Journal of Applied Mathematics
Volume 2009 (2009), Article ID 150420, 17 pages
doi:10.1155/2009/150420
Research Article

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

Delin Wu

College of Science, China Jiliang University, Hangzhou 310018, China

Received 3 July 2008; Accepted 17 January 2009

Academic Editor: Alberto Tesi

Copyright © 2009 Delin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domain Ω . Assuming 𝑓 = 𝑓 ( 𝑥 , 𝑡 ) ∈ 𝐿 2 l o c , we establish the existence of the uniform attractor in 𝐿 2 ( Ω ) and 𝐷 ( 𝐴 1 / 2 ) . The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

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, 𝜕 𝑢 1 𝜕 𝑡 − 𝜈 Δ 𝑢 + ( 𝑢 ⋅ ∇ ) 𝑢 + ∇ 𝑝 = 𝑓 i n Ω , 𝑔 ( ∇ ⋅ 𝑔 𝑢 ) = ∇ 𝑔 𝑔 ⋅ 𝑢 + ∇ ⋅ 𝑢 = 0 i n Ω , ( 1 . 1 ) where 𝑔 = 𝑔 ( 𝑥 1 , 𝑥 2 ) is a suitable smooth real-valued function defined on ( 𝑥 1 , 𝑥 2 ) ∈ Ω and Ω is a suitable bounded domain in ℝ 2 . Notice that if 𝑔 ( 𝑥 1 , 𝑥 2 ) = 1 , then (1.1) reduce to the standard Navier-Stokes equations.

In addition, we assume that the function 𝑓 ( ⋅ , 𝑡 ) = ∶ 𝑓 ( 𝑡 ) ∈ 𝐿 2 l o c ( ℝ ; 𝐸 ) is translation bounded, where 𝐸 = 𝐿 2 ( Ω ) o r 𝐻 − 1 ( Ω ) . This property implies that ‖ 𝑓 ‖ 2 𝐿 2 𝑏 = ‖ 𝑓 ‖ 2 𝐿 2 𝑏 ( ℝ ; 𝐸 ) = s u p 𝑡 ∈ ℝ 𝑡 𝑡 + 1 ‖ 𝑓 ( 𝑠 ) ‖ 2 𝐸 𝑑 𝑠 < ∞ . ( 1 . 2 )

We consider this equation in an appropriate Hilbert space and show that there is an attractor 𝔄 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 { 𝑈 𝜎 ( 𝑡 , 𝜏 ) , 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ } one should consider a family of processes { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Σ in some Banach space 𝐸 , where the functional parameter 𝜎 0 ( 𝑠 ) , 𝑠 ∈ ℝ is called the symbol and Σ is the symbol space including 𝜎 0 ( 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 conditions Ω . We apply measure of noncompactness method to nonautonomous g-Navier-Stokes equations equation with external forces 𝑓 ( 𝑥 , 𝑡 ) in 𝐿 2 l o c ( 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

Let Ω = ( 0 , 1 ) × ( 0 , 1 ) and we assume that the function 𝑔 ( 𝑥 ) = 𝑔 ( 𝑥 1 , 𝑥 2 ) satisfies the following properties:

(1) 𝑔 ( 𝑥 ) ∈ 𝐶 ∞ p e r ( Ω ) and

(2) there exist constants 𝑚 0 = 𝑚 0 ( 𝑔 ) and 𝑀 0 = 𝑀 0 ( 𝑔 ) such that, for all 𝑥 ∈ Ω , 0 < 𝑚 0 ≤ 𝑔 ( 𝑥 ) ≤ 𝑀 0 . Note that the constant function 𝑔 ≡ 1 satisfies these conditions.

We denote by 𝐿 2 ( Ω , 𝑔 ) the space with the scalar product and the norm given by ( 𝑢 , 𝑣 ) 𝑔 = Ω ( 𝑢 ⋅ 𝑣 ) 𝑔 𝑑 𝑥 , | 𝑢 | 2 𝑔 = ( 𝑢 , 𝑢 ) 𝑔 , ( 2 . 1 ) as well as 𝐻 1 ( Ω , 𝑔 ) with the norm ‖ 𝑢 ‖ 𝐻 1 ( Ω , 𝑔 ) = ( 𝑢 , 𝑢 ) 𝑔 + 2 𝑖 = 1 𝐷 𝑖 𝑢 , 𝐷 𝑖 𝑢 𝑔 1 / 2 , ( 2 . 2 ) where 𝜕 𝑢 / 𝜕 𝑥 𝑖 = 𝐷 𝑖 𝑢 .

