Finite dimensionality of 2-D micropolar fluid flow with periodic
  boundary conditions
Finite dimensionality of 2-D mi
ropolar �uid �ow with periodi
boundary 
onditions
P. Szopa
Institute of Applied Mathemati
s and Me
hani
s
Warsaw University,
Bana
ha 2, 02-097 Warsaw
szopa�du
h.mimuw.edu.pl
November 4, 2018
Abstra
t
This paper is devoted to des
ribe the �nite-dimensionality of a two-dimensional mi
ropolar �uid
�ow with periodi
 boundary 
onditions. We de�ne the notions of determining modes and nodes and
estimate the number of them, we also estimate the dimension of the global attra
tor. Finally we

ompare our results with analogous results for Navier-Stokes equation.
Keywords�mi
ropolar �uid, global solution, periodi
 boundary 
onditions, determining modes, de-
termining nodes, dimension of global attra
tor.
1 Introdu
tion
There are some heuristi
 as well as based on dimensional analysis arguments suggesting that the long-
time behavior of a turbulent �ow is determined by a �nite number of parameters. These arguments are
based on a 
onje
ture that rapidly varying, high-wavenumber 
omponents de
ay as fast as they leave
lower-wavenumber ingredients una�e
ted. It turns out from Kolmogorov's theory that in 3-dimensional
�ows only the wavenumbers up to the 
uto� value λK = (ε/ν
3)1/4 need to be 
onsidered. The question
is then redu
ed, as explained in [3℄, to �nd the number of resolution elements ne
essary to be 
onsidered
to des
ribe the behavior of a �uid in a volume - say a 
ube of length l0 on ea
h side. The smallest
resolved distan
e is ld = 1/λK and therefore the number of resolution elements is (l0/ld)
A theory of Krai
hnan [4℄, referring to a 2-dimensional turbulent �ow, allows us to estimate the
number of resolution elements to be 
onsidered as (l0/λKr)
, where λKr is the Krai
hnan length λKr =
(ν3/χ)1/6 and χ is the average enstrophy dissipation rate. We refer the reader to [5℄ for more detailed
dis
ussion of turbulent length s
ales.
http://arxiv.org/abs/0704.0687v1
The notion of determining modes arise naturally during 
onsideration of the Fourier de
omposition
of a �ow. There are some results 
on
erning determining modes in the 
ontext of 2-dimensional Navier-
Stokes equations. Foias and Prodi showed in [6℄ that if a number of Fourier modes of two di�erent
solutions have the same asymptoti
 behavior, then whole solutions have the same asymptoti
 behavior
too. Subsequent works have been aimed at estimating how many low modes are ne
essary to determine
the behavior of a �ow. The most re
ent results are in [7℄ for the 
ase of no-slip boundary 
onditions and
in [8℄ for the 
ase of periodi
 boundary 
onditions. The number of determining modes for 2-D mi
ropolar
�uid �ow with no-slip boundary 
onditions was estimated in [9℄.
In many pra
ti
al situations, for instan
e physi
al experiments, data are 
olle
ted from measurements
in �nite points in a domain of a �ow. Therefore o

urs a question how many measurement points are
ne
essary to determine the long-term behavior of a �ow. This leads to the notion of determining nodes,
whi
h was introdu
ed by Foias and Temam in [10℄. The most re
ent estimate for the lowest number of
determining nodes for Navier-Stokes equation in spa
e-periodi
 
ase was derived in [8℄.
Another approa
h to des
ribe the asymptoti
 behavior of a �ow with a �nite number of parameters
is to use a global attra
tor. For every traje
tory u in the phase spa
e we 
an 
hoose, due to "Shadowing
Lemma" (
.f. [11℄), a traje
tory uA lying on the attra
tor, that is arbitrary 
lose in an interval of
time, that is |u(t) − uA(t)| < ε for t ∈ (t0, t1). On the other hand the global attra
tor has �nite
Hausdor� and fra
tal dimension, so we 
an parametrize it by a �nite number of parameters (
.f. [12℄,
[13℄, [14℄). Therefore we 
an des
ribe approximately the long-term behavior of a �ow by a �nite number
of parameters.
There are many results 
on
erning the dimension of attra
tor for Navier-Stokes equation with a
variety of boundary 
onditions 
f. ie. periodi
 boundary 
onditions in [15℄, pipe-like domain with
arbitrary in�ow at in�nity in [16℄. The dimension of attra
tor for mi
ropolar �uid equations with
various boundary 
onditions was estimated in [17℄, [18℄.
In this paper we will 
onsider all of mentioned above ways of determining the long-time behavior of
a mi
ropolar �uid �ow by �nite number of parameters. We estimate the lowest number of determining
modes and nodes, and the dimension of the global attra
tor.
We will 
onsider the mi
ropolar �uid equations, whi
h in two-dimensional 
ase have form (
.f. [17℄)
− (ν + νr)∆u+ (u · ∇)u+∇p = 2νr rotω + f, (1.1)
div u = 0, (1.2)
− α∆ω + (u · ∇)ω + 4νrω = 2νr rot u+ g, (1.3)
where u = (u1, u2) is the velo
ity �eld, p is the pressure and ω is the mi
rorotation �eld interpreted as
the angular velo
ity of parti
les. In 2-dimensional 
ase we assume that the axe of rotation of parti
les
is perpendi
ular to x1, x2 plane. The �elds f = (f1, f2) and g are the external for
es and moments
respe
tively. Positive 
onstants ν, νr, α are the vis
osity 
oe�
ients and
rotu =
− ∂u1
, div u =
, rotω =
,− ∂ω
We supplement these equations with initial 
ondition
u(x, 0) = u0(x), ω(x, 0) = ω0(x)
and periodi
 boundary 
onditions
u(x+ Lei, t) = u(x, t), ω(x+ Lei, t) = ω(x, t) ∀x ∈ R2 ∀t > 0,
where e1, e2 is Eu
lidean base of R
and L is the period in the i-th dire
tion; Q = (0, L)2 is the square
of the period.
The existen
e and uniqueness of solutions for this model as well as existen
e of the global attra
tor
was proved in [19℄. We also assume (like in [19℄) that the spa
e averages of u, ω, f and g vanish.
This paper is organised as follows:
In se
tion 2 we introdu
e fun
tion spa
es and operators used throughout this arti
le.
Se
tion 3 presents some a priori estimates we need in the sequel.
In se
tion 4 we estimate the number of determining modes.
The number of determining nodes is estimatef in se
tion 5.
Se
tion 6 is devoted to re
all the notions of fra
tal and Hausdor� dimension and to estimate the
dimension of global atra
tor.
2 Mathemati
al setting of the problem
In this se
tion we introdu
e fun
tion spa
es and trilinear forms b and b1, Stokes and −∆ operators and
operator rot whi
h are ne
essary in our 
onsideration.
Fun
tion spa
es.
We will denote by X the spa
e X ×X with standard produ
t norm .
Lq is the usual Lebesque's spa
e Lq(Q) for q ∈ [1,∞]. We denote the s
alar produ
t in L2 by (·, ·)
and the norm in L2 by | · | when it doesn't produ
e 
onfusion.
Hm is usual Sobolev spa
es Hm(Q) for m ∈ N.
By Hmp (Q), m ∈ N we denote the spa
e of real fun
tions whi
h are in Hmloc(Rn) and periodi
 with
the period Q: u(x+ Lei) = u(x) i = 1, 2.
For an arbitrary m ∈ N, Hmp (Q) is a Hilbert spa
e with the s
alar produ
t
(u, v)m =
|α|≤m
Dαu(x)Dαv(x) dx
and the norm indu
ed by it. The fun
tions in Hmp (Q) are expli
itly 
hara
terized by their Fourier series
expansion
Hmp (Q) =
u : u =
2iπk/L·x, ūk = u−k, |u|m =
|k|2m|uk|2 <∞
where k/L = (k1/L, k2/L), the norm |u|m is equivalent to the norm {
k∈Zn(1 + |k|2m)|uk|2}1/2. We
also set
Ḣmp (Q) = {u ∈ Hmp (Q), c0 = 0}.
H and V are the divergen
e-free subsets of Ḣ0p(Q) and Ḣ
p(Q), respe
tively. We equip V with the
s
alar produ
t and the Hilbert norm
((u, v)) =
, ||u|| = {((u, u))}1/2 .
This norm is equivalent to the norm indu
ed by H1p (Q), and V is a Hilbert spa
e for this s
alar produ
t.
One 
an 
he
k that Ḣ−mp is the dual spa
e to Ḣ
p , we also denote the dual spa
e to V as V
Let H and V denote H × Ḣ0p and V × Ḣ1p , respe
tively, with standard produ
t norms.
Lq(0, T ;X) is the spa
e of strongly measurable fun
tions u : (0, T ) → X, where X is a Bana
h spa
e,
with the followning norm
||u||Lq(0,T ;X) =
||u(t)||qX dt
, 1 ≤ q <∞,
ess sup
t∈(0,T )
||u(t)||X , q = ∞.
C([0,T℄,X) is the spa
e of 
ontinuous fun
tions u : (0, T ) → X, where X is a Bana
h spa
e, with the
usual norm.
Stokes and −∆ operators.
Let us 
onsider Stokes problem, whi
h is obtained from the Navier-Stokes equation by negle
ting all
time-dependent and nonlinear terms. Thus it is the following one: for given f ∈ Ḣ0p or Ḣ−1p , �nd u ∈ Ḣ1p
and p ∈ L2 su
h that
−∆u+∇p = f, ∇ · u = 0. (2.1)
It is shown (
f., e.g. [11, 20℄) that the Stokes operator A asso
iated with spa
e periodi
ity 
ondition is
simply −∆ (provided that f ∈ H) with domain D(A) = Ḣ2p ∩H. A is one-to-one map form D(A) onto
The operator A−1 is linear, 
ontinuous from H into D(A). Sin
e the inje
tion of D(A) in H is

