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Structural stability for the Forchheimer equations interfacing with a Darcy fluid in a bounded region in \(\mathbb{R}^{3}\)
Boundary Value Problems volume 2021, Article number: 46 (2021)
Abstract
The structural stability for the Forchheimer fluid interfacing with a Darcy fluid in a bounded region in \(\mathbb{R}^{3}\) was studied. We assumed that the nonlinear fluid was governed by the Forchheimer equations in \(\Omega _{1}\), while in \(\Omega _{2}\), we supposed that the flow satisfies the Darcy equations. With the aid of some useful a priori bounds, we were able to demonstrate the continuous dependence results for the Forchheimer coefficient λ.
1 Introduction
Many papers in the literature studied the structural stability for the partial differential equations. They obtained the results of continuous dependence or convergence on the equations. Unlike the traditional stability study, they focused on the changes of the coefficients of the equations. This is to say, the structural stability mainly focuses on changes in the model itself, while the traditional stability focuses on the initial data. For a review of the nature of the structural stability, one could see the monograph of Ames and Straughan [3]. In continuum mechanics problems, it is important to obtain the continuous dependence result on the model itself. This problem is discussed for several different partial differential equations by Hirsch and Smale [8]. We usually want to know if a small change in the constructive coefficient in the equations themselves will lead to drastic changes in the solutions. If the answer is no, we can do further studies. It is very important for us to study the structural stability for the model.
There are many models that have been studied in a porous medium. Nield and Beijan [14] and Straughan [27, 28] discussed these models in their books. The authors of [2, 16, 17]studied these models in an unbounded domain and obtained some Saint-Venant-type results. They mainly focused on the studies of the Brinkman, Darcy, and Forchheimer equations in porous media.
Recently, some authors began to study the structural stability for equations in porous media. They obtained some continuous dependence results. For a review of these papers, one could see Payne and Straughan [19–22], Scott [23], Scott and Straughan [24], Straughan [26], Ames and Payne [1], Celebi, Kalantarov and Ugurlu [4, 5], Franchi and Straughan [6], Harfash [7], Kaloni and Guo [9], Li and Lin [10], Lin and Payne [11, 12], Payne, Song and Straughan [18], and Straughan and Hutter [30]. The Brinkman, Forchheimer, and Darcy equations are widely studied in these papers. They consider only one fluid in the domain. In reality, there typically exists more than one fluid in a domain. It is interesting to study two fluids interfacing with each other in one domain.
In [21], Payne and Straughan established the structural stability result for the Brinkman–Darcy interfacing equations. They studied the continuous dependence result for the interface boundary coefficient \(\alpha _{1}\). We change the Brinkman equations to the Forchheimer equations. However, if we use the same method as in [21], we cannot obtain a similar result. Since the equations do not contain the term △u, it is difficult to deal with the nonlinear term \(|u|u_{i}\). Recently, in [13] and [25], the authors studied the structural stability for the Forchheimer–Darcy interfacing problems in a bounded domain. In order to obtain their results, the authors obtained the results \({\mathop{\sup }_{ [ {0,\tau } ]} {{ \Vert T \Vert }_{\infty }}} \le T_{M} \) and \({\mathop{\sup }_{ [ {0,\tau } ]} {{ \Vert S \Vert }_{\infty }}} \le S_{M} \) for the temperatures T and S using the method proposed by Payne, Rodrigues, and Straughan in [15]. In the present paper, the equations for the temperatures are not the same as in [13] and [25]. We cannot get the same results by using the method proposed in [15]. We must seek a new method to get the results. How to get the maximum estimates and the related bounds for T and S is the biggest innovation of this paper. In our opinion, it is of great significance to study the structural stability for the Forchheimer–Darcy interfacing fluids.
The purpose of this paper is to study the manner in which a solution to a flow in a fluid which borders a porous medium depends on a coefficient in the Forchheimer equation. Thus, let an appropriate part of the plane \(z=x_{3}=0\) denote the boundary between a porous medium occupying a bounded region \(\Omega _{2}\) in \(\mathbb{R}^{3}\) and a nonlinear fluid occupying a bounded region \(\Omega _{1}\) in \(\mathbb{R}^{3}\), and the governing equations be Forchheimer equations. We denote the interface by L, and further denote the remaining parts of the boundaries of \(\Omega _{1}\) and \(\Omega _{2}\) by \(\Gamma _{1}\) and \(\Gamma _{2}\). We also denote \(\partial \Omega _{1}=\Gamma _{1}\cup L\) and \(\partial \Omega _{2}=\Gamma _{2}\cup L\).
