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1.   Introduction

The Forchheimer model equation describes the flow in a polar medium, which is widely used in fluid mechanics (see ^[1,2]). Increasing scholars have studied the spatial properties of the solution to the fluid equation defined on a semi-infinite cylinder, and a large number of results have emerged (see ^{[3,4,5,6,7,8,9]}).

In 2002, Payne and Song ^[3] have studied the following Forchheimer model

where $i = 1, 2, 3$. $u_i, p, T$ represent the velocity, pressure, and temperature of the flow, respectively. $g_i$ is a known function. $\Delta$ is the Laplace operator, $\gamma > 0$ is a constant, and $b$ is the Forchheimer coefficient. For simplicity, we assume that

In (1.1)–(1.3), $\Omega$ is defined as

where $D$ is a bounded simply-connected region on $(x_1, x_2)$-plane.

In this 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., $u_{i, j}u_{i, j} = \sum_{i, j = 1}^3\Big(\frac{\partial u_i }{\partial x_j}\Big)^2$. We also use the summation convention summed from 1 to 2, e.g., $u_{\alpha, \beta}u_{\alpha, \beta} = \sum_{\alpha, \beta = 1}^2\Big(\frac{\partial u_\alpha}{\partial x_\beta}\Big)^2$.

The Eqs (1.1)–(1.3) also satisfy the following initial-boundary conditions

where $f_i$ and $H$ are differentiable functions.

In this paper, we will study the structural stability of Eqs (1.1)–(1.8) on $\Omega$ by using the spatial decay results obtained in ^[3]. Since the concept of structural stability was proposed by Hirsch and Smale ^[10], the structural stability of various types of partial differential equations defined in a bounded domain has received sufficient attention(see ^{[11,12,13,14,15,16,17,18,19]}). Some perturbations are inevitable in the process of model establishment and simplification, so it is necessary to study that whether such small perturbations of the equations themselves will cause great changes in the solutions. This gives rise to the phenomenon of structural stability.

If the bounded domain is replaced by a semi-infinite pipe, the structural stability of the partial differential equations is very interesting and has begun to attract attention. Li and Lin ^[20] considered the continuous dependence on the Forchheimer coefficient of Forchheimer equations in a semi-infinite pipe. Different from the studies of^{[11,12,13,14,15,16,17,18,19]}, we should consider not only the time variable but also the space variable. Therefore, the methods in the literature cannot be directly applied to the semi-infinite region. Compared with ^[3], we not only reconfirmed the spatial decay result of ^[3], but also proved the structural stability of the solution to $b$ and $\gamma$.

We also introduce the notations:

where $z$ is a running variable along the $x_3$ axis.

2.   A prior bounds

First, to obtain the main result, we shall make frequent use of the following three inequalities.

Lemma 2.1.(see^[21]) If $\phi$ is a Dirichlet integrable function on $\Omega$ and $\int_\Omega\phi dx = 0$, then there exists a Dirichlet integrable function $\textbf{w} = (w_1, w_2, w_3)$ such that

and a positive constant $k_1$ depends only on the geometry of $\Omega$ such that

Lemma 2.2.(see ^[3,4]) If $\phi\Big|_{\partial D} = 0,$ then

where $\lambda$ is the smallest positive eigenvalue of

Here $\Delta_2$ is a two-dimensional Laplace operator.

Now, we give a lemma which has been proved by Horgan and Wheeler ^[4] and has been used by Payne and Song ^[6].

Lemma 2.3.(see ^[3,4]) If $\phi$ is a Dirichlet integrable function and $\phi\Big|_{\partial D} = 0, \phi\rightarrow \infty$ (as $x_3\rightarrow \infty$),

where $k_2 > 0$.

Lemma 2.4. If $\phi\in C_0^1(\Omega)$, then

where ^[22,23] have proved that the optimal value of $\Lambda$ is determined to be $\Lambda = \frac{1}{27}\Big(\frac{3}{4}\Big)^4$.

Using the maximum principle for the temperature $T$, we can have the following lemma which has been used in Song ^[5].

Lemma 2.5. Assume that $H\in L^\infty(\Omega)$, then

where $T_M = \sup_{\Omega\times\{t > 0\}}H$.

Second, we list some useful results which have been derived by Payne and Song ^[3].

