# The Exponential Attractors for the g-Navier-Stokes

Authors: Delin Wu; Jicheng Tao
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We consider the exponential attractors for the two-dimensional g-Navier-Stokes
equations in bounded domain Ω. We establish the existence of the
exponential attractor inL2(Ω).


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## Body

## 1. Introduction

In this paper, we study the behavior of solutions of the g-Navier-Stokes equations in
spatial dimension 2. These equations are a variation of the standard Navier-Stokes
equations, and they assume the form,(1.1)∂u∂t-νΔu+(u⋅∇)u+∇p=finΩ,1g(∇⋅gu)=∇gg⋅u+∇⋅u=0inΩ, where g=g(x1,x2) is a suitable smooth real-valued function defined
on Ω and Ω is a suitable bounded domain in
ℝ2. Notice that if g(x1,x2)=1, then (1.1) reduce to the standard Navier-Stokes equations.In Roh [1] the author established the global
regularity of solutions of the g-Navier-Stokes equations. One can refer to [2] for details. For the boundary conditions, we
will consider the periodic boundary conditions, while same results can be got for
the Dirichlet boundary conditions on the smooth bounded domain. Before we present
the derivation of the g-Navier-Stokes equations, it is convenient to recall some
relevant aspects of the classical theory of the Navier-Stokes equations. For many
years, the Navier-Stokes equations were investigated by many authors and the
existence of the attractors for 2D Navier-Stokes equations was first proved by
Ladyzhenskaya [3] and independently by
Foiaş and Temam [4]. The
finite-dimensional property of the global attractor for general dissipative
equations was first proved by Mallet-Paret [5]. For the analysis on the Navier-Stokes equations, one can refer to [6].In the past decades, many papers in the literature show that the long-time behavior
of dissipative systems can be understood through the concept of attractors, see
[7–14]. In addition, in [15] the
authors introduced the so-called exponential attractors, which is an interesting
intermediate object between the usual (global) attractors and an inertial manifold
and satisfies some nice properties like those of inertial manifolds (e.g., finite
fractal dimension, exponential attracting, stable with respect to some
perturbations). Indeed it now seems clear that the interesting object to investigate
is the exponential attractor, rather than the usual (global) attractor (which is
recovered as a byproduct). See [16, 17], and so forth. The exponential attractor is
a compact and positively invariant set having finite fractal dimension which
contains the global attractor and attracts every trajectory at an exponential rate.
It is also known that the exponential attractor enjoys stronger robustness than the
global attractor. When the semigroup of a dynamical system depends continuously on a
parameter, the global attractor is in general only upper-semicontinuous. In turn,
under some reasonable assumptions, if an exponential attractor exists, it can depend
continuously on the parameter. Such a continuous dependence was recently studied by
Efendiev and Yagi [18]. When the underlying
space is a Hilbert space, it is known by the same reference [15] quoted above that the squeezing property of semigroup
implies existence of exponential attractors and provides a sharp estimate of
attractor dimensions. When the underlying space is a Banach space, it is known by
Efendiev et al. [19] that the compact
smoothing property of semigroup implies existence of exponential attractors (Theorem
2.3). Another construction of
exponential attractors in Banach spaces was proposed by Dung and Nicolaenko in
[20]. We also refer to [17, 21–25] for more
details.In the paper, compared with the result obtained in [26], taking advantage of a recent result due to Efendiev et al. [19] (Theorem 2.3), we construct the exponential attractor. This paper is
organized as follows. In Section 2, we first
recall some basic results, and then, give an important technique tool [19], that is, Theorem 2.3. In Section 3, we
study the existence of compact exponential attractor for the two-dimensional
g-Navier-Stokes equations in the periodic boundary conditions
Ω.


