ABSTRACT

In this paper, we mainly study the existence and uniqueness of solutions and the asymptotic behavior of solutions for three-dimensional globally modified Bénard systems with delays under local Lipschitz conditions.

Subject Areas:

Fluid Mechanics, Partial Differential Equation

Keywords:

Bénard System, Delay, Galerkin Approximation, Asymptotic Behavior

1. Introduction

Let $\Omega \in {ℝ}^3$ be an open and bounded set with regular boundary $\Gamma$. We consider the following problem for the Bénard system with homogeneous Dirichlet boundary conditions on $\Omega$ :

$\{\begin{array}{ll}\frac{\partial u}{\partial t}-\nu \Delta u+\left(u\cdot \nabla \right)u+\xi \omega +\nabla p=f\mathrm{,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ \frac{\partial \omega }{\partial t}-\Delta \omega +\left(u\cdot \nabla \right)\omega =g\mathrm{,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ \nabla \cdot u=\mathrm{0,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ u=\mathrm{0,\; <mi>\; </mi>}\omega =\mathrm{0,}\hfill & \text{on}\left(\mathrm{0,}+\infty \right)\times \partial \Omega \mathrm{,}\hfill \\ u\left(x\mathrm{,0}\right)=u_0\left(x\right)\mathrm{,\; <mi>\; </mi>}\omega \left(x\mathrm{,0}\right)={\omega }_0\left(x\right)\mathrm{,}\hfill & x\in \Omega \mathrm{.}\hfill \end{array}$ (1)

Here $u\left(x\mathrm{,}t\right)\mathrm{,}\omega \left(x\mathrm{,}t\right)$ are respectively the velocity, temperature of the fluid, $p\left(x\mathrm{,}t\right)$ is the pressure of the fluid, $\nu$ is the viscous coefficient of the fluid, $\xi \in {ℝ}^3$ is a constant vector, $u_0\mathrm{,}{\omega }_0$ are the initial value. The external force terms $f\mathrm{<mo>\; :\; </mo>}ℝ\times \Omega \to {ℝ}^3$, $g\mathrm{<mo>\; :\; </mo>}ℝ\times \Omega \to ℝ$ are a given function.

As we all know, Bénard system consists of the Navier-Stokes equations coupled with a parabolic equation. This system is a well-known model in hydrodynamics. It describes the behavior of the velocity, the pressure and the temperature for an incompressible flow. A detailed description of the physical background can be found in [1] and [2] . However, the well-posedness of the Bénard system is still an unsolved problem in the three-dimensional case. Therefore, it is particularly important to modify equations for Bénard system. We consider the following globally modified Bénard system:

$\{\begin{array}{ll}\frac{\partial u}{\partial t}-\nu \Delta u+F_N\left(\Vert u\Vert \right)\left(u\cdot \nabla \right)u+\xi \omega +\nabla p=f\mathrm{,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ \frac{\partial \omega }{\partial t}-\Delta \omega +\left(u\cdot \nabla \right)\omega =g\mathrm{,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ \nabla \cdot u=\mathrm{0,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ u\mathrm{<mo>\; =\; </mo>0,\; <mi>\; </mi>}\omega =\mathrm{0,}\hfill & \text{on}\left(\mathrm{0,}+\infty \right)\times \partial \Omega \mathrm{,}\hfill \\ u\left(x\mathrm{,0}\right)=u_0\left(x\right)\mathrm{,\; <mi>\; </mi>}\omega \left(x\mathrm{,0}\right)={\omega }_0\left(x\right)\mathrm{,}\hfill & x\in \Omega \mathrm{.}\hfill \end{array}$ (2)

Here the function $F_N\left(r\right)\mathrm{<mo>\; :\; </mo>}{ℝ}^{+}\to {ℝ}^{+}$ is defined by

$F_N\left(r\right)\mathrm{<mo>\; :\; </mo>}=\min \left\{\mathrm{1,}\frac{N}{r}\right\}\mathrm{,}r\mathrm{,\; <mi>\; </mi>}N\in {ℝ}^{+}\mathrm{.}$

In the modified term of Equation (2), $F_N\left(\Vert u\Vert \right)$ depends on the norm $\Vert u\Vert ={\Vert \nabla u\Vert }_{{\left(L^2\left(\Omega \right)\right)}^{3\times 3}}$, which in turn depends on $\nabla u$ over the whole domain $\Omega$ and not just at or near the point $x\in \Omega$ under consideration. Essentially, it prevents large gradients dominating the dynamics and leading to explosions.

The globally modified Bénard system can be better described when we add the appropriate delay term to the system. In this way, it can take into account not only the present state of the system but the history of the solution. Namely, we consider the following equations:

