Random attractors for stochastic 2D hydrodynamical type systems
We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the...
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| author | Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da |
| author_facet | Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da |
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| description | We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray $\alpha$-model. We prove the existence of random attractors for the associated continuous random dynamical systems. Then we establish the upper semicontinuity of the random attractors as the parameter tends to zero. |
| first_indexed | 2026-03-24T02:02:29Z |
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| fulltext |
UDC 519.21
Cung The Anh (Hanoi Nat. Univ. Education, Vietnam),
Nguyen Tien Da (Hong Duc Univ., Thanh Hoa city, Vietnam)
RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE
SYSTEMS*
ВИПАДКОВI АТРАКТОРИ ДЛЯ СТОХАСТИЧНИХ ДВОВИМIРНИХ СИСТЕМ
ГIДРОДИНАМIЧНОГО ТИПУ
We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive
noise that covers numerous 2D hydrodynamical models, such as the 2D Navier – Stokes equations, 2D Boussinesq equations,
2D MHD equations, etc., and also some 3D models, like the 3D Leray \alpha -model. We prove the existence of random
attractors for the associated continuous random dynamical systems. Then we establish the upper semicontinuity of the
random attractors as the parameter tends to zero.
Вивчається асимптотична поведiнка розв’язкiв одного класу абстрактних нелiнiйних стохастичних рiвнянь еволюцiї
з адитивним шумом, що включає рiзноманiтнi двовимiрнi гiдродинамiчнi моделi, такi як двовимiрнi рiвняння Нав’є –
Стокса, двовимiрнi рiвняння Буссiнеска, двовимiрнi рiвняння магнiтогiдродинамiки тощо, а також деякi тривимiрнi
моделi типу тривимiрної \alpha -моделi Лерея. Доведено iснування випадкових атракторiв для вiдповiдних неперервних
випадкових динамiчних систем. Крiм того, встановлено напiвнеперервнiсть зверху випадкових атракторiв у випадку,
коли параметр прямує до нуля.
1. Introduction. The study of the asymptotic behavior of dynamical systems is one of the most
important problems of modern mathematical physics. In the deterministic case, the notion of global
attractors, a compact invariant and attracting set, plays a central role (see, for example, [11, 22]).
The concept of random attractors was introduced in [14, 15] as an extension to stochastic systems
of the concept of global attractors for deterministic systems. The theory of random attractors has
been shown to be very useful for the study of the long-time behavior of infinite-dimensional random
dynamical systems, see the recent survey [16] and references therein. Up to now, the existence of
random attractors has been proved for many classes of stochastic partial differential equations, see,
e.g., [2, 4, 6 – 8, 17 – 20, 24 – 28].
In this paper, we study the long-time behavior of solutions to the following abstract nonlinear
stochastic 2D hydrodynamical type system:
du+ (Au+B(u, u) +Ru)dt = fdt+ \varepsilon hd\omega . (1.1)
As pointed out in [11, 12], with suitable choices of A,B and R, this abstract model covers many 2D
hydrodynamical models such as 2D Navier – Stokes equations, 2D Boussinesq equations, 2D MHD
equations, 2D magnetic Bénard equations, and also some 3D models such as 3D Leray-\alpha model,
the shell models of turbulence. In the paper [12], the authors proved the existence and uniqueness
of weak solutions, and more importantly, the Wentzell – Freidlin type large deviation principle for
small multiplicative noise to this equation. The support of distribution of solutions to this abstract
* This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED)
(grant No. 101.02-2018.303).
c\bigcirc CUNG THE ANH, NGUYEN TIEN DA, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12 1647
1648 CUNG THE ANH, NGUYEN TIEN DA
model was described later in [13]. The stability and stabilization of solutions to the abstract model
(1.1) was investigated recently in [3]. It is also noticed that the existence and long-time behavior of
solutions in terms of existence of global attractors to the corresponding deterministic version of this
model (i.e., equation (1.1) without the stochastic term) was given in the monograph [11].
In this paper we consider the abstract equation (1.1) with additive noise. We first prove the exis-
tence of a random attractor for the continuous random dynamical system generated by the equation.
Then we establish the upper semicontinuity of the random attractor at \varepsilon = 0, that is, we compare
the random attractor of the stochastic equation (1.1) and the global attractor of the limit determi-
nistic equation, which is formally obtained when \varepsilon = 0. Under the assumptions in the paper (see
Subsection 2.2 for details), the associated random dynamical system is not necessary compact, and
therefore the pullback asymptotic compactness of the dynamical system cannot be obtained directly
by constructing random absorbing sets in a more regular space and using some compact embeddings.
In order to overcome this essential difficulty, we exploit the energy equations method to prove the
pullback asymptotic compactness. This method was first introduced by Ball in [5] for the deter-
ministic wave equation, and then extensively used by many authors for weakly dissipative equations
or equations in unbounded domains, both in deterministic and stochastic cases (see, for instance,
[6, 10, 21, 24] and references therein). It is worthy noticing that, as a direct consequence of the
abstract results obtained in this paper, we get the existence and upper semicontinuity of random
attractors for many 2D models in fluid mechanics, in both bounded domains and unbounded domains
satisfying the Poincaré inequality (see Remark 4.1 below).
The outline of this paper is as follows. In Section 2, we recall the theory of random attractors
and give a description of the problem. The existence of a random attractor for the associated random
dynamical system is proved in Section 3, while its upper semicontinuity is investigated in Section 4.
2. Preliminaries. 2.1. Random attractors. In this subsection, we recall some concepts and
results on theories of random dynamical systems and random attractors in [1, 9, 14, 18, 23].
Let (X, \| \cdot \| X) be a separable Banach space with Borel \sigma -algebra \scrB (X), and let (\Omega ,\scrF , P ) be a
complete probability space.
Definition 2.1. (\Omega ,\scrF , P, (\theta t)t\in \BbbR ) is called a metric dynamical system if \theta : \BbbR \times \Omega \rightarrow \Omega is
(\scrB (\BbbR )\times \scrF ,\scrF )-measurable, \theta 0 is the identity on \Omega , \theta s+t = \theta t\theta s for all s, t \in \BbbR , and \theta t(P ) = P
for all t \in \BbbR .
Definition 2.2. A continuous random dynamical system (RDS) on X over a metric dynamical
system (\Omega ,\scrF , P, (\theta t)t\in \BbbR ) is a mapping
\Phi : \BbbR + \times \Omega \times X \rightarrow X, (t, \omega , x) \mapsto \rightarrow \Phi (t, \omega , x),
which is (\scrB (\BbbR +)\times \scrF \times \scrB (X),\scrB (X))-measurable and satisfies, for P -a.e. \omega \in \Omega , the following
conditions:
(i) \Phi (0, \omega , \cdot ) is the identity of X;
(ii) \Phi (t+ s, \omega , x) = \Phi (t, \theta s\omega ,\Phi (s, \omega , x)) for all t, s \in \BbbR +, x \in X;
(iii) \Phi (t, \omega , \cdot ) : X \rightarrow X is continuous for all t \in \BbbR +.
Definition 2.3. A random bounded set \{ B(\omega )\} \omega \in \Omega of X is called tempered with respect to
(\theta t)t\in \BbbR if for P -a.e. \omega \in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
e - \beta t\| B(\theta - t\omega )\| = 0 for all \beta > 0,
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RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1649
where \| B\| X = \mathrm{s}\mathrm{u}\mathrm{p}x\in B \| x\| X .
