Abstract and Applied Analysis
Volume 2013, Issue 1 706091
Research Article
Open Access

H1-Random Attractors and Asymptotic Smoothing Effect of Solutions for Stochastic Boussinesq Equations with Fluctuating Dynamical Boundary Conditions

Yijin Zhang, 

Corresponding Author

Yijin Zhang

School of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China cqupt.edu.cn

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First published: 01 December 2013
https://doi.org/10.1155/2013/706091
Citations: 1
Academic Editor: Grzegorz Lukaszewicz

Abstract

This work is concerned with the random dynamics of two-dimensional stochastic Boussinesq system with dynamical boundary condition. The white noises affect the system through a dynamical boundary condition. Using a method based on the theory of omega-limit compactness of a random dynamical system, we prove that the L2-random attractor for the generated random dynamical system is exactly the H1-random attractor. This improves a recent conclusion derived by Brune et al. on the existence of the L2-random attractor for the same system.

1. Introduction

The Boussinesq equations are a coupled system of the Navier-Stokes equations and the scalar transport equation for fluid salinity, temperature, or density. These Boussinesq equations models various phenomena in environmental, geophysical and climate system, for example, oceanic density currents and the thermohaline circulation; see, for example, [1–3]. In this paper, the scalar quantity in the considered Boussinesq system is salinity, with dynamical boundary condition for the salinity.

Let  D ⊂ ℝ2  be a bounded domain with the  C1-smooth boundary  ∂D = Γ, in the vertical plane. Let  W0(t),  W1(t), and  W2(t)  be independent two-sided real-valued Wiener processes with values in appropriate function spaces. This paper is mainly concerned with the long time behavior of the solutions to the following Boussinesq equations with general additive noises and fluctuating dynamical boundary conditions [4]:
()
with velocity  u = u(x, t) = (u1(x, t), u2(x, t)) ∈ ℝ2, salinity  U = U(x, t) ∈ ℝ, and pressure  p = p(x, t) ∈ ℝ. In (1),  Fr  is the Froude number;  Re  is the Reynolds number; and  Pr  is the Prandtl number.  Δ  is the Laplacian operator;  ∇  is the gradient operator; div is the divergence operator;  γ  is the trace operator with respect to the boundary  Γ.  e2 ∈ ℝ2  is a unit vector in the upward vertical direction (opposite to the gravity). Finally,  f(x),  x ∈ Γ  is a given function describing the mean salinity flux through the boundary;  u0  and  U0  are the initial conditions;  ∂nUΓ  is the outer normal derivative. Without loss of generality, in this paper we take  Fr,  Re,  Pr,  c  and  α  to be 1.

Qualitatively, the Boussinesq system (1) emphasizes the random dynamical boundary condition which modes the interaction of boundary and the domain. In particular, if  U  describes the temperature then the heat exchange between physical domain and its boundary can be modeled; see, for example, [3]. For this Boussinesq system (1), the large deviation principle via a weak convergence approach is studied in [5], recently. In [4], they derived a priori estimates for the existence of random absorbing sets and showed that the random dynamical system (RDS) generated by the solution of (1) had a random attractor in  L2  space. When the random dynamical boundary condition in (1) is replaced by the nonhomogeneous boundary condition, [6, 7] proved the existences of random attractors for this system with multiplicative noise and additive noises in  L2  space, respectively. For the study of the random attractor about other models integrated the Navier-Stokes equations, we refer to [8] for the MHD equations.

In recent years, the theory of random attractors for some concrete dissipative stochastic partial differential equations has been studied by many authors; see, for example, [6, 8–12]; ever since [13, 14] launched their fundamental work on the RDS. Such an attractor, which generalizes nontrivially the global attractors well developed in [15–18] and so forth, is a compact invariant random set which attracts every orbit in the phase space. It is uniquely determined by attracting deterministic compact sets of phase space [19]. In order to obtain the existence of random attractors, one always need to show the existence of a compact random absorbing set in the sense of absorption [20]. This can be achieved by employing the standard Sobolev compact embedding of several functional spaces, when the generated RDSs are defined in some bounded domains; see, for example, [6–10, 20] and references cited there.

