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Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

Abstract

A priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in \(\Omega _{1}\), which was governed by the Boussinesq equations. For a porous medium in \(\Omega _{2}\), we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous dependence type for the Boussinesq coefficient λ. Following the method of a first-order differential inequality, we can further obtain the result that the solution depends continuously on the interface boundary coefficient α. These results showed that the structural stability is valid for the interfacing problem.

Introduction

Recently, people have become interested in obtaining stability results of solutions for physical problems of partial differential equations with changes in coefficients. Sometimes the equations themselves are changed. This stability was called the structural stability in order to distinguish it from the traditional stability on initial data and boundary data. These problems were widely studied in many papers by many authors. For the problems of continuum mechanics, it is important for the authors to establish the structural stability of the model. This importance is discussed by Hirsch and Smale [1] in the form of a differential equation. This stability estimation is basic. We want to know whether a slight change in the coefficients in the equations or boundary data, or even the equation itself, will lead to drastic changes in the solution. For a review of the nature of the structural stability, refer to the books written by Ames and Straughan [2] and Straughan [3].

There are many papers studying the structural stability on the coefficients in fluids equations in porous media. Representative is the work of Ames et al. [4, 5], Franchi and Straughan [6], Hoang and Ibragimov [7], Lin and Payne [810], Liu [11, 12], Liu et al. [1315], Scott [16], Scott and Straughan [17], Payne et al. [1821] and some related papers [2224]. The previous publications of structural stability usually study one fluid in a bounded domain. Usually, there exists more than one fluid in a domain. These fluids have some interactions. It is desirable to see what effect they can have on each other. So the study of two interfacing fluids may be interesting and meaningful. In [19], the authors studied the structural stability for a flow interfacing with a porous solid. They proved that the solution depends continuously on the coefficient of the interface boundary condition.

In this paper, we want to study the continuous dependence type results on the interface boundary coefficient and the Boussinesq coefficient for the solution of the Boussinesq–Darcy problem in \(R^{3}\). The Boussinesq equations interface with the Darcy equations through the mutual boundary. Thus, we suggest an appropriate part of the plane \(z=x_{1}=0\) is the mutual boundary for a porous fluid in a bounded region \(\Omega _{2}\) in and a nonlinear viscous fluid in \(\Omega _{1}\) in \(R^{3}\). We denote the interface by L. The remaining part of \(\partial \Omega _{1}\) is denoted by \(\Gamma _{1}\), and the remaining part of \(\partial \Omega _{2}\) is denoted by \(\Gamma _{2}\). We also denote \(\partial \Omega _{1}=\Gamma _{1}\cup L\) and \(\partial \Omega _{2}=\Gamma _{2}\cup L\).

Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) denote the velocity, temperature and pressure in \(\Omega _{1}\) and \(\Omega _{2}\), respectively. Then the Boussinesq flow equations are (see [2527])

$$\begin{aligned}& \frac{\partial u_{i}}{\partial t}-\mu \Delta u_{i}+ \lambda u_{j} u_{i,j}-g_{i}T+p_{,i} = 0,\quad \mbox{in } \Omega _{1}\times [0,\tau ], \\& \frac{\partial T}{\partial t}+u_{i}T_{,i} = k_{1} \Delta T,\quad \mbox{in } \Omega _{1}\times [0,\tau ], \\& u_{i,i} = 0, \quad \mbox{in } \Omega _{1}\times [0,\tau ], \end{aligned}$$
(1)

where \(g_{i}\) is the gravity force function; λ is the Boussinesq coefficient. The coefficients μ and \(k_{1}\) are kinematic viscosity and thermal conductivity, respectively. From [28, 29]), we can see that the Boussinesq equations are useful in studying fluid and geophysical fluid dynamics.

The Darcy equations can be written as (see Nield and Bejan [30])

$$\begin{aligned}& v_{i}-g_{i}\theta + \frac{\partial q}{\partial x_{i}} = 0,\quad \mbox{in } \Omega _{2}\times [0,\tau ], \\& \frac{\partial \theta }{\partial t}+v_{i} \frac{\partial \theta }{\partial x_{i}} = k_{2} \Delta \theta , \quad \mbox{in } \Omega _{2}\times [0,\tau ], \\& \frac{\partial v_{i}}{\partial x_{i}} = 0, \quad \mbox{in } \Omega _{2} \times [0,\tau ], \end{aligned}$$
(2)

where \(\Omega _{1}\) and \(\Omega _{2}\) are all bounded domains. They are all simply connected and star-shaped. The boundaries \(\partial \Omega _{1}\) and \(\partial \Omega _{2}\) are their boundaries, respectively. τ is a positive constant which satisfies \(0<\tau <\infty \). The following boundary conditions are satisfied:

$$\begin{aligned}& u_{i}=0;\qquad T=G(x,t),\quad \mbox{on } \Gamma _{1}\times [0,\tau ], \\& v_{i}n_{i}=0,\qquad \theta =\widetilde{G}(x,t),\quad \mbox{on } \Gamma _{2} \times [0,\tau ], \end{aligned}$$
(3)

for prescribed functions \(G(x,t)\) and \(\widetilde{G}(x,t)\) and \(n_{i}^{(1)}\), \(n_{i}^{(2)}\) denote the unit outward normals of \(\Omega _{1}\), \(\Omega _{2}\), respectively. Obviously, \(n_{3}^{(1)}=-n_{3}^{(2)}=-1\). The initial conditions are written as

$$\begin{aligned}& u_{i}(x,0)=f_{i}(x),\qquad T(x,0)=T_{0}(x),\quad \mbox{in } \Omega _{1}, \\& \theta (x,0)=\theta _{0}(x), \quad \mbox{in } \Omega _{2}, \end{aligned}$$
(4)

for prescribed functions \(f_{i}\), \(T_{0}\) and \(\theta _{0}\). The interface L conditions are

$$\begin{aligned}& u_{3}=v_{3}\leq 0,\qquad T=\theta , \qquad k_{1}T_{,3}=k_{2}\theta _{,3}, \\& q=p-2\mu u_{3,3},\qquad u_{\beta ,3}+u_{3,\beta }= \frac{\alpha }{\sqrt{k_{1}}}u_{\beta }, \end{aligned}$$
(5)

where α is a positive coefficient and the value of α can be defined by experiment. It is determined by the given fluid and porous solid. The boundary conditions (5) were given by Nield and Bejan in [30]. In [31], Jones deduced the last condition in (5).

In this paper, we want to obtain the continuous dependence on the Boussinesq coefficient λ and the interface boundary coefficient α for the Boussinesq–Darcy interfacing problems in a bounded domain. However, there are only a few papers studying this interfacing problem in a bounded domain (see Payne and Straughan [19] and Liu et al. [13]). For the unbounded domain, refer to Liu et al. [32]. However, compared with the above literature, in this paper, there is a nonlinear term \(u_{i}u_{i,j}\). In particular, the bound of \(\int _{\Omega }u_{i,j}u_{i,j}\,dx\) is needed in this paper. But the methods proposed in [13, 19, 32] cannot be used directly. Second, some well-known Sobolev inequalities cannot be held for the interfacing problem. Our biggest innovation is to overcome these difficulties. We are sure that we can obtain some new and interesting results. We will derive some useful a priori bounds by using different inequalities. With the aid of these a priori bounds, we derive the continuous dependence on the Boussinesq coefficient and the interface boundary coefficient.

In the following discussions, we use the comma to denote partial differentiation. We also use \(u_{i,k}\) to denote the partial differentiation with respect to the direction \(x_{k}\). This is to say \(u_{i,k}=\frac{\partial u_{i}}{\partial x_{k}}\). We also use the usual summation convection with repeated Latin subscripts summed from 1 to 3, and the Greek subscripts summed from 1 to 2. Therefore, \(u_{i,i}=\sum_{i=1}^{3} (\frac{\partial u_{i}}{\partial x_{i}} )^{2}\), \(u_{\beta ,\beta }=\sum_{\beta =1}^{2} ( \frac{\partial u_{\beta }}{\partial x_{\beta }} )^{2}\).

