Continuous Dependence on the Forchheimer Coefficient of the Forchheimer Fluid Interfacing with a Darcy Fluid

In this paper, we study two fluids in one bounded domain when they interface with each other. We want to know what effect they can give to each other. Let an appropriate part of the plane z = x3 = 0 denotes the boundary between a porous medium which occupies a bounded region Ω2 ⊆R3 and a nonlinear viscous fluid which occupies a bounded region Ω1 ⊆R3. The interface is denoted by L. The remaining parts of the boundaries of Ω1 and Ω2 are denoted, respectively, by Γ1 and Γ2. We also denote ∂Ω1 = Γ1 ∪ L and ∂Ω2 = Γ2 ∪ L. We also note that Ω1 is above the plane z = x3 = 0 and Ω2 is below the plane z = x3 = 0. Let ðui, T , pÞ and ðvi, θ, qÞ denote the velocity, temperature, and pressure in Ω1 and Ω2, respectively. The Forchheimer system consists of the partial differential equations (see [1])


Introduction
In this paper, we study two fluids in one bounded domain when they interface with each other. We want to know what effect they can give to each other. Let an appropriate part of the plane z = x 3 = 0 denotes the boundary between a porous medium which occupies a bounded region Ω 2 ⊆ ℝ 3 and a nonlinear viscous fluid which occupies a bounded region Ω 1 ⊆ ℝ 3 . The interface is denoted by L. The remaining parts of the boundaries of Ω 1 and Ω 2 are denoted, respectively, by Γ 1 and Γ 2 . We also denote ∂Ω 1 = Γ 1 ∪ L and ∂Ω 2 = Γ 2 ∪ L. We also note that Ω 1 is above the plane z = x 3 = 0 and Ω 2 is below the plane z = x 3 = 0.
Let ðu i , T, pÞ and ðv i , θ, qÞ denote the velocity, temperature, and pressure in Ω 1 and Ω 2 , respectively. The Forchheimer system consists of the partial differential equations (see [1] ∂T ∂t where g i is the gravity force function. The coefficient b is a positive constant which is named as the Forchheimer coefficient. The viscosity variation in (1) is accounted for by the term 1 + γT, i.e., we are considering a viscosity μ like μ = μ 1 ð1 + γTÞ, γ > 0. The Darcy equations are (see Nield and Bejan [2]) ∂θ ∂t where Ω 1 and Ω 2 are the bounded, simply connected, and star-shaped domains and τ is a given number satisfying 0 ≤ τ < ∞. We impose the following boundary conditions: for prescribed functions f i , T 0 , and θ 0 . On the interface L, the conditions are The purpose of this paper is to study the continuous dependence on the coefficients of problems (1)- (5). This type of stability is often called structural stability to distinguish it from continuous dependence on the initial data, on the boundary data, or even on the partial differential equation themselves. In continuum mechanics problems, it is necessary to be able to establish continuous dependence on the model; this is discussed in terms of differential equations by Hirsch and Smale [3]. Such stability estimates are fundamental in that one wishes to know if a small change in a coefficient in an equation or boundary data, or a small change in the equations themselves, will lead to a drastic change in the solution. When we study the continuous dependence or convergence, structural stability expresses the changes in the model itself rather than the original data. Many references to work of this nature are discussed in the monograph of Ames and Straughan [4] and the monograph of Straughan [5].
In this paper, we derive an a priori convergence result which compares the solution to the Forchheimer system of partial differential equations with that of the Darcy equations. The purpose of this paper is to study the continuous dependence of a solution to the Forchheimer system to a solution to Darcy equations on the Forchheimer coefficient b and the effective viscosity coefficient γ. Different from [14,26], there are two nonlinear terms γTu i and |u | u i and there is no Laplace term in (1). So, some Sobolev inequalities do not hold for our problem. This will bring great difficulty. To get our result, we must seek a new method to overcome the difficulty. In the next section, we derive a number of a priori bounds which will be used in establishing the continuous dependence result in Section 3.

A Priori Bounds
In this section, we want to drive bounds for the various norms of u, v, T, and θ.
2.1. Bounds for kuk 3 L 3 ðΩ 1 Þ , kuk 2 L 2 ðΩ 1 Þ , and kvk 2 L 2 ðΩ 2 Þ . Multiplying ð1:1Þ 1 with u i and integrating over Ω 1 , we obtain b u k k 3 Integrating by parts and using Young's inequality and the arithmetic-geometric mean inequality now lead to b u k k 3 where g 2 = max g i g i : By using the divergence theorem and Equation (2), we get 2b u k k 3 So, we have 2.2. Bounds for kTk 2 L 2 ðΩ 1 Þ and kθk 2 L 2 ðΩ 2 Þ . Payne et al. [27] have obtained the following result: with T M = max fkT 0 k ∞ , sup ½0,τ kGk ∞ , T LM g and T LM is the maximum temperature on the interface L. Similarly, in the Abstract and Applied Analysis with θ M = max fkθ 0 k ∞ , sup ½0,τ kGk ∞ , T LM g. However, in the area Ω 1 ∪ Ω 2 × ½0, τ, the maximum temperature cannot be reached on the interface L. Therefore, we observe that where To derive the bounds for kTk 2 L 2 ðΩ 1 Þ and kθk 2 L 2 ðΩ 2 Þ , we introduce another two functions φ andφ which for each t satisfy Then, from the identities we have So, (20) leads to that It follows by Lemma 1 that where Since we have In computing the bounds for kφk 2 Ω 1 and kφk 2 Ω 2 , we introduce another two functions h andh which for each t satisfy Then, from the identities 3 Abstract and Applied Analysis it follows from (26) that From (28), we have where Upon integration of (29), we have Inserting (31) into (28), we get The terms khk 2 L 2 ðΩ 1 Þ , Ð t 0 kh ,η k 2 L 2 ðΩ 1 Þ dη as well as khk 2 L 2 ðΩ 2 Þ , Ð t 0 kh ,η k 2 L 2 ðΩ 2 Þ dη may be bounded by boundary data by using a Rellich identity, cf. [6,7]. Combining (13), (22), (24), and (33), we may have where After integrating (34), we have Inserting (36) back into (34), we have the following lemma.

Continuous Dependence on the Forchheimer Coefficient
In this section, we want to derive an a priori estimate showing how ðu i , T, pÞ and ðv i , θ, qÞ depend continuously on the Forchheimer coefficient b. Let ðu i , T, pÞ and ðv i , θ, qÞ be solutions of (1)-(5) with b = b 1 , and ðu * i , T * , p * Þ and ðv * i , θ * , q * Þ be solutions of (1)-(5) with b = b 2 , respectively. We define Then, ðw i , Σ, πÞ satisfy the following equations and ðw m i , Σ m , π m Þ satisfy equations The boundary conditions are The initial conditions can be written as The interface L conditions are We observe for later convenience that ð4:3Þ 1 may be rearranged as Our main result is the following theorem.

Theorem 3.
Let ðu i , T, pÞ and ðv i , θ, qÞ be the the classical solutions to the initial-boundary value problem (1)-(5) corresponding to b 1 , and ðu * i , T * , p * Þ and ðv * i , θ * , q * Þ also be the the classical solutions to the initial-boundary value problem (1)-(5) but corresponding to b 2 . Then, for any t > 0, we have as b 1 ⟶ b 2 . The differences of velocities satisfy Furthermore, there is a positive function a 1 ðtÞ, given specifically in (58), such that where w, w m , Σ, Σ m , and σ have been defined in (40) and (41).
Proof. We begin with the identity ð From (51), it follows that

Data Availability
All data generated or analyzed during this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest.