Pullback $\mathcal{D}$-attractors for three-dimensional Navier-Stokes equations with nonlinear damping

We investigate the asymptotic behavior of solutions of the non-autonomous Navier-Stokes equation with nonlinear damping in three-dimensional bounded domain. When $3< \beta\leq5$, the existence of pullback attractors is proved in V and $\mathbf{H}^{2}(\Omega)$, respectively.

In this paper, we investigate the following non-autonomous Navier-Stokes equation with nonlinear damping:

$\mu>0$ is the kinematic viscosity of the fluid and $f=f(x,t)$ is the external body force. The unknown functions here are $u=u(x,t)=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))$ and $p=p(x,t)$, which stand for the velocity field and the pressure of the flow, respectively. In the damping term, $\beta\geq1$ and $\alpha>0$ are two constants. The given function $u_{\tau}=u_{\tau}(x)$ is the initial velocity.

When $\alpha=0$, problem (1) becomes the classical 3D Navier-Stokes equation with external force, which has been studied by many authors (see [1–8]). The damping arises from the resistance to the motion of the flow and describes various physical phenomena, such as porous media flow, drag or friction effects, and some dissipative mechanisms (see [9–11]). For the autonomous case, in [12], Cai and Jiu proved that Cauchy problem (1) possesses global strong solutions when $\beta\geq\frac{7}{2}$, and the global strong solution is unique when $\frac{7}{2}\leq\beta\leq5$. In [13], Zhang, Wu and Lu also investigated the uniqueness of strong solution of problem (1). They established that the strong solution exists when $\beta> 3$, and the global strong solution is unique when $3<\beta\leq5$. This improved the earlier results in [12]. In [14, 15], some authors discussed the $L^{2}$-decay rate of solutions of problem (1). In [16, 17], we studied the existence of global attractors and uniform attractors of strong solution of problem (1) when $\frac{7}{2}\leq\beta \leq5$.

In this paper, our aim is to study the long-time behavior of strong solution of problem (1) by the theory of pullback attractors. Pullback attractor theory is a natural generalization of the theory of global attractors developed to study autonomous dynamical systems, and it is well suited to study the non-autonomous dynamical systems. We shall prove the existence of pullback attractors in $(H_{0}^{1}(\Omega))^{3}$ and $(H^{2}(\Omega))^{3}$ under the assumption of an external force $f(x,t)$ satisfying a certain integrability condition. To attain our goal we use the methods introduced in [18–20], which will be explained in more detail in Section 2. Before formulating the main results of the paper, we shall introduce some function spaces and some notations.

We define the function spaces

where $\operatorname{cl}_{X}$ denotes the closure in the space X. It is well known that H, V are separable Hilbert spaces and identifying H and its dual $H'$, we have $V\hookrightarrow H\hookrightarrow V'$ with dense and continuous injections, and $V\hookrightarrow H$ is compact. H and V endowed, respectively, with the inner products

and norms $|\cdot|_{2}=(\cdot,\cdot)^{1/2}$, $\|\cdot\|=((\cdot,\cdot))^{1/2}$. In this paper, $\mathbf{H}^{2}(\Omega)=(H^{2}(\Omega))^{3}$, $\mathbf{L}^{p}(\Omega)=(L^{p}(\Omega ))^{3}$, and we use $|\cdot|_{p}$ to denote the norm in $\mathbf{L}^{p}(\Omega)$.

If $u\in L^{\infty}(\tau,T;H)\cap L^{2}(\tau,T;V)\cap L^{\beta+1}(\tau ,T;\mathbf{L}^{\beta+1}(\Omega))$ satisfies

then we say that u is a weak solution of (1) on $[\tau,T]$.

The weak formulation (2) is equivalent to the function equation

where $Au=-\tilde{P}\Delta u$ is the Stokes operator defined by $\langle Au,v\rangle=((u,v))$, and P̃ is the orthogonal projection of $(L^{2}(\Omega))^{3}$ onto H. $G(u)=\tilde{P}F(u)$ and $F(u)=\alpha|u|^{\beta-1}u$. $B:V\times V\rightarrow V'$ is a bilinear operator defined by $\langle B(u,v),w\rangle=b(u,v,w)$, $B(u)=B(u,u)$, where

and $\langle\cdot,\cdot\rangle$ is the duality product between V and $V'$.

