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Remarks on uniform attractors for the 3D non-autonomous Navier-Stokes-Voight equations

Yiwen Dou12, Xinguang Yang3* and Yuming Qin4

Author Affiliations

1 College of Information Sciences and Technology, Donghua University, Songjiang, Shanghai, 201620, People's Republic of China

2 College of Computer and Information Sciences, Anhui Polytechnique University, Wuhu, Anhui, 241000, People's Republic of China

3 College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People's Republic of China

4 Department of Applied Mathematics, Donghua University, Songjiang, Shanghai, 201620, People's Republic of China

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Boundary Value Problems 2011, 2011:49  doi:10.1186/1687-2770-2011-49

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/49


Received:7 July 2011
Accepted:28 November 2011
Published:28 November 2011

© 2011 Dou et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we show the existence of pullback attractors for the non-autonomous Navier-Stokes-Voight equations by using contractive functions, which is more simple than the weak continuous method to establish the uniformly asymptotical compactness in H1 and H2.

2010 Mathematics Subject Classification: 35D05; 35M10

Keywords:
Navier-Stokes-Voight equations; processes; contractive functions; uniform attractors

1 Introduction

Let Ω ⊂ R3 be a bounded domain with sufficiently smooth boundary ∂Ω. We consider the non-autonomous 3D Navier-Stokes-Voight (NSV) equations that govern the motion of a Klein-Voight linear viscoelastic incompressible fluid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M3">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M4">View MathML</a>

(1.4)

Here u = u(t, x) = (u1(t, x), u2(t, x), u3(t, x)) is the velocity vector field, p is the pressure, ν > 0 is the kinematic viscosity, and the length scale α is a characterizing parameter of the elasticity of the fluid.

When α = 0, the above system reduce to the well-known 3D incompressible Navier-Stokes system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M5">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M6">View MathML</a>

(1.6)

For the well-posedness of 3D incompressible Navier-Stokes equations, in 1934, Leray [1-3] derived the existence of weak solution by weak convergence method; Hopf [4] improved Leray's result and obtained the familiar Leray-Hopf weak solution in 1951. Since the 3D Navier-Stokes equations lack appropriate priori estimate and the strong nonlinear property, the existence of strong solution remains open. For the infinite-dimensional dynamical systems, Sell [5] constructed the semiflow generated by the weak solution which lacks the global regularity and obtained the existence of global attractor of the 3D incompressible Navier-Stokes equations on any bounded smooth domain. Chepyzhov and Vishik [6] investigated the trajectory attractors for 3D non-autonomous incompressible Navier-Stokes system which is based on the works of Leray and Hopf. Using the weak convergence topology of the space H (see below for the definition), Kapustyan and Valero [7] proved the existence of a weak attractor in both autonomous and non-autonomous cases and gave a existence result of strong attractors. Kapustyan, Kasyanov and Valero [8] considered a revised 3D incompressible Navier-Stokes equations generated by an optimal control problem and proved the existence of pullback attractors by constructing a dynamical multivalued process.

However, the infinite-dimensional systems for 3D incompressible Navier-Stokes equations have not yet completely resolved, so many mathematicians pay attention to this challenging problem. In this regard, Kalantarov and Titi [9] investigated the Navier-Stokes-Voight equations as an inviscid regularization of the 3D incompressible Navier-Stokes equations, and further obtained the existence of global attractors for Navier-Stokes-Voight equations. Recently, Qin, Yang and Liu [10] showed the existence of uniform attractors by uniform condition-(C) and weak continuous method to obtain uniformly asymptotical compactness in H1 and H2, Yue and Zhong [11] investigated the attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations in different methods. More details about the infinite-dimensional dynamics systems, we can refer to [12-27].

Using the contractive functions, we have in this paper established the uniformly asymptotical compactness of the processes {U(t, τ)}(t τ, τ R) to obtain the existence of the uniform attractor of the 3D non-autonomous NSV equations.

Main difficulties we encountered are as follows:

(1) how to obtain a contractive function,

(2) how to deduce the uniformly asymptotical compactness from a contractive function,

(3) how to obtain the convergence of contractive function.

