global attractor for the navier–stokes equations with the heat convection in a cylindrical pipe

Research Article

Received 14 November 2011 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2590MOS subject classification: 35Q35, 76D03, 76D05

Global attractor for the Navier–Stokesequations with the heat convection in acylindrical pipe

Piotr Kacprzyk*†

Communicated by M. A. Lachowicz

Global existence of regular solutions to the Navier–Stokes equations coupled with the heat convection in a cylindrical pipehas already been shown. In this paper, we prove the existence of the global attractor to the equations and convergence oftheir solutions to a stationary one. Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: Navier–Stokes equations slip boundary conditions; cylindrical domains global existence of regular solutions

1. Introduction

We consider the problem

v,t C v � rv � divT .v, p/D ˛.�/f in ��RC,

divv D 0 in ��RC,

�,t C v � r� � �� D 0 in ��RC,

Nn �D.v/ � N�˛ D 0, ˛ D 1, 2, on S�RC,

v � NnD 0, Nn � r� D 0 on S�RC,

vjtD0 D v0, � jtD0 D �0 in �,

(1.1)

where ˛ 2 C2.R/,� � R3 is a bounded domain, SD @�, v D .v1.x, t/, v2.x, t/, v3.x, t// 2 R3 is the velocity of the fluid, � D �.x, t/ 2 Rthe temperature, p D p.x, t/ 2 R the pressure, f D .f1.x, t/, f2.x, t/, f3.x, t// 2 R3 the external force, � > 0 the constant viscosity coeffi-cient, and � > 0 the constant heat coefficient. We introduce the Cartesian system x D .x1, x2, x3/ such that the cylinder� is parallel tothe x3 axis. We assume that SD S1[ S2, where S1 is the part of the boundary that is parallel to the axis x3, and S2 is perpendicular to x3.

Hence,

S1 Dn

x 2R3 : ' .x1, x2/D c0, �a� x3 � ao

,

S2.�a/Dn

x 2R3 : ' .x1, x2/ < c0, x3 D�ao

,

S2.a/Dn

x 2R3 : ' .x1, x2/ < c0, x3 D ao

,

where a, c0 are positive given numbers and '.x1, x2/D c0 describes a sufficiently smooth closed curve in the plane x3 D const and Nn isthe unit outward vector normal to the S, N�˛ , ˛ D 1, 2, are tangent vectors to S.

By T .v, p/, we denote the stress tensor

T .v, p/D �D.v/� pI,

Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland*Correspondence to: Piotr Kacprzyk, Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012

P. KACPRZYK

where I is the unit matrix and

D.v/D˚

vi,xj C vj,xi

�i,jD1,2,3

,

is the dilatation tensor.In [1], we prove global existence of solutions to Equation (1.1). Then we examine problem (1.1) step by step in time. Therefore, instead

of Equation (1.1), we consider the systems of problems

v,t C v � rv � divT .v, p/D ˛.�/f in �T.kC1/ D�� .kT , .kC 1/T/,

divv D 0 in �T.kC1/,

�,t C v � r� � �� D 0 in �T.kC1/,

Nn �D.v/ � N�˛ D 0, ˛ D 1, 2, on ST.kC1/ D S� .kT , .kC 1/T/,

v � NnD 0, Nn � r� D 0 on ST.kC1/,

vjtDkT D v.kT/, � jtDkT D �.kT/ in �,

(1.2)

where k 2N0 DN [ f0g and v.kT/, �.kT/ is calculated as a trace of v and � from the interval ..k � 1/T , kT�.Now, we formulate the main results from [1].

Theorem 1.1Assume that f 2 L1.RC, L6=5.�//, v.0/ 2 L2.�/. Then

kv.t/kL2.�/ � c�kfkL2.RC ,L6=5.�//

Ckv.0/kL2.�/

�for any t > 0.

Let us introduce the quantities

G1.kT/DkfkL1.kT ,.kC1/T ;H1.�//Ckv.kT/kH1.�/

Ck�.kT/kH1.�/Ckv,x3.kT/kH1.�/Ck�,x3 .kT/kH1.�/

and

.kT/Dkf,x3kL2.��.kT ,.kC1/T// Ckf3kL2.S2�.kT ,.kC1/T//

Ckv,x3.kT/kL2.�/Ck�,x3.kT/kL2.�/

.