We are interested in the solution of the following initial-boundary value problem. The governing equations for Forchheimer flow are (see [29])
where \({u_{i}}\), p, and T are the velocity, pressure, and temperature, κ is the thermal diffusivity. Here \({g_{i}} ( x )\) are gravity vector functions, and \(Q ( x,t )\) is a prescribed heat source. We assume that \({g_{i}}\) satisfy \(\vert g \vert \le {G_{1}}\). Here also Δ is the Laplace operator.
Equations (1.1) hold in the region \({\Omega _{1}} \times [ {0,\tau } ]\), where \(\Omega _{1}\) is a bounded, simply connected, and star-shaped domain with boundary \(\partial {\Omega _{1}}\) in \(\mathbb{R}^{3}\), and τ is a given number satisfying \(0 \le \tau < \infty \).
The Darcy equations governing the flow are (see [27])
where \({v_{i}}\), q, and S are the velocity, pressure, and temperature, while \(Q_{s} ( x,t )\) is a prescribed heat source.
Equations (1.2) hold in the region \({\Omega _{2}} \times [ {0,\tau } ]\), where \(\Omega _{2}\) is a bounded, simply connected, and star-shaped domain with boundary \(\partial {\Omega _{2}}\) in \(\mathbb{R}^{3}\), and τ is a given number satisfying \(0 \le \tau < \infty \).
We impose the boundary and initial conditions as follows:
We assume further that
Finally, the interfacing conditions are taken from [21] as
on \(L \times \{ {t > 0} \} \).
In the next section, we will derive some a priori bounds which will be used in deriving our main results. In Sect. 3, the convergence results for the Forchheimer coefficient are obtained.
In this present paper, the comma is used to indicate differentiation, and the differentiation with respect to the direction \(x_{k}\) is denoted as “,k”, thus \({u_{,i}} \) denotes \(\frac{{\partial u}}{{\partial {x_{i}}}}\). Hence, \({u_{i,i}} = \sum_{i = 1}^{3} { \frac{{\partial {u_{i}}}}{{\partial {x_{i}}}}}\).
2 A priori bounds
We now begin to derive a priori bounds for both T and S.
First, we introduce the function H, which takes the same boundary values as T:
Next, we introduce the function I, which takes the same boundary values as S:
On \(L \times \{ {t > 0} \} \),we let
If we let
we get
If we let
we know by maximum principle that \(\vert F \vert \le {F_{M}} \).
The following lemmas will be used in deriving our main result.
Lemma 1
For the temperatures T and S, we have the following estimates:
Proof
Multiplying (1.1)3 by \(2(T - H ) \) and integrating over \({\Omega _{1}} \times [ {0,t} ]\), we find
For the first function on the left-hand side of (2.8), using the divergence theorem and equations (1.4), (2.1), we find
For the second function on the left-hand side of (2.8), we have
For the first function on the right-hand side of (2.8), using the divergence theorem and equations (1.3), (1.5), and (2.1), we get
For the second function on the right-hand side of (2.8), we get
For the third function on the right-hand side of (2.8), using equations (1.4) and (2.1), we find
Combining (2.8)–(2.13), we obtain
Similarly, we get
Combining (2.14) and (2.15), we can get the desired result (2.7). □
Lemma 2
If
we have
where m and d are positive constants to be defined later.
Proof
Start with the identity
For the first function on the left-hand side of (2.17), using the divergence theorem and equations (2.1), (2.3), we get
For the fourth function on the right-hand side of (2.18), using the divergence theorem and equations (2.1), (2.3), we get
For the first function on the right-hand side of (2.17), we get
where \({d^{2}} = \mathop{\max }_{\Omega }{x_{i}}{x_{i}} \).