2 Main results

Notations: Throughout this paper, we set Rτ = [τ, +∞), τ R. C stands for a generic positive constant, depending on Ω, but independent of t. Lp(Ω)(1 ≤ p ≤ +∞) is the generic Lebesgue space, Hs(Ω) is the general Sobolev space. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M7">View MathML</a>, H, V, W is the closure of the set E in the topology of (L2(Ω))3, (H1(Ω))3, (H2(Ω))3 respectively. "⇀" stands for weak convergence of sequence.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M8">View MathML</a> be the hull of f0 as a symbol space:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M9">View MathML</a>

(2.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M10">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M11">View MathML</a> denotes the closure in the topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M12">View MathML</a>.

Under the assumptions of the initial data, the problem (1.1)-(1.4) has a global solution u C([τ, +∞), V). Uf(t, τ, uτ): V V denotes the processes generated by the global solutions and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M13">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M14">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M15">View MathML</a>

(2.4)

Let {T(s)} be the translation semigroup on Σ, we see that the family of processes {Uf(t, τ)} (f ∈ Σ) satisfies the translation identity if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M16">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M17">View MathML</a>

(2.6)

Next, we recall a simple method to derive uniformly asymptotical compactness which can be found in [28].

Definition 2.1 Let X be a Banach space and B be a bounded subset of X, Σ be a symbol space. We call a function ϕ(·,·;·,·) defined on (X × X) × (Σ × Σ) to be a contractive function on B × B if for any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M18">View MathML</a> and any {gn} ⊂ Σ, there are subsequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M20">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M21">View MathML</a>

(2.7)

We denote the set of all contractive functions on B × B by Contr(B, Σ).

Lemma 2.2 Let {Uf(t, τ)}(f ∈ Σ) be a family of processes satisfying the translation identity (2.5) and (2.6) on Banach space X and has a bounded uniform (w.r.t f ∈ Σ) absorbing set B0 X. Moreover, assume that for any ε > 0, there exist T = T(B0, ε) and ϕT Contr(B0, Σ) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M22">View MathML</a>

(2.8)

Then {Uf(t, τ)} (f ∈ Σ) is uniformly (w.r.t. f ∈ Σ) asymptotically compact in X.

Theorem 2.3 Assume that f ∈ Σ ⊆ L2(R, H), uτ V, then the problem (1.1)-(1.4) possesses uniform attractors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M23">View MathML</a> in V.

Theorem 2.4 Assume that f ∈ Σ ⊆ L2(R, H), uτ W, then the problem (1.1)-(1.4) possesses uniform attractors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M24">View MathML</a> in W.

3 Proof of Theorem 2.3

In this section, we shall prove Theorem 2.3 by two steps as follows, the first one is to get the existence of an absorbing ball, the second is to prove the asymptotical compactness by means of a contractive function.

From the property of solutions, we can easily derive that the set class {Uf(t, τ, uτ)} (τ, ≤ t) is a process in V for all τ t. Moreover, the mapping Uf(t, τ, uτ): V V is continuous.

Lemma 3.1 We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M26">View MathML</a>, fn f in L2(R, H), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M27">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M28">View MathML</a>

(3.2)

Proof. From the boundedness of the solutions in corresponding topological spaces, we easily conclude the results.   □

Lemma 3.2 Assume f L2(R, H), uτ V, then there exists a uniform (w.r.t. f ∈ Σ) absorbing set B0 of processes {Uf(t, τ, uτ)}.

Proof. For all u V, multiplying both sides of (1.1) with u and noting that ((u·∇)u, u) = 0, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M29">View MathML</a>

(3.3)

Consequently, for all τ R, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M30">View MathML</a>

(3.4)

Consider the property of the functional ⟨·,·⟩ + α1⟨∇·,∇·⟩, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M31">View MathML</a>

and there exists a constant C0 satisfying C1 C0 C2, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M32">View MathML</a>

Setting the radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M33">View MathML</a>, we easily get that there exists a constant C > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M34">View MathML</a>

(3.5)

for all uτ V, t τ.

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M35">View MathML</a>

then we denote R the nonnegative number given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M36">View MathML</a>

(3.6)

and consider the family of closed balls B0 in V defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M37">View MathML</a>

(3.7)

It is straightforward to check that B0 is a uniform absorbing ball for the processes {Uf(t, τ, uτ)}.   □

Lemma 3.3 Under the condition of f L2(R, H), the process {Uf(t, τ, uτ)} generated by the global solutions for problem (1.1)-(1.4) is uniformly asymptotically compact in V.

