Structural stability for a Brinkman-Forchheimer type model with temperature-dependent solubility

Abstract

We study the structural stability for the Brinkman-Forchheimer equations with temperature-dependent solubility. We prove both the convergence and continuous dependence results for the Forchheimer coefficient λ. We also demonstrate how to get the same results for the Forchheimer equations.

1 Introduction

In the last few years, some researchers have studied the question of the continuous dependence or convergence of solutions of problems in partial differential equations on the coefficients in the equations. It is called the question of structural stability. For one thing, when we study continuous dependence or convergence, the notion of structural stability is on changes in the model itself instead of the original data. The majority of references to work of this nature are given in the monograph of Ames and Straughan [1], which studies the structural stability about changes in the model itself. Hence, we tend to know that changes in the coefficients in the partial differential equations may be reflected physically by changes in the constitutive parameters. If we deeply study these equations by mathematical analysis, it is certainly giving us a helping hand to indicate their applicability in physics. For another, because of some inevitable errors, which may have occurred, continuous dependence or convergence results are significant. It is relevant to know the magnitude of the effect of such errors in the solutions. Consequently, we think it is valuable for us to study the subject of structural stability.

We tend to find a wide range of papers in the literature coping with the structural stability for varieties equations. Most of them focus on the Brinkman, Darcym, and Forchheimer models. These equations are discussed in the books of Nield and Bejan [2] and Straughan [3, 4]. In addition, several papers have dealt with Saint-Venant type spatial decay results for the Brinkman, Darcy, Forchheimer, and other equations for porous media. More recent work on the stability and continuous dependence questions in porous media problems has been carried out by Ames and Payne [5], Franchi and Straughan [6], Kaloni and Qin [7], Kaloni and Guo [8], Payne and Straughan [9–11], Payne et al. [12], Lin and Payne [13–15], Li and Lin [16], Celebi et al. [17, 18], Straughan [19], Scott [20], Scott and Straughan [21], and Harfash [22–24]. The fundamental model we study is based upon the equations of balance of momentum, balance of mass, conservation of energy, and conservation of salt concentration, adopting a Forchheimer approximation in the body force term in the momentum equation (see [3, 25]),

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{i}}{\partial t}+\lambda|u|u_{i}=-p_{,i}+\Delta u_{i}+g_{i}T-h_{i}C,\quad (x,t)\in\Omega\times[0,\tau], \\ \frac{\partial u_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T}{\partial t}+u_{i}\frac{\partial T}{\partial x_{i}}=\Delta T,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C}{\partial t}+u_{i}\frac{\partial C}{\partial x_{i}}=\Delta C+Lf(T)-kC,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . $$

(1.1)

where $u_{i}$ is the velocity, p denotes the pressure, T is the temperature, and C is the salt concentration. Here $g_{i}(x)$, $h_{i}(x)$ are gravity fields. Here also Δ is the Laplacian operator. a, b, L, and k are positive constants. Equations (1.1) follow in practice by employing a Forchheimer approximation which accounts for the variable C allowing the incompressibility condition to hold (see Fife [26]). The function f is at least $C^{1}$.

Equations (1.1) hold in the region $\Omega\times[0,\tau]$, where Ω is a bounded, simply connected, and star-shaped domain with boundary ∂Ω in $R^{3}$, and τ is a given number satisfying $0\leq\tau<\infty$. Associated with (1.1), we impose the boundary conditions

$$ u_{i}=0 ,\qquad \frac{\partial T}{\partial n}=0, \qquad \frac{\partial C}{\partial n}=0,\quad (x,t)\in \partial\Omega\times[0,\tau], $$

(1.2)

and additionally the concentration is given at $t=0$, i.e.,

$$\begin{aligned}& u_{i}(x,0)=u_{i0}(x), \end{aligned}$$

(1.3)

$$\begin{aligned}& T(x,0)=T_{0}(x),\qquad C(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(1.4)

We will derive both the convergence result and continuous result on the Forchheimer coefficient λ. In the present paper, a comma is used to indicate partial differentiation and the differentiation with respect to the direction $x_{k}$ is denoted as ,k, thus $u_{,i}$ denotes $\frac{\partial u}{\partial x_{i}}$. The usual summation convection is employed with repeated Latin subscripts summed from 1 to 3. Hence, $u_{i,i}=\sum_{i=1}^{3}\frac{\partial u_{i}}{\partial x_{i}}$, $\|\cdot\|$ denotes the norm of $L^{2}$, and $\|\cdot\|_{p}$ denotes the norm of $L^{p}$.

2 A priori bounds for the temperature T and the salt concentration C

Now we want an a priori bound or a maximum principle for T. Therefore multiplying (1.1)₃ by $T^{2p-1}$ and integrating by parts, we can obtain

$$ \int_{\Omega}T^{2p}\,dx- \int_{\Omega}T_{0}^{2p}\,dx=-\frac{2(2p-1)}{p} \int_{0}^{t} \int_{\Omega}\bigl(T^{p}\bigr)_{,i} \bigl(T^{p}\bigr)_{,i}\,dx\,d\eta\leq0. $$

(2.1)

Inequality (2.1) is now integrated and then we take the $\frac{1}{2p}$ power to find

$$ \biggl( \int_{0}^{t} \int_{\Omega}T^{2p}\,dx\,d\eta \biggr)^{\frac{1}{2p}}\leq \biggl( \int_{0}^{t} \int_{\Omega}T_{0}^{2p}\,dx\,d\eta \biggr)^{\frac{1}{2p}}. $$

(2.2)

We let $p\rightarrow\infty$ and then (2.2) leads to

$$ \sup_{[0,t]}\|T\|_{\infty}\leq \|T_{0}\|_{\infty}=T^{M}. $$

(2.3)

Since f is a $C^{1}$ function, and T is bounded, we can easily see that there exists a constant d such that

$$ f(T)\leq d. $$

(2.4)

For some $\xi\in(T,T^{*})$, we easily get

$$ \bigl\vert f'(\xi)\bigr\vert \leq k_{1}, $$

(2.5)

where $k_{1}$ is a positive constant.

Similarly we have

$$ \biggl( \int_{0}^{t} \int_{\Omega}C^{2p}\,dx\,d\eta \biggr)^{\frac{1}{2p}}\leq \biggl( \int_{0}^{t} \int_{\Omega}e^{(2p-1)(t-\eta)} \biggl(k(p)+ \int_{\Omega}C_{0}^{2p}\,dx \biggr)\,dx\,d\eta \biggr)^{\frac{1}{2p}}, $$

(2.6)

where $k(p)=\int^{\tau}_{0}\int_{\Omega}(Lf(T))^{2p}\,dx\,dt$.

We let $p\rightarrow\infty$, (2.6) leads to

$$ \sup_{[0,t]}\|C\|_{\infty}\leq C^{M}, $$

(2.7)

where $C^{M}=\max{\{e^{\tau}\|C_{0}\|_{\infty},L\, de^{\tau}\}}$.

3 Convergence and continuous dependence results for the Forchheimer coefficient λ

Now, let $(u_{i},T,C,p)$ be a solution to the boundary initial-value problem for the Brinkman-Forchheimer model,

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u_{i}}{\partial t}+\lambda|u|u_{i}=-p_{,i}+\Delta u_{i}+g_{i}T-h_{i}C,\quad (x,t)\in\Omega\times[0,\tau], \\ \frac{\partial u_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T}{\partial t}+u_{i}\frac{\partial T}{\partial x_{i}}=\Delta T,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C}{\partial t}+u_{i}\frac{\partial C}{\partial x_{i}}=\Delta C+Lf(T)-kC,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . \end{aligned}$$

(3.1)

$$\begin{aligned}& u_{i}=0 , \qquad \frac{\partial T}{\partial n}=0,\qquad \frac{\partial C}{\partial n}=0,\quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(3.2)

$$\begin{aligned}& u_{i}(x,0)=u_{i0}(x), \end{aligned}$$

(3.3)

$$\begin{aligned}& T(x,0)=T_{0}(x),\qquad C(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(3.4)

Moreover, let $(u^{*}_{i},T^{*},C^{*},p^{*})$ be a solution to the corresponding model with $\lambda=0$,

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u^{*}_{i}}{\partial t}=-p^{*}_{,i}+\Delta u^{*}_{i}+g_{i}T^{*}-h_{i}C^{*},\quad (x,t)\in\Omega\times[0,\tau], \\ \frac{\partial u^{*}_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T^{*}}{\partial t}+u^{*}_{i}\frac{\partial T^{*}}{\partial x_{i}}=\Delta T^{*},\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C^{*}}{\partial t}+u^{*}_{i}\frac{\partial C^{*}}{\partial x_{i}}=\Delta C^{*}+Lf(T^{*})-kC^{*},\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . \end{aligned}$$

(3.5)

$$\begin{aligned}& u^{*}_{i}=0 ,\qquad \frac{\partial T^{*}}{\partial n}=0,\qquad \frac{\partial C^{*}}{\partial n}=0, \quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(3.6)

$$\begin{aligned}& u^{*}_{i}(x,0)=u_{i0}(x), \end{aligned}$$

(3.7)

$$\begin{aligned}& T^{*}(x,0)=T_{0}(x),\qquad C^{*}(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(3.8)

The object of this section is to demonstrate that the solution of (3.1) converges to the solution of (3.5) as $\lambda\rightarrow0$. Now, we define the difference variables $\omega_{i}$, π, θ, and S by

$$ \omega_{i}=u_{i}-u^{*}_{i}, \qquad \pi=p-p^{*}, \qquad \theta=T-T^{*},\qquad S=C-C^{*}. $$

(3.9)

Then $(\omega_{i},\theta,S,\pi)$ is a solution of the problem

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial\omega_{i}}{\partial t}+\lambda|u|u_{i}=-\pi _{,i}+\Delta \omega_{i}+g_{i}\theta-h_{i}S,\quad (x,t)\in\Omega\times [0,\tau], \\ \frac{\partial\omega_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial\theta}{\partial t}+\omega_{i}\frac{\partial T}{\partial x_{i}}+u^{*}_{i}\frac{\partial\theta}{\partial x_{i}}=\Delta \theta,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial S}{\partial t}+\omega_{i}\frac{\partial C}{\partial x_{i}}+u^{*}_{i}S_{,i}=\Delta S+L(f(T)-f(T^{*}))-kS,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . $$

(3.10)

in $\Omega\times[0,\tau]$, subject to the boundary and initial conditions

$$\begin{aligned}& \omega_{i}=0 , \qquad \frac{\partial \theta}{\partial n}=0,\qquad \frac{\partial S}{\partial n}=0,\quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(3.11)

$$\begin{aligned}& \omega_{i}(x,0)=0, \end{aligned}$$

(3.12)

$$\begin{aligned}& \theta(x,0)=0, \qquad S(x,0)=0,\quad x \in\Omega. \end{aligned}$$

(3.13)

We will obtain the following result.

Theorem 1

Let $(u_{i},T,C,p)$ be the classical solution to the initial-boundary problem (3.1)-(3.4), $(u^{*}_{i},T^{*},C^{*},p^{*})$ be the classical solution to the initial-boundary problem (3.5)-(3.8) in $\Omega\times[0,\tau]$, and $(w_{i},\theta,S,\pi)$ be the difference of $(u_{i},T,C,p)$ and $(u^{*}_{i},T^{*},C^{*},p^{*})$, then the solution $(u_{i},T,C,p)$ converges to the solution $(u^{*}_{i},T^{*},C^{*},p^{*})$ as the Boussinesq coefficient λ tends to 0. The difference $(w_{i},\theta,S,\pi)$ satisfies

$$ \|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}\leq\frac{2\lambda}{3M_{1}} \biggl(\frac{3}{4}k_{2}^{2}k_{3}^{\frac{1}{2}}k_{4}^{\frac{3}{2}}+ \frac {1}{4}|\Omega| \biggr)e^{M_{1}\tau}. $$

(3.14)

Proof

We multiply (3.10)₁ by $\omega_{i}$ and integrate over Ω to find

$$\begin{aligned} \frac{d}{dt}\|\omega\|^{2}&=-2\lambda \int_{\Omega}|u|u_{i}\omega_{i}\,dx-2\| \nabla \omega\|^{2}+2 \int_{\Omega}g_{i}\theta\omega_{i}\,dx-2 \int_{\Omega}h_{i}S\omega_{i}\,dx \\ &\leq-2\lambda \int_{\Omega}|u|u_{i}u_{i}\,dx+2\lambda \int_{\Omega}|u|u_{i}u_{i}^{*}\,dx+2\|\omega \|^{2}+h^{2}\|S\|^{2}+g^{2}\|\theta\|^{2} \\ &\leq\frac{2}{3}\lambda \int_{\Omega}\bigl\vert u^{*}\bigr\vert u_{i}^{*}u_{i}^{*} \,dx+2\|\omega\|^{2}+h^{2}\|S\|^{2}+g^{2}\|\theta\|^{2}. \end{aligned}$$

(3.15)

We will use the following inequality in three dimensions:

$$ \|f\|_{4}\leq k_{2}^{\frac{1}{2}}\|f \|^{\frac{1}{4}}\|\nabla f\|^{\frac{3}{4}}, $$

(3.16)

where $k_{2}$ is a positive constant.

From the above inequality and the Young inequality, we can get

$$\begin{aligned} \int_{\Omega}\bigl\vert u^{*}\bigr\vert u_{i}^{*}u_{i}^{*} \,dx&\leq\frac{3}{4} \int_{\Omega}\bigl(u^{*}_{i}u^{*}_{i} \bigr)^{2}\,dx+ \int _{\Omega}\frac{1}{4}\,dx \\ &\leq\frac{3}{4}k_{2}^{2}\bigl\Vert u^{*}\bigr\Vert \bigl\Vert \nabla u^{*}\bigr\Vert ^{3}+\frac{1}{4}|\Omega|, \end{aligned}$$

(3.17)

where $|\Omega|$ is the measure of Ω.

We multiply (3.5)₁ by $u^{*}_{i}$ and integrate over Ω to find

$$\begin{aligned} \frac{1}{2}\frac{d}{d t}\bigl\Vert u^{*}\bigr\Vert ^{2}+\bigl\Vert \nabla u^{*}\bigr\Vert ^{2} =&7- \int_{\Omega}p^{*}_{,i}u^{*}_{i}\,dx+ \int_{\Omega}g_{i}T^{*}u^{*}_{i}\,dx- \int_{\Omega}h_{i}C^{*}u^{*}_{i}\,dx \\ \leq& \biggl(g^{2} \int_{\Omega}u^{*}_{i}u^{*}_{i}\,dx \int_{\Omega}\bigl(T^{*}\bigr)^{2}\,dx \biggr)^{\frac{1}{2}} \\ &{}+ \biggl(h^{2} \int_{\Omega}u^{*}_{i}u^{*}_{i}\,dx \int_{\Omega}\bigl(C^{*}\bigr)^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq&\bigl\Vert u^{*}\bigr\Vert ^{2}+\frac{1}{2}|\Omega | \bigl[g^{2}\bigl(T^{M}\bigr)^{2}+h^{2} \bigl(C^{M}\bigr)^{2} \bigr], \end{aligned}$$

(3.18)

where $g^{2}=\max_{\Omega}g_{i}g_{i}$, $h^{2}=\max_{\Omega}h_{i}h_{i}$.

Integration of (3.18) leads to

$$ \bigl\Vert u^{*}\bigr\Vert ^{2}\leq \Vert u_{0}\Vert ^{2}e^{2\tau}+\frac{1}{2} \bigl[g^{2}\bigl(T^{M}\bigr)^{2}+h^{2} \bigl(C^{M}\bigr)^{2} \bigr]|\Omega|\bigl(e^{2\tau}-1 \bigr)=k_{3}. $$

(3.19)

We now want to give a bound for $\|\nabla u^{*}\|^{2}$. Multiplying (3.5)₁ by $u^{*}_{i,t}$ and integrating over Ω, we have

$$ \int_{\Omega}u^{*}_{i,t}u^{*}_{i,t}\,dx- \int_{\Omega}u^{*}_{i,t}u^{*}_{i,jj}\,dx= \int_{\Omega}g_{i}u^{*}_{i,t}T^{*}\,dx- \int_{\Omega}h_{i}u^{*}_{i,t}C^{*}\,dx. $$

(3.20)

We can obtain

$$\begin{aligned}& \int_{0}^{t} \int_{\Omega}u^{*}_{i,t}u^{*}_{i,t}\,dx\,d\eta+ \frac{1}{2} \int_{\Omega}u^{*}_{i,j}u^{*}_{i,j} \,dx|_{\eta=t} \\& \quad \leq \int^{t}_{0} \int_{\Omega}u^{*}_{i,t}u^{*}_{i,t}\,dx\,d\eta+ \frac{1}{2} \int_{0}^{t} \int_{\Omega}\bigl(T^{*}\bigr)^{2}\,dx\,d\eta \\& \qquad {}+ \frac{1}{2} \int_{0}^{t} \int_{\Omega}\bigl(C^{*}\bigr)^{2}\,dx\,d\eta+ \frac{1}{2} \int _{\Omega}u_{i0,j}u_{i0,j}\,dx \end{aligned}$$

(3.21)

or

$$ \int_{\Omega}u^{*}_{i,j}u^{*}_{i,j} \,dx|_{\eta=t}\leq \bigl(T^{M}\bigr)^{2}|\Omega|\tau+ \bigl(C^{M}\bigr)^{2}|\Omega|\tau+ \int_{\Omega}u_{i0,j}u_{i0,j}\,dx=k_{4}. $$

(3.22)

Combining (3.15), (3.17), (3.19), and (3.22) gives

$$ \frac{d}{dt}\|\omega\|^{2}\leq\frac{2}{3} \lambda \biggl(\frac {3}{4}k_{2}^{2}k_{3}^{\frac{1}{2}}k_{4}^{\frac{3}{2}}+ \frac{1}{4}|\Omega | \biggr)+2\|\omega\|^{2}+h^{2}\|S\|^{2}+g^{2}\|\theta\|^{2}. $$

(3.23)

Multiplying (3.10)₃ by θ and integrating over Ω, we derive

$$\begin{aligned} \frac{d}{dt}\|\theta\|^{2}+2 \int_{\Omega}\theta_{,i}\theta _{,i}\,dx&=-2 \int_{\Omega}\theta\omega_{i}T_{,i}\,dx=2 \int_{\Omega}\theta _{,i}\omega_{i}T\,dx \\ &\leq2 \int_{\Omega}\theta_{,i}\theta_{,i}\,dx+ \frac{(T^{M})^{2}}{2}\| \omega\|^{2}. \end{aligned}$$

(3.24)

The Lagrange theorem says that

$$ f(T)-f\bigl(T^{*}\bigr)=\theta f'(\xi). $$

(3.25)

Multiplying (3.10)₄ by S and integrating over Ω and using (2.5), we find

$$\begin{aligned} \frac{d}{dt}\|S\|^{2}+2 \int_{\Omega}S_{,i}S_{,i}\,dx&=-2 \int_{\Omega}S\omega_{i}C_{,i}\,dx+2L \int_{\Omega}S\bigl(f(T)-f\bigl(T^{*}\bigr)\bigr)\,dx-2k \int_{\Omega}S^{2}\,dx \\ &\leq2 \int_{\Omega}S_{,i}\omega_{i}C \,dx+2Lk_{1} \int_{\Omega}S\theta \,dx \\ &\leq2 \int_{\Omega}S_{,i}S_{,i}\,dx+ \frac{(C^{M})^{2}}{2}\|\omega\|^{2}+L^{2}k_{1}^{2} \|S\|^{2}+\|\theta \|^{2}. \end{aligned}$$

(3.26)

From (3.23), (3.24), and (3.26), we get

$$\begin{aligned} \frac{d}{dt}\bigl(\|\omega\|^{2}+\|\theta \|^{2}+\|S\|^{2}\bigr) \leq&\frac {2}{3}\lambda \biggl( \frac{3}{4}k_{2}^{2}k_{3}^{\frac{1}{2}}k_{4}^{\frac {3}{2}}+ \frac{1}{4}|\Omega| \biggr) \\ &{}+ \biggl(2+\frac{(T^{M})^{2}}{2}+\frac {(C^{M})^{2}}{2} \biggr)\|\omega\|^{2} \\ &{}+\bigl(L^{2}k_{1}^{2}+h^{2}\bigr)\|S\|^{2}+\bigl(1+g^{2}\bigr)\|\theta\|^{2}. \end{aligned}$$

(3.27)

We set

$$ F_{1}=\|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}. $$

Then we have

$$ \frac{dF_{1}}{dt}\leq\frac{2}{3}\lambda \biggl( \frac {3}{4}k_{2}^{2}k_{3}^{\frac{1}{2}}k_{4}^{\frac{3}{2}}+ \frac{1}{4}|\Omega | \biggr)+M_{1}F_{1}, $$

(3.28)

where $M_{1}=\max{\{2+\frac{(T^{M})^{2}}{2}+\frac{(C^{M})^{2}}{2},L^{2}k_{1}^{2}+h^{2},1+g^{2}\}}$.

We easily see that

$$ \|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}\leq\frac{2\lambda}{3M_{1}} \biggl(\frac{3}{4}k_{2}^{2}k_{3}^{\frac{1}{2}}k_{4}^{\frac{3}{2}}+ \frac {1}{4}|\Omega| \biggr)e^{M_{1}t}. $$

(3.29)

Inequality (3.29) demonstrates the convergence of $u_{i}$ to $u^{*}_{i}$, T to $T^{*}$, and C to $C^{*}$ as $\lambda\rightarrow0$ in the indicated measure. □

Next, we will discuss the continuous dependence on the Forchheimer coefficient λ. Let $(u_{i},p,T,C)$ be a solution of the boundary initial-value problem for the thermal convection model,

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u_{i}}{\partial t}+\lambda_{1} |u|u_{i}=-p_{,i}+\Delta u_{i}+g_{i}T-h_{i}C,\quad (x,t)\in\Omega\times[0,\tau], \\ \frac{\partial u_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T}{\partial t}+u_{i}\frac{\partial T}{\partial x_{i}}=\Delta T,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C}{\partial t}+u_{i}\frac{\partial C}{\partial x_{i}}=\Delta C+Lf(T)-kC,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . \end{aligned}$$

(3.30)

$$\begin{aligned}& u_{i}=0 ,\qquad w \frac{\partial T}{\partial n}=0,\qquad \frac{\partial C}{\partial n}=0 ,\quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(3.31)

$$\begin{aligned}& u_{i}(x,0)=u{_{i0}}(x), \end{aligned}$$

(3.32)

$$\begin{aligned}& T(x,0)=T_{0}(x),\qquad C(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(3.33)

Furthermore, let $(u^{*}_{i},p^{*},T^{*},C^{*})$ be a solution to the following boundary initial-value problem:

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u^{*}_{i}}{\partial t}+\lambda _{2}|u^{*}|u^{*}_{i}=-p^{*}_{,i}+\Delta u^{*}_{i}+g_{i}T^{*}-h_{i}C^{*} ,\quad (x,t)\in\Omega \times[0,\tau], \\ \frac{\partial u^{*}_{i}}{\partial x_{i}}=0 ,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T^{*}}{\partial t}+u^{*}_{i}\frac{\partial T^{*}}{\partial x_{i}}=\Delta T^{*},\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C^{*}}{\partial t}+u^{*}_{i}\frac{\partial C^{*}}{\partial x_{i}}=\Delta C^{*}+Lf(T^{*})-kC^{*},\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . \end{aligned}$$

(3.34)

$$\begin{aligned}& u^{*}_{i}=0 ,\qquad \frac{\partial T^{*}}{\partial n}=0,\qquad \frac{\partial C^{*}}{\partial n}=0, \quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(3.35)

$$\begin{aligned}& u^{*}_{i}(x,0)=u{_{i0}}(x), \end{aligned}$$

(3.36)

$$\begin{aligned}& T^{*}(x,0)=T_{0}(x),\qquad C^{*}(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(3.37)

In this section, we establish the continuous dependence on the coefficient. To do this, let $(u_{i},T,C,p)$ and $(u^{*}_{i},T^{*},C^{*},p^{*})$ be solutions of (3.30) and (3.34) with the same boundary and initial conditions. Now, we define

$$ \omega_{i}=u_{i}-u_{i}^{*},\qquad \pi=p-p^{*},\qquad \theta=T-T^{*},\qquad S=C-C^{*}. $$

(3.38)

Then $(\omega_{i},\theta,S,\pi)$ is a solution of the problem

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial\omega_{i}}{\partial t}+(\lambda_{1}|u|u_{i}-\lambda _{2}|u^{*}|u^{*}_{i})=-\pi_{,i}+\Delta \omega_{i}+g_{i}\theta-h_{i}S,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial\omega_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial\theta}{\partial t}+\omega_{i}\frac{\partial T}{\partial x_{i}}+u^{*}_{i}\frac{\partial\theta}{\partial x_{i}}=\Delta \theta ,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial S}{\partial t}+\omega_{i}\frac{\partial C}{\partial x_{i}}+u^{*}_{i}S_{,i}=\Delta S+L(f(T)-f(T^{*}))-kS,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . $$

(3.39)

in $\Omega\times{t>0}$, subject to the boundary and initial conditions

$$\begin{aligned}& \omega_{i}=0 ,\qquad \frac{\partial \theta}{\partial n}=0, \qquad \frac{\partial S}{\partial n}=0,\quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(3.40)

$$\begin{aligned}& \omega_{i}(x,0)=0, \end{aligned}$$

(3.41)

$$\begin{aligned}& \theta(x,0)=0,\qquad S(x,0)=0,\quad x \in\Omega. \end{aligned}$$

(3.42)

We will obtain the following result.