$\{\begin{array}{ll}\frac{\partial u}{\partial t}-\nu \Delta u+F_N\left(\Vert u\Vert \right)\left(u\cdot \nabla \right)u+\xi \omega +\nabla p=G_1\left(t\mathrm{,}u\left(t-\rho \left(t\right)\right)\right)\mathrm{,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ \frac{\partial \omega }{\partial t}-\Delta \omega +\left(u\cdot \nabla \right)\omega =G_2\left(t\mathrm{,}\omega \left(t-\rho \left(t\right)\right)\right)\mathrm{,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ \nabla \cdot u=\mathrm{0,}\hfill & \text{in}\left(\mathrm{0,}+\infty \right)\times \Omega \mathrm{,}\hfill \\ u=\mathrm{0,\; <mi>\; </mi>}\omega =\mathrm{0,}\hfill & \text{on}\left(\mathrm{0,}+\infty \right)\times \partial \Omega \mathrm{,}\hfill \\ u\left(\tau \mathrm{,}x\right)=u_0\left(x\right)\mathrm{,\; <mi>\; </mi>}\omega \left(\tau \mathrm{,}x\right)={\omega }_0\left(x\right)\mathrm{,}\hfill & x\in \Omega \hfill \\ u\left(t\mathrm{,}x\right)={\varphi }_1\left(t-\tau \mathrm{,}x\right)\mathrm{,\; <mi>\; </mi>}\omega \left(t\mathrm{,}x\right)={\varphi }_2\left(t-\tau \mathrm{,}x\right)\mathrm{,}\hfill & \text{in}\left(\tau -h\mathrm{,}\tau \right)\times \Omega \mathrm{.}\hfill \end{array}$ (3)

Here $\tau \in ℝ$ is the initial time, $G_1\left(t\mathrm{,}u\left(t-\rho \left(t\right)\right)\right)\mathrm{,}{G}_2\left(t\mathrm{,}\omega \left(t-\rho \left(t\right)\right)\right)$ are the external force terms which depends on $u\left(t-\rho \left(t\right)\right)\mathrm{,}\omega \left(t-\rho \left(t\right)\right)$, respectively. $\rho \left(t\right)\ge 0$ is a delay function, ${\varphi }_1\mathrm{,}{\varphi }_2$ are respectively the velocity, temperature defined on $\left(-h\mathrm{,0}\right)$. In addition, $\rho \left(t\right)\le h,h>0$. When $\omega =0$, the Equation (3) is the Navier-Stokes equations with delays (see [3] [4] ).

This paper is inspired by the literature [3] . The [3] studies mainly the existence and uniqueness of strong solutions and the asymptotic behavior of solutions for three-dimensional globally modified Navier-Stokes equations with delays under local Lipschitz conditions. The purpose of this paper is to study the existence, uniqueness and asymptotic behavior of solutions of three-dimensional globally modified Bénard system with delays under local Lipschitz conditions. This paper is organized as follows. In Section 2, we recall some definitions of function spaces and some related properties. In Section 3, we give the proof of the existence and uniqueness of weak solutions. In Section 4, we consider the asymptotic behavior of solutions.

2. Preliminary

Let

$\begin{array}{l}\mathcal{V}=\left\{v\in {\left(C_0^{\infty }\left(\Omega \right)\right)}^3\mathrm{<mo>\; :\; </mo>}\nabla \cdot v=0\right\}\mathrm{.}\\ H=c{l}_{{\left(L^2\left(\Omega \right)\right)}^3}\mathcal{V}\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}V=c{l}_{{\left(H_0^1\left(\Omega \right)\right)}^3}\mathcal{V}\mathrm{.}\end{array}$

We denote by $\left(\cdot \mathrm{,}\cdot \right)$ the scalar product, ${\Vert \cdot \Vert }_2$ the norm in H. For $u\mathrm{,}v\in {\left(L^2\left(\Omega \right)\right)}^3$, $\left(u,v\right)={\displaystyle {\sum }_{j=1}^3}{\displaystyle {\int }_{\Omega }}{u}_j\left(x\right)v_j\left(x\right)\text{d}x$. We denote by $\left(\left(\cdot \mathrm{,}\cdot \right)\right)$ the scalar

product, $\Vert \cdot \Vert$ the norm in V. For $u\mathrm{,}v\in {\left(H_0^1\left(\Omega \right)\right)}^3$, $\left(\left(u\mathrm{,}v\right)\right)={\displaystyle {\sum }_{i\mathrm{,}j=1}^3}{\displaystyle {\int }_{\Omega }}\frac{\partial u_j}{\partial x_i}\frac{\partial v_j}{\partial x_i}\text{d}x$.

The dual space of V is $V^{\prime }$. It follows that $V\subset H\equiv H^{\prime }\subset V^{\prime }$, and the injections are dense and compact. We shall use ${\Vert \cdot \Vert }_{\ast }$ to denote the norm of $V^{\prime }$, $\langle \cdot \mathrm{,}\cdot \rangle$ represent the dual product between V and $V^{\prime }$. P is a Leray-Helmotz projection from ${\left(L^2\left(\Omega \right)\right)}^3$ to H, $\mathcal{T}$ is a Leray-Helmotz projection from ${\left(L^2\left(\Omega \right)\right)}^3$ to $L^2\left(\Omega \right)$, suppose $A_{1}u=-P\Delta u$, $A_2\omega =-\mathcal{T}\Delta \omega$, where $A_1\mathrm{,}{A}_2$ are linear continuous operators. Let $D\left(A_1\right)$ be the domain of the operator $A_1$ in H, $D\left(A_2\right)$ be the domain of the operator $A_2$ in $L^2\left(\Omega \right)$. In addition,

${\langle A_{1}u,v\rangle }_{V,V^{\prime }}=\left(\nabla u,\nabla v\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}{\langle A_2\omega ,\eta \rangle }_{H_0^1\left(\Omega \right),H^{-1}\left(\Omega \right)}=\left(\nabla \omega ,\nabla \eta \right),$

where $u\mathrm{,}v\in V\mathrm{,}\omega \mathrm{,}\eta \in H_0^1\left(\Omega \right)$.

