@v
@tþvk
@v
@xk
divTðDðvịị ẳ rPỵf;
div vẳ0;
vđ0;ỡ Ửv0đỡ; đ4:1ỡ
v;P are periodic with period L at each variable xi; iẳ1;. . .;d;
where vẳ ðv1;. . .;vdị and P are the velocity and the pressure; the initial velocityv0and the external body forcefare given:fis time-independent and v0 is at xi L-periodic and divergence-free.
We suppose that the constitutive relation between the stress tensorTand the symmetric velocity gradient Dðvị; defined asDðvị:ẳ12ðrvỵ ðrvịTị; is given through a scalar potentialF: Rdd
sym/Rþ
0;i.e., TðZị ẳ@FðZị
@Z for all Z2Rdd
sym:
Furthermore, we assume that for a certainp>1 there are constantsC>0 ande50 such that for allZ;x2Rdd
sym;
C1ðeỵ jxjịp2jZj24ZT r2FðxịZ4Cðeỵ jxjịp2jZj2; Fð0ị ẳ0; @Fð0ị
@Z ẳ0:
ð4:2ị
SetO:ẳ ð0;Lịd and define
L0ðOị:ẳ ujO;u2C1ðRdị; divuẳ0;
Z
O
udxẳ0; uis L-periodic at xi
:
Then, we set
L2DIV:ẳclosure of L0ðOị in L2ðOị norm;
WDIV1;p :ẳclosure of L0ðOị in W1;pðOị norm:
Finally, we introduce the operatorsNđỡandQđỡsetting hNðvị;ui:ẳ
Z
O
TijðDðvịịDijðuịdx;
hQðvị;ui:ẳ Z
O
vk
@vi
@xk
jidx:
We are going to apply the method of‘-trajectories to the following three situations:
(1) the case wheredẳ2;3 andp53dỵ2dỵ2 ð52ị;
(2) the case wheredẳ2 andp2 ð1;2ị;
(3) extensions to the Dirichlet problem.
Note that the first case involves the Navier–Stokes equations (NSEs) in two dimensions; however, the three-dimensional NSEs are not included.
The importance of the first case comes from the fact that three- dimensional NSEs are still not well understood, and the question of the global existence of a unique solution is an open matter. It is natural, as was suggested and confirmed by O.A. Ladyzhenskaya (see [8, 9]), to investigate
models with the constitutive law
TðDðvịị ẳmDðvị ỵnjDðvịjp2Dðvị; m;n>0; ð4:3ị which clearly satisfy (4.2) and which generalize the NSEs. It is interesting to ask for what range of parameterspis the model (4.1) with (4.3) well-posed. It turns out that it happens when the case (1) holds. For suchp’s, (4.1) with (4.3) can be considered as a well-posed alternative (perturbation) of the NSEs.
Interestingly, ifpẳ3 in (4.3) we obtain the Smagorinski model of turbulence (see [2] for further comments), which is thus involved in our analysis.
The importance of the second case comes from broad applications in various areas (see [18] for an illustrative list). Recall also that all power-law fluids fall into this category.
The beauty of these examples is also situated in the fact that properties of the solution change quite significantly. Even more, the choice of the phase spaceX differs from case to case.
Finally, we underline that the earlier methods of evaluating the fractal dimension of limit sets of two-dimensional NSEs seems inapplicable here; in the case of the method of Lyapunov exponents (see [3] and [24]) this is due to missing the regularity of the linearized problem to (4.1). The other scheme developed by Ladyzhenskaya in [10] requires that the leading elliptic operator commutes with orthogonal projectors.
Before starting the analysis of the first case we list several inequalities which are consequences of (4.2); see [15, pp. 198–199] for the proof.
