Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains

Abstract

The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor $u(t+\theta )$ in the delay term may lead to oscillations, hysteresis effects, and other phenomena in the solution, which increases the difficulty of studying the well-posedness and asymptotic behavior of the solution. This study investigates the global well-posedness and asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations incorporating infinite delays. To establish global well-posedness, we first construct several suitable function spaces and then prove them using the Galekin approximation method. Then, by accurately estimating the number of determining nodes, we reveal the asymptotic behavior of the solution. The results indicate that the long-term behavior of a strong solution can be determined by its values at a finite number of nodes.

1. Introduction

In this paper, we study the non-autonomous Navier–Stokes model with infinite delay in 2D bounded domains, which is given by the following equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}-\nu \Delta u+(u·\nabla )u+\nabla p=f\left(t\right)+G\left(t,u(t+\theta )\right)\mathrm{in}(x,t)\in \mathrm{\Omega }\times (\tau ,T),}\hfill \\ \mathrm{div}u=0\mathrm{in}(x,t)\in \mathrm{\Omega }\times (\tau ,T),\hfill \\ u=0\mathrm{on}(x,t)\in \partial \mathrm{\Omega }\times (\tau ,T),\hfill \\ u(x,\tau +\theta )=\varphi (x,\theta )x\in \mathrm{\Omega }\theta \in (-\infty ,0],\hfill \end{array}\right.\end{array}$$

(1)

where the unknown functions $u(x,t)=(u^{\left(1\right)}(x,t),u^{\left(2\right)}(x,t))$ and $p=p(x,t)$ represent the velocity field and the pressure of the fluid motion, respectively. The function $f\left(t\right)=(f^{\left(1\right)}\left(t\right),f^{\left(2\right)}\left(t\right))$ is the external force, which is a given vector function that varies with time t. The parameter $\nu >0$ denotes the viscosity coefficient. The function $u(t+\theta )$ represents the history of the state $u(·)$ at time t, where $\theta \in (-\infty ,0]$. The function $G(·,·)$ is a known continuous function, as required by the study.

In addition, $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is an open set with smooth boundaries, and the function $\varphi (x,\theta )$ is the initial value of Equation (1) in the delay time $(-\infty ,0]$. These assumptions typically serve as the basis for ensuring the well-posedness of the solution to Equation (1) in $\mathrm{\Omega }$. They also imply that the boundary has well-defined tangent vectors at each point and that the shape of the boundary does not exhibit sharp points or discontinuous regions. The assumption of regularity in the initial data is to satisfy the requirements of adaptability, regularity, and long-term behavior of the solution to Equation (1) in $\mathrm{\Omega }$.

The Navier–Stokes equations describe the motion of fluids, involving multiple factors such as velocity, pressure, and viscosity. Introducing delay factors allows for a more realistic simulation of inertia and time-delay effects in fluid flow, including interactions at fluid–solid interfaces and the impact of temperature variations on fluid states. In many physical phenomena, the response of fluids is not immediate but involves a time delay, which leads to the influence of time delay on the motion state of fluids and indirectly affects the stability of the system. Therefore, studying the Navier–Stokes equations with delay provides insights into flow stability across varying parameters and elucidates how delays influence flow transitions and turbulence onset. Incorporating delay factors enhances model accuracy, improving the prediction and understanding of fluid behavior. The study of Navier–Stokes equations with delay is not limited to traditional fluid mechanics but also involves multiple fields such as control theory, nonlinear dynamics, and mathematical physics, promoting interdisciplinary research and communication.

Owing to their wide applications, the Navier–Stokes equations have been extensively studied by mathematicians and physicists. In particular, the well-posedness and regularity of solutions are the most significant concerns. Several papers have been dedicated to investigating the well-posedness of solutions, including references [1,2,3,4]. Foias and Temam, among others, introduced the concepts of definite modules, degrees of freedom, and determining nodes of the system in [5]. These concepts describe how the asymptotic behavior of the system can be determined by a finite number of quantities, revealing the internal structural properties of the attractor. The term “determining nodes” refers to the process of numerically solving partial differential equations, where the solutions at certain specific grid points or nodes are uniquely determined by specific boundary conditions, initial conditions, or solutions at other nodes. In other words, the values of these nodes do not depend on the values of other unknown nodes but are directly calculated based on given conditions, thereby reducing uncertainty in the solving process. This concept is further elaborated in references [6,7,8,9,10,11,12,13,14,15]. Kakizawa used the energy method to prove the asymptotic behavior of strong solutions to the initial-boundary value problem for general semilinear parabolic equations in [12]. Additional related research can be found in references [6,7,8,9,11,14,16,17].

The phenomenon of time delay is ubiquitous and closely related to real life. For example, when we aim to control or change the original state of a system by applying external forces, we must consider not only the current state of the system but also the impact of its previous states, i.e., the lag factor. Therefore, time delay effects are often incorporated into models when dealing with practical problems. Examples include the delayed Navier–Stokes model (see [16,18]) and wave equations with time delay (see [19]). Hernandez and Wu studied the well-posedness and global attractors of abstract Cauchy problems with state-dependent delay and provided examples of PDEs with state-dependent delay in [20], which offers important insights into the application of state-dependent delay in specific PDEs and the study of their well-posedness and asymptotic solutions. Other related results can be found in references [21,22]. Regarding the Navier–Stokes equations, García-Luengo, Marín-Rubio, and Real investigated the asymptotic behavior of the Navier–Stokes model with finite delay in [8,14], while Chueshov studied the finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, using the Galerkin method to prove global well-posedness in [21]. For the study of infinite delay, Fu and Liu investigated the well-posedness of solutions to a class of second-order non-autonomous abstract time-delay functional differential equations with infinite delay and demonstrated the application of their conclusions through examples (see Section 6 in [23]), which can be referenced in [23,24,25]. However, these articles only studied the well-posedness of solutions for non-autonomous systems with infinite time delay. They did not address the asymptotic behavior and regularity of solutions, and the equations provided in the examples were not specifically applied to several specific differential equations. In this paper, we apply the above research conclusions to the Navier–Stokes equations. Based on the extensive foundational work on well-posedness established by Caraballo et al., we use the theory of determining nodes to characterize the asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations with infinite delay by estimating their number. The most crucial step in studying the process and methods of non-autonomous Navier–Stokes equations with infinite delay is to handle the infinite time delay term $u(t+\theta )$, which is particularly important for studying the well-posedness of nonlinear PDEs with time delay effects. In addition, dealing with the delay term is usually the most critical step in the research process of the well-posedness and asymptotic behavior of nonlinear PDE solutions with other delay effects.

The paper is arranged as follows: In Section 2, we first establish several basic function spaces and several operators, then introduce several relevant theorems and define the weak and strong solutions of Equation (1). In addition, to deal with the infinite delay term $u(t+\theta )$, we refer to descent function space ${\mathcal{C}}_{\gamma }\left(H\right)$. Unlike the finite delay term, we not only need to consider the influence of the system state at a certain time in the past, but also need to consider the comprehensive influence of the system state at the past time. In Section 3, we investigate the global well-posedness of the strong solutions for Equation (1) by the Galerkin method without considering the pressure term p. In Section 4, we commit our focus to proving Equation (1) has a finite number of determining nodes.

2. Preliminaries

In this section, we introduce several common conclusions that are essential applications that we used in this paper.

2.1. Basic Function Spaces and Conclusions

We introduce several basic function spaces, which are as follows:

$$\begin{array}{cc}\hfill & \mathcal{V}:=\left\{\phi \in {\left(C_0^{\infty }\left(\mathrm{\Omega }\right)\right)}^2:\mathrm{div}\phi =0,\phi =({\phi }^{\left(1\right)},{\phi }^{\left(2\right)})\right\};\hfill \\ \hfill & H:=\mathrm{closure}\mathrm{of}\mathcal{V}\mathrm{in}{\left(L^2\left(\mathrm{\Omega }\right)\right)}^2,\mathrm{with}\mathrm{norm}{\parallel ·\parallel }_H;\hfill \\ \hfill & V:=\mathrm{closure}\mathrm{of}\mathcal{V}\mathrm{in}{\left(H_0^1\left(\mathrm{\Omega }\right)\right)}^2,\mathrm{with}\mathrm{norm}{\parallel ·\parallel }_V;\hfill \\ \hfill & V^{\prime }:=\mathrm{dual}\mathrm{space}\mathrm{of}V\mathrm{with}\mathrm{norm}{\parallel ·\parallel }_{V^{\prime }}.\hfill \end{array}$$

The definitions of norms ${\parallel ·\parallel }_H$ and ${\parallel ·\parallel }_V$ are as follows:

$${\parallel \phi \parallel }_H={\left({\int }_{\mathrm{\Omega }}{\left|\phi \right|}^2\mathrm{d}x\right)}^{{\displaystyle \frac{1}{2}}},{\parallel \phi \parallel }_V={\left({\int }_{\mathrm{\Omega }}\left({\left|\phi \right|}^2+{|\nabla \phi |}^2\right)\mathrm{d}x\right)}^{{\displaystyle \frac{1}{2}}};$$

for convenience, we use “$\parallel ·\parallel$” as “${\parallel ·\parallel }_H$”. In addition, we denote the inner product in $L^2\left(\mathrm{\Omega }\right),H$ by $(·,·)$ and denote the inner product in V by $\left(\right(·,·\left)\right)$. In addition, we denote the dual product between V and $V^{\prime }$ by $⟨·,·⟩$.