In this paper, we are interested in the existence of random attractors of the Boussinesq system (1) in  H1  space which is stronger than  L2  space. It is pointed out that if the initial data belongs to  L2, the solution to the system (1) enters into the space  L2∩H1  and has no higher regularity, see [4]. Hence the Sobolev compact embedding cannot be employed in  H1. Here, we try to overcome the obstacle of compact imbedding by using the notion of omega-limits compactness, which was initiated in [12, 21] in the framework of RDS. This type of compactness is equivalent to the asymptotic compactness [22, 23] in some spaces and can be proved by check the flattening condition; see [21]. More precisely, we prove the existence of the  (L2, H1)-random attractor for this RDS (for the bispaces random attractors; the reader is referred to [11, 16–18]). To this end, a so qualitatively new method is necessary, for instance, to derive a priori estimates of the solutions such that the flattening condition holds in  H1  space and then the necessary omega-limits compactness for the RDS in  H1  space is followed; see Lemma 12 in Section 4. The main advantage of this technique is that we need not to estimate the solutions in functional spaces of higher regularity to demonstrate the existence of compact random absorbing set which does not work in this case. This method has been used recently to obtain the existence of random attractors in  H1  space for the stochastic reaction-diffusion equations [24, 25].

The conclusion in this study shows that the  L2-random attractor is qualitatively an  H1-random attractor. This implies that the solutions to (1) become eventually more smoothing than the initial data.

The outline of this paper is as follows. Section 2 presents some functional settings for the Boussinesq system. Section 3 lists the conditions and the main conclusion of this note. In Section 4, we prove some estimates for the solution orbits in  ℍ  and  𝕍and then prove our main conclusion.

2. Functional Settings

We recall some function spaces and operators that we will be used in the following discussion.

Let
()
endowed with the scalar inner productand with norm denoted by  ∥·∥. This notation is also used to denote the norm in  (L2(D)) 2  and  L2(D)  without any confusion.
We define a functional space  𝒱  integrated the boundary and also the divergence free condition:
()
Define
()
whereis the usual Sobolev space with equivalent norm  ∥∇·∥  and  H1(D)  is also the usual Sobolev space with the equivalent norm:
()
see [15, page 52], and  H1/2(Γ)  is given by  γ(H1(D))  endowed by a norm; see [15, page 48].
Let  ℍ  the closure of  𝒱  in  L2  and  𝕍  be the closure of  𝒱  in, and the norm in  𝒱  being denoted by ∥·∥𝕍. From the above argument, the space  𝕍  is equipped with the norm:
()
which is equivalent to the natural norm:.
For  U = (u, U, UΓ),  V = (v, V, VΓ), we define the operators
()
()
By Lemma  2.2 in [4], the operator  A  is a positive self-adjoint unbounded operator and has the Poincaré inequality:
()
Then  A−1  is also self-adjoint but compact operator in  ℍ, so we can utilize the elementary spectral theory in a Hilbert space. We infer that there exists a compete orthonormal family in  ℍ, also in  𝕍,of eigenvectors of  A. The corresponding spectrum of  A  is discrete and denoted bywhich are positive, increasing, and tend to infinity as  j → ∞.
In particular, we also can use the spectrum theory to allow us to define the operator  As, the power of  A. For  s > 0, the operator  As  is also an strictly positive and self-adjoint unbounded operator in  ℍ  with a dense domain  D(As) ⊂ ℍ. This allows us to introduce the function spaces,
()
This normon  D(As)  is equivalent to the usual norm induced by  H2s; see Temam [15] for details. In particular,  D(A0) = ℍ  and  D(A1/2) = 𝕍.
Based on the orthonormal basisof eigenfunctions of  A, we define the  m-dimensional subspace  𝕍m = span   {e1, e2, …, em} ⊂ 𝕍  and the canonical orthogonal projection  Pm : 𝕍 ↦ 𝕍m  such that for every  V ∈ 𝕍,  V  has a unique decomposition:  V = PmV + Vm, where
()
that is,.
When the projection  Pm  operates on the first component of  V = (v, V, VΓ), one can easily show that
()
for  m ∈ N+, the nature numbers set.
We also state the well-known Brezis-Gallouet’s inequality with Dirichlet boundary condition in two dimension case; see [26] or Proposition  3 in [27]; there exists a positive constant  c  such that
()
where  λ1  is the first eigenvalue of the Stokes operator  A1. By (12) we have
()
According to the above notations, we can write (1) as the following abstract evolution equation form:
()
where  U = (u, U, UΓ).