For anyp2 ð1;1ịit holds
TðZị Z 5K1ðeỵ jZjịp2jZj2;
jTðZịj4K2ðeỵ jZjịp2jZj: ð4:4ị Ifp52 then
ðTðZị TðZZịị ðZ* ZZị* 5K3jZZZj*2 Z 1
0
ðeỵ jlZỵ ð1lịZZjị* p2dl;
jTðZị Tð*ZZịj4K4jZZZj* Z 1
0
ðeỵ jlZỵ ð1lịZZjị* p2dl: ð4:5ị Ifp2 ð1;2ịthen
ðTðZị TðZZịị ðZ* ZZị* 5K5ðeỵ jZj ỵ jZZjị* p2jZZZj*2;
jTðZị TðZZịj* 4K6jZZZj:* (1)The Casedẳ2;3and p5ð3dỵ2ị=ðdỵ2ị
Let
e>0; f2L2DIV; and v0 2L2DIV: ð4:7ị We say that
v2L1ð0;T;L2DIVị \Lpð0;T;WDIV1;pị ð4:8ị with
v02Lp0ð0;T;ðWDIV1;pịnị ð4:9ị is a weak solution to (4.1) ifvð0ị ẳv0 and
hv0ðtị;ui ỵ hNðvðtịị;ui ỵ hQðvðtịị;ui ẳ hf;ui ð4:10ị holds for all u2WDIV1;p almost everywhere in ð0;Tị: Note that the initial condition is meaningful since (4.8) and (4.9) imply
v2Cđơ0;T;L2DIVỡ: đ4:11ỡ The following results have been obtained before.
Proposition4.1. Assume that(4.7)holds. Then there exists a weak solution to(4.1), (4.2)satisfying(4.8)–(4.11).Moreover,any weak solution satisfies
d
dtjjvjj22ỵc1jjrvjj22ỵc2jjrvjjpp4c3jjfjj22; ð4:12ị and its norm in the spaces of (4.8), (4.9) can be estimated by some CẳCðjjv0jjL2
DIVị:
Proof. The existence of a solution can be shown via monotone operator theory, and in fact it holds even foreẳ0:From the energy inequality (4.12) obtained from (4.10) by settingu:ẳvðtịthere follow the estimates (4.8) and (4.9) by using the duality argument. See [9] or [13] for details. ]
Propo sition4.2. Ifv02WDIV1;p;then there exists a weak solutionvto(4.1) satisfying
v2L1ð0;T;WDIV1;pị \L2ð0;T;W2;2ị \Lpð0;T;W1;3pị; ð4:13ị
v02L2ð0;T;L2DIVị; ð4:14ị andv;respectively v0;in these norms are bounded by someCẳCðjjrv0jjpị:
Moreover,ifv;uare two solutions of(4.1)corresponding to the initial values v0 and u0 then the difference w:ẳvu satisfies
d
dtjjwjj22 þc4 Z
O
jDðwịj2 Z 1
0
ðeỵ jlDðuị ỵ ð1lịDðvịjịp2dldx þc5jjrwjj224c6jjrujj
2p 2p1
3p jjwjj22: ð4:15ị
Proof. See [15, Secs. 5.3 and 5.4]. ]
Finally, we state the consequences of (4.15).
Propo sition4.3. Letu;vbe two weak solutions to(3.1)where moreover v2Lpð0;T;WDIV1;3pị:Then for04s4t4T;
jjuðtị vðtịjj24c7jjuðsị vðsịjj2; ð4:16ị wherec7 depends on the norm ofvinLpð0;T;WDIV1;3pị:Further,
jjuvjjL2ð‘;2‘;WDIV1;2ị4c8jjuvjjL2ð0;‘;L2DIVị; ð4:17ị
jju0v0jjL1ð0;‘;ðWDIV3;2ịnị4c9jjuvjjL2ð0;‘;WDIV1;2ị: ð4:18ị Proof. Since 24pimplies 2p=ð2p1ị4p;(4.16) follows from (4.15) – where we neglected the second and third terms on the left – by the Gronwall inequality.