In order to be able to express Equation (1) as an abstract equation, we define several operators. Firstly, we define the Stokes operator A as follows:

$$\begin{array}{c}\hfill ⟨Au,\phi ⟩=\nu \left(\right(u,\phi \left)\right)=\nu \sum _{i,j=1}^2{\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial u_i}{\partial x_j}}{\displaystyle \frac{\partial {\phi }_i}{\partial x_j}}\mathrm{d}x,\end{array}$$

(2)

for any $u=(u^{\left(1\right)},u^{\left(1\right)}),\phi =({\phi }^{\left(1\right)},{\phi }^{\left(2\right)})\in V$. Moreover, $D\left(A\right):=V\cap {\left(H^2\left(\mathrm{\Omega }\right)\right)}^2$, and the operator A is the linear continuous operator, which both form V to $V^{\prime }$ and $D\left(A\right)$ to H. Then, we denote the linear function

$$b(u,v,w)=\sum _{i,j=1}^2{\int }_{\mathrm{\Omega }}{u}_i{\displaystyle \frac{\partial v_j}{\partial x_i}}{w}_j\mathrm{d}x,\forall u,v,w\in V.$$

It is obvious that the linear function $b(·,·,·)$ is continuous trilinear on $V\times V\times V$ and it satisfies

$$b(u,v,w)=-b(u,w,v),b(u,v,v)=0,\forall u,v,w\in V.$$

In addition, we use the following lemma to provide estimators of the linear function $b(·,·,·)$, which can be found in references [26,27].

Lemma 1.

There exists a positive constant $C_{\mathrm{\Omega }}$ that depends only on Ω, such that

$$\begin{array}{cc}\hfill & \left|b(u,v,w)\right|\le C_{\mathrm{\Omega }}{\parallel u\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel Au\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel v\parallel }_V\parallel w\parallel ,\forall u\in D\left(A\right),v\in V,w\in H.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \left|b(u,v,w)\right|\le C_{\mathrm{\Omega }}{\parallel Au\parallel \parallel v\parallel }_V\parallel w\parallel ,\forall u\in D\left(A\right),v\in V,w\in H.\hfill \end{array}$$

(4)

If $u\in L^{\infty },v\in V,w\in H$, then the linear function $b(·,·,·)$ satisfies

$$\begin{array}{c}\hfill \left|b(u,v,w)\right|\le {\parallel u\parallel }_{L^{\infty }}{\parallel v\parallel }_V\parallel w\parallel .\end{array}$$

(5)

If $u,v,w\in V$, it satisfies

$$\begin{array}{c}\hfill \left|b(u,v,w)\right|\le {\parallel u\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel \nabla u\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel v\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel \nabla v\parallel }^{{\displaystyle \frac{1}{2}}}\parallel w\parallel .\end{array}$$

(6)

To handle the infinite delays $G(t,u(t+\theta \left)\right)$, we define the space ${\mathcal{C}}_{\gamma }\left(H\right)$ with a suitable constant $\gamma >0$, as follows:

$${\mathcal{C}}_{\gamma }\left(H\right)=\left\{\zeta \in \mathcal{C}((-\infty ,0);H)|\exists \underset{s\to -\infty }{lim}{e}^{\gamma s}\zeta \left(s\right)\in H\right\},$$

which is a Banach space with the norm

$${\parallel \zeta \parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}=\underset{s\in (-\infty ,0]}{sup}{e}^{\gamma s}\parallel \zeta \left(s\right)\parallel .$$

The space ${\mathcal{C}}_{\gamma }\left(H\right)$ represents the space of descent functions, where the parameter $\gamma >0$ is typically used to describe the growth rate of the function in ${\mathcal{C}}_{\gamma }\left(H\right)$. The purpose of introducing ${\mathcal{C}}_{\gamma }\left(H\right)$ is to ensure that the growth property of the function does not affect the long-term behavior of the system and also satisfies the descent conditions required by continuous functions with changing states due to time delay effects.

To establish the well-posedness of Equation (1), the continuous function $G:[\tau ,T]\times {\mathcal{C}}_{\gamma }\left(H\right)\mapsto {\left(L^2\left(\mathrm{\Omega }\right)\right)}^2$ is assumed to satisfy the following conditions.

(I) For any $\eta$, the mapping $t\mapsto G(t,\eta )$ is measurable;

(II) $G(t,0)=(0,0)$;

(III) There exists $L_G>0$, such that

$$\begin{array}{c}\hfill \parallel G(t,x_1)-G(t,x_2)\parallel \le L_G{\parallel x_1-x_2\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)},\forall x_1,x_2\in {\mathcal{C}}_{\gamma }\left(H\right).\end{array}$$

(7)

When studying differential equations with infinite time delay, Lipschitz conditions also help to investigate the asymptotic behavior of solutions. For example, when studying the steady-state or periodic solutions of a system after long-term operation, the Lipschitz condition can ensure that the solutions do not experience explosive growth in time. In addition, the Lipschitz condition ensures understanding of the continuous dependence on the initial conditions, as infinite delay terms may lead to temporal irregularities in the solution. By ensuring that the mapping is Lipschitz, it can be proven that small initial perturbations do not cause significant changes in the solution, thereby ensuring its stability.

Based on the above conclusion, we define the weak and strong solution of Equation (1) as follows.

2.2. The Weak Solution and Strong Solution

Definition 1.

Assume $f\in L^2(\tau ,T;V^{\prime })$ and $u\left(\tau \right)=\varphi \in {\mathcal{C}}_{\gamma }\left(H\right)$. A weak solution is any function $u\in {\mathcal{C}}_{\gamma }\left(H\right)$ of Equation (1) if it satisfies

(I) $u\in L^{\infty }(\tau ,T;H)\cap L^2(\tau ,T;V),$ for all $T>\tau$;

(II) For all $\sigma \in V$, such that

$$\begin{array}{c}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left(u\left(t\right),\sigma \right)-\nu ⟨Au\left(t\right),\sigma ⟩+b(u\left(t\right),u\left(t\right),\sigma )=⟨f\left(t\right),\sigma ⟩+\left(G(t,u(t+\theta \left)\right),\sigma \right),}\hfill \end{array}$$

(8)

$$\begin{array}{c}u(\tau +\theta )=\varphi \left(\theta \right)\in {\mathcal{C}}_{\gamma }\left(H\right),\theta \in (-\infty ,0] \hfill \end{array}$$

(9)

holds in the distribution sense of ${\mathcal{D}}^{\prime }(\tau ,T)$.

Assume $f\in L^2(\tau ,T;H)$; if $u\in {\mathcal{C}}_{\gamma }\left(H\right)$ is a weak solution of Equations (8) and () and $u\in L^{\infty }(\tau ,T;V)\cap L^2(\tau ,T;D\left(A\right))$ for all $T>\tau$, then $u\in {\mathcal{C}}_{\gamma }\left(H\right)$ is called a strong solution of Equations (8) and (9). Furthermore, the function $u^{\prime }\in L^2(\tau ,T;H)$.

Remark 1.

If $u\in {\mathcal{C}}_{\gamma }\left(H\right)$ is a weak solution to Equation (1), then in this case, from Equations (8) and (9), for any $\tau \le s\le t$, the following energy equality holds

$$\begin{array}{cc}\hfill & {\parallel u\left(t\right)\parallel }^2+2\nu {\int }_s^t{{\parallel u\left(r\right)\parallel }_V}^2\mathrm{d}r\hfill \\ \hfill =& {\parallel u\left(\tau \right)\parallel }^2+2{\int }_s^t\left(⟨f\left(r\right),u\left(r\right)⟩+\left(G(t,u(r+\theta \left)\right)\right),u\left(r\right)\right)\mathrm{d}r.\hfill \end{array}$$

(10)

In addition, we introduce a useful inequality in the differential equation theory. This will play an important role in making a priori estimates.

