Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability

1.   Introduction

The porous media fluid defined in a two-dimensional semi-infinite pipe has received extensive attention. Liu et al. ^[1] defined a semi-infinite strip pipe whose generatrix is parallel to the coordinate axis (see Figure 1) and obtained the Phragmén-Lindelöf alternative result of shallow water equations.

Payne and Schaefer ^[2] considered the case that the generatrix is not parallel to the coordinate axis (see Figure 2) and obtained the Phragmén-Lindelöf alternative for the biharmonic equation.

Recently, Li and Chen ^[3] considered the Darcy equations on a semi-infinite channel which was defined as

where $a > 0$ and $h(x_1)$ is a smooth curve in the plane. They proved that the solutions of Darcy equations grow polynomially or decay exponentially as a spatial variable $x_1\rightarrow \infty$. For more on such studies, one can see ^[2,4,5,6].

In this paper, we consider the convergence result on the Soret coefficient of the Darcy model in $R$. The importance of this type of convergence result was discussed by Hirsch and Smale ^[7] and there have been a lot of results. At first, people mainly focused on the structural stability of the solutions of various systems of partial differential equations in bounded domains (see ^{[8,9,10,11,12,13,14]}). Later, many scholars extended the study of structural stability to the case that there are two kinds of interface links in a bounded region (see ^{[15,16,17,18]}). Li et al. ^[19,20] considered the structural stability of Brinkman-Forechheimer equations and the thermoelastic equations of type III on a three-dimensional semi-infinite cylinder, respectively. The generatrix of the cylinder was parallel to the coordinate axis. However, the stability of partial differential equations on a two-dimensional pipe has not received enough attention. It is especially emphasized that the generatrix of the pipe considered in this paper is not parallel to the coordinate axis.

We investigate the following double-diffusive Darcy flow of a fluid through a porous medium in $R$ which can be written as (see ^[21])

where $\alpha = 1, 2$. $u_\alpha, p, T$ and $C$ represent the velocity, pressure, temperature, and concentration of the flow, respectively. $g_\alpha$ and $h_\alpha$ are bounded functions. $\sigma > 0$ is the Soret coefficient. For simplicity, we assume $\boldsymbol{g}$ and $\boldsymbol{h}$ satisfy $|\boldsymbol{g}|, \ |\boldsymbol{h}|\leq1.$ In this paper, we also use the summation convention summed from 1 to 2, and a comma is used to indicate differentiation. e.g., $u_{\alpha, \beta}u_{\alpha, \beta} = \sum_{\alpha, \beta = 1}^2\Big(\frac{\partial u_\alpha}{\partial x_\beta}\Big)^2$.

The initial-boundary conditions can be written as

where $T_0$ and $C_0$ are given functions. $F_\alpha$, $H$ and $\widetilde{H}$ are differentiable functions which are assumed to satisfy the appropriate compatibility conditions

Now, we let $v(x_1, x_ 2, t)$ denote a stream function which satisfies

Equations (1.1)–(1.8) can be converted to

where $v_n$ is the outward normal derivative of $v$, $\boldsymbol{g} = (g_1, g_2), \ \boldsymbol{h} = (h_1, h_2)$ and $\nabla^\bot = (\partial_{x_2}, -\partial_{x_1})$.

In the next section, we give several lemmas that have been derived in the literature. In Section 3, we derive an important lemma that can be used to derive our main result. In Section 4, we obtain the convergence result on the Soret coefficient and give some concrete examples. Section 5 shows the summary and outlook of this paper.

2.   Preliminary

We also introduce the notations

where $z$ is a running variable along the $x_1$ axis.

Here are some lemmas that will be often used in this paper.

Lemma 2.1 (see ^[5,22]) If $w(x_1, 0) = w(x_1, h) = 0$ and $w_n(x_1, 0) = w_n(x_1, h) = 0,$ then the following Wirtinger type inequality holds

where $z > 0$ is a moving point on the $x_1$ axis.

Lemma 2.2 If $\varphi(x_1, 0) = \varphi(x_1, h) = 0$ and $\varphi\rightarrow0,$ as $x_1\rightarrow \infty$, then

Proof. Since $\varphi(x_1, 0) = \varphi(x_1, h) = 0,$ we have

Since $\varphi\rightarrow0,$ as $x_1\rightarrow \infty$, we have

Combining Eqs (2.1) and (2.2) and integrating over $R_z$, we obtain

Using Young's inequality, we can obtain Lemma 2.2.