Because of the regularity of the boundary $\partial \Omega$, it is easy to prove that: $D\left(A_1\right)={\left(H^2\left(\Omega \right)\right)}^3\cap V$, $D\left(A_2\right)=H^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)$, furthermore, we obtain

$D\left(A_1\right)\subset V\subset H\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}D\left(A_2\right)\subset H_0^1\left(\Omega \right)\subset L^2\left(\Omega \right)\mathrm{.}$

The first eigenvalue of Stokes operator is defined by $\lambda =\underset{v\in V\backslash \left\{0\right\}}{{\displaystyle \inf }}\frac{{\Vert v\Vert }^2}{{\Vert v\Vert }_2^2}>0$, let ${\lambda }_1$ be the minimum of the first eigenvalues of the operators $A_1\mathrm{,}{A}_2$. For all $u\mathrm{,}v\mathrm{,}\omega \in V$, we put $b\left(u,v,\omega \right)={\displaystyle {\int }_{\Omega }}{\displaystyle {\sum }_{i,j=1}^3}{u}_i\frac{\partial v_j}{\partial x_i}{\omega }_j\text{d}x$. We know from [1] that

$b\left(u\mathrm{,}v\mathrm{,}\omega \right)$ is a trilinear continuous form on $V\times V\times V$, and $b\left(u,v,v\right)=0$ for $u\in V$, $v\in {\left(H_0^1\left(\Omega \right)\right)}^3$. For any $u\mathrm{,}v\mathrm{,}\omega \in V$, it follows that (see [5] )

$\left|b\left(u\mathrm{,}v\mathrm{,}\omega \right)\right|\le C\Vert u\Vert \Vert v\Vert {\left|\omega \right|}^{\frac{1}{2}}{\Vert \omega \Vert }^{\frac{1}{2}}\mathrm{.}$ (4)

Define

$b_N\left(u\mathrm{,}v\mathrm{,}w\right)=F_N\left(\Vert v\Vert \right)b\left(u\mathrm{,}v\mathrm{,}w\right)\mathrm{,\; <mi>\; </mi>}\forall u\mathrm{,}v\mathrm{,}w\in V\mathrm{,}$

where $b_N$ is linear in $u\mathrm{,}w$, but it is nonlinear in v. Obviously, $b_N\left(u,v,v\right)=0,\mathrm{<mi>\; </mi>}\forall u,v\in V$. According to the property of b (see [6] ) and the definition of $F_N$, we can get

$\left|b_N\left(u\mathrm{,}v\mathrm{,}\omega \right)\right|\le N{C}_1\Vert u\Vert \Vert \omega \Vert \mathrm{,}\forall u\mathrm{,}v\mathrm{,}\omega \in V\mathrm{.}$ (5)

The following is a lemma, which is specifically proved in [5] .

Lemma 2.1. For all $u\mathrm{,}v\in V$ and $N>0$, the following two conclusions hold:

1) $0\le \Vert u\Vert F_N\left(\Vert u\Vert \right)\le N$ ; 2) $\left|F_N\left(\Vert u\Vert \right)-F_N\left(\Vert v\Vert \right)\right|\le \frac{1}{N}{F}_N\left(\Vert u\Vert \right)F_N\left(\Vert v\Vert \right)\Vert u-v\Vert$.

For all $u\in V\mathrm{,}v\mathrm{,}\omega \in H_0^1\left(\Omega \right)$, let us define $c\left(u\mathrm{,}v\mathrm{,}\omega \right)={\displaystyle {\int }_{\Omega }}{\displaystyle {\sum }_{i=1}^3}{u}_i\frac{\partial v}{\partial x_i}\omega \text{d}x$. It has been given in [7] , and $c\left(u,v,v\right)=0$ for $u\in V\mathrm{,}v\in H_0^1\left(\Omega \right)$.

Let

$\langle B_N\left(u\mathrm{,}v\right)\mathrm{,}\omega \rangle =b_N\left(u\mathrm{,}v\mathrm{,}\omega \right)\mathrm{,}\forall u\mathrm{,}v\mathrm{,}\omega \in V\mathrm{,}$

$\langle C\left(u\mathrm{,}v\right)\mathrm{,}\omega \rangle =c\left(u\mathrm{,}v\mathrm{,}\omega \right)\mathrm{,}\forall u\in V\mathrm{,}v\mathrm{,}\omega \in H_0^1\left(\Omega \right)\mathrm{.}$

The following describes the properties of the external force terms $G_i:ℝ\times H\to H,\mathrm{<mi>\; </mi>}i=1,2$ :

(c1) for any $u\in H\mathrm{,\; <mi>\; </mi>}\omega \in L^2$, $G_1\left(\cdot \mathrm{,}u\right)\mathrm{<mo>\; :\; </mo>}ℝ\to H\mathrm{,}{G}_2\left(\cdot \mathrm{,}\omega \right)\mathrm{<mo>\; :\; </mo>}ℝ\to L^2$ are measurable;

(c2) there exists nonnegative functions $g_i\in L_{loc}^p\left(ℝ\right),\mathrm{<mi>\; </mi>}i=1,2,1\le p\le +\infty$, and a nondecreasing function $L\mathrm{<mo>\; :\; </mo>}\left(\mathrm{0,}\infty \right)\to \left(\mathrm{0,}\infty \right)$, such that for all $R_{T_i}>0$, if $\left|u\right|\mathrm{,\; <mi>\; </mi>}\left|v\right|\le R_{T_i}$, then:

$\left|G_i\left(t\mathrm{,}u\right)-G_i\left(t\mathrm{,}v\right)\right|\le L\left(R_{T_i}\right)g_i^{\frac{1}{2}}\left(t\right)\left|u-v\right|\mathrm{,}i=\mathrm{1,2,\; <mi>\; </mi>}t\in ℝ\mathrm{<mo>\; ;\; </mo>}$

(c3) there exists nonnegative functions $f_i\in L_{loc}^1\left(ℝ\right),\mathrm{<mi>\; </mi>}i=1,2$, such that for any $u\in H$ (or $L^2$ ),

$\left|G_i\left(t,u\right)\right|\le g_i\left(t\right){\left|u\right|}^2+f_i\left(t\right),\mathrm{<mi>\; </mi>}\forall \mathrm{<mi>\; </mi>}t\in ℝ.$

Supposing ${\varphi }_1\in L^{2p^{\prime }}\left(-h\mathrm{,0\; <mo>\; ;\; </mo>}H\right)\mathrm{,\; <mi>\; </mi>}{\varphi }_2\in L^{2p^{\prime }}\left(-h\mathrm{,0\; <mo>\; ;\; </mo>}{L}^2\right)$, $\left\{u_0\mathrm{,}{\omega }_0\right\}\in H\times L^2$, where $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. We consider a delay function $\rho \in C^1\left(ℝ\right)$ for any $t\in ℝ$ such that

$0\le \rho \left(t\right)\le h$, and there exists a constant ${\rho }_{\mathrm{<mo>\; *\; </mo>}}$ for any $t\in ℝ$ satisfying: ${\rho }^{\prime }\left(t\right)\le {\rho }_{*}<1$.

We define ${\tilde{g}}_i\left(t\right)=g_i\circ {\theta }^{-1}\left(t\right)$, where $\theta \left(s\right)=s-\rho \left(s\right)$, $\theta \mathrm{<mo>\; :\; </mo>}\left[\tau \mathrm{,}+\infty \right)\to \left[\tau -\rho \left(\tau \right)\mathrm{,}+\infty \right)$ is the differentiable and strictly increasing function, we obtain

$\begin{array}{l}{\displaystyle {\int }_{\tau }^T}{\left|G_1\left(t,u\left(t-\rho \left(t\right)\right)\right)\right|}^2\text{d}t\le {\displaystyle {\int }_{\tau }^T}{g}_1\left(t\right){\left|u\left(t-\rho \left(t\right)\right)\right|}^2\text{d}t+{\displaystyle {\int }_{\tau }^T}{f}_1\left(t\right)\text{d}t\\ ={\displaystyle {\int }_{\tau }^T}{\tilde{g}}_1\left(t-\rho \left(t\right)\right){\left|u\left(t-\rho \left(t\right)\right)\right|}^2\text{d}t+{\displaystyle {\int }_{\tau }^T}{f}_1\left(t\right)\text{d}t\\ ={\displaystyle {\int }_{\tau -\rho \left(\tau \right)}^{T-\rho \left(T\right)}}{\tilde{g}}_1(s){\left|u\left(s\right)\right|}^2\cdot \frac{1}{1-{\rho }^{\prime }\left(s\right)}\text{d}s+{\displaystyle {\int }_{\tau }^T}{f}_1\left(t\right)\text{d}t\\ \le \frac{1}{1-{\rho }_{*}}{\displaystyle {\int }_{\tau -\rho \left(\tau \right)}^{T-\rho \left(T\right)}}{\tilde{g}}_1\left(t\right){\left|u\left(t\right)\right|}^2\text{d}t+{\displaystyle {\int }_{\tau }^T}{f}_1\left(t\right)\text{d}t\\ \le \frac{1}{1-{\rho }_{*}}{\displaystyle {\int }_{\tau -\rho \left(\tau \right)}^T}{\tilde{g}}_1\left(t\right){\left|u\left(t\right)\right|}^2\text{d}t+{\displaystyle {\int }_{\tau }^T}{f}_1\left(t\right)\text{d}t,\end{array}$ (6)

Similarly, when $i=2$, there is also an analogy estimate for the $\omega$ term.

Moreover, we call $\left\{u\mathrm{,}\omega \right\}\in W_T$ as a set of solution for Equation (3) in $\left(\mathrm{0,}T\right)$, when

$W_T=\left(L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}V\right)\cap L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)\right)\times \left(L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}_0^1\left(\Omega \right)\right)\cap L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\right)\mathrm{,}$

and for any $v\in V\mathrm{,}\tilde{\omega }\in H_0^1\left(\Omega \right)$, the following is true:

$\{\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(u,v\right)+\nu \left(\left(u,v\right)\right)+b_N\left(u,u,v\right)+\left(\xi \omega ,v\right)=\left(G_1\left(s,u\left(s-\rho \left(s\right)\right)\right),v\right),\\ \frac{\text{d}}{\text{d}t}\left(\omega ,\tilde{\omega }\right)+\left(\left(\omega ,\tilde{\omega }\right)\right)+c\left(u,\omega ,\tilde{\omega }\right)=\left(G_2\left(s,\omega \left(s-\rho \left(s\right)\right)\right),\tilde{\omega }\right).\end{array}$

3. Existence and Uniqueness of Weak Solutions

Theorem 3.1. Under the conditions (c1)-(c3), assume that $\tau \in {ℝ}^{+}$, $\left\{u_0\mathrm{,}{\omega }_0\right\}\in H\times L^2$ and ${\varphi }_1\in L^{2p^{\prime }}\left(-h\mathrm{,0\; <mo>\; ;\; </mo>}H\right)$, ${\varphi }_2\in L^{2p^{\prime }}\left(-h\mathrm{,0\; <mo>\; ;\; </mo>}{L}^2\right)$. Then there exists a weak solution $\left\{u\mathrm{,}\omega \right\}$ of equation (3).

