Advertisement

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

  • Yuming QinEmail author
  • Xiuqing Wang
Open Access
Article
  • 204 Downloads

Abstract

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.

Keywords

Navier–Stokes–Voight equation Navier–Stokes equation Trajectory attractors 

Mathematics Subject Classification

35M13 35Q35 

1 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:
$$\begin{aligned} \left\{ \begin{array}{ll} u_{t} -\nu \triangle u+(u\cdot \nabla )u+\nabla p=g(x),~~x\in \Omega ,~~t\ge 0,\\ \nabla \cdot u=0, \end{array} \right. \end{aligned}$$
(1.1)
and the 3D Navier–Stokes–Voight equation:
$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}-\alpha ^{2}\triangle u_{t}-\nu \triangle u+(u\cdot \nabla )u+\nabla p=g(x),~~x\in \Omega ,~~t\ge 0,\\ \nabla \cdot u=0, \end{array} \right. \end{aligned}$$
(1.2)
subjecting to the noslip boundary condition
$$\begin{aligned} u|_{\partial \Omega }=0, \end{aligned}$$
(1.3)
and the initial conditions
$$\begin{aligned} u(x,0)=u_{0},~~x\in \Omega , \end{aligned}$$
(1.4)
where \(\Omega \subseteq {\mathbb {R}}^{3}\) 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(xt) is the pressure function, and \(g=(g^{1}(x),g^{2}(x),g^{3}(x))\) is an external term, \(\nu \) 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-\(\alpha \) model,
$$\begin{aligned} \left\{ \begin{array}{ll} v_{t} -\nu \triangle v+(u\cdot \nabla )v+\nabla P=g(x),~~x\in \Omega ,~~t\ge 0,\\ v=u-\alpha ^{2}\triangle u,~~\nabla \cdot u=0, \end{array} \right. \end{aligned}$$
(1.5)
and showed that the trajectory attractor of the Navier–Stokes-\(\alpha \) model converges to the trajectory attractor of the 3D Navier–Stokes system as \(\alpha \rightarrow 0^{+}\). The difference between [4] and [5] is that they chose different trajectory spaces.

There are also a lot of literature on Navier–Stokes-\(\alpha \) 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.

2 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.

2.1 Trajectory Space

We first introduce some functional spaces and operators. Set
$$\begin{aligned}&{\mathcal {V}}=\{\varphi (x)=(\varphi ^{1}(x),\varphi ^{2}(x),\varphi ^{3}(x))\in (C_{0}^{\infty }({\overline{\Omega }}))^{3},\quad \nabla \cdot \varphi =0,\quad \varphi |_{\partial \Omega }=0\};\\&H=\text {closure of } \mathcal {V~~ } \text {in } (L^{2}(\Omega ))^{3}~~ \text {with norm } |\cdot |,\quad H' ~\text {is the dual space of }H; \\&V=\text {closure of } \mathcal {V~~ } \text {in } (H^{1}(\Omega ))^{3}~~ \text {with norm } \Vert \cdot \Vert , \quad V' ~\text {is the dual space of }V, \end{aligned}$$
where \((\cdot ,\cdot )\), \(\langle \cdot ,\cdot \rangle \) denote the inner products in H and in V respectively. For any \(v\in V'\), the expression \(\langle u,v\rangle \) means the value of the functional v on a vector \(u\in V\). In the sequel, we identify H with its dual and we have the following inclusions,
$$\begin{aligned} D(A)\subset V\subset H=H'\subset V'\subset D(A)'. \end{aligned}$$
We set \(H^{\eta }=(-\triangle )^{\frac{\eta }{2}}H\) and use \(H^{-\eta }\) denote the dual space of \(H^{\eta }\). Clearly, the embedding \(H\hookrightarrow H^{-\eta }\) is compact.
The operator \(P:[L^{2}(\Omega )]^{3}\rightarrow H\) denotes the orthogonal projector, and A is the Stokes operator with the domain \(D(A)=(H^{2}(\Omega ))^{3}\cap V\), the operator A is self-adjoint and positive,
$$\begin{aligned} Au=-P\triangle u. \end{aligned}$$
The bilinear operator B(uu) on \(V\times V\rightarrow V'\) is defined as
$$\begin{aligned} \langle B(u,v),w\rangle =b(u,v,w), \end{aligned}$$
and the trilinear form \(b:V\times V\times V\),
$$\begin{aligned} b(u,v,w)=\int _{\Omega }(u\cdot \nabla )v\cdot wdx=\sum \limits _{i,j=1}^{3}\int _{\Omega }u_{i}\frac{\partial v_{j}}{\partial x_{i}}w_{j}dx, \end{aligned}$$
satisfies
$$\begin{aligned} b(u,v,w)=-b(u,w,v),\quad b(u,v,v)=0, \quad u,v,w \in V, \end{aligned}$$
(2.1)
and
$$\begin{aligned} |\langle B(u,v),w\rangle | \le c|u|_{3}|v||Aw| , \quad \Vert B(u,v)\Vert _{D(A)'}\le c|u|_{3}|v|. \end{aligned}$$
(2.2)
We recall the Poincar\(\acute{e}\) inequality,
$$\begin{aligned} |u |\le \frac{1}{\sqrt{\lambda _{1}}}\Vert u\Vert , ~~~~\forall u\in V, \end{aligned}$$
(2.3)
where \(\lambda _{k}~(k=1,2,\ldots )\) is the eigenvalue of Stokes operator A, and \(\lambda _{k}\) satisfies
$$\begin{aligned} 0<\lambda _{1}<\lambda _{2}< \cdots <\lambda _{k},\quad \lambda _{k}\rightarrow +\infty ,~k\rightarrow \infty . \end{aligned}$$
We consider the spaces \({\mathcal {F}}^{b}_{+}\) defined by
$$\begin{aligned} {\mathcal {F}}^{b}_{+}=\{z(\cdot )\mid z(\cdot )\in L^{2}_{b}({\mathbb {R}}_{+};V)\cap L^{\infty }({\mathbb {R}}_{+};H), \partial _{t}z(\cdot )\in L^{2}_{b}({\mathbb {R}}_{+};D(A)') \}, \end{aligned}$$
with norm
$$\begin{aligned} \Vert z\Vert _{{\mathcal {F}}^{b}_{+}}=\Vert z\Vert _{L^{2}_{b}({\mathbb {R}}_{+};V)}+\Vert z\Vert _{L^{\infty }({\mathbb {R}}_{+};H)}+\Vert \partial _{t}z\Vert _{L^{2}_{b}({\mathbb {R}}_{+};D(A)')}, \end{aligned}$$
where
$$\begin{aligned}&\Vert z\Vert ^{2}_{L^{2}_{b}({\mathbb {R}}_{+};V)}=\sup _{t\ge 0}\int _{t}^{t+1}\Vert z(s)\Vert ^{2}ds, \quad \Vert z\Vert _{L^{\infty }({\mathbb {R}}_{+};H)}=\text {ess}\sup _{t\ge 0}|z(t)| ,\\&\Vert \partial _{t}z\Vert ^{2}_{L^{2}_{b}({\mathbb {R}}_{+};D(A)')}=\sup _{t\ge 0}\int _{t}^{t+1}\Vert \partial _{t}z(s)\Vert ^{2}_{D(A)'}ds. \end{aligned}$$
We know that \({\mathcal {F}}^{b}_{+}\) with its norm \(\Vert \cdot \Vert _{{\mathcal {F}}^{b}_{+}}\) is a Banach space. Similarly, we define the space
$$\begin{aligned} {\mathcal {F}}^{loc}_{+}=\{z(\cdot )\mid z(\cdot )\in L^{2}_{loc}({\mathbb {R}}_{+};V)\cap L^{\infty }_{loc}({\mathbb {R}}_{+};H), \partial _{t}z(\cdot )\in L^{2}_{loc}({\mathbb {R}}_{+};D(A)') \}, \end{aligned}$$
and we define a topology \( \Theta ^{loc}_{+}\) on \({\mathcal {F}}^{loc}_{+}\), then we consider a topology sequence \(\{z_{n}(\cdot )\}\subset {\mathcal {F}}^{loc}_{+}\), \(z_{n}\rightarrow z\) in the topology \( \Theta ^{loc}_{+}\), i.e.,
$$\begin{aligned}&z_{n}\rightharpoondown z, \text { weakly in } L^{2}(0,M; V), \text { and weakly-star in } L^{\infty }(0,M;H),n\rightarrow \infty , \\&\partial _{t}z_{n}\rightharpoondown \partial _{t}z, \text { weakly in } L^{2}(0,M;D(A)'),n\rightarrow \infty . \end{aligned}$$
Let \(\{T(h)\mid \ge 0\}\) denote the time translation operator acting on the trajectory space,
$$\begin{aligned} T(h)z(t)=z(t+h). \end{aligned}$$