A priori bounds

In this section, we want to drive bounds for various norms of \(u_{i}\) in terms of known data which will be used in the next sections.

Lemma 2.1

If \(T_{0}, \theta _{0}, G, \widetilde{G}\in L^{\infty }\). Then the temperatures satisfy

$$ \sup_{[0,\tau ]} \Vert T \Vert _{\infty }, \sup_{[0,\tau ]} \Vert \theta \Vert _{\infty }\leq N_{M}, $$
(6)

where \(N_{M}=\max \{\|T_{0}\|_{\infty }, \sup_{[0,\tau ]}\|G\|_{\infty }, \| \theta _{0}\|_{\infty }, \sup_{[0,\tau ]}\|\widetilde{G}\|_{\infty }\}\).

Proof

First, we let \(T_{LM}\) denotes the maximum of the temperature on the interface L. Payne, Rodrigues and Straughan [33] have derived

$$ \sup_{[0,\tau ]} \Vert T \Vert _{\infty }\leq \max \Bigl\{ \Vert T_{0} \Vert _{\infty }, \sup_{[0, \tau ]}G_{\infty }, T_{LM}\Bigr\} $$

and

$$ \sup_{[0,\tau ]} \Vert \theta \Vert _{\infty }\leq \max \Bigl\{ \Vert \theta _{0} \Vert _{\infty }, \sup _{[0,\tau ]}\widetilde{G}_{\infty }, T_{LM}\Bigr\} . $$

However, in the area \(\Omega _{1}\cup \Omega _{2}\times [0, \tau ]\), the maximum of the temperature cannot be reached on the interface L. Therefore, we have the result (6). □

Lemma 2.2

If \(T_{0}, \theta _{0}, G, \widetilde{G}\in L^{\infty }\) and \(\Omega _{1}\), \(\Omega _{2}\) are bounded regions. Then

$$ \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx\leq e^{\tau } \int _{\Omega _{1}} \vert \boldsymbol{{f}} \vert ^{2}\,dx+g^{2}N_{M}^{2}\bigl( \vert \Omega _{1} \vert + \vert \Omega _{2} \vert \bigr) \bigl(e^{\tau }-1\bigr)\doteq A_{1}. $$
(7)

Proof

Multiplying (1)1 by \(u_{i}\), integrating over \(\Omega _{1}\) and using (6), we obtain

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \\& \quad = \mu \int _{\Omega _{1}}(u_{i,j}+u_{j,i})_{,j}u_{i}\,dx- \lambda \int _{ \Omega _{1}}u_{j} u_{i,j}u_{i}\,dx+ \int _{\Omega _{1}}g_{i}Tu_{i}\,dx- \int _{\Omega _{1}} p_{,i}u_{i}\,dx \\& \quad = -\mu \int _{\Omega _{1}}(u_{i,j}+u_{j,i})u_{i,j}\,dx+ \mu \int _{L}(u_{ \beta ,3}+u_{3,\beta })u_{\beta }n_{3}^{(1)}\,dA- \frac{1}{2}\lambda \oint _{ \partial \Omega _{1}} u_{3}u_{i}u_{i}n_{3}^{(1)}\,dA \\& \qquad {} + \frac{1}{2} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+\frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{1} \vert - \int _{L}(p-2\mu u_{3,3})u_{i}n_{i}^{(1)}\,dA \\& \quad \leq \frac{1}{2} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+ \frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{1} \vert + \int _{L}qv_{i}n_{i}^{(2)}\,dA. \end{aligned}$$

By the divergence theorem and (2) and the conditions in the interface, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \leq &-\mu \int _{\Omega _{1}}(u_{i,j}+u_{j,i})u_{i,j}\,dx \\ &{}+\frac{1}{2} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+\frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{1} \vert + \int _{\Omega _{2}}v_{i}(g_{i}\theta -v_{i})\,dx \\ \leq &\frac{1}{2} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+ \frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{1} \vert +\frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{2} \vert , \end{aligned}$$

or

$$ \frac{d}{dt} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \leq \int _{ \Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+g^{2}N_{M}^{2} \vert \Omega _{1} \vert +g^{2}N_{M}^{2} \vert \Omega _{2} \vert . $$
(8)

From (8) it follows that

$$ \frac{d}{dt} \biggl(e^{-t} \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \biggr) \leq \bigl(g^{2}N_{M}^{2} \vert \Omega _{1} \vert +g^{2}N_{M}^{2} \vert \Omega _{2} \vert \bigr)e^{-t}. $$

Upon integration, we can arrive at Lemma 2.2. □

Now we define

$$\begin{aligned}& F_{1}(t)= \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx, \qquad F_{2}(t)= \int _{0}^{t} \int _{\Omega _{2}} \vert \boldsymbol{{v}}_{t} \vert ^{2}\,dx\,d\eta , \\& F_{3}(t)= \int _{\Omega _{1}}u_{i,j}(u_{i,j}+u_{j,i})\,dx. \end{aligned}$$
(9)

Lemma 2.3

If \(T_{0}, \theta _{0}, G, \widetilde{G}\in L^{\infty }\) and \(\Omega _{1}\), \(\Omega _{2}\) are bounded regions, then

$$ F_{3}(t)\leq \frac{1}{2\mu }F_{1}(t)+a_{1}, $$
(10)

where \(a_{1}=\frac{1}{\mu }A_{1}+\frac{1}{2\mu }g^{2}N_{M}^{2}|\Omega _{1}|+ \frac{1}{4\mu }g^{2}N_{M}^{2}|\Omega _{2}|\).

Proof

Using the divergence theorem, we have

$$\begin{aligned}& \mu \int _{\Omega _{1}}u_{i,j}(u_{i,j} + u_{j,i})\,dx \\& \quad = \mu \int _{L}u_{ \beta }(u_{\beta ,3}+u_{3,\beta })n_{3}^{(1)}\,dA+2 \mu \int _{\Omega _{1}}u_{3}u_{3,3}n_{3}^{(1)}\,dA \\& \qquad {} - \int _{\Omega _{1}}u_{i}[u_{i,t}+\lambda u_{j}u_{i,j}-g_{i}T+p_{,i}]\,dx \\& \quad \leq \mu \int _{L}u_{\beta }(u_{\beta ,3}+u_{3,\beta })n_{3}^{(1)}\,dA- \int _{L}(p-2\mu u_{3,3})u_{3}n_{3}^{(1)}\,dA \\& \qquad {} + \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+\frac{1}{2} \int _{ \Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx -\frac{1}{2}\lambda \int _{ \partial \Omega _{1}}u_{3}u_{i}u_{i}n_{3}^{(1)}\,dA+ \frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{1} \vert \\& \quad \leq \frac{\mu \alpha }{\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }n_{3}^{(1)}\,dA- \int _{L}qv_{3}n_{3}^{(2)}\,dA \\& \qquad {} + \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx+\frac{1}{2} \int _{ \Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx +\frac{1}{2}g^{2}N_{M}^{2} \vert \Omega _{1} \vert . \end{aligned}$$
(11)

In the light of the condition on L, we compute

$$ \int _{L}qv_{3}n_{3}^{(2)}\,dA= \int _{\Omega _{2}}q_{,i}v_{i}\,dA= \int _{ \Omega _{2}}(g_{,i}\theta -v_{i})v_{i}\,dA \leq \frac{1}{4}g^{2}N_{M}^{2} \vert \Omega _{2} \vert . $$
(12)

Combining (11) and (12), we have Lemma 2.3. □

Lemma 2.4

If \(T_{0}, \theta _{0}, G, \widetilde{G}\in L^{\infty }\) and \(\Omega _{1}\), \(\Omega _{2}\) are bounded regions. Then

$$ \int _{\Omega _{1}} \vert \nabla \boldsymbol{u} \vert ^{2}\,dx\leq A_{4}(t), $$
(13)

where \(A_{4}(t)\) is a positive function which will be defined later.

Proof

We firstly compute

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}F_{1}(t) =& \mu \int _{\Omega _{1}}u_{i,t}(u_{i,j}+u_{j,i})_{,jt}\,dx - \int _{\Omega _{1}}u_{i,t}p_{i,t}\,dx-\lambda \int _{\Omega _{1}}u_{i,t}u_{i,j}u_{j,t}\,dx \\ &{}-\lambda \int _{\Omega _{1}}u_{i,t}u_{i,jt}u_{j}\,dx+ \int _{\Omega _{1}}u_{i,t}g_{i}T_{,t}\,dx \\ =&-\mu \int _{\Omega _{1}}u_{i,jt}(u_{i,jt}+u_{j,it})\,dx - \int _{L}u_{3,t}(p_{,t}-2 \mu u_{3,3t})n_{3}^{(1)}\,dA \\ &{}+\frac{\alpha \mu }{\sqrt{k_{1}}} \int _{L}u_{ \beta ,t}u_{\beta ,t}n_{3}^{(1)}\,dA \\ &{}-\lambda \int _{\Omega _{1}}u_{i,t}u_{i,j}u_{j,t}\,dx- \lambda \int _{ \Omega _{1}}u_{i,t}u_{i,jt}u_{j}\,dx+ \int _{\Omega _{1}}u_{i,t}g_{i}T_{,t}\,dx \\ =&-\mu \int _{\Omega _{1}}u_{i,jt}(u_{i,jt}+u_{j,it})\,dx + \int _{L}u_{3,t}q_{,t}n_{3}^{(2)}\,dA- \lambda \int _{\Omega _{1}}u_{i,t}u_{i,j}u_{j,t}\,dx \\ &{}-\lambda \int _{\Omega _{1}}u_{i,t}u_{i,jt}u_{j}\,dx+ \int _{\Omega _{1}}u_{i,t}g_{i}T_{,t}\,dx \\ =&-\mu \int _{\Omega _{1}}u_{i,jt}(u_{i,jt}+u_{j,it})\,dx + \int _{ \Omega _{2}}v_{i,t}(g_{i}\theta _{,t}-v_{i,t})\,dx-\lambda \int _{ \Omega _{1}}u_{i,t}u_{i,j}u_{j,t}\,dx \\ &{}-\frac{1}{2}\lambda \int _{\Omega _{1}}u_{i,t}u_{i,t}u_{3}n_{3}^{(1)}\,dx+ \int _{\Omega _{1}}u_{i,t}g_{i}T_{,t}\,dx \\ \leq &-\mu \int _{\Omega _{1}}u_{i,jt}(u_{i,jt}+u_{j,it})\,dx - \frac{1}{2} \int _{\Omega _{2}} \vert \boldsymbol{{v}}_{t} \vert ^{2}\,dx-\lambda \int _{\Omega _{1}}u_{i,t}u_{i,j}u_{j,t}\,dx \\ &{}+\frac{1}{2} \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx + \frac{1}{2}g^{2} \biggl( \int _{\Omega _{2}}\theta _{,t}^{2}\,dx+ \int _{ \Omega _{1}}T_{,t}^{2}\,dx \biggr). \end{aligned}$$
(14)

We find that the result given in Appendix B of Lin and Payne [10] for \(\|\boldsymbol{{u}}\|_{4}^{2}\)

$$ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{4}\,dx \biggr)^{\frac{1}{2}}\leq k \biggl[ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \biggr) + \biggl( \int _{ \Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{ \Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{3}{4}} \biggr] ,\quad k>0. $$
(15)

So, we have for an arbitrary constant \(\varepsilon _{1}>0\)

$$\begin{aligned}& -\int _{\Omega _{1}}u_{i,t}u_{i,j}u_{j,t}\,dx \\& \quad \leq \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{4}\,dx \biggr)^{\frac{1}{2}} \\& \quad \leq k \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl[ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr) + \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr)^{\frac{3}{4}} \biggr] \\& \quad \leq k \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr) \\& \qquad {}+k \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr)^{\frac{3}{4}} \\& \quad \leq k \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr) \\& \qquad {}+\frac{1}{4}k^{4}\varepsilon _{1}^{-3} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{2} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr)+\frac{3}{4}\varepsilon _{1} \biggl( \int _{ \Omega _{1}} \vert \nabla \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr). \end{aligned}$$
(16)

We notice that one has obtained the following results [13]:

$$ \gamma \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx\leq \int _{\Omega _{1}}u_{i,j}(u_{i,j}+u_{j,i})\,dx $$
(17)

and

$$ \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx\leq C \int _{\Omega _{1}}u_{i,j}(u_{i,j}+u_{j,i})\,dx, $$
(18)

where γ and C are positive constants which have been defined in [13]. Following the methods in [13], we can derive a similar result,

$$ \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}}_{t} \vert ^{2}\,dx\leq C \int _{ \Omega _{1}}u_{i,jt}(u_{i,jt}+u_{j,it})\,dx. $$
(19)

Combining (14), (16)–(19) and using (7), we obtain

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq & F_{1}(t)+2k\sqrt{C}F_{3}^{\frac{1}{2}}(t)F_{1}(t)+ \frac{1}{2}k^{4}C^{2}\varepsilon _{1}^{-3}F_{3}^{2}(t)F_{1}(t) \end{aligned}$$
(20)
$$\begin{aligned} &{}+g^{2} \biggl( \int _{\Omega _{2}}\theta _{,t}^{2}\,dx+ \int _{\Omega _{1}}T_{,t}^{2}\,dx \biggr), \end{aligned}$$
(21)

where we have chosen \(\varepsilon _{1}=\frac{4\mu }{3k_{5}}\).

We now use (1)2 and (2)2 with the boundary conditions (3) and (5) and the divergence theorem to obtain

$$\begin{aligned}& \frac{d}{dt} \biggl( \int _{\Omega _{1}}T_{t}^{2}\,dx+ \int _{\Omega _{2}} \theta _{t}^{2}\,dx \biggr) \\& \quad = 2 \int _{\Omega _{1}}T_{t}[-u_{i,t}T_{,i}-u_{i}T_{,it}+k_{1} \Delta T]\,dx+2 \int _{\Omega _{2}}\theta _{t}[-v_{i,t}\theta _{,i}-v_{i}\theta _{,it}+k_{2} \Delta \theta ]\,dx \\& \quad = -2k_{1} \int _{\Omega _{1}} \vert \nabla T_{t} \vert ^{2}\,dx-2k_{2} \int _{ \Omega _{2}} \vert \nabla \theta _{t} \vert ^{2}\,dx+2 \int _{\Omega _{1}}TT_{,it}u_{i,t}\,dx+2 \int _{\Omega _{2}}\theta \theta _{,it}v_{i,t}\,dx \\& \quad \leq \frac{N_{M}^{2}}{2} \biggl(\frac{1}{k_{1}}F_{1}(t)+ \frac{1}{k_{2}} \int _{\Omega _{2}} \vert \boldsymbol{{v}}_{t} \vert ^{2}\,dx \biggr). \end{aligned}$$
(22)

By integration of (24) we thus obtain

$$\begin{aligned}& \int _{\Omega _{1}}T_{t}^{2}\,dx+ \int _{\Omega _{2}}\theta _{t}^{2}\,dx \\& \quad \leq \frac{N_{M}^{2}}{2} \biggl(\frac{1}{k_{1}} \int _{0}^{t}F_{1}(\eta )\,d\eta + \frac{1}{k_{2}}F_{2}(t) \biggr) + \int _{\Omega _{1}}(T_{0,t})^{2}\,dx+ \int _{\Omega _{2}}(\theta _{0,t})^{2}\,dx. \end{aligned}$$
(23)

Inserting (25) into (20) and setting

$$\begin{aligned}& a_{1} = \frac{A_{1}k^{2}}{\varepsilon _{2}}+1+ \frac{1}{32\varepsilon _{2}^{4}}A_{1}^{4}k^{8} \varepsilon _{3}^{-3}, \qquad a_{2}=2k \sqrt{C},\qquad a_{3}=\frac{1}{2}k^{4}C^{2} \varepsilon _{1}^{-3}, \\& a_{4}= \frac{k^{2}}{\varepsilon _{2}}\sqrt[4]{A_{1}C^{3}},\qquad a_{5} = \frac{1}{32\varepsilon _{2}^{4}}k^{8}A_{1} \varepsilon _{4}^{-3}C^{3},\qquad a_{6}=\frac{N_{M}^{2}g^{2}}{2k_{1}},\qquad a_{7}= \frac{N_{M}^{2}g^{2}}{2k_{2}}, \\& a_{8} = \int _{\Omega _{1}}(T_{0,t})^{2}\,dx+ \int _{\Omega _{2}}( \theta _{0,t})^{2}\,dx, \end{aligned}$$
(24)

we have

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq & a_{1}F_{1}(t)+a_{2}F_{3}^{\frac{1}{2}}(t)F_{1}(t)+a_{3}F_{3}^{2}(t)F_{1}(t)+a_{4}F_{1}(t)F_{3}^{\frac{3}{4}}(t) \\ &{}+a_{5}F_{1}(t)F_{3}^{3}(t) +a_{6} \int _{0}^{t}F_{1}(\eta )\,d\eta +a_{7}F_{2}(t)+a_{8}. \end{aligned}$$
(25)

Then, using Lemma 2.3 and the Hölder inequality in (27), we get

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq & b_{1}F^{4}_{1}(t)+b_{2}F_{1}^{3}(t)+b_{3}F_{1}^{2}(t)+b_{4}F_{1}(t) \\ &{}+b_{5} +a_{6} \int _{0}^{t}F_{1}(\eta )\,d\eta +a_{7}F_{2}(t), \end{aligned}$$
(26)

for some computable positive constants \(b_{i}\) (\(i=1,2,3,4,5\)). Now, we define

$$ F(t)=F_{1}(t)+F_{2}(t)+M_{0} \int _{0}^{t}F_{1}(\eta )\,d\eta ,\quad M_{0}>0. $$
(27)

We have from (32)

$$ \frac{d}{dt}F(t)\leq b_{1}F^{4}(t)+b_{2}F^{3}(t)+b_{3}F^{2}(t)+b_{6}F(t)+b_{5}, $$
(28)

where \(b_{6}=\frac{1}{b_{4}+M_{0}}\max \{1, \frac{a_{7}}{b_{4}+M_{0}}, \frac{a_{6}}{(b_{4}+M_{0})M_{0}}\}\). Obviously, we have from (34)

$$ \frac{d}{dt}F(t)\leq b_{1} \bigl(F(t)+b_{7}\bigr)^{4}, $$
(29)

where

$$ b_{7}=\max \biggl\{ \frac{b_{2}}{4b_{1}}, \sqrt{\frac{b_{3}}{6b_{1}}}, \sqrt[3]{ \frac{b_{6}}{4b_{1}}}, \sqrt[4]{\frac{b_{5}}{b_{1}}}\biggr\} . $$
(30)

Therefore, we can get the result

$$ \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx+ \int _{0}^{t} \int _{ \Omega _{2}} \vert \boldsymbol{{v}}_{t} \vert ^{2}\,dx\,d\eta +M_{0} \int _{0}^{t} \int _{ \Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx\,d\eta \leq A_{2}(t), $$
(31)

where

$$ A_{2}(t)=\sqrt[3]{\frac{1}{(F(0)+b_{7})^{-3}-3b_{1}t}},\qquad F(0)= \int _{\Omega _{1}} \vert \boldsymbol{{f}}_{t} \vert ^{2}\,dx. $$
(32)

In view of (9), Lemma 3 and (18), we also have

$$ \int _{\Omega _{1}}u_{i,j}(u_{i,j}+u_{j,i})\,dx \leq A_{3}(t) $$
(33)

and

$$ \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx\leq A_{4}(t), $$
(34)

where

$$ A_{3}(t)=\frac{1}{2\mu }A_{2}(t)+a_{9},\qquad A_{4}(t)=CA_{3}(t). $$
(35)

Combining (13), (15) and (34), we may get the following lemma. □

Lemma 2.5

If \(T_{0}, \theta _{0}, G, \widetilde{G}\in L^{\infty }\) and \(\Omega _{1}\), \(\Omega _{2}\) are bounded regions. Then

$$ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{4}\,dx \biggr)^{\frac{1}{2}}\leq k \bigl[A_{1} +A_{1}^{\frac{1}{4}}A_{4}^{\frac{3}{4}}(t) \bigr] \doteq A_{5}(t). $$
(36)

Continuous dependence on λ

In this section, we want to establish the continuous dependence on \(g_{i}\). Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be solutions of (1)–(5) with \(\lambda =\lambda ^{(1)}\), and \((u^{*}_{i}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) be solutions of (1)–(5) with \(\lambda =\lambda ^{(2)}\), respectively.

We define

$$ w_{i}=u_{i}-u_{i}^{*},\qquad S=T-T^{*}, \qquad \pi =p-p^{*},\qquad \widetilde{\lambda }=\lambda ^{(1)}-\lambda ^{(2)}, $$
(37)

and

$$ w^{m}_{i}=v_{i}-v_{i}^{*},\qquad S^{m}=\theta -\theta ^{*}, \qquad \pi ^{m}=q-q^{*} .$$
(38)

Then \((w_{i}, T, \pi )\) satisfy the following equations:

$$\begin{aligned}& \frac{\partial w_{i}}{\partial t} -\mu \Delta w_{i}+\widetilde{ \lambda }u_{i,j}u_{j}+ \lambda ^{(2)}u_{j}w_{i,j}+\lambda ^{(2)}u^{*}_{i,j}w_{j}-g_{i}S+ \pi _{,i}=0,\quad \mbox{in } \Omega _{1}\times [0,\tau ], \\& \frac{\partial S}{\partial t} +w_{i}T_{,i}+u^{*}_{i}S_{,i}=k_{1} \Delta S,\quad \mbox{in } \Omega _{1}\times [0,\tau ], \\& w_{i,i} = 0, \quad \mbox{in } \Omega _{1}\times [0,\tau ], \end{aligned}$$
(39)

and \((w^{m}_{i}, S^{m}, \pi ^{m})\) satisfy the equations

$$\begin{aligned}& w^{m}_{i}-g_{i}S^{m}+ \pi ^{m}_{,i} = 0,\quad \mbox{in } \Omega _{2} \times [0, \tau ], \\& \frac{\partial S^{m}}{\partial t}+w_{i}^{m}\theta _{,i}+v^{*}_{i}S^{m}_{,i} = k_{2} \Delta S^{m},\quad \mbox{in } \Omega _{2}\times [0,\tau ], \\& w^{m}_{i,i} = 0,\quad \mbox{in } \Omega _{2} \times [0,\tau ]. \end{aligned}$$
(40)

The boundary conditions are

$$\begin{aligned}& w_{i} = 0;\qquad S=0, \quad \mbox{on } \Gamma _{1}\times [0,\tau ], \\& w^{m}_{i}n_{i} = 0, \qquad S^{m}=0,\quad \mbox{on } \Gamma _{2}\times [0,\tau ]. \end{aligned}$$
(41)

The initial conditions can be written as

$$ w_{i}(x,0)=0, \qquad S(x,0)=0, \quad \mbox{in } \Omega _{1},\qquad S^{m}(x,0)=0, \quad \mbox{in } \Omega _{2}. $$
(42)

The interface L conditions are

$$\begin{aligned}& w_{3}=w^{m}_{3},\qquad S=S^{m},\qquad k_{1}S_{,3}=k_{2}S^{m}_{,3}, \\& \pi ^{m}=\pi -2\mu w_{3,3}, \qquad w_{\beta ,3}+w_{3,\beta }= \frac{\alpha }{\sqrt{k_{1}}}w_{\beta }. \end{aligned}$$
(43)

We first give some useful lemmas.