Proof. For simplicity, and without loss of generality, we assume $\tau =0$. We consider the proof of the existence of weak solutions of Equation (3). We use Galerkin method to prove the existence of solutions, which is standard (see [8] [9] ).

Consider the following equations:

$\{\begin{array}{l}\frac{\partial u_m}{\partial t}+\nu A_1{u}_m+P_m{B}_N\left(u_m\mathrm{,}{u}_m\right)+\xi {\omega }_m=P_m{G}_1\left(t\mathrm{,}{u}_m\left(t-\rho \left(t\right)\right)\right)\mathrm{,}\\ \frac{\partial {\omega }_m}{\partial t}+A_2{\omega }_m+{\mathcal{T}}_{m}C\left(u_m\mathrm{,}{\omega }_m\right)={\mathcal{T}}_m{G}_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\mathrm{,}\\ u_m\left(0\right)=P_m{u}_0\mathrm{,}{\omega }_m\left(0\right)={\mathcal{T}}_m{\omega }_0\mathrm{,}\\ u_m=P_m{\varphi }_1\mathrm{,}{\omega }_m={\mathcal{T}}_m{\varphi }_2\mathrm{,}{u}_m\mathrm{,}{\omega }_m\in \left(-h\mathrm{,0}\right)\mathrm{.}\end{array}$ (7)

Next, we need to make some priori estimates for the approximate solution $\left\{u_m\mathrm{,}{\omega }_m\right\}$. So taking the inner product of the second equation of (7) with ${\omega }_m$, and using $c\left(u_m,{\omega }_m,{\omega }_m\right)=0$, we obtain

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\left|{\omega }_m\left(t\right)\right|}^2+{\Vert \omega \left(t\right)\Vert }^2=\left(G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\mathrm{,}{\omega }_m\right)\\ \le \frac{1}{2{\lambda }_1}{\left|G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{{\lambda }_1}{2}{\left|{\omega }_m\left(t\right)\right|}^2\\ \le \frac{1}{2{\lambda }_1}{\left|G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{1}{2}{\Vert {\omega }_m\left(t\right)\Vert }^2\mathrm{,}\end{array}$

that is

$\frac{\text{d}}{\text{d}t}{\left|{\omega }_m\right|}^2+{\Vert {\omega }_m\Vert }^2\le \frac{1}{{\lambda }_1}{\left|G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\right|}^2\mathrm{.}$

Integrating over $\left(\mathrm{0,}t\right)$, and using (6), we obtain for $t\in \left[\mathrm{0,}{T}_2\right]$

$\begin{array}{l}{\left|{\omega }_m\left(t\right)\right|}^2+{\displaystyle {\int }_0^t}{\Vert {\omega }_m\left(s\right)\Vert }^2\text{d}s\\ \le {\left|{\omega }_0\right|}^2+\frac{1}{{\lambda }_1}{\displaystyle {\int }_0^t}{\left|G_2\left(s,{\omega }_m\left(s-\rho \left(s\right)\right)\right)\right|}^2\text{d}s\\ \le K_{T_2}+\frac{1}{{\lambda }_1\left(1-{\rho }_{*}\right)}{\displaystyle {\int }_0^t}{\tilde{g}}_2\left(s\right){\left|{\omega }_m\left(s\right)\right|}^2\text{d}s,\end{array}$ (8)

where $K_{T_2}={\left|{\omega }_0\right|}^2+\frac{1}{{\lambda }_1\left(1-{\rho }_{*}\right)}{\displaystyle {\int }_{-\rho \left(0\right)}^0}{\tilde{g}}_2\left(s\right){\left|{\varphi }_2\left(s\right)\right|}^2\text{d}s+\frac{1}{{\lambda }_1}{\displaystyle {\int }_0^{T_2}}{f}_2\left(s\right)\text{d}s$. It follows from Gronwall lemma that

${\left|{\omega }_m\left(t\right)\right|}^2\le K_{T_2}\exp \left\{\frac{1}{{\lambda }_1\left(1-{\rho }_{\mathrm{<mo>\; *\; </mo>}}\right)}{\displaystyle {\int }_0^{T_2}}{\tilde{g}}_2\left(s\right)\text{d}s\right\}=C_{T_2}\mathrm{,}\forall \mathrm{<mi>\; </mi>}t\in \left[\mathrm{0,}{T}_2\right]\mathrm{.}$ (9)

Taking the inner product of the first equation of (7) with $u_m$, from (9) and $b_N\left(u_m,u_m,u_m\right)=0$, we obtain