Then for the functional setting of (1.1), we use the following functional spaces 𝐻 𝑔 = 𝐶 𝑙 𝐿 2 p e r ( Ω , 𝑔 ) 𝑢 ∈ 𝐶 ∞ p e r ∫ ( Ω ) ∶ ∇ ⋅ 𝑔 𝑢 = 0 , Ω , 𝑉 𝑢 𝑑 𝑥 = 0 𝑔 = 𝑢 ∈ 𝐻 1 p e r ∫ ( Ω , 𝑔 ) ∶ ∇ ⋅ 𝑔 𝑢 = 0 , Ω , 𝑢 𝑑 𝑥 = 0 ( 2 . 3 ) where 𝐻 𝑔 is endowed with the scalar product and the norm in 𝐿 2 ( Ω , 𝑔 ) , and 𝑉 𝑔 is the spaces with the scalar product and the norm given by ( ( 𝑢 , 𝑣 ) ) 𝑔 = Ω ( ∇ 𝑢 ⋅ ∇ 𝑣 ) 𝑔 𝑑 𝑥 , ‖ 𝑢 ‖ 𝑔 = ( ( 𝑢 , 𝑢 ) ) 𝑔 . ( 2 . 4 ) Also, we define the orthogonal projection 𝑃 𝑔 as 𝑃 𝑔 ∶ 𝐿 2 p e r ( Ω , 𝑔 ) ⟶ 𝐻 𝑔 ( 2 . 5 ) and we have that 𝑄 ⊆ 𝐻 ⟂ 𝑔 , where 𝑄 = 𝐶 𝑙 𝐿 2 p e r ( Ω , 𝑔 ) ∇ 𝜙 ∶ 𝜙 ∈ 𝐶 1 Ω , ℝ . ( 2 . 6 ) Then, we define the 𝑔 -Laplacian operator − Δ 𝑔 1 𝑢 ≡ 𝑔 1 ( ∇ ⋅ 𝑔 ∇ ) 𝑢 = − Δ 𝑢 − 𝑔 ( ∇ 𝑔 ⋅ ∇ ) 𝑢 ( 2 . 7 ) to have the linear operator 𝐴 𝑔 𝑢 = 𝑃 𝑔 − 1 𝑔 ( ∇ ⋅ ( 𝑔 ∇ 𝑢 ) ) . ( 2 . 8 ) For the linear operator 𝐴 𝑔 , the following hold (see [1, 2]):

(1) 𝐴 𝑔 is a positive, self-adjoint operator with compact inverse, where the domain of 𝐴 𝑔 , 𝐷 ( 𝐴 𝑔 ) = 𝑉 𝑔 ∩ 𝐻 2 ( Ω , 𝑔 ) .

(2) There exist countable eigenvalues of 𝐴 𝑔 satisfying 0 < 𝜆 𝑔 ≤ 𝜆 1 ≤ 𝜆 2 ≤ 𝜆 3 ≤ ⋯ , ( 2 . 9 ) where 𝜆 𝑔 = 4 𝜋 2 𝑚 / 𝑀 and 𝜆 1 is the smallest eigenvalue of 𝐴 𝑔 . In addition, there exists the corresponding collection of eigenfunctions { 𝑒 1 , 𝑒 2 , 𝑒 3 , … } which forms an orthonormal basis for 𝐻 𝑔 .

Next, we denote the bilinear operator 𝐵 𝑔 ( 𝑢 , 𝑣 ) = 𝑃 𝑔 ( 𝑢 ⋅ ∇ ) 𝑣 and the trilinear form 𝑏 𝑔 ( 𝑢 , 𝑣 , 𝑤 ) = 𝑛 𝑖 , 𝑗 = 1 Ω 𝑢 𝑖 𝐷 𝑖 𝑣 𝑗 𝑤 𝑗 𝑃 𝑔 𝑑 𝑥 = 𝑔 ( 𝑢 ⋅ ∇ ) 𝑣 , 𝑤 𝑔 , ( 2 . 1 0 ) where 𝑢 , 𝑣 , 𝑤 lie in appropriate subspaces of 𝐿 2 ( Ω , 𝑔 ) . Then, the form 𝑏 𝑔 satisfies 𝑏 𝑔 ( 𝑢 , 𝑣 , 𝑤 ) = − 𝑏 𝑔 ( 𝑢 , 𝑤 , 𝑣 ) f o r 𝑢 , 𝑣 , 𝑤 ∈ 𝐻 𝑔 . ( 2 . 1 1 )

We denote a linear operator 𝑅 on 𝑉 𝑔 by 𝑅 𝑢 = 𝑃 𝑔 1 𝑔 ( ∇ 𝑔 ⋅ ∇ ) 𝑢 f o r 𝑢 ∈ 𝑉 𝑔 , ( 2 . 1 2 ) and have 𝑅 as a continuous linear operator from 𝑉 𝑔 into 𝐻 𝑔 such that | ( 𝑅 𝑢 , 𝑢 ) | ≤ | ∇ 𝑔 | ∞ 𝑚 0 ‖ 𝑢 ‖ 𝑔 | 𝑢 | 𝑔 ≤ | ∇ 𝑔 | ∞ 𝑚 0 𝜆 𝑔 1 / 2 ‖ 𝑢 ‖ 𝑔 f o r 𝑢 ∈ 𝑉 𝑔 . ( 2 . 1 3 )

We now rewrite (1.1) as abstract evolution equations, 𝑑 𝑢 𝑑 𝑡 + 𝜈 𝐴 𝑔 𝑢 + 𝐵 𝑔 𝑢 + 𝜈 𝑅 𝑢 = 𝑃 𝑔 𝑓 , 𝑢 ( 𝜏 ) = 𝑢 𝜏 . ( 2 . 1 4 )

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

Let 𝐸 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 : 𝑈 ( 𝑡 , 𝜏 ) ∶ 𝐸 ⟶ 𝐸 , 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ . ( 3 . 1 ) Definition 3.1. A two-parameter family of mappings { 𝑈 ( 𝑡 , 𝜏 ) } is said to be a process in 𝐸 if 𝑈 ( 𝑡 , 𝑠 ) 𝑈 ( 𝑠 , 𝜏 ) = 𝑈 ( 𝑡 , 𝜏 ) , ∀ 𝑡 ≥ 𝑠 ≥ 𝜏 , 𝜏 ∈ ℝ , 𝑈 ( 𝜏 , 𝜏 ) = 𝐼 𝑑 , 𝜏 ∈ ℝ . ( 3 . 2 )

By ℬ ( 𝐸 ) we denote the collection of the bounded sets of 𝐸 . We consider a family of processes { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } depending on a parameter 𝜎 ∈ Σ . The parameter 𝜎 is said to be the symbol of the process { 𝑈 𝜎 ( 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 { 𝑈 𝜎 ( 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 𝜎 ∈ Σ 𝜏 ∈ 𝑅 𝑡 ≥ 𝜏 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝐵 ∈ ℬ ( 𝐸 ) . ( 3 . 3 )

A set 𝐵 0 ⊂ 𝐸 is said to be uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) absorbing for the family of processes { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Σ if for any 𝜏 ∈ 𝑅 and every 𝐵 ∈ ℬ ( 𝐸 ) there exists 𝑡 0 = 𝑡 0 ( 𝜏 , 𝐵 ) ≥ 𝜏 such that ⋃ 𝜎 ∈ Σ 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝐵 ⊆ 𝐵 0 for all 𝑡 ≥ 𝑡 0 .

A set 𝑃 ⊂ 𝐸 is said to be uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) attracting for the family of processes { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Σ if for an arbitrary fixed 𝜏 ∈ 𝑅 , l i m 𝑡 → + ∞ s u p 𝜎 ∈ Σ d i s t 𝐸 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝐵 , 𝑃 = 0 . ( 3 . 4 )

A family of processes possessing a compact uniformly absorbing set is called uniformly compact and a family of processes possessing a compact uniformly attracting set is called uniformly asymptotically compact.Definition 3.2. A closed set 𝒜 Σ ⊂ 𝐸 is said to be the uniform ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) attractor of the family of processes { 𝑈 𝜎 ( 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 { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Σ ∶ 𝒜 Σ ⊆ 𝒜 .

A family of processes { 𝑈 𝜎 ( 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 ) ↦ 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝑢 is continuous from 𝐸 × Σ into 𝐸 .

A curve 𝑢 ( 𝑠 ) , 𝑠 ∈ ℝ is said to be a complete trajectory of the process { 𝑈 ( 𝑡 , 𝜏 ) } if 𝑈 ( 𝑡 , 𝜏 ) 𝑢 ( 𝜏 ) = 𝑢 ( 𝑡 ) , ∀ 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ . ( 3 . 5 )

The kernel 𝒦 of the process { 𝑈 ( 𝑡 , 𝜏 ) } consists of all bounded complete trajectories of the process { 𝑈 ( 𝑡 , 𝜏 ) } : 𝒦 = 𝑢 ( ⋅ ) ∣ 𝑢 ( ⋅ ) s a t i s fi e s ( 3 . 6 ) , ‖ 𝑢 ( 𝑠 ) ‖ 𝐸 ≤ 𝑀 𝑢 f o r 𝑠 ∈ ℝ . ( 3 . 6 )

The set 𝒦 ( 𝑠 ) = { 𝑢 ( 𝑠 ) ∣ 𝑢 ( ⋅ ) ∈ 𝒦 } ⊆ 𝐸 ( 3 . 7 ) is said to be the kernel section at time 𝑡 = 𝑠 , 𝑠 ∈ ℝ .