Theorem 2

Let $(u_{i},T,C,p)$ be the classical solution to the initial-boundary problem (3.30)-(3.33), $(u^{*}_{i},T^{*},C^{*},p^{*})$ be the classical solution to the initial-boundary problem (3.34)-(3.37) in $\Omega\times(0,\tau)$, and $(w_{i},\theta,S,\pi)$ be the difference of $(u_{i},T,C,p)$ and $(u^{*}_{i}, T^{*},C^{*},p^{*})$, then the solution $(u_{i},T,C,p)$ converges to the solution $(u^{*}_{i},T^{*},C^{*},p^{*})$ as the Forchheimer coefficient $\lambda_{1}$ tends to $\lambda_{2}$. If we suppose that $\int_{\Omega}u_{i,t}(x,0)u_{i,t}(x,0)\,dx+\int_{\Omega}T_{i,t}(x,0)T_{i,t}(x,0)\,dx+\int _{\Omega}C_{i,t}(x,0)C_{i,t}(x,0)\,dx\leq R$, the difference $(w_{i},\theta ,S,\pi)$ satisfies

$$ \|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}\leq\lambda ^{2}te^{-M_{3}t}k^{2}_{2}k^{\frac{1}{2}}_{5}k_{6}^{\frac{3}{2}}, $$

(3.43)

where $\lambda=\lambda_{1}-\lambda_{2}$.

Proof

We first observe that

$$ \frac{1}{2}\frac{d}{dt}\|\omega\|^{2}+ \int_{\Omega}\bigl(\lambda_{1}|u|u_{i}- \lambda_{2}\bigl\vert u^{*}\bigr\vert u^{*}_{i}\bigr) \omega_{i}\,dx+\|\nabla \omega\|^{2}= \int_{\Omega}g_{i}\theta\omega_{i}\,dx- \int_{\Omega}h_{i}S\omega_{i}\,dx. $$

(3.44)

Moreover, we can get

$$\begin{aligned} \int_{\Omega}\bigl(\lambda_{1}|u|u_{i}- \lambda_{2}\bigl\vert u^{*}\bigr\vert u^{*}_{i}\bigr) \omega_{i}\,dx&\geq (\lambda+\lambda_{2}) \int_{\Omega}|u|u_{i}\omega_{i}\,dx- \lambda_{2} \int _{\Omega}\bigl\vert u^{*}\bigr\vert u^{*}_{i} \omega_{i}\,dx \\ &\geq\lambda \int_{\Omega}|u|u_{i}\omega_{i}\,dx+ \lambda_{2} \int_{\Omega}\bigl(|u|u_{i}\omega_{i}-\bigl\vert u^{*}\bigr\vert u^{*}_{i}\omega_{i}\bigr)\,dx. \end{aligned}$$

(3.45)

Since the operator $T(u)=|u|u$ is a monotonous operator, we get

$$ \int_{\Omega}\bigl(|u|u-\bigl\vert u^{*}\bigr\vert u^{*}\bigr) \omega \,dx\geq0. $$

(3.46)

From the above discussion, we can get

$$ \int_{\Omega}\bigl(\lambda_{1}|u|u_{i}- \lambda_{2}\bigl\vert u^{*}\bigr\vert u^{*}_{i}\bigr) \omega_{i}\,dx\geq \lambda \int_{\Omega}|u|u_{i}\omega_{i}\,dx. $$

(3.47)

Hence we get a similar inequality,

$$ \frac{1}{2}\frac{d}{dt}\|\omega\|^{2}+ \lambda \int_{\Omega}|u|u_{i}\omega_{i}\,dx+\| \nabla \omega\|^{2}\leq \int_{\Omega}g_{i}\theta\omega_{i}\,dx- \int_{\Omega}h_{i}S\omega_{i}\,dx. $$

(3.48)

Nevertheless, we use another method to get the bound for $\int_{\Omega}|u|u_{i}\omega_{i}\,dx$. We can use a similar method to give the bound for $\|u\|^{2}$,

$$ \|u\|^{2}\leq\|u_{0}\|^{2}e^{2\tau}+ \frac{1}{2} \bigl[g^{2}\bigl(T^{M}\bigr)^{2}+h^{2} \bigl(C^{M}\bigr)^{2} \bigr]|\Omega|\bigl(e^{2\tau}-1 \bigr)=k_{5}. $$

(3.49)

The next step is to give a bound for $\|\nabla u\|^{2}$. In [27], Liu used the similar method. Multiplying (1.1)₁ by $u_{i}$ and integrating over Ω, we have

$$\begin{aligned} \int_{\Omega}u_{i,j}u_{i,j}\,dx \leq&- \int_{\Omega}u_{i,t}u_{i}\,dx+ \int_{\Omega}g_{i}u_{i}T\,dx- \int_{\Omega}h_{i}u_{i}C\,dx \\ \leq& \biggl( \int_{\Omega}u_{i,t}u_{i,t}\,dx \biggr)^{\frac{1}{2}} \biggl( \int_{\Omega}u_{i}u_{i}\,dx \biggr)^{\frac{1}{2}}+\biggl(\int_{\Omega}u_{i}u_{i}\,dx\biggr)^{\frac{1}{2}}\biggl(g^{2}\int_{\Omega}T^{2}\,dx\biggr)^{\frac{1}{2}} \\ &{}+\biggl(\int_{\Omega}u_{i}u_{i}\,dx\biggr)^{\frac{1}{2}}\biggl(h^{2}\int_{\Omega}C^{2}\,dx\biggr)^{\frac{1}{2}} \\ \leq& k_{5}^{\frac{1}{2}}\biggl[\biggl(\int_{\Omega}u_{i,t}u_{i,t}\,dx\biggr)^{\frac{1}{2}}+|\Omega|^{\frac{1}{2}} gT^{M}+|\Omega|^{\frac{1}{2}}hC^{M}\biggr]. \end{aligned}$$

(3.50)