## 2. Preliminary Results

LetΩ=(0,1)×(0,1) and we assume that the function
g(x)=g(x1,x2) satisfies the following properties:(1)
g(x)∈Cper∞(Ω)(2)
there exist constantsm0=m0(g) and M0=M0(g) such that, for all
x∈Ω,0<m0≤g(x)≤M0. Note that the constant function
g≡1 satisfies these conditions.We denote byL2(Ω,g) the space with the scalar product and the norm
given by(2.1)(u,v)g=∫Ω(u⋅v)gdx,|u|g2=(u,u)g, as well as H1(Ω,g) with the norm(2.2)‖u‖H1(Ω,g)=[(u,u)g+∑i=12(Diu,Diu)g]1/2, where ∂u/∂xi=Diu.Then for the functional setting of the problems (1.1), we use the following functional spaces:(2.3)Hg=ClLper2(Ω,g){u∈Cper∞(Ω):∇⋅gu=0,∫Ωudx=0},Vg={u∈Hper1(Ω,g):∇⋅gu=0,∫Ωudx=0}, where Hg is endowed with the scalar product and the norm in
L2(Ω,g), and Vg is the spaces with the scalar product and the norm
given by(2.4)((u,v))g=∫Ω(∇u⋅∇v)gdx,‖u‖g=((u,u))g. Also, we define the orthogonal projection
Pg as(2.5)Pg:Lper2(Ω,g)⟶Hg and we have that Q⊆Hg⊥, where(2.6)Q=ClLper2(Ω,g){∇ϕ:ϕ∈C1(Ω¯,R)}. Then, we define the g-Laplacian operator(2.7)-Δgu≡1g(∇⋅g∇)u=-Δu-1g(∇g⋅∇)u to have the linear operator(2.8)Agu=Pg[-1g(∇⋅(g∇u))]. For the linear operator Ag, the following hold (see Roh [1]):(1)
Ag is a positive, self-adjoint operator
with compact inverse, where the domain of Ag,D(Ag)=Vg∩H2(Ω,g).(2)
There exist countable eigenvalues ofAg satisfying(2.9)0<λg≤λ1≤λ2≤λ3≤⋯, where λg=4π2m/M and λ1 is the smallest eigenvalue of
Ag. In addition, there exists the corresponding
collection of eigenfunctions {e1,e2,e3,…} which forms an orthonormal basis for
Hg.Next, we denote the bilinear operatorBg(u,v)=Pg(u·∇)v and the trilinear form(2.10)bg(u,v,w)=∑i,j=12∫Ωui(Divj)wjgdx=(Pg(u⋅∇)v,w)g, where u,v,w lie in appropriate subspaces of
L2(Ω,g). Then, the form bg satisfies(2.11)bg(u,v,w)=-bg(u,w,v)foru,v,w∈Hg.We denote a linear operatorR on Vg by(2.12)Ru=Pg[1g(∇g⋅∇)u]foru∈Vg, and have R as a continuous linear operator from
Vg into Hg such that(2.13)|(Ru,u)|≤|∇g|∞m0‖u‖g|u|g≤|∇g|∞m0λg1/2‖u‖gforu∈Vg.We now rewrite (1.1) as abstract
evolution equations,(2.14)dudt+νAgu+Bgu+νRu=Pgf,u(0)=u0.Let us first recall some basic matters on the dynamical system. LetE be a Banach space and let K be a subset of E,K being a metric space equipped with the distance
induced from the norm of E. Let S(t),0≤t<∞ be a family of mappings from
K into itself having the following properties: (i)
S(0)=I (the identity mapping); (ii)
