# The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations

Authors: Delin Wu
Journal: Journal of Applied Mathematics
 (2009)

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## Abstract

We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domainΩ. Assuming f=f(x,t)∈Lloc2, we establish the existence of the uniform attractor in L2(Ω) and D(A1/2). The fractal dimension is estimated for the
kernel sections of the uniform attractors obtained.


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## Body

## 1. Introduction

In this paper, we study the behavior of solutions of
the nonautonomous 2D g-Navier-Stokes equations. 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 (x1,x2)∈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 addition, we assume that the functionf(⋅,t)=:f(t)∈Lloc2(ℝ;E) is translation bounded, where E=L2(Ω)orH−1(Ω).
This property implies that(1.2)∥f∥Lb22=∥f∥Lb2(ℝ;E)2=supt∈ℝ∫tt+1∥f(s)∥E2ds<∞.We consider this equation in an appropriate Hilbert
space and show that there is an attractor𝔄 which all solutions approach as t→∞.
The basic idea of our construction, which is motivated by the works of [1, 2].In [1, 2] the author established the global
regularity of solutions of the g-Navier-Stokes equations. 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. 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, 4] and independently by Foias and Temam [5]. The
finite-dimensional property of the global attractor for general dissipative
equations was first proved by Mallet-Paret [6]. For the analysis on the
Navier-Stokes equations, one can refer to [7] and specially [8] for the periodic
boundary conditions.The book in  [9] considers some special classes of such
systems and studies systematically the notion of uniform attractor parallelling
to that of global attractor for autonomous systems. Later on, [10] presents a
general approach that is well suited to study equations arising in mathematical
physics. In this approach, to construct the uniform (or trajectory) attractors,
instead of the associated process {Uσ(t,τ),t≥τ,τ∈ℝ} one should consider a family of processes {Uσ(t,τ)}, σ∈Σ in some Banach space E,
where the functional parameter σ0(s),s∈ℝ is called the symbol and Σ is the symbol space including σ0(s).
The approach preserves the leading concept of invariance which implies the
structure of uniform attractor described by the representation as a union of
sections of all kernels of the family of processes. The kernel is the set of
all complete trajectories of a process.In the paper, we study the existence of compact
uniform attractor for the nonautonomous the two dimensional g-Navier-Stokes
equations in the periodic boundary conditionsΩ.
We apply measure of noncompactness method to nonautonomous g-Navier-Stokes
equations equation with external forces f(x,t) in Lloc2(ℝ;E) which is normal function (see Definition 4.2).
Last, the fractal dimension is estimated for the kernel sections of the uniform
attractors obtained.


## 2. Functional Setting

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) and(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 (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)→Hgand 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⋅∇)uto have the linear
operator(2.8)Agu=Pg[−1g(∇⋅(g∇u))].For the linear operator Ag,
the following hold (see [1, 2]):(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=1n∫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(τ)=uτ.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. Abstract Results