ompa
t, we 
an 
onsider A−1 as a 
ompa
t operator in H. A−1 is also self-adjoint as an operator in
H. Hen
e it possesses a sequen
e of eigenfun
tions wj , j ∈ N whi
h form an orthonormal basis of H
Awj = λjwj , wj ∈ D(A),
0 < λ1 ≤ λ2 ≤ . . . , λj → ∞ for j → ∞.
The operator −∆ has the same properties as the Stokes operator. The eigenvalues are the same but the
eigenfun
tions are di�erent, be
ause we 
onsider ω to be in R, and we denote them by ρk. For notation

onsisten
e we write A1 instead of −∆.
We 
an express every element u ∈ H and ω ∈ Ḣ0p as
u(x, t) =
uk(t)wk(x), ω(x, t) =
ωk(t)ρk(x).
The Galerkin proje
tors 
orresponding to the �rst m modes are:
Pmu(x, t) =
uk(t)wk(x) P
mωi(x, t) =
ωk(t)ρk(x).
We also denote proje
tions onto modes higher than m by Qm and Q
Qmu(x, t) =
k=m+1
uk(t)wk(x) Q
mω(x, t) =
k=m+1
ωk(t)ρk(x).
Trilinear forms.
We de�ne the trilinear forms b and b1 as follows
b(u, v, w) =
i,j=1
wj dx
for all u, v, w ∈ V and
b1(u, ω, ψ) =
for all u ∈ V, and all s
alar fun
tions ω,ψ ∈ Ḣ1p(Q). The forms b and b1 have the orthogonality property
b(u, v, v) = 0, b1(u, ω, ω) = 0.
In the 2-dimensional spa
e-periodi
 
