A Study of Structural Stability on the Bidispersive Flow in a Semi-Infinite Cylinder

Abstract

We consider the bidispersive flow with nonlinear boundary conditions in a bounded region. By using the differential inequality technique, we get the bound for the $L_4$-norm of the salinity which plays an important role. The continuous dependence and the convergence results on the Soret coefficient are established.

1. Introduction

In recent years, the stability of bidispersive porous medium has attracted extensive attention of scholars. One of the reasons why scholars did this is due to the important applications of bidispersive porous medium in many real life problems (see [1,2,3,4,5,6,7,8,9]).

Franchi et al. [10] considered a bidispersive medium diffusion model

$$\begin{array}{cc}\hfill \mu u_i+p_{,i}+a(u_i-v_i)& =g_{i}T,u_{i,i}=0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \nu v_i+q_{,i}-a(u_i-v_i)& =g_{i}T,v_{i,i}=0,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill T_{,t}-\Delta T+\alpha (u_i+v_i)T_{,i}& =0,\hfill \end{array}$$

(3)

where $u_i,v_i$ are the fluid velocities, T is the temperature, $p,q$ are the pressures, a is the interaction coefficient and $g_i$ is the gravity function. $\alpha$ and $\mu ,\nu$ denote the thermal expansion coefficient and the dynamic viscosities of the saturated fluid, respectively. In (1)–(3) and the whole paper, the comma is used to indicate partial differentiation and the usual summation convection is employed, with repeated Latin subscripts summed from 1 to 3, e.g., $u_{i,j}{u}_{i,j}={\sum }_{i,j=1}^3{\left(\frac{\partial u_i}{\partial x_j}\right)}^2$. We also use the summation convention summed from 1 to 2, e.g., $u_{\alpha ,\beta }{u}_{\alpha ,\beta }={\sum }_{\alpha ,\beta =1}^2{\left(\frac{\partial u_{\alpha }}{\partial x_{\beta }}\right)}^2$.

When we use models to simulate physical phenomena, errors are inevitably generated. The so-called structural stability is to study whether these errors will have a significant impact on the solution of the model. The study of bidispersive porous medium has attracted the attention of many researchers. The continuous dependence of the solution to (1)–(3) on the interaction coefficient, the viscosity coefficients and the radiation constant was established by Franchi et al. [10]. In another paper, Franchi et al. [11] developed a Brinkman theory for the bidispersive porous medium flow, where both salt and temperature field effects are present. Li et al. [12] further considered the structural stability of (1.1) in an infinite region. More results about structure stability of saturated porous medium can be found in [13,14,15,16,17,18,19,20,21,22,23,24,25,26].

In this paper, we consider a class of bidisperse porus media with salinity and temperature effects. Due to temperature gradient may directly influence salt concentration, we also consider Soret effect. The model can be written as

$$\begin{array}{cc}\hfill \mu u_i+p_{,i}+a(u_i-v_i)=g_{i}T+h_{i}C,u_{i,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \nu v_i+q_{,i}-a(u_i-v_i)=g_{i}T+h_{i}C,v_{i,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill T_{,t}-\Delta T+\alpha (u_i+v_i)T_{,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill C_{,t}-\Delta C+\alpha (u_i+v_i)C_{,i}=\sigma \Delta T,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(7)

where C is the concentration of salt, $h_i$ is the gravity function and $\sigma$ is the Soret coefficient. $\tilde{\mu }$ is the Brinkman viscosity coefficient. Without losing generality, we assume $g_i{g}_i,h_i{h}_i\le 1$ and $\mu =\nu =1$. The Soret effect is when a temperature gradient induces a change in solute concentration. This leads to complications in the ensuing stability analysis. To derive the continuous dependence on the Soret coefficient, we assume that the solutions to (4)–(7) satisfy

$$\begin{array}{cc}\hfill u_i{n}_i=v_i{n}_i=0,T=f_1(\mathit{x},t),C=f_2(\mathit{x},t),& \mathrm{on}\partial \Omega \times (0,\tau ),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill T(\mathit{x},0)=T_0\left(\mathit{x}\right),C(\mathit{x},0)=C_0\left(\mathit{x}\right),& \mathrm{in}\Omega ,\hfill \end{array}$$

(9)

where $f_1,f_2$ and $T_0,C_0$ are given functions greater than zero.

The main objective of this article is to investigate the structural stability of Equations (4)–(7) under different boundary conditions. We will use differential inequality technology to derive a prior estimates of the solution to Equations (4)–(7). Then, we use the “energy” method to establish the continuous dependence and the convergence result of the solution on Soret coefficient. Because we have considered the effect of Soret, the result derived by [27] no longer holds for C. We have to seek the bound of ${\left|\right|C\left|\right|}_4$ which will be derived in the next section.

2. Bound of ${\left|\right|\mathit{C}\left|\right|}_{\mathbf{4}}$

By using the method of Liu et al. [27], we can obtain the following lemmas.

Lemma 1. 

(see (2.11) and (2.40) in [27]) Assuming that $f_1,f_2\in L^{\infty }(\partial \Omega \times [0,\tau ])$ and $T_0\in L^{\infty }(\Omega )$, then

$$\begin{array}{c}\hfill \underset{[0,\tau ]}{sup}{\left|\right|T\left|\right|}_{\infty }\le T_M,\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_0^t{\int }_{\Omega }{T}_{,i}{T}_{,i}dxd\eta \le m_1\left(t\right),\end{array}$$

where $T_M=max\left\{\left|\right|f_1{\left|\right|}_{\infty },\left|\right|T_0{\left|\right|}_{\infty }\right\}$and$m_1\left(t\right)$is a positive function.