Now, we chooses2 ð0; ‘ịand integrate (4.15) overt2 ðs;2‘ị:Denoting Iðu;vị:ẳc4
Z
O
jDðwịj2 Z 1
0
ðeỵ jlDðuị ỵ ð1lịDðvịjịp2dldx;
we obtain jjwð2‘ịjj22ỵ
Z 2‘
s
Iðu;vị ỵc5 Z 2‘
s
jjrwjj224c6 Z 2‘
s
jjrujj
2p 2p1
3p jjwjj22ỵ jjwðsịjj22: Making use of (4.16) in the integral on the right we deduce after several obvious simplifications that
Z 2‘
‘
Iðu;vị ỵc5 Z 2‘
‘
jjrwjj224c10jjwðsịjj22;
which after integration overs2 ð0; ‘ịyields Z 2‘
‘
Iðu;vị ỵc5 Z 2‘
‘
jjrwjj224c10
‘ Z ‘
0
jjwjj22: ð4:19ị This in particular implies (4.18). The proof of (4.19) uses the duality argument. Taking w from the unit ball in L1ð‘;2‘;WDIV3;2ị ẳ ðL1ð‘;2‘;
ðWDIV3;2ịnịn;we observe that Z 2‘
‘
hw0;wi
4
Z 2‘
‘
jhNðuị Nðvị;wij ỵ Z 2‘
‘
jhQðuị Qðvị;wij:
Let us estimate the first term on the right; an estimate for the second term is straightforward. Using (4.5) and the fact thatrwis bounded by embedding we come to
c11 Z 2‘
‘
Z
O
jDðwịj Z 1
0
ðeỵ jlDðuị ỵ ð1lịDðvịjịp2dldx 4c12
Z 2‘
‘
Iðu;vị
1=2 Z 2‘
‘
Z
O
ðeỵ jruj ỵ jrvjịp2dx
1=2
:
The last integral is clearly bounded. Coming to the supremum over allwand using (4.19), we conclude (4.18). ]
As we will see below, the assertions of Propositions 4.1–4.3 are sufficient to apply the scheme developed in Section 2 to show the existence of an exponential attractor and the existence of a global attractor with finite fractal dimension.
The result concerning the exponential attractor is new; the second result has been proved earlier in [17]; however, the proof here is different: while in [17] (see also [14]) there is a restriction from above on the length of the trajectories ‘; we do not need any bound here. This flexibility can be useful in obtaining the optimal estimates on the fractal dimension.
Th e or em4.1. Assume that(4.7)holds and p53dþ2
dỵ2 ðd ẳ2;3ị:
Then system (4.1) possesses both a global attractor A with a finite fractal dimension and an exponential attractorE:
Proof. We use the general scheme of Section 2 withX ẳL2DIV;Y ẳWDIV1;p; and Zẳ ðWDIV1;pịn; the parameter p1 is set to be p and p2ẳp=ðp1ị:
In such a way
Y‘ẳ fu2Lpð0; ‘;WDIV1;pị; u02Lp0ð0; ‘;ðWDIV1;pịnịg and
X‘ẳL2ð0; ‘;L2DIVị;
where‘ >0 is arbitrary.
Step1. Assumption (A1) follows from Proposition 4.1.
Step2. Since (4.12) with the Poincaree inequality! ljjvjj224jjrvjj22 leads to d
dtjjvðtịjj22ỵc1ljjvðtịjj224c3jjfjj22;
we see thatB0:ẳ fv2X;jjvjj24rgsatisfies (A2) wheneverr2>c3jjfjj22=ðc1lị:
Step3. Consider a trajectoryw2X‘:Sincew2Lpð0; ‘;WDIV1;pịthere exists t2 ð0; ‘ị such that wðtị 2WDIV1;p:By Proposition 4.2, there exists a solution starting from wðtị which belongs to the spaces in (4.13), which is by Proposition 4.3 unique in the class of weak solutions. This implies (A3).
Step4. By the argument from Step 3, the setfwjẵ‘=2;‘;w2B0‘gis bounded in the space L1ð‘=2; ‘;WDIV1;pị: Hence (A4) follows from Lemma 2.1 with C‘ẳB0‘:
Step5. To verify (A5), it is enough to observe that the stronger assertion B0‘
X‘
B0‘ holds. Let wn 2B0‘ and wn!w0 in X‘: But wn are bounded in spaces (4.8) and (4.9), and by the arguments (based on the compactness ofQ and the monotonicity ofN) which are essentially the same as those that lead to the existence of the weak solution we see thatw0 is a weak solution. It remains to verify thatw0ð0ị 2B0: we can assume thatwnðtị !w0ðtịin X for almost everytand hencew0ðtị 2B0 for almost everyt:Butw0is continuous andB0 is closed.