2.2 Some Useful Lemmas

Lemma 2.1

[20] Let \(y(t)\in C^{1}[t_{0},t_{1}],~y\ge 0\) and the following inequality
$$\begin{aligned} y'(t)+ky(t)\le h(t) \end{aligned}$$
holds with \(k>0\), \(h(t)\in C [t_{0},t_{1}]\), then
$$\begin{aligned} y(t)\le y(0)e^{-kt}+\int _{0}^{t}e^{-k(t-s)}h(s)ds. \end{aligned}$$
In particular, if \(h(t)=C\), then
$$\begin{aligned} y(t)\le y(0)e^{-kt}+Ck^{-1}. \end{aligned}$$

Lemma 2.2

[6] Suppose that \(E_{1}\hookrightarrow E \subset E_{0}\), where E is a Banach space, and the embedding \(E_{1}\hookrightarrow E\) is compact. Set the space
$$\begin{aligned}&W_{p_{1},p_{0}}(0,T;E_{1},E_{0})\\&\qquad =\{\psi (s), s\in [0,T]\mid \psi (s)\in L^{p_{1}}(0,T;E_{1}),\psi '(s)\in L^{p_{0}}(0,T;E_{0}) \}, \end{aligned}$$
with the norm
$$\begin{aligned} \Vert \psi \Vert _{W_{p_{1},p_{0}}(0,T;E_{1},E_{0})}=\left( \int _{0}^{T}\Vert \psi (s)\Vert _{E_{1}}^{p_{1}}ds\right) ^{\frac{1}{p_{1}}}+ \left( \int _{0}^{T}\Vert \psi '(s)\Vert _{E_{0}}^{p_{0}}ds\right) ^{\frac{1}{p_{0}}}. \end{aligned}$$
We assume that \(p_{1}\ge 1\) and \(p_{0}\ge 1\). Then the following embedding is compact:
$$\begin{aligned} W_{p_{1},p_{0}}(0,T;E_{1},E_{0})\hookrightarrow L^{p_{1}}(0,T;E ). \end{aligned}$$
Moreover, when \(p_{1}=\infty \), then the following embedding is compact:
$$\begin{aligned} W_{\infty ,p_{0}}(0,T;E_{1},E_{0})\hookrightarrow C(0,T;E ). \end{aligned}$$

Lemma 2.3

[4] For any \(f(t)\in D(A)'\), assume the operator A is self-adjoint and positive, then the following inequality holds:
$$\begin{aligned} \Vert f\Vert ^{2}_{D(A)'}\le \Vert (1+\alpha ^{2}A)f\Vert ^{2}_{D(A)'}. \end{aligned}$$

Lemma 2.4

[4] For any \(f(t)\in L^{2}(0,M;D(A)')\), assume the operator A is self-adjoint and positive, then there holds that
$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0^{+}} \Vert (1+\alpha ^{2}A)f(t)\Vert _{L^{2}(0,M;D(A)')}=\Vert f(t)\Vert _{L^{2}(0,M;D(A)')}. \end{aligned}$$

2.3 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
$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t}u+\nu Au+B(u,u)=g(x), ~~x\in \Omega ,~~t>0, \\ \nabla \cdot u=0,\quad u|_{\partial \Omega }=0,\quad u(x,0)=u_{0}\in H. \end{array} \right. \end{aligned}$$
(2.4)

Lemma 2.5

If \(g\in H\), \(u_{0}\in H\), and \(M>0\), then problem (2.4) admits a weak solution \(u(t) \in L^{\infty }(0,M;H)\cap L^{2}(0,M;V)\) such that \(u(0)=u_{0}\) and there holds that
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} |u(t)|^{2} +\nu \Vert u(t)\Vert ^{2}=( g,u(t)),\quad \forall t\in [0, M]. \end{aligned}$$
(2.5)
Moreover, for any \(\psi (t)\in C_{0}^{\infty }(0,M)\), there holds that
$$\begin{aligned} -\frac{1}{2}\int _{0}^{M} |u(t)|^{2} \psi '(t)dt+\nu \int _{0}^{M}\Vert u(t)\Vert ^{2}\psi (t)dt\le \int _{0}^{M}( g,u(t))\psi '(t)dt. \end{aligned}$$
(2.6)

Definition 2.6

The trajectory space \({\mathcal {K}}_{0}\) is the union of weak solutions u(t) of problem (2.4) with an arbitrary \(u_{0}\in H\), i.e., u(t) satisfies (2.5)–(2.6). \({\mathcal {K}}_{0}\) is called the trajectory space of problem (2.4).

Lemma 2.7

If \(g\in H\), then the equation (2.4) has a trajectory attractor \({\mathcal {O}}_{0}\).

Proof

We refer to [6] for its proof. \(\square \)

3 Trajectory Attractors of 3D Navier–Stokes–Voight Equation

The 3D Navier–Stokes–Voight system can be rewritten as
$$\begin{aligned} \left\{ \begin{array}{ll} (1+\alpha ^{2}A)\partial _{t}u+\nu Au+B(u,u)=g(x),\quad x\in \Omega , \\ \nabla \cdot u=0,\quad u|_{\partial \Omega }=0,\quad u(x,0)=u_{0}\in V. \end{array} \right. \end{aligned}$$
(3.1)

Theorem 3.1

If \(g\in H\), \(u_{0}\in V\), and \(M>0\), then problem (3.1) admits a weak solution \(u(t)\in L^{\infty }(0,M;H)\cap L^{2}(0,M;V)\) such that \(u(0)=u_{0}\) and there holds that
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \left( |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\right) +\nu \Vert u(t)\Vert ^{2}=( g,u(t)),\quad \forall t\in [0, M]. \end{aligned}$$
(3.2)
Moreover, for any \(\psi (t)\in C_{0}^{\infty }(0,M)\), there holds that
$$\begin{aligned}&-\frac{1}{2}\int _{0}^{M} \left( |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\right) \psi '(t)dt\nonumber \\&\quad +\nu \int _{0}^{M}\Vert u(t)\Vert ^{2}\psi (t)dt=\int _{0}^{M}( g,u(t))\psi '(t)dt. \end{aligned}$$
(3.3)

Definition 3.1

The trajectory space \({\mathcal {K}}_{\alpha }^{+}\) is the union of weak solutions u(t) of problem (3.1) with an arbitrary \(u_{0}\in V\), i.e., u(t) satisfies (3.2)–(3.3). \({\mathcal {K}}_{\alpha }^{+}\) is called the trajectory space of problem (3.1).