For convenience, let 𝐵 𝑡 = ⋃ 𝜎 ∈ Σ ⋃ 𝑠 ≥ 𝑡 𝑈 𝜎 ( 𝑠 , 𝑡 ) 𝐵 , the closure 𝐵 of the set 𝐵 and ℝ 𝜏 = { 𝑡 ∈ ℝ ∣ 𝑡 ≥ 𝜏 } . Define the uniform ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) 𝜔 -limit set 𝜔 𝜏 , Σ ( 𝐵 ) of 𝐵 by 𝜔 𝜏 , Σ ⋂ ( 𝐵 ) = 𝑡 ≥ 𝜏 𝐵 𝑡 which can be characterized, analogously to that for semigroups, the following: 𝑦 ∈ 𝜔 𝜏 , Σ 𝑥 ( 𝐵 ) ⟺ t h e r e a r e s e q u e n c e s 𝑛 𝜎 ⊂ 𝐵 , 𝑛 𝑡 ⊂ Σ , 𝑛 ⊂ ℝ 𝜏 s u c h t h a t 𝑡 𝑛 ⟶ + ∞ , 𝑈 𝜎 𝑛 ( 𝑡 𝑛 , 𝜏 ) 𝑥 𝑛 ⟶ 𝑦 ( 𝑛 ⟶ ∞ ) . ( 3 . 8 )

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]).

Let 𝐵 ∈ ℬ ( 𝐸 ) . Its Kuratowski measure of noncompactness 𝜅 ( 𝐵 ) is defined by 𝜅 ( 𝐵 ) = i n f { 𝛿 > 0 ∣ 𝐵 a d m i t s a fi n i t e c o v e r i n g b y s e t s o f d i a m e t e r ≤ 𝛿 } . ( 3 . 9 ) Definition 3.3. A family of processes { 𝑈 𝜎 ( 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 𝐵 𝑡 is bounded for every 𝑡 and l i m 𝑡 → ∞ 𝜅 ( 𝐵 𝑡 ) = 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 { 𝑈 𝜎 ( 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 𝑡 0 = 𝑡 ( 𝜏 , 𝐵 , 𝜀 ) ≥ 𝜏 and a finite-dimensional subspace 𝐸 1 of 𝐸 such that
(i) ⋃ 𝑃 ( 𝜎 ∈ Σ ⋃ 𝑡 ≥ 𝑡 0 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝐵 ) is bounded; and(ii) ⋃ ‖ ( 𝐼 − 𝑃 ) ( 𝜎 ∈ Σ ⋃ 𝑡 ≥ 𝑡 0 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝑥 ) ‖ ≤ 𝜀 , ∀ 𝑥 ∈ 𝐵 , where 𝑃 ∶ 𝐸 → 𝐸 1 is a bounded projector.

Therefore we have the following results.Theorem 3.5. Let Σ be a metric space and let { 𝑇 ( 𝑡 ) } be a continuous invariant semigroup 𝑇 ( 𝑡 ) Σ = Σ on Σ . A family of processes { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Σ acting in 𝐸 is ( 𝐸 × Σ , 𝐸 ) -(weakly) continuous and possesses the compact uniform ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) attractor 𝐴 Σ satisfying 𝒜 Σ = 𝜔 0 , Σ 𝐵 0 = 𝜔 𝜏 , Σ 𝐵 0 = 𝜎 ∈ Σ 𝒦 𝜎 ( 0 ) , ∀ 𝜏 ∈ ℝ , ( 3 . 1 0 ) if it
(i)has a bounded uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) absorbing set 𝐵 0 , and(ii)satisfies uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝜎 ∈ Σ ) Condition ( C )
Moreover, if 𝐸 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 ) ∶ 𝐻 𝑔 → 𝐻 𝑔 ( 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 𝐻 𝑔 and 𝑉 𝑔 .

