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

Authors: Jincheng Shi; Lianhong Guo
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

The continuous dependence of the two fluids which interface with each other in a bounded domain is derived. It is first shown how to establish the apriori results for the solutions for the flow in a bounded domain occupied by a viscous fluid in contact with a porous solid. It is also proved that the solution depends continuously on the Forchheimer coefficient.


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## Body

## 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 planez=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.Letui,T,p and vi,θ,q denote the velocity, temperature, and pressure in Ω1 and Ω2, respectively. The Forchheimer system consists of the partial differential equations (see [1])
(1)buui+1+γTui−giT+∂p∂xi=0,inΩ1×0,τ,(2)∂T∂t+ui∂T∂xi=ΔT,inΩ1×0,τ,(3)∂ui∂xi=0,inΩ1×0,τ,where gi 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=μ11+γT,γ>0.The Darcy equations are (see Nield and Bejan [2])
(4)vi−giθ+∂q∂xi=0,inΩ2×0,τ,(5)∂θ∂t+vi∂θ∂xi=Δθ,inΩ2×0,τ,(6)∂vi∂xi=0,inΩ2×0,τ,where Ω1 and Ω2 are the bounded, simply connected, and star-shaped domains and τ is a given number satisfying 0≤τ<∞. We impose the following boundary conditions:
(7)ui=0;T=Gx,t,onΓ1×0,τ,vini2=0,θ=G~x,t,onΓ2×0,τ,for prescribed functions Gx,t and G~x,t, and ni2 denotes the unit outward normal of ∂Ω2. We also let ni1 be the unit outward normal of ∂Ω1. Obviously, n31=−n32 on L. The initial conditions are written as
(8)uix,0=fix,Tx,0=T0x,inΩ1,θx,0=θ0x,inΩ2,for prescribed functions fi, T0, and θ0. On the interface L, the conditions are
(9)u3=v3,T=θ,T,3=θ,3,q=p.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 the area of porous media, there have been many studies of continuous dependence, or structural stability, including those of Scott and Straughan [6, 7], Franchi and Straughan [8], Hoang and Ibragimov [9], Lin and Payne [10, 11], Liu [12, 13], Liu et al. [14–16], Scott [6], Scott and Straughan [7], Li [17], Cichon′ et al. [18], Ma and Liu [19], Ciarletta et al. [20], and Payne et al. [18–21]. These results about structural stability mainly focus on one fluid in the domain. But in reality, there are usually two or more fluids in the domain. They can interface with each other. So the study of such type of problems may be interesting and meaningful. It is worth drawing attention to this type of study. Several quantitative results in physical problems were obtained in [22–25].In this paper, we derive an apriori 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 γTui and ∣u∣ui 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.


## 2. APriori Bounds

In this section, we want to drive bounds for the various norms ofu,v,T, and θ.


### 2.1. Bounds foruL3Ω13,uL2Ω12, and vL2Ω22

Multiplying1.11 with ui and integrating over Ω1, we obtain
(10)buL3Ω13+1+γTuL2Ω12=∫Ω1giTuidx−∫Ω1∂p∂xiuidx.Integrating by parts and using Young’s inequality and the arithmetic-geometric mean inequality now lead to(11)buL3Ω13+1+γTuL2Ω12=∫Ω1giTuidx−∫Lpu3n31dA≤12uL2Ω12+12g2TL2Ω12+∫Lqvini2dA,where g2=maxgigi. By using the divergence theorem and Equation (2), we get
(12)2buL3Ω13+1+γTuL2Ω12≤g2TL2Ω12+2∫Ω2vigiθ−vidx≤g2TL2Ω12+g2θL2Ω22−vL2Ω22.So, we have(13)2buL3Ω13+1+γTuL2Ω12+vL2Ω22≤g2TL2Ω12+g2θL2Ω22.