ase the form b has one more orthogonality property [20℄
b(u, u,Au) = 0 ∀u ∈ D(A), (2.2)
whi
h the form b1 does not have � it is not true that b1(u, ω,A1ω) = 0 for all u ∈ D(A), ω ∈ D(A1).
The la
k of this orthogonality property 
auses that the a priori estimates, we obtain in the sequel, are
more involved than analogous estimates for NSE with periodi
 boundary 
onditions.
We do some estimates of the forms b and b1 using the Ladyzhenskaya inequality [21℄
||u||L4 ≤
|u|1/2||u||1/2, for all u ∈ Ḣ1p ,
and the Agmon inequality [22, 23℄
||u||L∞ ≤
|u|1/2|Au|1/2, for all u ∈ D(A).
The most relevant estimate of the 
onstants are obtained in [23℄. We also use the Hölder inequality:
|b(u, v, w)| ≤ c1|u|1/2‖u‖1/2‖v‖ · |w|1/2||w||1/2 u, v, w ∈ V,
|b(u, v, w)| ≤ c1|u|1/2||u||1/2||v||1/2|Av|1/2|w| u ∈ V, v, w ∈ D(A),
|b(u, v, w)| ≤ c1|u|1/2|Au|1/2||v|| · |w| u ∈ D(A), v ∈ V, w ∈ H,
|b1(u, ω, ψ)| ≤ c1|u|1/2||u||1/2|ψ|1/2||ψ||1/2||ω|| u, ω, ψ ∈ Ḣ1p ,
|b1(u, ω,A1ψ)| ≤ c1|u|1/2|Au|1/2||ω|| · |A1ψ| u ∈ D(A), ω ∈ Ḣ1p , ψ ∈ D(A1),
|b1(u, ω,A1ψ)| ≤ c1|u|1/2||u||1/2||ω||1/2|A1ω|1/2|A1ψ| u ∈ V, ω, ψ ∈ D(A1)
(2.3)
for an appropriate 
onstant c1.
The operator rot has the following properties,
rot u · ω dx =
rotω · u dx, (2.4)
| rotω|2 dx =
|∇ω|2 dx, (2.5)
| rot u|2 dx =
|∇u|2 dx, (2.6)
for all u ∈ V and ω ∈ Ḣ1p .
3 A priori estimates
In this se
tion we derive some a priori estimates, whi
h we use in the sequel. We will obtain these in
the terms of asymptoti
 strength of for
es and moments. To this end we set
F̃ = lim sup
|f(t)|2 + |g(t)|2
(3.1)
k1 = min{ν, α}, k2 = k1λ1. (3.2)
All inequalities, apart form the latter, we use throughout this se
tion 
ome from [19℄. Let us 
onsider
the following one
(|u(t)|2 + |ω(t)|2) + k2
|u(t)|2 + |ω(t)|2
≤ k−12
|f(t)|2 + |g(t)|2
. (3.3)
Integrating it with respe
t to t in the interval (0, t) we obtain in parti
ular
|u(t)|2+ |ω(t)|2 ≤ |u0|2 + |ω0|2 + k−12
|f(s)|2 + |g(s)|2
≤ |u0|2 + |ω0|2+k−12
||f ||2L2(0,T ;H) + ||g||
L2(0,T ;Ḣ0
≤ |u0|2 + |ω0|2 + k−12
||f ||2L2(0,∞;H) + ||g||
L2(0,∞;Ḣ0
(3.4)
whi
h implies the uniform bound of the norm of solution in H.
We apply Gronwall's inequality to (3.3) to obtain an uniform with respe
t to initial 
onditions bound
of the norm of solutions for large times. We pro
eed as follows
|u(t)|2 + |ω(t)|2 ≤e−k2(t−t0)
|u(t0)|2 + |ω(t0)|2
+ k−12
|f(s)|2 + |g(s)|2
ek2(s−t0) ds
≤e−k2(t−t0)
|u(t0)|2 + |ω(t0)|2
+ k−12
||f ||2L∞(t0,t;H) + ||g||
L∞(t0,t;Ḣ0p)
ek2(s−t0) ds
≤e−k2(t−t0)
|u(t0)|2 + |ω(t0)|2
+ k−22
||f ||2L∞(t0,t;H) + ||g||
L∞(t0,t;Ḣ0p)
ek2(t−t0) − 1
≤e−k2(t−t0)
|u(t0)|2 + |ω(t0)|2
+ k−22
1− e−k2(t−t0)
||f ||2L∞(t0,t;H) + ||g||
L∞(t0,t;Ḣ0p)
hen
e for t0 and t large enough
|u(t)|2 + |ω(t)|2 ≤ 2
F̃ 2. (3.5)
To estimate the average of the square of the norm of solutions in V we use the inequality
(|u(t)|2 + |ω(t)|2) + k1
||u(t)||2 + ||ω(t)||2
≤ k−12
|f(t)|2 + |g(t)|2
. (3.6)
By integration we obtain
|u(t+ T )|2 + |ω(t+ T )|2 + k1
||u(s)||2 + ||ω(s)||2
≤ k−12
|f(s)|2 + |g(s)|2
ds + |u(t)|2 + |ω(t)|2.
Be
ause |u(t)|2+ |ω(t)|2 is uniformly bounded with respe
t to t (
.f. (3.4)), then for t and T large enough
we have
∣|u(s)||2 + ||ω(s)||2
ds ≤ (k1k2)−1
|f(s)|2 + |g(s)|2 ds+
|u(t)|2 + |ω(t)|2
F̃ 2.
(3.7)
In order to derive two more estimates we 
onsider the inequality
||u(t)||2+ ||ω(t)||2
|Au(t)|2 + |A1ω(t)|2
|u(t)|2||ω(t)||2 +
||u(t)||2 + ||ω(t)||2
|f(t)|2 + |g(t)|2
(3.8)
Let us set
y(t) = ||u(t)||2 + ||ω(t)||2,
g̃(t) =
|u(t)|2||ω(t)||2 +
h̃(t) =
|f(t)|2 + |g(t)|2
then we 
an rewrite (3.8) in the form
≤ g̃y + h̃.
We 
he
k assumptions of uniform Gronwall's lemma (
.f. [15℄). For t and r large enough we have
g̃(s) ds ≤
|u(s)|2 + |ω(s)|2
||u(s)||2 + ||ω(s)||2
8ν2r r
α2νk22
||u(s)||2 + ||ω(s)||2
8ν2r r
α2νk1k
F̃ 4 ≡ a1,
h̃(s) ds ≤
F̃ 2 ≡ a2,
y(s) ds ≤2r
F̃ 2 ≡ a3.
Therefore by uniform Gronwall's lemma we obtain
||u(t)||2 + ||ω(t)||2 ≤
2 + 3k2r
F̃ 2 exp
8ν2r r
α2νk1k
, (3.9)
for all t > t0 + r, t0 su
h that above estimates holds. Let us denote for notation simpli
ity
ĉ1 =
2 + 3k2r
, ĉ2 =
8ν2r r
, ĉ3 =
α2νk1k
then (3.9) be
omes
||u(t)||2 + ||ω(t)||2 ≤ ĉ1F̃ 2 exp
ĉ2 + ĉ3F̃
. (3.10)
Now we want to derive the estimate on the average of square of the norm of solution in D(A)×D(A1).
Integrating (3.8) on the interval (t, t+ T ) with respe
t to t we get
||u(t+ T )||2+||ω(t+ T )||2 − ||u(t)||2 − ||ω(t)||2
|Au(s)|2 + |A1ω(s)|2
|u(t)|2||ω(t)||2 +
||u(t)||2 + ||ω(t)||2
|f(t)|2 + |g(t)|2
hen
e
|Au(s)|2+|A1ω(s)|2
ds ≤ 1
||u(t)||2 + ||ω(t)||2
|f(s)|2 + |g(s)|2
|u(s)|2 + |ω(s)|2
||u(s)||2 + ||ω(s)||2
||u(s)||2 + ||ω(s)||2
Be
ause solutions are uniformly bounded in the V norm for large t, then for t and T large enough
|Au(s)|2+ |A1ω(s)|2
ds ≤ 5
F̃ 2 +
16ν2r
||u(s)||2 + ||ω(s)||2
α2νk1
|u(s)|2 + |ω(s)|2
||u(s)||2 + ||ω(s)||2
32ν2r
αk21k2
F̃ 2 +
α2νk1k
||u(s)||2 + ||ω(s)||2
||u(s)||2 + ||ω(s)||2
32ν2r
αk21k2
F̃ 2 +
16Cĉ1
α2νk21k
F̃ 6 exp
ĉ2 + ĉ3F̃
(3.11)
We need one more to estimate the number of determining modes in terms of H−1 norm of for
es and
moments. Taking the s
alar produ
t of (1.1) with u in H we obtain
|u|2 + (ν + νr)||u||2 = 2νr(rotω, u) + (f, u) (3.12)
be
ause b(u, u, u) = 0. We estimate the terms of the RHS of (3.12) as follows
2νr(rotω, u) = 2νr(ω, rot u) ≤ 2νr|ω| · ||u|| ≤ 2νr|ω|2 +
||u||2,
(f, u) ≤ ||f ||H−1 ||u||H ≤
||u||2 +
||f ||2
(3.13)
We treat (1.3) in an analogous way. We multiply it by ω, integrate over Q and we have
|ω|2 + α||ω||2 + 4νr|ω|2 = 2νr(rot u, ω) + (g, ω). (3.14)
The terms of the RHS of (3.14) are estimaded
2νr(rot u, ω) ≤ 2νr|ω|2 +
||u||2,
(g, ω) ≤ ||g||H−1 ||ω|| ≤
||ω||2 +
||g||2H−1 .
(3.15)
Adding equations (3.12) and (3.14), using estimates (3.13) and (3.15) we arrive at
|u|2 + |ω|2
||u||2 + ||ω|2
||f ||2
+ ||g||2H−1
. (3.16)
Let us noti
e that (3.16) looks similar to (3.6). Therefore, pro
eeding in the same way we obtain
(||u(s)||2 + ||ω(s))||2) ds ≤ 2
F̃ 2−1 (3.17)
for t and T large enough. Finally we have all a priori estimates we need in the sequel.
4 Determining modes
In this se
tion we introdu
e the notion of determining modes for mi
ropolar �uids and estimate the
number of them in terms of parameters of the model. We follow the method des
ribed in [24℄.
Let us 
onsider two solutions of 2-D mi
ropolar �uid equations (u1, ω1) and (u2, ω2) (ui = (u
i (x, t), u
i (x, t))
and ωi = ωi(x, t)) 
orresponding to two possibly di�erent pairs of external for
es and moments (f1, g1)
and (f2, g2) respe
tively. More expli
itly, (u1, ω1) and (u2, ω2) satisfy the equations
− (ν + νr)∆u1 + (u1 · ∇)u1 +∇p = 2νr rotω1 + f1, (4.1)
div u1 = 0, (4.2)
− α∆ω1 + (u1 · ∇)ω1 + 4νrω1 = 2νr rot u1 + g1 (4.3)
− (ν + νr)∆u2 + (u2 · ∇)u2 +∇q = 2νr rotω2 + f2, (4.4)
div u2 = 0, (4.5)
− α∆ω2 + (u2 · ∇)ω2 + 4νrω2 = 2νr rotu2 + g2, (4.6)
with 
orresponding pressure terms p = p(x, t) and q = q(x, t). The boundary 
onditions are periodi
 for
both problems. It is assumed, that external for
es f1, f2 and moments g1, g2 have the same asymptoti
behavior for large time, that is,
|f1(x, t)− f2(x, t)|)2H−1 + |g1(x, t)− g2(x, t)|
dx→ 0 as t→ ∞. (4.7)
De�nition 1. The �rst m modes asso
iated with Pm and P
m are 
alled determining modes if the