Proof. For any initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M38">View MathML</a>, let ui(t, x) be the corresponding solutions to the symbols fi with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M39">View MathML</a>, that is, ui(t) is the solution of the problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M40">View MathML</a>

(3.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M41">View MathML</a>

(3.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M42">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M43">View MathML</a>

(3.11)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M44">View MathML</a>

(3.12)

then w(t) satisfies the equivalent abstract equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M45">View MathML</a>

(3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M46">View MathML</a>

(3.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M47">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M48">View MathML</a>

(3.16)

where B(u) = (u·∇)u, p has disappeared by the projection operator P.

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M49">View MathML</a>

(3.17)

Multiplying (3.13) by w and integrating over [s, T] × Ω, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M50">View MathML</a>

(3.18)

where τ s T. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M51">View MathML</a>

(3.19)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M52">View MathML</a>

(3.20)

Integrating (3.18) over [τ, T] with respect to s, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M53">View MathML</a>

(3.21)

If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M54">View MathML</a>

(3.22)

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M55">View MathML</a>

(3.23)

Since the family of processes has a uniformly bounded absorbing set, we choose T large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M56">View MathML</a>

(3.24)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M57">View MathML</a>.

Let un, um be the solutions with respect to the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M58">View MathML</a> and symbols fn(t), fm(t) ∈ Σ, m, n = 1, 2, . . . respectively. Then from Lemma 3.1, we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M59">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M60">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M61">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M62">View MathML</a> for the above T. By Lemma 2.2 and the property of the functional ⟨·,·⟩ + α2 ⟨∇·, ∇·⟩, the conclusion holds.   □

Proof of Theorem 2.3 From Lemmas 3.1-3.3, we can deduce the result easily.   □

4 Proof of Theorem 2.4

Similarly to the proof of Theorem 2.3, we easily obtain that the set class {Uf (t, τ, uτ)} (τ t) is a process in W for all τ t. Moreover, the mapping Uf (t, τ, uτ): W W is continuous. If we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M63">View MathML</a> is a sequence in W and weakly converges to uτ W, fn f in L2(R, H), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M64">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M65">View MathML</a>

(4.2)

Lemma 4.1 Assume f L2(R, H), then there exists a global uniform (w.r.t. f ∈ Σ) absorbing set B0 of the process {Uf (t, τ, uτ)}.

Proof. By the Faedo-Galerkin method, the standard elliptic operator theory and the Poincaré inequality, we get that u belongs to L2((τ, T); D(A)) ∩ L((τ, T); W), then using the Gronwall inequality and similar energy method to the proof of Theorem 3.1 in Qin, Yang and Liu [10], we can deduce the boundedness of u and the existence of absorbing set.   □

Lemma 4.2 Under the condition of f L2(R, H), uτ W, the process {Uf (t, τ, uτ)} generated by the global solutions for problem (1.1)-(1.4) is asymptotically compact in W.

Proof. For any initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M66">View MathML</a>, let ui(t, x) be the corresponding solutions to the symbols fi with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M67">View MathML</a>, that is, ui(t) is the solution of the problem (3.8)-(3.11). Denote A = -Δ and w(t) = u1(t) - u2(t), then w(t) satisfies the equivalent abstract equations (3.13)-(3.14).

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M68">View MathML</a>

(4.3)

Multiplying (3.13) by Aw and integrating over [s, T] × Ω, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M69">View MathML</a>

(4.4)

where τ s T. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M70">View MathML</a>

(4.5)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M71">View MathML</a>

(4.6)

Integrating (4.4) over [τ, T] with respect to s, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M72">View MathML</a>

(4.7)

If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M73">View MathML</a>

(4.8)

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M74">View MathML</a>

(4.9)

Since the family of the processes has a uniformly bounded absorbing set, we choose T large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M75">View MathML</a>

(4.10)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M76">View MathML</a>.

