PULLBACK ATTRACTORS IN Vg FOR NONAUTONOMOUS 2D g NAVIERSTOKES EQUATIONS IN SOME UNBOUNDED DOMAINS

PULLBACK ATTRACTORS IN Vg FOR NON-AUTONOMOUS 2D
g-NAVIER-STOKES EQUATIONS IN SOME UNBOUNDED
DOMAINS
CUNG THE ANH AND DAO TRONG QUYET

Abstract. We study the first initial boundary value problem for the nonautonomous 2D g-Navier-Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincar´
e inequality. We show the existence
of a pullback attractor for the process generated by strong solutions to the
problem with respect to a large class of non-autonomous forcing terms. To
overcome the difficulty caused by the unboundedness of the domain, the proof
is based on a pullback asymptotic compactness argument and the use of the
enstrophy equation.

1. Introduction
Let Ω be a (bounded or unbounded) domain in R2 with smooth boundary ∂Ω.
In this paper we study the long-time behavior of strong solutions to the following
non-autonomous 2D g-Navier-Stokes equations

ut − ν∆u + (u · ∇)u + ∇p = f, x ∈ Ω, t > τ,




∇ · (gu)
= 0, x ∈ Ω, t > τ,
(1.1)

u(x, t)
= 0, x ∈ ∂Ω, t > τ,




u(x, τ )
= u0 (x), x ∈ Ω,
where u = u(x, t) = (u1 , u2 ) is the unknown velocity vector, p = p(x, t) is the
unknown pressure, ν > 0 is the kinematic viscosity coefficient, f = f (x, t) is a
given force field and u0 is the initial velocity.
The g-Navier-Stokes equations is a variation of the standard Navier-Stokes equations and arises in a natural way when we study the standard 3D problem in the
thin domain Ωg = Ω × (0, g). We refer the reader to [16] for a derivation of the 2D
g-Navier-Stokes equations from the 3D Navier-Stokes equations and a relationship
between them. As mentioned in [16], good properties of the 2D g-Navier-Stokes
equations can initiate the study of the Navier-Stokes equations on the thin three
dimensional domain Ωg . In the last few years, the existence and asymptotic behavior of solutions to g-Navier-Stokes equations have been studied extensively (cf.
[1, 2, 4, 11, 12, 13, 14, 16]). Very recently, the existence and exponential growth of
pullback attractors for strong solutions to 2D g-Navier-Stokes equations in bounded
domains were proved in [15].
The aim of this paper is to continue studying the long-time behavior of strong
solutions to the non-autonomous 2D g-Navier-Stokes equations in domains that
are not necessarily bounded. To do this, we also use the theory of pullback attractors that has been developed recently and has shown to be very useful in the
understanding of the dynamics of non-autonomous dynamical systems (see e.g. the
2010 Mathematics Subject Classification. 35B41; 35Q30; 35D35.
Key words and phrases. g-Navier-Stokes equations; unbounded domain; strong solution; pullback attractor; pullback asymptotic compactness argument; enstrophy equation.
Corresponding author:
1


2D g-NAVIER-STOKES EQUATIONS

2

monograph [8]). The results obtained, in particular, extended the corresponding

results in bounded domains for the 2D Navier-Stokes equations [9] and for 2D gNavier-Stokes equations [15].
In order to study problem (1.1), we assume that the function g satisfies the
following hypothesis:
(G) g ∈ W 1,∞ (Ω) such that
1/2

0 < m0 ≤g(x)≤M0 for all x = (x1 , x2 ) ∈ Ω, and |∇g|∞ < m0 λ1 ,
where λ1 > 0 is the constant in the Poincar´e inequality (1.2) below.
We also assume that the domain Ω satisfies the Poincar´e inequality:
φ2 gdx ≤

1
λ1

|∇φ|2 gdx,

for all φ ∈ C0∞ (Ω).

(1.2)

Ω

Ω

Because the considered domain is unbounded, the compactness of the embeddings which plays an essential role when proving the existence of pullback attractors in [9, 15] is no longer valid here. To overcome this difficulty, we exploit the
asymptotic compactness argument introduced the first time by Ball [5] to prove
the pullback asymptotic compactness of the process, and as a consequence, we get
the existence of a pullback attractor. Such an approach has been used recently to
prove the existence of pullback attractors for some non-autonomous equations in
fluid mechanics in unbounded domains, such as the 2D Navier-Stokes equations [6],

the 2D g-Navier-Stokes equations [1], and the 3D Navier-Stokes-Voigt equations
[3]. In these works, because weak solutions are considered, the pullback asymptotic
compactness argument are applied for the energy equations. Here because we consider strong solutions, we apply the argument to the enstrophy equation instead of
the energy equation. Note that the nonlinear term of the g-Navier-Stokes equations
dissapears in the energy equation due to its antisymmetry, while the corresponding
term does not dissappear in the enstrophy equation. This introduce some new difficulty. In this paper, we deal with this difficulty by a careful analysis using some
ideas in [10].
The structure of the paper is as follows. In Section 2, for convenience of the
reader, we recall some results on functions spaces and inequalities related to gNavier-Stokes equations and abstract theory of pullback attractors. In Section 3,
we prove the existence of a pullback attractor for the process generated by strong
solutions to problem (1.1).
2. Preliminaries
2.1. Function spaces and inequalities for the nonlinear terms. Let L2 (Ω, g) =
(L2 (Ω))2 and H01 (Ω, g) = (H01 (Ω))2 be endowed, respectively, with the inner products
u · vgdx, u, v ∈ L2 (Ω, g),

(u, v)g =
Ω

and
2

∇uj · ∇vj gdx, u = (u1 , u2 ), v = (v1 , v2 ) ∈ H01 (Ω, g),

((u, v))g =
Ω j=1

and norms |u|2 = (u, u)g , ||u||2 = ((u, u))g . Thanks to assumption (G), the norms
|.| and ||.|| are equivalent to the usual ones in (L2 (Ω))2 and in (H01 (Ω))2 .
Let

V = {u ∈ (C0∞ (Ω))2 : ∇ · (gu) = 0}.


2D g-NAVIER-STOKES EQUATIONS

3

Denote by Hg the closure of V in L2 (Ω, g), and by Vg the closure of V in H01 (Ω, g).
It follows that Vg ⊂ Hg ≡ Hg ⊂ Vg , where the injections are dense and continuous.
We will use ||.||∗ for the norm in Vg , and ., . for duality pairing between Vg and
Vg .
We now define the trilinear form b by
2

b(u, v, w) =

ui
i,j=1

Ω

∂vj
wj gdx,
∂xi

whenever the integrals make sense. It is easy to check that if u, v, w ∈ Vg , then
b(u, v, w) = −b(u, w, v), and b(u, v, v) = 0.
Set A : Vg → Vg by Au, v = ((u, v))g , B : Vg × Vg → Vg by B(u, v), w =
b(u, v, w), and put Bu = B(u, u). Denote D(A) = {u ∈ Vg : Au ∈ Hg }, then
D(A) = H 2 (Ω, g) ∩ Vg and Au = −Pg ∆u, ∀u ∈ D(A), where Pg is the orthoprojector from L2 (Ω, g) onto Hg .

We have the following results.
Lemma 2.1. [1] If n = 2, then

c1 |u|1/2 u 1/2 v |w|1/2 w 1/2 , ∀u, v, w ∈ Vg ,



c |u|1/2 u 1/2 v 1/2 |Av|1/2 |w|, ∀u ∈ V , v ∈ D(A), w ∈ H ,
2
g
g
|b(u, v, w)| ≤
1/2
1/2

c
|u|
|Au|
v
|w|,
∀u
∈
D(A),
v
∈
V
,
w
∈
H

,
3
g
g



c4 |u| v |w|1/2 |Aw|1/2 , ∀u ∈ Hg , v ∈ Vg , w ∈ D(A),
where ci , i = 1, . . . , 4, are appropriate constants.
Lemma 2.2. [2] Let u ∈ L2 (0, T ; D(A)) ∩ L∞ (0, T ; Vg ), then the function Bu
defined by
(Bu(t), v)g = b(u(t), u(t), v), ∀v ∈ Hg , a.e. t ∈ [0, T ],
belongs to L4 (0, T ; Hg ), therefore also belongs to L2 (0, T ; Hg ).
Lemma 2.3. [4] Let u ∈ L2 (0, T ; Vg ), then the function Cu defined by
(Cu(t), v)g = ((

∇g
∇g
.∇)u, v)g = b(
, u, v), ∀v ∈ Vg ,
g
g

belongs to L2 (0, T ; Hg ), and therefore also belongs to L2 (0, T ; Vg ). Moreover,
|Cu(t)| ≤

|∇g|∞
. u(t) , for a.e. t ∈ (0, T ),
m0

and
||Cu(t)||∗ ≤

|∇g|∞
1/2

.||u(t)||, for a.e. t ∈ (0, T ).

m0 λ1

Since
1
∇g
− (∇.g∇)u = −∆u − (
.∇)u,
g
g
we have
(−∆u, v)g = ((u, v))g + ((

∇g
∇g
.∇)u, v)g = (Au, v)g + b(
, u, v), ∀u, v ∈ Vg .
g
g


2D g-NAVIER-STOKES EQUATIONS

4

2.2. Pullback attractors. Let X be a Banach space. Denote by B(X) the set of
all bounded subsets of X and . is the corresponding norm. For A, B ⊂ X, the
Hausdorff semi-distance between A and B is defined by
dist(A, B) = sup inf x − y .
x∈A y∈B

Let {U (t, τ ) : t ≥ τ, τ ∈ R} be a process in X, i.e., a two-parameter family of
mappings U (t, τ ) : X → X such that U (τ, τ ) = Id and U (t, s)U (s, τ ) = U (t, τ ) for
all t ≥ s ≥ τ, τ ∈ R.
Definition 2.1. The process {U (t, τ )} is said to be pullback asymptotically compact
if for any t ∈ R, any D ∈ B(X), any sequence τn → −∞, and any sequence xn ∈ D,
the sequence {U (t, τn )xn } is relatively compact in X.
Definition 2.2. A family of bounded sets Bˆ = {B(t) : t ∈ R} is called pullback
absorbing for the process {U (t, τ )} if for any t ∈ R, any D ∈ B(X), there exists
τ0 = τ0 (D, t) ≤ t such that
U (t, τ )D ⊂ B(t).
τ ≤τ0

Definition 2.3. A family Aˆ = {A(t) : t ∈ R} ⊂ B(X) is said to be a pullback
attractor for {U (t, τ )} if
(1) A(t) is compact for all t ∈ R;
(2) Aˆ is invariant, i.e.,
U (t, τ )A(τ ) = A(t), for all t ≥ τ ;
(3) Aˆ is pullback attracting, i.e.,
lim dist(U (t, τ )D, A(t)) = 0, for all D ∈ B(X), and all t ∈ R;

τ →−∞

(4) if {C(t) : t ∈ R} is another family of closed attracting sets then A(t) ⊂ C(t),
for all t ∈ R.
Theorem 2.1. [6, 7] Let {U (t, τ )} be a continuous process such that {U (t, τ )} is
pullback asymptotically compact. If there exists a family of pullback absorbing sets
Bˆ = {B(t) : t ∈ R}, then {U (t, τ )} has a unique pullback attractor Aˆ = {A(t) : t ∈
R} and
A(t) =

U (t, τ )B(τ ).
s≤t τ ≤s

3. Existence of a pullback attractor
We first recall the definition of strong solutions to problem (1.1).
Definition 3.1. A function u is called a strong solution to problem (1.1) on the
interval (τ, T ) if

2
2

 u ∈ C([τ, T ]; Vg ) ∩ L (τ, T ; D(A)), du/dt ∈ L (τ, T ; Hg ),
d
u(t) + νAu(t) + νCu(t) + B(u(t), u(t)) = f (t) in Hg , for a.e. t ∈ (τ, T ),

dt

u(τ ) = u0 .
Theorem 3.1. For any T > τ , u0 ∈ Vg , and f ∈ L2 (τ, T ; Hg ) given, problem (1.1)
has a unique strong solution u on (τ, T ). Moreover, the strong solution depends
continuously on the initial data in Vg .

2D g-NAVIER-STOKES EQUATIONS

5

Proof. The proof is standard and similar to the case of bounded domains (see [2]),
except some difficulty arising due to the unboundedness of the domain which can
be overcome by using techniques as for Navier-Stokes equations [17], so we omit
it here. However, in what follows we will recall some a priori estimates of strong
solutions which will be used later.
First, we have
d
|u(s)|2 + 2ν||u(s)||2 = 2 f (s), u(s) − 2ν(Cu(s), u(s))g .
ds
Using Lemma 2.3 and Cauchy’s inequality, we have
f 2∗
|∇g|∞
d
|u(s)|2 + 2ν||u(s)||2 ≤ 2 ν||u(s)||2 +
+ 2ν
||u(s)||2 ,
1/2
ds
2 ν
m0 λ
1

and hence

f 2∗

d
(3.1)
|u(s)|2 + 2ν(γ0 − )||u(s)||2 ≤
,
ds
2 ν
∞
> 0 and > 0 is chosen such that γ0 − > 0. Integrating
where γ0 = 1 − |∇g|1/2
m0 λ1

from τ to t, and applying Cauchy’s inequality we get
t

|u(t)|2 + 2ν(γ0 − )

u(s) 2 ds ≤ |u0 |2 +
τ

1
f
2 ν

2
L2 (τ,T ;Vg ) .

This implies that u is bounded in L∞ (τ, T ; Hg ) ∩ L2 (τ, T ; Vg ). Hence, it is easy to
check that Au and Bu are bounded in L2 (τ, T ; Vg ).
On the other hand, we have
1 d

u(t) 2 + ν|Au(t)|2 + ν(Cu(t), Au(t))g + b(u, u, Au) = f, Au .
2 dt
By Lemmas 2.1 and 2.3, we have
1 d
||u(t)||2 + ν|Au(t)|2
2 dt
ν
1
ν|∇g|∞
≤ |Au(t)|2 + |f (t)|2 + c3 |u(t)|1/2 |Au(t)|3/2 ||u(t)|| +
||u(t)|||Au(t)|.
4
ν
m0
Using Young’s inequality, we obtain
1 d
ν
1
||u(t)||2 + ν|Au(t)|2 ≤ |Au(t)|2 + |f (t)|2
2 dt
4
ν
ν
+ |Au(t)|2 + c3 |u(t)|2 |||u(t)||4
4
1/2
ν|∇g|∞ λ1
ν|∇g|∞
2
|Au(t)|

+
+
||u(t)||2 .
1/2
2m0
2m0 λ
1

Then, we have
|∇g|∞
d
)|Au(t)|2
||u(t)||2 + ν(1 −
1/2
dt
m0 λ1

(3.2)

1/2

2
ν|∇g|∞ λ1
≤ |f (t)|2 + 2c3 |u(t)|2 |||u(t)||4 +
||u(t)||2 .
ν
m0
From (3.1) we have
t+1

|u(t + 1)|2 + 2ν(γ0 − )

u 2 ds ≤ |u(t)|2 +
t

1
2ν

f

L2 (t,t+1;Vg ) .

It implies that
t+1

t+1

|u(s)|2 u(s) 2 ds ≤ u
t

2
L∞ (τ,T ;Hg )

u(s) 2 ds < +∞
t

for all t ≥ τ.


2D g-NAVIER-STOKES EQUATIONS

6

So, we can applying the uniform Gronwall inequality to obtain
u(t)

2

≤C

for all t ≥ τ + 1.