### 2.2. Bounds forTL2Ω12 and θL2Ω22

Payne et al. [27] have obtained the following result:
(14)sup0,τT∞≤TM,with TM=maxT0∞,sup0,τG∞,TLM and TLM is the maximum temperature on the interface L. Similarly, in the Ω2×0,τ,
(15)sup0,τθ∞≤θM,with θM=maxθ0∞,sup0,τG~∞,TLM. However, in the area Ω1∪Ω2×0,τ, the maximum temperature cannot be reached on the interface L. Therefore, we observe that
(16)maxsup0,τT∞,sup0,τθ∞,TLM≤NM,where NM=maxT0∞,sup0,τG∞,θ0∞,sup0,τG~∞. To derive the bounds for TL2Ω12 and θL2Ω22, we introduce another two functions φ and φ~ which for each t satisfy
(17)φ,η−Δφ=0,inΩ1,φ~,η−Δφ~=0,inΩ2,φ=G,onΓ1,φ~=G~,onΓ2,φx,0=T0,inΩ1,φ~x,0=θ0,inΩ2,φ=φ~,φ,3=φ~,3,onL.Then, from the identities(18)∫0t∫Ω1T−φT−φ,η+uiT,i−ΔT−φdxdη=0,∫0t∫Ω2θ−φ~θ−φ~,η+viθ,i−Δθ−φ~dxdη=0,we have
(19)T−φL2Ω12+θ−φ~L2Ω22+2∫0t∇T−φL2Ω12dη+2∫0t∇θ−φ~L2Ω22dη=2∫0t∫Ω1T−φ,iuiTdxdη+2∫0t∫Ω2θ−φ~,iviθdxdη.So, (20) leads to that
(20)T−φL2Ω12+θ−φ~L2Ω22+2∫0t∇T−φL2Ω12dη+2∫0t∇θ−φ~L2Ω22dη≤2NM∫0t∇T−φL2Ω12dη∫0tuL2Ω12dη1/2+2NM∫0t∇θ−φ~L2Ω22dη∫0tvL2Ω22dη1/2.It follows by Lemma1 that
(21)T−φΩ12+θ−φ~Ω22≤NM2∫0tuL2Ω12dη+∫0tvL2Ω22dη,where
(22)T−φΩ12=T−φL2Ω12+∫0t∇T−φL2Ω12dη,θ−φ~Ω22=θ−φ~L2Ω22+∫0t∇θ−φ~L2Ω22dη.Since(23)TΩ12≤T−φΩ12+φΩ12,θΩ22≤θ−φ~Ω22+φ~Ω22,we have
(24)TΩ12+θΩ22≤NM2∫0tuL2Ω12dη+∫0tvL2Ω22dη+φΩ12+φ~Ω22.In computing the bounds forφΩ12 and φ~Ω22, we introduce another two functions h and h~ which for each t satisfy
(25)Δh=0,inΩ1,Δh~=0,inΩ2,h=G,onΓ1,h~=G~,onΓ2,h=h~,h,3=h~,3,onL.Then, from the identities(26)∫0t∫Ω1φ−hφ−h,η−Δφ−h+h,ηdxdη=0,(27)∫0t∫Ω2φ~−h~φ~−h~,η−Δφ~−h~+h~,ηdxdη=0,it follows from (26) that
(28)φ−hL2Ω12+φ~−h~L2Ω22+2∫0t∇φ−hL2Ω12dη+2∫0t∇φ~−h~L2Ω22dη=T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+2∫0tφ−hL2Ω12dη∫0th,ηL2Ω12dη1/2+2∫0tφ~−h~L2Ω22dη∫0th~,ηL2Ω22dη1/2.From (28), we have
(29)φ−hL2Ω12+φ~−h~L2Ω22≤∫0tφ−hL2Ω12+φ~−h~L2Ω22dη+m1t,where
(30)m1t=T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.Upon integration of (29), we have
(31)∫0tφ−hL2Ω12+φ~−h~L2Ω22dη≤∫0tm1ηet−ηdη.Inserting (31) into (28), we get
(32)φ−hΩ12+φ~−h~≤T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0tm1ηet−ηdη+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.So(33)φΩ12+φ~Ω22≤φ−hΩ12+φ~−h~Ω22+hΩ12+h~Ω22≤hΩ12+h~Ω22+T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0tm1ηet−ηdη+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.The termshL2Ω12,∫0th,ηL2Ω12dη as well as h~L2Ω22,∫0th~,ηL2Ω22dη may be bounded by boundary data by using a Rellich identity, cf. [6, 7]. Combining (13), (22), (24), and (33), we may have
(34)TL2Ω12+θL2Ω22+∫0t∇TL2Ω12dη+∫0t∇θL2Ω22dη≤NM2g2∫0tTL2Ω12+θL2Ω22dη+m2t,where
(35)m2t=hΩ12+h~Ω22+T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0tm1ηet−ηdη+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.After integrating (34), we have
(36)∫0tTL2Ω12+θL2Ω22dη≤∫0tm2ηeNM2g2t−ηdη.Inserting (36) back into (34), we have the following lemma.Lemma 1.
Letui,T,p and vi,θ,q be the solutions of (1)-(5) with T0,θ0,G,G~∈L∞. Then,
(37)TL2Ω12+θL2Ω12+∫0t∇TL2Ω12dη+∫0t∇θL2Ω12dη≤P1t,where
(38)P1t=NM2g2∫0tm2ηeNM2g2t−ηdη+m2t.Combining (13) and (34), we can obtain the following lemma.Lemma 2.
Letui,T,p and vi,θ,q be the solutions of (1)-(5) with T0,θ0,G,G~∈L∞. Then,
(39)2buL3Ω13+1+γTuL2Ω12+vL2Ω22≤g2P1t≐P2t.