We consider the time translation semigroup \(\{T(h)\}\) acting on \({\mathcal {K}}_{\alpha }^{+}\).

Proposition 3.2

The trajectory space \({\mathcal {K}}_{\alpha }^{+}\) is invariant with the time translation semigroup \(\{T(h)\mid h\ge 0\}\):
$$\begin{aligned} T(h){\mathcal {K}}_{\alpha }^{+}\subseteq {\mathcal {K}}_{\alpha }^{+}, \forall h\ge 0. \end{aligned}$$

Proof

For any \(u(t)\in {\mathcal {K}}_{\alpha }^{+}\),
$$\begin{aligned} T(h)u(t)=u(h+t). \end{aligned}$$
Obviously, \(u(h+t)\) is also a weak solution of problem (3.1), hence \(u(h+t)\in {\mathcal {K}}_{\alpha }^{+}. \)\(\square \)

Lemma 3.2

If \(g\in H\) and u(t) is a weak solution of problem (3.1), then u(t) satisfies (3.2), and the following inequalities hold:
$$\begin{aligned}&|u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\le \left( |u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2}\right) e^{-\frac{\nu \lambda _{1}}{1+\lambda _{1}\alpha ^{2}}t} \nonumber \\&\quad +\frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}, \end{aligned}$$
(3.4)
$$\begin{aligned}&\nu \int _{t}^{t+1}\Vert u(s)\Vert ^{2}ds\le \left( |u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2}\right) e^{-\frac{\nu \lambda _{1}}{1+\lambda _{1}\alpha ^{2}}t} +\frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}\nonumber \\&\quad +\frac{ |g|^{2}}{\nu \lambda _{1} }. \end{aligned}$$
(3.5)

Proof

By the Cauchy inequality, it follows that
$$\begin{aligned} (g,u(t))\le \frac{\nu }{2}\Vert u(t)\Vert ^{2}+\frac{1}{2\nu \lambda _{1}}|g|^{2}. \end{aligned}$$
(3.6)
Then inserting (3.6) into (3.2) yields
$$\begin{aligned} \frac{d}{dt} \left( |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\right) +\nu \Vert u(t)\Vert ^{2}\le \frac{1}{ \nu \lambda _{1}}|g|^{2} . \end{aligned}$$
(3.7)
Using the Poincar\(\acute{e}\) inequality (2.3), we get
$$\begin{aligned} |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\le \frac{1}{\lambda _{1}}\Vert u(t)\Vert ^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}=\frac{1+\lambda _{1}\alpha ^{2}}{\lambda _{1}}\Vert u(t)\Vert ^{2}, \end{aligned}$$
i.e.,
$$\begin{aligned} \frac{ \lambda _{1} }{1+\lambda _{1}\alpha ^{2}}\left( |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\right) \le \Vert u(t)\Vert ^{2}. \end{aligned}$$
(3.8)
Inserting (3.8) into (3.7), we obtain
$$\begin{aligned} \frac{d}{dt} \left( |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\right) +\frac{\nu \lambda _{1} }{1+\lambda _{1}\alpha ^{2}} \left( |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\right) \le \frac{1}{ \nu \lambda _{1}}|g|^{2}. \quad \end{aligned}$$
(3.9)
Thus applying Gronwall Lemma 2.1 to (3.9), (3.4) follows. Integrating (3.7) on \([t,t+1]\), we find
$$\begin{aligned}&|u(t+1)|^{2}+\alpha ^{2}\Vert u(t+1)\Vert ^{2} +\nu \int _{t}^{t+1}\Vert u(s)\Vert ^{2}ds\nonumber \\&\quad \le |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2} +\frac{1}{ \nu \lambda _{1}}|g|^{2}, \end{aligned}$$
(3.10)
which, together with (3.4), gives (3.5). The proof is thus complete.

\(\square \)

Lemma 3.3

If \(g\in H\) and u(t) is a weak solution of problem (3.1), then the following inequality holds:
$$\begin{aligned} \int _{t}^{t+1}\Vert \partial _{t}u(s)\Vert _{D(A)'}^{2}ds\le C_{1}\left( |u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2}\right) e^{-\frac{\nu \lambda _{1}}{1+\lambda _{1}\alpha ^{2}}t} +R_{1}^{2}, \end{aligned}$$
(3.11)
where \( C_{1}, R_{1}\) only depend on \(\nu ,\lambda _{1},|g|\), but are independent of \(\alpha \).

