Necessary and Sufficient Conditions for
the Existence of Global Attractors for
Semigroups and Applications
Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
A BSTRACT. First we establish some necessary and sufficient conditions for the existence of the global attractor of an infinite
dimensional dynamical system, using the measure of noncompactness. Then we give a new method/recipe for proving the
existence of the global attractor. The main advantage of this
new method/recipe is that one needs only to verify a necessary
compactness condition with the same type of energy estimates
as those for establishing the absorbing set. In other words, one
doesn’t need to obtain estimates in function spaces of higher regularity. In particular, this property is useful when higher regularity
is not available, as demonstrated in the example on the NavierStokes equations on nonsmooth domains.
C ONTENTS
1.
2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 542
Measure of Noncompactness and Its Properties . . . . . . . . . . . . . . . . . . . . 1. 543
General Existence Theorems of Global Attractors . . . . . . . . . . . . . . . . . . 1. 544
Existence of Global Attractors for Semilinear Parabolic Equations in
H 1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1550
5. Global Attractors for the 2D Navier-Stokes Equations in H01 . . . . . . . . .1552
6. Attractors of the Navier-Stokes Equations in Nonsmooth Domains . . .1554
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1558
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1558
1541
c , Vol. 51, No. 6 (2002)
Indiana University Mathematics Journal 1542 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
1. I NTRODUCTION
As we know, many mathematical physics problems can be put into the perspective
of infinite dimensional systems, which can be equivalently described by C 0 semigroups in proper function spaces. One important object to describe the long time
dynamics of an infinite dimensional system is the global attractor, which is a connected and compact set in some function space, and which attracts all bounded
sets.
To show the existence of the global attractor, one normally needs to verify
(1) there exists an absorbing set, and
(2) the semigroup is uniformly compact.
Both conditions are usually proved with certain energy type of estimates, and the
uniform compactness is proved using estimates in a more regular function space.
Our main motivation of this article is to derive some conditions which are
weaker than the uniform compactness condition, and are easier to verify in view of
applications. Fortunately in this article we are able to establish some necessary and
sufficient conditions for the existence of the global attractor using the measure of
noncompactness. Furthermore, we establish a new method/recipe for proving the
existence of the global attractor. The main advantage of this new method/recipe
is that, in addition to the existence of an absorbing set, one only needs to verify a
necessary compactness condition with the same type of energy estimates as those
for establishing the absorbing set.
More precisely, first, using the measure of noncompactness, we introduce a
new concept of compactness called ω-limit compact. Then we show that there
exists a global attractor for a C 0 semigroup if and only if
(1) there is an absorbing set, and
(2) the semigroup is ω-limit compact.
Second, we observe that the measure γ(A) of noncompactness of a bounded
set A in a Banach space X = X1 ⊕ X2 with dim X1 < ∞ is determined by the
diameter of the projection of A to X2 , i.e., γ(A) ≤ diam(QA). Here Q : X → X2
is the canonical projection.
Using this property of the measure of noncompactness, we are able to show
that the ω-limit compact of a C 0 semigroup in a convex Banach space X is equivalent to the following condition.
Condition (C). For any bounded set B of X and for any ε > 0, there exist t(B) >
0 and a finite dimensional subspace X1 of X , such that {kP S(t)Bk} is bounded and
k(I − P )S(t)xk < ε
for t ≥ t(B), x ∈ B,
where P : X → X1 is a bounded projector.
Third, based on the above equivalent conditions for the ω-limit compactness,
we arrive at the following general method/recipe for the existence of the global
attractor of an infinite dimensional system:
Existence of Global Attractors and Applications
1543
Recipe. To verify the existence of a global attractor, one needs to show that
(1) there exists an absorbing set, and
(2) Condition (C) holds.
The main advantage of this new method/recipe is that one needs only to verify a necessary compactness condition with the same type of energy estimates as
those for establishing the absorbing set. In other words, one doesn’t need to obtain estimates in function spaces of higher regularity. In particular, this property is
useful when higher regularity is not available, as demonstrated in the example on
the Navier-Stokes equations on nonsmooth domains. It is easy to see that Condition (C) is related to the amplitude of the small eddies discussed in Foiaş, Manley
and Temam [2], and to the work of Foiaş and Prodi [6] on dependence of the
asymptotic behavior (in time) of the Navier-Stokes equations on the asymptotic
behavior of a certain finite number of numerical parameters.
In order to show how to use this method, we studied in Sections 4-6 the existence of global attractors in function spaces with stronger topology for a parabolic
equation, and for the Navier-Stokes equations in both smooth and nonsmooth
domains. This method can also be applied to obtain global attractors in stronger
topology for other equations such as wave equations with linear damping; this will
be reported in a forthcoming paper.