Combining (2.17)–(2.20), we obtain
Similarly, we get
Combining (2.21) and (2.22), we obtain
Since
where n and s are the normal and tangential vectors to ∂Ω, respectively, and \({\nabla _{s}}H \) and \({\nabla _{s}}I \) are the tangential derivatives, we have
We know Ω is star-shaped with respect to the region and, setting \(m = \mathop{\min }_{\partial \Omega } {x_{i}}{n_{i}} > 0 \), we then obtain
Multiplying (2.1)1 by \(2{H_{,t}} \) and integrating over \({\Omega _{1}} \), we find
Similarly, we get
Combining (2.26)–(2.28), we obtain
Therefore, integrating (2.29) yields
Gronwall inequality now implies (2.16). □
Lemma 3
For the functions H and I, we have the following estimates:
where \(m_{3}(t)=\frac{{{d^{2}}{m_{2}} ( t )}}{{2m}} + \frac{1}{2} ( {\int _{{\Omega _{1}}} {{H_{0,i}}{H_{0,i}}\,dx} + \int _{{\Omega _{2}}} {{I_{0,i}}{I_{0,i}}\,dx} } )+ \frac{m}{{2{d^{2}}}} ( \int _{0}^{t} {\int _{{\Gamma _{1}}} {{{ ( {\frac{{\partial H}}{{\partial n}}} )}^{2}}\,dS} \,d \eta } + \int _{0}^{t} {\int _{{\Gamma _{2}}} {{{ ( { \frac{{\partial I}}{{\partial n}}} )}^{2}}\,dS} \,d \eta } )\).
Proof
Multiplying (2.1)1 by \(2{H_{,t}} \) and integrating over \({\Omega _{1}} \), we find
Similarly, we get
Combining (2.26), (2.32), and (2.33), we obtain
Therefore, integrating (2.34) yields
Combining (2.32) and (2.33), we obtain
Therefore, integrating (2.36) yields the desired result (2.31). □
Lemma 4
For the functions H and I, we have the following estimates:
with \(m_{4}(t)=\int _{{\Omega _{1}}} {T_{0}^{2}\,dx} + \int _{{\Omega _{2}}} {S_{0}^{2}\,dx} + \int _{0}^{t} {\int _{{\Gamma _{1}}} {T_{U}^{2}\,dS} \,d \eta } + \int _{0}^{t} {\int _{{\Gamma _{2}}} {S_{U}^{2}\,dS} \,d \eta } + {m_{2}} ( t )\).
Proof
Multiplying (2.1)1 by 2H and integrating over \({\Omega _{1}} \), we find
Similarly, we get
Combining (2.38) and (2.39), we obtain
Therefore, integrating (2.40) yields the desired result (2.37). □
Lemma 5
For the temperatures T and S, we have the following estimates:
Proof
A combination of (2.7), (2.16), (2.31), and (2.37) leads to the desired result (2.41). □
Lemma 6
For the solutions \((u_{i},T)\) and \((v_{i},S)\) of equations (1.1) and (1.2), if we let \({F_{2}} ( t ) = \int _{{\Omega _{1}}} {{T^{2}}\,dx} + \int _{{\Omega _{2}}} {{S^{2}}\,dx} + \int _{{\Omega _{1}}} {{u_{i}}{u_{i}}\,dx} \), \({m_{5}} = \max \{ {4 + G_{1}^{2} + \frac{4}{\kappa }F_{M}^{2}G_{1}^{2},2 + \frac{8}{\kappa }F_{M}^{2}} \} \), \({D_{2}} ( t ) = ( {1 + \frac{4}{\kappa }F_{M}^{2}} )\int _{{\Omega _{1}}} {{f_{i}}{f_{i}}\,dx} + {m_{4}} ( t ) + 2\int _{0}^{t} {{m_{4}} ( \eta )\,d \eta } + 2 \kappa \int _{0}^{t} {{m_{1}} ( \eta )\,d \eta } + 2{m_{3}} ( t ) + 4\int _{0}^{t} {\int _{{\Omega _{1}}} {Q ^{2}\,dx} \,d \eta } + 4\int _{0}^{t} {\int _{{\Omega _{2}}} {Q_{s}^{2}\,dx} \,d \eta } \), we get
and
where \({m_{7}} ( t ) = \frac{{G_{1}^{2} + 1 + \vert {G_{1}^{2} - 1} \vert }}{4}{m_{6}} ( t ) + \frac{1}{2}\int _{{\Omega _{1}}} {{f_{i}}{f_{i}}\,dx} \).