Lemma 2.

(Gronwall Inequation) Assume $g,p,q\in \mathcal{C}([a,b];{\mathbb{R}}^{+})$ and for all $t\in [a,b]$ satisfy

$$g\left(t\right)\le g\left(a\right)+{\int }_a^t\left(p\left(s\right)g\left(s\right)+q\left(s\right)\right)\mathrm{d}s,$$

then any $t\in [a,b]$, and functions $g,p,q$ satisfy

$$\begin{array}{c}\hfill g\left(t\right)\le e^{{\int }_a^{t}p\left(s\right)\mathrm{d}s}\left(g\left(a\right)+{\int }_a^{t}q\left(s\right)\mathrm{d}s\right),\forall t\in [a,b].\end{array}$$

(11)

3. Global Well-Posedness

In this section, we study the global well-posedness of Equation (1). Therefore, we present the existence and uniqueness theorem for the weak solution of Equation (1). Therefore, we apply the treatment of the infinite delay term $u(t+\theta )$ to the proof of the global well-posedness of the non-autonomous Navier–Stokes equation, that is, adding the infinite delay term $u(t+\theta )$ to the proof of the global well-posedness of the non-autonomous Navier–Stokes equation.

3.1. The Existence of Solutions

Theorem 1.

Suppose $f\in L^2(\tau ,T;V^{\prime })$, and there exist constants $C=C(\nu ,{\lambda }_1,L_G)$ that satisfy

$${\displaystyle \frac{1}{\nu }}{\int }_{\tau }^t{e}^{C(t-r)}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2\mathrm{d}r<+\infty ,$$

and the function $G(·,·)$ satisfies the conditions (I)-(III). Then for all $T>\tau$ and $u\left(\tau \right)=\varphi \in {\mathcal{C}}_{\gamma }\left(H\right)$, the function $u\in {\mathcal{C}}_{\gamma }\left(H\right)\cap L^{\infty }(\tau ,T;H)\cap L^2(\tau ,T;V)$ is the weak solution of Equation (1).

Proof. 

For the proof of Theorem 1, we use the Galerkin approximation method. The proof is divided into the following three steps.

Step 1:

Construct the approximate solution.

Taking a set of canonical orthonormal basis $\left\{{\xi }_i\right\}\subset V$ on the Hilbert space H, denote

$$V_k=\mathrm{span}[{\xi }_1,{\xi }_2,\cdots ,{\xi }_k],k=1,2,\cdots ;$$

from the Gram–Schmidt theorem, the projection of H onto $V_k$ is defined by

$$P_{k}u=\sum _{i=1}^k(u,{\xi }_i){\xi }_i.$$

Since the Stokes operator A is compact operator on the V space, it satisfies

$$A{\xi }_i={\lambda }_i{\xi }_i,({\lambda }_i>0,i=1,2\cdots ,k),$$

where ${\lambda }_i$ is the eigenvalue of Stokes operator A. Sort them in ascending order and renumber them as follows:

$$0<{\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n\le \cdots ,$$

and they satisfy ${\displaystyle \underset{i\to \infty }{lim}{\lambda }_i=\infty }$. We take the minimum eigenvalue ${\lambda }_1$ of Stokes operator A as the first eigenvalue.

From this, we construct the approximate solution of Equation (1) from the space $V_k$ as follows:

$$u_k(x,t)=\sum _{i=1}^k{a}_{ki}\left(t\right){\xi }_i\left(x\right),$$

where $a_{ki}\left(t\right)=(u_k,{\xi }_i)$ and all $t\in (\tau ,T)$ satisfy

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(u_k\left(t\right),{\xi }_i\right)+\nu ⟨A{u}_k\left(t\right),{\xi }_i⟩+b(u_k,u_k,{\xi }_i)=⟨f\left(t\right),{\xi }_i⟩+\left(G(t,u_k(t+\theta )),{\xi }_i\right),\end{array}$$

(12)

and initial conditions

$$\begin{array}{c}\hfill u_k(\tau +\theta )=P_k\varphi \left(\theta \right),\theta \in (-\infty ,0].\end{array}$$

(13)

Therefore, Equations (12) and (13) satisfy the condition of existence and uniqueness of local solutions for the system of ordinary differential equation with infinite delay (e.g., [19]).

Step 2:

A priori estimate of the approximate solution.

Firstly, we estimate Equation (12), multiplied by $a_{ki}\left(t\right)$ to Equation (12) and sum it to obtain the ODE:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel u_k\left(t\right)\parallel }^2+\nu \parallel u_k{\left(t\right)\parallel }_V^2=⟨f\left(t\right),u_k\left(t\right)⟩+\left(G(t,u_k(t+\theta )),u_k\left(t\right)\right),\end{array}$$

(14)

which implies

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel u_k{\left(t\right)\parallel }^2+\nu \parallel u_k{\left(t\right)\parallel }_V^2\le {\parallel f\parallel }_{V^{\prime }}\parallel u_k{\left(t\right)\parallel }_V+\parallel G(t,u_k(t+\theta ))\parallel \parallel u_k{\left(t\right)\parallel }_V.\end{array}$$

(15)

Using Young and Poincare inequations, we choose suitable constants ${\displaystyle {\epsilon }_1={\epsilon }_2=\frac{\nu }{4}}$, then we obtain

$$\begin{array}{cc}\hfill & {\parallel f\parallel }_{V^{\prime }}\parallel u_k{\left(t\right)\parallel }_V+\parallel G(t,u_k(t+\theta ))\parallel \parallel u_k\left(t\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{2{\epsilon }_1}}{\parallel f\left(t\right)\parallel }_{V^{\prime }}^2+{\displaystyle \frac{\nu }{4}}\parallel u_k{\left(t\right)\parallel }_V^2+{\displaystyle \frac{L_G}{2{\epsilon }_2}}{\parallel u_k(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2,\hfill \end{array}$$

(16)

where ${\displaystyle \parallel u_k{(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}=\underset{s\in (-\infty ,0]}{sup}{e}^{\gamma s}\parallel u_k(t+s)\parallel \ge \parallel u_k\left(t\right)\parallel ,t\in [\tau ,T]}$. Substituting (16) into Equation (15), for $\tau \le r\le t\le T$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}{\parallel u_k\left(r\right)\parallel }^2+\nu {\lambda }_1\parallel u_k{\left(r\right)\parallel }^2+\frac{\nu }{2}\parallel u_k{\left(r\right)\parallel }_V^2\le \frac{4}{\nu }{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2+\frac{4L_G}{\nu }{\parallel u_k(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2.}\end{array}$$

(17)

Multiplying by $e^{-\nu {\lambda }_1(t-r)}$ and then integrating it for r over $[\tau ,t]$, we obtain

$$\begin{array}{cc}\hfill & \parallel u_k{\left(t\right)\parallel }^2+{\displaystyle \frac{\nu }{2}}{\int }_{\tau }^t{e}^{-\nu {\lambda }_1(t-r)}{\parallel u_k\left(r\right)\parallel }_V^2\mathrm{d}r\hfill \\ \hfill \le & e^{-\nu {\lambda }_1(t-\tau )}{\parallel u_k\left(\tau \right)\parallel }^2+{\int }_{\tau }^t{e}^{-\nu {\lambda }_1(t-r)}\left({\displaystyle \frac{4}{\nu }}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2+{\displaystyle \frac{4L_G}{\nu }}{\parallel u_k(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\right)\mathrm{d}r.\hfill \end{array}$$

(18)

According to the definition of ${\mathcal{C}}_{\gamma }\left(H\right)$, we obtain

$$\begin{array}{cc}\hfill \parallel u_k{(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2& ={\left(\underset{s\in (-\infty ,0]}{sup}{e}^{\gamma s}\parallel u_k(t+s)\parallel \right)}^2,\hfill \\ \hfill & \le max\left\{\underset{s\in (-\infty ,\tau -t]}{sup}{e}^{2\gamma s}\parallel u_k{(t+s)\parallel }^2,\underset{s\in (\tau -t,0]}{sup}{e}^{2\gamma s}{\parallel u_k(t+s)\parallel }^2\right\}\hfill \\ \hfill & =max\left\{J_1\left(t\right),J_2\left(t\right)+J_3\left(t\right)\right\},\hfill \end{array}$$