Proof

First, using the Poincar\(\acute{e}\) inequality (2.3), we have
$$\begin{aligned} \Vert g\Vert _{D(A)'}\le \frac{1}{\lambda _{1}}|g|, \end{aligned}$$
(3.12)
and
$$\begin{aligned} \Vert Au\Vert _{D(A)'}=|u|\le \frac{1}{\sqrt{\lambda _{1}}}\Vert u\Vert . \end{aligned}$$
Then using (3.5), we get
$$\begin{aligned}&\nu \int _{t}^{t+1}\Vert Au(s)\Vert _{D(A)'}^{2}ds \le \frac{\nu }{ \lambda _{1}}\int _{t}^{t+1}\Vert u(s)\Vert ^{2}ds\nonumber \\&\quad \le \frac{1}{ \lambda _{1}}\left( (|u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2} )e^{-\frac{\nu \lambda }{1+\lambda _{1}\alpha ^{2}}t} +\frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}+\frac{ |g|^{2}}{\nu \lambda _{1} }\right) . \qquad \end{aligned}$$
(3.13)
Second, we derive from inequality (2.2) that
$$\begin{aligned} \Vert B(u,u)\Vert _{D(A)'} \le c|u|\Vert u\Vert , \end{aligned}$$
(3.14)
which, along with (3.4)–(3.5), yields
$$\begin{aligned}&\int _{t}^{t+1}\Vert B(u,u)\Vert _{D(A)'}^{2}ds \le c^{2}\sup _{s\in [t,t+1]}|u(s)|^{2} \int _{t}^{t+1}\Vert u(s)\Vert ^{2}ds \nonumber \\&\quad \le \frac{c^{2}}{\nu } \left( |u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2} +\frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}} \right) \nonumber \\&\qquad \times \left( (|u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2} )e^{-\frac{\nu \lambda }{1+\lambda _{1}\alpha ^{2}}t} +\frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}+\frac{ |g|^{2}}{\nu \lambda _{1} }\right) . \end{aligned}$$
(3.15)
Obviously,
$$\begin{aligned}&\int _{t}^{t+1}\Vert (1+\alpha ^{2}A)\partial _{t}u(s)\Vert _{D(A)'}^{2}ds\le \nu \int _{t}^{t+1}\Vert Au(s)\Vert _{D(A)'}^{2}ds\nonumber \\&\quad +\int _{t}^{t+1}\Vert B(u,u)\Vert _{D(A)'}^{2}ds+\Vert g\Vert _{D(A)'}^{2}. \end{aligned}$$
(3.16)
Combining (3.12), (3.13), (3.15), then (3.16) becomes
$$\begin{aligned}&\int _{t}^{t+1}\Vert (1+\alpha ^{2}A)\partial _{t}u(s)\Vert _{D(A)'}^{2}ds \le C_{\alpha }(|u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2} )e^{-\frac{\nu \lambda }{1+\lambda _{1}\alpha ^{2}}t}\nonumber \\&\quad +R_{\alpha }^{2}, \end{aligned}$$
(3.17)
where
$$\begin{aligned}\left\{ \begin{array}{ll} C_{\alpha }=\frac{1}{\lambda _{1}}+\frac{c^{2}}{\nu } \left( |u(0)|^{2}+\alpha ^{2}\Vert u(0)\Vert ^{2} +\frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}} \right) ,\\ R_{\alpha }^{2}=C_{\alpha } \left( \frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}+\frac{ |g|^{2}}{\nu \lambda _{1} }\right) +\frac{1}{ \lambda _{1}^{2}}. \end{array} \right. \end{aligned}$$
Since \(\alpha \in (0,1)\), it follows
$$\begin{aligned}\left\{ \begin{array}{ll} C_{\alpha }\le C_{1}=\frac{1}{\lambda _{1}}+\frac{c^{2}}{\nu } \left( |u(0)|^{2}+ \Vert u(0)\Vert ^{2} +\frac{(1+\lambda _{1} )|g|^{2}}{\nu ^{2} } \right) ,\\ R_{\alpha }^{2}\le R_{1}^{2}=C_{1} \left( \frac{(1+\lambda _{1} )|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}+\frac{ |g|^{2}}{\nu \lambda _{1} }\right) +\frac{1}{ \lambda _{1}^{2}}. \end{array} \right. \end{aligned}$$
Now by applying Lemma 2.3 and (3.17), the proof is thus complete.

\(\square \)

Lemma 3.4

Let \(g\in H\), then
  1. (i)

    \({\mathcal {K}}_{\alpha }^{+} \subset {\mathcal {F}}^{b}_{+}\);

     
  2. (ii)
    for any function \(u(\cdot ) \in {\mathcal {K}}_{\alpha }^{+}\), the translation group \(\{T(h)\}\) is continuous in \({\mathcal {F}}^{b}_{+}\), and there holds that
    $$\begin{aligned} \Vert T(h)u(\cdot ) \Vert _{{\mathcal {F}}^{b}_{+}}^{2}\le C_{0}\left( |u(0)|^{2}+\Vert u(0)\Vert ^{2}\right) e^{-\frac{\nu \lambda _{1}}{1+\lambda _{1}\alpha ^{2}}h} +R^{2}_{0}, ~~\forall h\ge 0, \qquad \end{aligned}$$
    (3.18)
    where \(C_{0}, R_{0}\) only depend on \(\nu ,\lambda _{1},|g|\), but are independent of \(\alpha \).
     

Proof

The proof of (3.18) is clearly established by Lemmas 3.23.3. Let \(u_{1}(s)\rightarrow u_{2}(s)\) in \({\mathcal {F}}^{b}_{+}\). Then \(u_{1}(s+h)\rightarrow u_{2}(s+h)\) in \({\mathcal {F}}^{b}_{+}\) and the translation group \(\{T(h)\}\) is continuous in \({\mathcal {F}}^{b}_{+}\). \(\square \)

Proposition 3.3

The trajectory space \({\mathcal {K}}_{\alpha }^{+}\subseteq {\mathcal {F}}^{b}_{+}\) is compact in the topology \(\Theta _{+}^{loc}\).

Proof

We consider an arbitrary sequence \(\{u_{n}\}\subset {\mathcal {K}}_{\alpha }^{+}\), such that \(u_{n}\rightarrow u\) in the topology \(\Theta _{+}^{loc}\) as \(n\rightarrow \infty \), we need to prove \(u\in {\mathcal {K}}_{\alpha }^{+}\). Due to \(u_{n}\rightarrow u\) in the topology \(\Theta _{+}^{loc}\), by the definition of \(\Theta _{+}^{loc}\),
$$\begin{aligned}&u_{n}\rightharpoondown u, \text { weakly in } L^{2}(0,M; V), \text { and weakly-star in } L^{\infty }(0,M;H),n\rightarrow \infty , \\&\quad \partial _{t}u_{n}\rightharpoondown \partial _{t}u, \text { weakly in } L^{2}(0,M;D(A)'),n\rightarrow \infty . \end{aligned}$$
Noting \(\{u_{n}\}\subset {\mathcal {K}}_{\alpha }^{+}\) and using Lemma 3.2, \(u_{n}\) is bounded in \(L^{2}(0,M;V) \cap L^{\infty }(0,M;H)\), and \(u_{n} \rightarrow u\) strongly in \( L^{2}(0,M;H)\), then \(\{u_{n}\}\) contains a subsequence \(\{u_{n_{k}}\}\) such that
$$\begin{aligned}&B( u_{n_{k}}, u_{n_{k}})\rightharpoondown B(u,u)\text { weakly in } L^{\frac{4}{3}}(0,M;V' ), \end{aligned}$$
(3.19)
$$\begin{aligned}&u_{n_{k}}\rightharpoondown u \text { weakly in } L^{2}(0,M; V), \end{aligned}$$
(3.20)
$$\begin{aligned}&u_{n_{k}}\rightharpoondown u \text { weakly-star in } L^{\infty }(0,M;H). \end{aligned}$$
(3.21)
Since \(u_{n_{k}}(t) \in {\mathcal {K}}_{\alpha }^{+}\), we have
$$\begin{aligned} (1+\alpha ^{2}A)\partial _{t}u_{n_{k}}+\nu Au_{n_{k}}+B(u_{n_{k}},u_{n_{k}})=g(x). \end{aligned}$$
(3.22)
Taking limit of (3.22) and using (3.19)–(3.21), we have
$$\begin{aligned} (1+\alpha ^{2}A)\partial _{t}u +\nu Au +B(u ,u )=g(x), \end{aligned}$$
(3.23)
then u is a weak solution of problem (3.1), i.e., \(u\in {\mathcal {K}}_{\alpha }^{+}\). \(\square \)

Proposition 3.4

The ball \(B_{2R _{0}}=\{u(\cdot )\mid u(\cdot )\in {\mathcal {K}}_{\alpha }^{+}, \Vert T(h)u\Vert _{{\mathcal {F}}^{b}_{+}}\le 2R _{0}\} \) in \({\mathcal {F}}^{b}_{+}\) is an absorbing set of the semigroup \(\{T(h)\}\). \(B_{2R _{0}}\) is compact in the topology \(\Theta _{+}^{loc}\).

Proof

According to Lemma 3.4, for any bounded set \(B \subset {\mathcal {K}}_{\alpha }^{+}\), there exists a time \(h_{1}>0\), such that as \(h\ge h_{1}\),
$$\begin{aligned} T(h)B \subset B_{2R _{0}}, \end{aligned}$$
which indicates that \(B_{2R _{0}}\) is an absorbing set of semigroup T(h) on \({\mathcal {K}}_{\alpha }^{+}\). Next, since \(B_{2R _{0}}\subset {\mathcal {K}}_{\alpha }^{+}\subseteq {\mathcal {F}}^{b}_{+}\), and \(B_{2R _{0}}\) is closed, by Proposition 3.3, \(B_{2R _{0}}\) is compact in the topology \(\Theta _{+}^{loc}\). \(\square \)

Definition 3.5

[6] A set \({\mathcal {O}}_{\alpha }\subset {\mathcal {K}}_{\alpha }^{+}\) is said to be the trajectory attractor of equation, if
  1. (i)

    \({\mathcal {O}}_{\alpha }\) is compact in the topology \(\Theta _{+}^{loc}\),

     
  2. (ii)

    \({\mathcal {O}}_{\alpha }\) is strictly invariant, i.e., \(T(h){\mathcal {O}}_{\alpha }={\mathcal {O}}_{\alpha }\) for all \(h\ge 0\),

     
  3. (iii)

    \({\mathcal {O}}_{\alpha }\) is a uniformly attracting set for \({\mathcal {F}}^{b}_{+}\).

     

Theorem 3.5

If \(g\in H\), then the translation semigroup \(\{T(h)\}\) acting on \({\mathcal {K}}_{\alpha }^{+}\) has a trajectory attractor \({\mathcal {O}}_{\alpha }\). The set \({\mathcal {O}}_{\alpha }\) is bounded in \({\mathcal {F}}^{b}_{+}\) and compact in the topology \(\Theta _{+}^{loc}\).

Proof

Let
$$\begin{aligned} {\mathcal {O}}_{\alpha }=B_{2R _{0}}. \end{aligned}$$
  1. (i)

    By Proposition 3.4, we know that \({\mathcal {O}}_{\alpha }\) is compact in the topology of \(\Theta _{+}^{loc}\), and \(T(h){\mathcal {O}}_{\alpha }\subseteq {\mathcal {O}}_{\alpha }\) for all \(h\ge h_{1}\).

     
  2. (ii)
    On the other hand, let \(u(t)\in B_{2R _{0}}={\mathcal {O}}_{\alpha }\), by the definition of time translation semigroup \(\{T(h)\}\),
    $$\begin{aligned} u(t)=T(h)u(t-h), \end{aligned}$$
    (3.24)
    let \({\tilde{u}}(t)=u(t-h)\), obviously, \({\tilde{u}}\) is a weak solution of problem (3.1), i.e., \({\tilde{u}}\in {\mathcal {O}}_{\alpha }\), then there exists a time \(t_{2}>0\), such that as \(t\ge t_{2}\),
    $$\begin{aligned} \Vert {\tilde{u}}(t)\Vert _{{\mathcal {F}}^{b}_{+}}\le 2R _{0}, \end{aligned}$$
    (3.25)
    hence, \({\tilde{u}}\in B_{2R _{0}}={\mathcal {O}}_{\alpha }\). Then from (3.24)–(3.25), we derive \(u(t)\in T(h)B_{2R _{0}}\) as \(t\ge t_{2}\), i.e., \({\mathcal {O}}_{\alpha }\subseteq T(h){\mathcal {O}}_{\alpha }\) as \(t\ge t_{2}\). Letting \(t_{0}= \max \{h_{1},t_{2}\}\), as \(t>t_{0}\), then we have \({\mathcal {O}}_{\alpha }=T(h){\mathcal {O}}_{\alpha }\).
     
  3. (iii)

    By Proposition 3.4, we know that \({\mathcal {O}}_{\alpha }\) is an absorbing set of T(h). Then we prove that \({\mathcal {O}}_{\alpha }\) is a trajectory attractor of problem (3.1).

     
\(\square \)

4 Upper Semicontinuity of the Trajectory Attractors of 3D NS Equation

Theorem 4.1

Let a sequence \(\{u_{n}(t)\}\subset {\mathcal {K}}_{\alpha }^{+}\), \(\alpha _{n}\rightarrow 0^{+}~(n\rightarrow \infty )\), and \(u_{n}(t)\rightarrow u(t)\) in the topology \(\Theta _{+}^{loc}\) as \(n\rightarrow \infty \). Then u(t) is a weak solution of the 3D Navier–Stokes equation such that u satisfies the inequality (2.6), i.e., \(u\in {\mathcal {K}}_{0}^{+}\), where \({\mathcal {K}}_{0}^{+}\) is the trajectory space of problem (2.4).

Proof

Since \( u_{n}(t) \in {\mathcal {K}}_{\alpha }^{+}\), i.e., \( u_{n}(t) \) is a weak solution of equation
$$\begin{aligned} (1+\alpha _{n}^{2}A)\partial _{t}u_{n}+\nu Au_{n}+B(u_{n},u_{n})=g(x), \end{aligned}$$
(4.1)
and noticing (3.18), \( u_{n}(t)\) satisfies
$$\begin{aligned}&\text {ess}\sup \limits _{t\ge 0}|u_{n}(t)|^{2}\le C, \end{aligned}$$
(4.2)
$$\begin{aligned}&\sup \limits _{t\ge 0}\int _{t}^{t+1}\Vert u_{n}(\tau )\Vert ^{2}d\tau \le C, \end{aligned}$$
(4.3)
$$\begin{aligned}&\sup \limits _{t\ge 0}\int _{t}^{t+1}\Vert \partial _{t}u_{n}(\tau )\Vert ^{2}d\tau \le C. \end{aligned}$$
(4.4)
Since \(\partial _{t}u_{n}\rightharpoondown \partial _{t}u ~~ (n\rightarrow \infty )\text { weakly in } L^{2}(0,M;D(A)')\) and using Lemma 2.4, we have
$$\begin{aligned} (1+\alpha _{n}^{2}A)\partial _{t}u_{n}\rightharpoondown \partial _{t}u ~~ (n\rightarrow \infty )\text { weakly in } L^{2}(0,M;D(A)'). \end{aligned}$$
(4.5)
Since \(u_{n}\rightharpoondown u ~~ (n\rightarrow \infty )\text { weakly in } L^{2}(0,M;V )\), then \(Au_{n}\rightharpoondown Au ~~ (n\rightarrow \infty )\text { weakly in} L^{2}(0,M;V' )\). Obviously,
$$\begin{aligned} Au_{n}\rightharpoondown Au ~~ (n\rightarrow \infty )\text { weakly in } L^{2}(0,M;D(A)' ). \end{aligned}$$
(4.6)
By (2.2), (4.2) and (4.3), we have
$$\begin{aligned} \int _{0}^{M}\Vert B(u_{n},u_{n})\Vert _{D(A)'}^{2}ds\le c\int _{0}^{M}(|u_{n}|\Vert u_{n}\Vert )^{2}ds\le C, \end{aligned}$$
then
$$\begin{aligned} B(u_{n},u_{n})\rightharpoondown B(u ,u ) ~~ (n\rightarrow \infty )\text { weakly in } L^{2}(0,M;D(A)' ). \end{aligned}$$
(4.7)
Combining with (4.5)–(4.7), Eq. (4.1) converges to
$$\begin{aligned} \partial _{t}u +\nu Au +B(u ,u )=g(x), ~~ (n\rightarrow \infty )\text { weakly in } L^{2}(0,M;D(A)'). \end{aligned}$$
(4.8)
Next, we prove u satisfies the enery inequality (2.6). Since \(u_{n}(t)\rightarrow u(t)\) in the topology \(\Theta _{+}^{loc}\) as \(n\rightarrow \infty \), by the Aubin Theorem, there exists a subsequence \(u_{n}\) which we still denote by \(u_{n}\) such that
$$\begin{aligned} \Vert u_{n}(t)-u(t)\Vert _{L^{2}(0,M;H)}^{2}=\int _{0}^{M}|u_{n}(t)-u(t)|^{2}dt \rightarrow 0, ~~n\rightarrow \infty . \end{aligned}$$
(4.9)
Note that \(u_{n}(t)\) satisfies the energy equality
$$\begin{aligned}&-\frac{1}{2}\int _{0}^{M} \left( |u_{n}(t)|^{2}+\alpha _{n}^{2}\Vert u_{n}(t)\Vert ^{2}\right) \psi '(t)dt+\nu \int _{0}^{M}\Vert u_{n}(t)\Vert ^{2}\psi (t)dt\nonumber \\&\quad =\int _{0}^{M}( g,u_{n}(t))\psi '(t)dt,~~\forall \psi '(t)\in C_{0}^{\infty }(0,M). \end{aligned}$$
(4.10)
Applying (4.9) and the H\(\ddot{o}\)lder inequality, we have
$$\begin{aligned}&\int _{0}^{M} |u_{n}(t)|^{2} \psi '(t)dt- \int _{0}^{M} |u (t)|^{2} \psi '(t)dt \nonumber \\&\quad \le \sup \limits _{t\in [0,M]}|\psi '(t) |\int _{0}^{M} (|u_{n}(t)|+| u (t)|)(|u_{n}(t)|-| u (t)|) dt\nonumber \\&\quad \le \sup \limits _{t\in [0,M]}|\psi '(t)| \left( 2\int _{0}^{M}(|u_{n}(t)|^{2}+|u (t)|^{2} )dt\right) ^{\frac{1}{2}}\nonumber \\&\qquad \times \left( \int _{0}^{M}|u_{n}(t)-u(t)|^{2} dt\right) ^{\frac{1}{2}} \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$
(4.11)
Similarly,
$$\begin{aligned}&\left| \int _{0}^{M}( g,u_{n}(t))\psi '(t)dt-\int _{0}^{M}( g,u (t))\psi '(t)dt\right| \le \int _{0}^{M}|g||u_{n}(t)-u(t)|\psi '(t)dt\nonumber \\&\quad \le \left( \int _{0}^{M}(|g| \psi '(t))^{2}dt\right) ^{\frac{1}{2}} \left( \int _{0}^{M}|u_{n}(t)- u (t)|^{2} dt\right) ^{\frac{1}{2}}\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$
(4.12)
Since \(u_{n}\sqrt{\psi }\rightharpoondown u \sqrt{\psi }\) weakly in \(L^{2}(0,M;V)\), i.e., for any \(w\in V'\),
$$\begin{aligned} \int _{0}^{M}\langle u\sqrt{\psi },w\rangle dt= \lim \limits _{n\rightarrow \infty }\int _{0}^{M}\langle u_{n}\sqrt{\psi },w\rangle dt. \end{aligned}$$
Letting \(w=u\sqrt{\psi }\), then
$$\begin{aligned} \int _{0}^{M}\Vert u\sqrt{\psi }\Vert ^{2} dt\le \lim \limits _{n\rightarrow \infty }\left( \int _{0}^{M}\Vert u_{n}\sqrt{\psi }\Vert ^{2} dt\right) ^{\frac{1}{2}} \left( \int _{0}^{M}\Vert u\sqrt{\psi }\Vert ^{2} dt\right) ^{\frac{1}{2}}, \end{aligned}$$
and \(\psi (t)\) is independent of x, we obtain
$$\begin{aligned} \nu \int _{0}^{M}\Vert u(t)\Vert ^{2} \psi dt\le \nu \lim \limits _{n\rightarrow \infty } \int _{0}^{M}\Vert u_{n}(t)\Vert ^{2} \psi dt. \end{aligned}$$
(4.13)
Similarly, letting \(\psi '\) replace \(\psi \), and using (4.3), we have
$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \int _{0}^{M}\Vert u_{n}(t)\Vert ^{2} \psi '(t) dt\le C, \end{aligned}$$
then
$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \alpha _{n}^{2} \int _{0}^{M}\Vert u_{n}(t)\Vert ^{2} \psi '(t) dt=0. \end{aligned}$$
(4.14)
Taking limit on both sides of equation (4.10), and using (4.11)–(4.14), we have
$$\begin{aligned} -\frac{1}{2}\int _{0}^{M} |u(t)|^{2} \psi '(t)dt+\nu \int _{0}^{M}\Vert u(t)\Vert ^{2}\psi (t)dt\le \int _{0}^{M}( g,u(t))\psi '(t)dt. \end{aligned}$$
Thus \(u\in {\mathcal {K}}_{0}^{+}\), and the proof is now complete.

\(\square \)

Theorem 4.2

Let \({\mathbb {B}}_{\alpha }=\{u(t)\mid |u(t)|^{2}+\alpha ^{2}\Vert u(t)\Vert ^{2}\le \frac{(1+\lambda _{1}\alpha ^{2})|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}\}\), \(0<\alpha \le 1\), be the bounded sets of solutions of 3D Navier–Stokes–Voight equation (3.1). Then the following convergence holds:
$$\begin{aligned} T(h){\mathbb {B}}_{\alpha }\rightarrow {\mathcal {O}}_{0}~ \text {in the topology } ~ \Theta _{+}^{loc} ~\text {as}~ h\rightarrow +\infty ,~ \alpha \rightarrow 0^{+}, \end{aligned}$$
(4.15)
where \({\mathcal {O}}_{0} \) is the trajectory attractor of 3D incompressible Navier–Stokes equation (2.4).