(3.3)

This implies that u is bounded in L∞ (τ, T ; Vg ).
Integrating (3.2) from τ to t we get
t
2

u(t)

|Au(s)|2 ds

+ νγ0
τ

2

≤ u0

+

t

2
ν

t

|f (s)|2 ds + 2c3 |u|2L∞ (τ,T ;Hg ) u

τ
1/2
ν|∇g|∞ λ1

+

2
L∞ (τ,T ;Vg )

u(s) 2 ds
τ

t

||u(t)||2 .

m0

τ

This implies that u is bounded in L2 (τ, T ; D(A)). And then, we also have that Bu
is bounded in L2 (τ, T ; Hg ).
du
Now, we prove the boundedness of
. We have
dt
2

t

du
ds + ν
ds

τ

t

∇u
τ

Ω

t

∂∇u
dx ds + ν
∂s

t

(Cu, us )g ds +

b(u, u, us )ds

τ

τ
t

=

f, us ds.
τ

Using Cauchy’s and Ladyzhenskaya’s inequalities, we have
2

t

t

du
ν
ds +
ds
2

τ

τ
t

t

f, us ds −

=

t

b(u, u, us ) ds − ν

τ

≤ f

d
u 2 ds
ds

τ
L2 (τ,t;Hg ) .

us

L2 (τ,t;Hg )

t

|u|L4 |∇u|L4 |us |ds + ν

+
τ

≤ f

2
L2 (τ,T ;Hg )

+

1
us
4

t

|∇g|∞
1/2

m0 λ1

|u|1/2 |∇u||Au|1/2 |us |ds + ν
τ

≤ f

2

L2 (τ,T ;Hg )

+

1
us
4

1/2

|u||∇u|2 |Au|ds

||u||||us ||ds
τ

t

1/2

+ν

+

1
us
4

2
L2 (τ,t;Hg )

t

|u||∇u|2 |Au|ds +
τ

1
us
8

1
us
8

+

t

|∇g|∞
1/2

m0 λ1

τ

2
L2 (τ,T ;Hg )

+c

1/2

m0 λ1

|us |2 ds

·

τ

≤ f

t

|∇g|∞

2
L2 (τ,t;Hg )

t

+c

||u||||us ||ds
τ

2
L2 (τ,t;Hg )

t

+c

(Cu, us )g ds
τ

||u||||us ||ds
τ

2
L2 (τ,T ;Hg )
t

2
L2 (τ,T ;Hg )

||u(s)||2 ds.

+c
τ

Hence
t
τ

2

du
ds + ν|∇u(t)|2 ≤ 2 f
ds

2
L2 (τ,T ;Hg )

+ ν|∇u0 |2

t

t

|u||∇u|2 |Au|ds + c

+c
τ

||u(s)||2 ds,
τ


2D g-NAVIER-STOKES EQUATIONS

7

for all τ ≤ t ≤ T . Since u is bounded in L∞ (τ, T ; Vg ) ∩ L2 (τ, T ; D(A)),
bounded in L2 (τ, T ; Hg ).

du
is
dt

From now on we assume that f ∈ L2b (R, Hg ), i.e. f ∈ L2loc (R, Hg ) and satisfies

t+1
2
L2b

f

|f (s)|2 ds < +∞.

:= sup
t∈R

t

Thanks to Theorem 3.1, we can define a continuous process U (t, τ ) : Vg → Vg by
U (t, τ )u0 = u(t; τ, u0 ), τ ≤ t, u0 ∈ Vg ,
where u(t) = u(t; τ, u0 ) is the unique strong solution to problem (1.1) with the
initial datum u(τ ) = u0 .
We denote σ = λ1 νγ0 with λ1 is the constant in the Poincar´e inequality (1.2),
and introduce a new Hilbert norm in D(A) as follows
σ
u 2,
[[u]]2 : = νγ0 |Au|2 −
2
which is equivalent to the usual norm |Au| in D(A).
We now prove the weak continuity of the process U (t, τ ).
Lemma 3.1. Let {u0n } ⊂ Vg be a sequence converging weakly in Vg to an element
u0 in Vg . Then
U (t, τ )u0n
U (t, τ )u0n

U (t, τ )u0

weakly in Vg

for all τ ≤ t,

2

U (t, τ )u0

weakly in L (τ, T ; D(A))

(3.4)

for all τ ≤ t.

(3.5)

Proof. Let un (t) = U (t, τ, u0n ), u(t) = U (t, τ, u0 ). As in the proof of Theorem 3.1
we have, for all T ≥ τ ,
{un } is bounded in L∞ (τ, T ; Vg ) ∩ L2 (τ, T ; D(A)),

(3.6)

and
{un } is bounded in L2 (τ, T ; Hg ).
Then, for all v ∈ D(A),
t+a

((un (t + a) − un (t), v)) =

un (s), v ds

(3.7)

t
1/2

≤ v

D(A) a

un

L2 (τ,T,Hg )

≤ CT v

D(A) a

1/2

,

where CT is positive and independent of n. Then, for v = un (t + a) − un (t), which
belongs to D(A) for almost every t, from (3.6) we have
un (t + a) − un (t)

2
D(A)

≤ CT a1/2 un (t + a) − un (t)

D(A) .

Hence
T −a

T −a

un (t + a) − un (t)

2
D(A) dt

τ

≤ CT a1/2

un (t + a) − un (t)

D(A) dt.