Theorem 1.2Assume that G1 is finite, and is sufficiently small. Then there exists a solution to problem (1.2) such that

v, v,x3 , � , �,x3 2W2,12 .�� .kT , .kC 1/T//,

rp,rp,x3 2 L2.�� .kT , .kC 1/T//

and

1XiD0

��@ix3

v��

W2,12 .��.kT ,.kC1/T//

C��@i

x3��

W2,12 .��.kT ,.kC1/T//

C��@i

x3rp��

L2.��.kT ,.kC1/T//

�� ˇ.G1.kT//� A.k, T/,

(1.3)

where ˇ is an increasing positive function.

From the form of G1, we see that A.k, T/ does not increase with T in the interval ŒkT , .kC 1/T�.

Main Theorem 1Assume that

kf .t/kH1.�/ � e�ıtkf .0/kH1.�/,

kf,x3.t/kL2.�/� e�ıt kf,x3 .0/kL2.�/

,

f3jS2D 0, ı > 0.

(1.4)

Then

k˛..kC 1/T/kH1.�/ � k˛.kT/kH1.�/, (1.5)

where ˛ replaces v, � , v,x3 , �,x3 , for any k 2N [ f0g. Moreover, Theorem 1.2 implies global existence of solutions to problem (1.1).


P. KACPRZYK

Now, we formulate the main result of this paper. Let ı > 0 be fixed and NV D f.v, �/ 2 C1.�/ : divv D 0, v � NnjS D 0, r� � njS D0, jv,x3 j2,� < ı

�, then

H� closure of NV in the L2 � norm,

V � closure of NV in the H1 � norm.

Let v.t/, �.t/ be a weak global solution to problem (1.1) with initial data v0 D v.0/, �0 D �.0/. Then S.t/ : H ! H is a semiprocessdefined by .v.t/, �.t//D S.t/.v.0/, �.0//.

Main Theorem 2There exists a global attractor A in V for the semiprocess fS.t/gt�0 (see also (3.1)). The attractor is bounded in H1.�/ and compact andconnected in V . It attracts bounded sets in V .

The existence of global attractor for the Navier–Stokes equations was proved in [2]. Global existence of regular solutions to theNavier–Stokes equations guaranteeing the existence of a semiprocesses is guaranteed by results from papers [3–6]. In this paper, themethods for the Navier–Stokes equations were extended to the more general system (1.1). The existence of global regular solutions toproblem (1.1) guaranteeing the existence of semigroup is shown in [1]. We have to underline that the attractor has a restricted domainbecause ı in the definition of NV must be sufficiently small.

2. Notation and auxiliary results

We denote

�Tk D�� ..k � 1/T , kT/,

�T2,T1 D�� .T1, T2/ .(2.1)

Lemma 2.1 ( Korn inequality; see [7])Assume that

R� jD.v/j

2dx <1, v � NnjS D 0, divv D 0. If� in not axially symmetric, then there exists a constant c1 > 0 such that

kvk2H1.�/

� c1kD.v/k2L2.�/

. (2.2)

First, we need the estimates and the uniform Gronwall inequality.

Lemma 2.2 (see [8])A solution v 2 H2.�/ of the elliptic problem

divD.v/D f ,

v � NnjS D 0,

Nn �D.v/ � N�˛ jS D 0, ˛ D 1, 2,

satisfies the estimate

kvkH2.�/ � ckfkL2.�/. (2.3)

Lemma 2.3 (see [8])A solution .v, p/ 2 H2.�/� H1.�/ of the elliptic problem

divT .v, p/D f ,

divv D 0,

v � NnjS D 0,

Nn �D.v/ � N�˛ jS D 0, ˛ D 1, 2,

satisfies the estimate

kvkL2.�/CkrpkL2.�/ � ckfkL2.�/. (2.4)

Lemmas 2.2, 2.3 follow from the theory of boundary value problems for elliptic systems [8].