Next, we want to derive the $L_4$-norm of C by using the Lemma 1.

Letting that $\mathcal{C}$ is a solution to

$$\begin{array}{cc}\hfill {\mathcal{C}}_t+(u_i+v_i){\mathcal{C}}_{,i}=\Delta \mathcal{C},& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \mathcal{C}=f_2(\mathit{x},t),& \mathrm{on}\partial \Omega \times (0,\tau ),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathcal{C}(\mathit{x},0)=C_0\left(\mathit{x}\right),& \mathrm{in}\Omega .\hfill \end{array}$$

(12)

Then, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }{C}^{2}dx& \le {\int }_{\Omega }{(C-\mathcal{C})}^{2}dx+{\int }_{\Omega }{\mathcal{C}}^{2}dx,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\int }_{\Omega }{C}^{4}dx& \le {\int }_{\Omega }{(C-\mathcal{C})}^{4}dx+{\int }_{\Omega }{\mathcal{C}}^{4}dx.\hfill \end{array}$$

(14)

For $\mathcal{C}$, we can get the similar results to Lemma 1

$$\begin{array}{c}\hfill \underset{[0,\tau ]}{sup}{\left|\right|\mathcal{C}\left|\right|}_{\infty }\le {\mathcal{C}}_M,{\int }_0^t{\int }_{\Omega }{\mathcal{C}}_{,i}{\mathcal{C}}_{,i}dxd\eta \le m_2\left(t\right)\end{array}$$

(15)

where ${\mathcal{C}}_M=max\left\{\left|\right|f_2{\left|\right|}_{\infty },\left|\right|C_0{\left|\right|}_{\infty }\right\}$ and $m_2\left(t\right)$ is a positive function. Therefore, using (11), we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\mathcal{C}}^{4}dx& \le {\mathcal{C}}_M^4|\Omega |,\hfill \end{array}$$

(16)

where $|\Omega |$ is the measure of $\Omega$. Combining (7), (8), (9), (10), (11) and (12), we can obtain that $C-\mathcal{C}$ satisfies the following equations

$$\begin{array}{cc}\hfill {(C-\mathcal{C})}_t+(u_i+v_i){(C-\mathcal{C})}_{,i}=\sigma \Delta T+\Delta (C-\mathcal{C}),& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill (C-\mathcal{C})(\mathit{x},t)=0,& \mathrm{on}\partial \Omega \times (0,\tau ),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill (C-\mathcal{C})(\mathit{x},0)=0,& \mathrm{in}\Omega .\hfill \end{array}$$

(19)

Using (17)–(19), the Young inequality and Lemma 1, we obtain

$$\begin{array}{cc}\hfill \frac{1}{4}\frac{d}{dt}{\int }_{\Omega }{(C-\mathcal{C})}^{4}dx& =-3{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}^2{(C-\mathcal{C})}_{,i}dx-3\sigma {\int }_{\Omega }{T}_{,i}{(C-\mathcal{C})}^2{(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & =-3{\int }_{\Omega }{(C-\mathcal{C})}^2{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx+3\sigma {\int }_{\Omega }T(C-\mathcal{C}){(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & -3\sigma {\int }_{\Omega }\left[(C-\mathcal{C})T_{,i}+T{(C-\mathcal{C})}_{,i}\right](C-\mathcal{C}){(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & \le -\left[3-\frac{3}{2}\sigma {\epsilon }_1-\frac{3}{2}\sigma {\epsilon }_2\right]{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}^2{(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & +\frac{3}{2{\epsilon }_1}\sigma {\int }_{\Omega }\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]dx\hfill \\ \hfill & +\frac{3}{2{\epsilon }_2}\sigma T_M^2{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx,\hfill \end{array}$$

(20)

where ${\epsilon }_1$ and ${\epsilon }_2$ are positive constants. To bound the last term on the right of (20), using (6) and (17), we compute

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{T}^2{(C-\mathcal{C})}^{2}dx& ={\int }_{\Omega }T{T}_t{(C-\mathcal{C})}^{2}dx+{\int }_{\Omega }{T}^2(C-\mathcal{C}){(C-\mathcal{C})}_{t}dx\hfill \\ \hfill & ={\int }_{\Omega }T\left[\Delta T-(u_i+v_i)T_{,i}\right]{(C-\mathcal{C})}^{2}dx\hfill \\ \hfill & +{\int }_{\Omega }{T}^2(C-\mathcal{C})\left[\sigma \Delta T+\Delta (C-\mathcal{C})-(u_i+v_i){(C-\mathcal{C})}_{,i}\right]dx\hfill \\ \hfill & =-{\int }_{\Omega }\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]dx\hfill \\ \hfill & -2{\int }_{\Omega }T{(C-\mathcal{C})}_{,i}{T}_{,i}(C-\mathcal{C})dx+\sigma {\int }_{\Omega }{T}^2{T}_{,i}{(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & -2\sigma {\int }_{\Omega }T{T}_{,i}\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]dx\hfill \\ \hfill & \le \left[\sigma {\epsilon }_4-1\right]{\int }_{\Omega }\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]\left[T{(C-\mathcal{C})}_{,i}+(C-\mathcal{C})T_{,i}\right]dx\hfill \\ \hfill & +{\epsilon }_3{\int }_{\Omega }{(C-\mathcal{C})}^2{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx+\left[\frac{1}{{\epsilon }_3}+1+\frac{1}{{\epsilon }_4}\right]\sigma T_M^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx\hfill \\ \hfill & +\sigma T_M^2{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx,\hfill \end{array}$$

