The Uniform Attractors for the Nonhomogeneous 2 D Navier-Stokes Equations in Some Unbounded Domain

We consider the attractors for the two-dimensional nonautonomous Navier-Stokes equations in some unbounded domain Ω with nonhomogeneous boundary conditions. We apply the so-called uniformly ω-limit compact approach to nonhomogeneous Navier-Stokes equation as well as a method to verify it. Assuming f ∈ Lloc 0, T ;L2 Ω , which is translation compact and φ ∈ C1 b R ;H2 R1 × {±L} asymptotically almost periodic, we establish the existence of the uniform attractor inH1 Ω .


Introduction
We study the long-time behavior of a uniform flow past and infinitely long cylindrical obstacle.We will assume that the flow is uniform in the direction x of the axis of the cylindrical obstacle and the flow approaches U ∞ e x farther away from the obstacle.In this respect, we can consider a two-dimensional flow and assume that the obstacles are a disk with radius r more general obstacle can be treated in exactly the same way .A further simplification is to observe that since the flow is uniform at infinity, we may assume that the flow is in an infinitely long channel with width 2L L r and the obstacle is located at the center, while the flow at the boundary of the channel is almost the uniform flow at infinity.More precisely, we assume that the flow is governed by the following Navier-Stokes equations in Ω R 1 × −L, L \ B r 0 L r : where f ∈ L 2 loc 0, T ; L 2 Ω is translation compact and ϕ − U ∞ e x ∈ C 1 b R ; H 2 R 1 × {±L} is called asymptotically almost periodic, that is, for any ε ≥ 0, there is a number l l ε such that for each interval α, α l , α ∈ R , there exists a point τ ∈ α, α l such that μ ϕ s τ , ϕ s ≤ ε, ∀s ≥ l 1.2 the number τ is called the ε-period of the function ϕ see 1 , where μ is the metric in The basic idea of our construction is motivated by the works of 2 where an attractor A in space L 2 Ω to which all solutions approach as t → ∞ was shown.In this paper, we verify the existence of uniform attractors in space H 1 by using a noncompactness measure method.
We assume that the following Poincaré inequality holds: Throughout this paper, we introduce the spaces • , the norm in V, •, • the inner product in H or the dual product between V and V , •, • the inner product in V.
Here V is the dual of V V 1 .The constants C c are considered in a generic sense, which is independent of the physical parameters in the equations and may be different from line to line and even in the same line.

Setting of the problem
The first simplification is to introduce the new variables

2.1
Then u satisfies the equations

Delin Wu 3
We observe that ϕ ∈ C 1 b R ; H 2 R 1 × {±L} and is asymptotically almost periodic.Note that u and ϕ decay nicely near infinity.However, the boundary condition is not homogeneous, and thus we apply a modified Hopf's technique see 2-5 to homogenize the boundary condition.More specifically, we choose and we define, for ε < 1, we then define

2.5
Observe that ψ 1 ϕ at y ±L and ψ 2 −U ∞ e x at ∂B r .If we set

2.7
It is easy to check that for fixed ε, ν, U ∞ , r, and L, the right-hand side of 2.7 G ∈ C b R ; L 2 Ω see 2 is asymptotically almost periodic.
Boundary Value Problems

Abstract results
Let E be a Banach space, and let a two-parameter family of mappings {U t, τ The kernel K of the process {U t, τ } consists of all bounded complete trajectories of the process {U t, τ }: The set is said to be the kernel section at time t s, s ∈ R.
We consider the two projectors Π 1 and Π 2 from E × Σ onto E and Σ, respectively, Now we recall the basic results in 1 .
Theorem 3.2.Let a family of processes {U σ t, τ }, σ ∈ Σ, acting in the space E be uniformly w.r.t.σ ∈ Σ asymptotically compact and E × Σ, E -continuous.Also let Σ be a compact-metric space and let {T t } be a continuous-invariant T t Σ Σ semigroup on Σ satisfying the translation identity Then the semigroup {S t } corresponding to the family of processes {U σ t, τ }, σ ∈ Σ, and acting on possesses the compact attractor A which is strictly invariant with respect to {S t } : S t A A for all t ≥ 0.Moreover, iii the global attractor satisfies 3.9 iv the uniform attractor satisfies Here K σ 0 is the section at t 0 of the kernel K σ of the process {U σ t, τ } with symbol σ ∈ Σ.
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 with, analogously to that for semigroups, the following: 3.11 We will characterize the existence of the uniform attractor for a family of processes satisfying 3.7 in terms of the concept of measure of noncompactness that was put forward first by Kuratowski.
Let B ∈ B E .The Kuratowski measure of noncompactness κ B is defined by We present now a method to verify the uniform w.r.t.σ ∈ Σ ω-limit compactness see 6, 7 .
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; ii where P : E → E 1 is a bounded projector.

