Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents

1.   Introduction

The three dimensional incompressible Navier-Stokes equations is expressed as

which was proposed by Navier and Stokes respectively for the motion of incompressible viscous fluid with very small velocity gradient. Since last century, many mathematicians and physicists have studied the existence, uniqueness, regularity and long time behavior of Navier-Stokes equations deeply, and have obtained a series of significant results (see [1,3,19,21,24,29,33]). However, for the 3D Navier-Stokes model, the uniqueness of weak solution and the existence of strong solutions have not been solved as our best knowledge.

In the 1960s, Ladyzhenskaya [22,23] relaxed the limitation of small fluid velocity gradient, and derive several modified Navier-Stokes equations, one of which reads as

and reduced to

by Lions, see [27]. Note that it reduces into the classical Navier-Stokes equations (1) when $\mu_1 = 0$. Lions [27] proved the existence and uniqueness of global weak solutions for the initial boundary value problem of the model (3) by using the Faedo-Galerkin method. In 2008, Dong and Jiang [10] studied the optimal upper and lower bounds of the decay for higher order derivatives of the solution of (3). Under some assumptions on the external force and initial data, the existence and structure of uniform attractors of the system (3) were proved in [9]. Recently, Yang and Feng et al [37] considered the pullback dynamics of (3), and proved the existence of minimal and unique family of pullback attractors, and also presented the finite fractal dimension of pullback attractors. In addition, the upper semi-continuity of pullback attractor was also studied in [37] when the perturbed external force disappears as parameter tends to zero.

The Navier-Stokes equations with delays were firstly considered by Caraballo and Real in [5], then there are a lot of works concerning asymptotic behavior, stability, the existence of pullback attractors and the fractal dimensional of pullback attractors for time-delayed Navier-Stokes equations (see, e.g., [6,7,15,30,38]). It is worth to be pointed out that García-Luengo, Marín-Rubio and Real [15] obtained that the existence of pullback attractors for the 2D Navier-Stokes model with finite delay. Furthermore, the bounded fractal and Hausdorff dimension of the pullback attractors for 2D non-autonomous incompressible Navier-Stokes equations with delay was studied in [38]. The above work is to study the time delay which only exists in the external force. Later on, Planas et al [17,32] considered Navier-Stokes equations with double delays and proved the exponential stability of stationary solutions. For the double time-delayed 2D Navier-Stokes model, the existence of pullback attractors was proved in [12]. As far as we know, there are less results on the research for 3D modified Navier-Stokes equations with double time delays till we know now. Motivated by the results [12,15,37,39], this paper is concerned with the pullback dynamics for a three dimensional modified Navier-Stokes equation with double time delays defined on $(\tau,\infty)\times\Omega$, which can be described as

here $\Omega \in \mathbb{R}^3$ is a bounded domain with sufficiently smooth boundary $\partial\Omega$, $u = (u_1,u_2,u_3)$ is the velocity field of the fluid, $p$ is the pressure, $\mu_0>0$ and $\mu_1>0$ are the kinematic viscosities of the fluid, $f(x,t)$ is a generic external force, $g(t,u_t)$ is a external force with some hereditary characteristics and $h>0$ is a fixed positive constant. The function $u_t$ appeared in the delay term $g(t,u_t)$ is defined on $(-h,0)$ by the relation $u_t(s) = u(t+s),\ s \in (-h,0)$. Assume that the delay function $\rho$ in the convective term satisfies $\rho(t) \in C^1(\mathbb{R};[0,h])$ and $\rho'(t) \leq \rho^* < 1$ for all $t \in \mathbb{R}$, where $\rho^*$ is a constant.

The main features and results of this paper can be summarized as follows:

(Ⅰ) By using the Galerkin approximated technique and compact argument, we can prove the existence of global weak solution. Using the energy equation approach similar as in Ball [2], the pullback asymptotic compactness of the process can be shown, which leads to the existence of pullback attractors together with pullback dissipation of our problem. In addition, based on different universes, we can present the minimal family of pullback attractors on functional Banach space.

(Ⅱ) Since the topic in [16] only contains the delay on external force, the result in this paper is a further extension of our former result [16]. Moreover, the problem (4) has a delay function $\rho$ in the convection term, hence, there are more difficulties to achieve well-posedness and pullback asymptotic compactness, which also require appropriate restriction on the delay function $\rho$.

(Ⅲ) The upper semi-continuity of pullback attractors as perturbed delay can not be shown easily similar as in [16] since the lack of regular estimate in appropriate phase spaces.

The structure of this paper is arranged as follows. In Section 2, we give the definitions of some usual functional spaces and operators. Moreover, Some lemmas used later are also given in the end of the section. In Section 3, we first give the needed assumptions on the external forces, then show the abstract equivalent form of the system (4), and establish the well-posedness of the global weak solutions for system (4). Finally, the existence of the minimal pullback attractors for the abstract non-autonomous system is showed in Section 4.

2.   Mathematical setting

2.1.   Some usual functional spaces and operators

Denote

and $H$ is the closure of $E$ in $(L^2(\Omega))^3$ topology. The inner product and norm in $H$ are represented by $(\cdot,\cdot)$ and $|\cdot|$ respectively, which are defined as

$V$ is the closure of $E$ in $(H^1_0(\Omega))^3$ topology. The inner product and norm in $V$ are represented by $((\cdot,\cdot))$ and $\|\cdot\|$ respectively, which are defined as

Then we have $\|u\| = \|\nabla u\|_{(L^2(\Omega))^3} = |\nabla u|$ for all $u \in V$, and it is easy to verify that $H$ and $V$ are Hilbert spaces. Let $H'$ and $V'$ be dual spaces of $H$ and $V$ respectively. Then there is $V\hookrightarrow \hookrightarrow H \equiv H' \hookrightarrow V'$, where the injections are dense and continuous. Let $\|\cdot\|_*$ be the norm in $V'$, $\langle\cdot,\cdot\rangle$ be the dual product in $V$ and $V'$, where $\|\cdot\|_*$ is defined as

$P$ denotes the Helmholz-Leray orthogonal projection from $(L^2(\Omega))^3$ onto the space $H$ (see [11,34]). We define $A_1: = -P\Delta$ as the Stokes operator on $D(A_1) = (H^2(\Omega))^3 \cap V$, then $A_1:V\rightarrow V'$ satisfy $\langle A_1u,v\rangle = ((u,v))$, and $A_1$ is an isomorphism from $V$ into $V'$. There is $\|A_1u\|_* = \sup\limits_{v \in V,\|v\| = 1}|\langle A_1u,v\rangle| = \sup\limits_{v \in V,\|v\| = 1}|((u,v))|\leq\|u\|$, i.e., $\|A_1\|\leq 1$. Let $\{\lambda_i\}^\infty_{i = 1}$ be the eigenvalues of the operator $A_1$ with Dirichlet boundary condition, which satisfies $0<\lambda_1\leq \lambda_2 \leq\cdot\cdot\cdot$. By the property of the Stokes operator, the corresponding eigenfunctions $\{\omega_i\}_{i = 1}^\infty$ form an orthonormal complete basis in $H$. In addition, at this time we have the following Poincaré inequality

In order to deal with the nonlinear term $\mu_1{\|\nabla u\|}^2_{{L^2}(\Omega)}$, we define the operator $A_2: V\rightarrow V'$ as $A_2u: = -\mu_1\|\nabla u\|^2_{{L^2}(\Omega)}\Delta u$, which satisfies

It is easy to verify that $\langle A_2u-A_2v,u-v\rangle\geq 0$ for any $u,v \in V$, that is, $A_2$ is a monotone operator. We can obtain from (5) and (7) that

We also introduce the bilinear operator

and trilinear operator

2.2.   Some functional spaces with delay

We define some functional Banach space as

and some Lebesgue spaces on delayed interval as $L_H^p = L^p(-h,0;H)$ and $L_V^p = L^p(-h,0;V).$ The inner product and norm in $L^2_H$ are defined as

The inner product and norm in $L^2_V$ are defined as

2.3.   Some Lemmas

The following lemmas are used to prove the existence of weak solutions.

Lemma 2.1. ([24,34]) The bilinear operator $B(u,v)$ and trilinear operator $b(u,v,\omega)$ satisfy the properties

Lemma 2.2. ([41]) Suppose that $A$ be a nonlinear monotone operator from a separable Banach space $V$ to $V'$ satisfying conditions

(i) For all $v \in V$,

where $1 < p < \infty$ and $C$ is a positive constant independent of $v$.

(ii) (semi-continuous) for all $u,v,\omega \in V$ and $\lambda \in \mathbb{R}, \langle A(u +\lambda v),\omega\rangle$ is a continuous function of $\lambda$.

If $u_n \in L^p([0,T],V)$ with $1 < p < \infty$ such that

with $\frac 1p + \frac 1{p'} = 1$, and

then

3.   Global well-posedness

The following assumptions on the external forces are imposed for our results.

$(H_g)$ Let the function $g : \mathbb{R} \times C_H \rightarrow H$ satisfies the following properties:

(a) The function $g(\cdot,\xi)$ is measurable for any $\xi \in C_H$ and $g(\cdot,0) \equiv 0.$

(b) There exists a constant $L_g > 0$ such that for all $\xi, \eta \in C_H$,

(c) There exists a constant $C_g > 0$ such that for all $u, v \in C([\tau-h,t];H)$,

$(H_f)$ The function $f \in L^{\frac{4}{3}}_ {loc}(\mathbb{R},V')$ satisfies that there exists some $\sigma \in (0,\mu_0 \lambda_1 )$ such that

Based on the previous definitions of operators $P$, $A_1$ and $A_2$, the system (4) can be written as the following abstract equivalent form

Definition 3.1. Let $T>\tau$, assume that the initial data $(u^\tau,\phi) \in H\times (C_H\cap L^2_V) \triangleq M_H$, $f \in L^{\frac{4}{3}}_ {loc}(\mathbb{R},V')$ and $g$ satisfies the assumption $(H_g)$, a function $u = u(t,x) \in C([\tau-h,T];H) \cap L^4 (\tau,T;V)$ is called a weak solution to problem (4) if

holds for all $v \in V$ in the sense of $D'(\tau,T)$.

Theorem 3.2. For any $T > \tau$, if the initial data $(u^\tau,\phi) \in M_H = H\times (C_H\cap L^2_V)$, $f \in L^{\frac{4}{3}}_</italic> <italic> {loc}(\mathbb{R},V')$ and $g$ satisfies the assumption $(H_g)$, then problem (4) possesses a unique weak solution $u(t,x) \in C([\tau-h,T];H) \cap L^4(\tau,T;V)$.

Proof. Step 1. Local approximating sequence.

From the property of the Stokes operator $A_1$, we can see that the sequence of characteristic functions $\{\omega_i\}_{i = 1}^\infty \subset D(A_1)$, satisfying $A_1\omega_i = \lambda_i\omega_i$, constitutes a complete orthogonal basis in $H$. Let $H_m = \text {span}\{\omega_1,\cdot\cdot\cdot,\omega_m\}$, and projection $P_m:H\rightarrow H_m$ is defined as

Let $u_m(t) = \sum\limits_{i = 1}^{m}h_{im}(t)\omega_i$ be the approximated solutions satisfying the following Cauchy problem

The problem (9) is equivalent to a system of functional differential equation with respect to the unknown variables $\{h_{1m}(t),h_{2m}(t),\cdot \cdot \cdot ,h_{mm}(t)\}$. According to Theorem A1 in the appendix of [5] for the Cauchy problem of functional differential equation, the system (9) possesses a local solution on the interval $[\tau,t_m]$, that is, there exists a solution $u_m(t) = \sum\limits_{i = 1}^{m}h_{im}(t)\omega_i$ satisfying the approximation problem (9) on $[\tau,t_m].$

Step 2. The priori estimates for $\{u_m\}$ and $\{\frac{\partial u_m}{\partial t}\}$.

Multiplying (9) by $h_{im}(t)$, and then summing from $i = 1$ to $i = m$, in view of Young's inequality and the embedding $V\hookrightarrow H$, we obtain

Integrating (10) with respect to the variable $t$ from $\tau$ to $t$, using the definition of operator $P_m$, $f \in L^{\frac{4}{3}}_ {loc}(\mathbb{R},V')$ and the assumption $(H_g)$, we have

Thus, it follows from Gronwall's inequality that

which implies $t_m = T$ and

In particular, the sequence of functions $\{u_m(t-\rho(t))\}$ is bounded in $L^2 (\tau,T;V )$. Furthermore, approximate equations (9) can be rewritten as

We notice that $\|P_m\|_{\mathscr{L}(V,V)}\leq1$ and $P_m^* = P_m,$ then $\|P_m\|_{\mathscr{L}(V',V')}\leq1$. By the assumption $(H_g)$, in particular, we get

which implies that $g(s,u_{ms})$ is bounded in $L^{\frac43}(\tau,T;V')$.

For the convective term with delay, the desired estimation can be obtained by using Lemma 2.1 and Young's inequality. Indeed, it holds that

i.e., $B(u_m(t-\rho(t)),u_m)$ is bounded in $L^{\frac43}(\tau,T;V')$. By virtue of $\|A_1\| \leq 1$, it is easy to verify from (8) and (11) that

Combining (13)-(15), $f \in L^{\frac{4}{3}}_ {loc}(\mathbb{R},V')$ and (12), we can get

Step 3. Compact results and strong convergence.

Let

Now, applying the Aubin-Lions Lemma, we can derive

By using (11) and (15)-(17), we deduce that there exist a subsequence (still denote it by $\{u_m\}$) and $u \in L^\infty(\tau-h,T;H) \cap L^4 (\tau,T;V)$ such that

and

Next, we prove that

Let $u_m$ and $u_n$ be two solutions to the approximated system corresponding to initial value $(u_m^\tau,\phi_m)$ and $(u_n^\tau,\phi_n)$ respectively, and set $\omega(t) = u_m(t)-u_n(t)$, then we can get

We observe the fact that

Multiplying (20) by $\omega(t)$, then integrating the resultant over $\Omega$ and applying (21), we obtain

Thanks to the assumption $(H_g)$ and the monotonicity of operator $A_2$, we can deduce from Lemma 2.1 and Young's inequality that

Here $\varepsilon$ is a positive number, which is determined later. Note the fact that $|\omega(s)| \leq \|\omega_s\|_{C_H}$. Integrating (22) with respect to time $s$ on $[\tau,t]$, it holds that

With the help of a change of variable in the integral of $\omega(s-\rho(s))$ and $\rho'(t) \leq \rho^* < 1$, we can deduce by choosing $\varepsilon^2 = \frac{16(1-\rho*)}{9}\mu_0^2$ that

For simplicity, we set that $C' = (\frac {C^4}{2\varepsilon^3}+2L_g)$. If $t \geq \tau+ h$, it holds that $t+\theta \geq \tau$ for all $\theta \in[-h,0]$ and

Consequently, it follows that for all $t \geq \tau+ h$,

which, by the Gronwall inequality, implies

Since $\phi_m\rightarrow \phi$ in $C_H\cap L^2_V$, (24) indicates that $\{u_m\}$ is a Cauchy sequence in $C([\tau-h,T];H).$ Thus we complete the proof of (19).

Step 4. Passing the limit and uniqueness.

In order to pass the limit of (9), we need to discuss the convergence of nonlinear terms $b(u_m,u_m,\omega_i)$ and $\langle A_2u_m,\omega_i\rangle$. Using the properties of trilinear operator $b(u,v,\omega)$, Lemma 2.1, and the fact that $\omega_i$ is an eigenfunction of the Stokes operator, we obtain

We use the Hölder inequality to give

which, together with $(18)_1$, $(18)_3$ and (25), yields that

By a similar technique to the proof of Lemma 3.2 in [35], we can deduce

Moreover, it is easy to verify that the operator $A_2$ satisfies the conditions $(i)$ and $(ii)$ of Lemma 2.2. Combining $(18)_5$, ${(18)_7}$, (27) and Lemma 2.2, we get

Now, passing to the limit of (9), by combining (18)-(19), (26) and (28), we can infer that $u$ is indeed a weak solution to problem (4).

Last, we consider the uniqueness of solutions for our problem. Let $u,v$ be two weak solutions with the same initial $(u^\tau,\phi)$ and set $\omega = u -v$. By a similar technique as proving (23), we can obtain that for $t \in [\tau, T]$,

By Gronwall's inequality, it yields $\omega(t)\equiv 0,$ i.e., the solution is unique. Similarly, we can also verify that the solution is continuously dependent on the initial value.

4.   Pullback dynamics for (4)

In this section, we shall obtain the existence of pullback attractors for the process associated to (4).

4.1.   Preliminaries for pullback attractors

Let $(X,d_X)$ be a given metric space, and we denote $\mathbb{R}^2_d = \{(t,\tau)\in \mathbb{R}^2:\tau\leq t\}$.

A process on $X$ is a mapping $U(\cdot,\cdot)$ such that $\mathbb{R}^2_d \times X \ni(t,\tau,x)\mapsto U(t,\tau)x \in X$ with $U(\tau,\tau)x = x$ for any $(\tau,x)\in \mathbb{R}\times X$, and $U(t,r)U(r,\tau)x = U(t,\tau)x$ for any $\tau \leq r \leq t$ and all $x \in X$. A process $U(\cdot,\cdot)$ is said to be continuous if for any pair $\tau \leq t$, $U(t,\tau) :X \rightarrow X$ is continuous. A process $U(\cdot,\cdot)$ is said to be closed if for any $\tau \leq t$, and any sequence $\{x_n\}\subset X$, if $x_n \rightarrow x \in X$ and $U(t,\tau)x_n\rightarrow y \in X$, then $U(t,\tau)x = y$. Clearly, every continuous process is closed.