(3.8)

τ

Using Cauchy’s inequality and (3.6), we deduce from (3.8) that
T −a

un (t + a) − un (t) 2 dt ≤ CT a1/2 ,
τ

for another positive constant CT independent of n. Therefore
T −a

un (t + a) − un (t)

lim sup

a→0 n

τ

2
D(A)(Ωr ) dt

= 0,

for all r > 0, where Ωr = {x ∈ Ω : |x| < r}. Moreover, from (3.6),
{un |Ωr } is bounded in L∞ (τ, T ; H 1 (Ωr , g)) ∩ L2 (τ, T ; D(A)(Ωr ))

(3.9)


2D g-NAVIER-STOKES EQUATIONS

8

for all r > 0. Consider now a truncation function ρ ∈ C 1 (R+ ) with ρ(s) = 1 in

|x|2
un (x, t)
[0, 1], and ρ(s) = 0 in [2, +∞). For each r > 0, define vn,r (x, t) = ρ
r2
for x ∈ Ω√2r . Then, from (3.9), we have
T −a

vn,r (t + a) − vn,r (t)

lim sup

a→0 n

τ

2
D(A)(Ω√2r ) dt

= 0,

for all T > τ, r > 0,

while from (3.6) we deduce that vn,r is uniformly bounded in L∞ (τ, T ; H01 (Ω√2r , g))∩
L2 (τ, T ; D(A)(Ω√2r )) for all T > τ , r > 0. Thus, by applying Theorem 13.3 and
Remark 13.1 in [18], we obtain
{vn,r } is relatively compact in L2 (τ, T ; H01 (Ω√2r , g)),

for all T > τ, r > 0.

It follows that

{un |Ωr } is relatively compact in L2 (τ, T ; H01 (Ω√2r , g)),

for all T > τ, r > 0.

Then, by a diagonal process, we can extract a subsequence {un } such that
un → u weakly-* in L2loc (R; D(A)),
un → u strongly in L2loc (R; H01 (Ωr , g)), r > 0,

(3.10)

for some u ∈ L∞
loc (R, Ω). The convergences (3.10) allows us to pass to the limit in
the equation for un to find that u is a strong solution of (1.1) with u(τ ) = u0 . Since
the uniqueness of the strong solution, we must have u = u. Then by a contradiction
argument we deduce that the whole sequence {un } converges to u in the sense of
(3.10). This proves (3.5).
Now, from the strong convergence in (3.10) we also have that un (t) converges
strongly in H01 (Ωr , g) to u(t) for a.e. t ≥ τ and all r > 0. Hence for all v ∈ V,
((un (t), v))g → ((u(t), v))g for a.e. t ∈ R.
Moreover, from (3.6) and (3.7), we see that {(un (t), v)g } is equibounded and
equicontinuous on [τ, T ], for all T > τ . Therefore
((un (t), v))g → ((u(t), v))g , ∀ t ∈ R, ∀ v ∈ V.
Finally, (3.4) follows from the fact that V is dense in Vg .
Lemma 3.2. Let {u0n } ⊂ Hg be a sequence converging strongly in Hg to an element
u0 in Hg . Suppose u(t) = U (t, τ )u0 , un (t) = U (t, τ )u0n . Then, for all T > τ ,
un → u strongly in L2 (τ, T ; Vg ).
Proof. Suppose un and u are solutions to (1.1) with initial conditions u0n and u0 ,
we have
1 d
|un − u|2 + ν un − u 2

2 ds
= −b(un , un , un − u) + b(u, u, un − u) − ν(C(un − u), un − u)g
= −b(un − u, u, un − u) − ν(C(un − u), un − u)g
≤ c|un − u| un − u u + ν

|∇g|∞
1/2

m0 λ1

||un − u||2 ,

or
d
|un − u|2 + 2νγ0 un − u
ds

2

≤ 2c|un − u| un − u u
≤ νγ0 un − u

2

+ c u 2 |un − u|2 .


2D g-NAVIER-STOKES EQUATIONS

9

Thus,
d
|un − u|2 + νγ0 un − u
ds

2

≤ c u 2 |un − u|2 .

Therefore,
T

T

u(s) 2 |un (s) − u(s)|2 ds.

un (s) − u(s) 2 ds ≤ |u0n − u0 |2 + c

νγ0

τ

τ

Noting that u0n → u0 strongly in Hg , we have |un (t) − u(t)|2 → 0 for all t ∈ (τ, T ).
By Lebesgue’s dominant convergence theorem, we have
T

un (s) − u(s) 2 ds = 0,

lim

n→∞

τ

i.e. un → u strongly in L2 (τ, T ; Vg ).
Theorem 3.2. Suppose that f ∈ L2b (R; Hg ). Then, there exists a unique pullback
attractor Aˆ = {A(t) : t ∈ R} for the process U (t, τ ).
Proof. Let τ ∈ R, u0 ∈ Vg be fixed, and denote
for all t ≥ τ.

u(t) = u(t; τ, u0 ) = U (t, τ )u0

We will check the two conditions in Theorem 2.1.
ˆ of pullback absorbing sets.
i) The process U (t, τ ) has a family B
From
du
, v + ν((u, v))g + ν(Cu, v)g + b(u, u, v) = f, v ,
dt
choosing v = eσs u(s), we have
d σs
(e |u(s)|2 ) + 2νeσs u(s)
ds
≤

2

(3.11)

= σeσs |u(s)|2 + 2eσs f (s), u(s) − 2eσs ν(Cu(s), u(s))g

σ σs
e u(s)
λ1

2

|∇g|∞

+ 2eσs |f (s)||u(s)| + 2eσs ν

1/2

||u(s)||2 ,

m0 λ1

or
d σs
(e |u(s)|2 ) + 2νγ0 eσs u(s)
ds

2

≤

σ σs

e u(s)
λ1

2

≤ νγ0 eσs u(s)

2

+ 2eσs |f (s)||u(s)|
+

1 σs
e |f (s)|2 + σeσs |u(s)|2 .
σ

Hence
d σs
1
e |u(s)|2 ≤ eσs |f (s)|2 .
ds
σ
Integrating from τ to t we get
eσt |u(t)|2 ≤ eστ |u(τ )|2 +

