H2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains

Nonlinear Analysis 56 (2004) 1091 – 1103

www.elsevier.com/locate/na

H 2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains
Caidi Zhaoa;∗ , Yongsheng Lib a Department

of Mathematics and Information Science, Wenzhou Normal College, Wenzhou, Zhejiang 325027, China b Department of Applied Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

Received 29 May 2003; received in revised form 14 October 2003; accepted 18 November 2003

Abstract In this paper, the authors study the long time behavior of a non-Newtonian system in twodimensional unbounded domains. They prove the existence of H 2 -compact attractor for the system by showing the corresponding semigroup is asymptotically compact. ? 2003 Elsevier Ltd. All rights reserved. MSC: 35B41; 35Q35; 76D03 Keywords: H 2 -compact attractor; Non-Newtonian system; Asymptotical compactness; Unbounded domains

1. Introduction The motion of an isothermal, incompressible viscous 9uid can be described by the system ut + (u · ∇)u + ∇ ·  = f;

(1.1)

∇ · u = 0;

(1.2)

where u represents the velocity
This work is supported by the National Science Foundation of China under grant number 10001013 and ZheJiang Province Natural Science Foundation under grant number M103043. ∗

Corresponding author. Tel.: +86-577-86693175. E-mail address: [email protected] (C. Zhao).

0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.11.006

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C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

where p is the pressure, v = (vij ) is the viscous part of the stress tensor, f is the external body force. If the 9uid conforms to the Stokes law, that is, v depends linearly on e = (eij (u)), the symmetric deformation velocity tensor,
 @uj 1 @ui ; (1.3) + vij = eij (u); where eij (u) = @xi 2 @xj the 9uid is called a Newtonian one and the system turns into the well-known Navier– Stokes equations. In particular, if the viscosity of the 9uid can be neglected, v = 0, the system is reduced to the Euler equations. Many 9uid materials, for examples, liquid foams, polymeric 9uids such as oil in water, blood, etc., do not conform to the Stokes law. Their viscous stress tensors are nonlinear in e = (eij ) and may also depend on the derivatives of e. These 9uids are called non-Newtonian. A typical model is the bipolar viscous non-Newtonian 9uid whose constitutive relations have the form vij = 20 ( + |e|2 )−=2 eij − 21 Ieij ; ijk = 21

@eij ; @xk

0 ; 1 ;  ¿ 0;

k = 1; 2;

(1.4) (1.5)

 1=2 2 where  ¿ 0 is a constant and |e| = |e | . ij i; j In this paper, we are going to study the isothermal, incompressible, bipolar, viscous non-Newtonian 9uid on a two-dimensional in
t ¿ 0;

∇ · u = 0;

x ∈ ;

u = 0; u = u0 ;

(1.6) t ¿ 0;

ijk nj nk = 0; x ∈ ;

x ∈ @;

t = 0;

(1.7) t ¿ 0;

(1.8) (1.9)

where the summation convention of repeated indices is used, n = (n1 ; n2 ) is the exterior unit normal to the boundary @ of . We will concentrate our attention on 0 ¡  ¡ 1. We refer to [1,4,7–9,11] for detailed background. There is series works on the existence and uniqueness, regularity and long time behavior of weak solutions of the above equations (see [2,3,5,6,10,13,16]). In particular, Bloom and Hao [6] proved the existence of the global attractor. Because the domain of the spatial variables is unbounded, the embedding between the usual Sobolev spaces is no longer compact. To recover the compactness, [6] assumed f belongs to certain weighted space. Later on in [16], the authors avoided using the weighted space and proved the existence of global attractor in the H space. In this paper we use the ideas in [12,14] to show the existence of the global attractor in the V space. The new diNculty comes from the more nonlinear terms in the equation due to the lack of the compactness of the Sobolev embeddings. We use functional analysis to prove