2. M EASURE OF N ONCOMPACTNESS AND I TS P ROPERTIES
In this section, we recall the concept of measure of noncompactness and recapitulate its basic properties; see [3].
Definition 2.1. Let M be a metric space and A be a bounded subset of M .
The measure of noncompactness γ(A) of A is defined by
γ(A) = inf{δ > 0 | A admits a finite cover by sets of diameter ≤ δ}.
Lemma 2.2. Let M be a complete metric space, and γ be the measure of noncompactness.
(1) γ(B) = 0 if and only if B̄ is compact;
(2) If M is a Banach space, then γ(B1 + B2 ) ≤ γ(B1 ) + γ(B2 );
(3) γ(B1 ) ≤ γ(B2 ) whenever B1 ⊂ B2 ;
(4) γ(B1 ∪ B2 ) ≤ max{γ(B1 ), γ(B2 )};
(5) γ(B̄) = γ(B).
Lemma 2.3. Let M be an infinite dimensional Banach space, and B(ε) a ball of
radius ε; then γ(B(ε)) = 2ε.
Lemma 2.4. Let X be an infinite dimensional Banach space with the following
decomposition:
X = X1 ⊕ X2 , dim X1 < ∞.
Let P : X → X1 , Q : X → X2 be the canonical projectors, and A be a bounded subset
of X . If the diameter of QA is less then ε, then γ(A) < ε.
1544 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
Lemma 2.5. Let M be a complete metric space, and γ be the measure of noncompactness. Assume that {Fn } is a sequence of bounded and closed subsets of M ,
satisfying
(1) Fn 6= ϕ,
(2) Fn+1 ⊂ Fn , n = 1, 2, . . . , and
(3) γ(Fn ) → 0, as n → ∞.
T
Then F = ∞
n=1 Fn is a nonempty compact set.
3. G ENERAL E XISTENCE T HEOREMS OF G LOBAL ATTRACTORS
In this section, after recalling some definitions related to the global attractor of
a semigroup, we introduce a new concept called ω-limit compactness of a semigroup. Then we establish some necessary and sufficient conditions for the existence of the global attractor of an infinite dimensional dynamical system, using
the measure of noncompactness. Then we give a new method/recipe for proving
the existence of the global attractor.
Definition 3.1. Let M be a complete metric space. A one parameter family
{S(t)}t≥0 of maps S(t) : M → M , t ≥ 0 is called a C 0 semigroup if
(1) S(0) is the identity map on M ,
(2) S(t + s) = S(t)S(s) for all t , s ≥ 0,
(3) the function S(t)x is continuous at each point (t, x) ∈ [0, ∞) × M .
Definition 3.2. Let {S(t)}t≥0 be a C 0 semigroup in a complete metric space
M . For any subset B ⊂ M , the set ω(B) defined by
ω(B) =
\[
S(t)B
s≥0 t≥s
is called the ω-limit set of B .
It is easy to see that ϕ ∈ ω(B) if and only if there exists a sequence of elements
ϕn ∈ B and a sequence tn → ∞ such that
(3.1)
S(tn )ϕn → ϕ,
as n → ∞.
Definition 3.3. Let {S(t)}t≥0 be a C 0 semigroup in a complete metric space
M . A subset B0 of M is called an absorbing set in M if, for any bounded subset B
of M , there exists some t1 ≥ 0 such that S(t)B ⊂ B0 , for all t ≥ t1 .
Definition 3.4. Let {S(t)}t≥0 be a C 0 semigroup in a complete metric space
M . A subset A of M is called a global attractor for the semigroup if A is compact
and enjoys the following properties:
(1) A is an invariant set, i.e., S(t)A = A for any t ≥ 0;
(2) A attracts all bounded sets of M . That is, for any bounded subset B of M ,
Existence of Global Attractors and Applications
d(S(t)B, A) → 0,
(3.2)
1545
as t → ∞,
where d(B, A) is the semidistance of two sets B and A:
d(B, A) = sup inf d(x, y).
x∈B y∈A
Definition 3.5. A C 0 semigroup {S(t)}t≥0 in a complete metric space M is
called ω-limit compact if, for every bounded subset B of M and for any ε > 0,
there exists a t0 > 0 such that
γ
[
S(t)B ≤ ε.
t≥t0
Definition 3.6. A C 0 semigroup {S(t)}t≥0 in a complete metric space M is
called set-contractive if there exist an α ∈ [0, 1) and a t0 > 0 such that, for every
bounded subset B of M ,
γ(S(t0 )B) ≤ αγ(B).
Proposition 3.7. Assume that the C 0 semigroup {S(t)}t≥0 in a complete
metric
S
space M is set-contractive and that for every bounded subset B of M , t≥0 S(t)B is
also bounded. Then it is ω-limit compact.
