# On the Dimension of the Pullback Attractors for g-Navier-Stokes Equations

Authors: Delin Wu
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We consider the asymptotic behaviour of nonautonomous 2D g-Navier-Stokes equations in bounded domainΩ. Assuming that f∈Lloc2, which is translation bounded, the existence of the pullback attractor is
proved in L2(Ω) and H1(Ω). It is proved that the fractal dimension of the
pullback attractor is finite.


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## Body

## 1. Introduction

In this paper, we study the behavior of solutions of the nonautonomous 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 (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)  or  H-1(Ω). This property implies that(1.2)∥f∥Lb22=∥f∥Lb2(R;E)2=sup⁡t∈R∫tt+1∥f(s)∥E2ds<∞.We consider  this equation in an appropriate Hilbert space and show that there is a pullback attractor𝔄. This is the basic idea of our construction, which is motivated by the works of [1].LetΩ=(0,1)×(0,1). 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(1.3)(u,v)g=∫Ω(u⋅v)gdx,|u|g2=(u,u)g,
as well as H1(Ω,g) with the norm(1.4)∥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:(1.5)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(1.6)((u,v))g=∫Ω(∇u⋅∇v)gdx,∥u∥g=((u,u))g.
Also, we define the orthogonal projection Pg as(1.7)Pg:Lper2(Ω,g)⟶Hg,
and we have that Q⊆Hg⊥, where(1.8)Q=ClLper2(Ω,g){∇ϕ:ϕ∈C1(Ω¯,R)}.
Then, we define the g-Laplacian operator(1.9)-Δgu≡1g(∇⋅g∇)u=-Δu-1g(∇g⋅∇)u
to have the linear operator(1.10)Agu=Pg[-1g(∇⋅(g∇u))].
For the linear operator Ag, the following hold (see [1]).(1)Ag is a positive, self-adjoint operator with compact inverse, where the domain of Ag is  D(Ag)=Vg∩H2(Ω,g).(2) There exist countable eigenvalues ofAg satisfying(1.11)0<λg≤λ1≤λ2≤λ3≤⋯,
where λg=4π2m0/M0 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(1.12)bg(u,v,w)=∑i,j=12∫Ωui(Divj)wjgdx=(Pg(u⋅∇)v,w)g,
where u, v, and  w lie in appropriate subspaces of L2(Ω,g). Then, the form bg satisfies(1.13)bg(u,v,w)=-bg(u,w,v)foru,v,w∈Hg.
We denote a linear operator R on Vg by(1.14)Ru=Pg[1g(∇g⋅∇)u]foru∈Vg
and have R as a continuous linear operator from Vg into Hg such that(1.15)|(Ru,u)|≤|∇g|∞m0‖u‖g|u|g≤|∇g|∞m0λg1/2‖u‖gforu∈Vg.We now rewrite   (1.1) as abstract evolution equations: (1.16)dudt+νAgu+Bgu+νRu=Pgf,u(τ)=uτ.
In [1] the author established the global regularity of solutions of the g-Navier-Stokes equations. The Navier-Stokes equations were investigated by many authors, and the existence of the attractors for 2D Navier-Stokes equations was first proved in [2] and independently in [3]. The finite-dimensional property of the global attractor for general dissipative equations was first proved in [4]. For the analysis of the Navier-Stokes equations, one can refer to [5], specially [6] for the periodic boundary conditions.The theory of pullback (or cocycle) attractors has been developed for both the nonautonomous and random dynamical systems (see [7–13])  and has shown to be very useful in the understanding of the dynamics of nonautonomous dynamical systems.The understanding of the asymptotic behaviour of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem for a system having some dissipativity properties is to analyse the existence and structure of its global attractor, which,  in the autonomous case, is an invariant compact set which attracts all the trajectories of the system, uniformly on bounded sets. This set has, in general, a very complicated geometry which reflects the complexity of the long-time behaviour of the system (see [14–17]  and the references therein). However, nonautonomous systems are also of great importance and interest as they appear in many applications to natural sciences. In this situation, there are various options to deal with the problem of attractors for nonautonomous systems (kernel sections [18], skew-product formalism [16, 19], etc.); for our particular situation we have preferred to choose that of pullback attractor (see [9, 10, 13, 20]) which has also proved extremely fruitful, particularly in the case of random dynamical systems (see [11, 13]).In this paper, we study the existence of compact pullback attractor for the nonautonomous g-Navier-Stokes equations in bounded domainΩ with periodic boundary condition. It is proved that the fractal dimension of the pullback attractor is finite.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.