Lemma 3.1

Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be the classical solutions to the initial-boundary value problem (1)(5) corresponding to \(\lambda ^{(1)}\), and \((u_{i}^{*}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) also be the classical solutions to the initial-boundary value problem (1)(5) but corresponding to \(\lambda ^{(2)}\). Then for any \(t>0\) the differences of velocities satisfy

$$\begin{aligned}& \frac{d}{dt} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx + \int _{ \Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx \\& \quad \leq c_{1}(t) \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx + 2g^{2} \biggl[ \int _{\Omega _{1}}S^{2}\,dx+ \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \biggr]+2(\widetilde{\lambda })^{2}kA_{4}(t)A_{5}(t), \end{aligned}$$

where \(c_{1}(t)\) is a positive function which depends on t.

Proof

We begin with the identity

$$ \int _{\Omega _{1}}\biggl[\frac{\partial w_{i}}{\partial t}-\mu \Delta w_{i}+ \widetilde{\lambda }u_{i,j}u_{j}+ \lambda ^{(2)}u_{j}w_{i,j}+\lambda ^{(2)}u^{*}_{i,j}w_{j}-g_{i}S+ \pi _{,i}\biggr]w_{i}\,dx=0. $$
(44)

From (44) it follows that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx =& \mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})_{,j}w_{i}\,dx -\widetilde{ \lambda } \int _{\Omega _{1}}u_{i,j}u_{j}w_{i}\,dx- \lambda ^{(2)} \int _{ \Omega _{1}}u_{j}w_{i,j}w_{i}\,dx \\ &{}-\lambda ^{(2)} \int _{\Omega _{1}}u^{*}_{i,j}w_{j}w_{i}\,dx- \int _{ \Omega _{1}}\pi _{,i}w_{i}\,dx- \int _{\Omega _{1}}g_{i}Sw_{i}\,dx \\ \doteq & I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}. \end{aligned}$$
(45)

We now deal with \(I_{1}\) and \(I_{5}\). Using the divergence theorem, we have

$$\begin{aligned} I_{1}+I_{5} =&-\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx+ \mu \int _{\Omega _{1}}(w_{\beta ,3}+w_{3,\beta })w_{\beta }n_{3}^{(1)} \,dA \\ &{}- \int _{\Omega _{1}}(\pi -2\mu u_{3,3})w_{i}n_{i}^{(1)}\,dx \\ =&-\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx+ \frac{\alpha \mu }{\sqrt{k_{1}}} \int _{L}w_{\beta }w_{\beta }n_{3}^{(1)} \,dA + \int _{L}\pi ^{m}w^{m}_{i}n_{i}^{(2)}\,dA \\ \leq &-\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx+ \int _{ \Omega _{2}}\bigl(-w^{m}_{i}+ \widetilde{g}_{i}\theta +g_{i}^{(2)}S^{m} \bigr)w^{m}_{i}\,dx \\ \leq &-\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx- \frac{1}{2} \int _{\Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx+g^{2} \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx. \end{aligned}$$
(46)

Using the Hölder inequality, (15), Lemma 2.4, Lemma 2.5 and the Young inequality with \(\delta _{1}>0\), we have

$$\begin{aligned} I_{2} \leq & \vert \widetilde{\lambda } \vert \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{4}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{4}\,dx \biggr)^{\frac{1}{4}} \\ \leq & \vert \widetilde{\lambda } \vert \sqrt{kA_{4}(t)A_{5}(t)} \biggl[ \biggl( \int _{ \Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr) + \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{3}{4}} \biggr]^{\frac{1}{2}} \\ \leq &(\widetilde{\lambda })^{2}kA_{4}(t)A_{5}(t)+ \biggl[1+\frac{1}{4} \delta _{1}^{-3}\biggr] \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx + \frac{3}{4}\delta _{1} \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx. \end{aligned}$$
(47)

Following a similar procedure to deriving \(I_{3}\), we obtain

$$\begin{aligned} I_{3} \leq &\lambda ^{(2)} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{4}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{4}\,dx \biggr)^{\frac{1}{4}} \\ \leq &\lambda ^{(2)}\sqrt{kA_{5}(t)} \biggl\{ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr) \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr) \\ &{}+ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{7}{4}} \biggr\} ^{\frac{1}{2}} \\ \leq &\lambda ^{(2)}\sqrt{kA_{5}(t)} \biggl\{ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\ &{}+ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{8}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{7}{8}} \biggr\} \\ \leq & \biggl[\frac{(\lambda ^{(2)})^{2}}{2\delta _{2}}kA_{5}(t)+ \frac{(\lambda ^{(2)})^{8}}{8\delta _{3}^{7}}\bigl(kA_{5}(t)\bigr)^{4} \biggr] \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \\ &{}+\biggl(\frac{1}{2}\delta _{2}+ \frac{7}{8}\delta _{3}\biggr) \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx, \end{aligned}$$
(48)

where \(\delta _{2}\), \(\delta _{3}\) are positive constants to be determined later. Similarly, we have

$$\begin{aligned} I_{4} \leq &\lambda ^{(2)} \biggl( \int _{\Omega _{1}} \bigl\vert \nabla \boldsymbol{{u}}^{*} \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{4}\,dx \biggr)^{\frac{1}{2}} \\ \leq &\lambda ^{(2)}\sqrt{A_{4}(t)} \biggl[ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr) + \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{1}{4}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx \biggr)^{\frac{3}{4}} \biggr] \\ \leq & \biggl[\lambda ^{(2)}\sqrt{A_{4}(t)}+ \frac{1}{4\delta _{4}}\bigl( \lambda ^{(2)}\bigr)^{4}A^{2}_{4}(t) \biggr] \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \biggr) +\frac{3}{4}\delta _{4} \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx. \end{aligned}$$
(49)

We note that \(I_{6}\) can be bounded,

$$ I_{6}\leq g^{2} \int _{\Omega _{1}}S^{2}\,dx+\frac{1}{4} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx. $$
(50)

Similar to (19), we also have

$$ \int _{\Omega _{1}} \vert \nabla \boldsymbol{{w}} \vert ^{2}\,dx\leq k_{5} \int _{ \Omega _{1}}w_{i,j}(u_{i,j}+w_{j,i})\,dx. $$
(51)

We define the functions \(F_{1}(t)\) and \(F_{2}(t)\) by

$$ F_{1}(t)= \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx,\qquad F_{2}(t)= \int _{0}^{t} \int _{\Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx\,d\eta . $$
(52)

Inserting (45)–(51) into (44) and using (52), we have

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq &-2\biggl[\mu -\frac{3}{4}(\delta _{1}+ \delta _{4})k_{5} -\biggl(\frac{1}{2}\delta _{2}+\frac{7}{8}\delta _{3} \biggr)k_{5}\biggr] \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx \\ &{}+2\biggl[\frac{1}{4}\delta _{1}^{-3}+ \frac{(\lambda ^{(2)})^{2}}{2\delta _{2}}kA_{5}(t)+ \frac{(\lambda ^{(2)})^{8}}{8\delta _{3}^{7}} \bigl(kA_{5}(t)\bigr)^{4} + \lambda ^{(2)} \sqrt{A_{4}(t)} \\ &{}+\frac{1}{4\delta _{4}}\bigl(\lambda ^{(2)}\bigr)^{4}A^{2}_{4}(t) + \frac{5}{4}\biggr] \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx +2g^{2} \int _{ \Omega _{1}}S^{2}\,dx \\ &{}+2g^{2} \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx+2( \widetilde{\lambda })^{2}kA_{4}(t)A_{5}(t). \end{aligned}$$
(53)