S
Proof. For every bounded subset B of M , we know that t≥0 S(t)B is also
bounded, from the assumption. Hence, from the other assumption, that S(t) is
set-contractive, we have that there exist an α ∈ [0, 1) and a t1 > 0 such that
[
[
[
γ S(t1 )
S(t)B = γ
S(t)B ≤ αγ
S(t)B .
t≥0
t≥0
t≥t1
Inductively, we have
[
[
γ S(nt1 )
S(t)B = γ S(t1 )S((n − 1)t1 )
S(t)B
=γ
[
t≥nt1
t≥0
n
S(t)B ≤ α γ
[
S(t)B .
t≥0
S
n
Now, for any
S ε > 0, take n large enough such that α γ(
then we have γ( t≥nt1 S(t)B) ≤ ε. This completes the proof.
t≥0 S(t)B)
≤ ε;
❐
t≥0
Theorem 3.8. Let {S(t)}t≥0 be a C 0 semigroup in a complete metric space M .
Assume that
(1) S(t) is ω-limit compact;
(2) there exists a bounded absorbing subset B of M .
Then the ω-limit set of B , A = ω(B), is a compact attractor which attracts all
bounded subsets of M .
1546 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
Proof. Since S(t) is ω-limit compact and B is bounded, for any ε > 0, there
exists tε > 0 such that
[
γ
S(t)B < ε.
t≥tε
Take ε = 1/n, n = 1, 2, . . . ; we find a sequence {tn }, t1 < t2 < · · · < tn < · · · ,
such that
[
1
γ
S(t)B < , n = 1, 2, . . . .
n
t≥tn
Property (5) of the measure of noncompactness in Lemma 2.2 shows that
γ
[
S(t)B
<
t≥tn
By Lemma 2.5, we know that
is the ω-limit set of B . That is
T∞
A = ω(B) =
1
n
S
t≥tn
n=1
\[
n = 1, 2, . . . .
,
S(t)B is a nonempty compact set, and
∞ [
\
S(t)B =
s≥0 t≥s
S(t)B.
n=1 t≥tn
We now prove that A = ω(B) is invariant. In fact, if ψ ∈ S(t)ω(B), then
ψ = S(t)ϕ, for some ϕ ∈ ω(B). Hence there exist a sequence ϕn ∈ B and
tn → ∞ such that S(tn )ϕn → ϕ. Namely
S(t)S(tn )ϕn = S(t + tn )ϕn → S(t)ϕ = ψ,
which shows that ψ ∈ ω(B) and S(t)ω(B) ⊂ ω(B).
Conversely, if ϕ ∈ ω(B), by (3.1), we can find two sequences ϕn ∈ B and
tn → ∞ such that S(tn )ϕn → ϕ.
We need to prove that {S(tn − t)ϕn } has a subsequence which converges in
M . For any ε > 0, there exists a tε such that
γ
[
S(t 0 )B < ε,
t 0 ≥tε
which implies that
γ
[
S(t 0 − t)B < ε.
t 0 ≥tε +t
Hence there exists an N such that
[
n≥N
S(tn − t)ϕn ⊂
[
t 0 ≥t
ε +t
S(t 0 − t)B.
Existence of Global Attractors and Applications
It follows then that
γ
[
1547
S(tn − t)ϕn < ε.
n≥N
SN
Notice that n=N0 S(tn − t)ϕn contains only a finite number of elements,
where N0 is fixed such that tn − t ≥ 0 as n ≥ N0 . Using properties (1)-(4) for the
measure of noncompactness in Lemma 2.2, we have
γ
[
[
S(tn − t)ϕn = γ
S(tn − t)ϕn < ε.
n≥N0
n≥N
Let ε → 0. We then derive that
γ
[
S(tn − t)ϕn = 0.
n≥N0
This means that {S(tn − t)ϕn } is relatively compact. Therefore, there exist a
subsequence tnj → ∞ and ψ ∈ M such that
S(tnj − t)ϕnj → ψ
as tnj → ∞.
It’s readily to see that ψ ∈ ω(B) and
ϕ = lim S(tnj )ϕnj = lim S(t)S(tnj − t)ϕnj = S(t)ψ
j→∞
j→∞
belongs to S(t)ω(B).
It suffices to prove that A = ω(B) is an attractor in M and attracts all
bounded subsets of M . Assume otherwise; then there exists a bounded subset
B0 of M such that dist(S(t)B0 , A) does not tend to 0 as t → ∞. Thus there exist
δ > 0 and a sequence tn → ∞ such that
dist(S(tn )B0 , A) ≥ δ > 0,
∀n ∈ N.
For each n, there exists bn ∈ B0 satisfying
dist(S(tn )bn , A) ≥
(3.3)
δ
2
> 0.
