# Finite Dimensional Uniform Attractors for the Nonautonomous Camassa-Holm Equations

Authors: Delin Wu
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We consider the uniform attractors for the three-dimensional nonautonomous Camassa-Holm equations in the periodic boxΩ=[0,L]3. Assuming f=f(x,t)∈Lloc2((0,T);D(A−1/2)), we establish the existence of the uniform attractors in D(A1/2) and D(A). The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.


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## Body

## 1. Introduction

We consider the following viscous version of the three-dimensional Camassa-Holm equations in the periodic boxΩ=[0,L]3:
(1.1)∂∂t(α02u-α12Δu)-νΔ(α02u-α12Δu)-u×(∇×(α02u-α12Δu))+1ρ0∇p=f(x,t),∇·u=0,u(x,τ)=uτ(x),
where p/ρ0=π/ρ0+α02|u|2-α12(u·Δu) is the modified pressure, while π is the pressure, ν>0 is the constant viscosity, and ρ0>0 is a constant density. The function f is a given body forcing and α0>0,α1≥0 are scale parameters. Notice α0 is dimensionless while α1 has units of length. Also observe that at the limit α0=1,α1=0, we obtain the three-dimensional Navier-Stokes equations with periodic boundary conditions.We consider this equaton in an appropriate space and show that there is an attractor𝔄 which all solutions approach as t→∞. The basic idea of our construction, is motivated by the works of [1].In addition, we assume that the functionf(·,t)=:f(t)∈Lloc2(ℝ;E) is translation bounded, where E=D(A-1/2). This property implies that
(1.2)∥f∥Lb22=∥f∥Lb2(ℝ;E)2=supt∈R∫tt+1∥f(s)∥E2ds<∞.In [1] the authors established the global regularity of solutions of the autonomous Camassa-Holm, or Navier-Stokes-alpha (NS-α) equations, subject to periodic boundary conditions. The inviscid NS-α equations (Euler-α) were introduced in [2] as a natural mathematical generalization of the integrable inviscid 1D Camassa-Holm equation discovered in [3] through a variational formulation. An alternative more physical derivation for the inviscid NS-α equations (Euler-α) was introduced in [4–8].In the book [9], Haraux 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] present 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 three-dimensional Camassa-Holm equations in the periodic boxΩ=[0,L]3. We apply measure of noncompactness method to nonautonomous Camassa-Holm equations 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