At this stage we can conclude from Theorem 2.1 that the dynamical systemðLt;X‘ịhas global attractorA‘:
Step6. Note also that the setB1‘ has due to the smoothing arguments above better regularity; namely, it is bounded in the spaces in (4.13), and the time derivatives of its elements are bounded in L2ð0; ‘;L2DIVị: This implies that (A6) is satisfied with
W‘:ẳ fu2L2ð0; ‘;WDIV1;2ị;u02L1ð0; ‘;ðWDIV3;2ịnịg
andtẳ‘; as follows from (4.17) and (4.18) in Proposition 4.3. (Note that W‘ ++X‘by Lemma 1.6.) By Theorem 2.2,A‘X‘has a finite fractal dimension.
Step7. Assumptions (A7)–(A9) follow from Lemma 2.1 withC‘ ẳB0‘; cf. Steps 3 and 4 above. Finally, (A10) withbẳ1=2 follows from Lemma 2.2 withC‘ẳB1‘ since this set is bounded inL2ð0; ‘;L2DIVị:
Now, using Theorem 2.5 we conclude that the dynamical systemðLt;X‘ị has an exponential attractor. Finally, using Theorems 2.3, 2.4, and 2.6 we obtain a global attractor with finite fractal dimension and an exponential attractor also for the dynamics in the spaceX ẳL2DIV: ]
(2)Casedẳ2and p2 ð1;2ị In this part, we suppose that
e50; f 2Lp0ðOị; and v02WDIV1;2: ð4:20ị We define
Ipðzị:ẳ Z
O
ðeỵ jDðvịjịp2jDðzịj2dx and recall the assertions proved in [15].
Propo sition4.4. Let(4.20)be satisfied. Then there exists
v2L1ð0;T;WDIV1;2ị \L2ð0;T;W2;pðOịị ð4:21ị with
v02L2ð0;T;L2DIVị ð4:22ị such that the weak formulation
hv0ðtị;ui ỵ hNðvðtịị;ui ỵ hQðvðtịị;ui ẳ hf;ui ð4:23ị holds for everyu2WDIV1;2 and for almost allt2 ð0;Tị:
In addition,Galerkin approximationsvN that converge weakly tovin spaces (4.21)and (4.22)fulfill
d
dtjjrvNðtịjj22ỵc1IpðrvNðtịị4c2jjfjj2p0; Z T
0
jjðvNị0jj22ds4CẳCðjjvNð0ịjj1;2;jjfjjp0ị: ð4:24ị Proof. See Theorem 5.4.21 in [15] and its proof. ]
Next, we formulate immediate consequences of the previous proposition.
Propo sition 4.5. Let (4.20) hold. Then solution v; introduced in Proposition4.4,satisfies
v2L2ð0;T;W2;aðOịị with aẳ 4
4p: ð4:25ị
In addition to(4.24),Galerkin approximationsvN satisfy d
dtjjrvNjj22ỵc2jjr2vNjj2ajjeỵ jrvNjjjp22 4c3jjfjj2a0jjeỵ jrvNjjj2p2 ð4:26ị and(for alld2 ð0;Tị)
sup
t2ẵd;T
jjðvNị0ðtịjj22ỵ Z T
d
jjrðvNị0ðtịjj2adt4c4; ð4:27ị
wherec4ẳc4ðd;T;jjrv0jj2;jjfjjp0ị:
Moreover,ifu;v2L1ð0;T;WDIV1;2ịare two solutions of (4.1)corresponding to the initial valuesu0;v0 respectively,then the differencew:ẳuvsatisfies
d
dtjjwjj22ỵc5jjrwjj2a4c6jjwjj22; ð4:28ị where c5 depends on supt2ð0;Tịðjjruðtịjj2ỵ jjrvðtịjj2ị and c6 depends on supt2ð0;Tịjjruðtịjj2:
Proof. It is based on two inequalities,
Ipðzị5c7jjrzjj2ajjeỵ jrvjjjp22 ; ð4:29ị Z
O
ðeỵ jDðuịj ỵ jDðvịjịp2jDðwịj2dx5c8jjwjj2ajjeỵ jruj ỵ jrvjjjp22 ; ð4:30ị which are proved by means of Hoolder and Korn-like inequalities (see also. Lemma 5.3.24 in [15]). Sincev satisfies (4.21) we observe that (4.29) with zẳ rvimplies (4.25). Similarly, asvN satisfies (4.24) we see that (4.29) with zẳ rvN leads to (4.26).
