For the first term, we infer that



1

e− 2Re λt

χ2



y2



k2

E

1

|v0 (θ−t ω)|2 ≤ e− 2Re λt ∥v0 (θ−t ω)∥2 for all t ≥ T0 (ω).

(4.68)

Since v0 (θ−t ω) ∈ B(θ−t ω) and B = {B(ω)}ω∈Ω ∈ D , the right-hand side of (4.68) tend to zero as t → ∞. Therefore, given a T ′ = T ′ (B, ω, ε) > 0 such that for all t ≥ T ′ , e



1 − 2Re λt

χ

2



y2 k2

E



|v0 (θ−t ω)|2 ≤ ε.

(4.69)

Note that t



1

e 2Re λ(s−t ) (∥v(s, θ−t ω, v0 (θ−t ω)) + z (θs−t ω)∥6 + ∥v(s, θ−t ω, v0 (θ−t ω)) + z (θs−t ω)∥8 )ds

0 t

 ≤C

1

e 2Re λ(s−t ) (∥v(s, θ−t ω, v0 (θ−t ω))∥6 + ∥v(s, θ−t ω, v0 (θ−t ω))∥8 )ds

0 t

 +C 

1

e 2Re λ(s−t ) (∥z (θs−t ω)∥6 + ∥z (θs−t ω)∥8 )ds

0 t

≤C

1

e 2Re λ(s−t ) (∥v(s, θ−t ω, v0 (θ−t ω))∥6 + ∥v(s, θ−t ω, v0 (θ−t ω))∥8 )ds

0



0

+C

1

e− 2Re λs (∥z (θs ω)∥6 + ∥z (θs ω)∥8 )ds

−t

≤ ε(1 + r (ω)) as t → ∞.

(4.70)

Through a simple computation, we obtain that t



C k2

= ≤

1

e 2Re λ(s−t ) (∥z (θs−t ω)∥2 + ∥△z (θs−t ω)∥2 + ∥∇ z (θs−t ω)∥2 )ds

0

C



k2 C k2

0

1

e 2Re λs (∥z (θs ω)∥2 + ∥△z (θs ω)∥2 + ∥∇ z (θs ω)∥2 )ds

−t



0

−t

1

e 2Re λs |η(θs ω)|2 ds ≤ ε(r (ω)),

(4.71)

110

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

and t



C k2

1

e 2Re λ(s−t ) (∥v(s, θ−t ω, v0 (θ−t ω))∥2 + ∥∇v(s, θ−t ω, v0 (vω))∥

0

+ ∥∇v(s, θ−t ω, v0 (θ−t ω))∥2 + ∥∇v(s, θ−t ω, v0 (θ−t ω))∥4 )ds ≤ ε(r (ω), r0 (ω)).

(4.72)

According to (4.67)–(4.72), we infer that there exist positive random function r2 (ω), T2 = T2 (B, ω) and k0 (ε, ω) such that for all t ≥ T2 ,



χ2



E

y2 k2



|v(t , θ−t ω, v0 (θ−t ω))|2 ≤ ε(1 + r2 (ω)),

where r2 (ω) is a positive random function. So we deduce that



|v(t , θ−t ω, v0 (θ−t ω))| ≤ 2



|y|≥k0

χ

2

E

which completes the proof.



y2



k2

|v(t , θ−t ω, v0 (θ−t ω))|2 ≤ ε(1 + r2 (ω)),

(4.73)



In what follows, the asymptotic compactness of the solution of the problem (3.11)–(3.13) is proved. Lemma 4.4. Suppose that Φ ∈ L2 (E ). Let B = {B(ω)}ω∈Ω ∈ D , tn → ∞ and v0,n (ω) ∈ B(θ−tn ω). Then for P-a.e. ω ∈ Ω , the 2 sequence {v(tn , θ−tn ω, v0,n )}∞ n=1 has a convergent subsequence in L (E ). Proof. According to Lemma 4.1, we infer that, for P-a.e.ω ∈ Ω , 2 {v(tn , θ−tn ω, v0,n (θ−tn ω))}∞ n=1 is bounded in L (E ).

Hence, there exists v˜ ∈ L2 (E ) such that, up to a subsequence,

v(tn , θ−tn ω, v0,n (θ−tn ω)) ⇀ v˜ weakly in L2 (E ).

(4.74)

In what follows, we prove the weak convergence of (4.74) is actually strong convergence. Applying Lemma 4.3, we obtain that, given ε > 0, for P-a.e.ω ∈ Ω , there exist T ′ = T ′ (B, ω, ε) and k1 = k1 (ω, ε) such that for all t ≥ T ′ ,



|v(t , θ−t ω, v0 (θ−t ω))|2 ≤ ε. E \Ek

(4.75)

1

Due to tn → ∞, there is N1 = N1 (B, ω, ε) such that tn ≥ T ′ for every n ≥ N1 . Hence (4.75) implies that



|v(tn , θ−tn ω, v0,n (θ−tn ω))|2 ≤ ε, E \Ek

∀ n ≥ N1 .