Hereafter, we always assume that \Phi is a continuous RDS over (\Omega ,\scrF , P, (\theta t)t\in \BbbR ) and denote by
\scrD a collection of random subsets \{ B(\omega )\} \omega \in \Omega of X.
Definition 2.4. A random set \{ K(\omega )\} \omega \in \Omega \in \scrD is said to be a random absorbing set for \Phi in
\scrD if for every \{ B(\omega )\} \omega \in \Omega \in \scrD and P -a.e. \omega \in \Omega , there exists TB(\omega ) > 0 such that
\Phi (t, \theta - t\omega ,B(\theta - t\omega )) \subseteq K(\omega ) for all t \geq TB(\omega ).
Definition 2.5. A random dynamical system \Phi is called \scrD -pullback asymptotically compact in
X if for P -a.e. \omega \in \Omega , \{ \Phi (tn, \theta - tn\omega , xn)\} n\geq 1 has a convergent subsequence in X whenever
tn \rightarrow \infty , and xn \in B(\theta - tn\omega ), where \{ B(\omega )\} \omega \in \Omega \in \scrD .
Definition 2.6. A random set \{ \scrA (\omega )\} \omega \in \Omega of X is called a \scrD -random attractor for \Phi if the
following conditions are satisfied, for P -a.e. \omega \in \Omega :
(i) \scrA (\omega ) is compact, and \omega \mapsto \rightarrow d(x,\scrA (\omega )) is measurable for every x \in X;
(ii) \{ \scrA (\omega )\} \omega \in \Omega is invariant, that is, \Phi (t, \omega ,\scrA (\omega )) = \scrA (\theta t\omega ) for all t \geq 0;
(iii) \{ \scrA (\omega )\} \omega \in \Omega attracts every set in \scrD , that is, for every \{ B(\omega )\} \omega \in \Omega \in \scrD ,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow +\infty
\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (\Phi (t, \theta - t\omega ,B(\theta - t\omega )) ,\scrA (\omega )) = 0,
where \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} is the Hausdorff semidistance
\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(A,B) = \mathrm{s}\mathrm{u}\mathrm{p}
x\in A
\mathrm{i}\mathrm{n}\mathrm{f}
y\in B
\| x - y\| X for all A,B \subset X.
The following existence result for the random attractor for a continuous RDS can be found
in [7, 18, 23].
Theorem 2.1. Let \scrD be an inclusion-closed collection of random subsets of X and assume
that \Phi is a continuous RDS which has a random absorbing set \{ K(\omega )\} \omega \in \Omega . If \Phi is \scrD -pullback
asymptotically compact in X, then it has a unique \scrD -random attractor \{ \scrA (\omega )\} \omega \in \Omega which is given
by
\scrA (\omega ) =
\bigcap
\tau \geq 0
\bigcup
t\geq \tau
\Phi (t, \theta - t\omega ,K(\theta - t\omega )).
Let \Phi 0 be an autonomous dynamical system defined on the Banach space X. Given \varepsilon \in (0, 1],
suppose \Phi \varepsilon is a random dynamical system over (\Omega ,\scrF , P, (\theta t)t\in \BbbR ) which has a random absorbing set
K\varepsilon = \{ K\varepsilon (\omega )\} \omega \in \Omega and a random attractor \scrA \varepsilon = \{ \scrA \varepsilon (\omega )\} \omega \in \Omega . We suppose that the autonomous
dynamical system \Phi 0 : \BbbR + \times X \rightarrow X has a global attractor \scrA 0, which means that \scrA 0 is compact,
invariant and attracts every bounded subset of X uniformly (see, e.g., [11] for the theory of global
attractors).
Definition 2.7. The family of random attractors \{ \scrA \varepsilon \} \varepsilon \in (0,1] is said to be upper semicontinuous
at \varepsilon = 0 if
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0
\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (\scrA \varepsilon (\omega ),\scrA 0) = 0 for P -a.e. \omega \in \Omega .
Theorem 2.2 [23]. Suppose that the following conditions hold for P -a.e. \omega \in \Omega :
(i) \Phi \varepsilon n (t, \omega , xn) \rightarrow \Phi (t)x for all t \geq 0, provided \varepsilon n \rightarrow 0 and xn \rightarrow x in X;
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1650 CUNG THE ANH, NGUYEN TIEN DA
(ii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\| K\varepsilon (\omega )\| X \leq M, where \| K\varepsilon (\omega )\| X = \mathrm{s}\mathrm{u}\mathrm{p}
x\in K\varepsilon (\omega )
\| x\| X ;
(iii)
\bigcup
0<\varepsilon \leq 1
\scrA \varepsilon (\omega )is precompact in X.
Then the family of random attractors \{ \scrA \varepsilon \} \varepsilon \in (0,1] is upper semicontinuous at \varepsilon = 0.
2.2. Model description. Let H be a separable Hilbert space with the the norm | \cdot | and the
inner product (\cdot , \cdot ), let A be an (unbounded) self-adjoint positive linear operator on H. Set V =
= \mathrm{D}\mathrm{o}\mathrm{m}(A1/2). For v \in V set \| v\| = | A1/2v| . Let V \prime denote the dual of V (with respect to the inner
product (., .) of H ). Then we have the triple V \subset H \subset V \prime . Let \langle u, v\rangle denote the duality between
u \in V and v \in V \prime such that \langle u, v\rangle = (u, v) for u \in V, v \in H.
We assume B : V \times V \rightarrow V \prime and R : H \rightarrow H are continuous mappings satisfying the following
conditions:
Main assumptions:
B : V \times V \rightarrow V \prime is a bilinear continuous mapping.
For all u, v, w \in V,
\langle B(u, v), w\rangle = - \langle B(u,w), v\rangle . (2.1)
There exists a Banach (interpolation) space \scrH possessing the properties:
(i) V \subset \scrH \subset H ;
(ii) there exists a constant a0 > 0 such that
\| v\| 2\scrH \leq a0| v| \| v\| for any v \in V ; (2.2)
(iii) there exists a constant C > 0 such that
| \langle B(u, v), w\rangle | \leq C\| u\| \scrH \| v\| \| w\| \scrH \forall u, v, w \in V. (2.3)
R : H \rightarrow H is a bounded linear operator such that
\| R\| op < \lambda , (2.4)
where \lambda > 0 is the best constant in the inequality
\| u\| 2 \geq \lambda | u| 2 \forall u \in V. (2.5)
From (2.1) – (2.3), one can see that
for every \eta > 0 there exists C\eta > 0 such that
| \langle B(u, v), w\rangle | \leq \eta \| w\| 2 + C\eta \| u\| 2\scrH \| v\| 2\scrH for u, v, w \in V ; (2.6)
there exist a positive constant C0 such that
| \langle B(u, v), w\rangle | \leq C0\| u\| \| v\| \| w\| \forall u, v, w \in V,
\| B(u, u)\| V \prime \leq C0\| u\| 2 \forall u \in V.