(21)

where ${\epsilon }_3$ and ${\epsilon }_4$ are positive constants. Choosing ${\epsilon }_1={\epsilon }_2={\epsilon }_4=\frac{1}{2\sigma }$ and ${\epsilon }_3=\frac{1}{4{\sigma }^4}$ and combining (20) and (21),

$$\begin{array}{cc}\hfill \frac{d}{dt}\left[{\int }_{\Omega }{(C-\mathcal{C})}^{4}dx+12{\sigma }^2{\int }_{\Omega }{T}^2{(C-\mathcal{C})}^{2}dx\right]& \le \left[\frac{6}{{\epsilon }_2}+12{\sigma }^3\right]\sigma T_M^2{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & +12{\sigma }^3\left[\frac{1}{{\epsilon }_3}+1+\frac{1}{{\epsilon }_4}\right]T_M^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx.\hfill \end{array}$$

(22)

Integrating (22) from 0 to t, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{(C-\mathcal{C})}^{4}dx+12{\sigma }^2{\int }_{\Omega }{T}^2{(C-\mathcal{C})}^{2}dx& \le \left[\frac{6}{{\epsilon }_2}+12{\sigma }^3\right]\sigma T_M^2{\int }_0^t{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dxd\eta \hfill \\ \hfill & +12{\sigma }^3\left[\frac{1}{{\epsilon }_3}+1+\frac{1}{{\epsilon }_4}\right]T_M^2{\int }_0^t{\int }_{\Omega }{T}_{,i}{T}_{,i}dxd\eta .\hfill \end{array}$$

(23)

Using (17)–(19), we have

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{(C-\mathcal{C})}^{2}dx& =-{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx-\sigma {\int }_{\Omega }{T}_{,i}{(C-\mathcal{C})}_{,i}dx\hfill \\ \hfill & \le -\frac{1}{2}{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dx+\frac{{\sigma }^2}{2}{\int }_{\Omega }{T}_{,i}{T}_{,i}dx.\hfill \end{array}$$

(24)

Integrating (24) from 0 to t and using Lemma 1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{(C-\mathcal{C})}^{2}dx& +{\int }_0^t{\int }_{\Omega }{(C-\mathcal{C})}_{,i}{(C-\mathcal{C})}_{,i}dxd\eta \le {\sigma }^2{m}_1\left(t\right).\hfill \end{array}$$

(25)

Inserting (25) into (23) and using Lemma 1, we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }{(C-\mathcal{C})}^{4}dx\le \left[\frac{6}{{\epsilon }_2}+12{\sigma }^3\right]\sigma T_M^2{\sigma }^2{m}_1\left(t\right)+12{\sigma }^3\left[\frac{1}{{\epsilon }_3}+1+\frac{1}{{\epsilon }_4}\right]T_M^2{m}_2\left(t\right).\end{array}$$

(26)

Inserting (26) and (16) into (14), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{C}^{4}dx& \le \left[\frac{6}{{\epsilon }_2}+12{\sigma }^3\right]\sigma T_M^2{\sigma }^2{m}_1\left(t\right)+12{\sigma }^3\left[\frac{1}{{\epsilon }_3}+1+\frac{1}{{\epsilon }_4}\right]T_M^2{m}_2\left(t\right)+C_M^2|\Omega |\doteq n_1\left(t\right).\hfill \end{array}$$

(27)

3. Continuous Dependence Result

Assuming that $(u_i,v_i,T,C,p,q)$ and $(u_i^{*},v_i^{*},T^{*},C^{*},p^{*},q^{*})$ are solutions of (4)–(7) under the initial-boundary conditions (8) and (9), but with different Soret coefficients ${\sigma }_1$ and ${\sigma }_2$. If we let

$$w_i=u_i-u_i^{*},r_i=v_i-v_i^{*},{\Sigma }_1=T-T^{*},{\Sigma }_2=C-C^{*},{\pi }_1=p-p^{*},{\pi }_2=q-q^{*},\sigma ={\sigma }_1-{\sigma }_2,$$

then $(w_i,r_i,{\Sigma }_1,{\Sigma }_2,{\pi }_1,{\pi }_2)$ satisfies

$$\begin{array}{cc}\hfill w_i+a(w_i-r_i)=-{\pi }_{1,i}+g_i{\Sigma }_1+h_i{\Sigma }_2,w_{i,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill r_i-a(w_i-r_i)=-{\pi }_{2,i}+g_i{\Sigma }_1+h_i{\Sigma }_2,r_{i,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\Sigma }_{1,t}-\Delta {\Sigma }_1+\alpha (u_i+v_i){\Sigma }_{1,i}+\alpha (w_i+r_i)T_{,i}^{*}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\Sigma }_{2,t}-\Delta {\Sigma }_2+\alpha (u_i+v_i){\Sigma }_{2,i}+\alpha (w_i+r_i)C_{,i}^{*}=\sigma \Delta T+{\sigma }_2\Delta {\Sigma }_1,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill w_i{n}_i=r_i{n}_i=0,{\Sigma }_1={\Sigma }_2=0,& \mathrm{on}\partial \Omega \times (0,\tau ),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\Sigma }_1(\mathit{x},0)={\Sigma }_2(\mathit{x},0)=0,& \mathrm{in}\Omega .\hfill \end{array}$$

(33)

We can obtain the following theorem.

Theorem 1. 