In order to have a bound for $\int_{\Omega}u_{i,j}u_{i,j}\,dx$, we need only give a bound for $\int_{\Omega}u_{i,t}u_{i,t}\,dx$. We can observe that

$$\begin{aligned} \frac{d}{dt} \int_{\Omega}u_{i,t}u_{i,t}\,dx =&2 \int_{\Omega}u_{i,t}\bigl[-\lambda \vert u\vert u_{i}-p_{,i}+u_{i,jj}+g_{i}T-h_{i}C \bigr]_{,t}\,dx \\ \leq&-2\lambda \int_{\Omega}u_{i,t}|u|_{,t}u_{i} \,dx-2 \int_{\Omega}u_{i,jt}u_{i,jt}\,dx+2 \int_{\Omega}u_{i,t}(g_{i}T_{,t}-h_{i}C_{,t}) \,dx \\ \leq&-2\lambda \int_{\Omega}u_{i,t}u_{i}\frac{u_{k}u_{k,t}}{|u|} \,dx+2 \int_{\Omega}u_{i,t}u_{i,t}\,dx+ g^{2}\int_{\Omega}T_{,t}T_{,t}\,dx \\ &{}+h^{2}\int_{\Omega}C_{,t}C_{,t}\,dx \\ \leq&2 \int_{\Omega}u_{i,t}u_{i,t}\,dx+ g^{2}\int_{\Omega}T_{,t}T_{,t}\,dx+h^{2}\int_{\Omega}C_{,t}C_{,t}\,dx. \end{aligned}$$

(3.51)

Hence we should give the bound for $\int_{\Omega}T_{,t}T_{,t}\,dx$ and $\int_{\Omega}C_{,t}C_{,t}\,dx$. We find that

$$\begin{aligned} \frac{d}{dt} \int_{\Omega}T_{,t}T_{,t}\,dx&=2 \int_{\Omega}T_{,t}(T_{,iit}-u_{i,t}T_{,i}-u_{i}T_{,it}) \,dx =-2 \int_{\Omega}T_{,it}T_{,it}\,dx+2 \int_{\Omega}T_{,it}u_{i,t}T\,dx \\ &\leq\frac{(T^{M})^{2}}{2} \int_{\Omega}u_{i,t}u_{i,t}\,dx. \end{aligned}$$

(3.52)

Similarly we can get

$$\begin{aligned} \frac{d}{dt} \int_{\Omega}C_{,t}C_{,t}\,dx =&2 \int_{\Omega}C_{,t}\bigl(C_{,iit}-u_{i,t}C_{,i}-u_{i}C_{,it}+L \bigl[f(T)\bigr]_{,t}-kC_{,t}\bigr)\,dx \\ =&-2 \int_{\Omega}C_{,it}C_{,it}\,dx+2 \int_{\Omega}C_{,it}u_{i,t}C\,dx \\ &{}+2L \int_{\Omega}C_{,t}\bigl[f(T)\bigr]_{,t}-2k \int_{\Omega}C_{,t}C_{,t}\,dx \\ \leq&\frac{(C^{M})^{2}}{2} \int_{\Omega}u_{i,t}u_{i,t}\,dx+ \frac{L^{2}}{2k} \int_{\Omega}\bigl[f(T)\bigr]_{,t}\bigl[f(T) \bigr]_{,t}\,dx \\ \leq&\frac{(C^{M})^{2}}{2} \int_{\Omega}u_{i,t}u_{i,t}\,dx+ \frac{L^{2}k_{1}^{2}}{2k} \int_{\Omega}T_{,t}T_{,t}\,dx. \end{aligned}$$

(3.53)

We set

$$ F_{2}= \int_{\Omega}u_{i,t}u_{i,t}\,dx+ \int_{\Omega}T_{,t}T_{,t}\,dx+ \int_{\Omega}C_{,t}C_{,t}\,dx. $$

Combining (3.51)-(3.53), we can get

$$ \frac{dF_{2}}{dt}\leq M_{2}F_{2}, $$

(3.54)

where $M_{2}=\max{\{2+\frac{(C^{M})^{2}}{2}+\frac{(T^{M})^{2}}{2}, \frac{L^{2}k_{1}^{2}}{2k}+g^{2}, h^{2}\}}$.

Hence, from our assumption, we can get

$$\begin{aligned} F_{2}&\leq \biggl( \int_{\Omega}u_{i,t}(x,0)u_{i,t}(x,0)\,dx+ \int_{\Omega}T_{i,t}(x,0)T_{i,t}(x,0)\,dx+ \int_{\Omega}C_{i,t}(x,0)C_{i,t}(x,0)\,dx \biggr)e^{M_{2}t} \\ &\leq Re^{M_{2}\tau}. \end{aligned}$$

(3.55)

So we can draw the conclusion that

$$ \int_{\Omega}u_{i,j}u_{i,j}\,dx\leq k_{5}^{\frac{1}{2}}\biggl(Re^{M_{2}\tau}+ |\Omega|^{\frac{1}{2}}T^{M}g+|\Omega|^{\frac{1}{2}}C^{M}h\biggr)=k_{6}. $$

(3.56)

Using the inequalities (3.16), we multiply (3.39)₁ by $\omega_{i}$ and integrate over Ω to find

$$\begin{aligned} \frac{d}{dt}\|\omega\|^{2} =&-2\lambda \int_{\Omega}|u|u_{i}\omega_{i}\,dx-2\| \nabla \omega\|^{2} \\ &{}+2 \int_{\Omega}g_{i}\theta\omega_{i}\,dx-2 \int_{\Omega}h_{i}S\omega_{i}\,dx \\ \leq&\lambda^{2} \int_{\Omega}|u|^{4}\,dx+3\|\omega\|^{2}+g^{2} \|\theta\| ^{2}+h^{2}\|S\|^{2}. \end{aligned}$$

(3.57)

From (3.24), (3.26), and (3.57), we get

$$\begin{aligned} \frac{d}{dt}\bigl(\|\omega\|^{2}+\|\theta \|^{2}+\|S\|^{2}\bigr) \leq&\lambda^{2} \int _{\Omega}|u|^{4}\,dx+ \biggl(3+\frac{(T^{M})^{2}}{2}+ \frac{(C^{M})^{2}}{2} \biggr)\| \omega\|^{2} \\ &{}+\bigl(L^{2}k_{1}^{2}+h^{2}\bigr)\|S\|^{2}+\bigl(1+g^{2}\bigr)\|\theta\|^{2}. \end{aligned}$$

(3.58)

We set

$$ F_{3}=\|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}. $$

Then we have

$$ \frac{dF_{3}}{dt}\leq\lambda^{2}k^{2}_{2}k^{\frac{1}{2}}_{5}k_{6}^{\frac{3}{2}}+M_{3}F_{3}, $$

(3.59)

where $M_{3}=\max{\{3+\frac{(T^{M})^{2}}{2}+\frac{(C^{M})^{2}}{2}, L^{2}k_{1}^{2}+h^{2}, 1+g^{2}\}}$.