## 2.1. Bounds foruL3Ω13,uL2Ω12, and vL2Ω22

Multiplying1.11 with ui and integrating over Ω1, we obtain
(10)buL3Ω13+1+γTuL2Ω12=∫Ω1giTuidx−∫Ω1∂p∂xiuidx.Integrating by parts and using Young’s inequality and the arithmetic-geometric mean inequality now lead to(11)buL3Ω13+1+γTuL2Ω12=∫Ω1giTuidx−∫Lpu3n31dA≤12uL2Ω12+12g2TL2Ω12+∫Lqvini2dA,where g2=maxgigi. By using the divergence theorem and Equation (2), we get
(12)2buL3Ω13+1+γTuL2Ω12≤g2TL2Ω12+2∫Ω2vigiθ−vidx≤g2TL2Ω12+g2θL2Ω22−vL2Ω22.So, we have(13)2buL3Ω13+1+γTuL2Ω12+vL2Ω22≤g2TL2Ω12+g2θL2Ω22.


## 2.2. Bounds forTL2Ω12 and θL2Ω22

Payne et al. [27] have obtained the following result:
(14)sup0,τT∞≤TM,with TM=maxT0∞,sup0,τG∞,TLM and TLM is the maximum temperature on the interface L. Similarly, in the Ω2×0,τ,
(15)sup0,τθ∞≤θM,with θM=maxθ0∞,sup0,τG~∞,TLM. However, in the area Ω1∪Ω2×0,τ, the maximum temperature cannot be reached on the interface L. Therefore, we observe that
(16)maxsup0,τT∞,sup0,τθ∞,TLM≤NM,where NM=maxT0∞,sup0,τG∞,θ0∞,sup0,τG~∞. To derive the bounds for TL2Ω12 and θL2Ω22, we introduce another two functions φ and φ~ which for each t satisfy
(17)φ,η−Δφ=0,inΩ1,φ~,η−Δφ~=0,inΩ2,φ=G,onΓ1,φ~=G~,onΓ2,φx,0=T0,inΩ1,φ~x,0=θ0,inΩ2,φ=φ~,φ,3=φ~,3,onL.Then, from the identities(18)∫0t∫Ω1T−φT−φ,η+uiT,i−ΔT−φdxdη=0,∫0t∫Ω2θ−φ~θ−φ~,η+viθ,i−Δθ−φ~dxdη=0,we have
(19)T−φL2Ω12+θ−φ~L2Ω22+2∫0t∇T−φL2Ω12dη+2∫0t∇θ−φ~L2Ω22dη=2∫0t∫Ω1T−φ,iuiTdxdη+2∫0t∫Ω2θ−φ~,iviθdxdη.So, (20) leads to that
(20)T−φL2Ω12+θ−φ~L2Ω22+2∫0t∇T−φL2Ω12dη+2∫0t∇θ−φ~L2Ω22dη≤2NM∫0t∇T−φL2Ω12dη∫0tuL2Ω12dη1/2+2NM∫0t∇θ−φ~L2Ω22dη∫0tvL2Ω22dη1/2.It follows by Lemma1 that
(21)T−φΩ12+θ−φ~Ω22≤NM2∫0tuL2Ω12dη+∫0tvL2Ω22dη,where
(22)T−φΩ12=T−φL2Ω12+∫0t∇T−φL2Ω12dη,θ−φ~Ω22=θ−φ~L2Ω22+∫0t∇θ−φ~L2Ω22dη.Since(23)TΩ12≤T−φΩ12+φΩ12,θΩ22≤θ−φ~Ω22+φ~Ω22,we have
(24)TΩ12+θΩ22≤NM2∫0tuL2Ω12dη+∫0tvL2Ω22dη+φΩ12+φ~Ω22.In computing the bounds forφΩ12 and φ~Ω22, we introduce another two functions h and h~ which for each t satisfy
(25)Δh=0,inΩ1,Δh~=0,inΩ2,h=G,onΓ1,h~=G~,onΓ2,h=h~,h,3=h~,3,onL.Then, from the identities(26)∫0t∫Ω1φ−hφ−h,η−Δφ−h+h,ηdxdη=0,(27)∫0t∫Ω2φ~−h~φ~−h~,η−Δφ~−h~+h~,ηdxdη=0,it follows from (26) that
(28)φ−hL2Ω12+φ~−h~L2Ω22+2∫0t∇φ−hL2Ω12dη+2∫0t∇φ~−h~L2Ω22dη=T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+2∫0tφ−hL2Ω12dη∫0th,ηL2Ω12dη1/2+2∫0tφ~−h~L2Ω22dη∫0th~,ηL2Ω22dη1/2.From (28), we have
(29)φ−hL2Ω12+φ~−h~L2Ω22≤∫0tφ−hL2Ω12+φ~−h~L2Ω22dη+m1t,where
(30)m1t=T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.Upon integration of (29), we have
(31)∫0tφ−hL2Ω12+φ~−h~L2Ω22dη≤∫0tm1ηet−ηdη.Inserting (31) into (28), we get
(32)φ−hΩ12+φ~−h~≤T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0tm1ηet−ηdη+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.So(33)φΩ12+φ~Ω22≤φ−hΩ12+φ~−h~Ω22+hΩ12+h~Ω22≤hΩ12+h~Ω22+T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0tm1ηet−ηdη+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.The termshL2Ω12,∫0th,ηL2Ω12dη as well as h~L2Ω22,∫0th~,ηL2Ω22dη may be bounded by boundary data by using a Rellich identity, cf. [6, 7]. Combining (13), (22), (24), and (33), we may have
(34)TL2Ω12+θL2Ω22+∫0t∇TL2Ω12dη+∫0t∇θL2Ω22dη≤NM2g2∫0tTL2Ω12+θL2Ω22dη+m2t,where
(35)m2t=hΩ12+h~Ω22+T0−hx,0L2Ω12+θ0−h~x,0L2Ω22+∫0tm1ηet−ηdη+∫0th,ηL2Ω12dη+∫0th~,ηL2Ω22dη.After integrating (34), we have
(36)∫0tTL2Ω12+θL2Ω22dη≤∫0tm2ηeNM2g2t−ηdη.Inserting (36) back into (34), we have the following lemma.Lemma 1.
Letui,T,p and vi,θ,q be the solutions of (1)-(5) with T0,θ0,G,G~∈L∞. Then,
(37)TL2Ω12+θL2Ω12+∫0t∇TL2Ω12dη+∫0t∇θL2Ω12dη≤P1t,where
(38)P1t=NM2g2∫0tm2ηeNM2g2t−ηdη+m2t.Combining (13) and (34), we can obtain the following lemma.Lemma 2.
Letui,T,p and vi,θ,q be the solutions of (1)-(5) with T0,θ0,G,G~∈L∞. Then,
(39)2buL3Ω13+1+γTuL2Ω12+vL2Ω22≤g2P1t≐P2t.