Let $(w_i,r_i,{\Sigma }_1,{\Sigma }_2,{\pi }_1,{\pi }_2)$ be solution to (28)–(33). If $f_1,f_2\in L^{\infty }(\partial \Omega \times [0,\tau ])$ and $C_0,T_0\in L^{\infty }(\Omega )$, then

$$u_i\to u_i^{*},v_i\to v_i^{*},T\to T^{*},C\to C^{*},as{\sigma }_1\to {\sigma }_2.$$

Specifically,

$$\begin{array}{cc}\hfill {\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx& \le L_1\left(t\right){\sigma }^2,\hfill \\ \hfill {\int }_{\Omega }\left[w_i{w}_i+r_i{r}_i+2a(w_i-r_i)(w_i-r_i)\right]dx& \le L_2\left(t\right){\sigma }^2,\hfill \end{array}$$

where $L_1\left(t\right)$ and $L_2\left(t\right)$ are positive functions.

Proof. 

By multiplying (28) and (29) by $w_i$ and $r_i$, respectively, and integrating by parts in $\Omega$ we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }{w}_i\left[w_i+a(w_i-r_i)+{\pi }_{1,i}-g_i{\Sigma }_1-h_i{\Sigma }_2\right]dx& =0,\hfill \\ \hfill {\int }_{\Omega }{r}_i\left[r_i-a(w_i-r_i)+{\pi }_{2,i}-g_i{\Sigma }_1-h_i{\Sigma }_2\right]dx& =0.\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }[w_i{w}_i+r_i{r}_i& +a(w_i-r_i)(w_i-r_i)]dx\hfill \\ \hfill & ={\int }_{\Omega }{w}_i\left[g_i{\Sigma }_1+h_i{\Sigma }_2\right]dx+{\int }_{\Omega }{r}_i\left[g_i{\Sigma }_1+h_i{\Sigma }_2\right]dx\hfill \\ \hfill & \le \frac{1}{2}{\int }_{\Omega }\left[w_i{w}_i+r_i{r}_i\right]dx+2{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx.\hfill \end{array}$$

(34)

Using the Hölder inequality, we have from (34)

$$\begin{array}{cc}\hfill {\int }_{\Omega }[w_i{w}_i+r_i{r}_i& +2a(w_i-r_i)(w_i-r_i)]dx\le 4{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx.\hfill \end{array}$$

(35)

Multiplying (30) by ${\Sigma }_1$, integrating in $\Omega$ and using Lemma 1, we have

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{\Sigma }_1^{2}dx& +{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx=\alpha {\int }_{\Omega }(w_i+r_i)T^{*}{\Sigma }_{1,i}dx\hfill \\ \hfill & \le \alpha T_M{\left[{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx\right]}^{\frac{1}{2}}{\left[{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx\right]}^{\frac{1}{2}}.\hfill \\ \hfill & \le {ϵ}_1{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx+\frac{{\alpha }^2{T}_M^2}{4{ϵ}_1}\left[{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx\right],\hfill \end{array}$$

(36)

where ${ϵ}_1$ is a positive constant.

Multiplying (31) by ${\Sigma }_2$ and integrating in $\Omega$, we have

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{\Sigma }_2^{2}dx+{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx& =\alpha {\int }_{\Omega }(w_i+r_i)C^{*}{\Sigma }_{2,i}dx-{\sigma }_2{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{2,i}dx-\sigma {\int }_{\Omega }{T}_{,i}{\Sigma }_{2,i}dx\hfill \\ \hfill & \le {\sigma }_2{\left[{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx\right]}^{\frac{1}{2}}{\left[{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\right]}^{\frac{1}{2}}\hfill \\ \hfill & +\sigma {\left[{\int }_{\Omega }{T}_{,i}{T}_{,i}dx\right]}^{\frac{1}{2}}{\left[{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\right]}^{\frac{1}{2}}\hfill \\ \hfill & +\alpha {\left[{\int }_{\Omega }{\left(\right|\mathit{w}|}^4+{\left|\mathit{r}\right|}^4)dx\right]}^{\frac{1}{4}}{\left[{\int }_{\Omega }{\left(C^{*}\right)}^{4}dx\right]}^{\frac{1}{4}}{\left[{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\right]}^{\frac{1}{2}}.\hfill \end{array}$$

(37)

Using (27) and the result given in p674 of Scott [17] (as well as the result derived by Lin and Payne in Appendix B of [28])

$$\begin{array}{c}\hfill {\left({\int }_{\Omega }{\psi }^{4}dx\right)}^{\frac{1}{2}}\le k_1{\int }_{\Omega }{\psi }^{2}dx+k_2{\left({\int }_{\Omega }{\psi }^{2}dx\right)}^{\frac{1}{4}}{\left({\int }_{\Omega }{\psi }_{,i}{\psi }_{,i}dx\right)}^{\frac{3}{4}},k_1,k_2>0,\end{array}$$

(38)

we have from (37)