Next, we take the time derivative of the Galerkin system, multiply itsrth equation (see [15, p. 207, Eq. (2.21)]) bydtdcNr and sum it overrfrom 1 toN:
Thus we obtain 1
2 d
dtjjðvNị0jj22ỵc9 Z
O
ðeỵ jDðvNịjịp2 rdvN dt
2
dx4 Z
O
jrvNjjðvNị0j2dx;
which yields (with the help ofð4:24ị1 and (4.29) with zẳ ðvNị0) 1
2 d
dtjjðvNị0jj22ỵc10jjrðvNị0jj2a4jjrvNjj2jjðvNị0jj24 4c11jjrvNjj2jjðvNị0jj2
p1 p
2 jjrðvNị0jj
2 p a: Young’s inequality andð4:24ị1 then give
d
dtjjðvNị0jj22ỵc12jjrðvNị0jj2a4c13jjðvNị0jj22:
Multiplying the last inequality by a smooth cut-off function vanishing at tẳ0;integrating it with respect to time between 0 andt2 ð0;Tị;and using ð4:24ị2 lead then to (4.27).
Finally, using (4.23) and (4.6) we see that the differencewsatisfies 1
2 d
dtjjwjj22þK5 Z
O
ðeỵ jDðuịj ỵ jDðvịjịp2jDðwịj2dx4 Z
O
jwj2jrujdx:
Sinceu;v2L1ð0;T;WDIV1;2ị;and the right-hand side of (4.31) is bounded by jjrujj2jjwjj244c14jjrujj2jjwjj
2ðp1ị p 2 jjrwjj
2 p a;
we observe that the last two inequalities together with (4.30) lead to 1
2 d
dtjjwjj22þc15jjrwjj2a4c16jjwjj
2ðp1ị p 2 jjrwjj
2 p a: The Young inequality completes the proof of (4.28). ]
Now, we are ready to employ our general scheme to obtain the following theorem.
Tth e or em4.2. Let(4.20)hold andp2 ð1;2ị; dẳ2:Then the dynamical system(4.1)possesses a global attractor with finite fractal dimension and an exponential attractor.
Proof. First, we set X :ẳWDIV1;2; Y :ẳW2;aðOị;where aẳ4p4 and p1ẳ p2ẳ2:Hence we have
X‘ẳL2ð0; ‘;Xị;
Y‘ ẳ fv2L2ð0; ‘;Yị; v02L2ð0; ‘;L2DIVịg:
Also, we will work with the auxiliary spacesX0:ẳL2DIV; Y0:ẳW1;aðOị;and X‘0 ẳL2ð0; ‘;X0ị;
Y‘0 ẳ fv2L2ð0; ‘;Y0ị; v02L2ð0; ‘;ðY0ịnịg:
Note thatWDIV2;a +WDIV1;4=ð2pị:As 2p4 >2 for p2 ð1;2ị; one has Y‘++X‘ andY‘0+ +X‘0:
Step1. By Propositions 4.4 and 4.5, for everyv02WDIV1;2 and for allT >0 one can construct a uniquely defined solution satisfying (4.21)–(4.23).
It follows directly from (4.24) and (4.29) that Z T
0
jjr2vðtịjj2adtỵ Z T
0
jjv0ðtịjj22dt4CẳCðT;jjrv0jj2;jjfjjp0ị: ð4:31ị Finally, using (4.21)–(4.23), it is not difficult to observe that
v2Cđơ0;T;Xwỡ: đ4:32ỡ Thus, the assumption (A1) is verified.
