Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

A priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in Ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega _{1}$\end{document}, which was governed by the Boussinesq equations. For a porous medium in Ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega _{2}$\end{document}, we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous dependence type for the Boussinesq coefficient λ. Following the method of a first-order differential inequality, we can further obtain the result that the solution depends continuously on the interface boundary coefficient α. These results showed that the structural stability is valid for the interfacing problem.

, Scott [16], Scott and Straughan [17], Payne et al. [18][19][20][21] and some related papers [22][23][24]. The previous publications of structural stability usually study one fluid in a bounded domain. Usually, there exists more than one fluid in a domain. These fluids have some interactions. It is desirable to see what effect they can have on each other. So the study of two interfacing fluids may be interesting and meaningful. In [19], the authors studied the structural stability for a flow interfacing with a porous solid. They proved that the solution depends continuously on the coefficient of the interface boundary condition.
In this paper, we want to study the continuous dependence type results on the interface boundary coefficient and the Boussinesq coefficient for the solution of the Boussinesq-Darcy problem in R 3 . The Boussinesq equations interface with the Darcy equations through the mutual boundary. Thus, we suggest an appropriate part of the plane z = x 1 = 0 is the mutual boundary for a porous fluid in a bounded region 2 in and a nonlinear viscous fluid in 1 in R 3 . We denote the interface by L. The remaining part of ∂ 1 is denoted by 1 , and the remaining part of ∂ 2 is denoted by 2 . We also denote ∂ 1 = 1 ∪ L and Let (u i , T, p) and (v i , θ , q) denote the velocity, temperature and pressure in 1 and 2 , respectively. Then the Boussinesq flow equations are (see [25][26][27]) where g i is the gravity force function; λ is the Boussinesq coefficient. The coefficients μ and k 1 are kinematic viscosity and thermal conductivity, respectively. From [28,29]), we can see that the Boussinesq equations are useful in studying fluid and geophysical fluid dynamics.
The Darcy equations can be written as (see Nield and Bejan [30]) where 1 and 2 are all bounded domains. They are all simply connected and star-shaped.
The boundaries ∂ 1 and ∂ 2 are their boundaries, respectively. τ is a positive constant which satisfies 0 < τ < ∞. The following boundary conditions are satisfied: for prescribed functions G(x, t) and G(x, t) and n (1) i , n (2) i denote the unit outward normals of 1 , 2 , respectively. Obviously, n (1) 3 = -n (2) 3 = -1. The initial conditions are written as for prescribed functions f i , T 0 and θ 0 . The interface L conditions are where α is a positive coefficient and the value of α can be defined by experiment. It is determined by the given fluid and porous solid. The boundary conditions (5) were given by Nield and Bejan in [30]. In [31], Jones deduced the last condition in (5).
In this paper, we want to obtain the continuous dependence on the Boussinesq coefficient λ and the interface boundary coefficient α for the Boussinesq-Darcy interfacing problems in a bounded domain. However, there are only a few papers studying this interfacing problem in a bounded domain (see Payne and Straughan [19] and Liu et al. [13]). For the unbounded domain, refer to Liu et al. [32]. However, compared with the above literature, in this paper, there is a nonlinear term u i u i,j . In particular, the bound of u i,j u i,j dx is needed in this paper. But the methods proposed in [13,19,32] cannot be used directly. Second, some well-known Sobolev inequalities cannot be held for the interfacing problem. Our biggest innovation is to overcome these difficulties. We are sure that we can obtain some new and interesting results. We will derive some useful a priori bounds by using different inequalities. With the aid of these a priori bounds, we derive the continuous dependence on the Boussinesq coefficient and the interface boundary coefficient.
In the following discussions, we use the comma to denote partial differentiation. We also use u i,k to denote the partial differentiation with respect to the direction x k . This is to say u i,k = ∂u i ∂x k . We also use the usual summation convection with repeated Latin subscripts summed from 1 to 3, and the Greek subscripts summed from 1 to 2. Therefore,

A priori bounds
In this section, we want to drive bounds for various norms of u i in terms of known data which will be used in the next sections.
Then the temperatures satisfy where Proof First, we let T LM denotes the maximum of the temperature on the interface L. Payne, Rodrigues and Straughan [33] have derived However, in the area 1 ∪ 2 × [0, τ ], the maximum of the temperature cannot be reached on the interface L. Therefore, we have the result (6).