(2.7)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1651
Let f \in H and h \in D(A). We consider a small random perturbation of the 2D hydrodynamical
type systems given by
du+ (Au+B(u, u) +Ru)dt = fdt+ \varepsilon hd\omega , (2.8)
where \omega (t) is a two-sided real-valued Wiener process on a complete probability space (\Omega ,\scrF , P ) ,
where P is the Wiener distribution, \Omega is a subset of
\{ \omega \in C(\BbbR ,\BbbR ) : \omega (0) = 0\}
with P (\Omega ) = 1, \scrF is a \sigma -algebra. In addition, the space (\Omega ,\scrF , P ) is invariant under the Wiener
shift
\theta t\omega (\cdot ) = \omega (\cdot + t) - \omega (t), \omega \in \Omega , t \in \BbbR .
This means that (\Omega ,\scrF , P, (\theta t)t\in \BbbR ) is a metric dynamical system.
Let \sigma be a fixed positive constant such that
\sigma >
2\lambda C2a20
(\lambda - \| R\| op)2
\| h\| 2, (2.9)
where a0, C and \lambda are the constants in (2.2), (2.3) and (2.5), respectively. Consider the one-
dimensional Ornstein – Uhlenbeck equation
dy + \sigma ydt = d\omega (t).
One can check that a solution of this equation is given by
y(\theta t\omega ) = - \sigma
0\int
- \infty
e\sigma s(\theta t\omega )(s)ds.
Note that the random variable | y(\omega )| is tempered and y(\theta t\omega ) is P -a.e. continuous. Therefore, it
follows from [1] (Proposition 4.3.3) that there exists a tempered function r(\omega ) > 0 such that
| y(\omega )| 2 + | y(\omega )| 4 \leq r(\omega ) \forall \omega \in \Omega , (2.10)
where r(\omega ) satisfies, for P -a.e. \omega \in \Omega , r(\theta tw) \leq e
\sigma
2
| t| r(\omega ), t \in \BbbR .
Now, we need to transfer the stochastic equation (2.8) into a deterministic one with random
parameters. Let z(\theta t\omega ) = hy(\theta t\omega ) and v(t, \omega ) = u(t, \omega ) - \varepsilon z(\theta t\omega ), then v is a solution of the
equation
dv
dt
+Av +B(v, v) +Rv + \varepsilon B(v, z) + \varepsilon B(z, v) = f - \varepsilon Az - \varepsilon Rz - \varepsilon 2B(z, z) + \varepsilon \sigma z (2.11)
with v0(\omega ) = u0(\omega ) - \varepsilon z(\omega ).
Let \omega \in \Omega and v0 \in H. A mapping v(\cdot , \omega , v0) : [0,+\infty ) \rightarrow H is called a solution of problem
(2.11) if, for every T > 0,
v(\cdot , \omega , v0) \in C([0, T ];H) \cap L2(0, T ;V ),
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1652 CUNG THE ANH, NGUYEN TIEN DA
and v satisfies
(v(t), \xi ) +
t\int
0
\langle Av, \xi \rangle ds+
t\int
0
\langle \~B(v), \xi \rangle ds+
t\int
0
(Rv, \xi )ds =
t\int
0
( \~F , \xi )ds, (2.12)
where \~B(v) = \varepsilon B(v, z)+ \varepsilon B(z, v) and \~F = f - \varepsilon Az - \varepsilon Rz - \varepsilon 2B(z, z)+ \varepsilon \sigma z for every t \geq 0 and
\xi \in V. Since (2.11) is a deterministic equation, it follows from [11] (Section 4.4) that for every \omega \in \Omega
and v0 \in H given, problem (2.11) has a unique solution v in the sense of (2.12) which continuously
depends on v0 with the respect to the norm of H. Moreover, the solution v is (\scrF ,\scrB (H))-measurable
in \omega \in \Omega . This enables us to define a mapping \Phi : \BbbR + \times \Omega \times H \rightarrow H by
\Phi (t, \omega , u0(\omega )) = u(t, \omega , u0(\omega )) = v(t, \omega , v0(\omega )) + \varepsilon z(\theta t\omega ), (2.13)
then we see that \Phi is a continuous RDS associated with the stochastic equation (2.8).
Given a bounded nonempty subset B of H, we write | B| = \mathrm{s}\mathrm{u}\mathrm{p}\phi \in B | \phi | H . We denote by \scrD the
collection of random sets \{ B(\omega \} \omega \in \Omega of H, which satisfy for P -a.e. \omega \in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow +\infty
e -
\nu
2
t| B(\theta - t\omega )| H = 0,
where \nu := \lambda - \| R\| op > 0.
3. Existence of a random attractor. 3.1. Uniform estimates of solutions and existence
of a random absorbing set in \bfitH . In this subsection, we first establish the uniform estimates on
the solutions to problem (2.11), then we will show that the RDS \Phi associated with the stochastic
equation (2.8) has a random absorbing set in H.
Lemma 3.1. Let 0 < \varepsilon \leq 1, f \in H, h \in D(A), and (2.9) hold. Then for any \{ B(\omega )\} \omega \in \Omega \in \scrD
and for P -a.e. \omega , there exists T = T (B,\omega ) > 0 independent of \varepsilon such that for all t \geq T and
v0(\theta - t\omega ) \in B(\theta - t\omega ), the solution v of (2.11) satisfies
| v(t, \theta - t\omega , v0(\theta - t\omega ))| 2H \leq 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega )| 2dr
\right) g(\theta \tau \omega )d\tau , (3.1)
where
\nu = \lambda - \| R\| op > 0,
\beta =
\lambda C2a20
\nu
\| h\| 2,
c =
2(3 + \sigma 2)
\nu | f | 2
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
C2
0\| h\| 4, (\| R\| 2op + 1)\| h\| 2D(A)
\Bigr\}
,
g(\theta \tau \omega ) = | y(\theta \tau \omega )| 4 + | y(\theta \tau \omega )| 2 + 1.
(3.2)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1653
Proof. It follows from (2.11) that, for each \omega \in \Omega and v \in V,
1
2
d
dt
| v| 2 + \| v\| 2 \leq - \varepsilon \langle B(v, z), v\rangle + \| R\| op| v| 2 + \varepsilon (Rz, v) + (F, v) \leq
\leq - \varepsilon \langle B(v, z), v\rangle + \| R\| op
\lambda
\| v\| 2 + \| R\| op| z| | v| + (F, v),
where F = f - \varepsilon Az - \varepsilon 2B(z, z) + \varepsilon \sigma z. Therefore,
d
dt
| v| 2 + 2\beta 1\| v\| 2 \leq - 2\varepsilon \langle B(v, z), v\rangle + 2\| R\| op| z| | v| + 2(F, v), (3.3)
where \beta 1 =
\lambda - \| R\| op
\lambda
> 0. The right-hand side of (3.3) is bounded by
2| \varepsilon \langle B(v, z), v\rangle | + 2\| R\| op| z| \| v\| + 2(F, v) \leq
\leq C2a20
\beta 1
\| z\| 2| v| 2 + 2
\nu
(\| R\| 2op| z| 2 + | F | 2) + \beta 1\| v\| 2. (3.4)
From (3.3) and (3.4), we obtain
d
dt
| v| 2 +
\biggl(
\nu - \lambda C2a20
\nu
\| z\| 2
\biggr)
| v| 2 \leq 2
\nu
(\| R\| 2op| z| 2 + | F | 2),
where \nu = \lambda \beta 1 = \lambda - \| R\| op. On the other hand, using Schwarz’s inequality and noting that
0 < \varepsilon \leq 1, we have
| F | 2 = | f - \varepsilon Az - \varepsilon 2B(z, z) + \varepsilon \sigma z| 2 \leq
\leq (3 + \sigma 2)
\bigl[
C2
0\| z\| 4 + | Az| 2 + | f | 2 + | z| 2
\bigr]
.