(19)

where

$$\begin{array}{cc}& J_1\left(t\right)=\underset{s\in (-\infty ,\tau -t]}{sup}{e}^{2\gamma s}{\parallel u_k(t+s)\parallel }^2,\hfill \\ & J_2\left(t\right)=\underset{s\in (\tau -t,0]}{sup}\left[e^{2\gamma s-\nu {\lambda }_1(t+s-\tau )}{\parallel u_k\left(\tau \right)\parallel }^2\right],\hfill \\ & J_3\left(t\right)=\underset{s\in (\tau -t,0]}{sup}\left[e^{(2\gamma -\nu {\lambda }_1)s}{\int }_{\tau }^{t+s}{e}^{-\nu {\lambda }_1(t-r)}\left({\displaystyle \frac{4}{\nu }}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2+{\displaystyle \frac{4L_G}{\nu }}{\parallel u_k(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\right)\mathrm{d}r\right].\hfill \end{array}$$

Now, let us estimate $J_1\left(t\right),J_2\left(t\right)$, and $J_3\left(t\right)$. On one hand, since $\nu {\lambda }_1<2\gamma$,

$$\begin{array}{cc}\hfill J_1\left(t\right)& =\underset{s\in (-\infty ,\tau -t]}{sup}{e}^{2\gamma s}{∥P_k\varphi (t+s-\tau )∥}^2\le \underset{s\in (-\infty ,\tau -t]}{sup}{e}^{2\gamma s}{\parallel \varphi (t+s-\tau )\parallel }^2\hfill \\ \hfill & =\underset{s\in (-\infty ,0]}{sup}{e}^{2\gamma [s-(t-\tau \left)\right]}{\parallel \varphi \left(s\right)\parallel }^2=e^{2\gamma (\tau -t)}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\hfill \\ \hfill & \le e^{-\nu {\lambda }_1(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2.\hfill \end{array}$$

(20)

On the other hand, by direct computation, we have

$$\begin{array}{cc}\hfill & J_2\left(t\right)\le e^{-\nu {\lambda }_1(t-\tau )}\parallel u_k{\left(\tau \right)\parallel }^2\le e^{-\nu {\lambda }_1(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & J_3\left(t\right)\le {\int }_{\tau }^t{e}^{-\nu {\lambda }_1(t-r)}\left({\displaystyle \frac{4}{\nu }}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2+{\displaystyle \frac{4L_G}{\nu }}{\parallel u_k(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\right)\mathrm{d}r.\hfill \end{array}$$

(22)

Then, Equations (19)–(22) give

$$\begin{array}{cc}\hfill & \parallel u_k{(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\hfill \\ \hfill \le & e^{-\nu {\lambda }_1(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2+{\int }_{\tau }^t{e}^{-\nu {\lambda }_1(t-r)}\left({\displaystyle \frac{4}{\nu }}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2+{\displaystyle \frac{4L_G}{\nu }}{\parallel u_k(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\right)\mathrm{d}r.\hfill \end{array}$$

(23)

By Lemma 2, this implies

$$\begin{array}{c}\hfill \parallel u_k{(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\le e^{-\left(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }}\right)(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2+{\displaystyle \frac{4}{\nu }}{\int }_{\tau }^t{e}^{-\left(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }}\right)(t-r)}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2\mathrm{d}r,\end{array}$$

(24)

combining Equations (18) and (24); it can be concluded that the $\left\{u_k\left(t\right)\right\}$ is bounded in $L^{\infty }(\tau ,T;H)$. According to Equation (18), this means

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\nu }{2}}{e}^{-\nu {\lambda }_1(t-\tau )}{\int }_{\tau }^t{\parallel u_k\left(r\right)\parallel }_V^2\mathrm{d}r\hfill \\ \hfill \le & \parallel u_k{\left(\tau \right)\parallel }^2+{\int }_{\tau }^t{e}^{-\nu {\lambda }_1(t-r)}\left({\displaystyle \frac{4}{\nu }}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2+{\displaystyle \frac{4L_G}{\nu }}{\parallel u_k(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\right)\mathrm{d}r.\hfill \end{array}$$

(25)

By Equation (24), this implies that the $\left\{u_k\left(t\right)\right\}$ is bounded in $L^2(\tau ,T;V)$. In addition, for any $\psi \in V$, we obtain

$$\begin{array}{c}\hfill ⟨u_k^{\prime }\left(t\right),\psi ⟩+⟨A{u}_k\left(t\right),\psi ⟩+b(u_k,u_k,\psi )=⟨f\left(t\right),\psi ⟩+⟨G(t,u_k(t+\theta )),\psi ⟩.\end{array}$$

Using the Holder inequation, we obtain the following result:

$$\begin{array}{cc}\hfill \left|⟨u_k^{\prime }\left(t\right),\psi ⟩\right|& \le \left|⟨A{u}_k\left(t\right),\psi ⟩\right|+\left|b(u,u,\psi )\right|+|⟨f\left(t\right),\psi ⟩|+|⟨G(t,u_k(t+\theta )),\psi ⟩|\hfill \\ \hfill & \le {\parallel \psi \parallel }_V\left(\nu \parallel u_k\left(t\right){\parallel }_V+C_{\mathrm{\Omega }}\parallel u_k\left(t\right)\parallel \parallel \nabla u_k\left(t\right){\parallel +\parallel f\parallel }_{V^{\prime }}+\parallel G(t,u_k(t+\theta ))\parallel \right),\hfill \\ \hfill & \le {\parallel \psi \parallel }_V\left(\nu \parallel u_k\left(t\right){\parallel }_V+C_{\mathrm{\Omega }}\parallel u_k\left(t\right)\parallel \parallel u_k\left(t\right){\parallel }_V+{\parallel f\parallel }_{V^{\prime }}+\parallel G(t,u_k(t+\theta ))\parallel \right).\hfill \end{array}$$

(26)

Hence, it follows from Equations (7), (18), (24) and (25) that the sequence $\left\{u_k^{\prime }\left(t\right)\right\}$ is bounded in $L^2\left(\tau ,T;V^{\prime }\right)$.

Step 3:

Approximation of approximate solutions.

Now, based on the conclusions of the previous step, since the compactness theorem (see, e.g., [1,11,18]), for all $T>\tau$, there exist subsequences, which we still denote by $\left\{u_k\right\}$, such that as $k\to \infty$, the following conditions satisfy:

$$\begin{array}{cc}\hfill & u_k\stackrel{\ast }{\rightharpoonup }u\mathrm{in}{L}^{\infty }(\tau -h,T;H),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & u_k\rightharpoonup u\mathrm{in}{L}^2(\tau ,T;V), \hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & u_k^{\prime }\rightharpoonup u^{\prime }\mathrm{in}{L}^2(\tau ,T;V^{\prime }), \hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & u_k\to u\mathrm{in}{L}^2(\tau ,T;H), \hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & G(t,u_k(t+\theta ))\stackrel{\ast }{\rightharpoonup }\eta \left(t\right)\mathrm{in}{L}^{\infty }(\tau ,T;{\left(L^2\left(\mathrm{\Omega }\right)\right)}^2).\hfill \end{array}$$

(31)

where $\eta \left(t\right)\in L^{\infty }(\tau ,T;{\left(L^2\left(\mathrm{\Omega }\right)\right)}^2)$ is the limiting function of $G(t,u_k(t+\theta ))$. However, currently, we cannot yet obtain it:

$$\begin{array}{c}\hfill G(t,u_k(t+\theta ))\to G(t,u(t+\theta ))\in L^2(\tau ,T;{\left(L^2\left(\mathrm{\Omega }\right)\right)}^2).\end{array}$$

(32)

Next, we will prove Equation (32) through conclusions (27)–(31).