Lemma 2.4 (uniform Gronwall inequality; see [3, Ch. 3, Lemma 1.1])Let g, h, y : Œt0,1/! .0,1/ be continuous functions. Assume that for some r > 0 and all t > t0, the inequalities

y0.t/� g.t/y.t/C h.t/,

tCrZt

g.s/ds� a1,

tCrZt

h.s/ds� a2,

tCrZt

y.s/ds� a3


P. KACPRZYK

hold. Then y satisfies the uniform estimate

y.tC r/��a3

rC a2

�ea1 for t > t0. (2.5)

In [1], we proved the Lemmas

Lemma 2.5Assume that �.0/� c1 > 0. Then

�.t/� c1, t � 0. (2.6)

Lemma 2.6Assume that �.0/� c2 > 0. Then

�.t/� c2, t � 0. (2.7)

In this paper, we will show the existence of the global attractor for problem (1.1). We will apply methods from [9] and use the theoryof semiprocesses because in our case the external force f depends on time.

3. Existence of the global attractor

In this section, we prove the existence of the global attractor to the problem (1.1). We start by recalling some facts and definitionsfrom [9, Ch. 4].

Let us rewrite Equation .1.1/1 in the abstract form

v,t D A.v, t/D A�.t/.v/, t 2RC,

where the right-hand side depends explicitly on the time symbol .t/, which is the collection of all time-dependent coefficients of theequation (in the Navier–Stokes equations that will be the time-dependent external forces).

By ‰, we denote some metric of Banach space that contains the values of .t/ for a.e. t 2 RC. Moreover, we assume that .t/, as afunction of t, belongs to a topological space„ :D f�.t/, t 2RC : �.t/ 2‰ for a.e. t 2RCg.

Replacing the symbol .t/ by the shifted symbol .tCh/ should not change the attractor, hence we introduce a translation invariantsubspace†„ called the symbol space. Translation invariance means that for all .t/ 2† the relation T.h/.t/D .tC h/ 2† holds,where T.h/ : „! „ is the shift operator. In our case, it will be convenient to set † D †.0/ � f0.tC h/ : h 2 RCg, where 0 is thetime symbol of the initial equation and must be closure of the set f0.tC h/ : h 2RCg in the topology of„.

Let v.t/ be a weak global solution of problem (1.1) with initial data v0 D v.0/. We define a family of semiprocesses fU� .t, r/gt�r�0

acting on H, U.t, �/ : H! H by the formula

v.t/D U� .t, �/v.�/, (3.1)

where v.�/ is the initial condition and† 3 .t/D f .�, t/ is the external force.By B.H/, we denote the family of all bounded sets of H.

Definition 3.1 (see [9, Ch. 4, Definition 3.2])A family of semiprocesses, fU� .t, �/gt��0, 2†, is said to be uniformly bounded if for any B 2 B.H/we have[

�2†

[�2RC

[t��

U� .t, �/B 2 B.H/.

Definition 3.2 (see [9, Ch. 4, Definition 3.3])A set B0 2 H is said to be uniformly absorbing for the family of semiprocesses fU� .t, �/gt��0, 2 † if for any � 2 RC and forevery B 2 B.H/ there exists t0 D t0.� , B/, such that

S�2† U� .t, �/B B0 for all t � t0. If the set B0 is compact, we call the family of

semiprocesses uniformly compact.

Definition 3.3 (see [9, Ch. 4, Definition 3.4])A set P belonging to H is called uniformly attracting for the family of semiprocesses fU� .t, �/gt��0, 2 †, if for an arbitrary fixed� 2RC,

limt!1

�sup�2†

distH .U� .t, �/B, P/

�D 0.

If the set P is compact, we call the family of semiprocesses uniformly asymptotically compact.

Definition 3.4 (see [9, Ch. 4, Definition 3.5])A closed set A� � H is called the uniform attractor of the family of semiprocesses fU� .t, �/gt��0, 2 †, if it is uniformly attractingand contained in any closed uniformly attracting set of the family fU� .t, �/gt��0, 2†.


P. KACPRZYK

The existence of the global attractor follows from the following theorem:

Theorem 3.1 (see [9, Ch. 4, Theorem 3.1])If a family of semiprocesses fU� .t, �/gt��0, 2 †, is uniformly asymptotically compact, then it possesses a uniform global attractorA†. The set A† is compact in H.