From (1.1) one can easily see, after integration by parts, that
(2.1)ddt∫Ω(α02u-α12Δu)dx=∫Ωfdx.
On the other hand, because of the spatial periodicity of the solution, we have ∫ΩΔudx=0. As a result, we have d/dt∫Ωα02udx=∫Ωfdx, that is, the mean of the solution is invariant provided that the mean of the forcing term is zero. In this paper, we will consider forcing terms and initial values with spatial means that are zero, that is, we will assume ∫Ωuτ(x)dx=∫Ωfdx=0 and hence ∫Ωudx=0.Next, let us introduce some notation and background.(i)We denote𝒱={φ:φ is a vector-valued trigonometric polynomial defined on Ω, such that ∇·φ=0 and ∫Ωφ(x)dx=0}, and let H and V be the closures of 𝒱 in L2(Ω)3 and in H1(Ω)3, respectively, observe that H⊥, the orthogonal complement of H in L2(Ω)3, is {∇p:p∈H1(Ω)} (cf. [11] or [12]).(ii)We denoteP:L2(Ω)3→H the L2 orthogonal projection, usually referred as Helmholtz-Leray projector, and by A=-PΔ the Stokes operator with domain D(A)=(H2(Ω))3∩V. Notice that in the case of periodic boundary condition, A=-Δ|D(A) is a self-adjoint positive operator with compact inverse. Hence the space H has an orthonormal basis {wj}j=1∞ of eigenfunctions of A, that is, Awj=λjwj, with
(2.2)0<λ1≤λ2≤⋯≤λj≤⋯,λj→+∞,asj→∞,
in fact, these eigenvalues have the form |k|24π/L2 with k∈Z3∖{0}.(iii)We denote(·,·) the L2-inner product and by |·| the corresponding L2-norm. By virtue of Poincaré inequality, one can show that there is a constant c>0 such that
(2.3)c|Aw|≤∥w∥H2≤c-1|Aw|foreveryw∈D(A)
and that
(2.4)c|A1/2w|≤∥w∥H1≤c-1|A1/2w|foreveryw∈V.Moreover, one can show thatV=D(A1/2) (cf. [11, 12]). We denote ((·,·))=(A1/2·,A1/2·) and ∥·∥=|A1/2·| the inner product and norm on V, respectively. Notice that, based on the above, the inner product ((·,·)), restricted to V, is equivalent to the H1 inner product
(2.5)[u,v]=α02(u,v)+α12((u,v))foru,v∈V
provided α1>0. We denote V′ is the dual of V.Hereafter,c 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.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
(3.3)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)}:
(3.4)𝒦={u(·)∣u(·)satisfies(3.3)and∥u(s)∥E≤Mufors∈ℝ}.The set(3.5)𝒦(s)={u(s)∣u(·)∈𝒦}⊆E
is 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:
(3.6)y∈ωτ,Σ(B)⟺therearesequences{xn}⊂B,{σn}⊂Σ,{tn}⊂ℝτsuchthattn→+∞andUσn(tn,τ)xn→y(n→∞).We recall characterize the existence of the uniform attractor for a family of processes satisfying (3.6) in term of the concept of measure of noncompactness that was put forward first by Kuratowski (see [13, 14]).LetB∈ℬ(E) its Kuratowski measure of noncompactness κ(B) is defined by
(3.7)κ(B)=inf{δ>0∣Badmitsafinitecoveringbysetsofdiameter≤δ}.Definition 3.2.
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 [15, 16]).Definition 3.3.
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 3.4.
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)-continuous (weakly) and possesses the compact uniform (w.r.t.σ∈Σ) attractor AΣ satisfying
(3.8)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 Camassa-Holm Equations