Therefore,
| F (\theta t\omega | 2 \leq (3 + \sigma 2)[C2
0\| h\| 4| y(\theta t\omega )| 4 + | Ah| 2| y(\theta t\omega )| 2 + | f | 2 + | h| 2| y(\theta t\omega )| 2].
Hence, let \beta =
\lambda C2a20
\nu
\| h\| 2, we get
d
dt
| v| 2 + (\nu - \beta | y(\theta t\omega | 2)| v| 2 \leq c
\bigl(
| y(\theta t\omega )| 4 + | y(\theta t\omega )| 2 + 1
\bigr)
,
where
c =
2(3 + \sigma 2)
\nu | f | 2
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
C2
0\| h\| 4, (\| R\| 2op + 1)\| h\| 2D(A)
\Bigr\}
.
Multiplying (3.3) by \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\nu t - \beta
\int t
0
| y(\theta r\omega | 2dr
\biggr)
and integrating the resulting inequality on [0, s],
we obtain
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1654 CUNG THE ANH, NGUYEN TIEN DA
| v(s, \omega , v0(\omega )| 2H \leq \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \nu s+ \beta
s\int
0
| y(\theta r\omega | 2dr
\right) | v0(\omega )| 2+
+c
s\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu (\tau - s) - \beta
\tau \int
s
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau , (3.5)
where g(\theta \tau \omega ) = | y(\theta \tau \omega )| 4 + | y(\theta \tau \omega )| 2 + 1.
We now estimate the last term on the right-hand side of (3.5). To do this, in (3.5), we replace s
and \omega by t and \theta - t\omega , respectively,
| v(t, \theta - t\omega , v0(\theta - t\omega )| 2H \leq \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \nu t+ \beta
t\int
0
| y(\theta r - t\omega | 2dr
\right) | v0(\theta - t\omega )| 2H+
+c
t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu (\tau - t) - \beta
\tau \int
t
| y(\theta r - t\omega | 2dr
\right) g(\theta \tau - t\omega )dr =
= \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \nu t+ \beta
0\int
- t
| y(\theta r\omega | 2dr
\right) | v0(\theta - t\omega )| 2+
+c
0\int
- t
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau . (3.6)
Thanks to the Ergodic theorem, for any \omega \in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow +\infty
1
t
0\int
- t
| y(\theta r\omega | 2dr = E(| y(\omega )| 2).
On the other hand,
E(| y(\omega )| 2) =
\Gamma
\Bigl( 3
2
\Bigr)
\sigma
\surd
\pi
=
\surd
\pi
2\sigma
\surd
\pi
=
1
2\sigma
,
where \Gamma (\cdot ) is the Gamma function. Thus, there exists a number T1 = T1(B,\omega ) > 0 such that, for
all t \geq T1,
\beta
0\int
- t
| y(\theta r\omega | 2dr <
\beta
\sigma
t \leq \nu t
2
, (3.7)
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RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1655
where we have used the condition (2.9). By (3.6) and (3.7), we have
| v(t, \theta - t\omega , v0(\theta - t\omega )| 2H \leq e -
\nu
2
t| v0(\theta - t\omega )| 2H+
+c
0\int
- t
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau . (3.8)
Note that | y(\theta r\omega | is tempered, by using (2.10) and a few simple calculations, we can find that the
integrand of the second term on the right-hand side of (3.8) converges to zero exponentially when
t \rightarrow +\infty . Thus, the integral
R0 = c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )dr
is convergent. Moreover, since v0(\theta - t\omega ) \in B(\theta - t\omega ), for the first term on the right-hand side of
(3.8), we get
e -
\nu
2
t| v0(\theta - t\omega )| 2H \leq e -
\nu
2
t| B(\theta - t\omega )| 2H \rightarrow 0 ast \rightarrow +\infty .
This shows that there exists T2 = T2(B,\omega ) > 0 such that
e -
\nu
2
t| v0(\theta - t\omega )| 2H \leq 1 for all t \geq T2. (3.9)
From (3.8) and (3.9), we obtain
| v(t, \theta - t\omega , v0(\theta - t\omega ))| 2H \leq 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega )| 2dr
\right) g(\theta \tau \omega )d\tau
for all t \geq T = \mathrm{m}\mathrm{a}\mathrm{x} \{ T1, T2\} .
Lemma 3.1 is proved.
Lemma 3.2. The random dynamical system \Phi has a random absorbing set K = \{ K(\omega )\} \omega \in \Omega
in H, where K is independent of \varepsilon .
Proof. Let \{ B(\omega )\} \omega \in \Omega \in \scrD be fixed. For given u0(\omega ) \in H, let v be the solution of (2.11)
with the initial condition v(0) = u0(\omega ) - \varepsilon z(\omega ). Then we have
| v0(\omega )| 2H = | u0(\omega ) - \varepsilon z(\omega )| 2H \leq 2(| u0(\omega )| 2H + \varepsilon 2| z(\omega )| 2H) \leq
\leq 2(| B(\omega )| 2H + | z(\omega )| 2H).
This implies that v0(\omega ) \in \~B(\omega ) for all \omega \in \Omega , where
\~B(\omega ) =
\bigl\{
u \in H : | u| 2 \leq 2(| B(\omega )| 2H + | z(\omega )| 2H)
\bigr\}
. (3.10)
Moreover, since \{ B(\omega )\} \omega \in \Omega \in \scrD and | z(\omega )| is tempered, this follows
\Bigl\{
\~B(\omega )
\Bigr\}
\omega \in \Omega
\in \scrD . Then, by
(3.1), we obtain
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1656 CUNG THE ANH, NGUYEN TIEN DA
| v(t, \theta - t\omega , v0(\theta - t\omega ))| 2H \leq 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega )| 2dr
\right) g(\theta \tau \omega )d\tau , (3.11)
provided t \geq T and v0(\theta - t\omega ) \in \~B(\theta - t\omega ). On the other hand,
\Phi (t, \theta - t\omega , u0(\theta - t\omega )) = v(t, \theta - t\omega , v0(\theta - t\omega )) + \varepsilon z(\omega ). (3.12)
From (3.11) and (3.12), we get, for all t \geq T and u0(\theta - t\omega ) \in B(\theta - t\omega ),
| \Phi (t, \theta - t\omega , u0(\theta - t\omega ))| 2H \leq 2(r0(\omega ) + | z(\omega )| 2H),
where r0(\omega ) = 1 + c
\int 0
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\nu \tau + \beta
\int 0
\tau
| y(\theta r\omega )| 2dr
\biggr)
g(\theta \tau \omega )d\tau . This implies that \Phi possesses
a random absorbing set in H, which is independent of \varepsilon .
3.2. Pullback asymptotic compactness. In this subsection, we prove the \scrD -pullback asymptotic
compactness of solutions to problem (2.11). For this purpose, we need the following weak continuity
of solutions with respect to initial data, which can be established by the standard method as in [6, 21].