Proof

There exists a sequence \(\alpha _{n}\rightarrow 0^{+}\), as \(n\rightarrow +\infty \). Let \(u^{\alpha _{n}}(t)\in {\mathbb {B}}_{\alpha }\), \(u^{\alpha _{n}}(t)\) is the solutions of 3D Navier–Stokes–Voight equation (3.1), i.e., \(u^{\alpha _{n}}(t)\in {\mathcal {K}}_{\alpha }^{+}\),
$$\begin{aligned} T(h)u^{\alpha _{n}}(t)=u^{\alpha _{n}}(t+h), ~\text {since}~~T(h){\mathcal {K}}_{\alpha }^{+}\subseteq {\mathcal {K}}_{\alpha }^{+}, ~~\text {hence}~~ u^{\alpha _{n}}(t+h) \in {\mathcal {K}}_{\alpha }^{+}, \end{aligned}$$
i.e.,
$$\begin{aligned} (1+\alpha _{n}^{2}A)\partial _{t}u^{\alpha _{n}}(t+h)+\nu Au^{\alpha _{n}}(t+h)+B(u^{\alpha _{n}}(t+h),u^{\alpha _{n}}(t+h))=g(x). \nonumber \\ \end{aligned}$$
(4.16)
Since \(u^{\alpha _{n}}(t+h) \in {\mathcal {K}}_{\alpha }^{+}\), then
$$\begin{aligned} \Vert u^{\alpha _{n}}(t+h)\Vert _{{\mathcal {F}}^{b}_{+}}\le R. \end{aligned}$$
(4.17)
By Theorem 4.1, there exists \(u(t+h)\) in \({\mathcal {F}}_{+}^{loc}\), such that \(u^{\alpha _{n}}(t+h)\rightarrow u(t+h)\) in the topology \(\Theta _{+}^{loc} \) as \(n\rightarrow +\infty \), and \(u(t+h)\) is a weak solution of 3D Navier–Stokes equation, then \(u(t+h)\) solves
$$\begin{aligned} \partial _{t}u (t+h)+\nu Au (t+h)+B(u (t+h),u (t+h))=g(x). \end{aligned}$$
(4.18)
There exists a sequence \(h_{n}\rightarrow +\infty \), as \(n\rightarrow +\infty \). Next we prove \(\lim \limits _{n\rightarrow +\infty } u(t+h_{n})\in {\mathcal {O}}_{0}\). (4.17) means that
$$\begin{aligned} \Vert u (t+h_{n})\Vert _{{\mathcal {F}}^{b}_{+}}= & {} \sup _{t\ge -h_{n}}|u(t)| +\left( \sup _{t\ge -h_{n}}\int _{t}^{t+1}\Vert u(s)\Vert ^{2}ds\right) ^{\frac{1}{2}}\nonumber \\&+\left( \sup _{t\ge -h_{n}}\int _{t}^{t+1}\Vert \partial _{s}u(s)\Vert _{D(A)'}^{2}ds\right) ^{\frac{1}{2}} \le R, \end{aligned}$$
(4.19)
hence
$$\begin{aligned}&\lim \limits _{n\rightarrow +\infty } \Vert u (t+h_{n})\Vert _{{\mathcal {F}}^{b}_{+}}=\sup _{t\in {\mathbb {R}}}|u(t)| +\left( \sup _{t\in {\mathbb {R}}}\int _{t}^{t+1}\Vert u(s)\Vert ^{2}ds\right) ^{\frac{1}{2}}\nonumber \\&\quad +\left( \sup _{t\in {\mathbb {R}}}\int _{t}^{t+1}\Vert \partial _{s}u(s)\Vert _{D(A)'}^{2}ds\right) ^{\frac{1}{2}} \le R, \end{aligned}$$
(4.20)
i.e., for all \(t\in {\mathbb {R}}\),
$$\begin{aligned}&\lim \limits _{n\rightarrow +\infty } u(t+h_{n})=u(t)\in {\mathcal {F}}^{b}=L^{2}({\mathbb {R}},V)\nonumber \\&\quad \cap L^{\infty }({\mathbb {R}},H)\cap \{u\mid \partial _{t}u\in L^{2}({\mathbb {R}},D(A)')\}, \end{aligned}$$
(4.21)
and u(t) is a weak solution of 3D incompressible Navier–Stokes equation, i.e., \(u(t)\in {\mathcal {O}}_{0}\). \(\square \)

From the above argument, we have proved the following proposition.

Proposition 4.3

According to the definition of \({\mathcal {O}}_{\alpha }\), the following convergence holds:
$$\begin{aligned} {\mathcal {O}}_{\alpha }\rightarrow {\mathcal {O}}_{0}~\text {in the topology }~\Theta _{+}^{loc} ~\text {as} ~\alpha \rightarrow 0^{+}. \end{aligned}$$
(4.22)
The following theorem concerns the upper semicontinuity of trajectory attractors of 3D Navier–Stokes equation when the regular term \(\alpha ^{2}\triangle u_{t}\) 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.,
$$\begin{aligned} \text {dist}_{X}(B_{1},B_{2})=\sup \limits _{x\in B_{1}}\inf \limits _{y\in B_{2}}d_{X}(x,y),\quad B_{1},B_{2}\subset X. \end{aligned}$$

Theorem 4.4

Assume that \({\mathcal {O}}_{\alpha },{\mathcal {O}}_{0}\) are the trajectory attractors of 3D Navier–Stokes–Voight equation and Navier–Stokes equation respectively. Then the following convergence holds:
$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0^{+}}\text {dist}_{L^{2}(0,M; H^{ -\delta })}({\mathcal {O}}_{\alpha },{\mathcal {O}}_{0})= 0, \end{aligned}$$
(4.23)
here \(\delta \in (0,1]\).