LetE be a Banach space, and let a two-parameter
family of mappings {U(t,τ)}={U(t,τ)∣t≥τ,τ∈ℝ} act on E:(3.1)U(t,τ):E→E,t≥τ,τ∈ℝ.Definition 3.1.
A
two-parameter family of mappings{U(t,τ)} is said to be a process in E if(3.2)U(t,s)U(s,τ)=U(t,τ),∀t≥s≥τ,τ∈ℝ,U(τ,τ)=Id,τ∈ℝ.Byℬ(E) we denote the collection of the  bounded sets of E.
We consider a family of processes {Uσ(t,τ)} depending on a parameter σ∈Σ.
The parameter σ is said to be the  symbol  of the process {Uσ(t,τ)} and the set Σ is said to be the  symbol space.
In the sequel Σ is assumed to be a complete metric space.A family of processes{Uσ(t,τ)}, σ∈Σ is said to be   uniformly   (with respect to (w.r.t.)σ∈Σ)  bounded if for any B∈ℬ(E) the set(3.3)⋃σ∈Σ⋃τ∈R⋃t≥τUσ(t,τ)B∈ℬ(E).A setB0⊂E is said to be uniformly  (w.r.t.σ∈Σ) absorbing  for the family of processes {Uσ(t,τ)}, σ∈Σ if for any τ∈R and every B∈ℬ(E) there exists t0=t0(τ,B)≥τ such that ⋃σ∈ΣUσ(t,τ)B⊆B0 for all t≥t0.A setP⊂E is said to be uniformly (w.r.t.σ∈Σ) attracting for the family of processes {Uσ(t,τ)},σ∈Σ if for an arbitrary fixed τ∈R,(3.4)limt→+∞(supσ∈ΣdistE(Uσ(t,τ)B,P))=0.A family of processes possessing a compact uniformly
absorbing set is calleduniformly compact and a family of processes possessing a compact
uniformly attracting set is called uniformly asymptotically compact.Definition 3.2.
A
closed set𝒜Σ⊂E is said to be the uniform (w.r.t.σ∈Σ) attractor of the family of processes {Uσ(t,τ)},σ∈Σ if it is uniformly (w.r.t.σ∈Σ) attracting and it is contained in any closed
uniformly (w.r.t.σ∈Σ) attracting set 𝒜′ of the family of processes {Uσ(t,τ)},σ∈Σ:𝒜Σ⊆𝒜'.A family of processes{Uσ(t,τ)},σ∈Σ acting in E is said to be (E×Σ,E)-continuous,
if for all fixed t and τ,t≥τ,τ∈ℝ the mapping (u,σ)↦Uσ(t,τ)u is continuous from E×Σ into E.A curveu(s),s∈ℝ is said to be a complete trajectory of the process {U(t,τ)} if(3.5)U(t,τ)u(τ)=u(t),∀t≥τ,τ∈ℝ.Thekernel𝒦 of the process {U(t,τ)} consists of all bounded complete trajectories
of the process {U(t,τ)}:(3.6)𝒦={u(⋅)∣u(⋅)satisfies(3.6),∥u(s)∥E≤Mufors∈ℝ}.The set(3.7)𝒦(s)={u(s)∣u(⋅)∈𝒦}⊆Eis said to be the kernel section at time t=s,s∈ℝ.For convenience, letBt=⋃σ∈Σ⋃s≥tUσ(s,t)B,
the closure B¯ of the set B and ℝτ={t∈ℝ∣t≥τ}.
Define the uniform (w.r.t.σ∈Σ) ω-limit set ωτ,Σ(B) of B by ωτ,Σ(B)=⋂t≥τB¯t which can be characterized, analogously to
that for semigroups, the following:(3.8)y∈ωτ,Σ(B)⟺there are sequences{xn}⊂B,{σn}⊂Σ,{tn}⊂ℝτsuch thattn→+∞,Uσn(tn,τ)xn→y(n→∞).We recall characterize the existence of the uniform
attractor for a family of processes satisfying (3.8) in term of the concept of measure of
noncompactness that was put forward first by Kuratowski (see [11, 12]).LetB∈ℬ(E).
Its Kuratowski measure of noncompactness κ(B) is defined by(3.9)κ(B)=inf{δ>0∣Badmits a finite covering by sets of diameter≤δ}.Definition 3.3.
A
family of processes{Uσ(t,τ)},σ∈Σ is said to be uniformly (w.r.t.σ∈Σ)ω-limit compact if for any τ∈ℝ and B∈ℬ(E) the set Bt is bounded for every t and limt→∞κ(Bt)=0.We present now a method to verify the uniform(w.r.t.σ∈Σ)ω-limit compactness (see [13, 14]).Definition 3.4.
A family of processes{Uσ(t,τ)},σ∈Σ is said to satisfy uniformly (w.r.t.σ∈Σ) Condition (C)
if for any  fixed τ∈ℝ,B∈ℬ(E) and ε>0,
there exist t0=t(τ,B,ε)≥τ and a finite-dimensional subspace E1 of E such that
(i)
P(⋃σ∈Σ⋃t≥t0Uσ(t,τ)B) is bounded; and(ii)
∥(I−P)(⋃σ∈Σ⋃t≥t0Uσ(t,τ)x)∥≤ε,∀x∈B,
where P:E→E1 is a bounded projector.Therefore we have the following results.Theorem 3.5.
LetΣ be a metric space and let {T(t)} be a continuous invariant semigroup T(t)Σ=Σ on Σ.
A family of processes {Uσ(t,τ)},σ∈Σ acting in E is (E×Σ,E)-(weakly) continuous and possesses the compact
uniform (w.r.t.σ∈Σ) attractor AΣ satisfying(3.10)𝒜Σ=ω0,Σ(B0)=ωτ,Σ(B0)=⋃σ∈Σ𝒦σ(0),∀τ∈ℝ,if it
(i)
has a bounded uniformly(w.r.t.σ∈Σ) absorbing set B0,
and(ii)
satisfies uniformly(w.r.t.σ∈Σ) Condition (C)
Moreover, ifE is a uniformly convex Banach space then the
converse is true.