In this paper, we assume the external force $f(x,t)\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$, and the derivative $\frac{\mathrm{d}f}{\mathrm {d}t}\in L_{b}^{2}(\mathbb {R};H)$. Recall that a function $g(t)$ is said to be translation bounded (tr.b.) in $L_{\mathrm{loc}}^{2}(\mathbb {R};H)$ if

$L_{b}^{2}(\mathbb {R};H)$ denotes the collection of functions that are tr.b. in $L_{\mathrm{loc}}^{2}(\mathbb {R};H)$. Furthermore, we assume that $f(x,t)$ is uniformly bounded in H, i.e., there exists a positive constant K, such that

Throughout this paper, we assume that the external force $f(x,t)$ satisfies

where $0<\sigma<\frac{\mu\lambda_{1}}{2}$, and $\lambda_{1}$ is the first eigenvalue of the Stokes operator. Let $\mathcal{D}$ be the class of all families $\{D(t):t\in \mathbb {R}\}$ of nonempty subsets of $(H_{0}^{1}(\Omega))^{3}$ such that

where $[D(t)]=\sup\{\| u\|_{(H_{0}^{1}(\Omega))^{3}}^{2}: u\in D(t)\}$.

In this paper, the letter C is a generic positive constant, which may change its value from line to line, even in the same line.

In the next section, we provide basic definitions and results we shall use in this paper. In Section 3 we give some prior estimates of solutions. Based on these uniform estimation, in Section 4 we prove the existence of pullback attractors.

In this section, we will recall some basic definitions and abstract results about pullback attractor and state the theorems about the existence and uniqueness of global solutions of problem (1).

Let X be a complete metric space. A two-parameter family of mappings acting on X: $U(t,\tau):X\rightarrow X$, $t\geq\tau$, $\tau\in \mathbb {R}$, is said to be an evolutionary process if

1. (1)

   $U(t,\tau)=U(t,r)U(r,\tau)$, for all $\tau\leq r\leq t$,

2. (2)

   $U(\tau,\tau)=\mathrm{Id}$ is the identity operator, $\forall\tau\in \mathbb {R}$.

Let $\mathcal{D}$ be a nonempty class of families $\hat{D}=\{D(t):t\in \mathbb {R}\}$ of nonempty subsets of X.

Definition 2.1

A family $\hat{A}=\{A(t):t\in \mathbb {R}\}$ of nonempty subsets of X is said to be a pullback $\mathcal{D}$-attractor for the process $\{ U(t,\tau)\}_{t\geq\tau}$ in X, if

1. (1)

   $\hat{A}(t)$ is compact in X for any $t\in \mathbb {R}$,

2. (2)

   Â is invariant, i.e., $U(t,\tau)A(\tau)=A(t)$ for any $\tau\leq t$,

3. (3)

   Â is pullback $\mathcal{D}$-attracting, i.e.,

   $$ \lim_{\tau\rightarrow-\infty}\operatorname{dist}\bigl(U(t,\tau)D(\tau),A(t) \bigr)=0, $$

   for any $t\in \mathbb {R}$ and any $\hat{D}\in\mathcal{D}$.

Such a family $\hat{\mathcal{A}}$ is called minimal if $A(t)\subset C(t)$ for any family $\hat{C}=\{C(t):t\in \mathbb {R}\}$ of closed subsets of X such that $\lim_{\tau\rightarrow-\infty}\operatorname{dist}(U(t,\tau)B(\tau),C(t))=0$ for any $\hat{B}\in\mathcal{D}$.

Definition 2.2

It is said that $\hat{B}\in\mathcal{D}$ is pullback $\mathcal {D}$-absorbing for the process $\{U(t,\tau)\}_{t\geq\tau}$, if for any $\hat{D}\in\mathcal{D}$ and any $t\in \mathbb {R}$, there exists a $\tau _{0}(t,\hat{D})\leq t$ such that $U(t,\tau)D(\tau)\subset B(t)$ for any $\tau\leq\tau_{0}(t,\hat{D})$.