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{\Sigma }_2^{2}dx& +{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\hfill \\ \hfill & \le \frac{1}{4}{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx+2{\alpha }^2{n}_1\left(t\right)\left[k_1{\int }_{\Omega }{w}_i{w}_{i}dx+k_2{\left({\int }_{\Omega }{w}_i{w}_{i}dx\right)}^{\frac{1}{4}}{\left({\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx\right)}^{\frac{3}{4}}\right]\hfill \\ \hfill & +2{\alpha }^2{n}_1\left(t\right)\left[k_1{\int }_{\Omega }{r}_i{r}_{i}dx+k_2{\left({\int }_{\Omega }{r}_i{r}_{i}dx\right)}^{\frac{1}{4}}{\left({\int }_{\Omega }{r}_{i,j}{r}_{i,j}dx\right)}^{\frac{3}{4}}\right]\hfill \\ \hfill & +{ϵ}_3{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx+\frac{{\sigma }_2^2}{4{\epsilon }_3}{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx\hfill \\ \hfill & +{ϵ}_4{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx+\frac{{\sigma }^2}{4{\epsilon }_4}{\int }_{\Omega }{T}_{,i}{T}_{,i}dx\hfill \\ \hfill & \le \left[\frac{1}{4}+{ϵ}_3+{ϵ}_4\right]{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx+\frac{{\sigma }_2^2}{4{ϵ}_3}{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx+\frac{{\sigma }^2}{4{ϵ}_4}{\int }_{\Omega }{T}_{,i}{T}_{,i}dx\hfill \\ \hfill & +2{\alpha }^2{n}_1\left(t\right)\left[\left(k_1+\frac{1}{4}{ϵ}_2^{-3}{k}_2\right){\int }_{\Omega }{w}_i{w}_{i}dx+\frac{3}{4}{k}_2{ϵ}_2{\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx\right]\hfill \\ \hfill & +2{\alpha }^2{n}_1\left(t\right)\left[\left(k_1+\frac{1}{4}{ϵ}_2^{-3}{k}_2\right){\int }_{\Omega }{r}_i{r}_{i}dx+\frac{3}{4}{k}_2{ϵ}_2{\int }_{\Omega }{r}_{i,j}{r}_{i,j}dx\right],\hfill \end{array}$$

(39)

where ${ϵ}_2,{ϵ}_3$ and ${ϵ}_4$ are positive constants. Combining (36) and (39), we have

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx& +\left[\Gamma -\Gamma {ϵ}_1-\frac{{\sigma }_2^2}{4{ϵ}_3}\right]{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx+\left[\frac{3}{4}-{ϵ}_3-{ϵ}_4\right]{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\hfill \\ \hfill & \le k_3\left(t\right)\left[{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx\right]+k_4\left(t\right){ϵ}_2\left[{\int }_{\Omega }(w_{i,j}{w}_{i,j}+r_{i,j}{r}_{i,j})dx\right]\hfill \\ \hfill & +\frac{{\tilde{\sigma }}^2}{4{ϵ}_4}{\int }_{\Omega }{T}_{,i}{T}_{,i}dx,\hfill \end{array}$$

(40)

where $k_3\left(t\right)=\frac{{\alpha }^2{T}_M^2}{4{ϵ}_1}+2{\alpha }^2{n}_1\left(t\right)\left(k_1+\frac{1}{4}{ϵ}_2^{-3}{k}_2\right),k_4\left(t\right)=\frac{3}{2}{k}_2{\alpha }^2{n}_1\left(t\right)$.

Choosing ${ϵ}_1=\frac{1}{2},{ϵ}_3={ϵ}_4=\frac{1}{8}$, $\Gamma =6{\sigma }_2^2$ and using (35), we obtain

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx& +2{\sigma }_2^2{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx+{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\hfill \\ \hfill & \le 2k_3\left(t\right)max\{1,\frac{1}{\Gamma }\}{\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx\hfill \\ \hfill & +2k_4\left(t\right){ϵ}_2{\int }_{\Omega }(w_{i,j}{w}_{i,j}+r_{i,j}{r}_{i,j})dx+4{\sigma }^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx.\hfill \end{array}$$

(41)

Next, we derive bound of ${\int }_{\Omega }(w_{i,j}{w}_{i,j}+r_{i,j}{r}_{i,j})dx$. To do this, we follow the methods of [17,28]. Now, we give following identity

$$\begin{array}{c}\hfill {\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx={\int }_{\Omega }(w_{i,j}-w_{j,i})w_{i,j}dx+{\int }_{\Omega }{w}_{j,i}{w}_{i,j}dx.\end{array}$$

(42)

We compute

$$\begin{array}{cc}\hfill {\int }_{\Omega }(w_{i,j}-w_{j,i})w_{i,j}dx& =-a{\int }_{\Omega }\left[(w_{i,j}-r_{i,j})-(w_{j,i}-r_{j,i})\right]w_{i,j}dx\hfill \\ \hfill & +{\int }_{\Omega }(g_{i,j}-g_{j,i}){\Sigma }_1{w}_{i,j}dx+{\int }_{\Omega }(g_i{\Sigma }_{1,j}-g_j{\Sigma }_{1,i})w_{i,j}dx\hfill \\ \hfill & +{\int }_{\Omega }(h_{i,j}-h_{j,i}){\Sigma }_2{w}_{i,j}dx-{\int }_{\Omega }(h_i{\Sigma }_{2,j}-h_j{\Sigma }_{2,i})w_{i,j}dx.\hfill \end{array}$$

(43)

Using the Hölder inequality and the AG mean inequaity, we obtain from (43)

$$\begin{array}{cc}\hfill {\int }_{\Omega }(w_{i,j}-w_{j,i})w_{i,j}dx& \le -a{\int }_{\Omega }\left[(w_{i,j}-r_{i,j})-(w_{j,i}-r_{j,i})\right]w_{i,j}dx\hfill \\ \hfill & +4{\int }_{\Omega }{\Sigma }_1^{2}dx+\frac{1}{4}{\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx+4{\int }_{\Omega }{\Sigma }_2^{2}dx\hfill \\ \hfill & +4{\int }_{\Omega }{\Sigma }_{1,i}{\Sigma }_{1,i}dx+\frac{1}{4}{\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx+4{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx,\hfill \end{array}$$