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\left|u_m\left(t\right)\right|}^2+\nu {\Vert u_m\left(t\right)\Vert }^2\\ \le \left|\xi \right|\left({\omega }_m\left(t\right),u_m\left(t\right)\right)+\left(G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right),u_m\right)\\ \le \frac{1}{\nu {\lambda }_1}{\xi }^2{\left|{\omega }_m\left(t\right)\right|}^2+\frac{\nu {\lambda }_1}{4}{\left|u_m\left(t\right)\right|}^2+\frac{1}{\nu {\lambda }_1}{\left|G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{\nu {\lambda }_1}{4}{\left|u_m\left(t\right)\right|}^2\\ \le \frac{1}{\nu {\lambda }_1}{\left|G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{\nu }{2}{\Vert u_m\left(t\right)\Vert }^2+\frac{{\xi }^2}{\nu {\lambda }_1}{C}_{T_2},\end{array}$

that is

$\frac{\text{d}}{\text{d}t}{\left|u_m\right|}^2+\nu {\Vert u_m\Vert }^2\le \frac{2}{\nu {\lambda }_1}{\left|G_1\left(t\mathrm{,}{u}_m\left(t-\rho \right)\left(t\right)\right)\right|}^2+\frac{2{\xi }^2}{\nu {\lambda }_1}{C}_{T_2}\mathrm{.}$

Integrating over $\left(\mathrm{0,}t\right)$, and using (6), we obtain for $t\in \left[\mathrm{0,}{T}_1\right]$

$\begin{array}{l}{\left|u_m\left(t\right)\right|}^2+\nu {\displaystyle {\int }_0^t}{\Vert u_m\left(s\right)\Vert }^2\text{d}s\\ \le {\left|u_0\right|}^2+\frac{2}{\nu {\lambda }_1}{\displaystyle {\int }_0^t}{\left|G_1\left(s\mathrm{,}{u}_m\left(s-\rho \left(s\right)\right)\right)\right|}^2\text{d}s+\frac{2{\xi }^2}{\nu {\lambda }_1}{\displaystyle {\int }_0^t}{C}_{T_2}\text{d}s\\ \le K_{T_1}+\frac{2}{\nu {\lambda }_1\left(1-{\rho }_{\mathrm{<mo>\; *\; </mo>}}\right)}{\displaystyle {\int }_0^t}{\tilde{g}}_1\left(s\right){\left|u_m\left(s\right)\right|}^2\text{d}s\mathrm{,}\end{array}$ (10)

where $K_{T_1}={\left|u_0\right|}^2+\frac{2{\xi }^2}{\nu {\lambda }_1}{\displaystyle {\int }_0^{T_1}}{C}_{T_2}\text{d}s+\frac{2}{\nu {\lambda }_1\left(1-{\rho }_{*}\right)}{\displaystyle {\int }_{-\rho \left(0\right)}^0}{\tilde{g}}_1\left(s\right){\left|{\varphi }_1\left(s\right)\right|}^2\text{d}s+\frac{2}{\nu {\lambda }_1}{\displaystyle {\int }_0^{T_1}}{f}_1\left(s\right)\text{d}s$. It follows from Gronwall lemma that

${\left|u_m\left(t\right)\right|}^2\le K_{T_1}\exp \left\{\frac{2}{\nu {\lambda }_1\left(1-{\rho }_{\mathrm{<mo>\; *\; </mo>}}\right)}{\displaystyle {\int }_0^{T_1}}{\tilde{g}}_1\left(s\right)\text{d}s\right\}=C_{T_1}\mathrm{,}\forall t\in \left[\mathrm{0,}{T}_1\right]\mathrm{.}$ (11)

From (10) and (11), a subsequence of $\left\{u_m\right\}$ is bounded in $L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)$ and $L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}V\right)$ (where $T=\max \left\{T_1,T_2\right\}$ ). Next from the first equation of (7) and

(5), we obtain that $\frac{\text{d}{u}_m}{\text{d}t}$ is bounded in $L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{V}^{\prime }\right)$. Therefore, there exists an element $u\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}V\right)$ and $\frac{\text{d}u}{\text{d}t}\in L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{V}^{\prime }\right)$, such that

$\{\begin{array}{l}{u}_m\stackrel{\ast }{\rightharpoonup }u\text{in}\mathrm{<mi>\; </mi>}{L}^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)\mathrm{,}\\ u_m\rightharpoonup u\text{in}\mathrm{<mi>\; </mi>}{L}^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}V\right)\mathrm{,}\\ \frac{\text{d}{u}_m}{\text{d}t}\rightharpoonup \frac{\text{d}u}{\text{d}t}\text{in}\mathrm{<mi>\; </mi>}{L}^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{V}^{\prime }\right)\mathrm{.}\end{array}$

Obviously, $u\in C\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}H\right)$ (see [10] ). By the compactness Aubin-Lions (see [9] ), we can deduce

$\{\begin{array}{l}{u}_m\to u\text{in}\mathrm{<mi>\; </mi>}{L}^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)\mathrm{,}\\ u_m\to u\text{a}\text{.e}\text{.in}\mathrm{<mi>\; </mi>}\left(\mathrm{0,}T\right)\times \Omega \mathrm{.}\end{array}$

Similarly, from (8) and (9), there is also an analogy conclusion for the ${\omega }_m$ term.