## 3. Continuous Dependence on the Forchheimer Coefficient

In this section, we want to derive an a priori estimate showing howui,T,p and vi,θ,q depend continuously on the Forchheimer coefficient b. Let ui,T,p and vi,θ,q be solutions of (1)-(5) with b=b1, and ui∗,T∗,p∗ and vi∗,θ∗,q∗ be solutions of (1)-(5) with b=b2, respectively. We define
(40)wi=ui−ui∗,Σ=T−T∗,π=p−p∗,σ=b1−b2,(41)wim=vi−vi∗,Σm=θ−θ∗,πm=q−q∗Then,wi,Σ,π satisfy the following equations
(42)σuui+b2uiu−ui∗u∗+1+γT∗wi+γΣui−giΣ+π,i=0,inΩ1×0,τ,∂Σ∂t+wiT,i+ui∗Σ,i=ΔΣ,inΩ1×0,τ,∂wi∂xi=0,inΩ1×0,τ,and wim,Σm,πm satisfy equations
(43)wim−giΣm+π,im=0,inΩ2×0,τ,∂Σm∂t+wimθ,i+vi∗Σm,i=ΔΣm,inΩ2×0,τ,∂wim∂xi=0,inΩ2×0,τ.The boundary conditions are(44)wi=0,Σ=0,onΓ1×0,τ,wimni2p=0,Σm=0,onΓ2×0,τ.The initial conditions can be written as(45)wix,0=0,Σx,0=0,inΩ1,Σmx,0=0,inΩ2.The interfaceL conditions are
(46)w3=w3m,Σ=Σm,Σ,3=Σ,3m,πm=π.We observe for later convenience that4.31 may be rearranged as
(47)σu∗ui∗+b1uiu−ui∗u∗+1+γTwi+γΣui∗−giΣ+π,i=0.Our main result is the following theorem.Theorem 3.
Letui,T,p and vi,θ,q be the the classical solutions to the initial-boundary value problem (1)–(5) corresponding to b1, and ui∗,T∗,p∗ and vi∗,θ∗,q∗ also be the the classical solutions to the initial-boundary value problem (1)–(5) but corresponding to b2. Then, for any t>0, we have
(48)ui,T,p⟶ui∗,T∗,p∗,vi,θ,q→vi∗,θ∗,q∗,as b1⟶b2. The differences of velocities satisfy
(49)∫0twL2Ω12+wmL2Ω22dη≤σ2b1b2∫0tP2seNM2∫sta1ηdηds.
Furthermore, there is a positive functiona1t, given specifically in (58), such that
(50)ΣL2Ω12+ΣmL2Ω22≤σ2b1b2NM2∫0tP2seNM2∫sta1ηdηds,where w,wm,Σ,Σm,andσ have been defined in (40) and (41).Proof.
We begin with the identity(51)∫Ω1σuui+b2uiu−ui∗u∗+1+γT∗wi+γΣui−giΣ+π,iwidx=0.
From (51), it follows that
(52)12b2∣u∣+∣u∗∣wL2Ω12+1+γT∗wL2Ω12=−∫Ω1π,iwidx−σ∫Ω1uuiwidx−γ∫Ω1Σuiwidx+∫Ω1giwiΣdx.Using the divergence theorem, we have(53)12b2∣u∣+∣u∗∣wL2Ω12+1+γT∗wL2Ω12=−σ∫Ω1uuiwidx−∫Lπw3n31dA−γ∫Ω1Σuiwidx+∫Ω1giwiΣdx=−σ∫Ω1∣u∣uiwidx+∫Lπmw3mn32dA−γ∫Ω1Σuiwidx+∫Ω1giwiΣdx−σ∫Ω1∣u∣uiwidx+∫Ω2−wim+giΣmwimdx−γ∫Ω1Σuiwidx+∫Ω1giwiΣdx.We use the Cauchy-Schwarz inequality to have(54)−σ∫Ω1∣u∣uiwidx≤σuwL2Ω1uL3Ω13/2≤b1+b24uwL2Ω12+σ22b1b1+b2P2t.Using Hölder’s inequality and the A-G mean inequality, we have(55)∫Ω2−wim+giΣmwimdx≤−12wmL2Ω22+12g2ΣmL2Ω22.We find that the result given in Appendix B of Liu [13] for uL4Ω12(56)uL4Ω12≤kuL2Ω12+uL2Ω11/2∇uL2Ω13/2,k>0,or