From conclusions (27)–(31), we obtain

$$\begin{array}{c}\hfill u_k\left(t\right)\to u\left(t\right),a.e.t\in (\tau ,T),\mathrm{in}H,\end{array}$$

(33)

then for all $s,t\in [\tau ,T]$, it implies

$$\begin{array}{c}\hfill u_k\left(t\right)-u_k\left(s\right)={\int }_s^t{u_k}^{\prime }\left(r\right)\mathrm{d}r,\mathrm{in}{V}^{\prime };\end{array}$$

(34)

therefore, the $\left\{u_k\right\}$ is equicontinuous for all $[\tau ,T]$ and k in H. By the embedding relation $V\hookrightarrow \hookrightarrow H\hookrightarrow \hookrightarrow V^{\prime }$, and the Ascoli–Arzela theorem, it yields that $\left\{u_k\right\}$ is relatively compact in $V^{\prime }$, and for all $t\in [\tau ,T]$, which implies

$$\begin{array}{c}\hfill u_k\left(t\right)\to u\left(t\right),\mathrm{in}\mathcal{C}([\tau ,T];V^{\prime }).\end{array}$$

(35)

According to conclusions (27)–(31), for any sequence $\left\{t_k\right\}\subset [\tau ,T]$ that satisfies $t_k\to t$,

$$\begin{array}{c}\hfill u_k\left(t_k\right)\rightharpoonup u\left(t\right),\mathrm{in}H.\end{array}$$

(36)

Before proving Equation (32), we need to first prove

$$\begin{array}{c}\hfill u_k\left(t\right)\to u\left(t\right),\mathrm{in}\mathcal{C}([\tau ,T];H).\end{array}$$

(37)

Since for all $r\in [s,t]\subset [\tau ,T]$, the approximate solution $u_k$ satisfies

$$\begin{array}{cc}\hfill & \parallel u_k{\left(t\right)\parallel }^2+2\nu {\int }_s^t{\parallel u_k\left(r\right)\parallel }_V^2\mathrm{d}r\hfill \\ \hfill =& \parallel u_k{\left(s\right)\parallel }^2+2{\int }_s^t⟨f\left(r\right),u_k\left(r\right))⟩\mathrm{d}r+2{\int }_s^t\left(G(r,u_k(r+\theta ),u_k\left(r\right))\right)\mathrm{d}r,\hfill \end{array}$$

(38)

we obtain the following result by conclusions (27)–(31),

$$\begin{array}{c}\hfill {\int }_s^t{\parallel G(r,u_k(r+\theta ))\parallel }^2\mathrm{d}r\le C_1(t-s),\forall r\in [s,t]\subset [\tau ,T],\end{array}$$

(39)

where $C_1$ is a positive constant. According to the conclusions of (27)–(31), we verify that the function

$$u\in {\mathcal{C}}_{\gamma }\left(H\right)\cap L^{\infty }(\tau ,T;H)\cap L^2(\tau ,T;V),$$

is a weak solution of Equation (1), and it satisfies

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(u\left(t\right),w\right)+\nu ⟨Au\left(t\right),w⟩+b(u\left(t\right),u\left(t\right),w)=⟨f\left(t\right),w⟩+(\eta \left(t\right),w)\forall w\in V,\end{array}$$

(40)

in addition, for any $r\in [s,t]\subset [\tau ,T]$, the energy estimates also satisfy

$$\begin{array}{cc}\hfill & {\parallel u\left(t\right)\parallel }^2+2\nu {\int }_s^t{\parallel u\left(r\right)\parallel }_V^2\mathrm{d}r\hfill \\ \hfill =& {\parallel u\left(\tau \right)\parallel }^2+2{\int }_s^t\left(⟨f\left(r\right),u\left(r\right)⟩+\left(G(t,u(r+\theta )\right),u\left(r\right)\right)\mathrm{d}r.\hfill \end{array}$$

(41)

Due to the conclusions of (27)–(31) and Equation (39), we obtain

$$\begin{array}{c}\hfill {\int }_s^t{\parallel \eta \left(r\right)\parallel }^2\mathrm{d}r\le \underset{k\to \infty }{lim\; inf}{\int }_s^t{\parallel G(t,u(r+\theta ))\parallel }^2\mathrm{d}r\le C_1(t-s);\end{array}$$

(42)

therefore, consider the continuous functions $L_k,L:[\tau ,t]\to \mathbb{R}$, which are defined by

$$\begin{array}{cc}& L_k\left(t\right)={\displaystyle \frac{\nu }{2}}{\parallel u_k\left(t\right)\parallel }_V^2-{\int }_s^t⟨f\left(r\right),u_k\left(r\right)⟩\mathrm{d}r-C_{1}t,\hfill \\ & L\left(t\right)={\displaystyle \frac{\nu }{2}}{\parallel u\left(t\right)\parallel }_V^2-{\int }_s^t⟨f\left(r\right),u\left(r\right)⟩\mathrm{d}r-C_{1}t.\hfill \end{array}$$

As shown in Equations (38) and (41), it is evident that the functions $L_k\left(t\right),L\left(t\right)$ are non-increasing and continuous in $[\tau ,T]$. According to the conclusions of (27)–(30), we obtain

$$\begin{array}{c}\hfill L_k\left(t\right)\to L\left(t\right)a.e.t\in [\tau ,T],\end{array}$$

(43)

from the conclusions of (27)–(30), it is obvious that ${\displaystyle \parallel u\left(t_0\right)\parallel \le \underset{k\to \infty }{lim\; inf}\parallel u_k\left(t_k\right)\parallel }$. To prove

$$\begin{array}{c}\hfill {\displaystyle \underset{k\to \infty }{lim\; sup}\parallel u_k\left(t_k\right)\parallel \le \parallel u\left(t_0\right)\parallel ,}\end{array}$$

(44)

we assume $t_0>\tau$ and take the sequence $\left\{{\overline{t}}_n\right\}\subset \left\{t_k\right\}$ that satisfies (43), which is increasing, and approach the value $t_0$ from the left when $n\to \infty$. Since $L(·)$ is continuous, there exists $N_{\epsilon }$, and when $n>N_{\epsilon }$,

$$|L\left({\overline{t}}_n\right)-L\left(t_0\right)|<\epsilon .$$

Take $k>K\left(N_{\epsilon }\right)$ such that $t_k>{\overline{t}}_{N_{\epsilon }}$, by the sequence $L(·),L_k(·)$, is non-increasing and satisfies (43); we obtain

$$|L_k\left(t_k\right)-L\left(t_0\right)|\le |L_k\left({\overline{t}}_{N_{\epsilon }}\right)-L\left({\overline{t}}_{N_{\epsilon }}\right)|+|L\left({\overline{t}}_{N_{\epsilon }}\right)-L\left(t_0\right)|<2\epsilon .$$

At this time, the conclusions of (27)–(30) imply

$$\begin{array}{c}\hfill {\int }_{\tau }^{t_k}⟨f\left(r\right),u_k\left(r\right)⟩\mathrm{d}r\to {\int }_{\tau }^t⟨f\left(r\right),u\left(r\right)⟩\mathrm{d}r,(k\to \infty ).\end{array}$$

(45)

Due to these and functions $L_k,L$, we have proven Equation (44).

According to the definition of ${\mathcal{C}}_{\gamma }\left(H\right)$, we obtain

$$\begin{array}{cc}\hfill & \parallel u_k{(t+\theta )-u(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}\hfill \\ \hfill =& \underset{s\in (-\infty ,0]}{sup}{e}^{\gamma s}\parallel u_k(t+s)-u(t+s)|=max\left\{{\mathsf{\Phi }}_1\left(s\right),{\mathsf{\Phi }}_2\left(s\right)\right\}\hfill \\ \hfill \le & max\left\{e^{\gamma (\tau -t)}\parallel P_k{\varphi -\varphi \parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)},\underset{s\in (\tau -t,0]}{sup}{e}^{\gamma s}\parallel u_k(t+s)-u(t+s)\parallel \right\},\hfill \end{array}$$

(46)

where

$$\begin{array}{cc}& {\displaystyle {\mathsf{\Phi }}_1\left(s\right)=\underset{s\in (-\infty ,\tau -t]}{sup}{e}^{\gamma s}\parallel P_k\varphi (t+s-\tau )-\varphi (t+s-\tau )\parallel ,}\hfill \\ & {\displaystyle {\mathsf{\Phi }}_2\left(s\right)=\underset{s\in (\tau -t,0]}{sup}{e}^{\gamma s}\parallel u_k(t+s)-u(t+s)\parallel ,}\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\parallel u_k(t+\theta )-u(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}=0.\end{array}$$

(47)

In addition, the convergence results are obtained by Equations (7), (37) and (47), and conclusions of (27)–(31); therefore, we prove Equation (32). Thus, we have proven Theorem 1. □

3.2. The Uniqueness of Solutions

Theorem 2.

If all the conditions in the Theorem 1 are satisfied, then the function $u\in {\mathcal{C}}_{\gamma }\left(H\right)\cap L^{\infty }(\tau ,T;H)\cap L^2(\tau ,T;V)$ is the unique weak solution of Equation (1).

Proof. 

