Spatial Decay Bounds for the Brinkman Fluid Equations in Double-Diffusive Convection

: In this paper, we consider the Brinkman equations pipe ﬂow, which includes the salinity and the temperature. Assuming that the ﬂuid satisﬁes nonlinear boundary conditions at the ﬁnite end of the cylinder, using the symmetry of differential inequalities and the energy analysis methods, we establish the exponential decay estimates for homogeneous Brinkman equations. That is to prove that the solutions of the equation decay exponentially with the distance from the ﬁnite end of the cylinder. To make the estimate of decay explicit, the bound for the total energy is also derived.


Introduction
The Brinkman equations are one of the most important models in fluid mechanics. This model are mainly used to describe flow in a porous medium. For more details, one can refer to Nield and Bejan [1] and Straughan [2]. In the present paper, we define the Brinkman flow depending on the salinity and the temperature in a semi-infinite cylindrical pipe and derive the spatial decay properties. When the homogeneous initial-boundary conditions are applied on the lateral surface of the cylinder, We prove that the solutions of Brinkman equations decays exponentially with spatial variable.
In fact, the Brinkman equations have been studied by many papers in the literature. For example, Straughan [2] considered the mathematical properties of Brinkman equations as well as Darcy and Forchheimer equations, and stated how these equations describe the flow of porous media. Ames and Payne [3] studied the structural stability for the solutions to the viscoelasticity in an ill-posed problem. Franchi and Straughan [4] proved the structural stability for the solutions to the Brinkman equations in porous media in a bounded region. More relevant results one can see [5][6][7][8][9][10]. Paper [11] studied the double diffusive convection in porous medium and obtained the structural stability for the solutions. The continuous dependence for a thermal convection model with temperature-dependent solubility can be found in [12]. For more recent work about continuous dependence, one may refer to [13][14][15][16][17][18][19].
In this paper, let R be a semi-infinite cylinder and ∂R represents the boundary of R. D denotes the cross section of the cylinder with the smooth boundary ∂D (see Figure 1).
In this paper, we also use the following notations where z is a point along the x 3 axis. Clearly, R 0 = R and D 0 = D. Letting u i , T, C and p denote the fluid velocity, temperature, salt concentration and pressure, respectively. The Brinkman equations we study can be written as [20] where ν, σ > 0 denote the Brinkman coefficient, and the Soret coefficient, respectively. k 1 , k 2 , k 3 > 0. Without losing generality, we take them equal to 1. ∆ is the Laplacian operator. g i (x) and h i (x) are gravity field, which are given functions. We suppose that (1)-(4) have the following initial-boundary conditions u i = f i (x 1 , x 2 , t), T = F(x 1 , x 2 , t), C = G(x 1 , x 2 , t), on D 0 × {t ≥ 0}, In (1)- (8) and in the following, the usual summation convention is employed with repeated Latin subscripts summed from 1 to 3 and repeat Greek subscript summed from 1 to 2. The comma is used to indicate partial differentiation, i.e., u i, The purpose of this paper is to consider the spatial decay properties of the Equations (1)-(8) in a semi-infinite cylindrical pipe by using the symmetry of differential inequalities, that is, to prove that the solutions of the equations decay exponentially with the distance from the finite end of the cylindrical pipe.
In Section 2, some auxiliary inequalities are presented. We establish some useful lemmas in Section 3. The spatial exponential decay estimate for the solution is established in Section 4. Finally, in Section 5 we derive the bounds for the total energies.

Auxiliary Results
In this paper, we will use some inequalities in the following sections. Thus, we firstly list them as follows.

Lemma 1. Let D be a plane domain D with the smooth boundary
where λ 1 is the smallest eigenvalue of the problem Many papers have studied this inequality, e.g., one may see [21,22].
A representation theorem will be also used in next sections. We write this theorem as Lemma 2. Let D be a plane Lipschitz bound region and w be a differential function in D which satisfies D wdA = 0, then there exists a vector function ϕ α (x 1 , x 2 ) such that and a positive constant Λ depending only on the geometry of D such that The Lemma 2 was proofed by Babuška and Aziz [23] and Horgan and Wheeler [24] have used the Lemma 1 to viscous flow problems. The explicit upper bound of Λ can be found in Horgan and Payne [25]. In this paper, this Lemma 2 is used to eliminate the pressure function difference terms p, since we can prove that u 3 satisfy the hypothesis of this Lemma 2 later.
If w ∈ C 1 0 (D) and w ∈ C 1 0 (R), the following Sobolev inequalities hold For (11), we assume that w → 0 as x 3 → ∞. Payne [26] has given the derivation of (12). For a special case of the results one can see [27,28]. They have obtained the optimal value of Ω Ω = 1 27 In the following, we also use the following lemma.
We will also use the following lemmas which were derived in [29].

Lemma 4.
Let that the function ϕ is the solution of the problem where D z gdA = 0. Then

Some Useful Lemmas
In this section, we derive some useful lemmas which will be used in next section. First, we define a weighted energy expression where k, ρ 1 , ρ 2 are positive parameters and ξ > z > 0.

Lemma 5. Let u, T, C, p be solutions of Equations
Similar to Lemma 5, for E 2 (z, t) we can obtain the following lemma.