## 2. Abstract Results

We now recall the preliminary results of pullback attractors, as developed in [8–10, 13].The semigroupS(t) property is replaced by the process U(t,τ) composition property(2.1)U(t,τ)U(τ,s)=U(t,s)∀t≥τ≥s,
and, obviously, the initial condition implies that  U(τ,τ)=Id. As with the semigroup composition S(t)S(τ)=S(t+τ), this just expresses the uniqueness of solutions.It is also possible to present the theory within the more general framework of cocycle dynamical systems. In this case the second component ofU is viewed as an element of some parameter space J, so that the solution can be written as U(t,p)ϱ, and a shift map θt:J→J is defined so that the process composition becomes the cocycle property (2.2)U(t+τ,p)=U(t,θτp)U(τ,p).
However, when one tries to develop a theory under a unified abstract formulation, the context of cocycle (or skew-product flows) may not be the most appropriate to deal with the problem. In this paper, we apply a process U(t,τ) to  (1.16) by using the concept of measure of noncompactness to obtain pullback attractors.Byℬ(E) we denote the collection of the bounded sets of E.Definition 2.1.
LetU be a process on a complete metric space E. A family of compact sets {𝒜(t)}t∈ℝ is said to be a pullback attractor for U if, for all τ∈ℝ, it satisfies(i)
U(t,τ)𝒜(τ)=𝒜(t)  for  all  t≥τ,(ii)
lims→∞dist(U(t,t-s)D,𝒜(t))=0, for  D∈ℬ(E).The pullback attractor is said to be uniform if the attraction property is uniform in time, that is,
(2.3)lim⁡s→∞sup⁡t∈Rdist⁡(U(t,t-s)D,A(t))=0,forD∈B(E).Definition 2.2.
A family of compact sets{𝒜(t)}t∈ℝ is said to be a forward attractor for U if, for all τ∈ℝ, it satisfies(i)
U(t,τ)𝒜(τ)=𝒜(t) for  all t≥τ,(ii)
limt→∞dist(U(t,τ)D,𝒜(t))=0,  for  D∈ℬ(E).The forward attractor is said to be uniform if the attraction property is uniform in time, that is,
(2.4)lim⁡t→∞sup⁡τ∈Rdist⁡(U(t+τ,τ)D,A(t+τ))=0,forD∈B(E).
In the definition, dist(A,B) is the Hausdorff semidistance between A and B, defined as
(2.5)dist⁡(A,B)=sup⁡a∈Ainf⁡b∈Bd(a,b),forA,B⊆E.
Property (i) is a generalization of the invariance property for autonomous dynamical systems. The pullback attracting property  (ii)  considers the state of the system at time t when the initial time t-s goes to -∞.The notion of an attractor is closely related to that of an absorbing set.Definition 2.3.
The family{B(t)}t∈ℝ is said to be (pullback) absorbing with respect to the process U if, for all t∈ℝ and D∈ℬ(E), there exists S(D,t)>0 such that for all s≥S(D,t)(2.6)U(t,t-s)D⊂B(t).
The absorption is said to be uniform if S(D,t) does not depend on the time variable t.Now we recall the abstract results in [21].Definition 2.4.
The family of processes{U(t,t-s)} is said to be satisfying pullback Condition (C)  if, for any fixed B∈ℬ(E) and ɛ>0, there exist s0=s(B,t,ɛ)≥0 and a finite dimensional subspace E1 of E such that(i)
{∥P(⋃s≥s0U(t,t-s)B)∥E} is  bounded,(ii)
∥(I-P)(⋃s≥s0U(t,t-s)B)∥E≤ɛ, where P:E→E1 is a bounded projector.Theorem 2.5.
Let the family of processes{U(t,τ)} acting in E be continuous and possess  compact pullback attractor 𝒜(t) satisfying
(2.7)A(t)=⋃B∈Bω(B,t)¯,fort∈R,
if it(i)
has a bounded (pullback) absorbing setB,(ii)
satisfies pullback Condition (C).Moreover if E is a uniformly convex Banach space, then the converse is true.