Proof

Let \(u^{\alpha }\in {\mathcal {O}}_{\alpha },\) by Proposition 4.3, there exists \(u(t)\in {\mathcal {O}}_{0}\) such that \(u^{\alpha }\rightarrow u(t)\) weakly in \(L^{2}(0,M; V)\) as \(\alpha \rightarrow 0^{+}\), i.e., \(\forall \varphi \in V'\), \(u^{\alpha }, u\in V\), there holds that
$$\begin{aligned}&\lim \limits _{\alpha \rightarrow 0^{+}}\int _{0}^{M}\langle u^{\alpha }(t)-u(t),\varphi \rangle dt=0, \end{aligned}$$
(4.24)
$$\begin{aligned}&\int _{0}^{M}\langle u^{\alpha }(t)-u(t),\varphi \rangle dt= \int _{0}^{M}( A^{\frac{1}{2}}(u^{\alpha }(t)-u(t)),A^{ \frac{1}{2}}\varphi ) dt \nonumber \\&\quad =\int _{0}^{M}( u^{\alpha }(t)-u(t) ,A \varphi ) dt, \end{aligned}$$
(4.25)
Since \( \varphi \in V'\) without loss of generality, letting \(\varphi = A^{ - 1 }(u^{\alpha }(t)-u(t))\), it follows from (4.24), (4.25) that
$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0^{+}}\int _{0}^{M}|u^{\alpha }(t)-u(t)|_{H }^{2}dt =0, \end{aligned}$$
(4.26)
Let \(u^{\alpha }\in {\mathcal {O}}_{\alpha },\)\(u(t)\in {\mathcal {O}}_{0}\), \(w(t)=u^{\alpha }-u\), then w(t) satisfies
$$\begin{aligned} \partial _{t}w+\alpha ^{2}A u^{\alpha }_{t} +\nu Aw+B(u^{\alpha },w)+B(w,u)=0, \end{aligned}$$
(4.27)
then
$$\begin{aligned} \Vert \partial _{t}w\Vert _{L^{2}(0,M; D(A)')}\le \Vert \alpha ^{2} A u^{\alpha }_{t} +\nu Aw + B(u^{\alpha },w) +B(w,u)\Vert _{L^{2}(0,M; D(A)')}. \nonumber \\ \end{aligned}$$
(4.28)
According to (2.2) and the interpolation inequality, we have
$$\begin{aligned}&\Vert B(u^{\alpha },w)\Vert ^{2}_{L^{2}(0,M; D(A)')}\le c\int _{0}^{M}|u^{\alpha }|\Vert u^{\alpha }\Vert |w|^{2} ds \le c\left( \int _{0}^{M}|u^{\alpha }|^{2}|w|^{2} ds\right) ^{\frac{1}{2}}\nonumber \\&\qquad \times \left( \int _{0}^{M}\Vert u^{\alpha }\Vert ^{2}|w|^{2} ds\right) ^{\frac{1}{2}}\nonumber \\&\quad \le c\left( \sup _{s\in [0,M]}|u^{\alpha }|^{2}\int _{0}^{M} |w|^{2} ds\right) ^{\frac{1}{2}}\times \left( \sup _{s\in [0,M]}|w|^{2}\int _{0}^{M}\Vert u^{\alpha }\Vert ^{2} ds\right) ^{\frac{1}{2}}. \end{aligned}$$
(4.29)
Since \(|w|^{2}=|u^{\alpha }-u|^{2}\le 2( |u^{\alpha }|^{2}+|u|^{2})\), and using (3.4)–(3.5), we get
$$\begin{aligned}&\max \left\{ \sup _{s\in [0,M]}|u^{\alpha }|^{2},\sup _{s\in [0,M]}|w |^{2},\int _{0}^{M}\Vert u^{\alpha }\Vert ^{2} ds\right\} \nonumber \\&\quad \le 2\left( \left( |u(0)|^{2}+ \Vert u(0)\Vert ^{2}\right) +\frac{(1+\lambda _{1} )|g|^{2}}{\nu ^{2}\lambda _{1}^{2}}+\frac{ |g|^{2}}{\nu \lambda _{1} }\right) . \end{aligned}$$
(4.30)
Then (4.29) becomes
$$\begin{aligned}&\Vert B(u^{\alpha },w)\Vert ^{2}_{L^{2}(0,M; D(A)')}\le C\left( \int _{0}^{M} |w|^{2} ds\right) ^{\frac{1}{2}}. \end{aligned}$$
(4.31)
Similarly to B(wu), we obtain
$$\begin{aligned} \Vert B(w,u)\Vert ^{2}_{L^{2}(0,M; D(A)')}\le C\left( \int _{0}^{M} |w|^{2} ds\right) ^{\frac{1}{2}}, \end{aligned}$$
(4.32)
and
$$\begin{aligned} \nu \Vert Aw\Vert ^{2}_{L^{2}(0,M; D(A)')}=\nu \int _{0}^{M} |w|^{2} ds . \end{aligned}$$
(4.33)
By Lemma 2.4, taking \(f(t)=u^{\alpha }_{t}\), we have
$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0^{+}}\alpha ^{2}\Vert Au^{\alpha }_{t}\Vert _{L^{2}(0,M; H)}=0. \end{aligned}$$
Obviously, \(\Vert Au^{\alpha }_{t}\Vert _{D(A)'}=|u^{\alpha }_{t}|\le |Au^{\alpha }_{t}| \), then
$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0^{+}}\alpha ^{2}\Vert Au^{\alpha }_{t}\Vert _{L^{2}(0,M; D(A)')}=0. \end{aligned}$$
(4.34)
Combining (4.31)–(4.34) with (4.26), we obtain
$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0^{+}}\Vert \partial _{t}w\Vert _{L^{2}(0,M; D(A)')}=0. \end{aligned}$$
(4.35)
Letting \(E_{1}=H,~E =H^{-\delta }, ~E_{0}=D(A)'\), \(p_{1}=2, p_{0}=2\) , and using Lemma 2.2, we can prove (4.23).

\(\square \)

Notes

Acknowledgements

This paper was supported in part by the NNSF of China (No.11671075) and fundamental research funds for the central universities (No.2232019D3-43).

References

  1. 1.
    Bronzi, A.C., Rosa, R.M.S.: On the convergence of statistical solutions of the 3D Navier–Stokes-\(\alpha \) model as \(\alpha \) vanishes. Discret. Contin. Dyn. Syst. 34(1), 19–49 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, S., Holm, D.D., Margolin, L.G., Zhang, R.: Direct numerical simulations of the Navier–Stokes-\(\alpha \) model. Physica D 133, 66–83 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chepyzhov, V.V., Gatti, S., Grasselli, M., Miranville, A., Pata, V.: Trajectory and global attractors for evolution equations with memory. Appl. Math. Lett. 19, 87–96 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chepyzhov, V.V., Titi, E.S., Vishik, M.I.: On Convergence of Trajectory Attractors of 3D Navier–Stokes-\(\alpha \) Model as \(\alpha \) Approaches 0. Russian Academy of Sciences, (DoM) and London Mathematical Society (2007)Google Scholar
  5. 5.
    Chepyzhov, V.V., Titi, E.S., Vishik, M.I.: On the convergence of solutions of the Leray-\(\alpha \) model to the trajectory attractor of the 3D Navier–Stokes system. Discret. Contin. Dyn. Syst. 17, 481–500 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics, AMS, vol. 49. Collouium Publications, American Mathematical Society, Providence, RI (2002)Google Scholar
  7. 7.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory attractors for dissipative 2D Euler and Navier–Stokes equations. Russ. J. Math. Phys. 15(2), 156–170 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Deugoue, G.: Approximation of the trajectory attractor of the 3D MHD system. Commun. Pure Appl. Anal. 12(5), 2119–2144 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes-alpha model of fluid turbulence. Physica D 153, 505–519 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Foias, C., Holm, D.D., Titi, E.S.: The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14, 1–35 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Frigeri, S., Grasselli, M.: Global and trajectory attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Dyn. Differ. Equ. 24, 827–856 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Frigeri, S., Rocca, E.: Trajectory attractors for the Sun–Liu model for nematic liquid crystals in 3D. Nonlinearity 26(4), 933–958 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gal, C.G., Grassellt, M.: Trajectory attractors for binary fluid mixtures in 3D. Chin. Ann. Math. 31B(5), 655–678 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gorban, N.V., Kapustyan, O.V., Kasyanov, P.O.: Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carath odory’s nonlinearity. Nonlinear Anal. 98, 13–26 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grasselli, M., Schimperna, G., Zelik, S.: Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term. Nonlinearity 23(3), 707–738 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guo, B.L., Huang, D.W.: Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics. J. Math. Phys. 47(8), 083508, 1–23 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hetzer, G.: Trajectory attractors of energy balance climate models with Bio-Feedback. Differ. Equ. Appl. 3(4), 565–579 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kalantarov, V.K., Titi, E.S.: Global attractors and determining modes for the 3D Navier–Stokes–Voight equations. Chin. Ann. Math. 30B(6), 697–714 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mielke, A., Zelik, S.V.: Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains. Russ. Math. Surv. 57(4), 753–784 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Qin, Y.: Analytic Inequalities and their Applications in PDEs. Springer, Berlin (2017)CrossRefGoogle Scholar
  21. 21.
    Zelati, M.C., Gal, C.G.: Singular limits of Voight models in fluid dynamics. J. Math. Fluid Mech. 17, 233–259 (2005)CrossRefGoogle Scholar
  22. 22.
    Zhao, C.D., Zhou, S.F.: Pullback trajectory attractors for evolution equations and application to 3D incompressible non-Newtonian fluid. Nonlinearity 21, 1691–1717 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Sciences, Institute for Nonlinear SciencesDonghua UniversityShanghaiPeople’s Republic of China
  2. 2.College of Information Science and TechnologyDonghua UniversityShanghaiPeople’s Republic of China

Personalised recommendations