We choose \(\delta _{1}\), \(\delta _{2}\), \(\delta _{3}\) and \(\delta _{4}\) small enough such that

$$ \frac{3}{4}(\delta _{1}+\delta _{4})k_{5} +\biggl(\frac{1}{2}\delta _{2}+ \frac{7}{8}\delta _{3}\biggr)k_{5}=\mu . $$

Letting

$$ c_{1}(t)=2[\frac{1}{4}\delta _{1}^{-3}+ \frac{(\lambda ^{(2)})^{2}}{2\delta _{2}}kA_{5}(t)+ \frac{(\lambda ^{(2)})^{8}}{8\delta _{3}^{7}} \bigl(kA_{5}(t)\bigr)^{4} + \lambda ^{(2)} \sqrt{A_{4}(t)}+\frac{1}{4\delta _{4}}\bigl(\lambda ^{(2)} \bigr)^{4}A^{2}_{4}(t) + \frac{5}{4}, $$

we can get Lemma 3.1. □

Lemma 3.2

Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be the classical solutions to the initial-boundary value problem (1)(5) corresponding to \(\lambda ^{(1)}\), and \((u_{i}^{*}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) also be the classical solutions to the initial-boundary value problem (1)(5) but corresponding to \(\lambda ^{(2)}\). Then for any \(t>0\) we have

$$ \int _{\Omega _{1}}S^{2}\,dx+ \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \leq \frac{1}{2k_{1}}N_{M}^{2} \int _{0}^{t}F_{1}(\eta )\,d\eta + \frac{1}{2k_{2}}N_{M}^{2}F_{2}(t). $$

Proof

We multiply (39)2 and (40)2 by S and \(S^{m}\), respectively, and integrate by parts to find

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \biggl[ \int _{\Omega _{1}}S^{2}\,dx+ \int _{ \Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \biggr] \\& \quad = -k_{1} \int _{\Omega _{1}} \vert \nabla S \vert ^{2}\,dx-k_{2} \int _{\Omega _{2}} \bigl\vert \nabla S^{m} \bigr\vert ^{2}\,dx \\& \qquad {}+ \int _{\Omega _{1}}w_{i}TS_{,i}\,dx+ \int _{\Omega _{2}}w_{i}^{m} \theta S^{m}_{,i}\,dx \\& \quad \leq \frac{1}{4k_{1}}N_{M}^{2} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx+ \frac{1}{4k_{2}}N_{M}^{2} \int _{\Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx. \end{aligned}$$
(54)

Integrating (54) from 0 to t one may deduce

$$\begin{aligned}& \int _{\Omega _{1}}S^{2}\,dx+ \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \\& \quad \leq \frac{1}{2k_{1}}N_{M}^{2} \int _{0}^{t} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx\,d\eta +\frac{1}{2k_{2}}N_{M}^{2} \int _{0}^{t} \int _{\Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx\,d\eta . \end{aligned}$$
(55)

Combining (52) and (55), we may obtain Lemma 3.2. □

Now, we use Lemmas 3.1 and 3.2 to obtain

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq & c_{1}(t)F_{1}(t) + \frac{g^{2}N_{M}^{2}}{k_{1}} \int _{0}^{t}F_{1}(\eta )\,d\eta \\ &{}+\frac{g^{2}N_{M}^{2}}{k_{2}} F_{2}(t)+2(\widetilde{\lambda })^{2}kA_{4}(t)A_{5}(t). \end{aligned}$$
(56)

Setting

$$ F_{3}(t)=F_{1}(t)+F_{2}(t)+ \frac{k_{2}}{k_{1}} \int _{0}^{t}F_{1}( \eta )\,d\eta , $$
(57)

we obtain

$$ \frac{d}{dt}F_{3}(t)\leq c_{2}(t)F_{3}(t)+2(\widetilde{\lambda })^{2}kA_{4}(t)A_{5}(t), $$
(58)

where

$$ c_{2}(t)=\max \biggl\{ c_{1}(t)+\frac{k_{2}}{k_{1}}, \frac{g^{2}N_{M}^{2}}{k_{2}}\biggr\} . $$
(59)

Thus after integration we may derive from (58) the estimate

$$ F_{3}(t)\leq 2(\widetilde{\lambda })^{2}k \int _{0}^{t}A_{4}(\eta )A_{5}( \eta )e^{\int _{s}^{t}c_{2}(\eta )\,d\eta }\,ds. $$
(60)

Combining (57), Lemma 3.2 and (60), we have the following theorem.

Theorem 3.1

Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be the classical solutions to the initial-boundary value problem (1)(5) corresponding to \(\lambda ^{(1)}\), and \((u_{i}^{*}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) also be the classical solutions to the initial-boundary value problem (1)(5) but corresponding to \(\lambda ^{(2)}\). Then for any \(t>0\) we have

$$ (u_{i}, T, p)\rightarrow \bigl(u_{i}^{*}, T^{*}, p^{*} \bigr),\qquad (v_{i}, \theta , q)\rightarrow \bigl(v^{*}_{i}, \theta ^{*}, q^{*}\bigr), $$
(61)

as \(\lambda ^{(1)}\rightarrow \lambda ^{(2)}\). The differences of velocities satisfy

$$\begin{aligned}& \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx+ \int _{0}^{t} \int _{ \Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx\,d\eta +\frac{k_{2}}{k_{1}} \int _{0}^{t} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx\,d\eta \\& \quad \leq 2( \widetilde{\lambda })^{2}k \int _{0}^{t}A_{4}(\eta )A_{5}(\eta )e^{ \int _{s}^{t}c_{2}(\eta )\,d\eta }\,ds, \end{aligned}$$
(62)

where w, \(\boldsymbol{{w}}^{m}\), S, \(S^{m}\), have been defined in (37) and (38).

Furthermore, there are two positive \(c_{2}\), \(c_{3}(t)\), such that

$$\begin{aligned} \int _{\Omega _{1}}S^{2}\,dx+ \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \leq ( \widetilde{\lambda })^{2}k\frac{N_{M}^{2}}{k_{2}} \int _{0}^{t}A_{4}( \eta )A_{5}(\eta )e^{\int _{s}^{t}c_{2}(\eta )\,d\eta }\,ds. \end{aligned}$$
(63)

Inequalities (62) and (63) demonstrate the continuous dependence on λ in the indicated measure.

Continuous dependence on the interface coefficient

In this section, we want to establish the continuous dependence on the interface coefficient α. Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be solutions of (1)–(5) with \(\alpha =\alpha _{1}\), and \((u^{*}_{i}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) be solutions of (1)–(5) with \(\alpha =\alpha _{2}\), respectively.