Lemma 3.3. Let \omega \in \Omega and x0 \in H. If xn \rightharpoonup x0 weakly in H, then the solution v of problem
(2.11) has the following properties:
v (t, \omega , xn) \rightharpoonup v (t, \omega , x0)weakly in H for all t \geq 0,
v (\cdot , \omega , xn) \rightharpoonup v (\cdot , \omega , x0)weakly in L2 (0, T ;V ) for all T > 0,
u (\cdot , \omega , xn - \varepsilon z(\omega )) \rightharpoonup u (\cdot , \omega , x0 - \varepsilon z(\omega ))weakly in L2 (0, T ;V ) for all T > 0.
The next lemma is concerned with the pullback asymptotic compactness of problem (2.11).
Lemma 3.4. For every \omega \in \Omega , B = \{ B(\omega )\} \omega \in \Omega \in \scrD and tn \rightarrow +\infty , xn \in B(\theta - tn\omega ), the
sequence of solutions \{ v(tn, \theta - tn\omega , xn)\} to (2.11) has a convergent subsequence in H.
Proof. Since tn \rightarrow +\infty , there exists N0 \in \BbbN such that tn \geq T for all n \geq N0. Note that
xn \in B(\theta - tn\omega ), we get from (3.1) that, for all n \geq N0,
| v(tn, \theta - tn\omega , xn)| 2H \leq 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega )| 2dr
\right) g(\theta \tau \omega )d\tau .
Hence, there exist \~v \in H and a subsequence (which is not relabeled) such that
v(tn, \theta - tn\omega , xn) \rightharpoonup \~v in H. (3.13)
We now prove that the weak convergence of (3.13) is actually a strong convergence, which will
complete the proof. Note that (3.13) implies that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
| v(tn, \theta - tn\omega , xn)| H \geq | \~v| H .
So we only need to show that
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RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1657
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| v(tn, \theta - tn\omega , xn)| H \leq | \~v| H . (3.14)
We will establish (3.14) by the method of energy equations due to Ball [5]. Given k \in \BbbN , we have
v(tn, \theta - tn\omega , xn) = v(k + tn - k, \theta - tn\omega , xn) = v(k, \theta - k\omega , v(tn - k, \theta - tn\omega , xn)). (3.15)
Since tn \rightarrow \infty and xn \in B(\theta - tn\omega ), for each k, let Nk be large enough such that tn \geq T + k for
all n \geq Nk. Then it follows from Lemma 3.1 that, for n \geq Nk,
| v(tn - k, \theta - tn\omega , xn)| 2H \leq
\leq e\nu k
\left[ e - \nu
2
tn | xn| 2H + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau
\right] \leq
\leq e\nu k
\left( 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau
\right) , (3.16)
which shows that, for each fixed k \in \BbbN , the sequence v(tn - k, \theta - tn\omega , xn) is bounded in H. By a
diagonal process, one can find a subsequence (which we do not relabel) and a point \~vk \in H for each
k \in \BbbN such that
v(tn - k, \theta - tn\omega , xn) \rightharpoonup \~vk in H. (3.17)
By (3.15) – (3.17) and Lemma 3.3, we get that, for each k \in \BbbN ,
v(tn, \theta - tn\omega , xn) \rightharpoonup v(k, \theta - k\omega , \~vk) in H, (3.18)
and
v(\cdot , \theta - tn\omega , v(tn, \theta - tn\omega , xn)) \rightharpoonup v(\cdot , \theta - k\omega , \~vk) in L2(0, k;V ).
From (3.13) and (3.18), we obtain
v(k, \theta - k\omega , \~vk) = \~v for all k > 0. (3.19)
Denote \phi (v) = 2
\nu
\lambda
\| v\| 2 - 3
2
\nu | v| 2, we have
\nu
2\lambda
\| v\| 2 \leq \phi (v) \leq 2
\nu
\lambda
\| v\| 2 for all v \in V,
this indicates that \phi (\cdot ) is an equivalent norm of V. On the other hand, (2.11) implies that
d
dt
| v| 2 + 3
2
\nu | v| 2 + \phi (v) + 2\langle B(u, u), v\rangle \leq 2( \~F , v), (3.20)
where \nu is defined in (3.2) and \~F = f - \varepsilon Az - \varepsilon Rz + \varepsilon \sigma z.
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1658 CUNG THE ANH, NGUYEN TIEN DA
Multiplying (3.20) by e\eta t with \eta =
3\nu
2
and integrating the resulting equation on (0, t), we obtain
| v (t, \omega , v0(\omega )) | 2H + 2
t\int
0
e - \eta (t - s)\langle B(u (s, \omega , u0(\omega )) , u(s, \omega , u0(\omega ))), v(s, \omega , v0(\omega ))\rangle ds =
= e - \eta t| v0(\omega )| 2H + 2
t\int
0
e - \eta (t - s)(f + \varepsilon \sigma z(\theta s\omega ) - \varepsilon Az(\theta s\omega ) - \varepsilon Rz(\theta s\omega ), v (s, \omega , v0(\omega )) ds+
+
t\int
0
e - \nu (t - s)\phi (s, \omega , v0(\omega ))ds. (3.21)
Replacing t, \omega in (3.21) by k and \theta - t\omega , respectively, and by (3.19), we find
| \~v| 2H = | v(k, \theta - k\omega , \~vk)| 2H =
= 2
k\int
0
e - \eta (k - s)\langle B(u (s, \theta - k\omega , un,k) , u(s, \theta - k\omega , un,k)), v(s, \theta - k\omega , \~vk)\rangle ds+
+2
k\int
0
e - \eta (k - s)(f + \varepsilon \sigma z(\theta s - k\omega ) - \varepsilon Az(\theta s - k\omega ) - \varepsilon Rz(\theta s - k\omega ), v (s, \theta - k\omega , \~vk))ds+
+e - \eta k| \~vk| 2H +
k\int
0
e - \eta (k - s)\phi (s, \theta - k\omega , \~vk)ds, (3.22)
where un,k = \~vk + \varepsilon z(\theta - k\omega ). Similarly, by (3.15) and (3.21), we get
| v(k, \theta - k, vn,k)| 2H =
= 2
k\int
0
e - \eta (k - s)\langle B(u (s, \theta - k\omega , un,k) , u(s, \theta - k\omega , un,k)), v(s, \theta - k\omega , vn,k)\rangle ds+
+2
k\int
0
e - \eta (k - s)(f + \varepsilon \sigma z(\theta s - k\omega ) - \varepsilon Az(\theta s - k\omega ) - \varepsilon Rz(\theta s - k\omega ), v (s, \theta - k\omega , vn,k))ds+
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RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1659
+e - \eta k| vn,k| 2H +
k\int
0
e - \eta (k - s)\phi (s, \theta - k\omega , vn,k)ds, (3.23)
where
vn,k = v(tn - k, \theta - tn\omega , xn), un,k = vn,k + \varepsilon z(\theta - k\omega ).