The main result in this section reads.

Theorem 3.2There exists a global attractor A† in H for the family of semiprocesses fU� .t, �/gt��0, 2†, defined by (3.1). The attractor is boundedin V and compact and connected in H. It attracts bounded sets in H.

To prove this theorem, we need some estimates.

Lemma 3.1There exists a bounded and absorbing set in H for the family of semiprocesses fU� .t, �/gt��0, 2†.

ProofIn view of Theorem 1.1 and Lemmas 2.5 and 2.6, we see that if .v0, �0/ 2 B.0, r/ � H, where B.0, r/ is the ball centered at 0 with radiusr > 0, then

limt!1

sup k.v.t/, �.t//kL2.�/ � C.r/. (3.2)

Hence, for every .v0, �0/ 2 B.0, r/ there exists t0.r/ such that .v.t/, �.t// 2 B.0, %1.r// � H for all t > t0.r/, where C < %1 � 2C. Thisconcludes the proof. �

Lemma 3.2There exists a bounded and absorbing set V for the family of semiprocesses fU� .t, �/gt��0, 2†.

ProofWe multiply Equation (1.1)1 by divT .v, p/ and integrate over� to obtainZ

�

v,t � divT .v, p/dx �

Z�

jdivT .v, p/j2dxC

Z�

v � rv � divT .v, p/dx D

Z�

˛.�/f divT .v, p/dx. (3.3)

According to the definition of T .v, p/, we have

��

4

d

dt

Z�

jD.v/j2dx �

Z�

jdivT .v, p/j2dxC

Z�

v � rv � divT .v, p/dx D

Z�

˛.�/f divT .v, p/dx.

We estimate the integral I1 DR� v � rv � divT .v, p/dx. From Lemmas 2.2 and 2.3, we obtain

I1 � ckvkL6.�/krvkL3.�/

�kdivD.v/kL2.�/CkrpkL2.�/

� ckvkL6.�/krvkL3.�/

�kdivD.v/kL2.�/CkfkL2.�/

D I01.

Using interpolation inequality, the Young inequality and Lemma 2.2 yields

I01 � ckvkL6.�/krvk1=2L2.�/

krvk1=2L6.�/


� ckvk3=2

H1.�/kvk1=2

H2.�/


� ckvk3=2

H1.�/

�kdivD.v/k3=2

L2.�/Ckfk3=2

L2.�/

�� "

�kdivD.v/k2

L2.�/Ckfk2

L2.�/

�C c.1="/kvk6

H1.�/.

Hence, we obtain from Equation (3.3) the inequality

d

dt

Z�

jD.v/j2dxC

Z�

jdivT .v, p/j2dx �

ˇ̌̌ˇ̌̌Z�

˛.�/f � divT .v, p/dx

ˇ̌̌ˇ̌̌C c.1="/

�kvk6

H1.�/Ckfk2

L2.�/

�C "kdivD.v/k2

L2.�/. (3.4)

Using the inequality kdivD.v/kL2.�/ � ckdivT .v, p/kL2.�/ and Lemma 2.6, we obtain from Equation (3.4)

d

dtkD.v/k2

L2.�/CkdivD.v/k2

L2.�/� c.1="/

�kvk6

H1.�/Ckfk2

L2.�/

�. (3.5)


P. KACPRZYK

Next, using in Equation (3.5), the inequality

kD.v/kL2.�/ � ckdivD.v/kL2.�/,

we obtain

d

dtkD.v/k2

L2.�/CkD.v/k2

L2.�/� c

�kvk6

H1.�/Ckfk2

L2.�/

�D J.t/.

From Main Theorem 1, we obtain

.kC1/TZkT

kD.v.x, t//k2L2.�/

dt � a3,

.kC1/TZkT

J.t/dt � a2.