Let $\mathcal{P}(X)$ be the family of all nonempty subsets of $X$, and consider a family of nonempty sets parameterized in time $\widehat{D}_0 = \{D_0(t):t\in \mathbb{R}\}\subset \mathcal{P}(X)$. Let $\mathcal{D}$ be a nonempty class of families parameterized in time $\widehat{D} = \{D(t) : t \in \mathbb{R}\} \subset \mathcal P(X)$. The class $\mathcal{D}$ will be called a universe in $\mathcal P(X).$

Definition 4.1. (1) A process $U(\cdot,\cdot)$ on X is said to be pullback $\widehat{D}_0$-asymptotically compact if for any $t \in \mathbb{R}$ and any sequences $\{\tau_n \} \subset (-\infty,t]$ and $\{x_n \} \subset X$ satisfying $\tau_n \rightarrow -\infty$ and $x_n \in D_0(\tau_n)$ for all n, the sequence $\{U(t,\tau_n )x_n \}$ is relatively compact in $X$.

(2) Further, we say that a process $U(\cdot,\cdot)$ on X is pullback $\mathcal{D}$-asymptotically compact if it is pullback $\widehat{D}$-asymptotically compact for all $\widehat{D} \in \mathcal{D}.$

Definition 4.2. $\widehat{D}_0 = \{D_0(t):t\in \mathbb{R}\}\subset \mathcal{P}(X)$ is pullback $\mathcal{D}$-absorbing for the process $U(\cdot,\cdot)$ on X if for any $t \in \mathbb{R}$ and any $\widehat{D} = \{D(t) : t \in \mathbb{R}\} \in \mathcal{D}$, there exists a $\tau_0(t,\widehat{D})\leq t$ such that

Observe from the above definition that $\widehat{D}_0$ does not belong necessarily to the class $\mathcal{D}$.

Denote

where $\overline{\{\cdot\cdot\cdot\}}^X$ is the closure in $X$. We denote by $dist_{X}(X_1,X_2)$ the Hausdorff semi-distance in $X$ between two sets $X_1$ and $X_2$, defined as

In order to get our result, we need to use a classical theorem in [14,15].

Theorem 4.3. Consider a closed process $U:\mathbb{R}^2_d \times X \mapsto X$, a universe $\mathcal{D} \in \mathcal{P}(X)$, and a family $\widehat{D}_0 = \{D_0(t):t\in \mathbb{R}\}\subset \mathcal{P}(X)$ which is pullback $\mathcal{D}$-absorbing for U, and assume also that U is pullback $\widehat{D}_0$-asymptotically compact. Then, the family $\mathcal{A}_{\mathcal{D}} = \{\mathcal{A}_{\mathcal{D}} (t) : t \in \mathbb{R}\}$ is a family of pullback $\mathcal{D}$-attractors which is defined by

and has the following properties:

(a) for any $t \in {\mathbb{R}}$, the set $\mathcal{A}_{\mathcal{D}} (t)$ is a nonempty compact subset of $X$, and $\mathcal{A}_{\mathcal{D}}(t) \subset \Lambda(\widehat{D}_0,t)$.

(b)$\mathcal{A}_{\mathcal{D}}$ is pullback $\mathcal{D}$-attracting, i.e.,

(c)$\mathcal{A}_{\mathcal{D}}$ is invariant, i.e., $U(t,\tau)\mathcal{A}_{\mathcal{D}}(\tau) = \mathcal{A}_{\mathcal{D}} (t)$, $\forall \tau \leq t$.

(d) if $\widehat{D}_0 \in \mathcal{D}$, then $\mathcal{A}_{\mathcal{D}}(t) = \Lambda(\widehat{D}_0,t)\subset\overline{D_0(t)}^X$, $\forall t \in \mathbb{R}$.

Moreover, family $\mathcal{A}_{\mathcal{D}}$ is minimal in the sense that if $\widehat{C} = \{C(t):t \in \mathbb{R}\} \subset \mathcal{P}(X)$ is a family of closed sets such that for any $\widehat{D} = \{D(t):t \in \mathbb{R}\} \in \mathcal{D}$,

then $\mathcal{A}_{\mathcal{D}}(t) \subset C(t)$.