Proof
Multiplying (1.1)1 by \(2u_{i}\) and integrating over \(\Omega _{1}\), we find
For the second function on the right-hand side of (2.46), using the divergence theorem and equations (1.3), (1.5), we get
If we insert (1.2)1 and (2.47) into (2.46), we get
Therefore, integrating (2.48) yields
Similarly, we get
Combining (2.41), (2.49), and (2.50), we obtain
We can get
Gronwall inequality now implies the desired result (2.42). □
Similarly, we can also get the desired result (2.43).
Combining (2.51) and (2.42), we obtain the desired result (2.44).
Combining (2.50) and (2.42), we obtain the desired result (2.45).
Lemma 7
For the temperatures T and S, we have the following estimates:
Proof
Multiplying (1.1)3 by \(2r ( {{T^{2r - 1}} - {H^{2r - 1}}} ) \) and integrating over \({\Omega _{1}} \times [ {0,t} ] \), (where \(r>2\)), we find
For the first function on the right-hand side of (2.54), using the divergence theorem and equations (1.3), (1.5), we get
For the first function on the left-hand side of (2.54), using the divergence theorem and equations (1.4), (2.1), we find
For the second function on the left-hand side of (2.54), we get
For the third function on the left-hand side of (2.54), using Young inequality, we get
For the second function on the right-hand side of (2.54), using Young inequality and equations (1.4), (2.1), we find
Combining (2.54)–(2.59), we obtain
Similarly, we get
Combining (2.60) and (2.61), we get
Letting
we get
Gronwall inequality now implies
Raising to the power of \(\frac{1}{2r}\) both sides of (2.64), we have
From the definition of \(F_{3}(t)\), we have
Using the facts
and the equality
with \(a_{1},a_{2},\dots ,a_{p}\) all nonnegative constants, we can get the desired result (2.53). □
3 Continuous dependence results for the Forchheimer coefficient λ
In this section,we will discuss the continuous dependence on the Forchheimer coefficient λ. Let \(( {{u_{i}},T,p} )\) and \(( {{v_{i}},S,q} )\) be the solutions of (1.1)–(1.5) with \(\lambda = \lambda _{1} \) Similarly, we set \(( {u_{i}^{*} ,{T^{*} },{p^{*} }} )\) and \(( {v_{i}^{*} ,{S^{*} },{q^{*} }} )\) to be the solutions of (1.1)–(1.5) with \(\lambda = \lambda _{2}\).
We define \({\omega _{i}} = {u_{i}} - u_{i}^{*} \), \(\theta = T - {T^{*} }\), \(\pi = p - {p^{*} }\), \(\hat{\lambda }= {\lambda _{1}} - {\lambda _{2}}\), and \(\omega _{i}^{m} = {v_{i}} - v_{i}^{*} \), \({\theta ^{m}} = S - {S^{*} }\), \({\pi ^{m}} = q - {q^{*} }\).
We find that \(( {{\omega _{i}},\theta ,\pi } )\) satisfy the following equations:
and \(( {\omega _{i}^{m},{\theta ^{m}},{\pi ^{m}}} )\) satisfy
The boundary conditions are
and additionally the initial conditions are given at \(t=0\), i.e.,
The conditions on interface L ar
Theorem
Let \(( {{u_{i}},T,p} )\) and \(( {{v_{i}},S,q} )\) be the classical solutions to the initial-boundary value problem (1.1)–(1.5) with \(\lambda = \hat{\lambda }_{1} \), while \(( {u_{i}^{*} ,{T^{*} },{p^{*} }} )\) and \(( {v_{i}^{*} ,{S^{*} },{q^{*} }} )\) are the classical solutions to the initial-boundary value problem (1.1)–(1.5) with \(\lambda = {\hat{\lambda }_{2}}\). We define \(( {{\omega _{i}}, \theta ,\pi } )\) and \(( {\omega _{i}^{m},{\theta ^{m}},{\pi ^{m}}} )\) to be the differences of these two solutions, respectively. Then the solutions \(( {{u_{i}},T,p} )\) and \(( {{v_{i}},S,q} )\) converge to the solutions \(( {u_{i}^{*} ,{T^{*} },{p^{*} }} )\) and \(( {v_{i}^{*} ,{S^{*} },{q^{*} }} )\) as the Forchheimer coefficient λ̂ tends to 0. The differences of solutions satisfy
where \({m_{10}} = \max \{ {\frac{{N_{M}^{2}}}{\kappa }, \frac{{N_{M}^{2}G_{1}^{2}}}{{2\kappa }}} \} \), \({m_{11}} ( t ) = \frac{{N_{M}^{2}}}{{2\kappa {\lambda _{1}}{\lambda _{2}}}}{m_{7}} ( t ) + \frac{{N_{M}^{2}{m_{10}}}}{{2\kappa {\lambda _{1}}{\lambda _{2}}}}{e^{{m_{10}}t}} \int _{0}^{t} {{m_{7}} ( \eta ){e^{ - {m_{10}}\eta }}\,d \eta } \).