We now consider the limit of each term on the right-hand side of (3.23) as n \rightarrow \infty . For the first
term, by Lemma 3.2 and [6] (Corollary 5.3),
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
k\int
0
e - \eta (k - s)\langle B(u (s, \theta - k\omega , un,k) , u(s, \theta - k\omega , un,k)), v(s, \theta - k\omega , vn,k)\rangle ds =
=
k\int
0
e - \nu (k - s)\langle B(u (s, \theta - k\omega , \~uk) , u(s, \theta - k\omega , \~uk)), v(s, \theta - k\omega , \~vk)\rangle ds, (3.24)
where \~uk = \~vk + \varepsilon z(\theta - k\omega ). For the second term, note that
e - \eta (k - \cdot )(f + \varepsilon \sigma z(\theta \cdot - k\omega ) - \varepsilon Az(\theta \cdot - k\omega ) - \varepsilon Rz(\theta \cdot - k\omega )) \in L2(0, k;V \prime ), (3.25)
thus, we find
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
k\int
0
e - \eta (k - s)(f + \varepsilon \sigma z(\theta s - k\omega ) - \varepsilon Az(\theta s - k\omega ) - \varepsilon Rz(\theta s - k\omega ), v (s, \theta - k\omega , vn,k))ds =
=
k\int
0
e - \eta (k - s)(f + \varepsilon \sigma z(\theta s - k\omega ) - \varepsilon Az(\theta s - k\omega ) - \varepsilon Rz(\theta s - k\omega ), v (s, \theta - k\omega , \~vk))ds. (3.26)
Moreover,
\int k
0
e - \eta (k - s)\phi (\cdot )ds defines a norm in L2(0, k;V ) which is equivalent to the usual one,
thus, by (3.17), we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
k\int
0
e - \eta (k - s)\phi (s, \theta - k\omega , vn,k)ds =
k\int
0
e - \eta (k - s)\phi (s, \theta - k\omega , \~vk)ds. (3.27)
Finally, by (3.16) and \eta =
3\nu
2
, we get
e - \eta k| vn,k| 2H \leq e - (\eta - \nu )k
\left( 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau
\right) =
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1660 CUNG THE ANH, NGUYEN TIEN DA
= e -
\nu
2
k
\left( 1 + c
0\int
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \nu \tau + \beta
0\int
\tau
| y(\theta r\omega | 2dr
\right) g(\theta \tau \omega )d\tau
\right) . (3.28)
By (3.15) and (3.22) – (3.28), we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| v(tn, \theta - tn\omega , xn)| 2H - | \~v| 2H \leq e -
\nu
2
kr0(\omega ) - e - \nu k| \~vk| 2H \leq
\leq e -
\nu
2
kr0(\omega ) \rightarrow 0 as k \rightarrow +\infty ,
where r0(\omega ) = 1 + c
\int 0
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\nu \tau + \beta
\int 0
\tau
| y(\theta r\omega | 2dr
\biggr)
g(\theta \tau \omega )d\tau < +\infty . This implies that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| v(tn, \theta - tn\omega , xn)| 2H \leq | \~v| 2H ,
whence (3.14) follows.
Lemma 3.4 is proved.
Lemma 3.5. The RDS \Phi is pullback \scrD -asymptotically compact in H, that is, for every \omega \in \Omega ,
B = \{ B(\omega )\} \omega \in \Omega \in \scrD , and tn \rightarrow +\infty , xn \in B(\theta - tn\omega ), the sequence \Phi (tn, \theta - tn\omega , xn) has a
convergent subsequence in H.
Proof. Since B \in \scrD and xn \in B(\theta - tn\omega ), by the proof of Lemma 3.2, we find, for each n \in \BbbN ,
yn = xn - \varepsilon z(\omega ) \in \~B, where \~B \in \scrD is the family defined by (3.10). Then it follows from Lemma
3.4 that the sequence v(tn, \theta - tn\omega , yn) of solutions to problem (2.11) has a convergent subsequence
in H. On the other hand, by (2.13), we have
u(tn, \theta - tn\omega , xn) = v(tn, \theta - tn\omega , yn) + \varepsilon z(\omega ),
and hence, the sequence u(tn, \theta - tn\omega , xn) has a convergent subsequence in H. This implies that
\Phi (tn, \theta - tn\omega , xn) has a convergent subsequence in H.
Lemma 3.5 is proved.
3.3. Existence of a random attractor.
Theorem 3.1. For each \varepsilon > 0, the continuous RDS \Phi associated with problem (2.8) has a
unique \scrD -random attractor \scrA \varepsilon = \{ \scrA \varepsilon (\omega )\} \omega \in \Omega in H.
Proof. By Lemma 3.2, we know that the continuous RDS \Phi has a family of \scrD -random
absorbing sets \{ K\varepsilon (\omega )\} \omega \in \Omega in \scrD . On the other hand, by Lemma 3.5, we find that RDS \Phi is \scrD -
pullback asymptotically compact. Then it follows from Theorem 2.1 that \Phi has a unique \scrD -random
attractor \scrA \varepsilon in H and the structure of \scrA \varepsilon = \{ \scrA \varepsilon (\omega )\} \omega \in \Omega is given by
\scrA \varepsilon (\omega ) =
\bigcap
\tau \geq 0
\bigcup
t\geq \tau
\Phi (t, \theta - t\omega ,K\varepsilon (\theta - t\omega )).
4. Upper semicontinuity of the random attractor. In this section, we prove the upper semi-
continuity of random attractors for the 2D hydrodynamical type systems when the stochastic pertur-
bations approach zero. The existence of the global attractor \scrA 0 for the (deterministic) dynamical
system associated to (1.1) when \varepsilon = 0 has been proved in [11] (Section 4.5). To prove the upper
semicontinuity result, we first establish the convergence of solutions to problem (2.11) when \varepsilon \rightarrow 0,
and then show that the union of all perturbed random attractors is precompact in H.
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RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1661
Lemma 4.1. For every t \in \BbbR and \omega \in \Omega , if \varepsilon n \rightarrow 0 and xn \rightarrow x0 in H, then
v\varepsilon n(t, \omega , xn) \rightarrow v0(t, x0).
Proof. Let v0(\cdot ) = v0(\cdot , x0), for every n \in \BbbN , we denote vn(\cdot ) = v\varepsilon n(\cdot , \omega , xn) a solution of
equation (2.11) with \varepsilon is replaced by \varepsilon n. Denote wn = vn - v0, we obtain
\partial twn +Awn +Rwn +B(vn + \varepsilon nz, vn + \varepsilon nz) - B(v0, v0) = - \varepsilon nAz - \varepsilon nRz + \varepsilon n\sigma z,
with wn(0) = xn - x0. Hence, we have
1
2
d
dt
| wn| 2 + \| wn\| 2 = - (Rwn, wn) - \langle B(vn + \varepsilon nz, vn + \varepsilon nz) - B(v0, v0), wn\rangle -
- (\varepsilon nAz + \varepsilon n\sigma z,wn) + (\varepsilon nRz,wn) \leq
\leq \| R\| op
\lambda
\| wn\| 2 - \langle B(vn + \varepsilon nz, vn + \varepsilon nz) - B(v0, v0), wn\rangle -
- (\varepsilon nAz + \varepsilon n\sigma z,wn) + (\varepsilon nRz,wn),
thus,
d
dt
| wn| 2 +
2\nu
\lambda
\| wn\| 2 \leq - 2\langle B(vn + \varepsilon nz, vn + \varepsilon nz) - B(v0, v0), wn\rangle +
+2( - \varepsilon nAz + \varepsilon n\sigma z,wn) + 2( - \varepsilon nRz,wn). (4.1)
Now we will estimate the terms on the right-hand side of (4.1). For the first term, we get
| - 2\langle B(vn + \varepsilon nz, vn + \varepsilon nz) - B(v0, v0), wn\rangle | \leq
\leq 2| \langle B(wn, wn), v0\rangle | + \varepsilon n| \langle B(vn, z), wn\rangle | + \varepsilon n| \langle B(z, vn), wn\rangle | + \varepsilon 2n| \langle B(z, z), wn\rangle | , (4.2)
and by (2.2) – (2.5), we obtain
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1662 CUNG THE ANH, NGUYEN TIEN DA
2| \langle B(wn, wn), v0\rangle | \leq 2Ca0| wn| \| v0\| \| wn\| \leq
\leq 4\lambda C2a20
\nu
| wn| 2\| v0\| 2 +
\nu
4\lambda
\| wn\| 2,
2\varepsilon n| \langle B(vn, z), wn\rangle | \leq 2\varepsilon nC0\| vn\| \| z\| \| wn\| \leq
\leq 4\lambda C2
0
\nu
\varepsilon 2n\| z\| 2\| vn\| 2 +
\nu
4\lambda
\| wn\| 2,
2\varepsilon n| \langle B(z, vn), wn\rangle | \leq 2C0\varepsilon n\| vn\| \| z\| \| wn\| \leq
\leq 4\lambda C2
0
\nu
\varepsilon 2n\| z\| 2\| vn\| 2 +
\nu
4\lambda
\| wn\| 2,
2\varepsilon 2n| \langle B(z, z), wn\rangle | \leq 2C0\varepsilon
2
n\| z\| 2\| wn\| \leq
\leq 4\lambda C2
0
\nu
\varepsilon 4n\| z\| 4 +
\nu
4\lambda
\| wn\| 2.