Definition 2.3

Let X be a Banach space. A process $U(t,\tau)$ is said to be norm-to-weak continuous on X if for all $t,\tau\in \mathbb {R}$ with $t\geq\tau$ and for every sequence $x_{n}\in X$,

Obviously, a continuous process is a norm-to-weak continuous process. The following result is very useful to check that the process is norm-to-weak continuous.

Theorem 2.1

(see [19, 21, 22])

Let X, Y be two Banach spaces. $X^{*}$, $Y^{*}$ be, respectively, their dual spaces. Assume that X is dense in Y, the injection $i:X\rightarrow Y$ is continuous, its adjoint $i^{*}:Y^{*}\rightarrow X^{*}$ is dense, and U is a norm-to-weak continuous process on Y. Then U is a norm-to-weak continuous process on X if and only if for any $\tau\in \mathbb {R}$, $t\geq\tau$, $U(t,\tau)$ maps compact sets of X into bounded sets of X.

Definition 2.4

The process $\{U(t,\tau)\}_{t\geq\tau}$ is said to be pullback $\mathcal {D}$-asymptotically compact, if for any $t\in \mathbb {R}$ and any $\hat {D}\in\mathcal{D}$, any sequence $\tau_{n}\rightarrow-\infty$ and any sequence $x_{n}\in D(\tau _{n})$, the sequence $\{U(t,\tau_{n})x_{n}\}_{n=1}^{\infty}$ is relatively compact in X.

Lemma 2.1

(see [18–20])

Let $\{U(t,\tau)\}_{t\geq\tau}$ be a process in X satisfying the following conditions:

1. (1)

   $\{U(t,\tau)\}_{t\geq\tau}$ is norm-to-weak continuous in X;

2. (2)

   there exists a family B̂ of pullback $\mathcal{D}$-absorbing sets $\{B(t):t\in \mathbb {R}\}$ in X;

3. (3)

   $\{U(t,\tau)\}_{t\geq\tau}$ is pullback $\mathcal {D}$-asymptotically compact.

Then there exists a minimal pullback $\mathcal{D}$-attractor $\hat {\mathcal{A}}=\{A(t):t\in \mathbb {R}\}$ in X given by

Now, we recall the existence and uniqueness theorem for strong solution of problem (1).

Theorem 2.2

([17])

Suppose $f\in L_{b}^{2}(\mathbb {R};H)$, $u_{\tau}\in H$, and $\beta\geq1$. Then for any given $T>\tau$, there exists at least one solution u that satisfies (2). Moreover,

We say that $u(x,t)$ is a strong solution of (1), if it is a weak solution of (1), and satisfies

Theorem 2.3

Suppose $\beta>3$, $u_{\tau}\in V\cap\mathbf{L}^{\beta +1}(\Omega)$, and $f\in L_{b}^{2}(\mathbb {R};H)$. Then there exists a strong solution $u(x,t)$ that satisfies (2),

Moreover, when $3<\beta\leq5$, the strong solution is unique.

Proof

In [13], Zhang, Wu and Lu have proved this theorem in the autonomous case. For the non-autonomous case, it is similar to the proof of Theorem 3.1 in [13], so we omit it here. □

Because $\Omega\subset R^{3}$ is sufficiently regular, so $V\hookrightarrow\mathbf{L}^{6}(\Omega)$, and because $\mathbf {L}^{6}(\Omega)\hookrightarrow\mathbf{L}^{\beta+1}(\Omega)$ ($3<\beta\leq 5$), so $V\cap\mathbf{L}^{\beta+1}(\Omega)=V$. In the following, we use $u_{\tau}\in V$ to replace $u_{\tau}\in V\cap\mathbf{L}^{\beta+1}(\Omega)$.

In [16], we have proved that the strong solution $u(x,t)$ is continuous with respect to the initial value condition $u_{0}$ in the space V when $\frac{7}{2}\leq\beta\leq5$ (Proposition 7). From the proof, we can easily deduce that, when $3<\beta\leq5$, for the non-autonomous case, the strong solution $u(x,t)$ is also continuous with respect to the initial data $u_{\tau}$ in V.

In order to construct a process $\{U(t,\tau)\}_{t\geq\tau}$ for problem (1), we define $U(t,\tau): V\rightarrow V$ by $U(t,\tau)u_{\tau}=u(t)$, $t\geq\tau$. Obviously, the process $\{U(t,\tau)\}_{t\geq\tau}$ is a continuous process in V, so it is also a norm-to-weak continuous process in V.