1
σ

t

eσs |f (s)|2 ds.
τ

Hence it follows that
|U (t, τ, u0 )|2 ≤ eσ(τ −t) |u0 |2 +

e−σt σt
e
σ

t

t−1

|f (s)|2 ds + eσ(t−1)
t−1

e−σt
f 2L2 eσt + eσ(t−1) + · · ·
b
σ
σt
−σt
e
e
f 2L2
≤ eσ(τ −t) |u0 |2 +
b
σ 1 − e−σ
1+σ

σ(τ −t)
2
≤e
|u0 | +
f L2b .
σ2
≤ eσ(τ −t) |u0 |2 +

|f (s)|2 ds + · · ·
t−2


2D g-NAVIER-STOKES EQUATIONS

10

Denote
2(1 + σ)
f 2L2 .
b
σ2
there exists τ0 (B) such that

2
BHg = v ∈ Hg : |v|2 ≤ RH
:=
g

Then, for a given bounded set B ⊂ BHg
2

|U (t, τ )u0 |2 ≤ RH
g

for all τ ≤ τ0 (B).

On the other hand, from (3.11) choosing v(s) = u(s) and applying Cauchy’s inequality, we obtain
d
|u(s)|2 + νγ0 u(s)
ds
Integrating from t to t + 1

2

t+1

|u(t + 1)|2 + νγ0

|∇u|2 ds ≤ |u(t)|2 +
t

≤

1
|f (s)|2 .
σ

≤

1
σ

2(1 + σ)
f
σ2

t+1

|f (s)|2 ds
t
2
L2b

+

1
f
σ

2
L2b

2 + 3σ
f
σ2

=

2
L2b

for all t ≥ T0 .
On the other hand, from (3.11) choosing v(s) = Au(s) we have
1 d
|∇u(s)|2 + ν|Au(s)|2 + ν(Cu(s), Au(s))g + b(u, u, Au) = f, Au .
2 ds
It implies that
d
|∇u(s)|2 + 2ν|Au(s)|2 = − 2b(u, u, Au) + 2 f, Au − 2ν(Cu(s), Au(s))g
ds
|∇g|∞
≤c3 |u|1/2 u |Au|3/2 + 2|f ||Au| + 2ν
||u|||Au|
m0
≤c3 |u|1/2 u |Au|3/2 + 2|f ||Au|
+ 2ν

1/2

|∇g|∞

|Au|2 + ν
1/2

m0 λ1

|∇g|∞ λ1
||u||2 .
2m0

Then, we have

1/2

d
|∇g|∞ λ1
|∇u(s)|2 + 2νγ0 |Au(s)|2 ≤c3 |u|1/2 u |Au|3/2 + 2|f ||Au| + ν
||u||2
ds
2m0
≤νγ0 (|Au|3/2 )4/3 + c(|u|1/2 u )4
1/2

+ νγ0 |Au|2 +

1
|∇g|∞ λ1
|f |2 + ν
||u||2 ,
νγ0
2m0

or
d
u(s)
ds

2

≤

1

|f (s)|2 + (c|u|2 u
νγ0

1/2

2

+ν

|∇g|∞ λ1
)||u||2 .
2m0

Applying the uniform Gronwall inequality with
1/2

y(s) = u(s) 2 ;

a(s) = (c|u|2 u

2

+ν

|∇g|∞ λ1
);
2m0

b(s) =

1
|f (s)|2 ,
νγ0

for all u0 ∈ B ⊂ B(Vg ), t ≥ τ0 (t, B) ≥ τ , we obtain
U (t, τ, u0 )

2

≤

2 + 3σ
f
σ 2 νγ0

2
L2b

2 + 3σ + σ 2
=
f
σ 2 νγ0

1
+
f
νγ0
(
2
L2b e

2
L2b

e

(

2c(1+σ)(2+3σ)
νγ0 σ 4

2c(1+σ)(2+3σ)
νγ0 σ 4

f

f

1/2
|∇g|∞ λ1
4
+ν
2m0
L2
b

1/2
|∇g|∞ λ1
4
+ν

2m0
L2
b

)

2
:= RpV
,
g

)


2D g-NAVIER-STOKES EQUATIONS

11

for all u0 ∈ B ⊂ B(Vg ), t − 1 ≥ τ0 (t, B) ≥ τ , or
U (t + 1, τ )u0

2

2
≤ RpV
,
g

for all u0 ∈ B ⊂ B(Vg ), t ≥ τ0 (t, B) ≥ τ . It implies that with the following set
D0 = {v ∈ V : v ≤ RpVg },

then B = {D0 : t ∈ R} is a family of pullback absorbing sets for the process U (t, τ ).
ii) U (t, τ ) is pullback asymptotically compact.
Let us fix t ∈ R, a sequence τn → −∞ and a sequence u0n ∈ D0 . We have
to prove that from the sequence {U (t, τn )u0n }, we can extract a subsequence that
converges in Vg .
As the family D0 is pullback absorbing, for each integer k ≥ 0, there exists a
τDˆ 0 (k) such that
U (t − k, τ − k, D0 (τ − k)) ⊂ D0 (t − k) for all τ ≤ τDˆ 0 (k).
Observe now that for τ ≤ τDˆ 0 (k) + k,
U (t − k, τ, D0 (τ )) ⊂ D0 (t − k).
It is not difficult to conclude that there exist a subsequence {(τn , u0n )} ⊂ {(τn , u0n )},
and a sequence {wk ; k ≥ 0} ⊂ Vg , such that wk ∈ D0 (t − k) and for all k ≥ 0,
U (t − k, τn )u0n

wk weakly in Vg .