3. Existence of Random Attractor in 𝕍

In order to model the noise in the initial problem (1), we need to define a metric dynamical system (MDS) which is a group of measure preserving transformations on a probability space. For the definition of the MDS we refer to [28, 29] and so forth.

A standard model for a spatially correlated noise is the generalized time derivative of a two-sidedBrownian motion  ω = ω(x, t),  x ∈ ℝ2. Let  ℍ  be the separable Hilbert space defined in Section 2. As usual, we introduce the spatially valued Brownian motion MDS θ = (Ω, ℱ, ℙ, (θt) t∈ℝ), where Ω = {ω ∈ C0(ℝ, ℍ) : ω(0) = 0}  with compact open topology. This topology is metrizable by the complete metric:
()
where  dn(ω1, ω2) = max |t|≤n | ω1 − ω2 |   for  ω1  and  ω2  in  Ω.  ℱ = ℬ(C0(ℝ, ℍ))  is the Borel-σ-algebra induced by the compact open topology of  Ω. Suppose the Wiener process  ω  has covariance operator  Q. Let  ℙ  be the Wiener measure with respect to  Q. The Wiener shift is defined by
()
Then the measure  ℙ  is ergodic and invariant with respect to the shift  θ.
The associated probability space defines a canonical Wiener process  W. We also note that such a Wiener process  W  generates a filtration  (ℱt) t∈ℝ:
()
We introduce the following stochastic partial differential equation on  D:
()
Because  A  is a positive and self-adjoint operator, there exists a mild solution to this stochastic equation with the form
()
which is called an Ornstein-Uhlenbeck process; see [30]. For the Ornstein-Uhlenbeck process we have the regularity hypothesis; see also [4].

Lemma 1. Suppose that the covariance operator  Q  of the Wiener process  ω  has a finite trace; that is,  Q  satisfies that

()
for some  s ≥ 0  and some (arbitrary small)  δ > 0, where  tr ℍ  denotes the trace of the covariance operator. Then an  ℱ0-measurable Gaussian variable  Z = (z, Z) ∈ D(As)  exists, and the process (t, ω) → Z(θtω)  is a continuous stationary solution to the stochastic equation (19). Furthermore, the random variableis tempered and the expectation
()

Let  V = U − Z(θtω); by (15) and (19) we have the following deterministic equation with a parameter  ω:
()

Here we denote the solution to (23) by  V(t, ω, V0(ω)), V(t, ω)  or more briefly  V(t). By Lemma  4.7 in [4], the solution of the evolution equation (23) generates a continuous measurable RDS  ψ  in  ℍ  given by ψ(t, ω)V0(ω) = V(t, ω, V0(ω)). Put  U(t, ω, U0(ω)) = V(t, ω, U0(ω) − Z(ω)) + Z(θtω). Then U(t, ω, U0(ω))  or briefly  U(t)  is a solution to (23) with initial value  U0(ω).

Given
()
then  φ  is also a continuous measurable RDS in  ℍ  for the original equation (15), that is, (1).

As for the general theory of random dynamical systems we may refer to [28, 31] for details.

The main conclusion of this study states the following.

Theorem 2. One supposes that (21) holds. Set that

()
where  Kϵ  is a generic constant depending on the data of the problem and some  ϵ  that has to be chosen sufficiently small and  λ  is the same as in (9). Assume that the mathematical expectation of  M  
()
Then the RDS  φ  generated by (1) admits a unique random attractor  {𝒜𝕍(ω)} ω∈Ω  in  𝕍  in the sense that for  ℙ-a.s.ω ∈ Ω,
()
()
()
where dist 𝕍 denotes the Hausdorff semiistance in  𝕍  and  𝒟  is the collection of tempered subsets of  ℍ  as in [4]. Furthermore,  {𝒜𝕍(ω)} ω∈Ω  is identical with the random attractor  {𝒜ℍ(ω)} ω∈Ω  in  ℍ.

By the well-known abstract result of Theorem  3.5 in [20], the existence of a random attractor  {𝒜ℍ(ω)} ω∈Ω  in the following sense that
()
has been obtained in [4]. However, Our Theorem 2 shows the existence of random attractor which is compact in the space  𝕍, which is stronger than  ℍ. Thus the Sobolev compact embedding theorem is not available and it seems impossible to obtain a compact random absorbing set in the space  𝕍  when the initial data belongs to  ℍ. Then Theorem  3.5 in [20] is unapplicable to the proof of our Theorem 2.