Lemma 6. Let u, T, C, p be solutions of Equations
Proof. By the divergence theorem and Equations (1)-(8), we have Using the Schwarz inequality, the Poincaré inequality and the AG mean inequality, we can obtain Similar to (20) and (21), we have for B 2 and B 3 and where ε 3 , ε 4 are positive constants.
Next we may bound E 3 (z, t). First we let T M denotes that the maximum of T by using the maximum principle in R, i.e., Integrating by parts, using (3), (5), (6), (7) together with (9) and the AG mean inequality, we have Using Equations (3)- (7) and integrating by parts, we obtain By the Schwarz and the AG mean inequalities, it follows that from (47) for an arbitrary constant ε 5 > 0.
In order to bound the last term on the right of (48), using the Equations (9), (11) and (13), the Schwarz inequality and the AG mean inequality to obtain where the bound for max t ( R z C 2 dx) 1 2 will be derived later.
where ε 6 is a positive constant. Next, we use Lemmas 5-7 to prove our main result.

Main Result
First, we introduce a new function Using Lemmas 4-6 and in view of (51), we have where a 4 = a 1 k + Λ Choosing we can have from (52) From (53), we have and Combining (54), (55) and (56), we have Thus where Inequality (57) can be rewritten as Integrating (58) from z to ∞ leads to ∂Ψ ∂z + 2 Ψ ≤ 0, and hence Combining (53) and (59), we can obtain the following theorem.

Theorem 1. Let u, T, C, p be solutions of Equations
Remark 1. The result of Theorem 1 belongs to the study of Saint-Venant principle, which shows that the fluid decays exponentially with spatial variables on the cylinder.
To make the decay bound explicit, we have to derive the bounds for Ψ(0, t) and max t R C 2 dx in next section.

Bounds of Ψ(0, t) and max t R C 2 dx
From the previous section, we can see that a 3 involves the quantities max t R C 2 dx. To make our main result explicit, we have to derive bounds of Ψ(0, t) and max t R C 2 dx in term of the physical parameters σ, ν, g i , h i , the boundary data and so on. To do this, we begin with Now we assume that S is a sufficiently smooth function satisfying the same initial and boundary conditions as T. Thus, Using the Schwarz and the arithmetic-geometric mean inequalities, we can obtain where 1 , 2 , 3 are positive constants. Choosing we can obtain Obviously, the data terms in (65) involve 1 Similarly, we can bound t 0 R C ,i C ,i dxdη as well as max t R C 2 dx. Firstly, we introduce a function H: Then we have By the triangle inequality, we obtain that and max t R Then, which follows that Just as in the computation for T, we have the following inequality Thus, where 5 > 0 depends on 3 , 4 and σ. Next we have to derive a bound for t 0 R u i u i dxdη in term of data. To do this, we define a function for some positive constant i . Then, Obviously, we find that the last term of (75) is a data term. Now Noting that in R for ς 1 = f α,α f 3 , we can rewrite (76) as Inserting (78) back into (75), we may have a bound of the form for computable C 1 and C 2 . Combining (65) and (73) and by inequality (17), we have It is clear to see that (65) and (73), we can obtain Next we seek bound for the total energy Ψ(0, t). From (54) we can obtain for Ψ(0, t) We are left to derive bounds for t 0 R z u i,η u i,η dxdη and t 0 D u α,3 u α,3 dAdη. Multiplying (1) with u i,η and integrating in the region R × [0, t], we have which follows that where we have used the fact u 3,3 = −u α,α = − f α,α on D 0 and (83), and ε 6 is a positive constants. For the first term of (86), using the Schwarz and the AG mean inequalities we have To bound the second term on the right of (86), we define p to be the mean value of p over D 0 , i.e., where |D 0 | is the measure of D 0 . Since we obtain It follows by using Schwarz inequality that where 6 is a positive constant to be determined later.
To deal with the integral t 0 D 0 (p − p) 2 dAdη, we let an auxiliary function χ satisfying: From the definition of p in (88), it is clear that D 0 (p − p)dA = 0. Thus, the necessary condition for the existence of a solution is satisfied and we compute Since From (93), we can obtain Making use of (15), (16), (81) and (83) which follows that Obviously, from (97) we must establish a bound for the term t 0 D 0 u α,3 u α,3 dxdη. To do this, we begin with the identity t 0 R u i,3 νu i,jj − u i − p ,i − u i,η + g i T + h i C dxdη = 0.
where 7 is a positive constant.
Recalling (84) and using (104) and (107), we obtain which is to say that we have bounded the total energy.

Conclusions
In this paper, we consider the spatial decay bounds for the Brinkman equations in double-diffusive convection in a semi-infinite pipe. Using the results of this paper, we can continue to study the continuous dependence of the solution on the parameters in the system of equations. In addition, Using the results of this paper, we can continue to study the continuous dependence of the solution on the parameters in the system of equations. This research can refer to the method of [30,31]. In addition, if Equation (1) is replaced by a nonlinear problem (e.g., Forchheimer equations), it will be a more interesting topic.
Author Contributions: Conceptualization, and validation, Y.L.; formal analysis and investigation, X.C. and D.L. All authors have read and agreed to the published version of the manuscript.