By results in [1], we have had that
U (t − k, τn )u0n → wk strongly in Hg .
Observe that
w0 = weak − lim U (t, τn )u0n
n →∞

= weak − lim U (t, t − k)U (t − k, τn )u0n
n →∞

= U (t, t − k) weak − lim U (t − k, τn )u0n ) ,
n →∞

i.e.,
U (t, t − k)wk = w0 for all k ≥ 0.

Then, by the lower semi-continuity of the norm
w0 ≤ lim inf U (t, τn )u0n .
n →∞

So if we now also prove that
w0 ≥ lim sup U (t, τn )u0n ,
n →∞

then we will have
lim

n →∞

U (t, τn )u0n

= w0 ,

and this, together with the weak convergence, will imply the strong convergence in
Vg of U (t, τn )u0n to w0 .
From (3.11) choosing v = eσ(s−t) Au we get
d σ(s−t)
(e
u(s) 2 ) + 2νeσ(s−t) |Au(s)|2
ds
= σeσ(s−t) u(s) 2 + 2eσ(s−t) f (s), Au(s) − 2eσ(s−t) b(u, u, Au) − 2eσ(s−t) (Cu, Au)g .


2D g-NAVIER-STOKES EQUATIONS

12

Integrating from τ to t we obtain
t

u(t)

2

=eσ(τ −t) u(τ )

2

eσ(s−t) b(u, u, Au)ds

−2
τ

t

eσ(s−t) f (s), Au(s) − ν|Au(s)|2 −

+2
τ

σ
u(s)
2

2

ds

σ
u(s)
2

2

ds

t

eσ(s−t) (Cu, Au)g ds

−2
τ

t

≤e

σ(τ −t)

u(τ )

2

eσ(s−t) b(u, u, Au)ds

−2

τ

t

eσ(s−t) f (s), Au(s) − ν|Au(s)|2 −

+2
τ

+ 2ν

1/2

t

|∇g|∞

eσ(s−t) |Au(s)|2 ds + ν

1/2
m0 λ1

τ

|∇g|∞ λ1
2m0

t

eσ(s−t) ||u(s)||2 ds

τ

t

=eσ(τ −t) u(τ )

2

eσ(s−t) b(u, u, Au)ds

−2
τ

t

eσ(s−t) f (s), Au(s) − νγ0 |Au(s)|2 −

+2
τ

1/2

+ν

σ
u(s)
2

2

ds

t

|∇g|∞ λ1
2m0

eσ(s−t) ||u(s)||2 ds,
τ

or
t

U (t, τ )u0

2

≤eσ(τ −t) u0

2

eσ(s−t) b(U (s, τ )u0 , U (s, τ )u0 , AU (s, τ )u0 )ds

−2
τ

t

eσ(s−t) f (s), AU (s, τ )u0 − [[U (s, τ )u0 ]]2 ds

+2
τ

1/2

+ν

|∇g|∞ λ1
2m0

t

eσ(s−t) ||U (s, τ )u0 ||2 ds.
τ

Thus, for all k ≥ 0, τn ≤ k, we have
U (t, τn )u0n

2

= U (t, t − k)U (t − k, τn )u0n

≤ e−σk U (t − k, τn )u0n

2

2

t

−2
t−k

eσ(s−t) b(U (s, t − k)U (t − k, τn )u0n ,
U (s, t − k)U (t − k, τn )u0n , AU (s, t − k)U (t − k, τn )u0n )ds

t

+2
t−k
t

−2
t−k

eσ(s−t) f (s), AU (s, t − k)U (t − k, τn )u0n ds
eσ(s−t) [[U (s, t − k)U (t − k, τn )u0n ]]2 ds
1/2

|∇g|∞ λ1
+ν
2m0

t
τ

eσ(s−t) ||U (s, t − k)U (t − k, τn )u0n ||2 ds.

By τ ≤ τDˆ 0 (k) + k, k ≥ 0,
U (t − k, τ, D(τ )) ⊂ D0 (t − k),

we have
lim sup(e−σk U (t − k, τn )u0n
n →∞

2

2
) ≤ e−σk RpV
, ∀ k ≥ 0.
g


2D g-NAVIER-STOKES EQUATIONS

On the other hand, as U (t − k, τn )u0n
U (s, t − k)U (t − k, τn )u0n

13

wk weakly in Vg , we have

U (s, t − k)wk weakly in L2 (t − k, t; D(A)).

Taking into account that, in particular eσ(s−t) f (s) ∈ L2 (t − k, t; Hg ), we obtain
t

lim sup
n →∞

t−k

eσ(s−t) f (s), AU (s, t − k)U (t − k, τn )u0n ds
t

eσ(s−t) f (s), AU (s, t − k)wk ds.

=
t−k
t

eσ(s−t) [[v(s)]]2 ds

Moreover, as

1
2

defines a norm in L2 (t − k, t; D(A)), which

t−k

is equivalent to the usual one, we also obtain that
t

t

eσ(s−t) [[U (s, t − k)wk ]]2 ds ≤ lim inf
n →∞

t−k

t−k

eσ(s−t) [[U (s, t − k)U (t − k, τn )u0n ]]2 ds.