ondition
|Pmu1(x, t)− Pmu2(x, t)|2 + |P 1mω1(x, t)− P 1mω2(x, t)|2
dx→ 0 as t→ ∞ (4.8)
and the 
ondition for for
es and moments (4.7) imply
|Qmu1(x, t)−Qmu2(x, t)|2 + |Q1mω1(x, t)−Q1mω2(x, t)|2
dx→ 0 as t→ ∞. (4.9)
We will give the estimate of the value of the number m in terms of asymptoti
 strength of the for
es
and moments measured in terms of their L2 and H−1 norm, that is
F̃ = lim sup
|f1(t)|2 + |g1(t)|2
F̃−1 = lim sup
||f1(t)||2H−1 + ||g1(t)||
Theorem 1. Let fi ∈ L∞(0, T ;H), gi ∈ L∞(0, T ;L2) for i = 1, 2. If the for
es and moments satisfy

ondition (4.7), then the �rst m modes are determining in the sense of De�nition 1 provided that
16ν2r
dλ1αk1
dλ1k3k
F̃ 2−1.
Proof. This estimate is similar to that obtained in [9℄ but in this paper we have relaxed the 
onvergen
e
of for
es and moments to be only in H−1. Moreover, the obtained estimate is in terms of their H−1
norm. The same argumentation works in the 
ase of no-slip boundary 
oditions.
Our estimate is based on the following generalization of the 
lassi
al Gronwall's lemma (
.f. [24℄).
Lemma 1. Let γ = γ(t) and β = β(t) be lo
ally integrable real-valued fun
tions on [t0,∞) that satisfy
the following 
onditions for some T > 0:
lim inf
γ(τ) dτ > 0, (4.10)
lim sup
γ−(τ) dτ <∞, (4.11)
β+(τ) dτ = 0, (4.12)
where γ−(t) = max{−γ(t), 0} and β+(t) = max{β(t), 0}. Suppose that ξ = ξ(t) is an absolutely 
ontin-
uous nonnegative fun
tion on [t0,∞) that satis�es the following inequality almost everywhere on [t0,∞):
+ γξ ≤ β.
Then ξ(t) → 0 as t→ ∞.
Writing the equations of mi
ropolar �uid in fun
tional form for a pair of solutions (u1, ω1) and
(u2, ω2) and subtra
ting them we �nd
ut + (ν + νr)Au+B(u, u1) +B(u2, u) = 2νr rotω + f, (4.13)
ωt + αA1ω +B1(u, ω1) +B1(u2, ω) + 4νrω = 2νr rotu+ g, (4.14)
where u = u1 − u2, ω = ω1 − ω2, f = f1 − f2 and g = g1 − g2. First we will deal with the equation
(4.13), multiplying it by Qmu and integrating over Q we obtain
|Qmu|2 + (ν + νr)‖Qmu‖2 + b(u, u1, Qmu) +b(u2, u,Qmu)
= 2νr(rotω,Qmu) + (f,Qmu).
(4.15)
We estimate the linear terms of the RHS of (4.15) as follows
(f,Qmu) ≤ ||f ||H−1 ||Qmu||H1 ≤
ν + 2νr
||Qmu||2 +
ν + 2νr
||f ||2
2νr(rotω,Qmu) = 2νr[(Pm rotω,Qmu) + (Qm rotω,Qmu)] ≤ 2νr|Qm rotω| · |Qmu|
||Q1mω||2 +
|Qmu|2
be
ause (Pm rotω,Qmu) = 0. In order to estimate the form b we write
b(u, u1, Qmu) = b(Pmu, u1, Qmu) + b(Qmu, u1, Qmu) (4.16)
and (be
ause b(u, v, v) = 0)
b(u2, u,Qmu) = b(u2, Pmu,Qmu). (4.17)
Using (2.3) and the Young inequality we infer that
b(Pmu, u1, Qmu) ≤ c1|Pmu|1/2||Pmu||1/2|u1|1/2||u1||1/2||Qmu||,
b(Qmu, u1, Qmu) ≤ c1|Qmu| · ||Qmu|| · ||u1||
ν + νr
||Qmu||2 +
ν + νr
|Qmu|2||u1||2,
b(u2, Pmu,Qmu) ≤ c1|u2|1/2||u2||1/2|Pmu|1/2||Pmu||1/2||Qmu||.
(4.18)
Now we treat the equation (4.14) in a similar manner. Taking it's s
alar produ
t with Q1mω in Ḣ
we obtain
|Q1mω|2 + α||Q1mω||2 + b1(u, ω1, Q1mω) +b1(u2, ω,Q1mω)
= 2νr(rot u,Q
mω) + (g,Q
(4.19)
The terms of the RHS of (4.19) are estimated as follows
(g,Q1mω) ≤
||Q1mω||2 +
||g||2H−1 ,
2νr(rot u,Q
mω) = 2νr(Q
m rotu,Q
mω) ≤
||Qmu||2 + 4νr|Q1mω|2
and the form b1 as will be stated next
b1(Pmu, ω1, Q
mω) ≤ c1|Pmu|1/2||Pmu||1/2|ω1|1/2||ω1||1/2||Q1mω||,
b1(Qmu, ω1, Q
mω) ≤ c1|Qmu|1/2||Qmu||1/2||ω1|| · |Q1mω|1/2||Q1mω||1/2
c1||ω1||
|Qmu| · ||Qmu||+ |Q1mω| · ||Q1mω||
ν + νr
||Qmu||2 +
c21||ω1||2
2(ν + νr)
|Qmu|2 +
||Q1mω||2 +
c21||ω1||2
|Q1mω|2,
b1(u2, P
mω) ≤ c1|u2|1/2||u2||1/2||Q1mω|| · |P 1mω|1/2||P 1mω||1/2.
(4.20)
Adding (4.15) and (4.19) and using foregoing estimates we arrive at
|Qmu|2 + |Q1mω|2
||Qmu||2 + ||Q1mω||2
|Qmu|2 + |Q1mω|2
16ν2r
||u1||2 + ||ω1||2
≤ β(t)
(4.21)
where
β(t) = all terms 
onverging to 0 as t→ ∞,
k3 = min(ν + νr, α).
We make use of the following inequalities λm+1|Qmu|2 ≤ ||Qmu||2 and λm+1|Q1mω||2 ≤ ||Q1mω||2 in order
to write (4.21) in a form whi
h allows us to use the generalized Gronwall Lemma (Lemma 1)
|Qmu|2 +|Q1mω|2
|Qmu|2 + |Q1mω|2
k1λm+1 −
||u1||2 + ||ω1||2
16ν2r
(4.22)
Denoting
ξ(t) = |Qmu|2 + |Q1mω|2,
γ(t) = k1λm+1 −
||u1||2 + ||ω1||2
16ν2r
we 
an write (4.22) in the form
+ γξ ≤ β.
Now we have only to 
he
k the assumptions of Lemma 1. In se
tion 3 we have shown that (see (3.17))
||u1(s)||2 + ||ω1(s)||2 ds ≤
F̃ 2−1.
To 
he
k assumption (4.10) we write
lim inf
γ(s) ds ≥ k1λm+1 −
16ν2r
− lim sup
||u1||2 + ||ω1||2
≥ k1λm+1 −
16ν2r
k21k3
F̃ 2−1.
This assumption is satis�ed for
m ≥ 16ν
dλ1αk1
dλ1k3k
F̃ 2−1 (4.23)
It's easy to 
he
k that if m satis�es (4.23) then the other assumptions are also ful�lled. That ends the
proof.
Corollaries
This part of the paper was inspired by the paper of J.C. Robinson [25℄, in whi
h he shows how the
spatial distribution of a for
e in�uen
es the dimension of a global attra
tor for Navier-Stokes equation.
Suppose that the asymptoti
 amount of for
es f and moments g is equal to F̃ . We 
he
k how the
spatial distribution of external for
es and moments in�uen
es the number of determining modes. We