To this end, we first state some the following results of existence and uniqueness of solutions of (2.14).Proposition 4.1. Let 𝑓 ∈ 𝑉 be given. Then for every 𝑢 𝜏 ∈ 𝐻 𝑔 there exists a unique solution 𝑢 = 𝑢 ( 𝑡 ) on [ 0 , ∞ ) of (2.14), satisfying 𝑢 ( 𝜏 ) = 𝑢 𝜏 . Moreover,one has 𝑢 ( 𝑡 ) ∈ 𝐶 0 , 𝑇 ; 𝐻 𝑔 ∩ 𝐿 2 0 , 𝑇 ; 𝑉 𝑔 , ∀ 𝑇 > 0 . ( 4 . 1 ) Finally, if 𝑢 𝜏 ∈ 𝑉 𝑔 , then 𝑢 ( 𝑡 ) ∈ 𝐶 0 , 𝑇 ; 𝑉 𝑔 ∩ 𝐿 2 𝐴 0 , 𝑇 ; 𝐷 𝑔 , ∀ 𝑇 > 0 . ( 4 . 2 )

Proof. The Proof of Proposition 4.1 is similar to autonomous in [1, 15].

Now we will write (2.14) in the operator form 𝜕 𝑡 𝑢 = 𝐴 𝜎 ( 𝑡 ) ( 𝑢 ) , 𝑢 | 𝑡 = 𝜏 = 𝑢 𝜏 , ( 4 . 3 ) where 𝜎 ( 𝑠 ) = 𝑓 ( 𝑥 , 𝑠 ) is the symbol of (4.3). Thus, if 𝑢 𝜏 ∈ 𝐻 𝑔 , then problem (4.3) has a unique solution 𝑢 ( 𝑡 ) ∈ 𝐶 ( [ 0 , 𝑇 ] ; 𝐻 𝑔 ) ∩ 𝐿 2 ( [ 0 , 𝑇 ] ; 𝑉 𝑔 ) . This implies that the process { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } given by the formula 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝑢 𝜏 = 𝑢 ( 𝑡 ) is defined in 𝐻 𝑔 .

Now recall the following facts that can be found in [13].Definition 4.2. A function 𝜑 ∈ 𝐿 2 l o c ( ℝ ; 𝐸 ) is said to be normal if for any 𝜀 > 0 , there exists 𝜂 > 0 such that s u p 𝑡 ∈ ℝ 𝑡 𝑡 + 𝜂 ‖ 𝜑 ( 𝑠 ) ‖ 2 𝐸 𝑑 𝑠 ≤ 𝜀 . ( 4 . 4 ) Remark 4.3. Obviously, 𝐿 2 𝑛 ( ℝ ; 𝐸 ) ⊂ 𝐿 2 𝑏 ( ℝ ; 𝐸 ) . Denote by 𝐿 2 𝑐 ( ℝ ; 𝐸 ) the class of translation compact functions 𝑓 ( 𝑠 ) , 𝑠 ∈ ℝ , whose family of ℋ ( 𝑓 ) is precompact in 𝐿 2 l o c ( ℝ ; 𝐸 ) . It is proved in [13] that 𝐿 2 𝑛 ( ℝ ; 𝐸 ) and 𝐿 2 𝑐 ( ℝ ; 𝐸 ) are closed subspaces of 𝐿 2 𝑏 ( ℝ ; 𝐸 ) , but the latter is a proper subset of the former (for further details see [13]).

We now define the symbol space ℋ ( 𝜎 0 ) for (4.3). Let a fixed symbol 𝜎 0 ( 𝑠 ) = 𝑓 0 ( 𝑠 ) = 𝑓 0 ( ⋅ , 𝑠 ) be normal functions in 𝐿 2 l o c ( ℝ ; 𝐸 ) ; that is, the family of translation { 𝑓 0 ( 𝑠 + ℎ ) , ℎ ∈ ℝ } forms a normal function set in 𝐿 2 l o c ( [ 𝑇 1 , 𝑇 2 ] ; 𝐸 ) , where [ 𝑇 1 , 𝑇 2 ] is an arbitrary interval of the time axis ℝ . Therefore ℋ 𝜎 0 𝑓 = ℋ 0 = 𝑓 0 ( 𝑥 , 𝑠 + ℎ ) ∣ ℎ ∈ ℝ 𝐿 2 l o c ( ℝ ; 𝐸 ) . ( 4 . 5 )

Now, for any 𝑓 ( 𝑥 , 𝑡 ) ∈ ℋ ( 𝑓 0 ) , the problem (4.3) with 𝑓 instead of 𝑓 0 possesses a corresponding process { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } acting on 𝑉 𝑔 . As is proved in [10], the family { 𝑈 𝑓 ( 𝑡 , 𝜏 ) ∣ 𝑓 ∈ ℋ ( 𝑓 0 ) } of processes is ( 𝑉 𝑔 × ℋ ( 𝑓 0 ) ; 𝑉 𝑔 ) -continuous.