We define

$$ w_{i}=u_{i}-u_{i}^{*},\qquad S=T-T^{*}, \qquad \pi =p-p^{*},\qquad \sigma =\alpha _{1}-\alpha _{2}, $$
(64)

and

$$ w^{m}_{i}=v_{i}-v_{i}^{*},\qquad S^{m}=\theta -\theta ^{*},\qquad \pi ^{m}=q-q^{*}. $$
(65)

Then \((w_{i}, S, \pi )\) satisfy the following equation:

$$\begin{aligned}& \frac{\partial w_{i}}{\partial t}-\mu \Delta w_{i}+w_{j}u_{i,j}+u^{*}_{j}w_{i,j}-g_{i}S+ \pi _{,i} = 0,\quad \mbox{in } \Omega _{1}\times [0,\tau ], \\& \frac{\partial S}{\partial t}+w_{i}T_{,i}+u^{*}_{i}S_{,i} = k_{1} \Delta S,\quad \mbox{in } \Omega _{1}\times [0,\tau ], \\& w_{i,i} = 0, \quad \mbox{in } \Omega _{1}\times [0,\tau ], \end{aligned}$$
(66)

and \((w^{m}_{i}, S^{m}, \pi ^{m})\) satisfy equations

$$\begin{aligned}& w^{m}_{i}-g_{i}S^{m}+ \pi ^{m}_{,i} = 0, \quad \mbox{in } \Omega _{2} \times [0, \tau ], \\& \frac{\partial S^{m}}{\partial t}+w^{m}_{i}\theta _{,i}+v^{*}_{i}S^{m}_{,i} = k_{2} \Delta S^{m},\quad \mbox{in } \Omega _{2}\times [0,\tau ], \\& w^{m}_{i,i} = 0,\quad \mbox{in } \Omega _{2} \times [0,\tau ]. \end{aligned}$$
(67)

The boundary conditions are

$$\begin{aligned}& w_{i}=0; \qquad S=0,\quad \mbox{on } \Gamma _{1}\times [0,\tau ], \\& w^{m}_{i}n_{i}=0, \qquad S^{m}=0,\quad \mbox{on } \Gamma _{2}\times [0,\tau ]. \end{aligned}$$
(68)

The initial conditions can be written as

$$ w_{i}(x,0)=0,\qquad S(x,0)=0, \quad \mbox{in } \Omega _{1}, \qquad S^{m}(x,0)=0, \quad \mbox{in } \Omega _{2}. $$
(69)

The interface L conditions are

$$\begin{aligned}& w_{3}=w^{m}_{3},\qquad S=S^{m},\qquad k_{1}S_{,3}=k_{2}S^{m}_{,3}, \\& \pi ^{m}=\pi -2\mu w_{3,3}, \qquad w_{\beta ,3}+w_{3,\beta }= \frac{\sigma }{\sqrt{k_{1}}}u_{\beta }+\frac{\alpha _{2}}{\sqrt{k_{1}}}w_{\beta }. \end{aligned}$$
(70)

We give some useful lemmas.

Lemma 4.1

If \(T_{0}, \theta _{0}, G, \widetilde{G}\in L^{\infty }\) and \(\Omega _{1}\), \(\Omega _{2}\) are bounded regions, then

$$ \int _{L}u_{\beta }u_{\beta }\,dA\leq A_{6}(t), $$

where \(A_{6}(t)\) is a positive function which depends on t.

Proof

We use Eqs. (1), (2) to derive

$$\begin{aligned} \int _{\Omega _{1}}u_{i}u_{i,t}\,dx =&\mu \int _{\Omega _{1}}(u_{i,j}+u_{j,i})_{,j}u_{i}\,dx- \int _{\Omega _{1}}u_{j}u_{i,j}u_{i}\,dx+ \int _{\Omega _{1}}g_{i}Tu_{i}\,dx - \int _{\Omega _{1}} p_{,i}u_{i}\,dx \\ =&-\mu \int _{\Omega _{1}}(u_{i,j}+u_{j,i})u_{i,j}\,dx+ \mu \int _{L}(u_{ \beta ,3}+u_{3,\beta })u_{\beta }n_{3}^{(1)}\,dA \\ &{}- \int _{L}(p-2\mu u_{3,3})u_{i}n_{i}^{(1)}\,dA- \int _{\Omega _{1}}u_{j} u_{i,j}u_{i}\,dx+ \int _{\Omega _{1}}g_{i}Tu_{i}\,dx. \end{aligned}$$
(71)

Using the interface conditions, we obtain from (71)

$$\begin{aligned} \frac{\mu \alpha _{1}}{\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA =&- \mu \int _{\Omega _{1}}(u_{i,j}+u_{j,i})u_{i,j}\,dx - \int _{\Omega _{1}}u_{i}u_{i,t}\,dx \\ &{}+ \int _{L}qv_{i}n_{i}^{(2)}\,dA- \int _{\Omega _{1}}u_{j}u_{i,j}u_{i}\,dx+ \int _{\Omega _{1}}g_{i}Tu_{i}\,dx \\ =&- \int _{\Omega _{1}}u_{i}u_{i,t}\,dx+ \int _{\Omega _{2}}v_{i}(g_{i} \theta -v_{i})\,dx \\ &{}- \int _{\Omega _{1}}u_{j}u_{i,j}u_{i}\,dx+ \int _{\Omega _{1}}g_{i}Tu_{i}\,dx. \end{aligned}$$
(72)

By using the Hölder inequality, the AG mean inequality, (31), (34), Lemma 2.2 and Lemma 2.5, we have from (72)

$$\begin{aligned}& \frac{\mu \alpha _{1}}{\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA \\& \quad \leq \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}}_{t} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} + \frac{1}{4}g^{2}N_{M}^{2} \vert \Omega _{2} \vert \\& \qquad {}+ \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{4}\,dx\biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \nabla \boldsymbol{{u}} \vert ^{2}\,dx\biggr)^{\frac{1}{2}} + \biggl( \int _{\Omega _{1}}g_{i}g_{i}T^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega _{1}} \vert \boldsymbol{{u}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\& \quad \leq \sqrt{A_{1}A_{2}(t)} +\frac{1}{4}g^{2}N_{M}^{2} \vert \Omega _{2} \vert +A_{5}(t) \sqrt{A_{4}(t)}+\sqrt{g^{2}N_{M}^{2} \vert \Omega _{1} \vert A_{1}}. \end{aligned}$$
(73)

Therefore

$$ \int _{L}u_{\beta }u_{\beta }\,dA\leq A_{6}(t), $$
(74)

where

$$ A_{6}(t)=\frac{\sqrt{k_{1}}}{\mu \alpha _{1}} \biggl\{ \sqrt{A_{1}A_{2}(t)} +\frac{1}{4}g^{2}N_{M}^{2} \vert \Omega _{2} \vert +A_{5}(t) \sqrt{A_{4}(t)}+\sqrt{g^{2}N_{M}^{2} \vert \Omega _{1} \vert A_{1}} \biggr\} . $$
(75)

 □

Lemma 4.2

Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be the classical solutions to the initial-boundary value problem (1)(5) corresponding to \(\lambda ^{(1)}\), and \((u_{i}^{*}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) also be the classical solutions to the initial-boundary value problem (1)(5) but corresponding to \(\lambda ^{(2)}\). Then for any \(t>0\) we have

$$\begin{aligned}& \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx+ \int _{0}^{t} \int _{ \Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx\,d\eta +\frac{k_{2}}{k_{1}} \int _{0}^{t} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx\,d\eta \\& \quad \leq \frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{0}^{t} e^{\int _{s}^{t}c_{4}( \eta )\,d\eta } \int _{L}u_{\beta }u_{\beta }\,dA\,ds. \end{aligned}$$
(76)

Proof

We begin with the identity

$$ \int _{\Omega _{1}}\biggl[\frac{\partial w_{i}}{\partial t}-\mu \Delta w_{i}+ \lambda w_{j}u_{i,j}+\lambda u^{*}_{j}w_{i,j}-g_{i}S+\pi _{,i}\biggr]w_{i}\,dx=0. $$
(77)

From (77) it follows that

$$\begin{aligned} \frac{d}{dt} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx =&2\mu \int _{ \Omega _{1}}(w_{i,j}+w_{j,i})_{,j}w_{i}\,dx-2 \int _{\Omega _{1}}\pi _{,i}w_{i}\,dx -2\lambda \int _{\Omega _{1}}w_{j}u_{i,j}w_{i}\,dx \\ &{}-2\lambda \int _{\Omega _{1}}u^{*}_{j}w_{i,j}w_{i}\,dx+2 \int _{ \Omega _{1}}g_{i}Sw_{i}\,dx. \end{aligned}$$
(78)