## 4. Uniform Attractor of Nonautonomous g-Navier-Stokes Equations

This section deals with the existence of the attractor
for the two-dimensional nonautonomous g-Navier-Stokes equations with periodic
boundary condition (see [1, 2]).It is similar to autonomous case that we can establist
the existence of solution of (2.14) by the standard Faedo-Galerkin method.In [1, 2], the authors have shown that the
semigroup S(t):Hg→Hg(t≥0) associated with the autonomous systems (2.14) possesses a global attractor. The main
objective of this section is to prove that the nonautonomous system (2.14) has uniform attractors in Hg and Vg.To this end, we first state some the following results
of existence and uniqueness of solutions of (2.14).Proposition 4.1.
Letf∈V′ be given. Then for every uτ∈Hg there exists a unique solution u=u(t) on [0,∞) of (2.14), satisfying u(τ)=uτ.
Moreover,one has(4.1)u(t)∈C[0,T;Hg)∩L2(0,T;Vg),∀T>0.Finally, if uτ∈Vg,
then(4.2)u(t)∈C[0,T;Vg)∩L2(0,T;D(Ag)),∀T>0.Proof.
The
Proof of Proposition4.1 is similar to autonomous in [1, 15].Now we will write (2.14) in the operator form(4.3)∂tu=Aσ(t)(u),u|t=τ=uτ,where σ(s)=f(x,s) is the symbol of (4.3).
Thus, if uτ∈Hg,
then problem (4.3) has a unique solution u(t)∈C([0,T];Hg)∩L2([0,T];Vg).
This implies that the process {Uσ(t,τ)} given by the formula Uσ(t,τ)uτ=u(t) is defined in Hg.Now recall the following facts that can be found in
[13].Definition 4.2.
A functionφ∈Lloc2(ℝ;E) is said to be   normal  if for any ε>0,
there exists η>0 such that(4.4)supt∈ℝ∫tt+η∥φ(s)∥E2ds≤ε.Remark 4.3.
Obviously,Ln2(ℝ;E)⊂Lb2(ℝ;E).
Denote by Lc2(ℝ;E) the class of translation compact functions f(s),s∈ℝ,
whose family of ℋ(f) is precompact in Lloc2(ℝ;E).
It is proved in [13] that Ln2(ℝ;E) and Lc2(ℝ;E) are closed subspaces of Lb2(ℝ;E),
but the latter is a proper subset of the former (for further details see [13]).We now define thesymbol spaceℋ(σ0) for (4.3).
Let a fixed symbol σ0(s)=f0(s)=f0(⋅,s) be normal functions in Lloc2(ℝ;E);
that is, the family of translation {f0(s+h),h∈ℝ} forms a normal function set in Lloc2([T1,T2];E),
where [T1,T2] is an arbitrary interval of the time axis ℝ.
Therefore(4.5)ℋ(σ0)=ℋ(f0)=[f0(x,s+h)∣h∈ℝ]Lloc2(ℝ;E).Now, for anyf(x,t)∈ℋ(f0),
the problem (4.3) with f instead of f0 possesses a corresponding process {Uf(t,τ)} acting on Vg.
As is proved in [10], the family {Uf(t,τ)∣f∈ℋ(f0)} of processes is (Vg×ℋ(f0);Vg)-continuous.Let(4.6)𝒦f={uf(x,t)fort∈ℝ∣uf(x,t)issolutionof(4.3)satisfying∥uf(⋅,t)∥H≤Mf∀t∈ℝ}be the so-called kernel of the
process {Uf(t,τ)}.Proposition 4.4.
The