This section deals with the existence of the attractor for the three-dimensional nonautonomous Camassa-Holm equations with periodic boundary condition. To this end, we first state some the following results.Proposition 4.1.
Letf(x,t)∈Lloc2((0,T);D(A-1/2)) and let uτ∈V. Then problem (1.1) has a unique solution u(t) such that for any T>τ,
(4.1)u∈C([τ,T);V)∩L2([τ,T);D(A)),dudt∈L2([τ,T);H),
and such that for almost all t∈[τ,T) and for any w∈D(A),
(4.2)〈∂∂t(α02u+α12Δu),w〉D(A)′+ν〈A(α02u+α12Δu),w〉D(A)′+〈B̃(u,α02u+α12Δu,w〉D(A)′=(f,w),
here
(4.3)(B̃(u,v),w)=(B(u,v),w)-(B(w,v),u)=(B(v)u-B*(v)u,w)
for every u,v,w∈V.Proof.
We use the Galerkin procedure to prove global existence. The proof of Proposition4.1 is similar to autonomous Camassa-Holm in [1].If we denotev=α02u+α12Au, the system (1.1) can be written as
(4.4)dvdt+νAv+B(v)u+B*(v)u=Pf,v(x,τ)=vτ(x)∈H.
In [1] the authors have shown that the semigroup S(t):V→V(t≥0) associated with the autonomous system (4.4) possesses a global attractor in V and D(A). The main objective of this section is to prove that the nonautonomous system (4.4) have uniform attractors in V and D(A).Now recall the following facts that can be found in [15].Definition 4.2.
A functionφ∈Lloc2(ℝ;E) is said to be normal if for any ε>0, there exists η>0 such that
(4.5)supt∈ℝ∫tt+η∥φ(s)∥E2ds≤ε.We denote byLn2(ℝ;E) the set of all normal functions in Lloc2(ℝ;E).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 [15] 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 [15]).We now define thesymbol  space  ℋ(σ0) for (4.4). 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,
(4.6)ℋ(σ0)=ℋ(f0)=[f0(x,s+h)∣h∈ℝ]Lloc2(ℝ;E).Now, for anyf(x,t)∈ℋ(f0), problem (4.4) with f instead of f0 possesses a corresponding process {Uf(t,τ)} acting on V. As is proved in [10], the family {Uf(t,τ)∣f∈ℋ(f0)} of processes is (V×ℋ(f0);V)-continuous.Let(4.7)𝒦f={vf(x,t)fort∈ℝ∣vf(x,t)issolutionof(4.4)satisfying∥vf(·,t)∥H≤Mf∀t∈ℝ}
be the so-called kernel of the process {Uf(t,τ)}.Proposition 4.4.
The process{Uf(t,τ)} associated with (4.4) possesses absorbing sets
(4.8)ℬ0={v∣∥v∥≤r0},ℬ1={v∣|Av|≤r1}.Proof.
The proof of Proposition4.4 is similar to autonomous Camassa-Holm equation.The main results in this section are as follows.Theorem 4.5.
Iff0(x,s) is a normal function in Lloc2(ℝ;V′), then the processes {Uf0(t,τ)} corresponding to problem (1.1) possess compact uniform (w.r.t.τ∈ℝ) attractor 𝔄0 in V 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):
(4.9)𝔄0=𝔄ℋ(f0)=ω0,ℋ(f0)(ℬ0)=⋃f∈ℋ(f0)𝒦f(0),
where ℬ0 is the uniformly (w.r.t.f∈ℋ(f0)) absorbing set in V, 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 H. We write
(4.10)u=u1+u2;u1∈H1,u2∈H2foranyu∈H.
Now, we only have to verify Condition (C). Namely, we need to estimate|u2(t)|, where u(t)=u1(t)+u2(t) is a solution of (4.4) given in Proposition 4.1.
Lettingw=u2 in (4.2), we have
(4.11)12ddt(α02|u2|2+α12∥u2∥2)+ν(α02∥u2∥2+α12|Au2|2)+(B̃(u,α02u+α12Au),u2)=(Pf,u2),
Notice that
(4.12)|(Pf,u2)|≤|f|V′∥u2∥≤|f|V′2να02+ν4α02∥u2∥2.
From the above inequalities we get
(4.13)12ddt(α02|u2|2+α12∥u2∥2)+3ν4(α02∥u2∥2+α12|Au2|2)+(B̃(u,α02u+α12Au),u2)≤|f|V′2να02.
Since B̃ satisfies the following inequality (see [1, 12]):
(4.14)|(B̃(u,v),w)|≤c∥u∥∥v∥|w|1/2∥w∥1/2,∀u,v,w∈V,
then  by Young's inequality,
(4.15)|(B̃(u,α02u+α12Au),u2)|≤c(α02∥u∥2+α12∥u∥∥Au∥)|u2|1/2∥u2∥1/2≤ν4(α02∥u2∥2+α12|Au2|2)+M1(α0,α1,r0,r1).