Integrating by parts as in Sect. 3 now leads to

$$\begin{aligned}& 2\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})_{,j}w_{i}\,dx-2 \int _{ \Omega _{1}}\pi _{,i}w_{i}\,dx \\& \quad = -2\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx+2 \mu \int _{ \Omega _{1}}(w_{\beta ,3}+w_{3,\beta })w_{\beta }n_{3}^{(1)} \,dA \\& \qquad {}-2 \int _{\Omega _{1}}(\pi -2\mu u_{3,3})w_{i}n_{i}^{(1)}\,dx \\& \quad = -2\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx+ \frac{2\alpha _{1}\mu }{\sqrt{k_{1}}} \int _{L}w_{\beta }w_{\beta }n_{3}^{(1)}\,dA \\& \qquad {}+\frac{2\sigma \mu }{\sqrt{k_{1}}} \int _{L}u_{\beta }w_{\beta }n_{3}^{(1)}\,dA+2 \int _{L}\pi ^{m} w^{m}_{i}n_{i}^{(2)}\,dA \\& \quad \leq -2\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx+2 \int _{ \Omega _{2}}\bigl(-w^{m}_{i}+g_{i}S^{m} \bigr)w^{m}_{i}\,dx \\& \qquad {}+\frac{2\alpha _{1}\mu }{\sqrt{k_{1}}} \int _{L}w_{\beta }w_{\beta }n_{3}^{(1)}\,dA +\frac{2\sigma \mu }{\sqrt{k_{1}}} \int _{L}u_{\beta }w_{\beta }n_{3}^{(1)}\,dA \\& \quad \leq -2\mu \int _{\Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx- \int _{ \Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx+2g^{2} \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \\& \qquad {}+\frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA. \end{aligned}$$
(79)

Inserting (79), (48), (49), and (50) into (78) and using (51) and (52), we have

$$\begin{aligned}& \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \\& \quad \leq -2 \biggl[\mu -\frac{3}{4}\delta _{4}k_{5}- \biggl( \frac{1}{2}\delta _{2}+\frac{7}{8}\delta _{3}\biggr)k_{5} \biggr] \int _{ \Omega _{1}}(w_{i,j}+w_{j,i})w_{i,j}\,dx \\& \qquad {}+2\biggl[\frac{\lambda ^{2}}{2\delta _{2}}kA_{5}(t)+ \frac{\lambda ^{8}}{8\delta _{3}^{7}} \bigl(kA_{5}(t)\bigr)^{4} +\lambda \sqrt{A_{4}(t)}+ \frac{1}{4\delta _{4}}\lambda ^{4}A^{2}_{4}(t) +\frac{1}{4}\biggr] \int _{ \Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx \\& \qquad {}+2g^{2} \int _{\Omega _{1}}S^{2}\,dx+2g^{2} \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx+ \frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA. \end{aligned}$$
(80)

Choosing \(\delta _{2}\), \(\delta _{3}\), \(\delta _{4}\) such that

$$ \frac{3}{4}\delta _{4}k_{5}+\biggl( \frac{1}{2}\delta _{2}+\frac{7}{8} \delta _{3}\biggr)k_{5}=\mu , $$

and using Lemma 7 in (80), we have

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq & c_{3}(t) \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx+\frac{g^{2}}{k_{1}}N_{M}^{2} \int _{0}^{t} \int _{ \Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx\,d\eta \\ &{}+\frac{g^{2}}{k_{2}}N_{M}^{2} \int _{0}^{t} \int _{\Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx\,d\eta +\frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA, \end{aligned}$$
(81)

where

$$ c_{3}(t)=2\biggl[\frac{\lambda ^{2}}{2\delta _{2}}kA_{5}(t)+ \frac{\lambda ^{8}}{8\delta _{3}^{7}}\bigl(kA_{5}(t)\bigr)^{4} +\lambda \sqrt{A_{4}(t)}+ \frac{1}{4\delta _{4}}\lambda ^{4}A^{2}_{4}(t) +\frac{1}{4}\biggr]. $$
(82)

In view of (52), we have from (81)

$$\begin{aligned} \frac{d}{dt}\bigl[F_{1}(t)+F_{2}(t) \bigr] \leq & c_{3}(t)F_{1}(t)+ \frac{g^{2}}{k_{1}}N_{M}^{2} \int _{0}^{t}F_{1}(\eta )\,d\eta \\ &{}+\frac{g^{2}}{k_{2}}N_{M}^{2}F_{2}(t)+ \frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA. \end{aligned}$$
(83)

Defining \(F_{3}(t)\) as in (59), we have from (83)

$$ \frac{d}{dt}F_{3}(t)\leq c_{4}(t)F_{3}(t)+ \frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{L}u_{\beta }u_{\beta }\,dA, $$
(84)

where

$$ c_{4}(t)=\max \biggl\{ c_{3}(t)+\frac{k_{2}}{k_{1}}, \frac{g^{2}N_{M}^{2}}{k_{2}}\biggr\} . $$
(85)

Thus after integration we may derive Lemma 4.2. □

Combining Lemma 8 and Lemma 9, we have the following theorem.

Theorem 4.1

Let \((u_{i}, T, p)\) and \((v_{i}, \theta , q)\) be the classical solutions to the initial-boundary value problem (1)(5) corresponding to \(\alpha =\alpha _{1}\), and \((u_{i}^{*}, T^{*}, p^{*})\) and \((v^{*}_{i}, \theta ^{*}, q^{*})\) also be the classical solutions to the initial-boundary value problem (1)(5) but corresponding to \(\alpha =\alpha _{2}\). Then for any \(t>0\) we have

$$ (u_{i}, T, p)\rightarrow \bigl(u_{i}^{*}, T^{*}, p^{*} \bigr), \qquad (v_{i}, \theta , q)\rightarrow \bigl(v^{*}_{i}, \theta ^{*}, q^{*}\bigr), $$
(86)

as \(\alpha _{1}\rightarrow \alpha _{2}\). The differences of velocities satisfy

$$\begin{aligned}& \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx+ \int _{0}^{t} \int _{ \Omega _{2}} \bigl\vert \boldsymbol{{w}}^{m} \bigr\vert ^{2}\,dx\,d\eta +\frac{k_{2}}{k_{1}} \int _{0}^{t} \int _{\Omega _{1}} \vert \boldsymbol{{w}} \vert ^{2}\,dx\,d\eta \\& \quad \leq \frac{\sigma ^{2}\mu }{2\sqrt{k_{1}}} \int _{0}^{t} e^{\int _{s}^{t}c_{4}( \eta )\,d\eta }A_{6}(s)\,ds, \end{aligned}$$
(87)

where w, \(\boldsymbol{{w}}^{m}\), S, \(S^{m}\), σ have been defined in (64) and (65), and \(c_{5}\), \(c_{6}\) are positive constants which will be defined later.

Moreover, the differences of the temperatures satisfy

$$ \int _{\Omega _{1}}S^{2}\,dx+ \int _{\Omega _{2}}\bigl(S^{m}\bigr)^{2}\,dx \leq \frac{\sigma ^{2}N_{M}^{2}\mu }{4k_{2}\sqrt{k_{1}}} \int _{0}^{t}A_{6}(s)e^{ \int _{s}^{t}c_{4}(\eta )\,d\eta }\,ds. $$
(88)

Inequalities (87) and (88) are a priori bounds demonstrating the continuous dependence of the solution on the interface coefficient α.

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Acknowledgements

The authors would like to thank the referees for their helpful comments and suggestions.

Funding

This research was supported by the Foundation for natural Science in Higher Education of Guangdong (Grant No. 2019KZDXM042).

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YL proposed the main idea of this paper and wrote the whole paper. ZS prepared the manuscript initially. LC performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

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Correspondence to Yuanfei Li.

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Li, Y., Zhang, S. & Lin, C. Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain. Bound Value Probl 2021, 27 (2021). https://doi.org/10.1186/s13661-021-01501-0

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MSC

  • 35B40
  • 35Q30
  • 76D05

Keywords

  • Boussinesq equations
  • Continuous dependence
  • Boussinesq coefficient
  • Interfacing problem
  • A priori bounds
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