Fortunately, Theorem 2 can be proved by checking the omega-limit compactness in the space  𝕍, which is based on the viewpoint of Kuratowski measure of noncompactness. This kind of compactness can be easily obtained by showing the flattening condition, see [21].

For clarification, we state some general concepts used in the sequel. Let  X  be a Banach space with norm ∥·∥X  and  𝒟  be the collection of all random subsets of  X.

Definition 3 (see [21].)An RDS  φ  on  X  over an MDS is said to be omega-limit compact if for every  ε > 0  and  B = {B(ω)} ω∈Ω ∈ 𝒟, there exists  T = T(ε, B, ω)  such that for all  t ≥ T,

()
where  k(B)  is the Kuratowski measure of noncompactness of  B  defined by
()

Definition 4 (see [21].)An RDS  φ  on  X  over an MDS is said to possess the flattening condition if for  ℙ-a.s.ω ∈ Ω  and every  B = {B(ω)} ω∈Ω ∈ 𝒟, there exist  T = T(B, ω, ε) > 0  and a finite dimensional space  X1  of  X  such that for a bounded projector  P : X → X1,

()

Definition 5 (see [29].)(i) A random variable  X ∈ ℝ+  over an MDS is tempered if for  ℙ-a.s.ω ∈ Ω,

()

(ii) A random set  B = {B(ω)} ω∈Ω ∈ 𝒟  is called tempered if  R(ω) = sup x∈B(ω)∥x∥X  is a tempered random variable.

Definition 6. A random set  {K(ω)} ω∈Ω  is an  𝒟-random absorbing set for RDS  φ  over an MDS if for  ℙ-a.s.ω ∈ Ω  and every  B = {B(ω)} ω∈Ω ∈ 𝒟, there exists  T = T(B, ω) > 0  such that for all  t ≥ T,

()
where.

For our problem,  X = 𝕍  and  𝒟  is the collection of tempered subsets of  ℍ. The finite dimensional space X1 = 𝕍m = span   {e1, e2, …, em}  and the bounded operator  P = Pm  for a sufficient large  m, where ei  (i = 1,2, …, m)  and  Pm  are defined in Section 2.

4. The Proofs of Main Result

To prove Theorem 2, we need a series of lemmas. First we give a useful lemma similar to the classic Gronwall lemma (see [15]).

Lemma 7. Assume that  y,  y′, and  h  are three locally integrable functions and  y,  h  nonnegative on  [a, ∞)  such that for  s ≥ a,

()
Then for every  t ≥ a  and any positive constant  r, one has
()

Proof. Let  t ≤ s ≤ t + r. We multiply (36) by  eb(s−t)  and the resulting inequality reads

()
Hence, by integration from  s  to  t + r,
()
Integrating the above inequality with respect to  s  between  t  and  t + r, we get the desired inequality.

Lemma 8. There exist positive constants  K  and  Kϵ  such that the followings hold:

()
()
()
where
()
()

Proof. The inequality (40) is the same as the inequality  (18)  at page  1112 in [4]. The inequality (41) is a combination of the formulas  (15)  and  (16)  at pages 1110-1111 in [4] with a tiny modification. Following a same calculation as page  1114 in [4] we can obtain (42).

Lemma 9. Assume that  𝔼M < 0. Let  B = {B(ω)} ω∈Ω ∈ 𝒟. Then for  ℙ-a.s.ω ∈ Ω, there exist random radii  ϱ1(ω)  and  ϱ1(ω)  and constant  T = T(B, ω) > 0  such that for all  t ≥ T, the solution V(t, ω, U0(ω) − Z(ω))  of problem (23) with  U0 ∈ B(ω)  satisfies that for every  l ∈ [t, t + 1]:

()
()
where  𝒟  is the collection of tempered random subsets of  ℍ.