Remark 1. If $\widehat{D}_0 \in \mathcal{D},$ $D_0 (t)$ is closed for any $t \in \mathbb{R}$ and $\mathcal{D}$ is inclusion-closed, then $\mathcal{A}_{\mathcal{D}} \in \mathcal{D}$ and $\mathcal{A}_{\mathcal{D}}$ is unique family in $\mathcal{D}$ that satisfies (a), (b) and (c) above.

Let $\mathcal{D}_F^X$ be the universe of fixed nonempty bounded subsets of $X$, i.e., the class of all families $\widehat{D}$ of the form $\widehat{D} = \{D(t) = D : t \in \mathbb{R}\}$ with $D$ a fixed nonempty bounded subset of $X$.

Lemma 4.4. ([14]) Under the assumptions of Theorem 4.3, if the universe $\mathcal{D}$ contains the universe $\mathcal{D}_F^X$, then both attractors, $\mathcal{A}_{\mathcal{D}_F^X}$ and $\mathcal{A}_{\mathcal{D}}$, exist, and the following relation holds:

4.2.   The process and pullback dissipation

In view of Theorem 3.2, here we take the phase space $M_H = H\times(C_H\cap L_V^2)$ equipped with the norm $\|(\xi,\phi)\|_{M_H} = \|\xi\|_H+\|\phi\|_{C_H}+\|\phi\|_{L_V^2}$ for $(\xi,\phi) \in M_H$. It is easy to verify the following proposition.

Proposition 1. Consider given $g:\mathbb{R}\times C_H \rightarrow H$ and $f \in L_{loc}^{\frac{4}{3}}(\mathbb{R};V')$ satisfying assumptions $(H_g)$ and $(H_f)$. Then the solution of problem (1.1) generates a bipara-metric family of mappings $U(t,\tau):M_H \rightarrow M_H$ by $U(t,\tau)(u^\tau,\phi) = (u(t;\tau,u^\tau,\phi),u_t(\cdot;\tau,u^\tau,\phi))$, which is a continuous process.

In order to prove the existence of pullback attractors for the process $U$, we need the following assumption:

$(H_1)$ For every $u \in L^2(\tau-h,T;V)$, there exists a value $\sigma \in (0,\mu_0\lambda_1 )$ which is independent of $u$ such that

Lemma 4.5. $g:\mathbb{R}\times C_H \rightarrow H$ and $f \in L_{loc}^{\frac{4}{3}}(\mathbb{R};V^\prime)$ satisfy assumptions $(H_g)$, $(H_1)$ and $(H_f)$. Then, for any $(u^\tau,\phi) \in M_H$, there exists a value $\sigma \in (0,\mu_0\lambda_1 )$ such that the solution u of (4) holds the estimates

and

Proof. Multiplying (4) by $u$ and integrating the resultant over $\Omega$, using integration by parts and Young's inequality, we have

Now multiplying (32) by 2$e^{\sigma t}$ and applying Poincaré's inequality (6), one has

We integrate (33) over the interval $[\tau,t]$ with respect to $t$, use $\sigma \in (0,\mu_0\lambda_1 )$ and assumption $(H_1)$, to conclude

On account of the assumption $(H_f)$, we apply the Gronwall inequality to (34) to derive

which means

Consequently, the estimation (29) is proved. Thanks to (32) and (6), we obtain

Integrating (35) over the interval $(s,t)$, and using the assumptions on $g$, we can get

Thus, (30) and (31) are obtained immediately from (36). Now we complete the proof.

Definition 4.6. (Universe) We will denote by $\mathcal{D}_\sigma^{M_H}$ the class of all families of nonempty subsets $\widehat{D} = \{D(t):t \in \mathbb{R}\} \subset \mathcal{P}(M_H)$ such that

Remark 2. According to the above definition and the notation $\mathcal{D}_F^X$, it is obvious that $\mathcal{D}_F^{M_H}\subset \mathcal{D}_\sigma^{M_H}$ and that both are inclusion-closed.

Based on the above universe and Lemma 4.5, we can present the pullback dissipation in $M_H$. Let $\overline{B}_X(0,R)$ be the closed ball with zero as the center and $R$ as the radius in $X$.