We easily see that

$$ \|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}\leq\lambda ^{2}te^{-M_{3}t}k^{2}_{2}k^{\frac{1}{2}}_{5}k_{6}^{\frac{3}{2}}. $$

(3.60)

Inequality (3.60) demonstrates the convergence of $u_{i}$ to $u^{*}_{i}$, T to $T^{*}$, and C to $C^{*}$ as $\lambda_{1}\rightarrow\lambda_{2}$ in the indicated measure. □

4 The case for the Forchheimer equations

The above equations we discussed are of the Brinkman-Forchheimer equations type. If we consider the Forchheimer equations if Δu is deleted, we will demonstrate another theorem. Now, let $(u_{i},p,T,C)$ be a solution to the boundary initial-value problem for the Forchheimer model,

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u_{i}}{\partial t}+\lambda|u|u_{i}=-p_{,i}+g_{i}T-h_{i}C,\quad (x,t)\in\Omega\times[0,\tau], \\ \frac{\partial u_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T}{\partial t}+u_{i}\frac{\partial T}{\partial x_{i}}=\Delta T,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C}{\partial t}+u_{i}\frac{\partial C}{\partial x_{i}}=\Delta C+Lf(T)-kC,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . \end{aligned}$$

(4.1)

$$\begin{aligned}& u_{i}=0 ,\qquad \frac{\partial T}{\partial n}=0,\qquad \frac{\partial C}{\partial n}=0,\quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(4.2)

$$\begin{aligned}& u_{i}(x,0)=u{_{i0}}(x), \end{aligned}$$

(4.3)

$$\begin{aligned}& T(x,0)=T_{0}(x),\qquad C(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(4.4)

Furthermore, let $(u^{*}_{i},p^{*},T^{*},C^{*})$ be a solution to the following boundary initial-value problem:

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} \frac{\partial u^{*}_{i}}{\partial t}=-p^{*}_{,i}+g_{i}T^{*}-h_{i}C^{*},\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial u^{*}_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial T^{*}}{\partial t}+u^{*}_{i}\frac{\partial T^{*}}{\partial x_{i}}=\Delta T^{*},\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial C^{*}}{\partial t}+u_{i}^{*}\frac{\partial C^{*}}{\partial x_{i}}=\Delta C^{*}+Lf(T^{*})-kC^{*},\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . \end{aligned}$$

(4.5)

$$\begin{aligned}& u^{*}_{i}=0 , \qquad \frac{\partial T^{*}}{\partial n}=0,\qquad \frac{\partial C^{*}}{\partial n}=0, \quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(4.6)

$$\begin{aligned}& u^{*}_{i}(x,0)=u{_{i0}}(x), \end{aligned}$$

(4.7)

$$\begin{aligned}& T^{*}(x,0)=T_{0}(x),\qquad C^{*}(x,0)=C_{0}(x),\quad x \in \Omega. \end{aligned}$$

(4.8)

In this section, we establish convergence on the coefficient λ. To do this, let $(u_{i},T,C,P)$ and $(u^{*}_{i},T^{*},C^{*},P^{*})$ be solutions of (4.1) and (4.5) with the same boundary and initial conditions. Now we define

$$ \omega_{i}=u_{i}-u_{i}^{*},\qquad \pi=p-p^{*}, \qquad \theta=T-T^{*}, \qquad S=C-C^{*}. $$

(4.9)

Then $(\omega_{i},\theta,S,\pi)$ is a solution of the problem

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial\omega_{i}}{\partial t}+\lambda|u|u_{i}=-\pi _{,i}+g_{i}\theta-h_{i}S,\quad (x,t)\in\Omega\times[0,\tau], \\ \frac{\partial\omega_{i}}{\partial x_{i}}=0,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial\theta}{\partial t}+\omega_{i}\frac{\partial T}{\partial x_{i}}+u^{*}_{i}\frac{\partial\theta}{\partial x_{i}}=\Delta \theta,\quad (x,t)\in \Omega\times[0,\tau], \\ \frac{\partial S}{\partial t}+\omega_{i}\frac{\partial C}{\partial x_{i}}+u^{*}_{i}S_{,i}=\Delta S+L(f(T)-f(T^{*}))-kS,\quad (x,t)\in \Omega\times[0,\tau], \end{array}\displaystyle \right . $$

(4.10)

in $\Omega\times[0,\tau]$, subject to the boundary and initial conditions

$$\begin{aligned}& \omega_{i}=0 ,\qquad \frac{\partial \theta}{\partial n}=0, \qquad \frac{\partial S}{\partial n}=0,\quad (x,t)\in \partial\Omega\times[0,\tau], \end{aligned}$$

(4.11)

$$\begin{aligned}& \omega_{i}(x,0)=0, \end{aligned}$$

(4.12)

$$\begin{aligned}& \theta(x,0)=0, \qquad S(x,0)=0,\quad x \in\Omega. \end{aligned}$$

(4.13)

We will obtain the following result.

Theorem 3

Let $(u_{i},T,C,p)$ be the classical solution to the initial-boundary problem (4.1)-(4.4), $(u^{*}_{i},T^{*},C^{*},p^{*})$ be the classical solution to the initial-boundary problem (4.5)-(4.8) in $\Omega\times(0,\tau)$, and $(w_{i},\theta,S,\pi)$ be the difference of $(u_{i},T,C,p)$ and $(u^{*}_{i},T^{*},C^{*},p^{*})$, then the solution $(u_{i},T,C,p)$ converges to the solution $(u^{*}_{i},T^{*},C^{*},p^{*})$ as the Forchheimer coefficient λ tends to 0. The difference $(w_{i},\theta,S,\pi)$ satisfies

$$ \|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}\leq\lambda ^{2}te^{-M_{3}t}k_{2}^{2}k_{5}^{\frac{1}{2}}k_{7}^{\frac{3}{2}}. $$

(4.14)

We can also get the continuous dependence result for different Forchheimer coefficients $\lambda\rightarrow0$.