Payne and Song have established a function

where $a_1$ and $a_2$ are positive constants. From Eqs (3.27) and (3.36) of ^[3], we know that

where $k_3$ is a positive constant and $k_4(t)$ is a function related to the boundary values.

Combining Eqs (2.1) and (2.2), we have the following lemma.

Lemma 2.6. Assume that $H\in L^\infty(\Omega)$ and $\int_DfdA = 0$, then

In order to derive the main result, we need bounds for $||\textbf{u}||_{L^2(\Omega)}^2$ and $||\textbf{u}||_{L^2(\Omega)}^3$.

Lemma 2.7. Assume that $f_i\in H^1(\Omega), H, \widetilde{H}\in L^\infty(\Omega)$, $\int_Df_3dA = 0$ and $f_{\alpha, \alpha}-\gamma f_3 = 0$ then

where $k_6(t)$ is a positive function.

Proof. To deal with boundary terms, we set $\textbf{S} = (S_1, S_2, S_3)$, where

Using Eq (1.1), we have

Using the divergence theorem, we have

Using the Hölder inequality and Young's inequality, we have

Inserting Eqs (2.5)–(2.8) into Eq (2.4) and choosing that $\varepsilon_1 = \frac{3}{4}$, we obtain

After choosing

we can complete the proof of Lemma 2.7.

3.   Important lemma

In this section, we derive an important lemma which leads to our main result.

Assume that $(u_i^*, T^*, p^*)$ is a solution of Eqs (1.1)–(1.8) when $b = b^*$. If we let

then $(\mathcal{D}_i, \Sigma, \pi)$ satisfies

We can have the following lemma.

Lemma 3.1. Assume that $(\mathcal{D}_i, \Sigma, \pi)$ is a solution to Eqs (3.1)–(3.6) with $\int_Df_3dA = 0, H\in L^\infty(\Omega)$ and the boundary data (e.g., $H$) satisfies Eq (3.21), then

where $n_6^*$ is the maximum of $n_6(t)$ and $n_6(t), n_7(t)$ will be defined in Eq (3.39).

Proof. We define an auxiliary function

where $\omega > 0$.

Using the divergence theorem and Eq (3.1), we have

Since

from Eq (3.9) we have

Using the Hölder inequality, Young's inequality and Lemma 2.6, we obtain

Using the Hölder inequality, Young's inequality and Lemmas 2.3 and 2.7, we obtain

Calculating the differential of Eq (3.10) and then inserting Eqs (3.11)–(3.13) into Eq (3.10), we have

where we have dropped the fourth term of Eq (3.10).

Similarly, we have

Using the divergence theorem and Eq (3.2), we have

Using the Hölder inequality and Lemma 2.5, we have

Calculating the differential of Eq (3.16) and then inserting Eq (3.17) into Eq (3.16), we have

Now, we define

Combining Eqs (3.14) and (3.18), we have

Choosing $\omega > \frac{4}{\gamma^2}T_M^2$ and the boundary data (e.g., $H$) satisfies

from Eq (3.20) we obtain

Integrating Eq (3.22) from $z$ to $\infty$, we obtain

We note that

According to Lemma 2.1, there exists a vector function $\textbf{w} = (w_1, w_2, w_3)$ such that

Therefore, using Eq (3.1) we obtain

Since

we have

Using the Hölder inequality, Lemmas 2.2, 2.3, 2.1, 2.7 and 2.6, and Young's inequality, we obtain

Using the Hölder inequality, Lemmas 2.3, 2.1 and 2.7, and Young's inequality, we obtain

Using the Hölder inequality, Lemmas 2.4, 2.1 and 2.7, and Young's inequality, we obtain

Inserting Eqs (3.26)–(3.28) into Eq (3.25), we have

Using the Hölder inequality, Young's inequality and Lemmas 2.5, 2.2, 2.1, 2.7 and 2.3, we have

Inserting Eqs (3.29)–(3.32) into Eq (3.24), we obtain

where

Now, we begin to derive a bound of $\Phi_2(z, t)$ which has been defined in Eq (3.15). Using the Hölder inequality, Young's inequality, Lemmas 2.7 and 2.3, we have

Inserting Eqs (3.34)–(3.36) into Eq (3.15), we obtain

Combining Eqs (3.19), (3.22), (3.33) and (3.37), we obtain

where

4.   Continuous dependence on the coefficient $b$

In this section, we will analysis Lemma 3.1 to derive the following theorem.