process{Uf(t,τ)}:Hg→Hg(Vg) associated with the (4.3) possesses absorbing sets(4.7)ℬ0={u∈Hg∣|u|g≤ρ0},ℬ1={u∈Vg∣∥u∥g≤ρ1}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.Proof.
The
proof of Proposition4.4 is similar to autonomous g-Navier-Stokes
equation. We can obtain absorbing sets in Hg and Vg the following from [1] and the proof of the
main results as follow.The main results in this section are as follows.Theorem 4.5.
Iff0(x,s) is normal function in Lloc2(ℝ;Vg'),
then the processes {Uf0(t,τ)} corresponding to problem (2.14) possess compact uniform (w.r.t.τ∈ℝ) attractor 𝔄0 in Hg which coincides with the uniform (w.r.t.f∈ℋ(f0)) attractor 𝔄ℋ(f0) of the family of processes {Uf(t,τ)},f∈ℋ(f0):(4.8)𝔄0=𝔄ℋ(f0)=ω0,ℋ(f0)(ℬ0)=⋃f∈ℋ(f0)𝒦f(0),where ℬ0 is the uniformly (w.r.t.f∈ℋg(f0)) absorbing set in Hg and 𝒦f is the kernel of the process {Uf(t,τ)}.
Furthermore, the kernel 𝒦f is nonempty for all f∈ℋ(f0).Proof.
As in
the previous section, for fixedN,
let H1 be the subspace spanned by w1;…;wN,
and H2 the orthogonal complement of H1 in Hg.
We write(4.9)u=u1+u2;u1∈H1,u2∈H2for anyu∈Hg.
Now, we only have to verify Condition(C).
Namely, we
need to estimate |u2(t)|2,
where u(t)=u1(t)+u2(t) is a solution of (2.14) given in Proposition 4.1.
Multiplying (2.14) by u2,
we have(4.10)(dudt,u2)g+(νAgu,u2)g+(B(u,u),u2)g=(f,u2)g−(Ru,u2)g.It follows that(4.11)12ddt|u2|g2+ν∥ug∥g2≤|(B(u,u),u2)g|+|(f,u2)g|+(Ru,u2)g.Since bg satisfies the following inequality (see
[15]):(4.12)|bg(u,v,w)|≤c|u|g1/2∥u∥g1/2∥v∥g|w|g1/2∥w∥g1/2,∀u,v,w∈Vg,thus,(4.13)|(B(u,u),u2)g|≤c|u|g1/2∥u∥g3/2|u2|g1/2∥u2∥g1/2≤cλm+1|u|g1/2∥u∥g3/2∥u2∥g≤ν6∥u2∥g2+cρ0ρ13.Next, the Cauchy
inequality,(4.14)|(Ru,u2)g|=|(νg(∇g⋅∇)u,u2)g|≤νm0|∇g|∞∥u∥g|u2|g≤ν6∥u2∥g2+3νρ12|∇g|∞22m02λgλm+1.Finally, we have(4.15)|(f,u2)g|≤|f|Vg'∥u2∥≤ν6∥u2∥g2+32ν|f|Vg'2.Putting (4.13)–(4.15) together,
there exist constant M1=M1(m0,|∇g|∞,ρ0,ρ1) such that(4.16)12ddt|u2|g2+12ν∥u2∥g2≤3|f|Vg'2ν+M1.Therefore, we deduce
that(4.17)ddt|u2|g2+νλm+1|u2|22≤2M1+3ν|f|Vg'2.Here M1 depends on λm+1,
is not increasing as λm+1 increasing.
By the Gronwall inequality, the above inequality
implies(4.18)|u2(t)|g2≤|u2(t0+1)|22e−νλm+1(t−(t0+1))+2M1νλm+1+3ν∫t0+1te−νλm+1(t−s)|f|V′2ds.Applying (4.4) for any ε(4.19)3ν∫t0+1te−νλm+1(t−s)|f|Vg'2ds<ε3.Using (2.9) and let t1=t0+1+(1/νλm+1)ln(3ρ02/ε),
then t≥t1 implies(4.20)2Mνλm+1<ε3;|u2(t0+1)|22e−νλm+1(t−(t0+1))≤ρ02e−νλm+1(t−(t0+1))/2<ε3.Therefore, we deduce from
(4.18) that(4.21)|u2|22≤ε,∀t≥t1,f∈ℋ(f0),which indicates {Uf(t,τ)},f∈ℋ(f0) satisfying uniform (w.t.r.f∈ℋ(f0))  Condition (C)
in Hg.
Applying Theorem 3.5 the proof is complete.Theorem 4.6.
Iff0(x,s) is normal function in Lloc2(ℝ;Hg),