In this section, we derive uniform estimates on solutions of problem (1) when $\tau\rightarrow-\infty$. These estimates are necessary to prove the existence of pullback $\mathcal{D}$-absorbing sets and the pullback asymptotic compactness of process $\{U(t,\tau)\}_{t\geq\tau}$ associated with the system.

Lemma 3.1

Under the assumptions (4)-(5) and $f\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal {D}$ and $t\in \mathbb {R}$, there exists $\tau_{0}=\tau_{0}(t,\hat{D})< t$, such that

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{0}(t,\hat{D})$.

Proof

Taking the inner product of (1)₁ with u, we obtain

where $\lambda_{1}$ is the first eigenvalue of the Stokes operator. Thus,

Multiplying (6) by $e^{\sigma t}$ and then integrating over $(\tau, t)$, we derive that

Since $0<\sigma<\frac{\mu\lambda_{1}}{2}$, we have

Since $u(\tau)\in D(\tau)$, for every $t\in \mathbb {R}$, there exists $\tau_{0}=\tau_{0}(t,\hat{D})< t$ such that, for all $\tau\leq\tau_{0}$,

By (7) and (8), we find that

 □

Lemma 3.2

Under the assumptions (4)-(5) and $f\in L_{\mathrm{loc}}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal{D}$ and $t\in \mathbb {R}$, there exists $\tau_{1}=\tau_{1}(t,\hat{D})< t-2$, such that

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$.

Proof

It follows from (6) that

Let $s\in[t-2,t]$. Multiplying (9) by $e^{\sigma t}$, then relabeling t as ξ and integrating with respect to ξ over $(\tau,s)$, we get

Since $u(\tau)\in D(\tau)$, for every $t\in \mathbb {R}$, there exists $\tau_{1}=\tau_{1}(t,\hat{D})< t-2$, such that, for all $\tau\leq\tau_{1}$,

By (10) and (11), we have, for $s\in[t-2,t]$,

Integrating (12) with respect to s over the interval $(t-2,t)$ produces

Multiplying (6) by $e^{\sigma t}$, then relabeling t as ξ and integrating with respect to ξ over $(t-2,t)$, by (12) we obtain, for all $\tau\leq\tau_{1}$,

which along with (13) completes the proof. □

Corollary 3.1

Under the assumptions (4)-(5) and $f\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal {D}$ and $t\in \mathbb {R}$,

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$.

Proof

It is straightforward from Lemma 3.2. □

Lemma 3.3

Under the assumptions (4)-(5) and $f\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then, for any $\hat{D}\in\mathcal {D}$ and $t\in \mathbb {R}$,

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$.

Proof

Multiplying the first equation of (1) by $u_{t}$, $-\Delta u$, respectively, and then integrating the resulting equation on Ω, we obtain

From (15) we have

From (16) we get

Taking (17), (18) together, it follows that

From the proof of Theorem 3.1 in [13], we can find that, when $3<\beta\leq5$,

Substituting (20) into (19), we find that

Applying the uniform Gronwall lemma to (21) on interval $[t-1,t]$, we have

Noticing

according to Corollary 3.1, we have

 □

Lemma 3.4

Under the assumptions (4)-(5) and $f\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal {D}$ and $t\in \mathbb {R}$,

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$.

Proof

Similar to the proof of Lemma 3.3, applying the uniform Gronwall lemma to (21) on interval $[t-2,t-1]$, we can also prove

Thanks to (18), (20), we have

Integrating (23) from $t-1$ to t, according to Corollary 3.1 and (22), we can obtain

which completes the proof. □

Lemma 3.5

Under the assumptions (4)-(5) and $f\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$, $\frac{\mathrm{d}f}{\mathrm{d}t}\in L_{b}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal {D}$ and $t\in \mathbb {R}$, there exists a family of positive constants $\{r_{1}(t):t\in \mathbb {R}\}$ such that

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$, where $r_{1}(t)$ is a positive constant which is independent of the initial data.