S(t)S(s)=S(t+s),0≤t,s<∞ (the semigroup property); (iii) the mapping
G:[0,∞)×K→K,(t,u0)→S(t)u0, is continuous. Such a family is called a
continuous (nonlinear) semigroup acting on K. The image of S(·)u0 drawn in K is called the trajectory starting from
K. The whole of such trajectories is the dynamical
system (S(t),K,E), where K and E are called the phase-space and the universal space,
respectively.A subset𝒜 of the phase-space K is the global attractor of
(S(t),K,E) if the following conditions are satisfied: (i)
𝒜 is a compact subset of E; (ii) 𝒜 is an invariant set, that is,
S(t)𝒜=𝒜 for every 0<t<∞; (iii) 𝒜 attracts every bounded subset of
K, namely, for any bounded subset
B⊂K, it holds that limt→∞dist(S(t)B,𝒜)=0, where dist(A,B)=supx∈Ainfy∈B∥x-y∥E denotes the Hausdorff pseudodistance between two
sets A and B.We recall the definition of an exponential attractor (see, e.g., [15, 17]).Definition 2.1.
A compact set𝔄⊂E is an exponential attractor for
S(t) if
(1)
it has finite fractal dimension,dimF𝔄<+∞,
(2)
it is positively invariant,S(t)𝔄⊆𝔄,for allt≥0,
(3)
it attracts exponentially the bounded subsets ofE in the following sense:
(2.15)
∀
B
⊂
E
bounded
,
dist
⁡
(
S
(
t
)
B
,
A
)
≤
Q
(
∥
B
∥
E
)
e
-
α
t
,
t
≥
0
, where the positive constant
α, the monotonic function
Q are independent of B.Remark 2.2.
We note that the existence of an exponential attractor𝔄 for the semigroup S(t) automatically implies the existence of the
global attractor 𝒜 and the embedding 𝒜⊂𝔄. We note however that, in contrast to the
global attractor, an exponential attractor is not uniquely defined.To construct an exponential attractor, we make use of the following result due to
Efendiev et al. [19].Theorem 2.3.
LetX,Y be two Banach spaces such that
Y is compactly embedded in
X. Let Z be a bounded closed subset of
X. Assume that a semigroup
(S(t))t>0  on X satisfies the following conditions: there
exists a time t*>0, constants L1,L2>0, and exponents γ1,γ2>0 such that S(t*) maps Z into itself and (2.16)‖S(t*)u0-S(t*)v0‖Y≤L1‖u0-v0‖X,‖S(s)u0-S(t)v0‖X≤L2(|s-t|γ1+‖u0-v0‖Xγ2) hold for any u0,v0∈Z and s,t∈[0,t*]. Then the dynamical system
((S(t))t>0,Z) admits an exponential attractor.Hereafterc will denote a generic scale invariant positive
constant, which is independent of the physical parameters in the equation and may be
different from line to line and even in the same line.