Applying now Lemma 2.4 yields

kD.v.t//k2L2.�/

�a3

TC a2, (3.6)

where t � T .Finally, using the Korn inequality we obtain from (3.6)

kv.t/k2H1.�/

�a3

TC a2, for t > T . (3.7)

Now, we multiply Equation (1.1)3 by�� and integrate over� to obtain

d

dtkr�k2

L2.�/C �k��k2

L2.�/� ckvkL6.�/kr�kL3.�/k��kL2.�/

� kvkH1.�/kr�k1=2L2.�/

kr�k1=2L6.�/

k��kL2.�/.(3.8)

Then from Equation (3.8) we obtain

d

dtkr�kL2.�/Ck��k

2L2.�/

� "k��k2L2.�/

C c.1="/kvk4H1.�/

k�k2H1.�/

. (3.9)

From Equation (3.9) and inequality ddt k�k

2L2.�/

� 0, we obtain

d

dtk�k2

H1.�/Ck�k2

H1.�/� ckvk4

H1.�/k�k2

H1.�/. (3.10)

Now, using Main Theorem 1 in Equation (3.10), we obtain

.kC1/TZkT

k�k2H1.�/

dt � a03,

.kC1/TZkT

kvk4H1.�/

dt � a01.

Applying Lemma 2.4 yields

k�.t/k2H1.�/

�a03T

ea01 , where t > T . (3.11)

In view of Equations (3.7) and (3.11), we see that if .v0, �0/ 2 B.0, r/ � H, then exists t1.r/ such that .v.t/, �.t// 2 B.0, %2.r// � V for allt > t1.r/, where

a3

TC a2C

a03T

ea01 < %2 � 2

�a3

TC a2C

a03T

ea01

�.

This concludes the proof. �

Proof of Theorem 3.2. We take %Dmaxf%1, %2g. Then due to Lemmas 3.1 and 3.2, there exists an absorbing set B.0, %/ that is boundedin V and bounded and compact in H. From Theorem 3.1, we conclude the proof.


P. KACPRZYK

4. Convergence to stationary solutions

First, we examine the stationary problem (1.1)

v1 � rv1 � divT .v1, p1/D ˛ .�1/ f1 in �,

divv1 D 0 in �,

v1 � r�1 � ��1 D 0 in �,

Nn �D .v1/ � N�˛ D 0, ˛ D 1, 2 on S,

Nv1 � NnD 0, Nn � r�1 D 0, on S.

(4.1)

Multiplying Equation (4.1)1 by v1, integrating the result over �, next by parts and using the Korn inequality and Equation (4.3),we obtain

kv1k2H1.�/

�4c2

1

�2˛2 .�1/ kf1k

2L6=5.�/

. (4.2)

Multiplying (4.1)3 by �1 integrating the result over� next by parts we obtain

kr�1k2L2.�/

D 0, then �1 D const. (4.3)

Let V D v.t/� v1, pD p.t/� p1, HD �.t/� �1. Then .V , P, H/ satisfies the system of equations

V,t � divT .V , P/D�V � rv1 � v � rVC˛.�/f � ˛ .�1/ f1 in �T ,

divV D 0 in �T ,

H,t � ��HC v � rHC V � r�1 D 0 in �T ,

Nn �D.V/ � N�˛ D 0, ˛ D 1, 2 on ST ,

V � NnD 0, Nn � rHD 0 on ST ,

VjtD0 D V.0/� v1, HjtD0 D �.0/� �1 in �.

(4.4)

Multiplying Equation (4.4)1 by V and integrating over� and using Korn inequality yields

1

2

d

dtkVkL2.�/C

�

c1kVk2

H1.�/� 2kVk2

L4.�/kv1kH1.�/

Ck˛.�/f � ˛ .�1/ f1k2L2.�/

� 4kVk1=2L2.�/

krVk3=2L2.�/

kv1kH1.�/

Ck˛.�/f � ˛ .�1/ f1k2L2.�/

.

(4.5)

The first term on the right-hand side is estimated as follows:

4kVk1=2L2.�/

krVk3=2L2.�/

kv1kH1.�/

��

2c1kVk2

H1.�/C 8

c1

�kVk2

L2.�/kv1k

4H1.�/

.(4.6)

Then from Equations (4.5) and (4.6), we obtain

d

dt

�kVk2

L2.�/e

��c1�16

c1� kv1k

4H1.�/

�t�

� k˛.�/f � ˛ .�1/ f1k2L2.�/

e

��c1�16

c1� kv1k

4H1.�/

�t.