Now, we need prove that
t

lim

n →∞

t−k

eσ(s−t) b(U (s, t − k)U (t − k, τn )u0n , U (s, t − k)U (t − k, τn )u0n ,

AU (s, t − k)U (t − k, τn )u0n )ds
t

eσ(s−t) b(U (s, t − k)wk , U (s, t − k)wk , AU (s, t − k)wk )ds.

=
t−k

(3.12)
Let v0,n = U (t−k, τn )u0n , wk = v0 . Then, to show (3.12), we present the following
lemma.
Lemma 3.3. Suppose
v0,n

v0

v0,n → v0

weakly in Vg
strongly in Hg .

Then
t

eσ(s−t) b(U (s, t − k)v0,n , U (s, t − k)v0,n , AU (s, t − k)v0,n )ds

lim

n →∞

t−k

t

eσ(s−t) b(U (s, t − k)v0 , U (s, t − k)v0 , AU (s, t − k)v0 )ds.

=
t−k

Proof. Set U (t, t − k)v0,n = vn (t), U (t, t − k)v0 = v(t). Then
v(s) = U (s, t − k)wk ,

vn (s) = U (s, t − k)U (t − k, τn )u0n .

Thus, we need to prove that
t

t

eσ(s−t) b(vn (s), vn (s), Avn (s))ds =

lim

n →∞

eσ(s−t) b(v(s), v(s), Av(s))ds.

t−k

t−k

We have
t

t

eσ(s−t) b(vn , vn , Avn )ds −
t−k
t

eσ(s−t) b(v, v, Av)ds
t−k
t

eσ(s−t) b(vn − v, vn , Avn )ds +

≤
t−k

eσ(s−t) b(vn , vn − v, Avn )ds
t−k
t

eσ(s−t) b(v, v, Avn − Av)ds

+
t−k

:=I1 + I2 + I3 ,


2D g-NAVIER-STOKES EQUATIONS

14

and we estimate I1 , I2 , I3 one by one.
First,
t

I1 ≤ C

|vn − v|L4 |∇vn |L4 |Avn |L2 ds
t−k
t

|vn − v|1/2 vn − v

≤C

1/2

vn

1/2

|Avn |3/2 ds

t−k
1/4

t

|vn − v|2 vn − v

≤C

2

3/4

t

vn 2 ds

|Avn |2 ds
t−k

t−k
1/4

t

|vn − v|2 ds

≤C

→ 0 (n → ∞).

t−k

Second,
t

|vn |L4 |∇vn − ∇v|L4 |Avn |L2 ds

I2 ≤ C
t−k
t

|vn |1/2 vn

≤C

1/2

vn − v

1/2

|Avn − Av|1/2 |Avn |ds

t−k
t
2

≤C

2

|vn | vn

2

1
4

t
2

vn − v ds

t−k

1

4

t
2

|Avn − Av| ds
t−k

1
2

|Avn | ds
t−k

1/4

t

vn − v 2 ds

≤C

→ 0 (n → ∞).

t−k

Since Bv ∈ L2 (τ, T ; Hg ), we also have
vn

v0

weakly in D(A).

So,
t

eσ(s−t) | B(v), A(vn − v) | → 0 (n → ∞).

I3 ≤
t−k

Return to the proof of the pullback asymptotic compactness of {U (t, τ )} in Vg ,
from above estimates, we have
lim sup U (t, τn )u0n

2

n →∞

t
2
≤ e−σk RpV
+2
g

eσ(s−t) f (s), AU (s, t − k)wk
t−k

− b(U (s, t − k)wk , U (s, t − k)wk , AU (s, t − k)wk ) − [[U (s, t − k)wk ]]2 ds
1/2

+ν

|∇g|∞ λ1
2m0

t

eσ(s−t) U (s, t − k)wk

2

ds.

t−k

On the other hand, we have
w0

2

= U (t, t − k)wk

2
t

=e−σk wk

2

eσ(s−t) b(U (s, t − k)wk , U (s, t − k)wk , AU (s, t − k)wk )

−2
t−k

t

t

eσ(s−t) f (s), AU (s, t − k)wk ds − 2

+2
t−k

1/2

+ν

eσ(s−t) [[U (s, t − k)wk )]]2 ds
t−k

|∇g|∞ λ1
2m0

t

eσ(s−t) U (s, t − k)wk
t−k

2

ds.


2D g-NAVIER-STOKES EQUATIONS

15

Then
2

lim sup U (t, t − τn )u0n
n →∞

2
≤ e−σk RpV
+ w0
g

2

− e−σk wk

2

2
≤ e−σk RpV
+ w0 2 ,
g

and thus, taking into account that
−σk

e

2
RpV
g

2 + 3σ + σ 2
=
f
σ 2 νγ0

(
2
L2b e

2c(1+σ)(2+3σ)
νγ0 σ 4

f

1/2
|∇g|∞ λ1
4
+ν
2m0
L2
b

) −σk

e

→ 0,

when k → +∞, we easily obtain
lim sup U (t, τn )u0n

2

≤ w0 2 .

n →∞

This completes the proof.
Acknowledgements. This work was done while the authors were staying at the
Vietnam Institute of Advanced Study in Mathematics (VIASM) as research fellows.
The authors would like to thank the Institute for its hospitality and support.

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2D g-NAVIER-STOKES EQUATIONS

Cung The Anh
Department of Mathematics, Hanoi National University of Education

136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
E-mail address:
Dao Trong Quyet
Faculty of Information Technology, Le Quy Don Technical University
100 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
E-mail address:

16