## 3. Exponential Attractor of g-Navier-Stokes Equations

This section deals with the existence of the exponential attractor for the
two-dimensional g-Navier-Stokes equations with periodic boundary condition.In Roh [1], the authors have shown that the
semigroup S(t):Hg→Hg(t≥0) associated with the systems (2.14) possesses a global attractor in
Hg and Vg. The main objective of this section is to prove
that the system (2.14) has
exponential attractors in Hg.To this end, we first state some of the following results of existence and uniqueness
of solutions of (2.14).Theorem 3.1.
Letf∈Vg′ be given. Then for every
u0∈Hg there exists a unique solution
u=u(t) on [0,∞) of (2.14). Moreover, one has (3.1)u(t)∈C[0,T;Hg)∩L2(0,T;Vg),∀T>0. Finally, if u0∈Vg, then (3.2)u(t)∈C[0,T;Vg)∩L2(0,T;D(Ag)),∀T>0.Proof.
The Proof of Theorem3.1 is
similar to Roh [1] and Kwaket al. [26] and Temam [12].In a similar manner as in [13, 14], we can establish the following a priori
estimate for for (2.14).Lemma 3.2.
Letℬ be a bounded subset of
Hg. The semigroup {S(t)}:Hg(Vg)→Hg(Vg) associated with (2.14) possesses absorbing sets (3.3)B0={u∈Hg∣|u|g≤ρ0}∀t≥t0(B),B1={u∈Vg∣‖u‖g≤ρ1∀t≥t1(B)=t0(B)+1} which absorb all bounded sets of
Hg. Moreover ℬ0 and ℬ1 absorb all bounded sets of
Hg and Vg in the norms of Hg and Vg, respectively.Let(S(t))t≥0 be the semigroup associated with (2.14). Since Ω is bounded, Vg is compactly embedded in Hg. Then we consider Hg,Vg as X,Y in Theorem 2.3, respectively. The crucial point is the choice of the bounded subset
Z. Let(3.4)A=⋃t≥τS(t)B1¯, where B¯ denote the closure of B in Hg and τ is the time when ℬ1 absorbs itself. We claim that
𝔄 has all properties required for
Z. In fact, it is easy to see that
𝔄 is positively invariant under the semiflow
S(t). In order to see that 𝔄 has the other required properties, we begin with
constructing uniform a priori estimates in time t for the solution u to (2.14).Now we consider difference of two solutions of (2.14) starting from ℬ0.Proposition 3.3.
Let the assumptions of Theorem2.3
hold. Then, there exists a time t*>0, constants L1>0, and exponents γ1,γ2>0 such that S(t*) maps 𝔄 into itself and (3.5)|S(s)u0-S(t)v0|g≤L1(|s-t|γ1+|u0-v0|gγ2) holds for any u0,v0∈𝔄 and s,t∈[0,t*].Proof.
Letu10,u20∈𝔄 and let u1 and u2 be two solutions to (2.14) with
u1(0)=u10,u2(0)=u20, respectively.
Letũ=u1-u2 which satisfies(3.6)dũdt+νAgũ+Bgu1-Bgu2+νRũ=0. Multiplying (3.6) by ũ, we have (3.7)(dũdt,ũ)g+(νAgũ,ũ)g+(B(u1,u1)-B(u2,u2),ũ)g+(Rũ,ũ)g=0,(3.8)12ddt|ũ|g2+ν‖ũg‖g2+bg(ũ,u2,ũ)+(Rũ,ũ)g=0. It follows that (3.9)12ddt|ũ|g2+ν‖ũg‖g2≤|bg(ũ,u2,ũ)|+(Rũ,ũ)g. Since bg satisfies the following inequality (see Temam
[12]): (3.10)|bg(u,v,w)|≤c|u|g1/2‖u‖g1/2‖v‖g|w|g1/2‖w‖g1/2,∀u,v,w∈Vg, thus, (3.11)|bg(ũ,u2,ũ)|≤c‖u2‖g‖ũ‖g|ũ|g≤ν4‖ũ‖g2+c‖u2‖g2|ũ|g2. Next, the Cauchy inequality, (3.12)|(Rũ,ũ)g|=|(νg(∇g⋅∇)ũ,ũ)g|≤νm0|∇g|∞‖ũ‖g|ũ|g≤ν4‖ũ‖g2+c|ũ|g2. Finally, putting (3.11)-(3.12) together, there exist constant M0=M0(m0,|∇g|∞,ρ0,ρ1) such that (3.13)12ddt|ũ|g2+12ν‖ũ‖g2≤M0|ũ|g2. Therefore, we deduce that (3.14)ddt|ũ|g2≤c|ũ|g2. By the Gronwall inequality, the above inequality