(4.76)

1

Furthermore, by Lemmas 4.1 and 4.2, there exists T ′′ = T ′′ (B, ω) such that for all t ≥ T ′′ ,

∥v(t , θ−t ω, v0 (θ−t ω))∥2H 1 (E ) ≤ C (1 + r ′ (ω)).

(4.77)

Let N2 = N2 (B, ω) be large enough such that tn ≥ T ′′ for n ≥ N2 . Then by (4.77) we infer that

∥v(tn , θ−tn ω, v0,n (θ−tn ω))∥2H 1 (E ) ≤ C (1 + r ′ (ω)) ∀ n ≥ N2 .

(4.78)

By the compactness of embedding H 1 (Ek1 ) ↩→ L2 (Ek1 ), it follows from (4.78) that, up to a subsequence,

v(tn , θ−tn ω, v0,n (θ−tn ω)) → v˜ strongly in L2 (Ek1 ), which shows that for the given ε > 0, there exists N3 = N3 (B, ω, ε) such that for all n ≥ N3 ,

∥v(tn , θ−tn ω, v0,n (θ−tn ω)) − v˜ ∥2L2 (E

k1 )

≤ ε.

(4.79)

Note that v˜ ∈ L2 (E ). Therefore there exists k2 = k2 (ε) such that



|˜v |2 ≤ ε. E \Ek

2

(4.80)

H. Lu et al. / Nonlinear Analysis 90 (2013) 96–112

111

Let k3 = max{k1 , k2 } and N4 = max{N1 , N3 }. By (4.76), (4.79) and (4.80), we obtain that for all n ≥ N4 ,

∥v(tn , θ−tn ω, v0,n (θ−tn ω)) − v˜ ∥2L2 (E ) ≤



|v(tn , θ−tn ω, v0,n (θ−tn ω)) − v˜ |2 Ek

3 

|v(tn , θ−tn ω, v0,n (θ−tn ω)) − v˜ |2 ≤ 5ε,

+ E \E k

3

which shows that

v(tn , θ−tn ω, v0,n (θ−tn ω)) → v˜ strongly in L2 (E ).  5. Random attractors In this section, we prove the existence of a D -random attractor for the random dynamical system Ψ associated with the stochastic quasi-geostrophic equation on the unbounded domain E. By (3.14) and (3.15), Ψ satisfies

Ψ (t , θ−t ω, u0 (θ−t ω)) = u(t , θ−t ω, u0 (θ−t ω)) = v(t , θ−t ω, v0 (θ−t ω)) + z (ω),

(5.1)

where v0 (θ−t ω) = u0 (θ−t ω) − z (θ−t ω). Let B = {B(ω)}ω∈Ω ∈ D and define B˜ (ω) = v ∈ L2 (E ) : ∥v∥ ≤ ∥u(ω)∥ + ∥z (ω)∥, u(ω) ∈ B(ω) .





(5.2)

We claim that B˜ = {B˜ (ω)}ω∈Ω ∈ D provided B = {B(ω)}ω∈Ω ∈ D . Note that B = {B(ω)}ω∈Ω ∈ D implies that lim e−δ t d(B(θ−t ω)) = 0.

(5.3)

t →∞

Since z (ω) is tempered, by (5.2)–(5.3) we deduce that lim e−δ t d(B˜ (θ−t ω)) ≤ lim e−δ t d(B(θ−t ω)) + lim e−δ t ∥z (θ−t ω)∥ = 0,

t →∞

t →∞

t →∞

which shows B˜ = {B˜ (ω)}ω∈Ω ∈ D . Then by Lemma 4.1, for P-a.e.ω ∈ Ω , if v0 (ω) ∈ B˜ (ω), there is T1 = T1 (B˜ , ω) such that for all t ≥ T1 ,

∥v(t , θ−t ω, v0 (θ−t ω))∥ ≤ r (ω),

(5.4)

where r (ω) is a positive random function satisfies e−δ t r (θ−t ω) → 0

as t → ∞.

(5.5)

Denote by K (ω) = {u ∈ L2 (E ) : ∥u∥ ≤ r (ω) + ∥z (ω)∥}.