Since B is an absorbing set, S(tn )B0 and S(tn )bn belong to B for n sufficiently
large. As in the proof above, we know that S(tn )bn is relatively compact and
possesses at least one cluster point β,
β = lim S(tni )bni = lim S(tni − t1 )S(t1 )bni ,
ni →∞
ni →∞
❐
where t1 is such that S(t1 )B0 ⊂ B . Hence β belongs to A = ω(B) and this
contradicts (3.3). The proof is complete.
1548 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
Theorem 3.9. Let {S(t)}t≥0 be a C 0 semigroup in a complete metric space M .
Then S(t) has a global attractor A in M if and only if
(1) {S(t)}t≥0 is ω-limit compact, and
(2) there is a bounded absorbing set B ⊂ M .
Proof. By Theorem 3.8, we only have to prove the necessity. Since A is an
attractor, the ε-neighborhood of A is the absorbing set. Hence it suffices to prove
that the semigroup is ω-limit compact. To this end, for every bounded subset B
of M and any ε > 0, there exists tε (B) ≥ 0 such that
[
S(t)B ⊂ Nε/4 (A) = x ∈ M | dist(x, A) <
t≥tε (B)
ε
4
.
On the other hand, since A is compact, there exists a finite number of elements x1 , x2 , . . . , xn ∈ M such that
A⊂
n
[
ε
.
N xi ,
4
i=1
Therefore
Nε/4 (A) ⊂
n
[
ε
N xi ,
,
i=1
2
which implies that
γ
[
S(t)B ≤ γ(Nε/4 (A)) ≤ ε.
t≥tε (B)
❐
The proof is complete.
Theorem 3.10. Let X be a Banach space and {S(t)}t≥0 be a C 0 semigroup in
X.
(1) If Condition (C) in the Introduction for {S(t)}t≥0 holds true, then {S(t)}t≥0 is
ω-limit compact.
(2) Let X be a uniformly convex Banach space. Then {S(t)}t≥0 is ω-limit compact
if and only if Condition (C) holds true.
(1)
Proof.
From the second conclusion of Lemma 2.2 and Lemmas 2.3-2.5, we have
γ
[
t≥t(B)
[
[
S(t)B ≤ γ P
S(t)B + γ (I − P )
S(t)B
t≥t(B)
≤ γ(N(0, ε)) = 2ε.
t≥t(B)
Existence of Global Attractors and Applications
1549
Therefore, S(t) is ω-limit compact.
(2) Let X be a uniformly convex Banach space, and S(t) be ω-limit compact.
Then for every bounded subset B of X and for any ε > 0, there exists t(B) > 0
such that
[
γ
S(t)B <
t≥t(B)
ε
2
.
Namely, there exists a finite number of subsets A1 , A2 , . . . , An with diameter less
than ε/2, such that
[
S(t)B ⊂
[
S(t)B ⊂
t≥t(B)
Ai .
i=1
t≥t(B)
Let xi ∈ Ai ; then
n
[
n
[
ε
.
N xi ,
2
i=1
Let X1 = span{xi , . . . , xn }; since X is uniformly convex, there exists a projection
P : X → X1 such that, for any x ∈ X , kx − P xk = dist(x, X1 ). Hence
ε
2
< ε,
namely Condition (C) holds true.
❐
k(I − P )S(t)xk ≤
The following theorem follows from Theorems 3.9 and 3.10.
Theorem 3.11. Let X be a Banach space and {S(t)}t≥0 be a C 0 semigroup in
X . Then there is a global attractor for {S(t)}t≥0 in X if the following conditions hold
true:
(1) {S(t)}t≥0 satisfies Condition (C), and
(2) there is a bounded absorbing set B ⊂ X .
Remark 3.12. The equivalent condition for the ω-limit compactness, Condition (C), enables us to establish a general method/recipe for the existence of the
global attractor of an infinite dimensional system as described in the Introduction.
We iterate that, in order to prove the existence of the global attractor for an infinite
dimensional dynamical system, we need only to verify
(1) the existence of an absorbing set, and
(2) Condition (C).
The main advantage to the uniform compactness condition by Condition (C) is
that Condition (C) can be verified with energy estimates in the same function
space as for the existence of absorbing set. In other words, one doesn’t need to
obtain estimates in function spaces with stronger topology. This property is useful
when higher regularity is not available, as demonstrated in the examples in the
next three sections. In particular, for the Navier-Stokes equations on nonsmooth
1550 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
domains, where higher regularity is not available, our method becomes necessary
for establishing the existence of a global attractor in D(A1/4 ).
4. E XISTENCE OF G LOBAL ATTRACTORS FOR S EMILINEAR PARABOLIC
E QUATIONS IN H 1 S OBOLEV S PACES
Let Ω be an open bounded subset of Rn , n ≥ 3, with smooth boundary ∂Ω, and
let g(x, u) be a Carathéodory function of the following form:
(4.1)
g(x, u) = a|u|p−1 · u + h(x, u).