(4.7)

Integrating Equation (4.7) with respect to time, we obtain

kV.t/k2L2.�/

� ce��c1�16

c1� kv1k

4H1.�/

�t

tZ0

��f�

t0��2

L2.�/

C��f�

t0� f1

��2L2.�/

�� e

��c1�16

c1� kv1k

4H1.�/

�t0

dt0

C e��c1�16

c1� kv1k

4H1.�/

�tkV.0/k2

L2.�/.

(4.8)

Assuming the decay

kf .t/k2L2.�/

Ckf .t/� f1k2L2.�/

� e�ıt�kf .0/k2

L2.�/Ckf .0/� f1k

2L2.�/

�,


P. KACPRZYK

where ı > 0 some constant, we obtain from Equation (4.8)

kV.t/k2L2.�/

� ce��c1�16

c1� kv1k

4H1.�/

�t

[email protected]/k2

L2.�/C

tZ0

e

��ıC �c1

�16c1� kv1k

4H1.�/

�t0

dt0�kf .0/k2

L2.�/Ckf .0/� f1k

2L2.�/

�1A

� ce��c1�16

c1� kv1k

4H1.�/

�t

kV.0/k2

L2.�/C

1

�ıC �c1� 16 c1

� kv1k4H1.�/

��kf .0/k2

L2.�/Ckf .0/� f1k

2L2.�/

��e

��ıC �c1

�16c1� kv1k

4H1.�/

�t� 1

�!.

(4.9)

Next, multiplying Equation (4.5)3 by H and integrating over� yields

d

dtkHk2

L2.�/C �krHk2

L2.�/D 0.

Then from Equation (4.3), we obtain kH.t/k2L2.�/

� ckH.0/k2L2.�/

.

Therefore, from Equations (4.2) and (4.9), we obtain

Theorem 4.1Let f , f1 2 L2.�/ and ı > 0 some constant. Assume that the viscosity � is large compared with the external force field f1, that is,

�

c1� 256

c51

�5˛4 .�1/ kf1k

4L2.�/

> 0

and

kf .t/k2L2.�/

Ckf .t/� f1k2L2.�/

� e�ıt�kf .0/k2

L2.�/Ckf .0/� f1k

2L2.�/

�,

where t � 0. Then

kv.t/� v1k2L2.�/

� c

e��c1�16

c1� kv1k

4H1.�/

�tkv.0/� v1k

2L2.�/

C1

�ıC �c1� 16 c1

� kv1k4H1.�/

��kf .0/k2

L2.�/Ckf .0/� f1k

2L2.�/

��e�ıt � e

��c1�16

c1� kv1k

4H1.�/

�t��!

t!10

and

k�.t/� �1k2L2.�/

� c k�.0/� �1k2L2.�/

,

where t � 0.

5. Regularity of the attractor

In Section 3, we have proved that there exists a bounded compact and absorbing set B in H. Now, we will show that this set is in factcompact in V . It suffices to bound this set in H2.�/.

Lemma 5.1Assume that f 2 L2.kT , .kC 1/T ; L1.�//, f,t 2 L2.�� ŒkT , .kC 1/T�/. Then for the family of semiprocesses fU� .t, �/gt��0, 2†, thereexists a bounded and absorbing set in H2.�/.

ProofWe differentiate Equation (1.1)1 with respect to time and take the inner product with v,t so that

d

dtkv,tk

2L2.�/

Ckv,tk2H1.�/

� " kv,tk2H1.�/

Z�

jv,tj2 jrvjdxC ckfkL1.�/

Z�

j�,tj jv,tjdx

C c.1="/ kf,tk2L6=5.�/

� " kv,tk2H1.�/

C ckvkH1.�/ kv,tk1=2L2.�/

kv,tk3=2H1.�/

C c.1="/�kfk2

L1.�/k�,tk

2L2.�/

Ckf,tk2L6=5.�/

�.