## 3. Pullback 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 in a bounded domainΩ with periodic boundary condition.In [1], the author  has shown that the semigroup S(t):Hg→Hg(t≥0) associated with the autonomous systems (1.16) possesses a global attractor in Hg and Vg. The main objective of this section is to prove that the nonautonomous system (1.16) has uniform attractors in Hg and Vg.To this end, we first state  the following results of existence and uniqueness of solutions of   (1.16).Proposition 3.1.
Letf∈V′ be given. Then for every uτ∈Hg there exists a unique solution u=u(t) on [0,∞) of   (1.16), satisfying u(τ)=uτ. Moreover, one has
(3.1)u(t)∈C[τ,T;Hg)∩L2(τ,T;Vg),∀T>τ.
Finally, if uτ∈Vg, then
(3.2)u(t)∈C[τ,T;Vg)∩L2(τ,T;D(Ag)),∀T>τ.Proof.
The Proof of Proposition3.1 is similar to autonomous case in [1, 17].Proposition 3.2.
The process{U(t,t-s)}:Vg→Vg associated with the system (1.16) possesses (pullback) absorbing sets, that is, there exists a family {B(t)}t∈R of bounded (pullback) absorbing sets in Hg and Vg for the process U, which is given by
(3.3)B0=B(t)={u∈Hg∣|u|g≤ρ0},B1=B(t)={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 Proposition3.2 is similar to autonomous g-Navier-Stokes equation. We can obtain absorbing sets in Hg and Vg   from [1].The main results in this section are as follows.Theorem 3.3.
Iff(x,t)∈Lb2(R;V′) and uτ∈Hg, then the processes  {U(t,t-s)}  corresponding to problem (1.16) possess  compact pullback attractor 𝒜0(t) in Hg which coincides with the pullback attractor:
(3.4)A0(t)=⋃B0∈Bω(B0,t)¯,
where ℬ0 is the (pullback) absorbing set in Hg.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
(3.5)u=u1+u2,u1∈H1,u2∈H2foranyu∈Hg.
Now, we only have to verify Condition (C). Namely, we need to estimate |u2(t)|g, where u(t)=u1(t)+u2(t) is a solution of   (1.16) given in Proposition 3.1.
Multiplying  (1.16) by u2, we have
(3.6)(dudt,u2)g+(νAgu,u2)g+(Bg(u,u),u2)g=(f,u2)g-(νRu,u2)g.
It follows that
(3.7)12ddt|u2|g2+ν‖ug‖g2≤|(B(u,u),u2)g|+|(f,u2)g|+(Ru,u2)g.
Since bg satisfies the following inequality (see [6]):
(3.8)|bg(u,v,w)|≤c|u|g1/2‖u‖g1/2‖v‖g|w|g1/2‖w‖g1/2,∀u,v,w∈Vg,
thus,
(3.9)|(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, using the Cauchy inequality,
(3.10)|(νRu,u2)g|=|(νg(∇g⋅∇)u,u2)g|≤νm0|∇g|∞∥u∥g|u2|g≤ν6∥u2∥g2+3νρ12|∇g|∞22m02λgλm+1.
Finally, we have
(3.11)|(f,u2)g|≤|f|Vg′∥u2∥≤ν6∥u2∥g2+32ν|f|Vg′2.
Putting (3.9)–(3.11) together, there exists constant M1=M1(m0,|∇g|∞,ρ0,ρ1) such that
(3.12)12ddt|u2|g2+12ν∥u2∥g2≤3|f|Vg′2ν+M1.
Therefore, we deduce  that
(3.13)ddt|u2|g2+νλm+1|u2|g2≤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 that(3.14)|u2(t)|g2≤|u2(τ)|g2e-νλm+1(t-τ)+2M1νλm+1+3ν∫τte-νλm+1(t-s)|f|V′2ds.
If we consider the time t-s instead of τ (so that we can use more easily the definition of pullback attractors), we have
(3.15)3ν∫τte-νλm+1(t-σ)|f(σ)|V′2dσ=3ν∫t-ste-νλm+1(t-σ)|f(σ)|V′2dσ.