(44)

where we have assumed that $|\nabla g|,|\nabla h|\le 1$ for convenience. Inserting (44) into (42), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx& +2a{\int }_{\Omega }(w_{i,j}-r_{i,j})w_{i,j}dx\hfill \\ \hfill & \le 2a{\int }_{\Omega }(w_{j,i}-r_{j,i})w_{i,j}dx+2{\int }_{\Omega }{w}_{j,i}{w}_{i,j}dx\hfill \\ \hfill & +8{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx+8{\int }_{\Omega }\left[{\Sigma }_{1,i}{\Sigma }_{1,i}+{\Sigma }_{2,i}{\Sigma }_{2,i}\right]dx.\hfill \end{array}$$

(45)

Similar to (45), we have for ${\int }_{\Omega }{r}_{i,j}{r}_{i,j}dx$

$$\begin{array}{cc}\hfill {\int }_{\Omega }{r}_{i,j}{r}_{i,j}dx& -2a{\int }_{\Omega }(w_{i,j}-r_{i,j})r_{i,j}dx\hfill \\ \hfill & \le -2a{\int }_{\Omega }(w_{j,i}-r_{j,i})r_{i,j}dx+2{\int }_{\Omega }{r}_{j,i}{r}_{i,j}dx\hfill \\ \hfill & +8{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx+8{\int }_{\Omega }\left[{\Sigma }_{1,i}{\Sigma }_{1,i}+{\Sigma }_{2,i}{\Sigma }_{2,i}\right]dx.\hfill \end{array}$$

(46)

Combining (45) and (46), we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }(w_{i,j}{w}_{i,j}+r_{i,j}{r}_{i,j})dx& +2a{\int }_{\Omega }(w_{i,j}-r_{i,j})(w_{i,j}-r_{i,j})dx\hfill \\ \hfill & \le 2a{\int }_{\Omega }(w_{j,i}-r_{j,i})(w_{i,j}-r_{i,j})dx+2{\int }_{\Omega }{w}_{j,i}{w}_{i,j}dx+2{\int }_{\Omega }{r}_{j,i}{r}_{i,j}dx\hfill \\ \hfill & +16{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx+16{\int }_{\Omega }\left[{\Sigma }_{1,i}{\Sigma }_{1,i}+{\Sigma }_{2,i}{\Sigma }_{2,i}\right]dx.\hfill \end{array}$$

(47)

Lin and Payne [28] have proved that ${\int }_{\Omega }{w}_{j,i}{w}_{i,j}dx\le 0$ if $\Omega$ is a convex region. if $\Omega$ is a non-convex region, we adopt the methods which have been used by Scott [10]. We let ${\psi }_i$ to satisfy that

$$|{\psi }_i|,|{\psi }_{i,j}\le M,in\Omega ,{\psi }_i{n}_i\le {\psi }_0>0,on\partial \Omega .$$

We have

$$\begin{array}{cc}{\psi }_0{\int }_{\partial \Omega }{w}_i{w}_{i}dS& \le {\int }_{\partial \Omega }{\psi }_j{n}_j{w}_i{w}_{i}dS={\int }_{\Omega }{\psi }_{j,j}{w}_i{w}_{i}dx+2{\int }_{\Omega }{\psi }_j{w}_i{w}_{i,j}dx\hfill \\ & \le M\left\{2{\int }_{\Omega }{w}_i{w}_{i}dx+{\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx\right\}\hfill \end{array}$$

(48)

In view of (8) and since $\Omega$ is a non-convex region, we get

$$\begin{array}{cc}{\int }_{\Omega }{w}_{i,j}{w}_{j,i}dx={\int }_{\partial \Omega }{w}_{i,j}{w}_j{n}_{i}dS& ={\int }_{\partial \Omega }{\left(w_i{n}_i\right)}_{,j}{w}_{j}dS-{\int }_{\partial \Omega }{w}_i{n}_{i,j}{w}_{j}dS\hfill \\ & \le m{\int }_{\partial \Omega }{w}_i{w}_{i}dS.\hfill \end{array}$$

Combining (48) and (49), we have

$$\begin{array}{c}\hfill {\int }_{\Omega }{w}_{j,i}{w}_{i,j}dx\le k_5{\int }_{\Omega }{w}_i{w}_{i}dx+k_6{\delta }_1{\int }_{\Omega }{w}_{i,j}{w}_{i,j}dx,k_5,k_6>0,\end{array}$$

(49)

if $\Omega$ is a non-convex region, where ${\delta }_1$ is a positive arbitrary constant. Obviously, (49) is also valid for ${\int }_{\Omega }{r}_{j,i}{r}_{i,j}dx$ and ${\int }_{\Omega }(w_{j,i}-r_{j,i})(w_{i,j}-r_{i,j})dx$, i.e.,

$$\begin{array}{cc}{\int }_{\Omega }{r}_{j,i}{r}_{i,j}dx& \le k_5{\int }_{\Omega }{r}_i{r}_{i}dx+k_6{\delta }_2{\int }_{\Omega }{r}_{i,j}{r}_{i,j}dx,\hfill \end{array}$$

(50)

$$\begin{array}{cc}{\int }_{\Omega }(w_{j,i}-r_{j,i})(w_{i,j}-r_{i,j})dx& \le k_5{\int }_{\Omega }(w_i-r_i)(w_i-r_i)dx\hfill \\ & +k_6{\delta }_3{\int }_{\Omega }(w_{i,j}-r_{i,j})(w_{i,j}-r_{i,j})dx,\hfill \end{array}$$