Let 𝒦 𝑓 = 𝑢 𝑓 ( 𝑥 , 𝑡 ) f o r 𝑡 ∈ ℝ ∣ 𝑢 𝑓 ‖ ‖ 𝑢 ( 𝑥 , 𝑡 ) i s s o l u t i o n o f ( 4 . 3 ) s a t i s f y i n g 𝑓 ‖ ‖ ( ⋅ , 𝑡 ) 𝐻 ≤ 𝑀 𝑓 ∀ 𝑡 ∈ ℝ ( 4 . 6 ) be the so-called kernel of the process { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } .Proposition 4.4. The process { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } ∶ 𝐻 𝑔 → 𝐻 𝑔 ( 𝑉 𝑔 ) associated with the (4.3) possesses absorbing sets ℬ 0 = 𝑢 ∈ 𝐻 𝑔 ∣ | 𝑢 | 𝑔 ≤ 𝜌 0 , ℬ 1 = 𝑢 ∈ 𝑉 𝑔 ∣ ‖ 𝑢 ‖ 𝑔 ≤ 𝜌 1 ( 4 . 7 ) which absorb all bounded sets of 𝐻 𝑔 . Moreover ℬ 0 and ℬ 1 absorb all bounded sets of 𝐻 𝑔 and 𝑉 𝑔 in the norms of 𝐻 𝑔 and 𝑉 𝑔 , respectively.

Proof. The proof of Proposition 4.4 is similar to autonomous g-Navier-Stokes equation. We can obtain absorbing sets in 𝐻 𝑔 and 𝑉 𝑔 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. If 𝑓 0 ( 𝑥 , 𝑠 ) is normal function in 𝐿 2 l o c ( ℝ ; 𝑉 𝑔 ) , then the processes { 𝑈 𝑓 0 ( 𝑡 , 𝜏 ) } corresponding to problem (2.14) possess compact uniform ( 𝑤 . 𝑟 . 𝑡 . 𝜏 ∈ ℝ ) attractor 𝔄 0 in 𝐻 𝑔 which coincides with the uniform ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) attractor 𝔄 ℋ ( 𝑓 0 ) of the family of processes { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } , 𝑓 ∈ ℋ ( 𝑓 0 ) : 𝔄 0 = 𝔄 ℋ ( 𝑓 0 ) = 𝜔 0 , ℋ ( 𝑓 0 ) ℬ 0 = 𝑓 ∈ ℋ ( 𝑓 0 ) 𝒦 𝑓 ( 0 ) , ( 4 . 8 ) where ℬ 0 is the uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ 𝑔 ( 𝑓 0 ) ) absorbing set in 𝐻 𝑔 and 𝒦 𝑓 is the kernel of the process { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } . Furthermore, the kernel 𝒦 𝑓 is nonempty for all 𝑓 ∈ ℋ ( 𝑓 0 ) .