Here 1 ≤ p ≤ n/(n − 2) and h(x, u) satisfies:
(4.2)
(4.3)
h(x, s)
=0
|s|p
h0 (x, s)
lim
=0
s→∞ |s|p−1
lim
s→∞
uniformly for x ∈ Ω,
uniformly for x ∈ Ω.
We are interested in the following initial boundary value problem involving a
scalar function u = u(x, t):
(4.4)
(4.5)
(4.6)
∂u
− d∆u + g(x, u) = 0
∂t
u=0
in Ω × R+ ,
u(x, 0) = u0 (x)
x ∈ Ω,
on ∂Ω,
where d > 0 is a constant and u0 (x) is given.
For this initial boundary value problem, we know from [14] and [12] that,
for u0 given in H01 (Ω) and for any T > 0, there exists a unique solution u of
(4.4)-(4.6) satisfying
u ∈ C([0, T ]; H01 (Ω)) ∩ L2 ([0, T ]; H 2 (Ω)).
Thanks to this existence theorem, the initial boundary value problem is equivalent to a semigroup {S(t)}t≥0 defined by
S(t) : H01 (Ω) → H01 (Ω),
S(t)u0 = u(·, t).
Then the main objective of this section is to use the method developed in Section 3
to show the existence of the global attractor in H01 (Ω) of the semigroup {S(t)}t≥0
associated with the above initial boundary value problem.
Theorem 4.1. Let Ω denote an open bounded subset of Rn , n ≥ 3, and let
g(x, u) be a Carathéodory function satisfying (4.1)-(4.3). Then the semigroup {S(t)}t≥0
associated with the initial boundary value problem (4.4)-(4.6) possesses a maximal attractor A, which is compact and connected in H01 (Ω).
Existence of Global Attractors and Applications
1551
Proof. Using the same method as in [14], it is easy to know that the semigroup S(t) associated with the initial boundary value problem (4.4)-(4.6) possesses an absorbing ball B(0, ρ) in H01 (Ω). Therefore, there exists some t0 > 0
such that, for any t ≥ t0 , S(t)B(0, ρ) ⊂ B(0, ρ).
In addition, by the assumption of g , there exists a constant M > 0, such that
Z
(4.7)
Ω
|g(u)|2 dx ≤ M,
∀u ∈ B(0, ρ).
Then we need to prove that the ω-limit set of B(0, ρ) is compact in H01 (Ω).
It suffices then to verify Condition (C) in H01 (Ω). Namely, we need to show that,
for any ε > 0, there is a finitely dimensional subspace H1 of H01 (Ω) and some
t ∗ > 0 such that
ε
γ((I − P )S(t)B(0, ρ)) <
2
as t ≥ t ∗ ,
where P : H01 (Ω) → H1 is the projection. Let λj and ωj be eigenvalues and
eigenvectors of the Laplacian operator −∆ in H01 (Ω), which form an orthonormal
basis of L2 (Ω). Let H1 = span{ω1 , . . . , ωN }, P : L2 → H1 be the projection, and
Q = Id − P .
Hereafter, we use | · | for the L2 norm, and k · k for the H01 norm given by
kuk = |∇u|. For every u0 ∈ B(0, ρ), we take the L2 inner product between (4.4)
and −∆u2 = −∆Qu:
1 d 2
1
+ d|∆u2 |2 −
u
2 dt 2 H0 (Ω)
(4.8)
Z
Ω
g(u)∆u2 dx = 0,
which implies
(4.9)
1 d 2
1
+ d|∆u2 |2 ≤
u
2 dt 2 H0 (Ω)
Z
|g(u)| |∆u2 | dx
Z
Z
1
d
≤
|g(u)|2 dx +
|∆u2 |2 dx.
2d Ω
2 Ω
Ω
By (4.7), the above inequality implies that
(4.10)
d
1
ku2 k2 ≤ −dλN+1 ku2 k2 + M.
dt
d
By the Gronwall inequality, we have
ku2 (t)k2 ≤ e−dλN+1 (t−t0 ) ku2 (t0 )k2 +
M
d2 λ
N+1
[1 − e−dλN+1 (t−t0 ) ] <
ε2
4
,
1552 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
provided that
8M
λN+1 > 2 2 ,
ε d
t > t0 +
2
dλN+1
√
ln
2 2ρ
ε
!
.
❐
The proof is complete.
5. G LOBAL ATTRACTORS FOR THE 2D N AVIER -S TOKES E QUATIONS IN
H01
As in the previous section, the main objective of this section is to use the theory developed in Section 3 to prove the existence of global attractors for the 2dimensional Navier-Stokes equations in H01 . Again, the verification of the uniform
compactness needed for the existence of global attactors is replaced by the weak
condition, Condition (C), which can be verified with energy estimates in H01 itself.