In summary, when $t_m=+\infty ,m\ge 1$, there exists $\left\{u,\omega \right\}\in \left(L^{\infty }\left(0,T;H\right)\cap L^2\left(0,T;V\right)\right)\times \left(L^{\infty }(0,T;L^2\left(\Omega \right)\right)\cap L^2\left(0,T;H_0^1\left(\Omega \right)\right)$ for any $T>0$, such that

$\{\begin{array}{l}\left\{u_m\mathrm{,}{\omega }_m\right\}\to \left\{u\mathrm{,}\omega \right\}\text{in}\mathrm{<mi>\; </mi>}{L}^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)\times L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\mathrm{,}\\ \left\{u_m\mathrm{,}{\omega }_m\right\}\to \left\{u\mathrm{,}\omega \right\}\text{in}\mathrm{<mi>\; </mi>}H\times L^2\left(\Omega \right)\mathrm{<mi>\; </mi>}a\mathrm{.}e\mathrm{.\; <mi>\; </mi>}t\in \left(\mathrm{0,}T\right)\mathrm{.}\end{array}$

Since $\left\{u_m\mathrm{,}{\omega }_m\right\}$ weak converges to $\left\{u\mathrm{,}\omega \right\}$ in $V\times H_0^1\left(\Omega \right)$, we cannot deduce that $\Vert u_m\Vert \to \Vert u\Vert \mathrm{,\; <mi>\; </mi>}\Vert {\omega }_m\Vert \to \Vert \omega \Vert$ or $F_N\left(\Vert u_m\left(t\right)\Vert \right)\to F_N\left(\Vert u\left(t\right)\Vert \right)$ and $C\left(u_m\mathrm{,}{\omega }_m\right)\to C\left(u\mathrm{,}\omega \right)$ at least almost everywhere. So we need make a stronger estimate.

Taking the inner product of the first equation of (7) with $A_1{u}_m$, and using (9), we obtain

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert u_m\Vert }^2+\nu {\left|A_1{u}_m\right|}^2\\ \le b_N\left(u_m,u_m,A_1{u}_m\right)+\left|\xi \right|\left({\omega }_m,A_1{u}_m\right)+\left(G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right),A_1{u}_m\right)\\ \le \frac{N}{\Vert u_m\Vert }C{\Vert u_m\Vert }^{\frac{3}{2}}{\left|A_1{u}_m\right|}^{\frac{3}{2}}+\left|\xi \right|\left({\omega }_m,A_1{u}_m\right)+\left(G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right),A_1{u}_m\right)\\ \le C{N}^4{\Vert u_m\Vert }^2+\frac{\nu }{8}{\left|A_1{u}_m\right|}^2+\frac{1}{\nu }{\xi }^2{\left|{\omega }_m\right|}^2+\frac{\nu }{4}{\left|A_1{u}_m\right|}^2\\ +\frac{2}{\nu }{\left|G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{\nu }{8}{\left|A_1{u}_m\right|}^2\\ \le \frac{\nu }{2}{\left|A_1{u}_m\right|}^2+C{N}^4{\Vert u_m\Vert }^2+\frac{{\xi }^2}{\nu }{C}_{T_2}+\frac{2}{\nu }{\left|G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right)\right|}^2,\end{array}$

that is

$\frac{\text{d}}{\text{d}t}{\Vert u_m\Vert }^2+\nu {\left|A_1{u}_m\right|}^2\le C{N}^4{\Vert u_m\Vert }^2+\frac{2{\xi }^2}{\nu }{C}_{T_2}+\frac{4}{\nu }{\left|G_1\left(t,u_m\left(t-\rho \left(t\right)\right)\right)\right|}^2\mathrm{.}$ (12)

Integrating over $\left(s\mathrm{,}t\right)$, and using (6), we obtain for all $0\le s\le t\le T$

$\begin{array}{l}{\Vert u_m\left(t\right)\Vert }^2\\ \le {\Vert u_m\left(s\right)\Vert }^2+C{N}^4{\displaystyle {\int }_0^T}{\Vert u_m\left(r\right)\Vert }^2\text{d}r+\frac{2{\xi }^2}{\nu }{\displaystyle {\int }_0^T}{C}_{T_2}\text{d}r+\frac{4}{\nu }{\displaystyle {\int }_0^T}{\left|G_1\left(r\mathrm{,}{u}_m\left(r-\rho \left(r\right)\right)\right)\right|}^2\text{d}r\\ \le {\Vert u_m\left(s\right)\Vert }^2+C{N}^4{\displaystyle {\int }_0^T}{\Vert u_m\left(r\right)\Vert }^2\text{d}r+\frac{2{\xi }^2}{\nu }{\displaystyle {\int }_0^T}{C}_{T_2}\text{d}r+\frac{4}{\nu \left(1-{\rho }_{\mathrm{<mo>\; *\; </mo>}}\right)}{\displaystyle {\int }_{-\rho \left(0\right)}^0}{\tilde{g}}_1\left(r\right){\left|{\varphi }_1\left(r\right)\right|}^2\text{d}r\\ +\frac{4}{\nu \mathrm{<mo\; stretchy="false">\; (\; </mo>1}-{\rho }_{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}{\displaystyle {\int }_0^T}{\tilde{g}}_1\left(r\right){\left|u_m\left(r\right)\right|}^2\text{d}r+\frac{4}{\nu }{\displaystyle {\int }_0^T}{f}_1\left(r\right)\text{d}r\\ :={\Vert u_m\left(s\right)\Vert }^2+\Phi .\end{array}$ (13)

The following estimate $\Phi$. Thanks to (10) and (11), we know that $u_m$ is bounded in $L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}V\right)\cap L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}H\right)$, therefore, there exists ${\tilde{K}}_{T_1}>0$, such that for all integer $m\ge 1$