Using a similar method of papers ^[9,23,24], we can have the following lemma.

Lemma 2.3 Assume that $T_0, H\in L^\infty$, then

where $T_m = \max\Big\{||T_0||_\infty, \ \sup_{[0, \tau]}H_\infty(\eta)\Big\}.$

If $C_0, \widetilde{H}\in L^\infty$ and $\sigma = 0$ in (1.4), we can also have

where $C_m = \max\Big\{||C_0||_\infty, \ \sup_{[0, \tau]}\widetilde{H}_\infty(\eta)\Big\}.$

To obtain our main result, we shall use the following result which can be written as follows.

Lemma 2.4 (see ^[3]) Let $(v, T, C)$ be a solution of the Eqs (1.1)–(1.10) in $R$ and $\forall\ z\geq a$ such that $F(z, t) < 0$. Then for any fixed $t$

holds, where $\beta_1, \beta_2, \omega, m_1, m_2$ and $m_3$ are positive constants which depends on the middle parameter $\sigma$ and boundary conditions of the equation; also $F(z, t)$ has been defined as

Remark 2.1 Lemma 2.4 shows that the solution of Eqs (1.11)–(1.21) decays exponentially with $z\rightarrow \infty$. Only in this case, the study of structural stability is meaningful. Lemma 2.4 will also provide a priori bounds for the estimate of nonlinear terms (see e.g., (3.55) and (3.56)).

Remark 2.2 Li and Chen ^[3] considered several special cases of $h(z)$, e.g., $h(z) = h$ a constant, $h(z)\leq k_1z^{\tau_1}$ and $h(z)\leq k_2z(lnz)^{\tau_2}$, where $k_1, k_2 > 0$ and $0 < \tau_1, \tau_2\leq1.$ In the fourth section, we will also consider several special cases with $h(z) = h$ as a constant, $\tau_1 = \frac{1}{2}, k_1 = 1$ and $\tau_1 = \frac{2}{3}, k_1 = 1$.

Lemma 2.5 (see ^[3]) Assume that $(v, T, C)$ are solutions of Eqs (1.1)–(1.10). If $F(z, t) < 0$ for any $z\geq a$, then

where $r(t)$ is a positive known function which depends only on $t$.

Now, we derive a bound of $\int_Rv_{, \alpha}v_{, \alpha}dx_2d\xi$.

Lemma 2.6 Assume that $\widetilde{F}_1, \widetilde{F}_2, H, \widetilde{H}\in L^\infty(R)$, then

where

Additionally, $r^*$ is the maximum value of $r(t)$ in $[0, \tau]$.

Proof. Using Eq (1.11), we have

Using Eqs (1.18)–(1.20) it follows that

Using Lemma 2.5 in Eq (2.5), we can get Lemma 2.6.

3.   Important lemma

In this section, we derive the convergence result when the Soret coefficient $\sigma\rightarrow0$. To do this, we let $(v, T, C)$ be the solution of Eqs (1.11)–(1.21). Furthermore, let $(v^*, T^*, C^*)$ be the solution to the following equations

Remark 3.1 We note that Lemmas 2.4 and 2.5 also hold for $(v^*, T^*, C^*)$.

Now, we let

Then $(w, \theta, \Sigma)$ satisfies

We establish the following auxiliary functions

where $\omega$ is an arbitrary positive constant.

Using the divergence theorem and Eqs (3.12)-(3.21), we have

Similarly, we have

and

We also define

where $\delta_2$ and $\delta_3$ are positive constants.

Using the Cauchy-Schwarz inequality, we have

Inserting Eqs (3.29) and (3.30) into Eq (3.28) and choosing $\omega = \max\{\frac{4}{\delta_3}, \frac{4}{\delta_2}\}$, we have

From Eq (3.28), we have

Using the Cauchy-Schwarz inequality, we have

Inserting Eqs (3.33) and (3.34) into Eq (3.32), we have

and

Based on Eqs (3.22)–(3.24) and using Eq (3.36), we have the following lemma.

Lemma 3.1. Assume that $T_0, C_0, H, \widetilde{H}\in C^\infty$, $\frac{1}{h(z)}\in C(R)$ and the function $E(z, t)$ is defined in Eq (3.28). Then $E(z, t)$ satisfies

where $n_1$ is a positive constant.