Boundary Value Problems
Therefore we have the following results.
Theorem 3.5.Let Σ be a compact metric space and let {T t } be a continuous invariant semigroup T t Σ Σ on Σ satisfying the translation identity 3.7 .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 i has a bounded uniformly w.r.t.σ ∈ Σ absorbing set B 0 ; ii satisfies uniformly w.r.t.σ ∈ Σ condition C .
Moreover, if E is a uniformly convex Banach space then the converse is true.

The attractor of the nonhomogeneous Navier-Stokes equations
We say that v is a weak solution of 2.7 if in the distributional sense, where a v, w Av, w V , V .The well-posedness of 4.2 can be derived using a standard Faedo-Galerkin approach.It can be viewed as a family of semiprocesses on H with the symbol space Σ defined as endowed with the product norm of the supremum norm on C b R ; H 2 Ω for ψ and the supremum norm of C b R ; L 2 Ω for G or the norm of L 2 loc 0, T ; L 2 Ω for f .The symbol space Σ is a compact space by our assumptions on ϕ and f, and the explicit construction of ψ and F. For each

4.4
where A : V → V is the Stokes operator defined by and P is the Leray-Hopf projection from L 2 Ω onto H.We can also define on Σ the semigroup {T s } s≥0 given by T s σ t T s σ t σ t s , ∀t ≥ 0, ∀s ≥ 0, ∀σ ∈ Σ.Since the symbol space Σ is compact, the semigroup {T s } s≥0 is continuous and compact and in particular asymptotically compact.It is then obvious that this family of semiprocesses satisfies the translation invariance property.Now recall the following facts that can be found in 6 .
Lemma 4.1.Assume that f s ∈ L 2 c R; E is translation compact, then for any ε > 0, there exists η > 0 such that Since A −1 is a continuous-compact operator in H, by the classical spectral theorem, there exists a sequence {λ j } ∞ j 1 , Now we will write 4.4 in the operator form where σ s ∈ Σ is the symbol of 4.9 .Thus, if v τ ∈ H, then problem 4.9 has a unique solution v t ∈ C 0, T ; H ∩ L 2 0, T ; V .This implies that the process {U σ t, τ } given by the formula be the so-called kernel of the process {U σ t, τ }.
Proposition 4.2.The process {U σ t, τ } : V → V associated with 4.9 possesses absorbing sets which absorb all bounded sets of V in the norm of V .
Proof.Multiplying 4.4 by Av, we have

4.14
We have to estimate each term in the right-hand side of 4.14 .First, we recall some inequalities 8

4.15
By using Young's inequality, we have

4.17
Thanks to Hardy's inequality and 2.3 -2.5 , we have
The main results in this section are as follows.
Theorem 4.3.If f ∈ L 2 loc 0, T ; L 2 Ω is translation compact, then the processes {U σ t, τ } corresponding to problem 4.9 possess the compact uniform w.r.t.τ ∈ R attractor A 0 in V which coincides with the uniform w.r.t.σ ∈ Σ attractor A Σ of the family of processes {U σ t, τ }, σ ∈ Σ, where B is a uniformly w.r.t.σ ∈ Σ absorbing set in V and K σ is the kernel of the process {U σ t, τ }.Furthermore, the kernel K σ is nonempty for all σ ∈ Σ.
Proof.Using Proposition 4.2, the family of processes {U σ t, τ }, σ ∈ Σ, corresponding to 4.9 possesses the uniformly w.r.t.σ ∈ Σ absorbing set in V .Now we prove the existence of the compact uniform w.r.t.σ ∈ Σ attractor in V by applying the method established in Section 3, that is, we prove that the family of processes {U σ t, τ }, σ ∈ Σ, corresponding to 4.9 satisfies uniformly w.r.t.σ ∈ Σ condition C .
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.We write

4.25
To estimate B v, v , Av 2 , we recall some inequalities 10 4.26 from which we deduce that 4.27 and using 4.26

4.28
Expanding and using Young's inequality, together with the first one of 4.28 and the second one of 4.15 , we have ∂u ∂t − νΔu u•∇ u ∇p f, div u 0, Boundary Value Problems u ϕ on ∂Ω 1 {y ±L}, u 0 on ∂Ω 2 ∂B r , u U ∞ e x if x −→ ±∞, 1.1 H 2 for any v ∈ H,