Proof

From (15) and (20) we deduce that

Thus

Integrating (26) from $t-1$ to t, according to Corollary 3.1, Lemma 3.4, (22), and (24), we can obtain

We now differentiate (3)₁ with respect to t and then take the inner product with $u_{t}$ in H to obtain

According to Lemma 2.4 in [16], $(F'(u)u_{t})\cdot u_{t}$ is positive definite, hence we have

Thus,

Thanks to (14),we have

where $r_{0}(s)=Ce^{-\sigma s}\int_{-\infty}^{s} e^{\sigma\xi}|f(\xi )|_{2}^{2}\,\mathrm{d}\xi$.

Applying the uniform Gronwall lemma to (29) on interval $[t-1,t]$, we can get

 □

Lemma 3.6

Under the assumptions (4)-(5) and $f\in L_{\mathrm {loc}}^{2}(\mathbb {R};H)$, $\frac{\mathrm{d}f}{\mathrm{d}t}\in L_{b}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal{D}$ and $t\in \mathbb {R}$, there exists a family of positive constants $\{r_{2}(t):t\in \mathbb {R}\}$ such that

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$, where $r_{2}(t)$ is a positive constant which is independent of the initial data.

Proof

Like the proof of Proposition 5 in [16], we can obtain

 □

Lemma 3.7

Under the assumptions (4)-(5) and $f\in L_{\mathrm{loc}}^{2}(\mathbb {R};H)$, $\frac{\mathrm{d}f}{\mathrm{d}t}\in L_{b}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$, $\tau\in \mathbb {R}$, and $u(t)$ be the solution of problem (1). Then for any $\hat{D}\in\mathcal{D}$ and $t\in \mathbb {R}$, there exists a family of positive constants $\{r_{3}(t):t\in \mathbb {R}\}$ such that

for any $u_{\tau}\in D(\tau)$ and $\tau\leq\tau_{1}(t,\hat{D})$, where $r_{3}(t)$ is a positive constant which is independent of the initial data.

Proof

From inequality (28) we have

Integrating the above inequality from t to $t+1$, then we have

By Lemma 3.6, we know that

so using the Agmon inequality we obtain

We now differentiate (3)₁ with respect to t, then taking the inner product with $Au_{t}$ in H to obtain

According to Lemma 2.4 in [16], for any $u,v,w\in \mathbb {R}^{3}$, $|(F'(u)v)\cdot w|\leq C|u|^{\beta-1}|v||w|$, so

Because

combining (34)-(37) with (33), we get

Thanks to (32), applying the uniform Gronwall lemma to (38) on interval $[t,t+1]$, we get

 □

In Section 2, we have known that the process $\{U(t,\tau)\}_{t\geq\tau }$ associated with (1) is norm-to-weak continuous in V. In this section, we prove the existence of pullback attractors in V and $\mathbf{H}^{2}(\Omega)$ for the non-autonomous Navier-Stokes equation with nonlinear damping.

Theorem 4.1

Under the assumptions (4)-(5) and $f\in L_{\mathrm{loc}}^{2}(\mathbb {R};H)$, $\frac{\mathrm{d}f}{\mathrm{d}t}\in L_{b}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$ and $\tau\in \mathbb {R}$, then the process $\{U(t,\tau )\}_{t\geq\tau}$ associated with (1) has a pullback $\mathcal {D}$-attractor $\mathcal{A}_{1}$ in V.

Proof

Let $B_{0}=\{B(t): t\in \mathbb {R}\}$ and $C_{0}=\{C(t):t\in \mathbb {R}\}$ be pullback $\mathcal{D}$-absorbing sets in V and in $D(A)$ obtained by Lemma 3.3 and Lemma 3.6, respectively. Since $D(A)\hookrightarrow V$ is compact, we have $\{U(t,\tau)\}_{t\geq\tau}$ is pullback $\mathcal{D}$-asymptotically compact in V. Then by Lemma 2.1, $\{U(t,\tau)\}_{t\geq\tau}$ has a minimal pullback $\mathcal {D}$-attractor $\mathcal{A}_{1}$ in V. □

Lemma 4.1

The process $\{U(t,\tau)\}_{t\geq\tau}$ is norm-to-weak continuous in $\mathbf{H}^{2}(\Omega)$.