Let un, um be the solutions with respect to the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M77">View MathML</a> and symbols fn(t), fm(t) ∈ Σ, m, n = 1, 2, . . . respectively. Then we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M78">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M79">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M80">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/49/mathml/M81">View MathML</a> for the above T. By Lemma 2.2 and the property of the functional ⟨·, A·⟩ + α2A·, A·⟩, the conclusion holds.   □

Proof of Theorem 2.4 From Lemmas 4.1-4.2, we can deduce the result easily.   □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The work in part was supported by the NNSF of China (No. 11031003 and 10871040).

References

  1. Leray, J: Etude de diverses equations integrales nonlineaires et de quelques problemes que pose l'hydrodynamique. J Math Pures Appl. 12, 1–82 (1933)

  2. Leray, J: Essai sur les mouvements plans d'un liquide visqueux que limitent des parois. J Math Pures Appl. 13, 331–418 (1934)

  3. Leray, J: Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63, 193–248 (1934). Publisher Full Text OpenURL

  4. Hopf, E: Ueber die Anfangswertaufgable fur die hydrodynamischen Grundgleichungen. Math Nachr. 4, 213–231 (1951)

  5. Sell, GR: Global attractors for the three-dimensional Navier-Stokes equations. J Dyn Differ Equ. 8, 1–33 (1996). Publisher Full Text OpenURL

  6. Chepyzhov, VV, Vishik, MI: Evolution equations and their trajectory attractors. J Math Pures Appl. 76, 664–913 (1997)

  7. Kapustyan, OV, Valero, J: Weak and strong attractors for the 3D Navier-Stokes system. J Differ Equ. 240, 249–278 (2007). Publisher Full Text OpenURL

  8. Kapustyan, OV, Kasyanov, PO, Valero, J: Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system. J Math Anal Appl. 373, 535–547 (2011). Publisher Full Text OpenURL

  9. Kalantarov, VK, Titi, ES: Global attractors and determining models for the 3D Navier-Stokes-Voight equations. Chin Ann Math. 30(B), 697–714 (2009)

  10. Qin, Y, Yang, X, Xin, Liu: Uniform attractors for a 3D non-autonomous Navier-Stokes-Voight Equations, Preprint. (2010, in press)

  11. Yue, G, Zhong, C: Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations. Discret Contin Dyn Syst Ser B. 2011(6-3), 985–1002 (2011)

  12. Babin, AV, Vishik, MI: Attractors of Evolutionary Equations, Studies in Mathematics and Its Applications. North-Holland, Amesterdam, London, New York, Tokyo (1992)

  13. Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematical Physics. Providence, RI: American Mathematical Society (2001)

  14. Chueshov, I: Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA. (2002)

  15. Feifeisl, E: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

  16. Hale, JK: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence, RI (1988)

  17. Hou, Y, Li, K: The uniform attractors for the 2D non-autonomous Navier-Stokes flow in some unbounded domain. In: Nonlinear Anal TMA. 58, 609–630

  18. Ladyzhenskaya, OA: The mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)

  19. Ladyzhenskaya, OA: Attractors for Semigroup and Evolution Equations. Cambridge University Press, Cambridge (1991)

  20. Lu, SS, Wu, H, Zhong, CK: Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces. Disc Cont Dyn Syst. 13(3), 701–719 (2005)

  21. Miranville, A, Wang, X: Attractors for Non-autonomous nonhomogenerous Navier-Stokes equations. pp. 1047–01061. Nonlinearity (1997)

  22. Qin, Y: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors. Operator Theory, Advances and Applications. Birkhäuser, Basel-Boston-Berlin

  23. Qin, Y: Universal attractor in H4 for the nonlinear one-dimensional compressible Navier-Stokes equations. J Differ Equ. 207(1), 21–72 (2004). Publisher Full Text OpenURL

  24. Robinson, JC: Infinite-Dimensional Dynamics Systems. Cambridge University Press, Cambridge (2001)

  25. Sell, GR, You, Y: Dynamics of Evolutionary Equations. Springer, New York (2002)

  26. Temam, R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1997)

  27. Temam, R: Navier-Stokes Equations, Theory and Numerical Analysis. North Holland, Amsterdam (1979)

  28. Yang, L: Uniform attractor for non-autonomous hyperbolic equation with critical exponent. Appl Math Comput. 203, 895–902 (2008). Publisher Full Text OpenURL