Set ${\overline{u}}_1$ and ${\overline{u}}_2$ be the weak solutions of Equation (1). Denote the function $U\left(t\right)={\overline{u}}_1\left(t\right)-{\overline{u}}_2\left(t\right)$; for any $t>\tau$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel U\left(t\right)\parallel }^2+\nu {\parallel \nabla U\left(t\right)\parallel }^2+b(U,{\overline{u}}_1,U)=\left(G(t,{\overline{u}}_1(t+\theta ))-G(t,{\overline{u}}_2(t+\theta )),U\left(t\right)\right).\end{array}$$

(48)

Furthermore, according to Lemma 1, and Equations (14) and (39), using Young, Holder, and Poincare inequations, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel U\left(t\right)\parallel }^2+\nu {\parallel U\left(t\right)\parallel }_V^2\hfill \\ \hfill \le & C_{\mathrm{\Omega }}\left({\displaystyle \frac{{\parallel U\left(t\right)\parallel }^2{\parallel {\overline{u}}_1\left(t\right)\parallel }_V^2}{2{\epsilon }_3}}+{\displaystyle \frac{{\epsilon }_3}{2}}{\parallel U\left(t\right)\parallel }_V^2\right)+L_G{\parallel U(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}\parallel U\left(t\right)\parallel .\hfill \end{array}$$

(49)

Integrating Equation (49), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\parallel U\left(t\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\parallel U\left(\tau \right)\parallel }^2+\nu {\int }_{\tau }^t{\parallel U\left(r\right)\parallel }_V^2\mathrm{d}r\hfill \\ \hfill \le & C_{\mathrm{\Omega }}{\int }_{\tau }^t\left({\displaystyle \frac{{\parallel U\left(r\right)\parallel }^2{\parallel {\overline{u}}_1\left(r\right)\parallel }_V^2}{2{\epsilon }_3}}+{\displaystyle \frac{{\epsilon }_3}{2}}{\parallel U\left(r\right)\parallel }_V^2\right)\mathrm{d}r+{\int }_{\tau }^t{L}_G{\parallel U(r+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}\parallel U\left(r\right)\parallel \mathrm{d}r.\hfill \end{array}$$

(50)

Take ${\displaystyle {\epsilon }_3=\frac{2\nu }{C_{\mathrm{\Omega }}}}$; then, according to the definition of ${\mathcal{C}}_{\gamma }\left(H\right)$, we obtain

$$\parallel U(r+\theta )\parallel =\underset{s\in (-\infty ,0]}{sup}{e}^{\gamma s}\parallel U(r+s)\parallel \le \underset{z\in (\tau ,r)}{sup}\parallel U\left(z\right)\parallel ,r\in [\tau ,t].$$

Therefore, we have

$$\begin{array}{c}\hfill \underset{z\in (\tau ,t)}{sup}{\parallel U\left(z\right)\parallel }^2\le {\int }_{\tau }^t\left({\displaystyle \frac{C_{\mathrm{\Omega }}{\parallel {\overline{u}}_1\left(r\right)\parallel }_V^2}{2\nu }}+2L_G\right)\underset{z\in (\tau ,r)}{sup}{\parallel U\left(z\right)\parallel }^2\mathrm{d}r,\end{array}$$

(51)

where ${\parallel U\left(\tau \right)\parallel }^2=0$. By Lemma 2, we conclude that ${\parallel U\left(t\right)\parallel }^2=0$; therefore, we have proven Theorem 2. □

4. Estimate the Number of Determining Nodes

In this section, we prove that Equation (1) has a finite number of determining nodes.

4.1. Introduction to Relevant Lemmas

Let ${\overline{u}}_1$ and ${\overline{u}}_2$ be the two strong solutions of Equation (1) corresponding to the external force and torque $f_1$ and $f_2$, respectively, and $f_1,f_2\in L^2(\tau ,T;H)$. The asymptotic strength of the external force and moment is described by its $L^2$ norm, i.e.,

$$\begin{array}{c}\hfill F:=\underset{t\to +\infty }{lim\; sup}{\left({\int }_{\mathrm{\Omega }}{\left|f_i(x,t)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{d}x,i=1,2.\end{array}$$

(52)

which can be found in references [12,28]. Therefore, we have the following definition.

Definition 2

([12,28,29]). Consider N measuring points $x_i$ in the space $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$, where $i=1,2,\cdots ,N$. Denoted $\Lambda =\left\{x^1,x^2,\cdots ,x^N\right\}$. Suppose that ${\overline{u}}_1$ and ${\overline{u}}_2$ are two strong solutions of Equation (1). If by

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}\underset{i=1,2,\cdots ,N}{max}|{\overline{u}}_1(x^i,t)-{\overline{u}}_2(x^i,t)|=0,\end{array}$$

(53)

we can obtain

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\int }_{\mathrm{\Omega }}{|{\overline{u}}_1(x,t)-{\overline{u}}_2(x,t)|}^2\mathrm{d}x=0,\end{array}$$

(54)

then the point set Λ is called the definite node set of Equation (1), and the point in Λ is the definite node of Equation (1).

In addition, in order to prove the main conclusions of this section, the following important lemmas (see [11,26,29,30,31]) should also be used.

Lemma 3.

Let Ω be covered by N identical squares. Remember the point set

$$\mathsf{\Lambda }=\left\{x^1,x^2,\cdots ,x^N\right\}\subset \mathrm{\Omega },$$

where the point $x^i$ belongs to and only belongs to one of the squares, then for $\forall u\in D\left(A\right)$, there exists the normal number $C_{\mathrm{\Omega }}$, which is only dependent on Ω, such that

$$\begin{array}{c}\hfill {\parallel \nabla u\parallel }^2\le C_{\mathrm{\Omega }}N{\eta }^2\left(u\right)+{\displaystyle \frac{C_{\mathrm{\Omega }}}{{\lambda }_{1}N}}{\parallel Au\parallel }^2,\end{array}$$

(55)

where $\eta \left(u\right)=\underset{i=1,2,\cdots ,N}{max}\left|u\left(x^i\right)\right|.$

Lemma 4.

If $P\left(t\right)$ and $Q\left(t\right)$ are real-valued functions on $[\tau ,+\infty )$ and there exist $T>0$ and constant $C_2$, such that

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; inf}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}P\left(r\right)\mathrm{d}r=C_2>0,\underset{t\to +\infty }{lim\; sup}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{P}^{-}\left(r\right)\mathrm{d}r<\infty ,\end{array}$$

(56)

and

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{Q}^{+}\left(r\right)\mathrm{d}r=0,\end{array}$$

(57)

where $P^{-}\left(t\right)=max\left\{-P\left(t\right),0\right\},Q^{+}\left(t\right)=max\left\{Q\left(t\right),0\right\}$. Suppose $\xi \left(t\right)$ is the non-negative absolutely continuous function on $[\tau ,+\infty )$. If $\xi \left(t\right)$ satisfies

$${\displaystyle \frac{\mathrm{d}\xi \left(t\right)}{\mathrm{d}t}}+P\left(t\right)\xi \left(t\right)\le Q\left(t\right),$$

then ${\displaystyle \underset{t\to +\infty }{lim}\xi \left(t\right)=0}$.

Next, we prove that if any two strong solutions of Equation (1) have the same asymptotic behavior at a finite number of points in space, then these two solutions will have the same asymptotic behavior almost everywhere in the entire space.

4.2. Main Results

Theorem 3.

Let Ω be covered by N identical squares. The set Λ is defined in Lemma 3, ${\displaystyle N>\frac{16C_{\mathrm{\Omega }}^3{C}_5}{5{\nu }^2{\lambda }_1}}$, where $C_5:=C_4{\parallel \varphi \left(s\right)\parallel }^2+C_3{F}^2$ and each point $x^i$ belongs to and only belongs to one of the squares. Let ${\overline{u}}_1$ and ${\overline{u}}_2$ be two strong solutions of Equation (1) corresponding to external forces $f_1$ and $f_2$, respectively, and $f_1$ and $f_2$ have the same asymptotic strength, i.e.,

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\int }_{\mathrm{\Omega }}{\left|f_i(x,t)\right|}^2\mathrm{d}x=0.\end{array}$$

(58)

Then, the point set Λ is the definite node set of Equation (1).

Proof. 