(51)

where ${\delta }_2$ and ${\delta }_3$ are positive arbitrary constants. Inserting (49)–(51) into (47) and using (35), we have

$$\begin{array}{cc}{\int }_{\Omega }(w_{i,j}{w}_{i,j}+r_{i,j}{r}_{i,j})dx& +2a{\int }_{\Omega }(w_{i,j}-r_{i,j})(w_{i,j}-r_{i,j})dx\hfill \\ & \le 4a{k}_5{\int }_{\Omega }(w_i-r_i)(w_i-r_i)dx+4k_5{\int }_{\Omega }{w}_i{w}_{i}dx+4k_5{\int }_{\Omega }{r}_i{r}_{i}dx\hfill \\ & +32{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx+32{\int }_{\Omega }\left[{\Sigma }_{1,i}{\Sigma }_{1,i}+{\Sigma }_{2,i}{\Sigma }_{2,i}\right]dx\hfill \\ & \le 16(k_5+2){\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx+32{\int }_{\Omega }\left[{\Sigma }_{1,i}{\Sigma }_{1,i}+{\Sigma }_{2,i}{\Sigma }_{2,i}\right]dx,\hfill \end{array}$$

(52)

where we have chosen that ${\delta }_1={\delta }_2={\delta }_3=\frac{1}{4k_6}.$ Inserting (52) into (41) and choosing that ${ϵ}_2\le min\{\frac{{\sigma }_2^2}{32k_4},\frac{1}{64k_4}\}$, we obtain

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx& \le k_7\left(t\right){\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx+4{\sigma }^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx,\hfill \end{array}$$

(53)

where $k_7\left(t\right)=\left[2k_3+32k_4{ϵ}_2(k_5+2)\right]max\{1,\frac{1}{\Gamma }\}$. Integrating (53) from 0 to t and using Lemma 1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }\left[\Gamma {\Sigma }_1^2+{\Sigma }_2^2\right]dx& \le 4{\sigma }^2{e}^{{\int }_0^t{k}_7\left(\eta \right)d\eta }{\int }_0^t{e}^{-{\int }_0^s{k}_7\left(\eta \right)d\eta }{\int }_{\Omega }{T}_{,i}{T}_{,i}dxds\hfill \\ \hfill & \le 4{\sigma }^2{e}^{{\int }_0^t{k}_7\left(\eta \right)d\eta }{\int }_0^t{\int }_{\Omega }{T}_{,i}{T}_{,i}dxds\hfill \\ \hfill & \le 4m_1\left(t\right){\sigma }^2{e}^{{\int }_0^t{k}_7\left(\eta \right)d\eta }.\hfill \end{array}$$

(54)

Inserting (54) into (35), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }[w_i{w}_i+r_i{r}_i& +2a(w_i-r_i)(w_i-r_i)]dx\le 16max\{1,\frac{1}{\Gamma }\}{m}_1\left(t\right){\sigma }^2{e}^{{\int }_0^t{k}_7\left(\eta \right)d\eta }.\hfill \end{array}$$

(55)

Choosing

$$\begin{array}{cc}\hfill L_1\left(t\right)& =4m_1\left(t\right)e^{{\int }_0^t{k}_7\left(\eta \right)d\eta },\hfill \\ \hfill L_2\left(t\right)& =16max\{1,\frac{1}{\Gamma }\}{m}_1\left(t\right)e^{{\int }_0^t{k}_7\left(\eta \right)d\eta },\hfill \end{array}$$

we can complete the proof of Theorem 1. □

4. Convergence Result as $\sigma$ Tends to Zero

Convergence is to study the trend of solution changes when the parameter of the equation tends zero. It is different from continuous dependence of solution. Franchi et al. [10,11] derived the continuous dependence of the solutions to the double diffusive bidispersive convection. However, there have been no studies on the convergence of solutions to the double diffusive bidispersive convection by scholars. In this section, we study convergence result as $\sigma$ tends to zero. To do this, we let that $(u_i^{*},v_i^{*},T^{*},C^{*},p^{*},q^{*})$ is the solution to (1)–(9) with $\sigma =0$, i.e.,

$$\begin{array}{cc}\hfill u_i^{*}+a(u_i^{*}-v_i^{*})=-p_{,i}^{*}+g_i{T}^{*}+h_i{C}^{*},u_{i,i}^{*}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill v_i^{*}-a(u_i^{*}-v_i^{*})=-q_{,i}^{*}+g_i{T}^{*}+h_i{C}^{*},v_{i,i}^{*}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill T_{,t}^{*}-\Delta T^{*}+\alpha (u_i^{*}+v_i^{*})T_{,i}^{*}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill C_{,t}^{*}-\Delta C^{*}+\alpha (u_i^{*}+v_i^{*})C_{,i}^{*}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill u_i{n}_i=v_i{n}_i=0,T=f_1(\mathit{x},t),C=f_2(\mathit{x},t),& \mathrm{on}\partial \Omega \times (0,\tau ),\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill T(\mathit{x},0)=T_0\left(\mathit{x}\right),C(\mathit{x},0)=C_0\left(\mathit{x}\right),& \mathrm{in}\Omega .\hfill \end{array}$$

(61)

Obviously, Lemma 1 is also valid for $T^{*}$ and $C^{*}$, i.e.,

$$\begin{array}{c}\hfill \underset{[0,\tau ]}{sup}|T^{*}|\le T_M,\underset{[0,\tau ]}{sup}\left|C^{*}\right|\le C_M,\end{array}$$

(62)

where $C_M=max\left\{\left|\right|f_2{\left|\right|}_{\infty },\left|\right|C_0{\left|\right|}_{\infty }\right\}$.