Applying continuous integral and Lemma II   1.3 in [18]  for any ɛ, there exists η=η(ɛ)>0 such that
(3.16)∫t-ηt|f(σ)|V′2dσ<νɛ18;
thus, we have
(3.17)3ν∫t-ηte-νλm+1(t-σ)|f(σ)|V′2dσ≤ɛ6,(3.18)3ν∫t-st-ηe-νλm+1(t-σ)|f(σ)|V′2dσ≤3ν∫t-η-1t-ηe-νλm+1(t-σ)|f(σ)|V′2dσ+3ν∫t-η-2t-η-1e-νλm+1(t-σ)|f(σ)|V′2dσ+⋅⋅⋅≤3νe-νλm+1η(∫t-η-1t-η|f(σ)|V′2dσ+e-νλm+1∫t-η-2t-η-1|f(σ)|V′2dσ+⋅⋅⋅)≤3νe-νλm+1η(1+e-νλm+1+⋅⋅⋅)sup⁡s∈R∫s-1s|f(σ)|V′2dσ≤(3/ν)e-νλm+1η1-e-νλm+1‖f‖Lb22.
Using (1.11) and letting  s1=(1/νλm+1)ln(3ρ02/ɛ), then s≥s1 implies that
(3.19)3ν∫t-st-ηe-νλm+1(t-σ)|f(σ)|V′2dσ≤(3/ν)e-νλm+1η1-e-νλm+1‖f‖Lb2(R;V′)2≤ɛ6,(3.20)2M1νλm+1≤ɛ3,|u2(τ)|g2e-νλm+1(t-τ)≤ρ02e-νλm+1s1≤ɛ3.
Therefore, we deduce from (3.14) that
(3.21)|u2|g2≤ɛ,∀s≥s1,
which indicates {U(t,τ)} satisfying pullback Condition (C) in Hg. Applying Theorem 2.5, the proof is complete.According to Propositions3.1-3.2, we can now regard that the families of processes {U(t,τ)} are defined in Vg and ℬ1 is a pullback absorbing set in Vg.Theorem 3.4.
Iff(x,t)∈Lb2(R;Hg), then the processes {U(t,τ)} corresponding to problem (1.16) possess  compact pullback attractor 𝒜1(t) in Vg:
(3.22)A1(t)=⋃B1∈Bω(B1,t)¯,
where ℬ1 is the absorbing set in Vg.Proof.
Using Proposition3.2, we have that the family of processes {U(t,τ)} corresponding to (1.16) possess  the pullback absorbing set in Vg.
Now we testify that the family of processes{U(t,τ)} corresponding to (1.16) satisfies pullback Condition (C).
Multiplying   (1.16) by Agu2(t), we have
(3.23)(dvdt,Agu2)+(νAgu,Agu2)+(Bg(u,u),Agu2)g=(f,Agu2)-(νRu,Agu2)g.
It follows that
(3.24)12ddt∥u2∥g2+ν|Agu2|g2≤|(Bg(u,u),Agu2)g|+|(f,Agu2)g|+|(νRu,Agu2)g|.
To estimate (Bg(u,u),Au2)g, we recall some inequalities (see [22]), 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,(3.26)|w|L∞(Ω)2≤c∥w∥g(1+log⁡|Agw|λg∥w∥g2)1/2,
from which we deduce that
(3.27)|Bg(u,v)|≤c|u|L∞(Ω)|∇v|g|u|g|∇v|L∞(Ω),
and using (3.26),
(3.28)|Bg(u,v)|≤c{∥u∥g∥v∥g(1+log⁡|Agu|2λg∥u∥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 (3.28) and the second one of (3.25), we have
(3.29)|(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
(3.30)|Agu1|g2≤λm∥u1∥g2
and set
(3.31)L=1+log⁡λm+1λg.
Next, using the Cauchy inequality,
(3.32)|(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
(3.33)|(f,Agu2)|≤|f|g|Agu2|2≤ν6|Agu2|g2+32ν|f|g2.
Putting (3.29)–(3.33) together, there exists a constant M2 such that
(3.34)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 that
(3.35)∥u2∥g2≤∥u2(τ)∥g2e-νλm+1(t-τ)+2M2νλm+1+3ν∫τte-νλm+1(t-s)|f|g2ds.
We consider the time t-s instead of τ. The following result is similar to (3.17)–(3.19), for any ɛ:
(3.36)2cν∫τte-νλm+1(t-σ)/2|f|g2dσ≤ɛ3.
Using (1.11) and letting  s2=(2/νλm+1)ln(3ρ12/ɛ), then s≥s2 implies that
(3.37)2M2νλm+1≤ɛ3,∥u2(τ)∥g2e-νλm+1(t-τ)≤ρ12e-νλm+1s<ɛ3.
Therefore, we deduce from (3.35) that
(3.38)∥u2∥g2≤ɛ,∀s≥s1,
which indicates {U(t,τ)} satisfying pullback Condition (C) in Vg.