Proof. Using the Hölder inequality, Young's inequality and Lemma 2.1, we have

Combining Eqs (3.22) and (3.38)–(3.40), we obtain

Using the Hölder inequality, Young's inequality, Lemmas 2.1, 2.3, 2.5 and Eq (2.5), we have

Using Lemma 2.3, we have

Inserting Eqs (3.42) and (3.45) into Eq (3.23), we obtain

Similarly, we have

and

Inserting Eqs (3.47)–(3.52) into Eq (3.24), we obtain

Now, inserting Eqs (3.41), (3.46) and (3.53) into Eq (3.28), choosing $\delta_2 < \frac{1}{2 T_m^2}, \delta_3 < \frac{1}{4C_m^2}$ and noting Eqs (3.36) and (3.31), we obtain

where

From Lemma 2.4, we have

Integrating Eq (3.55) from $z$ to $\infty$, we get

Combining Eqs (3.54)–(3.56), we can obtain Lemma 3.1.

4.   Convergence results on the Soret coefficient

Based on Lemma 3.1, we derive our main results in this section. To do this, integrating Eq (3.37) from $a$ to $z$, we obtain

To get the convergence results on the Soret coefficient, we have to derive the upper bound for $E(a, t)$. To do this, we choose $z = a$ in Eq (3.37) to get

By differentiating Eqs (3.22)–(3.24), then choosing $z = a$ and using Eqs (3.19) and (3.20), it can be obtained that

Using the Hölder inequality, Young's inequality and Lemmas 2.3 and 2.5, we have

On the other hand, we choose $z = a$ in Eq (3.36) to get

Inserting Eqs (4.4)–(4.6) into Eq (4.3) and noting Eq (4.7), we get

or

Inserting Eq (4.8) into Eq (4.2), we have

where $n_2 = n_1(h(a)+h^\frac{3}{2}(a))\frac{4}{\beta_2}\delta_3r(t)+\frac{4\delta_3}{\beta_3\beta_2\omega}m_1 +\frac{4\delta_3}{\beta_3\beta_2\omega}m_2$.

Now, inserting Eq (4.9) into Eq (4.1) and in light of Eq (3.31), we can obtain the following theorem.

Theorem 4.1 Letting $(w, \theta, \Sigma)$ is solution of Eqs (3.12)–(3.21) with $T_0, C_0, H, \widetilde{H}\in C^\infty$, then

Specifically

Next, we give some examples.

Remark 4.1 If $h(z) = h$ is a positive constant, then

Therefore, we have

Choosing $n_1$ such that $\frac{1}{n_1}\neq2m_3$ and $\frac{1}{n_1}\neq m_3$, then

and

Inserting Eqs (4.11)–(4.15) into Eq (4.10), we get

where

From Eq (4.16), we can conclude that Theorem 4.1 not only shows the convergence of the solutions of Eqs (3.12)–(3.21) on the coefficient $\sigma$, but it also shows a exponentially decay result as $z\rightarrow \infty$.

Remark 4.2. If $h(z) = \sqrt{z}$, then

Therefore

Moreover, we also have

and

Inserting Eqs (4.19)–(4.22) into Eq (4.10), we obtain

where

From Eq (4.23), we can conclude that Theorem 4.1 not only shows the convergence of the solutions of Eqs (3.12)–(3.21) on the coefficient $\sigma$, but it also shows a exponentially decay result as $z\rightarrow \infty$. Obviously, the decay rate is slightly slower than that in Remark 3.1.

Remark 4.3. If $h(z)$ satisfies

then

Therefore, we have

Choosing $n_1 > 3$, we get

and

Inserting Eqs (4.27)–(4.30) into Eq (4.10), we obtain

where

In this case, the inequality (Eq 4.31) not only shows the convergence of the solutions of Eqs (3.12)–(3.21) on the coefficient $\sigma$, but it also shows a exponentially decay result as $z\rightarrow \infty$.

5.   Conclusions

In this paper, the convergence of the solutions of Eqs (3.12)–(3.21) on the coefficient $\sigma$ has been obtained and three examples have been given. Obviously, the convergence of various systems of partial differential equations defined on $R$ is rare. But Eq (1.1) is linear. It will be meaningful to study nonlinear equations (e.g., Brinkman equations, Forchheimer equations) by using the method in this paper.

Acknowledgments

The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work is supported by the Tutor System Rroject of Guangzhou Huashang College (2021HSDS16).

Conflict of interest

All authors declare no conflicts of interest in this paper.