$\begin{array}{l}C{N}^4{\displaystyle {\int }_0^T}{\Vert u_m\left(r\right)\Vert }^2\text{d}r+\frac{2{\xi }^2}{\nu }{\displaystyle {\int }_0^T}{C}_{T_2}\text{d}r+\frac{4}{\nu \left(1-{\rho }_{*}\right)}{\displaystyle {\int }_{-\rho \left(0\right)}^0}{\tilde{g}}_1\left(r\right){\left|{\varphi }_1\left(r\right)\right|}^2\text{d}r\\ +\frac{4}{\nu \left(1-{\rho }_{*}\right)}{\displaystyle {\int }_0^T}{\tilde{g}}_1\left(r\right){\left|u_m\left(r\right)\right|}^2\text{d}r+\frac{4}{\nu }{\displaystyle {\int }_0^T}{f}_1\left(r\right)\text{d}r\le {\tilde{K}}_{T_1}.\end{array}$

Integrating (13) for s over $\left[\mathrm{0,}t\right]$, we obtain

$t{\Vert u_m\left(t\right)\Vert }^2\le {\displaystyle {\int }_0^T}{\Vert u_m\left(s\right)\Vert }^2\text{d}s+T{\tilde{K}}_{T_1}\le \underset{m\ge 1}{\sup }\left({\displaystyle {\int }_0^T}{\Vert u_m\left(s\right)\Vert }^2\text{d}s\right)+T{\tilde{K}}_{T_1}\mathrm{<mo>\; :\; </mo>}={\hat{K}}_{T_1}\mathrm{.}$

Consequently, ${\Vert u_m\left(t\right)\Vert }^2\le \frac{1}{\epsilon }{\hat{K}}_{T_1}$ for any $t\in \left[\epsilon \mathrm{,}T\right]$ and any $0<\epsilon <T$.

On the other hand, using the uniformly Gronwall inequality for (12), we have

${\Vert u_m\Vert }^2\le \left(\frac{{\tilde{K}}_{T_1}}{C{N}^4}+\frac{2{\xi }^2}{\nu }{C}_{T_2}+\frac{4}{\nu }{\left|G_1\left(t\mathrm{,}{u}_m\left(t-\rho \left(t\right)\right)\right)\right|}^2\right){\text{e}}^{C{N}^4}={C^{\prime }}_{T_1}\mathrm{.}$ (14)

Taking the inner product of the second equation of (7) with $A_2{\omega }_m$, we obtain

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert {\omega }_m\Vert }^2+{\left|A_2{\omega }_m\right|}^2\\ \le c\left(u_m\mathrm{,}{\omega }_m\mathrm{,}{A}_2{\omega }_m\right)+\left(G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\mathrm{,}{A}_2{\omega }_m\right)\\ \le C{\Vert u_m\Vert }^2{\Vert {\omega }_m\Vert }^2+\frac{1}{4}{\left|A_2{\omega }_m\right|}^2+{\left|G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{1}{4}{\left|A_2{\omega }_m\right|}^2\\ \le C{C^{\prime }}_{T_1}{\Vert {\omega }_m\Vert }^2+\frac{1}{4}{\left|A_2{\omega }_m\right|}^2+{\left|G_2\left(t\mathrm{,}{\omega }_m\left(t-\rho \left(t\right)\right)\right)\right|}^2+\frac{1}{4}{\left|A_2{\omega }_m\right|}^2\mathrm{,}\end{array}$

where

$c\left(u\mathrm{,}\omega \mathrm{,}{A}_2\omega \right)\le {\left|u\right|}_{\infty }\left|\nabla \omega \right|\left|A_2\omega \right|\le C\Vert u\Vert \Vert \omega \Vert \left|A_2\omega \right|\le C{\Vert u\Vert }^2{\Vert \omega \Vert }^2+\frac{1}{4}{\left|A_2\omega \right|}^2\mathrm{.}$

That is

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\Vert {\omega }_m\Vert }^2+{\left|A_2{\omega }_m\right|}^2\le C{C^{\prime }}_{T_1}{\Vert {\omega }_m\Vert }^2+2{\left|G_2\left(t,{\omega }_m\left(t-\rho \left(t\right)\right)\right)\right|}^2.\hfill \end{array}$

Similarly, integrating over $\left(s\mathrm{,}t\right)$, we obtain for all $0\le s\le t\le T$

${\Vert {\omega }_m\left(t\right)\Vert }^2\le {\Vert {\omega }_m\left(s\right)\Vert }^2+{\tilde{K}}_{T_2}\mathrm{,}$ (15)

where

$\begin{array}{l}C{C^{\prime }}_{T_1}{\displaystyle {\int }_0^T}{\Vert {\omega }_m\left(r\right)\Vert }^2\text{d}r+\frac{2}{1-{\rho }_{\mathrm{<mo>\; *\; </mo>}}}{\displaystyle {\int }_{-\rho \left(0\right)}^0}{\tilde{g}}_2\left(r\right){\left|{\varphi }_2\left(r\right)\right|}^2\text{d}r\\ +\frac{2}{1-{\rho }_{\mathrm{<mo>\; *\; </mo>}}}{\displaystyle {\int }_0^T}{\tilde{g}}_2\left(r\right){\left|{\omega }_m\left(r\right)\right|}^2\text{d}r+2{\displaystyle {\int }_0^T}{f}_1\left(r\right)\text{d}r\le {\tilde{K}}_{T_2}\mathrm{.}\end{array}$

Integrating (15) for s over $\left[\mathrm{0,}t\right]$, we obtain

$t{\Vert {\omega }_m\left(t\right)\Vert }^2\le \underset{m\ge 1}{\sup }\left({\displaystyle {\int }_0^T}{\Vert {\omega }_m\left(s\right)\Vert }^2\text{d}s\right)+T{\tilde{K}}_{T_2}:={\hat{K}}_{T_2}.$