Proposition 2. Suppose that $g:\mathbb{R}\times C_H \rightarrow H$ and $f \in L_{loc}^{\frac{4}{3}}(\mathbb{R};V^\prime)$ satisfy assumptions $(H_g)$, $(H_f)$ and $(H_1)$. Then, the family $\widehat{D}_0 = \{D_0(t):t\in \mathbb{R}\}\subset M_H$ is defined by

is pullback $\mathcal{D}_\sigma^{M_H}$-absorbing for the process $U(t,\tau)$ on $M_H$ and $\widehat{D}_0 \in \mathcal{D}_\sigma^{M_H}$, where

Proof. Fix $t \in \mathbb{R}$, we derive from (29) that there exists a pullback time $\tau(\widehat{D},t) \leq t-2h$ such that

holds for all $t \geq \tau$ with $\tau \leq \tau(\widehat{D},t)$ and $(u^\tau,\phi) \in D(\tau)$.

In particular, we observe that $\|u_t\|_{C_H}\leq R_H^2(t).$ Now, putting $s = t - h$ in the estimate (30) and using (29), we deduce immediately that $\|u_t\|_{L^2_V}^2 \leq R_{L^2_V}^2(t).$ Moreover, in view of the above estimates, $(H_f)$ and the definition of universe, we can see clearly the fact that $\widehat{D}_0$ belongs to $\mathcal{D}_\sigma^{M_H}.$ Therefore, the proof is done.

4.3.   Pullback asymptotic compactness

To use Theorem 4.3, we also need to establish the asymptotically compact of the process. We give the following result.

Theorem 4.7. Assume that $g:\mathbb{R}\times C_H \rightarrow H$ and $f \in L_{loc}^{\frac{4}{3}}(\mathbb{R};V^\prime)$ satisfy assumptions $(H_g)$ and $(H_1)$. Then, the process $U(t,\tau)$ defined in proposition 1 is pullback $\mathcal{D}_\sigma^{M_H}$-asymptotically compact.

Proof. Fix a value $t \in \mathbb{R}$ and consider a family $\widehat{D} \in \mathcal{D}_\sigma^{M_H}$, let $\{\tau_n \} \subset (-\infty,t]$ with $\tau_n\rightarrow -\infty$ and $\{(u^{\tau_n},\phi^n\})$ with $\{(u^{\tau_n},\phi^n)\} \in D(\tau_n)$ be two sequences for all $n$, then we denote $\{(u^n,u^n_t )\} \in \widehat{D}$ as a sequence with $u^n (\cdot) = u(\cdot;\tau_n ,u^{\tau_n} ,\phi^n)$.

In the same way as Proposition 2, by using the estimations in Lemma 4.5, we obtain that there exists a pullback time $\tau_1(\widehat{D},t) \leq t-4h-1$ such that the subsequence $\{u^n:\tau_n \leq \tau_1(\widehat{D},t)\}$ is bounded in $L^\infty(t-4h-1,t;H) \cap L^4 (t-3h-1,t;V).$ By virtue of the equation (12) and the embedding theorem, we have that

By applying a similar technique as proving (16) in Theorem 3.2, we derive from the assumption $(H_g)$ that $\{(u^n)'\}$ is uniformly bounded in $L^\frac43(t-2h-1,t;V')$. Thanks to the Aubin-Lions Lemma, the assumptions on $g$ and the diagonal procedure, there exists a subsequence (still denote it by $\{u^n\}$) and a function $u \in L^\infty(t-4h-1,t;H) \cap L^4 (t-3h-1,t;V)$ such that

From the above convergences, we derive that $u \in C([t-2h-1,t];H)$ is a weak solution for problem (1.1) in the interval $(t-h-1,t)$ with $u_{t-h-1}$ as initial data.

By using the same technique as proving (19) in Theorem 3.2, we can get

Consequently, we obtain that for any sequence $\{s_n\} \subset [t-h-1,t]$ with $s_n\rightarrow s_*$,

Now, our goal is to obtain that

First of all, we can conclude from the weak convergence (38) that

Furthermore, in view of the energy equality (32), we infer that for all $t-h-1 \leq s_1 \leq s_2 \leq t$,

where $y$ can be $u$ and all $u^n$. Hence we can define the continuous functions on the interval $[t-h-1,t]$ as

and

It is clear that $\mathcal{J}$ and $\mathcal{J}_n$ are non-increasing functions. In addition, the convergences (37) indicates that

Therefore, it is possible to choose a sequence $\{s_k\} \subset (t-h-1,t)$ satisfying that $\lim\limits_{k\rightarrow \infty}s_k = s_*$ and

Since $\mathcal{J}(s)$ is continuous, for $\forall \varepsilon>0$, there exists $k_\varepsilon \in \mathbb{N}$ such that

Because $\mathcal{J}_n(s)$ is uniformly continuous with respect to time $s$, there exists $n(k_\varepsilon)$ such that