Proof. As in the previous section, for fixed 𝑁 , let 𝐻 1 be the subspace spanned by 𝑤 1 ; … ; 𝑤 𝑁 , and 𝐻 2 the orthogonal complement of 𝐻 1 in 𝐻 𝑔 . We write 𝑢 = 𝑢 1 + 𝑢 2 ; 𝑢 1 ∈ 𝐻 1 , 𝑢 2 ∈ 𝐻 2 f o r a n y 𝑢 ∈ 𝐻 𝑔 . ( 4 . 9 )
Now, we only have to verify Condition ( C ) . Namely, we need to estimate | 𝑢 2 ( 𝑡 ) | 2 , where 𝑢 ( 𝑡 ) = 𝑢 1 ( 𝑡 ) + 𝑢 2 ( 𝑡 ) is a solution of (2.14) given in Proposition 4.1.
Multiplying (2.14) by 𝑢 2 , we have 𝑑 𝑢 𝑑 𝑡 , 𝑢 2 𝑔 + 𝜈 𝐴 𝑔 𝑢 , 𝑢 2 𝑔 + 𝐵 ( 𝑢 , 𝑢 ) , 𝑢 2 𝑔 = 𝑓 , 𝑢 2 𝑔 − 𝑅 𝑢 , 𝑢 2 𝑔 . ( 4 . 1 0 ) It follows that 1 2 𝑑 | | 𝑢 𝑑 𝑡 2 | | 2 𝑔 ‖ ‖ 𝑢 + 𝜈 𝑔 ‖ ‖ 2 𝑔 ≤ | | 𝐵 ( 𝑢 , 𝑢 ) , 𝑢 2 𝑔 | | + | | 𝑓 , 𝑢 2 𝑔 | | + 𝑅 𝑢 , 𝑢 2 𝑔 . ( 4 . 1 1 ) Since 𝑏 𝑔 satisfies the following inequality (see [15]): | | 𝑏 𝑔 | | ( 𝑢 , 𝑣 , 𝑤 ) ≤ 𝑐 | 𝑢 | 𝑔 1 / 2 ‖ 𝑢 ‖ 𝑔 1 / 2 ‖ 𝑣 ‖ 𝑔 | 𝑤 | 𝑔 1 / 2 ‖ 𝑤 ‖ 𝑔 1 / 2 , ∀ 𝑢 , 𝑣 , 𝑤 ∈ 𝑉 𝑔 , ( 4 . 1 2 ) thus, | | 𝐵 ( 𝑢 , 𝑢 ) , 𝑢 2 𝑔 | | ≤ 𝑐 | 𝑢 | 𝑔 1 / 2 ‖ 𝑢 ‖ 𝑔 3 / 2 | | 𝑢 2 | | 𝑔 1 / 2 ‖ ‖ 𝑢 2 ‖ ‖ 𝑔 1 / 2 ≤ 𝑐 𝜆 𝑚 + 1 | 𝑢 | 𝑔 1 / 2 ‖ 𝑢 ‖ 𝑔 3 / 2 ‖ ‖ 𝑢 2 ‖ ‖ 𝑔 ≤ 𝜈 6 ‖ ‖ 𝑢 2 ‖ ‖ 2 𝑔 + 𝑐 𝜌 0 𝜌 3 1 . ( 4 . 1 3 ) Next, the Cauchy inequality, | | 𝑅 𝑢 , 𝑢 2 𝑔 | | = | | | 𝜈 𝑔 ( ∇ 𝑔 ⋅ ∇ ) 𝑢 , 𝑢 2 𝑔 | | | ≤ 𝜈 𝑚 0 | ∇ 𝑔 | ∞ ‖ 𝑢 ‖ 𝑔 | | 𝑢 2 | | 𝑔 ≤ 𝜈 6 ‖ ‖ 𝑢 2 ‖ ‖ 2 𝑔 + 3 𝜈 𝜌 2 1 | ∇ 𝑔 | 2 ∞ 2 𝑚 2 0 𝜆 𝑔 𝜆 𝑚 + 1 . ( 4 . 1 4 ) Finally, we have | | 𝑓 , 𝑢 2 𝑔 | | ≤ | 𝑓 | 𝑉 ′ 𝑔 ‖ ‖ 𝑢 2 ‖ ‖ ≤ 𝜈 6 ‖ ‖ 𝑢 2 ‖ ‖ 2 𝑔 + 3 2 𝜈 | 𝑓 | 2 𝑉 ′ 𝑔 . ( 4 . 1 5 ) Putting (4.13)–(4.15) together, there exist constant 𝑀 1 = 𝑀 1 ( 𝑚 0 , | ∇ 𝑔 | ∞ , 𝜌 0 , 𝜌 1 ) such that 1 2 𝑑 | | 𝑢 𝑑 𝑡 2 | | 2 𝑔 + 1 2 𝜈 ‖ ‖ 𝑢 2 ‖ ‖ 2 𝑔 ≤ 3 | 𝑓 | 𝑉 ′ 𝑔 2 𝜈 + 𝑀 1 . ( 4 . 1 6 ) Therefore, we deduce that 𝑑 | | 𝑢 𝑑 𝑡 2 | | 2 𝑔 + 𝜈 𝜆 𝑚 + 1 | | 𝑢 2 | | 2 2 ≤ 2 𝑀 1 + 3 𝜈 | 𝑓 | 2 𝑉 ′ 𝑔 . ( 4 . 1 7 ) Here 𝑀 1 depends on 𝜆 𝑚 + 1 , is not increasing as 𝜆 𝑚 + 1 increasing.
By the Gronwall inequality, the above inequality implies | | 𝑢 2 | | ( 𝑡 ) 2 𝑔 ≤ | | 𝑢 2 𝑡 0 | | + 1 2 2 𝑒 − 𝜈 𝜆 𝑚 + 1 ( 𝑡 − ( 𝑡 0 + 1 ) ) + 2 𝑀 1 𝜈 𝜆 𝑚 + 1 + 3 𝜈 ∫ 𝑡 𝑡 0 + 1 𝑒 − 𝜈 𝜆 𝑚 + 1 ( 𝑡 − 𝑠 ) | 𝑓 | 2 𝑉 ′ 𝑑 𝑠 . ( 4 . 1 8 ) Applying (4.4) for any 𝜀 3 𝜈 𝑡 𝑡 0 + 1 𝑒 − 𝜈 𝜆 𝑚 + 1 ( 𝑡 − 𝑠 ) | 𝑓 | 2 𝑉 ′ 𝑔 𝜀 𝑑 𝑠 < 3 . ( 4 . 1 9 ) Using (2.9) and let 𝑡 1 = 𝑡 0 + 1 + ( 1 / 𝜈 𝜆 𝑚 + 1 ) l n ( 3 𝜌 2 0 / 𝜀 ) , then 𝑡 ≥ 𝑡 1 implies 2 𝑀 𝜈 𝜆 𝑚 + 1 < 𝜀 3 ; | | 𝑢 2 ( 𝑡 0 | | + 1 ) 2 2 𝑒 − 𝜈 𝜆 𝑚 + 1 ( 𝑡 − ( 𝑡 0 + 1 ) ) ≤ 𝜌 2 0 𝑒 − 𝜈 𝜆 𝑚 + 1 ( 𝑡 − ( 𝑡 0 + 1 ) ) / 2 < 𝜀 3 . ( 4 . 2 0 ) Therefore, we deduce from (4.18) that | | 𝑢 2 | | 2 2 ≤ 𝜀 , ∀ 𝑡 ≥ 𝑡 1 𝑓 , 𝑓 ∈ ℋ 0 , ( 4 . 2 1 ) which indicates { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } , 𝑓 ∈ ℋ ( 𝑓 0 ) satisfying uniform ( 𝑤 . 𝑡 . 𝑟 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) Condition ( C ) in 𝐻 𝑔 . Applying Theorem 3.5 the proof is complete.