onsider some 
ases and write the 
al
ulations only for f be
ause these for g are exa
tly the same.
1. Let us 
onsider that the for
es and moments a
t only in two s
ales i.e.
|fn(t)|2 = |fN (t)|2 =
|f(t)|2
Then the H
norm of f satis�es
||f ||2
|f |2
∼ |f |2
That, after inserting to (4.23), gives us
16ν2r
dλ1αk1
dλ1k3k
Therefore the number of determining modes depends on inverse of the number of modes in whi
h
for
es and moments are a
ting.
2. Suppose that for
es and moments are a
ting only in some s
ales and we know nothing about the
distribution of it's magnitude through modes
fkwk.
The following inequalities are straightforward 
onsequen
e of the de�nition of the norm in H
|f |2 ≤ ||f ||2
|f |2
and we arrive at
F̃ 2 ≤ F̃ 2−1 ≤
F̃ 2. (4.24)
Inserting (4.24) to (4.23) we get
16ν2r
dλ1αk1
dλ1k3k
F̃ 2−1 ≥
16ν2r
dλ1αk1
dλ1k3k
3. Assume that the magniude of for
es is uniormly distributed in the �rst N modes that is |fk|2 =
(1/N)|f |2
. Then the following equalities hold
||f ||2
|f |2
Sin
e for large k is λk ∼ c · k then F̃−1 ∼ N−1/2(lnN)1/2F̃ and we have the following estimate of
the number of determining modes
m ≥ 16ν
dλ1k3k1
dλ1k3k
F̃ 2N−1/2(lnN)1/2.
4. Let us 
onsider that the for
es and moments a
t in �rst N s
ales and the norm of ea
h mode
in
erases linearly with it's number
|fk|2 =
2||f ||2
N(N + 1)
for k = 1, . . . N . Then we have
||f ||2H−1 =
2k||f ||2
n(n+ 1)
2||f ||2
n(n+ 1)
Sin
e λk ∼ k then
||f ||2H−1 ∼
||f ||2
(n+ 1)
and the (4.23) implies
m ≥ 16ν
dλ1k3k1
dλ1k3k
cF̃ 2
5. Now we assume that the norm of ea
h mode de
erases linearly with it's number, that is
|fk|2 =
2||f ||2
n(n+ 1)
· (n+ 1− k).
Thus we have
||f ||2H−1 =
2||f ||2
n(n+ 1)
n+ 1− k
2||f ||2
n(n+ 1)
2||f ||2
n(n+ 1)
[(n+ 1) ln n− n].
From (4.23) and the foregoing we infer that
16ν2r
dλ1k3k1
dλ1k3k
2F̃ 2
n(n+ 1)
[(n+ 1) ln n− n].
Above 
onsiderations show that if we in
rease the number of modes in whi
h the for
es and moments
a
t or we a
t only in modes with high-wavenumber then the number of modes ne
essary to determine
the �ow de
reases. It 
ould be so be
ause in small s
ales, 
orresponding to high-wavenumber modes, the
damping e�e
t of vis
osity is stronger than in large s
ales. Moreover the number of determining modes
depends on how are the for
es and moments distributed throughout the modes.
5 Determining nodes
In this se
tion we de�ne the notion of determining nodes and estimate the number of them in terms of
asymptoti
 strength of for
es and moments F̃ .
We 
onsider two solutions of mi
ropolar �uid equations (u1, ω1) and (u2, ω2):
− (ν + νr)∆u1 + (u1 · ∇)u1 +∇p = 2νr rotω1 + f1, (5.1)
div u1 = 0, (5.2)
− α∆ω1 + (u1 · ∇)ω1 + 4νrω1 = 2νr rot u1 + g1 (5.3)
− (ν + νr)∆u2 + (u2 · ∇)u2 +∇q = 2νr rotω2 + f2, (5.4)
div u2 = 0, (5.5)
− α∆ω2 + (u2 · ∇)ω2 + 4νrω2 = 2νr rotu2 + g2, (5.6)

orresponding to two possibly di�erent pairs of external for
es and moments (f1, g1) and (f2, g2) respe
-
tively, the 
orresponding pressure terms are p = p(x, t) and q = q(x, t). The boundary 
onditions are
periodi
 with vanishing spa
e averages for both problems. We assume, as in previous se
tion, that for
es
and moments have the same asymptoti
 behavior but now we want them to 
onverge in the L2 norm,
that is,
|f1(x, t)− f2(x, t)|2 + |g1(x, t)− g2(x, t)|2
dx→ 0 as t→ ∞. (5.7)
We 
onsider a set of N measurement points (
alled nodes), denoted by Σ, Σ = {x1, x2, . . . , xN}. We
assume, that these points are uniformly distributed within the domain Q in the sense that Q may be

overed by N identi
al squares Q1, . . . , QN su
h that there is one and only one given point in ea
h
square: xi ∈ Qi.
We assume, that the �ows have the same time-asymptoti
 behavior at measurement points. This