(5.6)

Then we have the following result. Lemma 5.1. Suppose that Φ ∈ L2 (E ). Let K = {K (ω)}ω∈Ω be given by (5.6). Then K = {K (ω)}ω∈Ω ∈ D is a closed absorbing set of Ψ in D . Proof. According to (5.5) we deduce that lim e−δ t d(K (θ−t ω)) ≤ lim e−δ t r (θ−t ω) + lim e−δ t ∥z (θ−t ω)∥ = 0,

t →∞

t →∞

t →∞

which implies that K = {K (ω)}ω∈Ω ∈ D . We now show that K is also an absorbing set of Ψ in D . Given B = {B(ω)}ω∈Ω ∈ D and u0 (ω) ∈ B(ω), by (5.1) and (5.4) we obtain that, for all t ≥ T1 , u(t , θ−t ω, v0 (θ−t ω)) ≤ ∥v(t , θ−t ω, v0 (θ−t ω))∥ + ∥z (ω)∥ ≤ r (ω) + ∥z (ω)∥, which along with (5.1) and (5.6) implies that

Ψ (t , θ−t ω, B(θ−t ω)) ⊆ K (ω),

∀ t ≥ T1 ,

and hence K = {K (ω)}ω∈Ω ∈ D is a closed absorbing set of Ψ in D .

(5.7) 

The D -pullback asymptotic compactness of Ψ is given by the following lemma. Lemma 5.2. Suppose that Φ ∈ L2 (E ). Then the random dynamical system Ψ is D -pullback asymptotically compact in L2 (E ); that is, for P-a.e. ω ∈ Ω , the sequence {Ψ (tn , θ−tn ω, u0,n )} has a convergent subsequence in L2 (E ) provided tn → ∞, B = {B(ω)}ω∈Ω ∈ D and u0,n ∈ B(θ−tn ω).

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Proof. Since B = {B(ω)}ω∈Ω belongs to D , so does B˜ = {B˜ (ω)}ω∈Ω which is given by (5.2). Then according to Lemma 4.4, it follows that, for P-a.e.ω ∈ Ω , up to a subsequence {v(tn , θ−tn ω, v0,n )} is convergent in L2 (E ), where v0,n = u0,n − z (θ−tn ω) ∈ B˜ (θ−tn ω). This along with (5.1) shows that, up to a subsequence {Ψ (tn , θ−tn ω, u0,n )} is convergent in L2 (E ).  Applying Lemmas 5.1, 5.2 and Proposition 2.1, we directly establish the existence of a D -random attractor for Ψ . Theorem 5.1. Suppose that Φ ∈ H01 (E ). Let D be the collection of random sets given by (3.16). Then the random dynamical system Ψ has a unique D -random attractor in L2 (E ). Acknowledgments The second author’s research was supported in part by the NSF of China (Nos 10972018, and 11272024). The third author’s research was supported in part by the NSF of China (Nos 11271050, and 11071162), and the Project of Shandong Province Higher Educational Science and Technology Program (No. J10LA09). The fourth author’s research was supported in part by the NSF of China (No. 11271052). References [1] J. Pedlosky, Geophysical Fluid Dynamics, seconnd ed., Springer-Verlag, Berlin, New York, 1987. [2] S. Wang, Attractors for the 3-D baroclinic quasi-geostrophic equations of largescale atmosphere, Journal of Mathematical Analysis and Applications 165 (1992) 266–283. [3] J. Wang, Global solutions of the 2D dissipative quasi-geostrophic equations in Besov space, SIAM Journal on Mathematical Analysis 36 (2004) 1014–1030. [4] D. Huang, B. Guo, Y. Han, Random attractors for a quasi-geostrophic dynamical system under stochastic forcing, International Journal of Dynamical Systems and Differential Equations 3 (2008) 147–154. [5] A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [6] J.K. Hale, Asymptotic Behavior of Dissipative Systems, in: American Surveys and Monographs, vol. 25, AMS, Providence, 1988. [7] J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. [8] R. Sell, Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002. [9] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer-Verlag, Berlin, 1997. [10] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields 100 (1994) 365–393. [11] F. Flandoli, B. Schmalfuss, Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise, Stochastics and Stochastic Reports 59 (1996) 21–45. [12] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. [13] T. Caraballo, J.A. Langa, J.C. Robinson, A stochastic pitchfork bifurcation in a reaction–diffusion equation, Proceedings of the Royal Society A 457 (2001) 2041–2061. [14] H. Crauel, A. Debussche, F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations 9 (1997) 307–341. [15] P.W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction–diffusion equations on unbounded domains, Journal of Differential Equations 246 (2009) 845–869. [16] B. Wang, Random attractors for stochastic FitzHugh–Nagumo system on unbounded domains, Nonlinear Analysis 71 (2009) 2811–2828. [17] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on R3 , arXiv:0810.1988v1 [math.AP]. [18] B. Wang, X. Gao, Random attractors for stochastic wave equations on unbounded domains, Discrete and Continuous Dynamical Systems (Supplement) (2009) 800–809. [19] Z. Brzezniak, Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains, Transactions of the American Mathematical Society 358 (2006) 5587–5629. [20] B. Wang, Random attractors for stochastic Benjamin–Bona–Mahony equations on unbounded domains, Journal of Differential Equations 246 (2009) 2506–2537. [21] L. Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa 13 (1959) 115–162. [22] P.W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical system, Stochastics and Dynamics 6 (2006) 1–21.