then the processes {Uf0(t,τ)} corresponding to problem (2.14) possesses compact uniform (w.r.t.τ∈ℝ) attractor 𝔄1 in Vg which coincides with the uniform (w.r.t.f∈ℋ(f0)) attractor 𝔄ℋ(f0) of the family of processes {Uf(t,τ)},f∈ℋ(f0):(4.22)𝔄1=𝔄ℋ(f0)=ω0,ℋ(f0)(ℬ1)=⋃f∈ℋ(f0)𝒦f(0),where ℬ1 is the uniformly (w.r.t.f∈ℋ(f0)) absorbing set in Vg and 𝒦f is the kernel of the process {Uf(t,τ)}.
Furthermore, the kernel 𝒦f is nonempty for all f∈ℋ(f0).Proof.
Using
Proposition4.4,
we have the family of processes {Uf(t,τ)}, f∈ℋ(f0) corresponding to (4.3) possesses the uniformly (w.r.t.f∈ℋ(f0)) absorbing set in Vg.
Now we prove the existence of compact uniform(w.r.t.f∈ℋ(f0)) attractor in Vg by applying the method established in Section
3, that is, we testify that the family of processes {Uf(t,τ)},f∈ℋ(f0) corresponding to (4.3) satisfies uniform (w.r.t.f∈ℋ(f0)) Condition (C).
Multiplying (2.14) by Agu2(t),
we have(4.23)(dvdt,Agu2)+(νAgu,Agu2)+(Bg(u,u),Agu2)g=(f,Agu2)−(Ru,Agu2)g.It follows that(4.24)12ddt∥u2∥g2+ν|Agu2|g2≤|(Bg(u,u),Agu2)g|+|(f,Agu2)g|+|(Ru,Agu2)g|.To estimate (Bg(u,u),Au2)g,
we recall some inequalities [16]: for every u,v∈D(Ag):(4.25)|Bg(u,v)|≤c{|u|g1/2∥u∥g1/2∥v∥g1/2|Agv|g1/2,|u|g1/2|Agu|g1/2∥v∥g(see [16])(4.26)|w|L∞(Ω)2≤c∥w∥g(1+log|Agw|λg∥w∥g2)1/2from which we deduce
that(4.27)|Bg(u,v)|≤c|u|L∞(Ω)|∇v|g|u|g|∇v|L∞(Ω),and
using, (4.26)(4.28)|Bg(u,v)|≤c{∥u∥g∥v∥g(1+log|Agu|2λg∥w∥g2)1/2,|u|g|Agv|g(1+log|Ag3/2v|2λg∥Agv∥g2)1/2.Expanding and using Young's
inequality, together with the first one of (4.28) and the second one of (4.25), we
have(4.29)|(Bg(u,u),Agu2)|≤|(Bg(u1,u1+u2),Agu2)|+|(Bg(u2,u1+u2),Agu2)|≤cL1/2∥u1∥g|Agu2|g(∥u1∥g+∥u2∥g)+c|u2|g1/2|Agu2|g3/2≤ν6|Agu2|g2+cνρ14L+cν3ρ02ρ14,t≥t0+1,where we use(4.30)|Agu1|g2≤λm∥u1∥g2and set(4.31)L=1+logλm+1λg.Next, using the Cauchy
inequality,(4.32)|(Ru,Agu2)g|=|(νg(∇g⋅∇)u,Agu2)g|≤νm0|∇g|∞∥u∥g|Agu2|g≤ν6|Agu2|g2+3ν2|∇g|∞2ρ12.Finally, we estimate |(f,Agu2)| by(4.33)|(f,Agu2)|≤|f|g|Agu2|2≤ν6|Agu2|g2+32ν|f|g2.Putting (4.29)–(4.33) together, there exists a constant M2 such that(4.34)ddt∥u2∥g2+νλm+1∥u2∥g2≤3ν|f|g+M2.Here M2=M2(ρ0,ρ1,L,ν,|∇g|) depends on λm+1,
is not increasing as λm+1 increasing. Therefore, by the Gronwall
inequality, the above inequality implies(4.35)∥u2∥g2≤∥u2(t0+1)∥g2e−νλm+1(t−(t0+1))+2M2νλm+1+3ν∫t0+1te−νλm+1(t−s)|f|g2ds.
Applying (4.4) for any ε(4.36)3ν∫t0+1te−νλm+1(t−s)|f|g2ds<ε3.
Using (2.9) and let t1=t0+1+(1/νλm+1)ln(3ρ12/ε),
then t≥t1 implies(4.37)2M2νλm+1<ε3;∥u2(t0+1)∥g2e−νλm+1(t−(t0+1))≤ρ12e−νλm+1(t−(t0+1))<ε3.Therefore, we deduce
from (4.35) that(4.38)∥u2∥g2≤ε,∀t≥t1,f∈ℋ(f0),which indicates {Uf(t,τ)},f∈ℋ(f0) satisfying uniform (w.t.r.f∈ℋ(f0)) Condition (C)
in Vg.