Proof. We replace  t  with  l, and  ω  with  θ−t−1ω  in (40) to produce that for  l ∈ [t, t + 1],

()
By noting that for  l ∈ [t, t + 1],  l − t − 1 ∈ [−1,0], then we find that for  s ∈ [−t − 1,0],
()
from which and (47) it follows that for every  l ∈ [t, t + 1],
()
By the Birkhoff’s ergodic theorem and along with our assumption (26), it yields that
()
and then it implies that
()
As  G(θτ)  is subexponential growth, so
()
Note that  ∥Z(ω)∥2  is also tempered random variable and  U0(ω) ∈ B(ω)  with  B ∈ 𝒟, whence there exists constant  T = T(B, ω) > 0  such that for all  t ≥ T  with  l ∈ [t, t + 1],
()
Furthermore,  ϱ1(ω)  is tempered; see [4]. This completes the proof of (45).

Next, we show (46). Integrating (41) from  t  to  t + 1  with  t ≥ T, where  T  is in (53), we get

()
Then by (45) we get that for all  t ≥ T,
()
which gives an expression for  ϱ2(ω).

Lemma 10. Assume that  𝔼M < 0. Let  B = {B(ω)}ω∈Ω ∈ 𝒟. Then for  ℙ-a.s.ω ∈ Ω, there exist random radium  R(ω)  and constant  T = T(B, ω) > 0  such that for all  t ≥ T, the solution  V(t, ω, U0(ω) − Z(ω))  of problem (23) with  U0 ∈ B(ω)  satisfies that for every  l ∈ [t, t + 1],

()
where  𝒟  is the collection of tempered random subsets of  ℍ.

Proof. Using the classic Gronwall’s lemma (see [15]) to the inequality (42) on the interval  [s, l]  with t ≤ s ≤ l ≤ t + 1, we get that

()
Note that by Lemma 9 there exists  T = T(B, ω)  such that for all  t ≥ T,
()
Then by (57) and (58) it gives that for all  t ≥ T  and  l ∈ [t, t + 1],
()
which gives an expression for  R(ω).

Lemma 11. Assume that  𝔼M < 0. Let  B = {B(ω)}ω∈Ω ∈ 𝒟. Then for  ℙ-a.s.ω ∈ Ω  and every  ε > 0, there are  N = N(ε, ω), K = K(ω), and  T = T(ε, B, ω) > 0  such that for all  t ≥ T  and  m ≥ N, the solution V(t, ω, U0(ω) − Z(ω))  of problem (23) with  U0 ∈ B(ω)  satisfies that

()
where  𝒟  is the collection of tempered random subsets of  ℍ.

Proof . Multiplying (23) by  AVm  with respect to the  L2  inner product leads us to

()
where
()
In order to estimate  I1, we rewrite it as
()
where by utilizing the inequality (14) and Agmon’s inequality in  ℝ2  (see [15]), it gives that
()
()
Similarly by utilizing the Agmon’s inequality, we deduce that
()
Then by (63)–(66) we get that
()
We then estimate  I2  in (61). We first have
()
where
()
Then it follows from (68) and (69) that
()
Moreover,
()
Then by (67), (70), and (71), formula (61) becomes
()
where
()
Then by utilizing Lemma 7 to (72) we deduce that
()
By Lemma 11, there exists  T = T(B, ω) > 0  such that for all  t ≥ T,
()
where
()
is independent of  λm+1. By a similar calculation as (75), we find that there exist an random variablesuch that for all  t ≥ T,
()
where  R(ω)  is in Lemma 9. Then (74) together with (75) and (77) implies that for all  t ≥ T,
()
as  m → +∞. Consequently, for every  ε > 0, there exists a integer  N  and positive constants  K = K(ω)  such that for all  t ≥ T + 1  and  m ≥ N,
()
This leads to the desirable conclusion.

Lemma 12. Assume that  𝔼M < 0. Then the RDS  φ  corresponding to the Boussinesq system (1) is omega-limit compact in  𝕍; that is, for every  ε > 0  and an arbitrary  B = {B(ω)} ω∈Ω ∈ 𝒟, there is an T = T(ε, B, ω) > 0  such that for  ℙ-a.s.ω ∈ Ω,

()
where  𝒟  is the collection of tempered random subsets of  ℍ.