Lemma 2.2
If T 0 , θ 0 , G, G ∈ L ∞ and 1 , 2 are bounded regions. Then Proof Multiplying (1) 1 by u i , integrating over 1 and using (6), we obtain By the divergence theorem and (2) and the conditions in the interface, we have Upon integration, we can arrive at Lemma 2.2.

Now we define
Lemma 2.3 If T 0 , θ 0 , G, G ∈ L ∞ and 1 , 2 are bounded regions, then where a 1 = 1 Proof Using the divergence theorem, we have In the light of the condition on L, we compute Combining (11) and (12), we have Lemma 2.3.

Lemma 2.4
If T 0 , θ 0 , G, G ∈ L ∞ and 1 , 2 are bounded regions. Then where A 4 (t) is a positive function which will be defined later.
Proof We firstly compute We find that the result given in Appendix B of Lin and Payne [10] for u 2 , k > 0. (15) So, we have for an arbitrary constant ε 1 > 0 We notice that one has obtained the following results [13]: where γ and C are positive constants which have been defined in [13]. Following the methods in [13], we can derive a similar result, Combining (14), (16)- (19) and using (7), we obtain where we have chosen ε 1 = 4μ 3k 5 . We now use (1) 2 and (2) 2 with the boundary conditions (3) and (5) and the divergence theorem to obtain d dt By integration of (24) we thus obtain Inserting (25) into (20) and setting we have Then, using Lemma 2.3 and the Hölder inequality in (27), we get for some computable positive constants b i (i = 1, 2, 3, 4, 5). Now, we define We have from (32) Therefore, we can get the result where In view of (9), Lemma 3 and (18), we also have where Combining (13), (15) and (34), we may get the following lemma.
We define and Then (w i , T, π) satisfy the following equations: and (w m i , S m , π m ) satisfy the equations The boundary conditions are The initial conditions can be written as The interface L conditions are We first give some useful lemmas. (u i , T, p) and (v i , θ , q) be the classical solutions to the initial-boundary value problem (1)-(5) corresponding to λ (1) , and (u * i , T * , p * ) and (v * i , θ * , q * ) also be the classical solutions to the initial-boundary value problem (1)-(5) but corresponding to λ (2) . Then for any t > 0 the differences of velocities satisfy

Lemma 3.1 Let
where c 1 (t) is a positive function which depends on t.
Proof We begin with the identity From (44) it follows that We now deal with I 1 and I 5 . Using the divergence theorem, we have Using the Hölder inequality, (15), Lemma 2.4, Lemma 2.5 and the Young inequality with δ 1 > 0, we have Following a similar procedure to deriving I 3 , we obtain where δ 2 , δ 3 are positive constants to be determined later. Similarly, we have We note that I 6 can be bounded, Similar to (19), we also have We define the functions F 1 (t) and F 2 (t) by Inserting (45)-(51) into (44) and using (52), we have We choose δ 1 , δ 2 , δ 3 and δ 4 small enough such that Letting we can get Lemma 3.1.

Lemma 3.2 Let (u i , T, p) and
(v i , θ , q) be the classical solutions to the initial-boundary value problem (1)-(5) corresponding to λ (1) , and (u * i , T * , p * ) and (v * i , θ * , q * ) also be the classical solutions to the initial-boundary value problem (1)-(5) but corresponding to λ (2) . Then for any t > 0 we have Proof We multiply (39) 2 and (40) 2 by S and S m , respectively, and integrate by parts to find Integrating (54) from 0 to t one may deduce Combining (52) and (55), we may obtain Lemma 3.2.
Now, we use Lemmas 3.1 and 3.2 to obtain Setting we obtain where Thus after integration we may derive from (58) the estimate Combining (57), Lemma 3.2 and (60), we have the following theorem.
Furthermore, there are two positive c 2 , c 3 (t), such that