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the convergence of these nonlinear terms and prove the corresponding semigroup is asymptotically compact. The main result of this paper is Theorem 1.1. Let f ∈ H . Then the semigroup {S(t)}t¿0 associated with (1.6)–(1.9) possesses a global attractor A ⊂ V satisfying (1) (Compactness) A is compact in V ; (2) (Invariance) S(t)A = A, t ¿ 0; (3) (Attractivity) ∀ B ⊂ V bounded, limt→∞ dist V (S(t) B; A ) = 0, where distV is the distance in the norm topology of V (see below for the notations). The rest of the paper is organized as follows. Section 2 is preliminary. We present the functional setting and existence results of solutions. In Section 3 we show the asymptotic compactness property of the semigroup and prove our main result. Throughout this paper we denote by Lp (); W m; p () both the scalar and vectorvalued Lebesgue and Sobolev spaces, by  · p and  · m; p their norms, especially  ·  =  · 2 ; H m () = W m; 2 (). That a vector is in some space X means all its components are in X . Denote O ∇ · ’ = 0 in ; ’ = 0 on @} V = {’ = (’1 ; ’2 ) ∈ C0∞ (); H is the closure of V in L2 (), V is the closure of V in H 2 (), V  and H  the dual spaces of V and H , respectively. If we identify the dual space H  with H itself, then V ,→ H = H  ,→ V  , with dense and continuous injections. We use “,→” Denote the imbedding between space, “*” weakly convergent and “→” strongly convergent, C the generic constant. The summation convention of repeated indices is used in the whole paper. 2. Preliminaries Let

 a(u; v) =



@eij (u) @eij (v) d x; @xk @xk

u; v ∈ V:

(2.1)

Lemma 2.1 (Bloom and Hao [5]). There exist positive constants C1 and C2 such that 
@eij (u) @eij (u) 2 6 C2 u2H 2 () ∀u ∈ V: ; (2.2) C1 uH 2 () 6 @xk @xk We see that a(·; ·) de
Au; v = a(u; v)

∀v ∈ V

where ·; · is the dual product between V  and V .

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C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

Let D(A) = {’ ∈ V; A’ ∈ H }, then D(A) is a Hilbert space and A is also an isometry from D(A) to H . In fact, A = PI2 , where P is the projector from L2 () into H . We also de
∀u; v; w ∈ V:

For any u ∈ V ,

B(u); w = b(u; u; w)

∀w ∈ V

(2.4)

de
N (u); v = 

Then N (u) ∈ V and N (·) : V → V  is continuous. When u ∈ D(A), N (u) can be extended to H via 

N (u); v = − {∇ · [(u)e(u)]} · v d x ∀v ∈ H: 



2

Moreover, ∀u ∈ L (0; T ; V ), N (u(t)) ∈ L2 (0; T ; V  ) (see [5]). When u ∈ L2 (0; T ; D(A))∩ L∞ (0; T ; H ), N (u(t)) ∈ L2 (0; T ; H ). With the above notations, the weak formula of (1.6)–(1.9) can be equivalently put into the following functional equations in V  : ut + 21 Au + B(u) + N (u) = f;

t ¿ 0;

u(0) = u0 :

(2.6) (2.7)

Lemma 2.2 (Bloom and Hao [6]). (i) For given f ∈ L2 (0; T ; H ) and u0 ∈ V; T ¿ 0, there exists a unique weak solution to (2.6) and (2.7) satisfying u ∈ C([0; T ]; V ) ∩ L2 (0; T ; D(A));

ut ∈ L2 (0; T ; H )

(2.8)

and the solution operator S(t) : u0 → u(t) is continuous from V into itself. (ii) Let f ∈ L∞ (0; ∞; H ). Then there exists a - ¿ 0 such that, for any B ⊂ V bounded, there exists a TB ¿ 0 such that S(t)u0 V 6 -

∀u0 ∈ B;

t ¿ TB :

Recalling the de
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Thus, we obtain C1 u2H 2 6 (Au; u) 6 Au u 6 Au uH 2 ; C1 uH 2 6 Au;

(2.9)

∀u ∈ D(A):

(2.10)

At the same time, (2.8) implies that, for u(t) = S(t)u0 ; u0 ∈ V ,  T sup u(t)H 2 ¡ + ∞; Au(t)2 dt ¡ ∞:

(2.11)

t∈[0;T ]

0

Lemma 2.3. Let u0; n → u0 in H , u(t) = S(t)u0 , un = S(t)u0; n be the corresponding solutions. Then ∀T ¿ 0, we have un (t) → u(t)

in L2 (0; T ; V ):

(2.12)

Proof. From (2.9) we obtain un − u2H 2 6 Cun − u A(un − u):

(2.13)

Integrating (2.13) from 0 to T , we have by Young inequality that 1=2  T 1=2  T  T 2 2 2 un − u dt A(un − u) dt : (2.14) un − uH 2 dt 6 C 0