ondition 
an be written in the form:
j=1,...,N
|u1(xj , t)− u2(xj , t)| → 0, as t→ ∞ (5.8)
j=1,...,N
|ω1(xj, t)− ω2(xj , t)| → 0, as t→ ∞. (5.9)
We want to estimate how many points of observation are ne
essary to determine the asymptoti
 behavior
of a �ow in the following sense:
De�nition 2. The set Σ is 
alled a set of determining nodes if (5.8), (5.9) together with 
ondition for
for
es and moments (5.7) implies
|u1(x, t)− u2(x, t)|2 + |ω1(x, t)− ω2(x, t)|2
dx→ 0 as t→ ∞. (5.10)
We will a
tually show, that (5.8), (5.9) and (5.7) imply that solutions 
onverge to ea
h other in a
stronger norm asso
iated with enstrophy, that is
|∇u1(x, t)−∇u2(x, t)|2 + |∇ω1(x, t)−∇ω2(x, t)|2
dx→ 0 as t→ ∞.
Let us denote
η(w) = max
1≤j≤N
|w(xj)|
for ea
h velo
ity or mi
rorotation �eld w.
There are two lemmas used in the proof of existen
e of a �nite set of determining nodes. One of
them is generalized Gronwall's lemma, already used in previous se
tion, the other one is the following
lemma from [8℄.
Lemma 2. Let the domain Q be 
overed by N identi
al squares. Consider the set Σ = {x1, . . . xN} of
points in Q, distributed one in ea
h square. Then for ea
h ve
tor �eld in Ḣ2p, the following quantities
hold:
|w|2 ≤ c
η(w)2 +
|∆w|2, (5.11)
||w||2 ≤ cNη(w)2 + c
|∆w|2, (5.12)
||w||2L∞(Q) ≤ cNη(w)
|∆w|2, (5.13)
where the 
onstant c depends only on the shape of Q i.e. L1/L2.
Theorem 2. Let Q be domain 
overed by N identi
al squares Q1, . . . , QN and 
onsider a set Σ =
{x1, . . . , xN} of points in Q distributed one in ea
h square xi ∈ Qi, for 1 ≤ i ≤ N . Let f1 and f2 be two
for
ing terms in L∞(0,∞;H), g1 and g2 be two moments in L∞(0,∞; Ḣ0p ), satisfying (5.7).
Then the set Σ is a set of determining nodes in the sense of De�nition 2 for the 2-dimensional
mi
ropolar �uid equations with periodi
 boundary 
onditions provided that
− 2νr +
5αk2 + 32ν
αk21k2
F̃ 2 +
16Cĉ1
ανk21k
F̃ 6 exp(ĉ2 + ĉ3F̃
16c41ĉ1
λ1ανk1k2
F̃ 4 exp(ĉ2 + ĉ3F̃
where F̃ is de�ned in (3.1).
Proof. Let us denote u = u1 − u2, f = f1 − f2 et
. Subtra
ting equations (5.1) and (5.4) we �nd
+ (ν + νr)∆u+ (u1 · ∇)u1 − (u2 · ∇)u2 = 2νr rotω + f. (5.14)
By taking the inner produ
t of (5.14) and Au in H, we get
||u(t)||2 + (ν + νr)|Au|2 +b(u, u1, Au) + b(u2, u,Au)
= 2νr(rotω,Au) + (f,Au).
(5.15)
Exploiting the orthogonality property (2.2) we obtain (
.f. [24℄)
b(u, u1, Au) + b(u2, u,Au) = −b(u, u,Au1),
thus we 
an write (5.15) in the form
||u(t)||2 + (ν + νr)|Au|2 = 2νr(rotω,Au) + (f,Au) + b(u, u,Au1). (5.16)
We estimate the terms in the RHS of (5.16) using (2.3), Lemma 2 and Young's inequality
2νr(rotω,Au) ≤
|Au|2 + 2νr||ω||2,
(f,Au) ≤
|Au|2 +
|f |2,
b(u, u,Au1) ≤ c1|u|1/2||u|| · |Au|1/2|Au1|
c1/4c1
η(u)1/2||u|| · |Au|1/2|Au1|
c1/4c1
|Au| · ||u|| · |Au1|
1/4c1
η(u)1/2||u|| · |Au|1/2|Au1|
c1/2c21
||u||2|Au1|2 +
|Au|2.
Using above estimates we �nd from (5.16) that
||u||2 +
ν|Au|2 −
c1/2c21
||u||2|Au1|2
≤ 2νr||ω||2 +
|f |2 + c
1/4c1
η(u)1/2||u|| · |Au|1/2|Au1|.
(5.17)
Now we treat equations for mi
rorotation (5.3) and (5.6) in similar way. Subtra
ting them we �nd
+ αA1ω + (u1 · ∇)ω1 − (u2 · ∇)ω2 + 4νrω = 2νr rot u+ g. (5.18)
Multiplying (5.18) by A1ω and integrating over Q we get
||ω||2 + α|A1ω|2 +b1(u, ω,A1ω) + b1(u2, ω,A1ω) + 4νr||ω||2
= 2νr(rot u,A1ω) + (g,A1ω).
(5.19)
We estimate the nonlinear terms as follows
b1(u, ω1, A1ω) ≤ c1|u|1/2|Au|1/2||ω1|| · |A1ω|
|A1ω|2 +
|u| · |Au| · ||ω1|2
|A1ω|2 +
|Au|2 +
|u|2||ω1||4,
b1(u2, ω,A1ω) ≤ c1|u2|1/2|Au2|1/2||ω|| · |A1ω|
|A1ω|2 +
|u2| · |Au2| · ||ω||2
and the terms in the RHS of (5.19) similar to the terms in the RHS of (5.16), that is
2νr(rot u,A1ω) ≤
|A1ω|2 +
||u||2,
(g,A1ω) ≤
|A1ω|2 +
|g|2.
Using above estimates we infer from (5.19) that
||ω||2 + α
|A1ω|2 +4νr||ω||2 −
||u||2 − 2c
|u2| · |Au2| · ||ω||2
|g|2 +
|Au|2 +
|u|2||ω1||4.
(5.20)
Adding (5.17) and (5.20) we have
||u||2 + ||ω||2
|Au|2 + |A1ω|2
−||u||2
|Au1|2 +
−||ω||2
|u2| · |Au2| − 2νr
|u|2||ω1||4
|f |2 + |g|2
η(u)1/2||u|| · |Au|1/2|Au1|,
(5.21)
where k3 = min{νr, α}. By Lemma 2 we 
on
lude that
|Au|2 ≥
||u||2 − λ1N2η(u)2, (5.22)
and analogous inequality for ω
|A1ω|2 ≥
||ω||2 − λ1N2η(ω)2. (5.23)
By the Poin
aré inequality we get
|u|2||ω1||4 ≤
α2νλ1
||u||2||ω1||4. (5.24)
Taking into a

ount (5.22) � (5.24) we infer from (5.21) that
||u||2 +||ω||2
k1λ1N
|Au1|2 −
− 2c1
|u2| · |Au2|
+2νr −
α2νλ1
||ω1||4
||u||2 + ||ω||2
|f |2 + |g|2
η(u)1/2||u|| · |Au|1/2|Au1|
k1λ1N
(η(u)2 + η(ω)2).
(5.25)
Denoting
γ(t) = 2
k1λ1N
|Au1|2 −
|u2| · |Au2|+ 2νr −
α2νλ1
||ω1||4
β(t) = 2
|f |2 + |g|2
η(u)1/2||u|| · |Au|1/2|Au1|+
k1λ1N
(η(u)2 + η(ω)2)
ξ(t) = ||u||2 + ||ω||2
we 
an write (5.25) in the form
+ γξ ≤ β.
The time average of β = β(t) goes to zero as time goes to in�nity be
ause for t bounded away from zero,
time averages of the square of norms of u1 and u2 in D(A) are uniformly bounded (see se
tion 3), so the