Denote $U={\overline{u}}_1-{\overline{u}}_2,f=f_1-f_2$ and

$$G(t,U(t+\theta ))=G_1(t,{\overline{u}}_1(t+\theta ))-G_2(t,{\overline{u}}_2(t+\theta )).$$

Since ${\overline{u}}_1$ and ${\overline{u}}_2$ are the two solutions of Equation (1), in the sense of distribution, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial U}{\partial t}}-\nu \Delta U+(U·\nabla ){\overline{u}}_1+({\overline{u}}_2·\nabla )u=f(t,x)+G(t,U(t+\theta )).\end{array}$$

(59)

Next, we prove that under the condition of Theorem 1, if Equation (53) holds, then there is

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\parallel U\left(t\right)\parallel }^2=0.\end{array}$$

(60)

Firstly, the inner product of $AU$ with Equation (59), respectively, is obtained with

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel \nabla U\left(t\right)\parallel }^2+\nu {\parallel AU\parallel }^2\hfill \\ \hfill =& -b(U,{\overline{u}}_1,AU)-b({\overline{u}}_2,U,AU)+(f,AU)+(G(t,U(t+\theta )),AU),\hfill \end{array}$$

(61)

by the Lemma 1 and Young inequality. Take ${\displaystyle {\epsilon }_3=\frac{\nu }{4}}$, then we can obtain

$$\begin{array}{c}\hfill \left|b\right(U,u_1,AU\left)\right|\le C_{\mathrm{\Omega }}\parallel \nabla U\parallel \parallel \nabla {\overline{u}}_1\parallel \parallel AU\parallel \le {\displaystyle \frac{2C_{\mathrm{\Omega }}^2}{\nu }}{\parallel \nabla U\parallel }^2\parallel \nabla {\overline{u}}_1{\parallel }^2+{\displaystyle \frac{\nu }{8}}{\parallel AU\parallel }^2,\end{array}$$

(62)

$$\begin{array}{c}\hfill \left|b\right({\overline{u}}_2,U,AU\left)\right|\le C_{\mathrm{\Omega }}\parallel \nabla {\overline{u}}_2\parallel \parallel \nabla U\parallel \parallel AU\parallel \le {\displaystyle \frac{2C_{\mathrm{\Omega }}^2}{\nu }}\parallel \nabla {\overline{u}}_2{\parallel }^2{\parallel \nabla U\parallel }^2+{\displaystyle \frac{\nu }{8}}{\parallel AU\parallel }^2.\end{array}$$

(63)

Similarly, it is obvious that

$$\begin{array}{c}\hfill \left|(f,Au)\right|\le {\displaystyle \frac{\nu }{16}}{\parallel AU\parallel }^2+{\displaystyle \frac{4}{\nu }}{\parallel f\parallel }^2,\end{array}$$

(64)

and

$$\begin{array}{c}\hfill \left|\left(G(t,U(t+\theta \left)\right),AU\right)\right|\le {\displaystyle \frac{\nu }{16}}{\parallel AU\parallel }^2+{\displaystyle \frac{4L_G^2}{\nu }}{\parallel U(t+\theta )\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2.\end{array}$$

(65)

By Equation (24), substituting (62)–(65) into Equation (61), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel \nabla U\left(t\right)\parallel }^2+{\displaystyle \frac{5\nu }{4}}{\parallel Au\parallel }^2-{\displaystyle \frac{4C_{\mathrm{\Omega }}^2}{\nu }}{\parallel \nabla U\left(t\right)\parallel }^2\left(\parallel \nabla {\overline{u}}_1{\parallel }^2+{\parallel \nabla {\overline{u}}_2\parallel }^2\right)\hfill \\ \hfill \le & {\displaystyle \frac{8L_G^2}{\nu }}{e}^{-\left(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }}\right)(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2+{\displaystyle \frac{8}{\nu }}{\parallel f\parallel }^2+2{\left({\displaystyle \frac{4L_G}{\nu }}\right)}^2{\int }_{\tau }^t{e}^{-\left(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }}\right)(t-r)}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2\mathrm{d}r.\hfill \end{array}$$

(66)

By Equation (55) in Lemma 3, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel \nabla U\left(t\right)\parallel }^2+\left[{\displaystyle \frac{5\nu {\lambda }_{1}N}{4C_{\mathrm{\Omega }}}}-{\displaystyle \frac{4C_{\mathrm{\Omega }}^2}{\nu }}\left(\parallel \nabla {\overline{u}}_1{\parallel }^2+{\parallel \nabla {\overline{u}}_2\parallel }^2\right)\right]{\parallel \nabla U\left(t\right)\parallel }^2\hfill \\ \hfill \le & {\displaystyle \frac{8L_G^2}{\nu }}{e}^{-\left(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }}\right)(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2+{\displaystyle \frac{8}{\nu }}{\parallel f\parallel }^2+2{\left({\displaystyle \frac{4L_G}{\nu }}\right)}^2{\int }_{\tau }^t{e}^{-\left(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }}\right)(t-r)}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2\mathrm{d}r\hfill \\ \hfill & +{\displaystyle \frac{5\nu {\lambda }_1{N}^2}{4}}{\eta }^2\left(U\right).\hfill \end{array}$$

(67)

Secondly, we define

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\xi \left(t\right):={\parallel \nabla U\left(t\right)\parallel }^2,\hfill \\ {\displaystyle P\left(t\right):=\frac{5\nu {\lambda }_{1}N}{4C_{\mathrm{\Omega }}}-\frac{4C_{\mathrm{\Omega }}^2}{\nu }\left(\parallel \nabla {\overline{u}}_1{\parallel }^2+{\parallel \nabla {\overline{u}}_2\parallel }^2\right).}\hfill \\ {\displaystyle Q\left(t\right):=\frac{8L_G^2}{\nu }{e}^{-\left(\nu {\lambda }_1-\frac{4L_G}{\nu }\right)(t-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2+\frac{8}{\nu }{\parallel f\parallel }^2+\frac{5\nu {\lambda }_1{N}^2}{4}{\eta }^2\left(U\right)}\hfill \\ {\displaystyle +2{\left(\frac{4L_G}{\nu }\right)}^2{\int }_{\tau }^t{e}^{-\left(\nu {\lambda }_1-\frac{4L_G}{\nu }\right)(t-r)}{\parallel f\left(r\right)\parallel }_{V^{\prime }}^2\mathrm{d}r.}\hfill \end{array}\right.\end{array}$$

Then, the inequality (67) can be expressed as

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}\xi \left(t\right)}{\mathrm{d}t}}+P\left(t\right)\xi \left(t\right)\le Q\left(t\right).\end{array}$$

(68)

Next, we verify that $P\left(t\right)$ and $Q\left(t\right)$ satisfy the conditions of Lemma 4. There exist constants $C_3$ and $C_4$ such that

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; sup}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{\parallel \nabla {\overline{u}}_i\left(r\right)\parallel }^2\mathrm{d}r\le C_5,i=1,2,\end{array}$$

(69)

where $C_5:=C_4{\parallel \varphi \left(s\right)\parallel }^2+C_3{F}^2$. Therefore, when ${\displaystyle N>\frac{16C_{\mathrm{\Omega }}^3{C}_5}{5{\nu }^2{\lambda }_1}}$, $P\left(r\right)$ and $P^{-}\left(r\right)$ satisfy

$$\begin{array}{cc}\hfill \underset{t\to +\infty }{lim\; inf}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}P\left(r\right)\mathrm{d}r\ge & {\displaystyle \frac{5\nu {\lambda }_{1}N}{4C_{\mathrm{\Omega }}}}-{\displaystyle \frac{4C_{\mathrm{\Omega }}^2}{\nu }}\underset{t\to +\infty }{lim\; sup}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{\parallel \nabla {\overline{u}}_i\left(r\right)\parallel }^2\mathrm{d}r\hfill \\ \hfill \ge & {\displaystyle \frac{5\nu {\lambda }_{1}N}{4C_{\mathrm{\Omega }}}}-{\displaystyle \frac{4C_{\mathrm{\Omega }}^2{C}_5}{\nu }}>0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{t\to +\infty }{lim\; sup}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{P}^{-}\left(r\right)\mathrm{d}r\le & {\displaystyle \frac{5\nu {\lambda }_{1}N}{4C_{\mathrm{\Omega }}}}+{\displaystyle \frac{4C_{\mathrm{\Omega }}^2}{\nu }}\underset{t\to +\infty }{lim\; sup}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{\parallel \nabla {\overline{u}}_i\left(r\right)\parallel }^2\mathrm{d}r\hfill \\ \hfill \le & {\displaystyle \frac{5\nu {\lambda }_{1}N}{4C_{\mathrm{\Omega }}}}+{\displaystyle \frac{4C_{\mathrm{\Omega }}^2{C}_5}{\nu }}<+\infty ,\hfill \end{array}$$

they satisfy Equation (56).