(57)uL4Ω12≤k1+δ4uL2Ω12+34δ−1/3∇uL2Ω12,k>0,for arbitrary positive constant δ>0. We use this inequality for ΣL4Ω12 with δ=1 and (2.30), (2.32) in (55) to get
(58)−γ∫Ω1Σuiwidx≤γuwL2Ω1ΣL4Ω1uL2Ω11/2≤b1+b24uwL2Ω12+kγ2b1+b2P21/2tΣL2Ω12+ΣL2Ω11/2∇ΣL2Ω13/2≤b1+b24uwL2Ω12+34δ1∇ΣL2Ω12+kγ2b1+b2P21/2t+k4P22tγ864b1+b24δ13ΣL2Ω12,for arbitrary δ1>0. We also have
(59)∫Ω1giwiΣdx≤12wL2Ω12+12g2ΣL2Ω12.Combining (53), (54), (55), (58), and (59), we obtain
(60)12b2u+u∗wL2Ω12+1+γT∗wL2Ω12+12wmL2Ω22≤b1+b22u∣wL2Ω12+σ22b1b1+b2P2t+12g2ΣmL2Ω22+kγ2b1+b2P21/2t+k4γ864b1+b24δ13P22t+12g2ΣL2Ω12+34δ1∇ΣL2Ω12+12wL2Ω12.We follow a similar procedure starting from (47) to obtain
(61)12b1u+u∗wL2Ω12+1+γTwL2Ω12+12wmL2Ω22≤b1+b22u∗∣wL2Ω12+σ22b2b1+b2P2t+12g2ΣmL2Ω22+kγ2b1+b2P21/2t+k4γ864b1+b24δ13P22t+12g2ΣL2Ω12+34δ1∇ΣL2Ω12+12wL2Ω12.We add (60) and (61) to find
(62)1+γT+γT∗wL2Ω12+wmL2Ω22≤σ22b1b2P2t+a1tΣL2Ω12+ΣmL2Ω22+32δ1∇ΣL2Ω12,where
(63)a1t=2kγ2b1+b2P21/2t+k4γ832b1+b24δ13P22t+g2.To boundΣL2Ω12 and ΣmL2Ω22, we multiply 3.32 and 3.42 by Σ and Σm, respectively, and integrate by parts to find
(64)ΣL2Ω12+ΣmL2Ω22=−2∫0t∇ΣL2Ω12dη−2∫0t∇ΣmL2Ω22dη+2∫0t∫Ω1wiTΣ,idxdη+2∫0t∫Ω2wimθΣ,imdxdη≤−∫0t∇ΣL2Ω12dη−∫0t∇ΣmL2Ω22dη+NM2∫0twL2Ω12dη+NM2∫0twmL2Ω22dη,or
(65)ΣL2Ω12+ΣmL2Ω22+∫0t∇ΣL2Ω12dη+∫0t∇ΣmL2Ω22dη≤NM2∫0twL2Ω12dη+NM2∫0twmL2Ω22dη.Combining (62) and (4.26), we find that
(66)wL2Ω12+wmL2Ω22≤σ22b1b2P2t+32δ1∇ΣL2Ω12+a1tNM2∫0twL2Ω12dη+∫0twmL2Ω22dη,or
(67)⊑′t≤σ22b1b2P2t+32δ1∇ΣL2Ω12+a1tNM2⊑t,where
(68)⊑t=∫0twL2Ω12dη+∫0twmL2Ω22dη.Thus, after integration, we may derive from (67) the estimate:
(69)⊑t≤σ22b1b2∫0tP2seNM2∫sta1ηdηds+32δ1eNM2∫0τa1ηdη∫0t∇ΣL2Ω12dη.Using (65) and (66) and choosing δ1 in (69), we have
(70)∫0twL2Ω12dη+∫0twmL2Ω22dη≤σ2b1b2∫0tP2seNM2∫sta1ηdηds,(71)ΣL2Ω12+ΣmL2Ω22≤σ2b1b2NM2∫0tP2seNM2∫sta1ηdηds.Inequalities (70) and (71) are a priori bounds demonstrating continuous dependence of the solution on the Forchheimer coefficient b.



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*Source: 7971038-2020-08-28.xml*