ondition (4.12) of Lemma 1 is satis�ed. Using again the same argument one 
an 
he
k that assumption
(4.11) is satis�ed.
In order to 
he
k assumption (4.10) we write
lim inf
γ(s) ds ≥2
k1λ1N
+ 2νr −
− lim sup
|Au1(s)|2
|u2(s)| · |Au2(s)|+
||ω1(s)||4
k1λ1N
+ 2νr −
− lim sup
|Au1(s)|2 +
|u2(s)|2
|Au2(s)|2 +
||ω1(s)||4
k1λ1N
+ 2νr −
5αk2 + 32ν
αk21k2
F̃ 2 +
16Cĉ1
ανk21k
F̃ 6 exp(ĉ2 + ĉ3F̃
− 16c
λ1ανk1k2
F̃ 4 exp(ĉ2 + ĉ3F̃
Therefore if
N ≥ c
− 2νr +
5αk2 + 32ν
αk21k2
F̃ 2 +
16Cĉ1
ανk21k
F̃ 6 exp(ĉ2 + ĉ3F̃
16c41ĉ1
λ1ανk1k2
F̃ 4 exp(ĉ2 + ĉ3F̃
(5.26)
then (4.10) holds, other assumptions are also ful�lled and we infer from uniform Gronwall lemma that
||u(t)||2 + ||ω(t)||2 → 0 as t→ ∞. That ends the proof.
6 Hausdor� and fra
tal dimension of attra
tor
In this se
tion we re
all the notions of Hausdor� and fra
tal dimension of an invariant set. Then we show
that the attra
tors Aνr , νr ≥ 0 asso
iated with the system (1.1)-(1.3) have �nite Hausdor� and fra
tal
dimensions, whi
h 
an be estimated by 
onstants depending on the data: f, g, ν, α and the domain of
�ow Q but independent of the mi
rorotation vis
osity νr.
Let (X, d) be a metri
 spa
e and Y ⊂ X be a subset of X. For every d ∈ R+ and ǫ > 0 we set
µH(Y, d, ǫ) = inf
rdi ,
where the in�mum is for all 
overings of Y by a family (Bi)i∈I of balls of radii ri ≤ ǫ. µH(Y, d, ǫ) is non
in
reasing fun
tion of ǫ. The number µH(Y, d), 
alled the d-dimensional Hausdor� measure of set Y , is
de�ned as
µH(Y, d) = lim
µH(Y, d, ǫ) = sup
µH(Y, d, ǫ).
If µH(Y, d
′) <∞ for some d′ then µH(Y, d) = 0 for every d > d′. Then there exist d0 ∈ [0,∞] su
h that
µH(Y, d) = 0 for every d > d0 and µH(Y, d) = ∞ for d < d0. The number d0 is 
alled the Hausdor�
dimension of Y � dH(Y ).
Now we de�ne the fra
tal dimension of Y . Let ǫ > 0, we denote by nY (ǫ) the minimum number of
balls of X of radius ǫ ne
essary to 
over the set Y . The fra
tal dimension of Y , 
alled also the 
apa
ity
of Y , is the number dF de�ned as follows
dF (Y ) = lim sup
log nY (ǫ)
log 1/ǫ
We refer the reader to [26℄ for more details.
Theorem 3. There exist a 
onstant C0 depending only on the shape of Q, whi
h has following property.
If N is an integer su
h that
N − 1 < 2C0(k31k2)−1/2
|f |2 + |g|2
)1/2 ≤ N, (6.1)
where k1, k2 are as in (3.2), then the N -dimensional volume element in phase spa
e H is exponentially
de
aying, moreover the Hausdor� dimensions of Aνr , νr ≥ 0 are less or equal to N and the fra
tal
dimensions are less or equal to 2N .
Proof. The relation (6.1) pro
eeds from estimates for the uniform Lyapunov exponents asso
iated with
attra
tors Aνr .
First, we rewrite our equations (1.1) - (1.3) in a more suitable form. In order to do this we introdu
e
some notations. Let ūi = (ui, ωi) ∈ H (or V ) for i = 1, 2. We introdu
e s
alar produ
ts and norms in H
and V as follows
[ū1, ū2] = (u1, u2) + (ω1, ω2), [ū] = [ū, ū]
for all ū, ū1, ū2 ∈ H and
[[ū1, ū2]] = (∇u1,∇u2) + (∇ω1,∇ω2), [[ū]] = [[ū, ū]]1/2
for all ū, ū1, ū2 ∈ V. The notation seems to be 
onfusing, but it will be always 
lear from 
ontext if (·, ·)
denotes the s
alar produ
t in L2 or a ve
tor in H or V.
We de�ne the trilinear form B on V × V × V by
B(ū1, ū2, ū3) = b(u1, u2, u3) + b1(u1, ω2, ω3)
and asso
iate with B the bilinear 
ontinuous operator B from V × V to V ′ as follows
〈B(ū1, ū2), φ〉 = B(ū1, ū2, φ) ū1, ū2, φ ∈ V.
We de�ne bilinear forms R and a from V × V by
R(ū1, ū2) = −2νr(rotω1, u2)− 2νr(rot u1, ω2) + 4νr(ω1, ω2),
a(ū1, ū2) = (ν + νr)(∇u1,∇u2) + α(∇ω1,∇ω2)
and asso
iate linear, 
ontinuous operators on V to V ′ by
〈R(ū), φ〉 = R(ū, φ), 〈A(ū), φ〉 = a(ū, φ), ū, φ ∈ V.
Let G = (f, g) ∈ H, then the weak form of (1.1)-(1.3), whi
h has form (
.f. [19℄)
(u(t), ϕ(t)) + (ν + νr)(∇u(t),∇ϕ(t)) +b(u(t), u(t), ϕ)
= 2νr(rotω(t), ϕ) + (f, ϕ)
(6.2)
for all ϕ ∈ V ,and
(ω(t), ψ) + α(∇ω(t),∇ψ) +b1(u(t), ω(t), ψ) + 4νr(ω(t), ψ)
= 2νr(rot u, ψ) + (g(t), ψ)
(6.3)
for all ψ ∈ Ḣ1p (Q), in the sense of s
alar distributions on (0,∞), 
an be rewritten as
[ū, φ] + a(ū, φ) +B(ū, ū, φ) +R(ū, φ) = [G,φ],
where ū = (u, ω), φ = (ϕ,ψ) or in the fun
tional form
ū+A(ū) + B(ū, ū) +R(ū) = G.
We will 
on
ern ourself with the 
orresponding, linearized about ū problem
U = F ′(ū)U,
where F ′(ū) = −A(U)− B(ū, U)− B(U, ū) +R(U).
Our aim now is to estimate form above the tra
e
TrF ′(ū) ◦ PN =
[F ′(ū)ϕj , ϕj ],
where ϕj = ϕj(τ), j = 1, . . . , N is an orthonormal in H basis of PN (τ)H = Span{U1(τ), . . . , UN (τ)},
PN (τ, ξ1, . . . , ξN ) being the orthogonal proje
tor in H on the spa
e spanned by Uj(τ), j = 1, . . . , N ,
where Uj satisfy
Uj = F
′(ū)Uj , Uj(0) = ξj , j = 1, . . . , N.
Be
ause B(ū, ϕj , ϕj) = 0, we have
[F ′(ū)ϕj , ϕj ] = −a(ϕj , ϕj)−B(ϕj , ū, ϕj)−R(ϕj , ϕj).
Let us write ū = (u, ω) and ϕj = (vj , zj), then we obtain
a(ϕj , ϕj) = −(ν + νr)
||vj ||2 − α
||zj ||2,
B(ϕj , ū, ϕj) = −
b(vj , u, vj)−
b1(vj , ω, zj)
R(ϕj , ϕj) = 2νr
(rot zj , vj) + 2νr
(rot vj , zj)− 4νr
|zj |2.
Now let us 
onsider the operator a+R
a(ϕj , ϕj) +R(ϕj , ϕj) = (ν + νr)((vj , vj))
2 + α((zj , zj))
−2νr(rot zj , vj)− 2νr(rot vj, zj) + 4νr|zj |2.
Be
ause of (2.4), S
hwartz and Young's inequalities we have
2νr(rot zj , vj) + 2νr(rot vj, zj) = 4νr(rot zj , vj) ≤ νr||vj ||2 + 4νr|zj |2,
therefore
a(ϕj , ϕj) +R(ϕj , ϕj) ≥ ν((vj , vj)) + α((zj , zj)) ≥ k1[[ϕj , ϕj ]] ∀ϕj ∈ V,
where k1 = min{ν, α} as in (3.2), when
e
a(ϕj , ϕj) +R(ϕj , ϕj)
≤ −k1
[[ϕj , ϕj ]]
We estimate the trilinear form b as follows
b(vj , u, vj)
(vj · ∇)uvj dx
|∇u(x, t)|ρ1(x, t) dx,
where
ρ1(x, t) =
|vj(x, t)|2,
and similarly the form b1
b1(vj , u, zj)
(vj · ∇)uzj dx
|∇u(x, t)|ρ1(x, t)1/2ρ2(x, t)1/2 dx,
where
ρ1(x, t) =
|zj(x, t)|2.
Therefore we 
an estimate the form B as follows
B(ϕj , ū, ϕj)
|∇u|ρ1 + |∇ω|ρ
Setting ρ = ρ1 + ρ2 and by Cau
hy and S
hwartz inequalities we get
B(ϕj , ū, ϕj)
(ρ|∇u|+ ρ|∇ω|) dx
ρ2 dx
(|∇u|+ |∇ω|)2 dx
≤ |ρ|
|∇u|2 + |∇ω|2
2|ρ| · [[ū]].
From the above estimates we infer that
TrF ′(ū(τ)) ◦ PN (τ) ≤ −k1
[[ϕj(τ)]]
2|ρ(τ)| · [[ū(τ)]]. (6.4)
Be
ause the family {ϕj(τ)}Nj=1 is orthonormal in H, then the family of 
orresponding pairs (vj , zj) is
orthonormal in L2(Q)2×2 and we 
an use a generalization of the Sobolev-Lieb-Thirring inequality ([15℄)
and write
|ρ(τ)|2 ≤ C0
||vj ||2 + ||zj ||2
[[ϕj(τ)]]
2, (6.5)
where the 
onstant C0 depends only on the shape of Q.
Let us per
eive that
ρ(x, t) dx = N , then by S
hwartz inequality we get
N2 ≤ |Q| · |ρ|2,
and taking into a