By Lemma 4, this implies

$${\displaystyle \underset{t\to +\infty }{lim}\eta \left(U\left(t\right)\right)=0.}$$

Combining them, then we have

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}\left({\displaystyle \frac{5\nu {\lambda }_1{N}^2}{4}}{\eta }^2\left(U\left(r\right)\right)+{\displaystyle \frac{8}{\nu }}{\parallel f\left(r\right)\parallel }^2\right)\mathrm{d}r=0.\end{array}$$

(70)

Since ${\nu }^2{\lambda }_1>4L_G$, we obtain

$$\begin{array}{cc}\hfill & \underset{t\to +\infty }{lim}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{e}^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(r-\tau )}{\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\mathrm{d}r\hfill \\ \hfill =& \underset{t\to +\infty }{lim}{\displaystyle \frac{1}{T(\nu {\lambda }_1-\frac{4L_G}{\nu })}}\left(e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-\tau )}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-\tau )}\right){\parallel \varphi \left(s\right)\parallel }_{{\mathcal{C}}_{\gamma }\left(H\right)}^2\hfill \\ \hfill =& 0.\hfill \end{array}$$

According to Equation (58), for all $\epsilon >0$ there exists $T_1>\tau$ such that when $t>T_1$, it satisfies

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{T(\nu {\lambda }_1-\frac{4L_G}{\nu })}}\underset{z\in [T_1,+\infty )}{sup}{\parallel f\left(z\right)\parallel }^2{\int }_{\tau }^{t+T}\left[e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-r)}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-r)}\right]\mathrm{d}r<{\displaystyle \frac{\epsilon }{2}};\end{array}$$

(71)

therefore, for $\forall T_1>\tau$, we have

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}\left(e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-r)}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-r)}\right)=0.\end{array}$$

(72)

Hence, for the $\epsilon$ of Equation (71), there exists $T_2$ such that when $t>T_2$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{T(\nu {\lambda }_1-\frac{4L_G}{\nu })}}{\int }_{\tau }^{T_0}{\parallel f\left(r\right)\parallel }^2\mathrm{d}r\left[e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-T_0)}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-T_0)}\right]<{\displaystyle \frac{\epsilon }{2}}.\end{array}$$

(73)

Denote $T_0=max\{T_1,T_2\}$, then for $\epsilon >0$, when $t>T_0$, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{T}}{\int }_t^{t+T}{\int }_{\tau }^r{e}^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(\rho -r)}{\parallel f\left(r\right)\parallel }^2\mathrm{d}\rho \mathrm{d}r\hfill \\ \hfill \le & {\displaystyle \frac{1}{T}}{\int }_{\tau }^{t+T}{\int }_t^{t+T}{e}^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(\rho -r)}{\parallel f\left(r\right)\parallel }^2\mathrm{d}\rho \mathrm{d}r\hfill \\ \hfill =& {\displaystyle \frac{1}{T(\frac{4L_G}{\nu }-\nu {\lambda }_1)}}{\int }_{\tau }^{t+T}{\parallel f\left(r\right)\parallel }^2·\left[e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-r)}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-r)}\right]\mathrm{d}r\hfill \\ \hfill \le & {\displaystyle \frac{1}{T(\nu {\lambda }_1-\frac{4L_G}{\nu })}}{\int }_{\tau }^{T_0}{\parallel f\left(r\right)\parallel }^2\mathrm{d}r·\left[e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-T_0)}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-T_0)}\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{T(\nu {\lambda }_1-\frac{4L_G}{\nu })}}\underset{z\in [T_0,+\infty )}{sup}{\parallel f\left(z\right)\parallel }^2{\int }_{T_0}^{t+T}\left[e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t-r)}-e^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(t+T-r)}\right]\mathrm{d}r\hfill \\ \hfill <& \epsilon ,\hfill \end{array}$$

(74)

which implies

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{\displaystyle \frac{1}{T}}{\int }_t^{t+T}{\int }_{\tau }^{\rho }{e}^{-(\nu {\lambda }_1-{\displaystyle \frac{4L_G}{\nu }})(\rho -r)}{\parallel f\left(r\right)\parallel }^2\mathrm{d}\rho \mathrm{d}r=0.\end{array}$$

(75)

By Equations (70) and (75), we can see that $Q^{+}\left(t\right)$ satisfies Equation (57).

Finally, by Lemma 4, we obtain

$$\underset{t\to +\infty }{lim}\xi \left(t\right)=\underset{t\to +\infty }{lim}{\parallel \nabla U\left(t\right)\parallel }^2=0.$$

According to the Poincare inequation, we have proven Equation (60). According to Definition 2, Theorem 3 is proven. □

5. Conclusions

The problem addressed in this paper is the global well-posedness and asymptotic behavior of solutions to non autonomous Navier–Stokes equations with infinite time delay and node determination. This study establishes a mathematical framework based on the function space ${\mathcal{C}}_{\gamma }\left(H\right)$ and demonstrates the well-posedness of Equation (1) under the assumption that the function $G(·,·)$ satisfies Lipschitz continuity with respect to time t. Furthermore, the long-term behavior of strong solutions is shown to be characterized by their values at a finite number of spatial nodes. The result of this paper is to apply the infinite delay term $u(t+\theta )$ to the non-autonomous Navier–Stokes equations and to study the well-posedness and asymptotic behavior of the Navier–Stokes equations with delay effects using relevant existing theoretical results. This result provides theoretical support and specific examples for studying nonlinear PDEs with time delay effects to a certain extent. It not only validates the relevant conclusions obtained from time delay differential equations but also reveals the research methods for nonlinear PDEs of other time delay effects, further verifying some examples given by time delay differential equations (see [23,24,25]).

However, this method also has some limitations, as the construction of ${\mathcal{C}}_{\gamma }\left(H\right)$ is only a function space established for handling the infinite delay term $u(t+\theta )$, as the asymptotic behavior of function u becomes particularly important at $t\to \infty$ and creates a convergence relationship between the infinite delay terms $u(t+\theta )$ and $u\left(t\right)$. In addition, the function $G(·,·)$ itself must satisfy Lipschitz properties, otherwise it will greatly affect the well-posedness of the solution of Equation (1). For the physical meaning of the Navier–Stokes equation itself, we only consider the case where the Reynolds number is particularly small to study the global well-posedness and determining nodes of Equation (1). For other cases, due to the need to consider the pressure p term, the method used in this paper may not be applicable.

The research method presented in this paper has high universality and can be applied to various types of nonlinear PDEs. By adding an infinite delay term $u(t+\theta )$ to the existing conditions of a nonlinear PDE, it becomes a nonlinear PDE with infinite delay, which has a more profound impact on the well-posedness of solutions and other related issues. This provides theoretical support and technical means for solving other complex systems, especially for the study of differential equations with infinite delay. In the current research results, only the system of non-autonomous micro polar fluid flow has been studied for global well-posedness and asymptotic behavior of solutions by adding an infinite delay term $u(t+\theta )$ on the original basis (see [32]).

Due to the existence of time delay factors, the introduction and application of numerical methods need to consider the influence of historical states, which leads to a more complex implementation of the algorithm. In order to solve Navier–Stokes equations with time delay, it is generally necessary to discretize them. Furthermore, determining nodes are the discrete points determined during this process. In the process of analyzing problems, it is usually necessary to calculate analytical and numerical solutions at determined nodes and to strictly control the time step based on the introduced time delay factors in order to avoid the instability of numerical solutions and evaluate the accuracy of numerical methods, ensuring the stability and convergence of the algorithm. By comparing these solutions, we can assess the effectiveness of the selected nodes and discretization methods, thereby improving the efficiency and accuracy of solving Navier–Stokes equations with time delays, ensuring that researchers can better simulate and control fluid systems, and enhance the reliability and performance of engineering design.

Advancements in computational technology open new avenues for research on Navier–Stokes equations with time delay, particularly in exploring time-delay effects in high-dimensional spaces and complex geometries. This includes the behavior of fluids under complex boundary conditions or multiphase flow, particularly in applications such as biomedical engineering and environmental science. In recent years, machine learning has gained prominence in fluid dynamics, offering novel approaches to modeling and real-time control of fluid systems. Integrating time-delay factors with machine learning methods presents a promising direction for developing more accurate fluid models and real-time control strategies, advancing both theoretical understanding and practical applications. These methods can be used to learn fluid behavior from experimental data and predict the system’s response under different conditions.

In addition, for delay differential equations, there are various types of delay effects such as discrete delay and state-dependent delay. Future research will focus on investigating the global well-posedness and asymptotic behavior of solutions to differential equations with state-dependent delay $u(t-\sigma (t,u(t+\theta )\left)\right)$ and generalize the relevant conclusions of abstract functional differential equations with state-dependent delay in reference [20] through several specific partial differential equations. The manuscript on the study of partial differential equations with state-dependent delay is currently in progress.