0

0

From the continuity of S(t) in H (see [16]), we see that un → u holds in H . We also have from (2.11)  T 1=2 A(un − u)2 dt ¡ ∞: 0

Therefore, we obtain from (2.14) that  T lim un − u2H 2 dt = 0; n→∞

0

and thus un (t) → u(t) holds in L2 (0; T ; V ). The proof of the lemma is completed. Lemma 2.4. Let u0; n * u0 holds in V . Then S(t)u0; n * S(t)u0

∀t ¿ 0

(2.15)

∀T ¿ 0

(2.16)

holds in V ; S(t)u0; n * S(t)u0 holds in L2 (0; T ; D(A)). The proof of this lemma is very similar with that of Lemma 2.2 in [12] and is omitted here.

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C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

3. Asymptotical compactness of S(t) We assume in the sequel that f ∈ H is independent of t. Then S(t) is a semigroup in H . In this section, we prove the asymptotic compactness of S(t). First de
(3.1)

Let k1 = C12 =C2 , u(t) = S(t)u0 ; u0 ∈ V . Multiplying both sides of (2.6) by Au we obtain

 @eij (u) @eij (u) d @eij (u) @eij (u) + k 1 1 ; ; dt @xk @xk @xk @xk = (f; Au) − B(u); Au − N (u); Au −
(3.2)

(3.3)

where O K(f; u) = (f; Au) − b(u; u; Au) − N (u); Au −
(3.4)

Thus {S(tn )un } is weakly precompact in V and there exists a subsequence (still denote by {S(tn )un }) such that S(tn )un * w in V

(3.5)

for some w ∈ B(0; -). Similarly for each T ∈ N, we also have S(tn − T )un ∈ B(0; -) for tn ¿ T + TB ;

(3.6)

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Thus {S(tn − T )un } is weakly precompact in V and there exists a subsequence (still denote by {S(tn − T )un }) such that S(tn − T )un * wT in V

∀T ∈ N;

(3.7)

and wT ∈ B(0; -). By Lemma 2.4 we obtain S(T )S(tn − T )un = S(tn )un * S(T )wT in V

∀T ∈ N

and thus from the uniqueness of limit we have w = S(T )wT

∀T ∈ N:

From (3.5) and Lemma 2.1 we
(3.8)

(3.9)

(3.10)

For T ∈ N and tn ¿ T we have by (3.3) that
 @eij (S(tn )un ) @eij (S(tn )un ) ; @xk @xk
 @eij (S(T )S(tn − T )un ) @eij (S(T )S(tn − T )un ) = ; @xk @xk
 @eij (wn (0; T )) @eij (wn (0; T )) −k1 1 T = e ; @xk @xk  T − e−k1 1 (T −s) b(wn (s; T ); wn (s; T ); Awn (s; T )) ds  +

0

T

0

 −  +

T

0

0

T

e−k1 1 (T −s) (f; Awn (s; T )) ds e−k1 1 (T −s)
where wn (s; T ) = S(s)S(tn − T )un . From (3.6) and Lemma 2.1 we
(3.11)

(3.12)

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C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

Also by Lemma 2.4, we deduce from (3.7) that wn (·; T ) = S(·)S(tn − T )un * S(·)wT in V: −k1 1 (T −s)

2

f ∈ L (0; T ; V ), we
f; wn (s; T ) ds = lim n→∞

(3.13)



0

0

(3.14)

Moreover, since < · = is a norm in V equivalent to  · V and 0 ¡ e−k1 1 T 6 e−k1 1 (T −s) 6 1 ∀s ∈ [0; T ]; T we see that { 0 e−k1 1 (T −s) < · =2 ds}1=2 is a norm in L2 (0; T ; V ) equivalent to the usual norm. Thus from (3.13) we deduce that  T  T e−k1 1 (T −s)
0

0

Now, we are going to show  T e−k1 1 (T −s) b(wn (s; T ); wn (s; T ); Awn (s; T )) ds lim n→∞

0

 =  lim

n→∞

T

0 T

0

 =

e−k1 1 (T −s) b(S(s)wT ; S(s)wT ; AS(s)wT ) ds;  

T

0





(3.16)

e−1 k1 (T −s) {∇ · [(wn (s; T ))e(wn (s; T ))]} · Awn (s; T ) d x ds e−k1 1 (T −s) {∇ · [(S(s)wT )e(S(s)wT )]} · AS(s)wT d x ds:

(3.17)

Denote u0; n = S(tn − T )un ; u0 = wT . Recall that, by the boundedness of S(t) in V and the asymptotic compactness of S(t) in H (see [16]), u0; n is bounded in V and converges in H . To prove (3.16) and (3.17), we need to prove the following two lemmas. Lemma 3.1. Let u0; n * u0 in V and u0; n → u0 in H . Then ∀T ¿ 0,  T e−k1 1 (T −s) b(S(t)u0; n ; S(t)u0; n ; AS(t)u0; n ) ds lim n→∞

0

 =

T

0

e−k1 1 (T −s) b(S(t)u0 ; S(t)v0 ; AS(t)u0 ) ds:

(3.18)

Lemma 3.2. Suppose the conditions in Lemma 3.1 hold. Then ∀T ¿ 0,  T e−k1 1 (T −t) {∇ · [(S(t)u0; n )e(S(t)u0; n )]} · AS(t)u0; n d x ds lim n→∞

0

 =

0



T





e−k1 1 (T −t) {∇ · [(S(t)wT )e(S(t)wT )]} · AS(t)wT d x ds:

(3.19)

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Proof of Lemma 3.1. Denote S(t)u0; n = un (t); S(t)u0 = u(t), then we need to show  T e−k1 1 (T −s) {b(un (t); un (t); Aun (t)) − b(u(t); u(t); Au(t))} ds = 0: (3.20) lim n→∞

0

In fact, we have

 T



k1 1 (T −s)



e {b(u (t); u (t); Au (t)) − b(u(t); u(t); Au(t))} ds n n n



0



6

T

e

0



+

−k1 1 (T −s)

T

0



+

T

0



b(un − u; un ; Aun ) ds





e−k1 1 (T −s) b(u; un − u; Au) ds

e

−k1 1 (T −s)



b(u; u; A(un − u)) ds

= K1 + K2 + K3 :

(3.21)

Firstly, since H 2 () ,→ L∞ (), we have by HRolder inequality that  T  T un − u un L∞ Aun  dt 6 C un − u un H 2 Aun  dt K1 6 C 0

0

 6C  6C

T

0 T

0

un − u2 un 2H 2 dt

1=2  0

T

Aun 2 dt

1=2

un − u2 dt → 0 as n → ∞;

(3.22)

here (2.11) and (2.12) are used. Secondly, we have in a similar way that  T un − u uL∞ Au dt K2 6 C 0

 6C

T

0

 6C

T

0

 6C

uH 2 un − u Au dt

0

T

u2H 2 Au2 un −

u2H 2

1=2  dt

0

T

2

1=2

un − u dt

1=2 dt

→ 0;

n → ∞:

Finally, from (2.11), we see that B(u) ∈ L2 (0; T ; H ). By Lemma 2.4 we have



 T



B(u); Aun − Au dt

= 0: lim K3 =

lim n→∞ n→∞ 0

From (3.21)–(3.24), we have (3.20) and thus the lemma.

(3.23)

(3.24)

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C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

Proof of Lemma 3.2. Denote S(t)u0; n = un (t); S(t)u0 = u(t). We need to prove the following:  T lim e−k1 1 (T −s) {∇ · [(un )e(un )]} · Aun d x ds n→∞

=

0



0



 T

e−k1 1 (T −s) {∇ · [(u)e(u)]} · Au d x ds:

(3.25)

In fact, we have  T e−k1 1 (T −s) {∇ · [(un )e(un )]} · Aun − {∇ · [(u)e(u)]} · Au d x ds 0



=

 T 0

+



e−k1 1 (T −s) {∇ · [(un )e(un ) − (u)e(u)]} · Aun d x ds

 T 0



e−k1 1 (T −s) {∇ · [(u)e(u)]} · (Aun − Au) d x ds

= M1 + M2 :

(3.26)

Now set F(s) = ( + |s|2 )−=2 s;

s = (s1 ; s2 ; s3 ; s4 ) ∈ R4 ;

whose


F(b) − F(a) =

1

0

∀s ∈ R4 ; ∀s ∈ R4 :

DF(a + (b − a))(b − a) d:

Taking a = e(u) = (eij (u)); b = e(un ) = (eij (un )), applying the integration by parts
 T