Moreover, the differences of velocities satisfy the following estimates:
where \({m_{12}} = \max \{ {G_{1}^{2},\frac{{2\kappa }}{{N_{M}^{2}}}} \} \).
Proof
Multiplying (3.1)1 by \({2\omega _{i}}\) and integrating over \({\Omega _{i}}\), we see
For the third function on the right-hand side of (3.8), using the divergence theorem and Eqs. (3.3), (3.5), we get
For the second function on the right-hand side of (3.8), we have
For the first function on the right-hand side of (3.8), we have
Combining (3.8)–(3.11), we have
In order to estimate \(\int _{{\Omega _{1}}} {\theta \theta \,dx} + \int _{{\Omega _{2}}} {{ \theta ^{\mathrm{{m}}}}{\theta ^{m}}\,dx} \), we multiply (3.1)3 by 2θ and get
For the first function on the right-hand side of (3.13), using the divergence theorem and Eqs. (3.3), (3.5), we get
For the second function on the right-hand side of (3.13), using the divergence theorem and Eqs. (1.3), (3.5), we get
For the third function on the right-hand side of (3.13), using the divergence theorem and Eqs. (1.5), (3.3), and (3.5), we get
Combining (3.13)–(3.16), we get
Similarly, we multiply (3.2)3 by \(2\theta ^{m}\), we have
Combining (3.17) and (3.18), we have
Combining (3.12) and (3.19), we have
If we let \({F_{4}} ( t ) = \int _{{\Omega _{1}}} {{\theta ^{2}}\,dx} + \int _{{\Omega _{2}}} {{{ ( {{\theta ^{\mathrm{{m}}}}} )}^{2}}\,dx} + \frac{{N_{M}^{2}}}{{2\kappa }}\int _{{\Omega _{1}}} {{\omega _{i}}{ \omega _{i}}\,dx} \), \({m_{10}} = \max \{ {\frac{{N_{M}^{2}}}{\kappa }, \frac{{N_{M}^{2}G_{1}^{2}}}{{2\kappa }}} \} \).
Therefore, integrating (3.20) yields
Gronwall inequality implies
where \({m_{11}} ( t ) = \frac{{N_{M}^{2}}}{{2\kappa {\lambda _{1}}{\lambda _{2}}}}{m_{7}} ( t ) + \frac{{N_{M}^{2}{m_{10}}}}{{2\kappa {\lambda _{1}}{\lambda _{2}}}}{e^{{m_{10}}t}} \int _{0}^{t} {{m_{7}} ( \eta ){e^{ - {m_{10}}\eta }}\,d \eta } \).
Inserting (3.22) into (3.12), we have
where \({m_{12}} = \max \{ {G_{1}^{2},\frac{{2\kappa }}{{N_{M}^{2}}}} \} \). □
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Acknowledgements
The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions.
Funding
The work was supported National Natural Science Foundation of China (Grant ♯ 61907010), Natural Science Foundation in Higher Education of Guangdong, China (Grant ♯ 2018KZDXM048; ♯ 2019KZDXM036; ♯ 2019KZDXM042; ♯2020ZDZX3051), the General Project of Science Research of Guangzhou (Grant ♯ 201707010126), and the science foundation of Huashang College Guangdong University of Finance & Economics (Grant ♯ 2019HSDS28).
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Shi, J., Liu, Y. Structural stability for the Forchheimer equations interfacing with a Darcy fluid in a bounded region in \(\mathbb{R}^{3}\). Bound Value Probl 2021, 46 (2021). https://doi.org/10.1186/s13661-021-01525-6
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DOI: https://doi.org/10.1186/s13661-021-01525-6