Upper Semicontinuity of Trajectory Attractors for 3D Incompressible Navier–Stokes Equation

In this paper, we first establish the existence of a trajectory attractor for the Navier–Stokes–Voight (NSV) equation and then prove upper semicontinuity of trajectory attractors of 3D incompressible Navier–Stokes equation when 3D NSV equation is considered as a perturbative equation of 3D incompressible Navier–Stokes equation.


Introduction
It is very significant and difficult to study the uniqueness and asymptotic behaviour of the evolution equations and their long-time behavior of solutions can be described by attractors. But, we all know that the uniquences of weak solutions for the 3D Navier-Stokes equation is still open until now. To this end, Chepyzhov and Vishik in [6] proposed the trajectory attractor theory, which can describe the long-time behavior of solutions whose uniqueness is not known. We study the relationship between the following 3D Navier-Stokes equation: B Yuming Qin yuming_qin@hotmail.com Xiuqing Wang dqwangdq@foxmail.com 1 u t − ν u + (u · ∇)u + ∇ p = g(x), x ∈ , t ≥ 0, ∇ · u = 0, (1.1) and the 3D Navier-Stokes-Voight equation: (1.2) subjecting to the noslip boundary condition u| ∂ = 0, (1.3) and the initial conditions where ⊆ R 3 is a smooth bounded domain, u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) is the velocity vector, p(x, t) is the pressure function, and g = (g 1 (x), g 2 (x), g 3 (x)) is an external term, ν is a positive viscous constant.
There has been many literature on the trajectory attractors [3,7,[14][15][16][17]19]. First, Chepyzhow et al. [4,5] study the following 3D Navier-Stokes-α model, (1.5) and showed that the trajectory attractor of the Navier-Stokes-α model converges to the trajectory attractor of the 3D Navier-Stokes system as α → 0 + . The difference between [4] and [5] is that they chose different trajectory spaces. There are also a lot of literature on Navier-Stokes-α model [1,2,9,10,18]. Zelati and Gal [21] proved the existence of global and exponential attractors, then they prove the convergence of the (strong) global attractor of the 3D Navier-Stokes-Voight model to the (weak) global attractor of the 3D Navier-Stokes equation, i.e., they prove the convergence of the (strong) global attractor of the 3D Navier-Stokes-Voight model to the trajectory attractor of the 3D Navier-Stokes equation. But, they did not prove upper semicontinuity of trajectory attractors. In addition, trajectory attractor was also obtained for the fluid dynamics systems such as the MHD system [8], liquid crystal flow [12], binary fluid mixtures [13], and Cahn-Hilliard-Navier-Stokes equations [11]. These 3D systems all have trajectory attractors but not necessarily have global attractors. Zhao and Zhou [22] first proposed the concept of pullback trajectory attractors, and proved the existence of pullback trajectory attractors to 3D incompressible non-Newtonian fluid.

Preliminaries
In this section, we will first introduce some Sobolev spaces which will be used, and their dual spaces. It is also necessary to introduce some notations related to trajectory attractors. At last, we give a result on the trajectory attractors of 3D incompressible Navier-Stokes equation.

Trajectory Space
We first introduce some functional spaces and operators. Set where (·, ·), ·, · denote the inner products in H and in V respectively. For any v ∈ V , the expression u, v means the value of the functional v on a vector u ∈ V . In the sequel, we identify H with its dual and we have the following inclusions, We set H η = (− ) η 2 H and use H −η denote the dual space of H η . Clearly, the embedding H → H −η is compact.
The operator P : [L 2 ( )] 3 → H denotes the orthogonal projector, and A is the Stokes operator with the domain D(A) = (H 2 ( )) 3 ∩ V , the operator A is self-adjoint and positive, and We recall the Poincaré inequality, where λ k (k = 1, 2, . . .) is the eigenvalue of Stokes operator A, and λ k satisfies We consider the spaces F b + defined by with norm We know that F b + with its norm · F b + is a Banach space. Similarly, we define the space and we define a topology loc + on F loc + , then we consider a topology sequence {z n (·)} ⊂ F loc + , z n → z in the topology loc + , i.e., Let {T (h) |≥ 0} denote the time translation operator acting on the trajectory space,