Let Ω denote an open bounded domain of R2 with smooth boundary Γ .1 The
unknowns are the velocity field u = (u1 , u2 ) and the pressure p . The equations
are
(5.1)
(5.2)
∂u
+ (u · ∇)u − ν∆u + ∇p = f ,
∂t
div u = 0,
supplemented with the following initial value and boundary conditions:
(5.3)
(5.4)
u=0
on Γ ,
u(x, 0) = u0 (x),
x ∈ Ω.
Let H and V the Hilbert spaces defined respectively by
(5.5)
H = {u ∈ L2 (Ω)n , div u = 0, u · n = 0 on Γ },
(5.6)
V = {u ∈ H01 (Ω), div u = 0}.
The space H is endowed with the scalar product and the norm of L2 (Ω)n denoted
by (·, ·) and | · |, and V is endowed with the scalar product
((u, v)) =
2
X
i,j
∂ui ∂vi
,
∂xj ∂xj
!
1 When Ω is sufficiently smooth, there will be no higher regularity for the soultions of the NavierStokes equations. For instance, when Ω is a rectangular region, H 3 regularity for the Stokes problem
is not available. Consequently, one does not appear to have energy estimates in H 2 space, needed for
uniform compactness for global attractors in H 1 space. Therefore our method in this section appears
to be necessary. Of course, for simplicity, we still assume that Ω smooth boundary Γ , but the analysis
applies certainly to other not so smooth domains such as rectangles.
Existence of Global Attractors and Applications
1553
and the norm kuk = {((u, u))}1/2 .
Multiplying (5.1) by a test function v in V and integrating over Ω, using the
Green formula and the boundary conditions, we find that the term involving p
disappears and there remains
(5.7)
d
(u, v) + ν(Au, v) + (B(u), v) = (f , v),
dt
∀v ∈ V .
where A = −P ∆u, P is the Leray projector from L2 (Ω)n onto H , and B(u) =
B(u, u) is defined by
(5.8)
(B(u, v), w) =
2 Z
X
i,j=1
Ω
ui
∂vj
wj dx.
∂xj
Thus in weak form, the equations (5.1)-(5.4) are equivalent to the following operator equations
(5.9)
(5.10)
du
+ νAu + B(u, u) = f ,
dt
u(0) = u0 .
The linear self-adjoint operator A is an isomorphism from its domain D(A) =
H 2 (Ω)n ∩ V onto H . Since the embedding of H 1 (Ω) in L2 (Ω) is compact, the
embedding of V in H is compact. Thus A−1 is a continuous compact operator in
H , and by the classical spectral theorem there exists a sequence λj ,
0 < λ1 ≤ λ2 ≤ · · · ,
λj → ∞,
and a family of elements ωj of D(A), which are orthonormal in H , such that
(5.11)
Aωj = λj ωj ,
∀j.
We summarize here some classical results related to the existence of solutions
for the 2D Navier-Stokes equations; see among others [14].
Theorem 5.1.
(1) Let f , u0 ∈ H . Then there exists a unique solution u of (5.9) and (5.10) such
that, for any T > 0, u ∈ C([0, T ]; H) ∩ L2 (0, T ; V ). Furthermore, for t > 0, u
is analytic in t with values in D(A), and the mapping u0 → u(t) is continuous
from H into D(A).
(2) If u0 ∈ V , then for any T > 0, u ∈ C([0, T ]; V ) ∩ L2 (0, T ; D(A)).
The above theorem allows us to define a C 0 semigroup {S(t)}t≥0 by S(t) :
u0 → u(t). Also the following existence theorem of global attractors in H was
proved in [7]; see also [14].
1554 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
Theorem 5.2.
(1) The semigroup {S(t)}t≥0 associated with (5.9) and (5.10) possesses a global attractor A ⊂ V that is compact, connected and maximal in H , and attracts all
bounded set of H .
(2) The semigroup {S(t)}t≥0 possesses an absorbing set BV (0, ρ1 ) of V , centered at
0, with radius ρ1 , satisfying that, for any bounded set B of H , there exists some
t0 > 0 such that S(t)B ⊂ BV (0, ρ1 ) for t ≥ t0 .
The main result in this section is as follows.
Theorem 5.3. The global attractor A of the semigroup {S(t)}t≥0 given in Theorem 5.2 above is compact, connected and maximal in V , and attracts all bounded set
of V . Consequently it attracts all bounded sets of H in V norm.
❐
Proof. We only have to verify Condition (C), which is a direct consequence
of (1.23) in [4].
6. ATTRACTORS OF THE N AVIER -S TOKES E QUATIONS IN
N ONSMOOTH D OMAINS
This section deals with the existence of the attractor for the two-dimensional
Navier-Stokes equations in a bounded Lipschitz domain Ω with nonhomogenous
boundary condition, see [1].