Step 2. We want to construct an absorbing, invariant set B0: We will work with Galerkin approximations first: the embedding inequality jjrvjj2 4Cjjr2vjjaand some elementary inequalities enable us to rewrite (4.26) into the form
d
dtjjrvNjj22ỵk1jjrvNjjp2 k2ðjjfjja0;eị40; ð4:33ị wherek1;k2 >0 are independent ofN:Let Rbe large enough so that
k1Rpk2ðjjfjja0;eị>0: ð4:34ị Now, it is elementary to see that a closed ball B0:ẳBXð0;RịX with R satisfying (4.34) is due to (4.33) uniformly (also w.r.t. N) absorbing and positively invariant in the sense of (A2) forvN:
We claim that such a set is also invariant and positively absorbing for the solutions themselves. Let us check the invariance (the absorbing property is verified analogously). If v02B0 then the initial conditions vN0 :ẳPNv0 for Galerkin approximations vN (where PN are orthogonal projectors) belong also to B0: Hence vN remain pointwise in B0: We can assume that vNðtị !vðtịinX for almost allt;wherevis the unique solution with initial condition v0: Hence vðtị belongs to B0 for almost all t: But v is weakly continuous andB0is closed and convex, hence weakly closed, which finishes the proof.
Step 3. Due to the uniqueness of solutions, (A3) is satisfied and, moreover, we have also the solution operatorsSt defined inX:
Step4. By (4.28) and the Gronwall inequality, the operatorsSt: X0/X0 are uniformly Lipschitz continuous for t2 ẵ0;T on the set B0: Hence, by Lemma 2.1 with C‘ ẳB0‘ where we replace X with X0; operators Lt are uniformly Lipschitz continuous for t2 ẵ0;T on B0‘ with respect to X‘0 topology. So if wn2B0‘ and wn!w in X‘; then Ltwn!Ltw in X‘0: But by (4.21) and (4.22) the sequenceLtwn is compact inX‘;henceLtwn!LtwinX‘ and (A4) holds.
Step5. Defining for some fixedt>0;
B1‘ :ẳLtB0‘X
0
‘; ð4:35ị
we see that the definition (4.35) differs from (2.3) in that the closure is taken with respect toX‘0:SinceLtB0‘ is bounded in the spaces (4.21), (4.22), and (4.25), so isB1‘; and this regularity also enables us to limit passage in the equation, henceB1‘ X‘:Moreover, ifw2B1‘ thenwðtị 2B0for almost allt;
and sinceB0is weakly closed andwis weakly continuous, we havewð0ị 2B0; i.e.,B1‘ B0‘:Finally,B1‘ is compact both inX‘andX‘0since it is closed in the weaker of both topologies and totally bounded by regularity.
Hence we can use Theorem 2.1 to obtain the global attractorA‘ toðLt;X‘ị:
Step6. We will now show that dfX‘ðA‘ị is finite. First of all, we claim that forw1;w22B1‘;
jjL‘w1L‘w2jjL2ð0;‘;W1;aðOịị4c1jjw1w2jjX0
‘; ð4:36ị
jjðL‘w1ị0 ðL‘w2ị0jjL1ð0;‘;ðWDIV3;2ịnị4c2jjw1w2jjX0
‘: ð4:37ị
Now, (4.36) follows from (4.28) and (4.37) follows from (4.36) and (4.6) by the duality argument, the proof being similar to a proof of (4.17),(4.18) in Proposition 4.3 above.