According to the non-increasing property of all $\mathcal{J}_n$ and (41)-(42), we derive that for all $n \geq n(k_\varepsilon)$,

Because of the arbitrariness of $\varepsilon$, we have that $\limsup\limits_{n\rightarrow \infty} \mathcal{J}_n(s_n) \leq \mathcal{J}(s_*)$, which, by virtue of (37) again, implies

Therefore, combining (38), (39) and (43), we conclude that

By using again the energy equality (40) satisfied by $u$ and $u^n$, the convergences in (37) and (44), we can deduce that

which, together with the weak convergence already proved in (37), gives

Combining (44) with (45), we derive that the process is pullback $\mathcal{D}_\sigma^{M_H}$-asymptotically compact. Thus we finish the proof.

4.4.   Existence of pullback attractors

In this subsection, by using the results obtained in subsection 4.2 and subsection 4.3, we shall establish the main result of the paper as follows.

Theorem 4.8. Assume that $g:\mathbb{R}\times C_H \rightarrow H$ satisfies $(H_g)$ and $f \in L_{loc}^{\frac{4}{3}}(\mathbb{R};V^\prime)$ fulfills conditions $(H_f)$ and $(H_1)$. Then, there exist the minimal pullback $\mathcal{D}_F^{M_H}$-attractor

and the minimal pullback $\mathcal{D}_\sigma^{M_H}$-attractor

for the process defined in Proposition 1. The family $\mathcal{A}_{\mathcal{D}_\sigma^{M_H}} \in \mathcal{D}_\sigma^{M_H}$ and the following relation holds:

Moreover, the pullback attractor $\mathcal{A}_{\mathcal{D}_\sigma^{M_H}}$ is unique (in the sense of Remark 1).

Proof. From Proposition 1, we observe that the process $U$ is continuous in $M_H$. Furthermore, we can also obtain that there exists a pullback absorbing family $\widehat{D}_0 \in \mathcal{D}_\sigma^{M_H}$ from Proposition 2 and the process $U$ is pullback $\mathcal{D}_\sigma^{M_H}$-asymptotically compact from Theorem 4.7. Consequently, by using Theorem 4.3 and Lemma 4.4, we derive that the pullback attractors $\mathcal{A}_{\mathcal{D}_\sigma^{M_H}}$ and $\mathcal{A}_{\mathcal{D}_F^{M_H}}$ exist and

Since $\widehat{D}_0 \in \mathcal{D}_\sigma^{M_H},$ $D_0 (t)$ is closed for any $t \in \mathbb{R}$ and $\mathcal{D}_\sigma^{M_H}$ is inclusion-closed, we conclude from Remark 1 that $\mathcal{A}_{\mathcal{D}_\sigma^{M_H}}$ belongs to $\mathcal{D}_\sigma^{M_H}$ and $\mathcal{A}_{\mathcal{D}_\sigma^{M_H}}$ is unique. Moreover, in view of the property $(d)$ in Theorem 4.3, we can get

which, along with (47), gives (46). The proof is complete.

Appendix: Background of the modified fluid flow models

In the Appendix of Ladyzhenskaya [23], the classical incompressible Navier-Srokes equations is approximated by using a class of regular Navier-Stokes systems which are described as

and its special case (2), which reflects the physical phenomena that $\|\nabla u(x,t)\|_{L^2(\Omega)}$ should not be too large or infinite. In this line of work, Smagorinsky in 1960s proposed a similar approximating equation, known as Ladyzhenskaya-Smagorinsky model

However, even for these systems, the uniqueness and stability are still open questions when Reynold number is large. To overcome this difficulty and simplify Ladyzhenskaya models, Lions [27] replaced $\mathcal{D}u$ by $\nabla u$, and thus deduced another two systems as

and

Lions [27] proved the existence of weak solutions to systems (50) and (51) for $p \geq 1 + 2n/(n+2)$ and the uniqueness of solutions to system (50) for $p \geq (n+2)/2$. However, the uniqueness of the system (51) is still an open problem.

Acknowledgments

The authors are grateful to the referees for their helpful suggestions which improved the presentation of this paper. Lan Huang was partially supported by the NSFC (No. 11501199 and No. 11871212) and the Young Key Teachers Project in Higher Vocational Colleges of Henan Province (No. 2020GZGG109). Xin-Guang Yang was partially supported by the Fund of Young Backbone Teachers in Henan Province (No. 2018GGJS039), Incubation Fund Project of Henan Normal University (No. 2020PL17) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003).