Proof. By Lemma 11, for every  U0(ω) ∈ B(ω), there exist constantsand  T = T(ε, B, ω)  and N1 ∈ ℕ  such that for all  t ≥ T  and  m ≥ N1,and

()
Note that
()
as  m → ∞, and then there exists  N2 ∈ ℕ  such that for every  m ≥ N2,
()
Put  N = max   {N1, N2}. By the definition of the RDS  φ, along with (81) and (82), we find that there exist such that for all  t ≥ T  
()
where  T = T(ε, B, ω) > 0  is the same as (81). That is to say, the RDS  φ  satisfies the flattening conditions in  𝕍. By utilizing the additive property of Kuratowski measure of noncompactness; see Lemma  2.5 (iii) in [12]; it follows from (84) that for  ℙ-a.s.ω ∈ Ω,
()
where  B𝕍(0, ε)  is the  ε-neighborhood at centre  0  in  𝕍. This completes the proof.

Proof of Theorem 2. By Theorem  5.1 in [4], the RDS  φ  associated with the Boussinesq system (1) admits a unique compact random attractorin  ℍ. Furthermore  {𝒜H(ω)} ω∈Ω ∈ 𝒟. Put

()
where  ϱ1(ω)  is in (42). Observe that  ∥Z(ω)∥  is tempered, and then Lemma 9 implies thatis a random absorbing set for the RDS  φ  in  𝒟. By Theorem  3.5 in [20], we know that  {𝒜H(ω)} ω∈Ω  is the  Ω-limit set of  {K0(ω)} ω∈Ω  (see [19]); that is,
()

For  ω ∈ Ω, the following is given:

()
In the sequel we will show thatis a random attractor in  𝕍  in the sense that  {𝒜𝕍(ω)} ω∈Ω  satisfies (27)–(29).

We divide the proof into three steps.

Step  1 (compactness). By Lemma  4.5 (v) in [12] and the omega-limit compactness of  φ  in  𝕍  (from Lemma 12), we have

()
Sinceis norm-closed in  𝕍, thanks to the nested property of the Kuratowski measure of noncompactness (see again Lemma  2.5 (iv) in [12]), we know that for  ω ∈ Ω,  𝒜𝕍(ω)  is nonempty and compact as required, which shows (27).

Step  2 (invariant property). By (87) and (88) it is easy to see that for  ω ∈ Ω,

()
()
It is obvious that  𝒜𝕍(ω)⊆𝒜ℍ(ω)  for  ω ∈ Ω. Conversely, if  x ∈ 𝒜ℍ(ω), by the equivalent regime (90), there exist two sequences  tn → ∞  and  xn ∈ K0(θ−tω)  such that
()
Note thatand  φ  is omega-limit compact in  𝕍  (from Lemma 12). Then there exist an y ∈ 𝕍  and a subsequence  {nk}  withsuch that
()
and by the nested relation of  𝕍  and  ℍ, we have  x = y, whence connection with (91) and (93) we have  x ∈ 𝒜𝕍(ω). Then  𝒜ℍ(ω)⊆𝒜𝕍(ω)  for  ω ∈ Ω, which indicates that  {𝒜𝕍(ω)} ω∈Ω  possesses the invariance property (28).

Step  3 (attracting property). Suppose that (29) does not hold. Then there exists  δ > 0,and  tn → ∞  such that

()
According to Lemma 12,  φ  is omega-limit compact in  𝕍. Then we can extract a subsequence from the sequence(denoted by its original form) satisfying that there exists an  y ∈ 𝕍  such that when  tn → ∞,
()
We then need to show that  y ∈ 𝒜𝕍(ω)  for  ω ∈ Ω. Indeed, since  {K0(ω)} ω∈Ω  is a random absorbing set for  φ  in  ℍ, then for, there exists  T = T(B, ω) > 0  such that for all  t ≥ T,
()
At the same time, by the cocycle property of the RDS  φ, for  tn ≥ t  we have that
()
We now choose fixedsuch that. DenoteandThen (96) indicates that. It is obvious that if  tn → ∞  then. Thus it follows from (97) that the limit in (95) can be rewritten as
()
as, whereas by (91) we get that  y ∈ 𝒜𝕍(ω)  for  ω ∈ Ω, which contradicts (94). This proves that  {𝒜𝕍(ω)} ω∈Ω  possesses attracting property (29).

Note that the collection  𝒟  considered also includes all deterministic bounded subsets in  𝕍. Then the uniqueness for  {𝒜𝕍(ω)} ω∈Ω  is followed by Theorem  4.3 in [19].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

    Acknowledgments

    This work was supported by the China Natural Science Fund Grant (no. 11071199) and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ120515).

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