It then follows from Lemma 1.3 withXẳX‘;LẳL‘;CẳA‘;and Y:ẳ fw2L2ð0; ‘;W1;aðOịị;w02L1ð0; ‘;ðWDIV3;2ịnịg ð4:38ị (note that Y++XẳX‘ by Lemma 1.6) that dX
0
‘
f ðA‘ị51: Due to the interpolation inequality,
jjujj1;24cjjujj
p pþ2 2 jjujj
2 pþ2
2;a with aẳ 4
4p;
(4.25), and the H.oolder inequality we have onB0‘ that Z ‘
0
jjw1ðsị w2ðsịjj21;2ds4C Z ‘
0
jjw1ðsị w2ðsịjj22ds
pþ2p
: ð4:39ị In other words, the identity mappingI: X‘0/X‘ isa-Hoolder continuous on. B1‘ withaẳp=ðpỵ2ị:Thus, by Lemma 1.2,
dfX‘ðA‘ị4 1ỵ2 p
dfX‘0ðA‘ị51: ð4:40ị
Step 7. By Lemma 2.1 with C‘ ẳB1‘ and with X replaced with X0 one obtains that the mapping e: X‘0/X0 is Lipschitz continuous on B1‘: We need, however, the Hoolder continuity from. X‘ into X: We compute
jjruð‘ịjj22 ẳ Z
O
jruðx; ‘ịj2dxẳ1
‘ Z ‘
0
Z
O
d
dsðsjruðx;sịj2ịdxds 41
‘ Z ‘
0
jjruðsịjj22dsỵc
‘ Z ‘
0
Z
O
jruðx;sịj jru0ðx;sịjdxds 41
‘jjujj2X‘þc
‘ Z ‘
0
jjru0ðsịjjajjruðsịjja0ds 41
‘jjujj2X
‘þc
‘ Z ‘
0
jjru0ðsịjjajjruðsịjj
2ðp1ị p
2 jjr2uðsịjj
2p p a ds
41
‘jjujj2X
‘þc
‘ Z ‘
0
jjruðsị0jj2ads
12
Z ‘
0
jjr2uðsịjj2ads
2p2p Z ‘
0
jjruðsịjj22ds
p1p
:
With the help of (4.25) and (4.27) this implies that jjuð‘ịjj1;24Cjjujj
p1 p X‘ ;
and puttinguẳw1w2 we see that (A7) and (A8) hold. In particular, we obtain the existence of a global attractor with a finite fractal dimension in the spaceX ẳWDIV1;2:
Step8. For the sake of the construction of an exponential attractor, we will start with the topologyX‘0instead ofX‘:Then (A9) holds by Lemma 2.1
withC‘ ẳB1‘ (X replaced withX0), and (4.27) gives (A10) withaẳ1=2 by Lemma 2.2. Hence by Theorem 2.5 we obtainE‘;an exponential attractor for the dynamical systemðLt;B1‘ịwith all metric notions with respect toX‘0: However, since the identity mapping I: X‘0/X‘ is on B1‘ Hoolder. continuous, it is straightforward to see that E‘ is also an exponential attractor with respect to the metric ofX‘:From Theorem 2.6 then follows the existence of an exponential attractor also in the spaceX ẳWDIV1;2: (3)Extensions to the Dirichlet Problem
The final part of this section is aimed at extending the results presented above by considering instead of the space-periodicity ofvandP (seeð4:1ị4) the Dirichlet boundary condition
vẳ0 at ð0;1ị @O; ð4:41ị
whereOis a bounded, open set inRd;dẳ2;3:Forp52;we can formulate the following two theorems.
Th e or em4.3. LetOhave the Lipschitz boundary@O:Let(4.7)hold and let
p5dþ2
2 : ð4:42ị
Then the dynamical system(4.1)–(4.2)with(4.41)possesses a global attractor with finite fractal dimension and an exponential attractor.
Proof. The proof uses three ingredients: First, the validity of Proposi- tion 4.1 for the analyzed system (4.1) with (4.41); second, the uniqueness theorem for weak solutions if (4.42) holds; and third, the L1ð0;T;WDIV1;pị regularity proved by ‘‘testing the equation by time derivativev0:’’ Both this regularity and the uniqueness require that v2Lpð0;T;WDIV1;pị only. See [12, 13, 14, 15] for details.
The rest of the proof is analogous to the proof of Theorem 4.1. In fact, because of the uniqueness for initial data in X ẳL2DIV; some steps of the proof are simpler. ]
Th e or em4.4. LetOR3have aC2-boundary and let(4.7)hold. Assume that
p59
4 and dẳ3: ð4:43ị
Then the assertion of Theorem4.3holds.
Proof. The steps of the proof are completely analogous to the steps of the proof of Theorem 4.1. The only difference comes from the fact that regularity results of type (4.13) are proved for the Dirichlet problem only if (4.43) holds; see [16]. ]
Let us finally remark that nothing is known if p2 ð1;2ị; dẳ2; and (4.1) is considered with Dirichlet boundary conditions (4.41). This is connected with the fact that no global (this means up to the boundary) regularity result of the type (4.13) has been proved for (4.1) with (4.41) for p52;to date.