ount (6.5)
[[ϕj(τ)]]
C0|Q|
. (6.6)
We estimate the se
ond term of the RHS of (6.4) using Young's inequality, (6.5) and (6.6)
2|ρ| · [[ū]] ≤ k1
|ρ|2 + C0
[[ū]]2 ≤ k1
[[ϕj ]]
[[ū]]2.
From the above 
onsiderations, we have
TrF ′(ū(τ)) ◦ PN (τ) ≤ −
C0|Q|
[[ū]]2. (6.7)
Let ū0 = (u0, ω0) ∈ A, ū(τ) = S(τ)ū0. Let us set
qN (t) = sup
ū0∈A
TrF ′(S(τ)ū0) ◦ PN (τ) dτ : ξi ∈ H, [ξi] ≤ 1, i = 1, . . . , N
qN = lim sup
qN(t)
In view of (6.7), we 
on
lude that
qN ≤ −
2C0|Q|
γ, (6.8)
where
γ = lim
[[S(τ)ū0]]
2 dτ.
We 
an estimate γ in terms of the data using (3.6). Sin
e
[[S(t)ū0]]
[G]2 +
[ū0]
and by (6.8)
qN ≤ −
2C0|Q|
k21k2
[G]2. (6.9)
Setting
2C0|Q|
, κ2 =
k21k2
[G]2,
we 
an write (6.9) in the form
qN ≤ −κ1N2 + κ2.
The general theory provided in [15℄ allows us to 
on
lude that the uniform Lyapunov exponents µj
asso
iated with the attra
tor satisfy
µ1 + · · ·+ µj ≤ −κ1j2 + κ2, ∀j ∈ N,
and for N satisfying (
.f.. Lemma VI, 2.2 in [15℄)
N − 1 <
the thesis of Theorem 3 holds.
The estimate of the dimension of the global attra
tor, whi
h we obtained, is similar to analogous
estimate for mi
ropolar �uid equations with no-slip boundary 
onditions (
.f. [17℄), but of the higher
order than analogous estimate for NSE with periodi
 boundary 
onditions.
7 Con
lusions
The la
k of orthogonality property of the form b1, whi
h was mentioned is subse
tion 2, 
auses that
the estimates of numbers of determining modes, nodes and the dimension of global attra
tor are more
involved than 
orresponding estimates for Navier-Stokes equation in spa
e-periodi
 
ase. The reason is
following: estimates of the square of the norm of solution in V and the average of the square of the norm
in D(A) for NSE, whi
h are analogous to (3.9) and (3.11), are square with respe
t to F (
.f. [24℄), where
F = lim sup
|f(x, t)|2 dx
The estimates (3.9) and (3.11) are exponential with respe
t to F̃ , whi
h implies exponential estimate
of the number of determining nodes. We 
an't obtain linear dependen
e of the number of determining
modes on F̃ be
ause the form b1 doesn't have the orthogonality property (2.2).
We 
he
k how the estimates we obtained depend on νr. The 
ase when νr is small is parti
ularly
interesting be
ause if νr = 0 the mi
ropolar �uid system redu
es to the Navier-Stokes system and the
velo
ity �eld u be
omes independent on mi
rorotation �eld ω
ut − ν∆u+ (u · ∇)u+∇p = f,
div u = 0,
ωt − α∆ω + (u · ∇)ω = g.
1. The estimate of the number of determining modes m ≈ c1F̃ 2−1+c2 (
.f. (4.23)) is similar to that for
Navier-Stokes equation ([24℄) and mi
ropolar �uid equations ([9℄), both with Diri
hlet boundary

onditions. The 
oe�
ient c1 doesn't depend on νr but if νr → 0 then c2 → 0 so, if νr = 0 our
estimate agrees with that for Navier-Stokes equation.
2. The estimate of the number of determining nodes m ≈ P (F̃ )+Q(F̃ ) exp(R(F̃ )) (
.f. (5.26)), where
P,Q and R are polynomials is similar to that for NSE with Diri
hlet boundary 
onditions ([24℄).
If νr → 0 the above estimate remains exponential with respe
t to F̃ . Our result is mu
h worse
than analogous result for Navier-Stokes equation in [24℄, where it was shown that the dependen
e
is linear.
3. The estimate of the dimension of a global attra
tor (6.1) doesn't depend on νr and is similar to
su
h estimates for NSE [24℄ with Diri
hlet boundary 
onditions and for mi
ropolar �uid equations
with Diri
hlet boundary 
ondition [17℄
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	Introduction
	Mathematical setting of the problem
	A priori estimates
	Determining modes
	Determining nodes
	Hausdorff and fractal dimension of attractor
	Conclusions
	Bibliography