Thus, we obtain
(4.16)12ddt(α02|u2|2+α12∥u2∥2)+ν2(α02∥u2∥2+α12|Au2|2)≤M1(α0,α1,r0,r1)+|f|V′2να02.
Therefore, we deduce that
(4.17)12ddt(α02|u2|2+α12∥u2∥2)+ν2λm+1(α02|u2|2+α12∥u2∥2)≤M1+cν|f|V′2.
Here, M1=M1(α0,α1,r0,r1) depends on λm+1, and is not increasing as λm+1 increasing.
By the Gronwall inequality, the above inequality implies(4.18)α02|u2|2+α12∥u2∥2≤(α02|u2(t0+1)|2+α12∥u2(t0+1)∥2)e-νλm+1(t-(t0+1))+2M1νλm+1+2cν∫t0+1te-νλm+1(t-s)|f|V′2ds.
Applying [10, Definition 4.1 and Lemma II  1.3] for any ε1>0,ε=ε1/α12,
(4.19)2cν∫t0+1te-νλm+1(t-s)|f|V′2ds<ε13.
Using (2.2) and letting t1=t0+1+1/νλm+1ln3r02/ε1, then t≥t1 implies
(4.20)2M1νλm+1<ε13,(α02|u2(t0+1)|2+α12∥u2(t0+1)∥2)e-νλm+1(t-(t0+1))≤r02e-νλm+1(t-(t0+1))<ε13.
Therefore, we deduce from (4.18) that
(4.21)∥u2∥2≤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 V. Applying Theorem 3.4, the proof is complete.According to Propositions4.1 and 4.4, we can now regard that the families of processes {Uf(t,τ)},f∈ℋ(f0) for (1.1) are defined in D(A) and ℬ1 is a uniformly (w. r. t.f∈ℋ(f0)) absorbing set in D(A).Theorem 4.6.
Iff0(x,s) is normal function in Lloc2(ℝ;V′), then the processes {Uf0(t,τ)} corresponding to problem (1.1) possesses compact uniform (w.r.t.τ∈ℝ) attractor 𝔄1 in Hper2=D(A) 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 D(A) 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 (4.4) possesses the uniformly (w.r.t.f∈ℋ(f0)) absorbing set in D(A).
Now we testify that the family of processes{Uf(t,τ)},f∈ℋ(f0) corresponding to (4.4) satisfies uniform (w.r.t.f∈ℋ(f0)) Condition (C).
Lettingw=Au2 in (4.2), we have
(4.23)12ddt(α02∥u2∥2+α12|Au2|2)+ν(α02|Au2|2+α12|A3/2u2|2)+(B̃(u,α02u+α12Au),Au2)=(Pf,Au2).
Notice that
(4.24)|(Pf,Au2)|≤|f|V′|A3/2u2|≤|f|V′2να12+ν4α12|A3/2u2|2.
Therefore, we get
(4.25)12ddt(α02∥u2∥2+α12|Au2|2)+3ν4(α02|Au2|2+α12|A3/2u2|2)+(B̃(u,α02u+α12Au),u2)≤|f|V′2να12.
To estimate (B̃(u,α02u+α12Au)),Au2)g, we recall some inequalities ([1, 11, 12, 14]): for every u,v∈D(Ag),
(4.26)|B(u,v)|≤c{|u|1/2∥u∥1/2∥v∥1/2|Av|1/2|u|1/2|Av|1/2∥v∥;
and [12]
(4.27)|w|L∞(Ω)≤c∥w∥(1+log|Aw|λ1∥w∥2)1/2
from which we deduce that
(4.28)|B(u,v)|≤c|u|L∞(Ω)|∇v||u||∇v|L∞(Ω),
and using (4.27),
(4.29)|B(u,v)|≤c{∥u∥∥v∥(1+log|Au|2λ1∥u∥2)1/2|u||Av|(1+log|A3/2v|2λ1∥Av∥2)1/2.
Expanding and using Young's inequality, together with the first one of (4.29) and the second one of (4.26), we have
(4.30)|(B̃(u,α02u+α12Au)),Au2)|≤|(B̃(u1,α02u+α12Au),Au2)|+|(B̃(u2,α02u+α12Au),Au2)|≤cL1/2∥u1∥|Au2|(α02∥u∥+α12|Au|)+c|u2|1/2|Au2|3/2(α02∥u∥+α12|Au|)≤ν4(α02|Au2|2+α12|A3/2u2|2)+M2(α0,α1,r0,r1),t≥t0+1,
where we use
(4.31)|Au1|2≤λm∥u1∥2,
and set
(4.32)L=1+logλm+1λ1.
Thus, we have
(4.33)12ddt(α02∥u2∥2+α12|Au2|2)+ν2(α02|Au2|2+α12|A3/2u2|2)≤M2+|f|V′2να12.
Here M2=M2(α0,α1,r0,r1) depends on λm+1, and is not increasing as λm+1 increasing.
By the Gronwall inequality, the above inequality implies(4.34)α02∥u2∥2+α12|Au2|2≤(α02∥u2(t0+1)∥2+α12|Au2(t0+1)|2)e-νλm+1(t-(t0+1))+2M2νλm+1+2cν∫t0+1te-νλm+1(t-s)|f|V′2ds.
Applying [10, Definition 4.1 and Lemma II  1.3] for any ε1>0,ε=ε1/α12,
(4.35)2cν∫t0+1te-νλm+1(t-s)|f|V′2ds<ε13.
Using (2.2) and let t1=t0+1+1/νλm+1ln3r12/ε1, then t≥t1 implies
(4.36)2M2νλm+1<ε13;(α02∥u2(t0+1)∥2+α12|Au2(t0+1)|2)e-νλm+1(t-(t0+1))≤r12e-νλm+1(t-(t0+1))<ε13.
Therefore, we deduce from (4.34) that
(4.37)|Au2|2≤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 D(A).