## 5. Dimension of the Uniform Attractor

In this section we estimate the fractal dimension (for
definition see, e.g., [2, 10, 15]) of the kernel sections of the uniform
attractors obtained in Section 4 by applying the methods in [17].Process{U(t,τ)} is said to be uniformly quasidifferentiable on {𝒦(s)}τ∈ℝ,
if there is a family of bounded linear operators {L(t,τ;u)∣u∈𝒦(s),t≥τ,τ∈ℝ},L(t,τ;u):E→E such that(5.1)limsupε→0τ∈ℝsupu,v∈𝒦(s)0<|u−v|≤ε|U(t,τ)v−U(t,τ)u−L(t,τ;u)(v−u)||v−u|=0.We want to estimate the fractal dimension of the
kernel sections𝒦(s) of the process {U(t,τ)} generated by the abstract evolutionary
(2.14). Assume that {L(t,τ;u)} is generated by the variational equation
corresponding to (2.14)(5.2)∂tw=F′(u,t)w,w|t=τ=wτ∈E,t≥τ,τ∈ℝ,that is, L(t,τ;uτ)wτ=w(t) is the solution of (5.2), and u(t)=U(t,τ)uτ is the solution of (2.14) with initial value uτ∈𝒦(τ).
For natural number j∈N,
we set(5.3)q˜j=limT→+∞supτ∈Rsupuτ∈𝒦(τ)(1T∫ττ+TTr(F′(u(s),s))ds),where Tr is trace of the operator.We will need the following Theorem VIII.3.1 in [10,] and [2].
Theorem 5.1.
Under the assumptions above, let us suppose thatUτ∈ℝ𝒦(τ) is relatively compact in E,
and there exists qj,j=1,2,…,
such that(5.4)q˜j≤qj,foranyj≥1,qn0≥0,qn0+1<0,forsomen0≥1,qj≤qn0+(qn0−qn0+1)(n0−j),∀j=1,2,….Then,(5.5)dF(𝒦(τ))≤d0:=n0+qn0qn0−qn0+1,∀τ∈ℝ.We now consider (2.14) with f∈Ln2(ℝ;Vg′).
The equations possess a compact uniform (w.r.t.f∈ℋ(f)) attractor 𝔄ℋ(f) and ⋃τ∈ℝ𝒦f(τ)⊂𝔄ℋ(f).
By [2, 10, 15], we know that the associated process {Uf(t,τ)} is uniformly quasidifferentiable on {𝒦f(τ)}τ∈ℝ and the quasidifferential is Hölder-continuous with respect to uτ∈𝒦f(τ).
The corresponding variational equation is(5.6)∂tw=−νAgu−Bgu−νRu+Pgf≡F′(u(t),t)w,w|t=τ=wτ∈E,τ∈ℝ.We have the main results in this section.Theorem 5.2.
Suppose
thatf(t) satisfies the assumptions of Theorem 4.5.