Theorem 4.1. Let $(u_i, T, p)$ and $(u_i^*, T^*, p^*)$ be solutions of the Eqs (1.1)–(1.8) in $\Omega$, corresponding to $b_1$ and $b_2$, respectively. If $\int_Df_3dA = 0$, Equation (3.21) holds and $f_{\alpha, \alpha}-\gamma_1f_3 = 0, H\in L^\infty(\Omega\times\{t > 0\})$, then

Specifically, either the inequality

holds, or the inequality

holds.

Proof. Using Lemma 3.1, we have

Now, we consider (4.1) for two cases.

Ⅰ. If $n_6^* = k_3$, we integrate Eq (4.1) from $0$ to $z$ to obtain

Ⅱ. If $n_6^*\neq k_3$, we integrate Eq (4.1) from $0$ to $z$ to obtain

From Eqs (4.2) and (4.3), to obtain the main result, we can conclude that we have to derive a bound for $\Phi(0, t)$. We choose $z = 0$ in Lemma 3.1 to obtain

Clearly, if we want to derive a bound for $\Phi(0, t)$, we only need derive a bound for $-\frac{\partial \Phi}{\partial z}(0, t)$. To do this, choosing $z = 0$ in Eq (3.19) and combining Eqs (3.8) and (3.15), we have

In light of the boundary conditions (3.4)–(3.6), from Eq (4.5) we can know that

Inserting Eq (4.6) into Eq (4.4), we obtain

Therefore, from Eqs (4.2), (4.3) and (4.7) we have

Combining Eqs (3.24), (4.8) and (4.9) we can complete the proof of Theorem 4.1.

Remark 4.1 Theorem 1 shows that the small perturbation of Forchheimer coefficient will not cause great changes to the solution of Eqs (1.1)–(1.8). Meanwhile, Theorem 1 also shows that the solutions of Eqs (2.12)–(2.21) decay exponentially as the space variable $z\rightarrow \infty$.

5.   Continuous dependence on the coefficient $\gamma$

This section shows how to use the prior estimates in Section 2 and the method in Section 3 to derive the continuous dependence of the solution on $\gamma$. Assume that $(u_i^*, T^*, p^*)$ is a solution of Eqs (1.1)–(1.8) with $\gamma = \gamma^*$.

If we also let

then $(\mathcal{D}_i, \Sigma, \pi)$ satisfies

We also define $\Phi_1(z, t)$ as that in Eq (3.8). Similar to Eq (3.10), we have

Using the Hölder inequality, Young's inequality and Lemmas 2.5 and 2.6, we obtain

Combining Eqs (3.12), (3.13), (5.8) and (5.9), we obtain

Inserting Eqs (3.18) and (5.10) into Eq (3.19), choosing $\omega > \frac{4T_M^2}{\gamma^2}$ and the boundary data satisfies

we have

Integrating Eq (5.12) from $z$ to $\infty$, we obtain

Similar to the calculation in Eqs (3.33) and (3.37), we can get

for $n_6'(t), n_7'(t) > 0$.

After similar analysis as in the previous section, we can get the following theorem from Eq (5.14).

Theorem 5.1. Let $(u_i, T, p)$ and $(u_i^*, T^*, p^*)$ be solutions of the Eqs (1.1)–(1.8) in $\Omega$, corresponding to $b_1$ and $b_2$, respectively. If $\int_Df_3dA = 0$, Equation (5.11) holds and $f_{\alpha, \alpha}-\gamma_1f_3 = 0, H\in L^\infty(\Omega\times\{t > 0\})$, then

Specifically, either the inequality

holds, or the inequality

holds.

6.   Conclusions

In this paper, using a priori estimates of the solutions, we show how to control the nonlinear term, and obtain the structural stability of the solution of the Forchheimer equation in a semi-infinite cylinder. Meanwhile, the spatial decay results of the solution are also obtained. The methods in this paper can bring some inspiration for the structural stability of other nonlinear partial differential equations.

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 (2021HSDS13) and the Key projects of universities in Guangdong Province (NATURAL SCIENCE) (2019KZDXM042).

Conflict of interest

The authors declare there is no conflict of interest. Conceptualization, and validation, Z. Li.; formal analysis, Z W. Zhang; investigation, Y. Li. All authors have read and agreed to the published version of the manuscript.