In operator form, the Navier-Stokes equations can be written as follows:
(6.1)
(6.2)
du
+ νAu + B(u, u) + B(u, ψ) + B(ψ, u) = f − B(ψ, ψ),
dt
u(0) = u0 ∈ H.
Here the operators A, B , the spaces H , V are the same as those defined in Section
5, and the forcing function f ∈ H . The function ψ ∈ C 2 (Ω) ∩ L∞ (Ω̄) is the
lifting function for the nonhomogenuous boundary condition as constructed in
[1]; it satisfies the following inequalities:
(6.3)
(6.4)
|ψ(x)| + |∇ψ(x)| dist(x, ∂Ω) ≤ C1 ,
Z
|∇ψ(x)|2 · dist(x, ∂Ω) dx ≤ C2 .
∀x ∈ Ω,
Ω
In [1], the authors have shown that the semigroup S(t) : H → H (t ≥ 0)
associated with the equations (6.1) and (6.2) possesses a global attractor in H and
a bounded absorbing set in D(A1/4 ). The main objective of this section is to prove
that the semigroup has a global attractor in D(A1/4 ).
To this end, we first state some results selected from [1].
Existence of Global Attractors and Applications
1555
Lemma 6.1 (Hardy’s inequality). There exists a constant C3 such that, for any
u ∈ H01 (Ω),
Z
(6.5)
Ω
|u(x)|2
dx ≤ C3
[dist(x, ∂Ω)]2
Z
Ω
|∇u(x)|2 dx.
Lemma 6.2. There exists a constant C4 such that, for any u ∈ D(A1/4 ),
Z
(6.6)
Ω
|u(x)|2
dx ≤ C4
dist(x, ∂Ω)
Z
Ω
|A1/4 u|2 dx,
|u|4 ≤ C4 |A1/4 u|2 .
(6.7)
Lemma 6.3. Let f ∈ H and let ψ ∈ C 2 (Ω) ∩ L∞ (Ω̄) satisfying (6.3) and
(6.4). Then the problem (6.1)-(6.2) has a unique solution u(t) such that, for any
T > 0,
u ∈ C([0, T ]; H) ∩ L2 ((0, T ); V ),
(6.8)
du
∈ L2 ((0, T ), V 0 ),
dt
and such that, for almost all t ∈ [0, T ] and for any v ∈ V ,
(6.9)
du
, v + ν(Au(t), v) + (B(u, u), v) + (B(u, ψ), v) + (B(ψ, u), v)
dt
= (f , v) − (B(ψ, ψ), v) .
Lemma 6.4. The semigroup S(t) : H → H associated with the equation (6.1)
possesses an absorbing set
B = {u ∈ D(A1/4 ) : |A1/4 u|2 ≤ ρ}
which absorbs all bounded sets of H . Moreover, B absorbs all bounded sets of D(A1/4 )
in the norm of D(A1/4 ).
By Lemma 6.4, for any bounded subset K in H , there exists a t0 > 0 such that
S(t)K ⊂ B as t ≥ t0 .
The main result in this section is as follows.
Theorem 6.5. The semigroup S(t) : H → H associated with the equation (6.1)
possesses a global attractor in D(A1/4 ).
Proof. As in the previous section, for fixed N , let H1 be the subspace spanned
by w1 , . . . , wN , and H2 the orthogonal complement of H1 in H . We write
u = u1 + u2 ,
u1 ∈ H1 , u2 ∈ H2 ,
for any u ∈ H.
1556 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
Again, we only have to verify Condition (C). Namely, we need to estimate
|A1/4 u2 (t)|2 , where u(t) = u1 (t) + u2 (t) is a solution of the equations (6.1)
and (6.2) given in Lemma 6.3.
Multiplying the equation (6.1) by A1/2 u2 , we have
du 1/2
, A u2 + ν(Au, A1/2 u2 ) + (B(u, u), A1/2 u2 ) + (B(u, ψ), A1/2 u2 )
dt
+ (B(ψ, u), A1/2 u2 ) = (f , A1/2 u2 ) − (B(ψ, ψ), A1/2 u2 ).
It follows that
(6.10)
2
1 d 1/4 2
A u2 2 + ν A3/4 u2 2
2 dt
≤ |(B(u, u), A1/2 u2 )| + |(B(u, ψ), A1/2 u2 )|
+ |(B(ψ, u), A1/2 u2 )| + |(f , A1/2 u2 )|
+ |(B(ψ, ψ), A1/2 u2 )|.
We have to estimate each term in the right-hand side of (6.10).
First, by Holder’s inequality and Lemma 6.2,
Z
(6.11)
|(B(u, u), A1/2 u2 )| ≤
Ω
|u| · |∇u| · |A1/2 u2 | dx
≤ |u|4 · |A1/2 u|4 · |A1/2 u2 |2
≤ C42 |A1/4 u| · |A3/4 u|2 ·
≤
≤
C42 ρ
1/4
λm+1
1
1/4
λm+1
|A3/4 u2 |2
· |A3/4 u|2 · |A3/4 u2 |2
4
2
ν
A3/4 u2 2 + 2C4 · ρ A3/4 u2 .
2
2
1
/2
8
νλm+1
Next, using(6.3), (6.5) and the Cauchy inequality,
Z
(6.12)
|(B(u, ψ), A1/2 u2 )| ≤
|u| · |∇ψ| · |A1/2 u2 | dx
Z
|u|
≤ C1
|A1/2 u2 | dx
Ω dist(x, ∂Ω)
Z
1/2
|u|2
≤ C1
dx
· |A1/2 u2 |2
2
Ω [dist(x, ∂Ω)]
Ω
≤ C1 · C3 · |A1/2 u|2 · |A1/2 u2 |2
≤
C1 C3
3/4
u|2
1/4 1/4 |A
λ1 λm+1
· |A3/4 u2 |2
≤
Existence of Global Attractors and Applications
≤
ν
A3/4 u2 2 +
2
8
2C12 C32
1/2 1/2
λ1 λm+1 ν
1557
3/4 2
A u .
2
Similarly by (6.3),
Z
(6.13)
|(B(ψ, u), A
1/2
u2 )| ≤
Ω
|ψ| · |∇u| · |A1/2 u2 | dx
Z
≤ C1
Ω
|∇u| · |A1/2 u2 | dx
≤ C1 |∇u|2 · |A1/2 u2 |2
≤
C1
3/4
u|2
1/4 1/4 |A
λ1 λm+1
≤
3/4 2
2C12
ν
A u .
A3/4 u2 2 +
2
2
1
/
2
1/2
8
λ1 λm+1 ν
· |A3/4 u2 |2
We now estimate (f , A1/2 u2 ) by
(6.14)
1
|(f , A1/2 u2 )| ≤ |f |2 · |A1/2 u2 |2 ≤
≤
ν
A3/4 u2 2 +
2
8
1/4
λm+1
2
1/2
νλm+1
|f |2 · |A3/4 u2 |2
2
f .
2
Finally, we estimate |(B(ψ, ψ), A1/2 u2 )|; by (6.3), (6.4) and Lemma 6.2,
Z
(6.15)
|(B(ψ, ψ), A1/2 u2 )| ≤
Ω
|ψ| · |∇ψ| · |A1/2 u2 | dx
Z
≤ C1
Ω
|∇ψ|2 dist(x, ∂Ω) dx
Z
·
Ω
1/2
≤ C1 · C2
≤
|A
1/2
u2 |
1/2
· C4
2
1/2
1
dist(x, ∂Ω)
· |A3/4 u2 |2
2
ν
A3/4 u2 2 + 2C1 C2 C4 .
2
8
ν
Putting (6.10)-(6.14) together, there exists a constant C7 such that
1/2
dx
1558 Q INGFENG M A , S HOUHONG WANG & C HENGKUI Z HONG
d
A1/4 u2 2 + 3 ν|A3/4 u2 |2 ≤ C7 A3/4 u2 + C7
2
2
1/2
dt
8
λm+1
≤
2 2 A3/4 u1 2 + A3/4 u2 2 + C7
4C7 1/2
λm+1
2
2
4C7 1/2 ≤ C8 λm+1 A1/4 u1 2 + 1/2 A3/4 u2 2 + C7 .
λm+1
Here C7 depends on λm+1 , but it is not increasing as λm+1 increases. Therefore,
we deduce that
(6.16)
d
A1/4 u2 2 + νλm+1 A1/4 u2 2 ≤ C9 (λ1/2 + 1).
m+1
2
2
dt
4
By the Gronwall inequality, the above inequality implies
|A1/4 u2 (t)|2 ≤ |A1/4 u2 (t0 )|2 e−νλm+1 (t−t0 )/4 +
1/2
C10 (λm+1 + 1)
λm+1
−1/2
1
≤ |A1/4 u2 (t0 )|2 e−νλm+1 (t−t0 )/4 + C10 (λm+1 + λ−
m+1 ).
❐
The proof is complete.
Acknowledgement. The work was supported in part by the National Science
Foundation of China under Grant 19971036, by the Office of Naval Research
under Grant N00014-96-1-0425, and by the National Science Foundation under
Grant DMS-0072612. Part of this paper was done while CZ was visiting Indiana
University.
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Q INGFENG M A , C HENGKUI Z HONG :
Department of Mathematics
Lanzhou University
Lanzhou, 730000, P. R. of China.
E- MAIL: [email protected]
S HOUHONG WANG :
Department of Mathematics
Indiana University
Bloomington IN 47405, U. S. A.
E- MAIL: [email protected]
Received : November 26th, 2001.