(4.3)
For the second term, after a few simple computations, we get
2| (\varepsilon nAz + \varepsilon n\sigma z,wn)| \leq
2
\nu
(1 + \sigma 2)\varepsilon 2n(| Az| 2 + | z| 2) + \nu
2\lambda
\| wn\| 2 =
= c1\varepsilon
2
n(| Az| 2 + | z| 2) + \nu
2\lambda
\| wn\| 2, (4.4)
where c1 =
2
\nu
(1 + \sigma 2). Moreover,
2| ( - \varepsilon nRz,wn)| = 2\varepsilon n| (Rz,wn)| \leq
2
\nu
\varepsilon 2n\| R\| 2op| z| 2 +
\nu
2\lambda
\| wn\| 2 \leq
\leq c2\varepsilon
2
n| z| 2 +
\nu
2\lambda
\| wn\| 2, (4.5)
where c2 =
2\| R\| 2op
\nu
. From (4.1) – (4.5), we have
d
dt
| wn| 2 \leq c3\rho
2| wn| 2 +
\Bigl(
c4\rho
2 + (c1 + c2)\| h\| 2D(A)
\Bigr)
\varepsilon 2n| y(\theta t\omega )| 2 + c5\varepsilon
4
n| y(\theta t\omega )| 4,
where
c3 =
4\lambda C2a20
\nu
\rho 2,
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RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1663
c4 =
8\lambda C2
0
\nu
\| h\| 2,
c5 =
4\lambda C2
0
\nu
\| h\| 4,
\rho = \rho (t) = \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,t]
\| vn(s)\| .
By the Gronwall lemma, we deduce that
| wn(t)| 2H \leq (| wn(0)| 2H + \varepsilon 2nr(t))e
c3\rho 2t \rightarrow 0 as n \rightarrow \infty ,
where
r(t) =
t\int
0
\Bigl[
(c4\rho
2 + (c1 + c2)\| h\| 2D(A))| y(\theta s\omega )|
2 + c5| y(\theta s\omega )| 4
\Bigr]
ds < +\infty .
This implies that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty | wn(t)| H = 0.
Lemma 4.1 is proved.
The above lemma enables us to obtain the second condition in Theorem 2.2.
Lemma 4.2. For P -a.e. \omega \in \Omega ,
\Phi \varepsilon n (t, \omega , xn) \rightarrow \Phi (t)x for all t \geq 0, provided \varepsilon n \rightarrow 0 and xn \rightarrow x in H.
Proof. By (2.13), we have
\Phi \varepsilon n(t, \omega , xn) = v\varepsilon n(t, \omega , xn - \varepsilon nz(\omega )) + \varepsilon nz(\theta t\omega ).
Note that xn \rightarrow x0 in H and \varepsilon n \rightarrow 0, we find
| xn - \varepsilon nz(\omega ) - x0| H \leq | xn - x0| H + \varepsilon n| z(\omega )| H \rightarrow 0.
Therefore, by Lemma 4.1, we obtain
| v\varepsilon n(t, \omega , xn - \varepsilon nz(\omega )) - v0(t, x0)| H \rightarrow 0 as n \rightarrow \infty .
But \Phi (t, x0) = v0(t, x0), then
| \Phi \varepsilon n(t, \omega , xn) - \Phi (t, x0)| H = | v\varepsilon n(t, \omega , xn - \varepsilon nz(\omega )) + \varepsilon nz(\theta t\omega ) - v0(t, x0)| H \leq
\leq | v\varepsilon n(t, \omega , xn - \varepsilon nz(\omega )) - v0(t, x0)| H + \varepsilon n| z(\theta t\omega )| H \rightarrow 0.
Lemma 4.2 is proved.
The proof of the following lemma is similar to that given in [6], so we only state the result.
Lemma 4.3. Let \{ B(\omega )\} \omega \in \Omega \in \scrD . Suppose that \varepsilon n \rightarrow \varepsilon 0, tn \rightarrow +\infty and yn \in B(\theta - tn\omega ), then
\{ \Phi \varepsilon n(tn, \theta - tn\omega , yn)\}
is precompact in H.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1664 CUNG THE ANH, NGUYEN TIEN DA
Now, given 0 < \varepsilon \leq 1, it follows from Lemma 3.2 that, for every \{ B(\omega )\} \omega \in \Omega \in \scrD and P -a.e.
\omega \in \Omega , there exists T = T (B,\omega ) > 0, independent of \varepsilon , such that, for all t \geq T,
| \Phi \varepsilon (t, \theta - t\omega , u0(\theta - t\omega ))| 2H = | v(t, \theta - t\omega , v0(\theta - t\omega )) + \varepsilon z(\omega )| 2H \leq
\leq 2(| v(t, \theta - t\omega , v0(\theta - t\omega ))| 2H + \varepsilon 2| z(\omega )| 2H) \leq
\leq 2(r0(\omega ) + \varepsilon 2| z(\omega )| 2H),
where r0(\omega ) = 1 + c
\int 0
- \infty
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\nu \tau + \beta
\int 0
\tau
| y(\theta r\omega )| 2dr
\biggr)
g(\theta \tau \omega )d\tau . Denote
K\varepsilon (\omega ) =
\bigl\{
u \in H : | u| 2 \leq 2(r0(\omega ) + \varepsilon 2| z(\omega )| 2H)
\bigr\}
(4.6)
and
K(\omega ) =
\bigl\{
u \in H : | u| 2 \leq 2(r0(\omega ) + | z(\omega )| 2H)
\bigr\}
.