## 4. The Dimension of the Pullback Attractor

To estimate the dimension of the pullback attractor𝒜0(t), we will apply the abstract machinery in [18, 23]. Let F:Vg×ℝ→Vg′ be a given family of nonlinear operators such that, for all τ∈ℝ and any uτ∈Hg, there exists a unique function u(t)=u(t;τ,u0) satisfying(4.1)u∈L2(τ,T;Vg)∩C[τ,T;Hg),F(u(t),t)∈L1(τ,T;Vg′),∀T>τ,dudt=F(u(t),t),t>τ,u(τ)=uτ,
where F(u)=-νAgu-Bgu-νRu+Pgf.Using the standard methods (see [17, 18]), we can show that  {U(t,τ)} is uniformly quasidifferentiable on {B(t)}t∈ℝ. Then, for all τ≤T and any uτ,vτ∈Hg, there exists a unique v(t)=v(t;τ,uτ,vτ), which is a solution of(4.2)v∈L2(τ,T;Vg)∩C[τ,T;Hg),dvdt=F′(U(t,τ)uτ,t)v,v(τ)=vτ.
For all τ<T, we define the linear operator L(t,uτ):Hg→Hg by(4.3)v(t;τ,uτ,vτ)=L(t,τ,uτ)vτ.Theorem 4.1.
Suppose thatf(t) satisfies the assumptions of Theorem 3.3. Then, if γ=1-(2|∇g|∞/m0λg1/2)>0, the Pullback attractor (uniformly in the past) 𝒜0 defined by (3.4) satisfies
(4.4)dF(A0)≤βα,
where
(4.5)α=c2νm0λ1′γ2M0,β=c1d12ν3m0γsup⁡φj∈Hg,|φj|≤1j=1,2,…,m1T∫τ-Tτ∥f(s)∥Vg′2ds,
with the constant c1,c2 of (3.29) and (3.32) of Chapter VI in  [17], λ1′ is the first eigenvalue of the Stokes operator and  d1=|∇g|∞2/4m0+|∇g|∞+M0.Proof.
With Theorem3.3 at our disposal we may apply the abstract framework in [17, 18, 23, 24].
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 v(s;τ,uτ,vτj)∈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 3.3 and 3.4, we may obtain
(4.6)∑i=1m〈F′(U(s,τ)uτ,s)φi,φi〉=-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 [17, 18, 23, 24]), one obtains
(4.7)∑i=1m∥φ∥g2≥λ1+⋅⋅⋅+λm≥m0M0(λ1′+⋯+λm′)≥m0M0c2λ1′m2,Trj(F′(U(s,τ)uτ,s)≤-ν(1-|∇g|∞m0λ11/2)∑i=1m∥φj∥g2+∥U(s,τ)uτ∥(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 that
(4.8)ddt|U(s,τ)uτ|g2+ν∥U(s,τ)uτ∥g2≤∥f∥Vg′2ν+2νm0λg1/2|∇g|∞∥U(s,τ)uτ∥g2
for λg=4π2m0/M0, and then
(4.9)∫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(4.10)qm=sup⁡φj∈Hg,|φj|≤1j=1,2,…,m(1T∫τ-TτTrj(F′(U(s,τ)uτ,s)ds)),q̃m≤-νm02M0(1-2|∇g|∞m0λ11/2)c2λ1′m2+c1d12νm0(sup⁡φj∈Hg,|φj|≤1j=1,2,…,m(1T∫τ-Tτ∥U(s,τ)uτ∥g2ds))≤-νm02M0(1-2|∇g|∞m0λ11/2)c2λ1′m2+c1d12νm0(1ν2sup⁡φj∈Hg,|φj|≤1j=1,2,…,m1T∫τ-Tτ∥f(s)∥Vg′2ds+|uτ|2νT)(1-2|∇g|∞m0λg1/2)-1),qm=limsup⁡T→∞q̃m≤-αm2+β.
Hence,
(4.11)dim⁡F⁡A0(τ)≤βα.



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*Source: 893240-2010-12-20.xml*