Proof

We know $i:D(A)\hookrightarrow V$, $i^{*}: V^{*}\hookrightarrow (D(A))^{*}$ and i, $i^{*}$ are dense. From Section 2, we know that $\{ U(t,\tau)\}_{t\geq\tau}: V\rightarrow V$ is norm-to-weak continuous. From Lemma 3.6, we find that $\{U(t,\tau)\}_{t\geq\tau}$ has a pullback $\mathcal{D}$-absorbing set in $D(A)$. That is to say, $\{U(t,\tau)\} _{t\geq\tau}$ maps a bounded set in V into a bounded set in $D(A)$, therefore $\{U(t,\tau)\}_{t\geq\tau}$ maps a compact set in $D(A)$ into a bounded set in $D(A)$. By Theorem 2.1, the proof is completed. □

Theorem 4.2

Under the assumptions (4)-(5) and $f\in L_{\mathrm{loc}}^{2}(\mathbb {R};H)$, $\frac{\mathrm{d}f}{\mathrm{d}t}\in L_{b}^{2}(\mathbb {R};H)$. Let $3<\beta\leq5$ and $\tau\in \mathbb {R}$, then the process $\{U(t,\tau )\}_{t\geq\tau}$ associated with (1) has a pullback $\mathcal {D}$-attractor $\mathcal{A}_{2}$ in $\mathbf{H}^{2}(\Omega)$.

Proof

Let $C=\{C(t):t\in \mathbb {R}\}$ be a pullback $\mathcal {D}$-absorbing set in $D(A)$ obtained in Lemma 3.6. Then we need only to show that for any $t\in \mathbb {R}$, any $\tau_{n}\rightarrow-\infty$, and $u_{0n}\in C(\tau _{n})$, $\{u_{n}(\tau_{n})\}_{n=0}^{\infty}$ is precompact in $\mathbf {H}^{2}(\Omega)$, where $u_{n}(\tau_{n})=u(t;\tau_{n},u_{0n})=U(t,\tau_{n})u_{0n}$.

Because $V\hookrightarrow H$ is compact, from Lemma 3.7 we know that

In the following, we prove that $\{u_{n}(\tau_{n})\}_{n=0}^{\infty}$ is a Cauchy sequence in $\mathbf{H}^{2}(\Omega)$. From (3) we have

Taking the inner product of (41) with $Au_{nk}(\tau_{nk})-Au_{nj}(\tau _{nj})$ we can obtain

Like the proof of Lemma 4.2 in [16], we can also prove

Taking into account (40), (42), and (43), we have

Now, because

we have

Equation (45) implies that the process $\{U(t,\tau)\}_{t\geq\tau}$ is pullback $\mathcal{D}$-asymptotically compact in $\mathbf{H}^{2}(\Omega )$. Combining Lemma 3.6, Lemma 4.1, and Theorem 2.1, yields Theorem 4.2 immediately. □

In this paper, we consider the 3D Navier-Stokes equations with nonlinear damping $\alpha|u|^{\beta-1}u$ with initial and Dirichlet boundary conditions which arises in the fluid dynamics. Under suitable assumptions on the external force function f, we obtain the existence of pullback $\mathcal{D}$-attractors of solutions in V and $\mathbf {H}^{2}(\Omega)$, respectively. In [16] and [17], we have discussed the existence of global attractors and uniform attractors of such 3D NSEs in V and $\mathbf{H}^{2}(\Omega)$ with $\frac{7}{2}\leq \beta\leq5$. From this paper, we find that the pullback $\mathcal {D}$-attractors can exist in V and $\mathbf{H}^{2}(\Omega)$ with $\beta \in(3,5]$. Obviously, this improves the results in [16] and [17].

The authors of this paper would like to express their sincere thanks to the reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. This work is supported in part by the National Natural Science Foundation of China (Nos. 11426171, 111501442, 11402194) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2016JM1025).

Affiliations

1. College of Mathematics and Information Science, Shannxi Normal University, Xi’an, 710062, China

   • Xue-li Song
   •  & Jian-hua Wu

2. College of Science, Xi’an University of Science and Technology, Xi’an, 710054, China

   • Xue-li Song
   •  & Fei Liang

Corresponding author

Correspondence to Xue-li Song.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of the three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Reprints and Permissions