(5.1)

Then from the Young’s inequality, we obtain

d

dtkv,tk

2L2.�/

Ckv,tk2H1.�/

� c�kv,tk

2L2.�/

kvk4H1.�/

Ckf,tk2L6=5.�/

Ckfk2L1.�/

k�,tk2L2.�/

�. (5.2)


P. KACPRZYK

Now, we differentiate Equation (1.1)3 with respect to time and take the inner product with �,t , so that

d

dtk�,tk

2L2.�/

Ckr�,tk2L2.�/

� c

Z�

jv,tj jr� j j�,tjdx � c kv,tkL2.�/kr�kL3.�/ k�,tkL6.�/

. (5.3)

Adding Equations (5.2) and (5.3), we obtain

d

dt

�kv,tk

2L2.�/

Ck�,tk2L2.�/

�Ckv,tk

2H1.�/

Ckr�,tk2L2.�/

� c�kv,tk

2L2.�/

�kvk4

H1.�/Ckr�k2

L3.�/

�Ckfk2

L1.�/k�,tk

2L2.�/

Ckf,tk2L6=5.�/

�.

(5.4)In view of Equation (5.4) and using Lemma 2.4, we obtain

kv,tk2L2.�/

Ck�,tk2L2.�/

�

�b3

TC b2

�eb1 , t > T , (5.5)

where

.kC1/TZkT

�kvk4

H1.�/Ckr�k2

L3.�/Ckfk2

L1.�/

�dt � b1,

.kC1/TZkT

kf,tk2L6=5.�/

dt � b2,

.kC1/TZkT

�kv,tk

2L2.�/

Ck�,tk2L2.�/

�dt � b3.

Now, we multiply Equation (1.1)1 by divT .v, p/, integrate over� and use the Hölder inequality, so that

kdivT .v, p/k2L2.�/

�c�kv,tkL2.�/

kdivT .v, p/kL2.�/CkvkL6.�/krvkL3.�/kdivT .v, p/kL2.�/CkdivT .v, p/kL2.�/kfkL2.�/

��"kdivT .v, p/k2

L2.�/C c.1="/

�kv,tk

2L2.�/

Ckfk2L2.�/

�C ckvkL6.�/krvk1=2

L2.�/krvk1=2

L6.�/kdivT .v, p/kL2.�/

�"kdivT .v, p/k2L2.�/

C c.1="/�kv,tk

2L2.�/

Ckfk2L2.�/

�Ckvk3=2

H1.�/kdivT .v, p/k3=2

L2.�/� "kdivT .v, p/k2

L2.�/

C c.1="/�kv,tk

2L2.�/

Ckfk2L2.�/

Ckvk6H1.�/

�.

(5.6)

Then Equations (3.7) and (5.5) yields

kv.t/k2H2.�/

� c�kv,tk

2L2.�/

Ckvk6H1.�/

Ckfk2L2.�/

�, t > T .

In view of Equations (5.5) and (5.6), we conclude that

kv.t/k2H2.�/

<1, t > T . (5.7)

Next, we multiply Equation (1.1)3 by�� , integrate over� and use the Hölder inequality, so that

k��k2L2.�/

� k�,tkL2.�/k��kL2.�/CkvkL6.�/kr�kL3.�/k��kL2.�/

� "k��k2L2.�/

C c.1="/ k�,tk2L2.�/

CkvkL6.�/kr�k1=2L2.�/

k��k3=2L2.�/

� "k��k2L2.�/

C c.1="/�k�,tk

2L2.�/

Ckvk4H1.�/

k�k2H1.�/

�.

(5.8)

Then from Equations (5.5) and (5.8), we obtain

k�.t/kH2.�/ �1, t > T . (5.9)

Hence, there exists a ball B.0, %3/� H2.�/ centered at 0 with sufficiently large radius %3 so that

.v.t/, �.t// 2 B.0, %3/ for all t > t0 D t0 .v0, �0/ .

�

In view of Main Theorem 1 and considerations from Section 3, there exists a global attractor in V . Thus, we have proved the following:

Main Theorem 2There exists a global attractor A in V for the family of semiprocesses fU� .t, �/gt��0 defined by Equation (3.1). The attractor is boundedin H2.�/, compact and connected in V . It attracts bounded sets in V .


P. KACPRZYK

Acknowledgement

The paper is partially supported by the MNiSW Grant NN 201 396 937.

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global attractor for the navier–stokes equations with the heat convection in a cylindrical pipe

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