−k1 1 (T −s)

|M1 | =

e {∇ · [F(eij (un )) − F(eij (u))]} · Aun d x ds

0



6 C(; )

 T 0

 6 C(; )

0

 6 C(; )

 T

0

T

((|D2 u| + |D2 un |)|eij (un − u)| + |D2 (un − v)|)|Aun | d x ds

((D2 un  + D2 u)e(un − u)∞ + D2 un − D2 u)Aun  ds (e(un − u)∞ + D2 un − D2 u)Aun  ds;

where (2.11) is used. By Gagliardo–Nirenberg inequality e(un − u)∞ + D2 un − D2 u 6 Cun − u1=2 (un D(A) + uD(A) )1=2 ;

(3.27)

C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

we have



|M1 | 6 C(; )

T

0


 6 C(; )

0

→ 0;

1101

un − u1=2 (un D(A) + uD(A) )3=2 ds T

un − u2 ds

n → ∞;

1=4
 0

T

(un D(A) + uD(A) )2 ds

3=4

∀T ∈ N:

Note that, e−k1 1 (T −s) {∇ · [(u)e(u)]} ∈ L2 (0; T ; L2 ()); By Lemma 2.4, we have  T M2 = e−k1 1 (T −s) {∇ · [(u)e(u)]} · (Aun − Au) d x ds → 0 0

(3.28)



as n → ∞; ∀T ∈ N . The proof of Lemma 3.2 is completed. Note that S(tn − T )un is bounded in V and converges in H , by Lemmas 3.1 and 3.2 immediately we have (3.16) and (3.17). Now we continue to prove (3.10). Taking (3.12) and (3.14)–(3.17) into account, we obtain
 @eij (S(tn )un ) @eij (S(tn )un ) lim sup ; @xk @xk n→∞  T 6 (C2 -)2 e−k1 1 T + e−k1 1 (T −s) (f; AS(s)wT ) ds  −  +

0

T

0 T

e−k1 1 (T1 −s) {
0



e−k1 1 (T −s) {∇ · [(S(s)wT )e(S(s)wT )]} · AS(s)wT d x ds:

(3.29)

At the same time, by applying (3.3) to w = S(T )wT we obtain
 @eij (w) @eij (w) ; @xk @xk  T = e−k1 1 (T −s) {(f; AS(s)wT ) − b(S(s)wT ; S(s)wT ; AS(s)wT )} ds 0

− +

 0

T

e

−k1 1 (T −s)

 T 0



2


−k1 1 T




@eij (wT ) @eij (wT ) ; @xk @xk



e−k1 1 (T −s) {∇ · [(S(s)wT )e(S(s)wT )]} · AS(s)wT d x ds:

(3.30)

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C. Zhao, Y. Li / Nonlinear Analysis 56 (2004) 1091 – 1103

From (3.29) and (3.30) we have
 @eij (S(tn )un ) @eij (S(tn )un ) ; lim sup @xk @xk n→∞

@eij (wT ) @eij (wT ) 2 6 (C2 -) − e−k1 1 T ; @xk @xk 
@eij (w) @eij (w) ∀T ∈ N: + ; @xk @xk Recall that wT ∈ B(0; -) is bounded, let T → ∞ in (3.31), we obtain

 @eij (S(tn )un ) @eij (S(tn )un ) @eij (w) @eij (w) 6 ; ; lim sup @xk @xk @xk @xk n→∞ and (3.10) holds. From (3.9) and (3.10) we have

 @eij (S(tn )un ) @eij (S(tn )un ) @eij (w) @eij (w) = : ; ; lim n→∞ @xk @xk @xk @xk

(3.31)

(3.32)

(3.33)

Therefore, we deduce from Lemma 2.1 and (3.33) that lim S(tn )un H 2 = wH 2 :

n→∞

(3.34)

Because V is a Hilbert space, from (3.34) and (3.5) we see that lim S(tn )un − wH 2 = 0:

n→∞

Therefore, we have proved the following theorem. Theorem 3.3. Let f ∈ H . Then the semigroup {S(t)}t¿0 associated with (1.6)–(1.9) is asymptotically compact in V . Proof of Theorem 1.1. From Lemma 2.2 we see that the solution operator of (1.6)– (1.9) generates a continuous semigroup {S(t)}t¿0 in V which has a bounded absorbing set B(0; -) in V . From Theorem 3.3 we see the semigroup is asymptotically compact in V . Therefore, by the standard theory (see [15, Chapter I.1.1]), we
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