## 5. Dimension of the Uniform Attractor

In this section, we estimate the fractal dimension (for definition see, e.g., [10, 12, 17, 18]) of the kernel sections of the uniform attractors obtained in Section 4 by applying the methods in [19].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)lim supε→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 equation (4.4). Assume that {L(t,τ;u)} is generated by the variational equation corresponding to (4.4),
(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 (1.1) with initial value uτ∈𝒦(τ). For natural number j∈N, we set
(5.3)q̃j=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  [10, Theorem VIII.3.1].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)q̃j≤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 (4.4) with f∈Ln2(ℝ;V′). The equations possess a compact uniform (w.r.t.f∈ℋ(f)) attractor 𝒜ℋ(f) and ⋃τ∈ℝ𝒦f(τ)⊂𝒜ℋ(f). By [10, 12], 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=-Aw-B̃(u,w)-B̃(w,u)≡F′(u(t),t)w,w|t=τ=wτ∈E,τ∈ℝ,
where B̃(u,w)=-P(u×(∇×w)), one can easily show that
(5.7)(B̃(u,w),θ)=(B(u,w),θ)-(B(θ,w),u),foreveryu,w,θ∈V.We have the main results in this section.Theorem 5.2.
Letf∈Ln2(ℝ;V′). Then the fractal dimension of the kernel sections of the uniform attractor, dF𝒦f(s), satisfy
(5.8)dF𝒦f(s)≤cmax{1ν4/3λ1(Gα12)2/3,r03/10G3/5ν3/2λ13/5α03/4α19/20},
where
(5.9)G=1ν2γlim supT→∞supτ∈R1T∫ττ+T|f|V′2ds,1γ=min{1να02,1νλ1α12}.Proof.
Now we estimate them-dimensional trace of F′(u(t),t). Let Qm is the orthogonal projector from H to QmH with orthonormal basis φ1,…,φm∈V. Similar to that of [1, 19], we have
(5.10)Tr(F′(u(t),t)·Qm)=∑i=1m(F′(u(t),t)φi,φi)≤-cν2λ1m5/3+cν3/2∥u∥1/2α05/4α13/4(α02∥u∥2+α12|Au|2)+c|Au|4/3(m2ν)1/3.
It follows from (4.2) that
(5.11)lim supT→∞supτ∈ℝ1T∫ττ+T(α02∥u∥2+α12|Au|2)ds≤1ν2γlimsupT→∞supτ∈ℝ1T∫ττ+T|f|V′2ds.
By HÖlder’s inequality we have
(5.12)lim supT→∞supτ∈ℝ1T∫ττ+T|Au|4/3ds≤limsupT→∞supτ∈ℝ(1T∫ττ+T|Au|2ds)2/3,
and we get
(5.13)limsupT→∞supτ∈ℝ1T∫ττ+T|Au|4/3ds≤(Gα12)2/3.
Therefore, from the above, we have
(5.14)dF𝒦f(s)≤cmax{1ν4/3λ1(Gα12)2/3,r03/10G3/5ν3/2λ13/5α03/4α19/20},
which concludes our proof.



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*Source: 952657-2009-08-06.xml*