Some Useful Lemmas
Lemma 2.1 [20] Let y(t) ∈ C 1 [t 0 , t 1 ], y ≥ 0 and the following inequality In particular, if h(t) = C, then with the norm We assume that p 1 ≥ 1 and p 0 ≥ 1. Then the following embedding is compact: Moreover, when p 1 = ∞, then the following embedding is compact:

Lemma 2.3 [4]
For any f (t) ∈ D(A) , assume the operator A is self-adjoint and positive, then the following inequality holds:

Lemma 2.4 [4]
For any f (t) ∈ L 2 (0, M; D(A) ), assume the operator A is selfadjoint and positive, then there holds that

Trajectory Attractors of 3D Incompressible Navier-Stokes Equation
In this subsection, we will present the theory of trajectory attractors of 3D incompressible Navier-Stokes equation, which can be found in [6]. With the orthogonal projector P, 3D incompressible Navier-Stokes system can be rewritten as  Proof We refer to [6] for its proof.

Lemma 3.3 If g ∈ H and u(t) is a weak solution of problem
where C 1 , R 1 only depend on ν, λ 1 , |g|, but are independent of α.
Proof First, using the Poincaré inequality (2.3), we have and Then using (3.5), we get Second, we derive from inequality (2.2) that

Now by applying Lemma 2.3 and (3.17)
, the proof is thus complete.
Proof The proof of (3.18) is clearly established by Lemmas 3.

Proposition 3.3 The trajectory space
Proof We consider an arbitrary sequence {u n } ⊂ K + α , such that u n → u in the topology loc + as n → ∞, we need to prove u ∈ K + α . Due to u n → u in the topology loc + , by the definition of loc + , Since u n k (t) ∈ K + α , we have Taking limit of (3.22) and using (3.19)-(3.21), we have then u is a weak solution of problem (3.1), i.e., u ∈ K + α .

Proposition 3.4 The ball B
(i) By Proposition 3.4, we know that O α is compact in the topology of loc + , and , obviously,ũ is a weak solution of problem (3.1), i.e.,ũ ∈ O α , then there exists a time t 2 > 0, such that as t ≥ t 2 ,

iii) By Proposition 3.4, we know that O α is an absorbing set of T (h). Then we prove
that O α is a trajectory attractor of problem (3.1).

Upper Semicontinuity of the Trajectory Attractors of 3D NS Equation
Theorem 4.1 Let a sequence {u n (t)} ⊂ K + α , α n → 0 + (n → ∞), and u n (t) → u(t) in the topology loc + as n → ∞. Then u(t) is a weak solution of the 3D Navier-Stokes equation such that u satisfies the inequality (2.6), i.e., u ∈ K + 0 , where K + 0 is the trajectory space of problem (2.4).
Proof Since u n (t) ∈ K + α , i.e., u n (t) is a weak solution of equation Next, we prove u satisfies the enery inequality (2.6). Since u n (t) → u(t) in the topology loc + as n → ∞, by the Aubin Theorem, there exists a subsequence u n which we still denote by u n such that Note that u n (t) satisfies the energy equality (4.10) Taking limit on both sides of equation (4.10), and using (4.11)-(4.14), we have Thus u ∈ K + 0 , and the proof is now complete.
the bounded sets of solutions of 3D Navier-Stokes-Voight equation (3.1). Then the following convergence holds: where O 0 is the trajectory attractor of 3D incompressible Navier-Stokes equation (2.4).
From the above argument, we have proved the following proposition. The following theorem concerns the upper semicontinuity of trajectory attractors of 3D Navier-Stokes equation when the regular term α 2 u t in NSV equation is considered as its a perturbative term. Denote by dist X (B 1 , B 2 ) the Hausdorff semidistance in space X between B 1 and B 2 , i.e., dist X (B 1 , B 2 ) = sup