Then, if γ=1−(2|∇g|∞)/(m0λg1/2)>0,
the Uniform attractor 𝔄0 defined by (4.8) satisfies(5.7)dF(𝔄0)≤βα,where(5.8)α=c2νm0λ1′γ2M0,β=c1d12ν3m0γsupφj∈Hg|φj|≤1j=1,2,…,m1T∫ττ+T∥f(s)∥Vg′2ds,the constant c1,c2 of (3.29) and (3.32) of Chapter VI in [15] and [2], λ1′ is the first eigenvalue of the Stokes operator
and d1=|∇g|∞2/4m0+|∇g|∞+M0.Proof.
With
Theorem4.5 at our disposal we may apply the abstract framework in [2, 10, 15, 17].
Forξ1,ξ2,…,ξm∈Hg,
let vj(t)=L(t,uτ)⋅ξj,
where uτ∈Hg.
Let {φj(s);j=1,2,…,m} be an orthonormal basis for span {vj;j=1,2,…,m}.
Since vj∈Vg almost everywhere s≥τ,
we can also assume that φj(s)∈Vg almost everywhere s≥τ.
Then, similar to the Proof process of Theorems 4.5 and 4.6, we may obtain(5.9)∑i=1m(F′(U(s,τ)uτ,s)φi,φi)g=−ν∑i=1m∥φj∥g2−∑i=1mbg(φj,U(s,τ)uτ,φj)−∑i=1m(νg(∇g⋅∇)φj,φj)g,almost everywhere s≥τ.
From this equality, and in particular using the Schwarz and Lieb-Thirring
inequality (see [2, 10, 15, 17]), one obtains(5.10)∑i=1m∥φ∥g2≥λ1+⋯+λm≥m0M0(λ1′+⋯+λm′)≥m0M0c2λ1′m2,Trj(F′(U(s,τ)uτ,s)g≤−ν(1−|∇g|∞m0λ11/2)∑i=1m∥φj∥g2+∥U(s,τ)uτ∥g(c1d1m0∑i=1m∥φj∥g2)1/2≤−ν2(1−2|∇g|∞m0λ11/2)∑i=1m∥φj∥g2+c1d12νm0∥U(s,τ)uτ∥g2≤−νm02M0(1−2|∇g|∞m0λ11/2)c2λ1′m2+c1d12νm0∥U(s,τ)uτ∥g2,on the other hand, we can deduce (2.14) that(5.11)ddt|U(s,τ)uτ|g2+ν∥U(s,τ)uτ∥g2≤∥f∥Vg′2ν+2νm0λg1/2|∇g|∞∥U(s,τ)uτ∥g2for λg=4π2m0/M0,
and then(5.12)∫τt∥U(s,τ)uτ∥g2ds≤(1ν2∫τt∥f(s)∥Vg′2ds+|uτ|2ν)(1−2|∇g|∞m0λg1/2)−1,t≥τ.
Now we define(5.13)qm=supφj∈Hg|φj|≤1j=1,2,…,m(1T∫ττ+TTrj(F′(U(s,τ)uτ,s)ds))g,Using  Theorem 5.1, we
have(5.14)q˜m≤−νm02M0(1−2|∇g|∞m0λ11/2)c2λ1′m2+c1d12νm0(supφj∈Hg|φj|≤1j=1,2,…,m(1T∫ττ+T∥U(s,τ)uτ∥g2ds))≤−νm02M0(1−2|∇g|∞m0λ11/2)c2λ1′m2+c1d12νm0(1ν2supφj∈Hg|φj|≤1j=1,2,…,m(1T∫ττ+T∥f(s)∥Vg′2ds+|uτ|2νT)(1−2|∇g|∞m0λg1/2)−1),qm=limsupT→∞q˜m≤−αm2+β,Hence(5.15)dimF𝒜0(τ)≤βα.



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*Source: 150420-2009-03-04.xml*