implies that (3.15)|ũ|g≤ect|u10-u20|g.
Next we multiply (2.14) by
u, and we have (3.16)(dudt,u)g+(νAgu,u)g+(B(u,u),u)g=(f,u)g-(Ru,u)g. Since Ω is bounded, the Poincaré inequality holds:
(3.17)λg|u|g2≤‖u‖g2∀u∈Vg. There exist constant M1=M1(m0,|∇g|∞,ρ0,ρ1) such that we deduce that (3.18)ddt|u|g2+νλ|u|g2≤2M1+3ν|f|Vg′2. Multiplying (2.14) by Agu, we have (3.19)12ddt‖u‖g2+ν|Agu|g2≤|(Bg(u,u),Agu)g|+|(f,Agu)g|+|(Ru,Agu)g|. Expanding and using Young’s inequality,
together with bg satisfying inequalities [12], there exists a constant M2=M2(m0,|∇g|∞,ρ0,ρ1) such that (3.20)ddt‖u‖g2+νλg‖u‖g2≤3ν|f|g+M2. Since if u0∈𝔄⊂Vg then the solution u with u(0)=u0 satisfies du/dt∈L2(0,T;Hg), it holds that (3.21)|u(s)-u(t)|g≤|∫ts|dudr(r)|dr|≤|s-t|1/2‖dudt‖L2(0,T;Hg). From (3.18), (3.20) and
Lemma 3.2 we can show that there
exists a constant M>0 which satisfies ∥du/dt∥L2(0,T;Hg)≤M and depends on T but not on u0. Putting (3.15) and (3.21) together, therefore (3.5) turns out to be valid with exponents γ1=1/2 and γ2=1.Proposition 3.4.
Let the assumptions of Theorem2.3
hold. Then, there exists a time t*>0 and constants L2>0 such that S(t*) maps 𝔄 into itself and (3.22)‖S(t*)u0-S(t*)v0‖g≤L2|u0-v0|g hold for any u0,v0∈𝔄 and s,t∈[0,t*].Proof.
Multiplying (3.6) by
tAgũ, we have (3.23)(dũdt,tAgũ)+(νAgũ,tAgũ)+bg(ũ,u2,tAgũ)g+bg(u1,ũ,tAgũ)g+(Rũ,tAgũ)g=0. It follows that (3.24)12ddtt‖ũ‖g2+νt|Agũ|g2-12‖ũ‖g2≤t|bg(ũ,u2,Agũ)g|+t|bg(u1,ũ,Agũ)g|+t|(Rũ,Agũ)g|. To estimate bg, we recall some inequalities [12]: for every u,v∈D(Ag), (3.25)|Bg(u,v)|≤c{|u|g1/2∥u∥g1/2‖v‖g1/2|Agv|g1/2,|u|g1/2|Agu|g1/2‖v‖g. Expanding and using Young’s inequality,
together with (3.25), we have
(3.26)|bg(ũ,u2,Agũ)g|≤|ũ|g1/2|Agũ|g1/2‖u2‖g|Agũ|g≤ν3|Agũ|g2+cν|ũ|g2‖u2‖g4,(3.27)|bg(u1,ũ,Agũ)g|≤|u1|g1/2‖u1‖g1/2‖ũ‖g1/2|Agũ|g1/2|Agũ|g≤ν3|Agũ|g2+cν|u1|g2‖u1‖g2‖ũ‖g2. Next, using the Cauchy inequality, (3.28)|(Rũ,Agũ)g|=|(1g(∇g⋅∇)ũ,Agũ)g|≤|∇g|∞m0‖ũ‖g|Agũ|g≤ν3|Agũ|g2+34ν|∇g|∞2‖ũ‖g2. Since (3.13), we have (3.29)‖ũ‖g2≤c|ũ|g2, Putting (3.26)–(3.29) together, therefore we have (3.30)ddtt‖ũ‖g2≤c(1+|u1|g2‖u1‖g2)t‖ũ‖g2+c(t‖u2‖g4+1)|ũ|g2. By the Gronwall inequality and (3.15), the above inequality
implies (3.31)t‖ũ‖g2≤C(t)|u10-u20|g2, where (3.32)C(t)=∫0tcexp⁡(s+∫st(1+|u1|g2‖u1‖g2)dr)(s‖u2‖g4+1)ds. By taking t=t1(ℬ1), we complete the proof.Now, we give our main theorem which relies on the Propositions3.3 and 3.4 to construct an exponential attractor.Theorem 3.5.
There exists a subset𝔄 of Hg such that S(t) maps 𝔄 into itself and the dynamical system
((S(t))t>0,𝔄) admits an exponential attractor.Based on the abve results (Propositions3.3 and 3.4) and
applying Theorem 2.3, we can deduce
Theorem 3.5.



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