If we let

$$w_i=u_i-u_i^{*},r_i=v_i-v_i^{*},{\Sigma }_1=T-T^{*},{\Sigma }_2=C-C^{*},{\pi }_1=p-p^{*},{\pi }_2=q-q^{*},$$

then $(w_i,r_i,{\Sigma }_1,{\Sigma }_2,{\pi }_1,{\pi }_2)$ satisfies

$$\begin{array}{cc}\hfill w_i+a(w_i-r_i)=-{\pi }_{1,i}+g_i{\Sigma }_1+h_i{\Sigma }_2,w_{i,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill r_i-a(w_i-r_i)=-{\pi }_{2,i}+g_i{\Sigma }_1+h_i{\Sigma }_2,r_{i,i}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill {\Sigma }_{1,t}-\Delta {\Sigma }_1+\alpha (u_i+v_i){\Sigma }_{1,i}+\alpha (w_i+r_i)T_{,i}^{*}=0,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill {\Sigma }_{2,t}-\Delta {\Sigma }_2+\alpha (u_i+v_i){\Sigma }_{2,i}+\alpha (w_i+r_i)C_{,i}^{*}=\sigma \Delta T,& \mathrm{in}\Omega \times (0,\tau ),\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill w_i{n}_i=r_i{n}_i=0,{\Sigma }_1={\Sigma }_2=0,& \mathrm{on}\partial \Omega \times (0,\tau ),\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill {\Sigma }_1(\mathit{x},0)={\Sigma }_2(\mathit{x},0)=0,& \mathrm{in}\Omega .\hfill \end{array}$$

(68)

We can have the following result.

Theorem 2. 

Let $(w_i,r_i,{\Sigma }_1,{\Sigma }_2,{\pi }_1,{\pi }_2)$ be solution to (63)–(68). If $f_1,f_2\in L^{\infty }(\partial \Omega \times [0,\tau ])$ and $C_0,T_0\in L^{\infty }(\Omega )$, then

$$u_i\to u_i^{*},v_i\to v_i^{*},T\to T^{*},C\to C^{*},as\sigma \to 0.$$

Specifically,

$$\begin{array}{cc}\hfill {\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx& \le L_3\left(t\right){\sigma }^2,\hfill \\ \hfill {\int }_{\Omega }\left[w_i{w}_i+r_i{r}_i+2a(w_i-r_i)(w_i-r_i)\right]dx& \le 4L_3\left(t\right){\sigma }^2,\hfill \end{array}$$

where $L_3\left(t\right)>0$ is a function.

Proof. 

We multiply (65) by ${\Sigma }_2$, integrate in $\Omega$ and use (62) to have

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{\Sigma }_2^{2}dx& +{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx=\alpha {\int }_{\Omega }(w_i+r_i)C^{*}{\Sigma }_{2,i}dx-\sigma {\int }_{\Omega }{T}_{,i}{\Sigma }_{2,i}dx\hfill \\ \hfill & \le \alpha C_M{\left[{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx\right]}^{\frac{1}{2}}{\left[{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\right]}^{\frac{1}{2}}\hfill \\ \hfill & +\sigma {\left[{\int }_{\Omega }{T}_{,i}{T}_{,i}dx\right]}^{\frac{1}{2}}{\left[{\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx\right]}^{\frac{1}{2}}\hfill \\ \hfill & \le {\int }_{\Omega }{\Sigma }_{2,i}{\Sigma }_{2,i}dx+\frac{1}{2}{\alpha }^2{C}_M^2{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx+\frac{1}{2}{\sigma }^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx,\hfill \end{array}$$

where we have used (62).

Therefore, we have

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\Omega }{\Sigma }_2^{2}dx& \le {\alpha }^2{C}_M^2{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx+{\sigma }^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx.\hfill \end{array}$$

(69)

Combining (36) and (69), we have

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx& \le \frac{1}{4}{b}_1{\int }_{\Omega }(w_i{w}_i+r_i{r}_i)dx+{\sigma }^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx,\hfill \end{array}$$

(70)

where $b_1=4{\alpha }^2{C}_M^2+2{\alpha }^2{T}_M^2$. Inserting (35) into (70), we have

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx& \le b_1{\int }_{\Omega }({\Sigma }_1^2+{\Sigma }_2^2)dx+{\sigma }^2{\int }_{\Omega }{T}_{,i}{T}_{,i}dx,\hfill \end{array}$$

(71)

where we have chosen that ${\epsilon }_1=1$.

Integrating (71) from 0 to t and using Lemma 1, from (71), we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }\left[{\Sigma }_1^2+{\Sigma }_2^2\right]dx\le {\sigma }^2{e}^{b_{1}t}{m}_2\left(t\right).\end{array}$$

(72)

Inserting (72) into (35), we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }\left[w_i{w}_i+r_i{r}_i+2a(w_i-r_i)(w_i-r_i)\right]dx\le 4{\sigma }^2{e}^{b_{1}t}{m}_2\left(t\right).\end{array}$$

(73)

Choosing $L_3\left(t\right)=e^{b_{1}t}{m}_2\left(t\right)$, we can obtain Theorem 2. □

5. Conclusions

In this article, we consider the bidispersive fluid affected by salinity. On the basis of prior estimation, the continuous dependence of the Soret coefficient is derived. If the region is replaced by a semi-infinite cylinder, considering the influence of the Soret coefficient on the model would be a meaningful topic. Next, we will investigate this issue.