Proof

First of all, we may calculate $\int_{0}^{t}\int_{\Omega}C_{,i}C_{,i}\,dx\,d\eta$ and $\int_{0}^{t}\int_{\Omega}T_{,i}T_{,i}\,dx\,d\eta$,

$$\begin{aligned} \frac{\partial}{\partial t} \int_{\Omega}C^{2}\,dx&=2 \int_{\Omega}C\bigl(\Delta C+Lf(T)-kC-u_{i}C_{,i} \bigr)\,dx \\ &=-2 \int_{\Omega}C_{,i}C_{,i}\,dx+2L \int_{\Omega}Cf(T)\,dx-2k \int_{\Omega}C^{2}\,dx \\ &\leq-2 \int_{\Omega}C_{,i}C_{,i}\,dx+ \frac{L^{2}d^{2}}{2k}|\Omega|. \end{aligned}$$

(4.15)

Integrating (4.15) over Ω, we find

$$ 2 \int^{t}_{0} \int_{\Omega}C_{,i}C_{,i}\,dx\,d\eta\leq \int_{\Omega}C_{0}^{2}\,dx+ \frac{1}{2k}L^{2}d^{2}\tau|\Omega|. $$

(4.16)

Similarly, we can obtain

$$ 2 \int^{t}_{0} \int_{\Omega}T_{,i}T_{,i}\,dx\,d\eta\leq \int_{\Omega}T_{0}^{2}\,dx. $$

(4.17)

Then we want to give a bound for $\|\nabla u\|^{2}$. We know that

$$ \int_{\Omega}u_{i,j}u_{i,j}\,dx= \int_{\Omega}u_{i,j}(u_{i,j}-u_{j,i}) \,dx+ \int_{\Omega}u_{i,j}u_{j,i}\,dx. $$

(4.18)

After some integration by parts, it implies

$$\begin{aligned} \int_{\Omega}u_{i,j}u_{j,i}\,dx&= \oint_{\partial\Omega}u_{i,j}u_{j}n_{i}\, ds- \int_{\Omega}u_{i,ij}u_{j}\,dx \\ &=- \oint_{\partial\Omega}u_{i,i}u_{j}n_{j}\,ds+ \int_{\Omega}u_{i,i}u_{j,j}\,dx=0. \end{aligned}$$

(4.19)

We set

$$ J(t)= \int_{\Omega}u_{i,j}(u_{i,j}-u_{j,i}) \,dx. $$

(4.20)

We note that

$$ k_{g}^{2}=\max_{\Omega} {g_{i,j}g_{i,j}},\qquad k_{h}^{2}=\max _{\Omega}{h_{i,j}h_{i,j}}. $$

(4.21)

We can obtain

$$\begin{aligned} \frac{dJ}{dt} =&2 \int_{\Omega}u_{i,jt}u_{i,j}\,dx- \int_{\Omega}u_{i,jt}u_{j,i}\,dx- \int_{\Omega}u_{j,it}u_{i,j}\,dx \\ =&2 \int_{\Omega}u_{i,jt}(u_{i,j}-u_{j,i}) \,dx \\ =&2 \int_{\Omega}(u_{i,j}-u_{j,i})\bigl[-\lambda \bigl(|u|u_{i}\bigr)_{,j}-p_{i,j}+(g_{i}T)_{,j}-(h_{i}C)_{,j} \bigr]\,dx \\ =&-2\lambda \int_{\Omega}|u|u_{i,j}u_{i,j}\,dx-2\lambda \int_{\Omega}u_{i,j}u_{i}\frac{u_{k}u_{k,j}}{|u|} \,dx+2\lambda \oint_{\partial\Omega }u_{j,i}|u|u_{i}n_{j} \,ds \\ &{}+2 \int_{\Omega}(u_{i,j}-u_{j,i}) (g_{i,j}T-h_{i,j}C)\,dx+2 \int_{\Omega}(u_{i,j}-u_{j,i}) (g_{i}T_{,j}-h_{i}C_{,j})\,dx \\ \leq&2 \int_{\Omega}(u_{i,j}-u_{j,i}) (u_{i,j}-u_{j,i})\,dx+2k^{2}_{m} \int _{\Omega}\bigl(T^{2}+C^{2}\bigr)\,dx \\ &{}+2k^{2}_{n}\int_{\Omega}(T_{,j}T_{,j}+C_{,j}C_{,j})\,dx, \end{aligned}$$

(4.22)

where $k^{2}_{m}=\max{\{k^{2}_{g},k^{2}_{h}\}}$, $k^{2}_{n}=\max{\{g^{2},h^{2}\}}$.

We know

$$ \int_{\Omega}(u_{i,j}-u_{j,i}) (u_{i,j}-u_{j,i})\,dx=2 \int_{\Omega}(u_{i,j}-u_{j,i})u_{i,j} \,dx=2J(t). $$

(4.23)

From (4.22) and (4.23), we can get

$$ \frac{dJ}{dt}\leq4J+2k^{2}_{m}|\Omega|\bigl[\bigl(T^{M}\bigr)^{2}+\bigl(C^{M}\bigr)^{2}\bigr] +2k^{2}_{n}\int_{\Omega}(T_{,j}T_{,j}+C_{,j}C_{,j})\,dx. $$

(4.24)

Combining (4.16), (4.17), and (4.24), we have

$$\begin{aligned} \|\nabla u\|^{2} \leq& 2k^{2}_{m}e^{4\tau}|\Omega|\tau\bigl[\bigl(T^{M}\bigr)^{2}+\bigl(C^{M}\bigr)^{2}\bigr] \\ &{}+e^{4\tau}k_{n}^{2}\biggl(\int_{\Omega}T_{0}^{2}+C_{0}^{2}\,dx+\frac{1}{2k}L^{2}d^{2}\tau|\Omega|\biggr) \\ =&k_{7}. \end{aligned}$$

(4.25)

Similarly we can obtain

$$ \|u\|_{4}^{4}\leq k_{2}^{2}k_{5}^{\frac{1}{2}}k_{7}^{\frac{3}{2}}. $$

(4.26)

We can use a similar method to get the result that

$$ \|\omega\|^{2}+\|\theta\|^{2}+\|S \|^{2}\leq\lambda ^{2}te^{-M_{3}t}k_{2}^{2}k_{5}^{\frac{1}{2}}k_{7}^{\frac{3}{2}}. $$

(4.27)

In inequalities (4.27) we demonstrate the convergence of $u_{i}$ to $u^{*}_{i}$, T to $T^{*}$, and C to $C^{*}$ as $\lambda\rightarrow0$ in the indicated measure. We can also get the continuous dependence result. □