Then, for every 0 < \varepsilon \leq 1, \{ K\varepsilon (\omega )\} \omega \in \Omega \in \scrD is a closed absorbing set for \Phi \varepsilon in \scrD and\bigcup
0<\varepsilon \leq 1
K\varepsilon (\omega ) \subset K(\omega ). (4.7)
It follows from the invariance of the random attractor \{ \scrA \varepsilon (\omega )\} \omega \in \Omega \in \scrD and (4.7) that\bigcup
0<\varepsilon \leq 1
\scrA \varepsilon (\omega ) \subset
\bigcup
0<\varepsilon \leq 1
K\varepsilon (\omega ) \subset K(\omega ). (4.8)
Lemma 4.4. For every \omega \in \Omega , the union
\bigcup
0<\varepsilon \leq 1\scrA \varepsilon (\omega ) is precompact in H.
Proof. First, we take an sequence \{ xn\} \subset
\bigcup
\varepsilon \in (0,1]\scrA \varepsilon (\omega ), then there exists \varepsilon 0 such that \scrA \varepsilon 0
contains infinitely many elements of xn. From the compactness of \scrA \varepsilon 0 , we find that \{ xn\} has a
convergent subsequence. On the other hand, we can assume that xn \in \scrA \varepsilon n(\omega ) with \varepsilon n \in (0, 1]
and \varepsilon n \not = \varepsilon m when m \not = n. Due to \{ \varepsilon n\} \subset (0, 1], without loss of generality, we can assume that
\varepsilon n \rightarrow \varepsilon 0 \in [0, 1] as n \rightarrow +\infty . Fix a sequence tn such that tn \rightarrow +\infty . By the invariance of \scrA \varepsilon n ,
we find that
\scrA \varepsilon n(\omega ) = \Phi \varepsilon n (tn, \theta - tn\omega ,\scrA \varepsilon n (\theta - tn\omega )) .
Since xn \in \Phi \varepsilon n (tn, \theta - tn\omega ,\scrA \varepsilon n (\theta - tn\omega )) , there exists an element yn \in \scrA \varepsilon n (\theta - tn\omega ) such that
xn = \Phi \varepsilon n (tn, \theta - tn\omega , yn) . Note that by (4.8), we have \varepsilon n \rightarrow \varepsilon 0, tn \rightarrow +\infty , and yn \in K(\theta - tn\omega ),
thus applying Lemma 4.3, we obtain \{ \Phi \varepsilon n (tn, \theta - tnw, yn)\} is precompact in H, i.e., \{ xn\} has a
convergent subsequence.
Theorem 4.1. For P -a.e. \omega \in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0
\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (\scrA \varepsilon (\omega ),\scrA 0) = 0. (4.9)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
RANDOM ATTRACTORS FOR STOCHASTIC 2D HYDRODYNAMICAL TYPE SYSTEMS 1665
Proof. Recall that \{ K\varepsilon (\omega )\} \omega \in \Omega is a closed absorbing set for \Phi \varepsilon in \scrD , where K\varepsilon (\omega ) is given
by (4.6). By (4.6) we find that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
| K\varepsilon (\omega )| \leq
\sqrt{}
2r0(\omega ) := M. (4.10)
Let \varepsilon n \rightarrow 0 and xn \rightarrow x0 in H, then by Lemma 4.1 we find that, for P -a.e. \omega \in \Omega and t \geq 0,
\Phi \varepsilon n(t, \omega , xn) \rightarrow \Phi (t, x0). (4.11)
Note that (4.10), (4.11) and Lemma 4.2 indicate all conditions in Theorem 2.2 are satisfied, and
hence, (4.9) follows.
Remark 4.1. As a direct consequence of the abstract results obtained in the paper, we get the
existence and upper semicontinuity of random attractors for many 2D partial differential equations
in fluid mechanics with additive noise in bounded domains or unbounded domains satisfying the
Poincaré inequality, including 2D Navier – Stokes equations, 2D MHD equations, 2D Boussinesq
equations, 2D magnetic Bénard equations, and also some 3D models such as 3D Leray-\alpha model, the
shell models of turbulence. To do this, it only need to verify the abstract conditions for each concrete
model (see [11] (Section 4.6) or [12] for details).
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Received 02.11.16,
after revision — 22.01.19
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
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| id | umjimathkievua-article-361 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:29Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b8/a3312507574cc1779f4167dfb518edb8.pdf |
| spelling | umjimathkievua-article-3612020-05-07T13:35:14Z Random attractors for stochastic 2D hydrodynamical type systems Випадковi атрактори для стохастичних двовимiрних систем гiдродинамiчного типу Випадковi атрактори для стохастичних двовимiрних систем гiдродинамiчного типу Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da Випадкові атрактори; стохастичні 2D системи гідродинамічного типу; верхня напівперервність; Аргументи рівняння енергії Random attractors; stochastic 2D hydrodynamical type systems; upper semicontinuity; energy equation arguments We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray $\alpha$-model. We prove the existence of random attractors for the associated continuous random dynamical systems. Then we establish the upper semicontinuity of the random attractors as the parameter tends to zero. Вивчається асимптотична поведінка розв'язків одного класу абстрактних нелінійних стохастичних рівнянь еволюції з адитивним шумом, що включає різноманітні двовимірні гідродинамічні моделі, такі як двовимірні рівняння Нав'є–Стокса, двовимірні рівняння Буссінеска, двовимірні рівняння магнітогідродинаміки тощо, а також деякі тривимірні моделі типу тривимірної $\alpha$-моделі Лерея.Доведено існування випадкових атракторів для відповідних неперервних випадкових динамічних систем.Крім того, встановлено напівнеперервність зверху випадкових атракторів у випадку, коли параметр прямує до нуля. Вивчається асимптотична поведінка розв'язків одного класу абстрактних нелінійних стохастичних рівнянь еволюції з адитивним шумом, що включає різноманітні двовимірні гідродинамічні моделі, такі як двовимірні рівняння Нав'є–Стокса, двовимірні рівняння Буссінеска, двовимірні рівняння магнітогідродинаміки тощо, а також деякі тривимірні моделі типу тривимірної $\alpha$-моделі Лерея.Доведено існування випадкових атракторів для відповідних неперервних випадкових динамічних систем.Крім того, встановлено напівнеперервність зверху випадкових атракторів у випадку, коли параметр прямує до нуля. Institute of Mathematics, NAS of Ukraine 2019-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/361 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 12 (2019); 1647-1666 Український математичний журнал; Том 71 № 12 (2019); 1647-1666 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/361/1524 |
| spellingShingle | Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da Cung, The Anh Nguyen, Tien Da Random attractors for stochastic 2D hydrodynamical type systems |
| title | Random attractors for stochastic 2D hydrodynamical type systems |
| title_alt | Випадковi атрактори для стохастичних двовимiрних систем гiдродинамiчного типу Випадковi атрактори для стохастичних двовимiрних систем гiдродинамiчного типу |
| title_full | Random attractors for stochastic 2D hydrodynamical type systems |
| title_fullStr | Random attractors for stochastic 2D hydrodynamical type systems |
| title_full_unstemmed | Random attractors for stochastic 2D hydrodynamical type systems |
| title_short | Random attractors for stochastic 2D hydrodynamical type systems |
| title_sort | random attractors for stochastic 2d hydrodynamical type systems |
| topic_facet | Випадкові атрактори стохастичні 2D системи гідродинамічного типу верхня напівперервність Аргументи рівняння енергії Random attractors stochastic 2D hydrodynamical type systems upper semicontinuity energy equation arguments |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/361 |
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