Theorem 4.6. If 𝑓 0 ( 𝑥 , 𝑠 ) is normal function in 𝐿 2 𝑙 𝑜 𝑐 ( ℝ ; 𝐻 𝑔 ) , then the processes { 𝑈 𝑓 0 ( 𝑡 , 𝜏 ) } corresponding to problem (2.14) possesses compact uniform ( 𝑤 . 𝑟 . 𝑡 . 𝜏 ∈ ℝ ) attractor 𝔄 1 in 𝑉 𝑔 which coincides with the uniform ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) attractor 𝔄 ℋ ( 𝑓 0 ) of the family of processes { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } , 𝑓 ∈ ℋ ( 𝑓 0 ) : 𝔄 1 = 𝔄 ℋ ( 𝑓 0 ) = 𝜔 0 , ℋ ( 𝑓 0 ) ℬ 1 = 𝑓 ∈ ℋ ( 𝑓 0 ) 𝒦 𝑓 ( 0 ) , ( 4 . 2 2 ) where ℬ 1 is the uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) absorbing set in 𝑉 𝑔 and 𝒦 𝑓 is the kernel of the process { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } . Furthermore, the kernel 𝒦 𝑓 is nonempty for all 𝑓 ∈ ℋ ( 𝑓 0 ) .

Proof. Using Proposition 4.4, we have the family of processes { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } , 𝑓 ∈ ℋ ( 𝑓 0 ) corresponding to (4.3) possesses the uniformly ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) absorbing set in 𝑉 𝑔 .
Now we prove the existence of compact uniform ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) attractor in 𝑉 𝑔 by applying the method established in Section 3, that is, we testify that the family of processes { 𝑈 𝑓 ( 𝑡 , 𝜏 ) } , 𝑓 ∈ ℋ ( 𝑓 0 ) corresponding to (4.3) satisfies uniform ( 𝑤 . 𝑟 . 𝑡 . 𝑓 ∈ ℋ ( 𝑓 0 ) ) Condition ( C ) .
Multiplying (2.14) by 𝐴 𝑔 𝑢 2 ( 𝑡 ) , we have 𝑑 𝑣 𝑑 𝑡 , 𝐴 𝑔 𝑢 2 + 𝜈 𝐴 𝑔 𝑢 , 𝐴 𝑔 𝑢 2 + 𝐵 𝑔 ( 𝑢 , 𝑢 ) , 𝐴 𝑔 𝑢 2 𝑔 = 𝑓 , 𝐴 𝑔 𝑢 2 − 𝑅 𝑢 , 𝐴 𝑔 𝑢 2 𝑔 . ( 4 . 2 3 ) It follows that 1 2 𝑑 ‖ ‖ 𝑢 𝑑 𝑡 2 ‖ ‖ 2 𝑔 | | 𝐴 + 𝜈 𝑔 𝑢 2 | | 2 𝑔 ≤ | | 𝐵 𝑔 ( 𝑢 , 𝑢