An approach to the global well-posedness of a coupled 3-dimensional Navier-Stokes-Darcy model with Beavers-Joseph-Sa ﬀ man-Jones interface boundary condition

: This study focused on investigating the global well-posedness of a coupled Navier-Stokes-Darcy model with the Beavers-Joseph-Sa ﬀ man-Jones interface boundary condition in the three-dimensional Euclidean space. By utilizing this approach, we successfully obtained the global strong solution of the system in the three-dimensional space. Furthermore, we demonstrated the exponential stability of this strong solution. The signiﬁcance of such coupled systems lies in their pivotal role in the analysis of subsurface ﬂow problems, particularly in the context of karst aquifers


Introduction
The study of the coupling between free flow and porous media flow has garnered widespread attention in recent years, owing to its diverse applications in geosciences (e.g., karst aquifers, hyporheic flow, contaminant transport), health sciences (e.g., blood flow), and industrial processes; see [1][2][3][4][5] and the references therein.Insights derived from a comprehensive understanding of the Navier-Stokes-Darcy equations can be readily employed to tackle various engineering challenges.The typical mathematical analysis on the well-posedness of the associated initial boundary value problem has been done by Layton et al. [6] and Discacciati et al. [7].The mathematical analysis of the miscible displacement problem in the subsurface was done in a seminal paper by Alt-Luckhaus [8] and by others such as Fabrie-Langlais [9], Fabrie-Gallouët [10], and Marpeau-Saad [11].
We select a microelement in the fluid-structure coupling system to consider a plane as the research object, which means that the fluid flow on the cross-section of the interface is isotropic.The schematic diagram is shown in Figure 1: For the fluid flow in the porous medium, we employ the mass conversation law of the porosity medium and Darcy's law [12,13] to describe the system as follows: where v = v(x, t) ∈ R 3 denotes the velocity of the flow in the porosity medium.Obviously, we can get that where P 2 = P 2 (x, t) ∈ R 3 , µ 2 > 0 represents the pressure and the viscosity of the flow in Ω m , respectively, and Π denotes the permeability tensor.We use the incompressible Navier-Stokes equations with constant viscosity µ 1 > 0 to describe the flow in Ω f as the following equations: where Ω f ⊂ R 3 covers the domain of the free flow, u = u(x, t) = u 1 , u 2 , u 3 ∈ R 3 is the velocity of the free flow, P 1 = P 1 (x, t) represents the pressure of the flow in Ω f , and µ 1 is the viscosity of the free flow.
The interface-boundary and initial conditions are given by where α is an empirically determined coefficient, τ i , i = {1, 2} represents two orthogonal tangent vectors in the horizontal direction, and n 1 denotes the exterior unit vector normal of ∂Ω f satisfying (1.4) 1 is derived by the balance of force in the normal direction and Beavers-Joseph-Saffman-Jones interface boundary condition (1.4) 2 states the shear force to the tangential stress of the fluid velocity along Γ i .We acknowledge the pioneering work of researchers who have contributed to the fields of fluid dynamics in porous media, computational methods for solving coupled equations, and the development of interface-boundary conditions [14][15][16][17][18].For the conditions at the sharp interface, a comprehensive review of these interface selections is provided in [19].It is known that there are three options for the shear stress conditions in the tangential velocity: The BJ (Beavers-Joseph) condition [14], the BJJ (Beavers-Joseph-Jones) condition [17], and the BJSJ (Beavers-Joseph-Saffman-Jones) condition [18], which is equivalent to what is known as BJS interface condition in some other literature such as [20].Additionally, two choices are available for the balance of force in the normal direction at the interface: The Lions interface condition and the Rankine-Hudoniot condition.
However, there have been few achievements in the mathematical analysis, especially in the case of well-posedness of strong solutions [41,42].This is mainly due to the strong coupling of the interface, which makes it difficult to obtain high-order estimates of the system.The convection phenomenon under consideration is notably more intricate than that in a single fluid (see [43] for the free-flow and [44,45] for fluids in a porous medium).
In recent years, researchers have obtained results on non-stationary weak solutions [30,[46][47][48].Cui, Dong, and Guo [41] have studied the strong solutions and exponential decay in the two-dimensional case, and we have extended these results to the problem in the 3-D Euclidean space in this paper.Our primary objective is to conduct an initial analysis on the global well-posedness of a coupled Navier-Stokes-Darcy model in the context of the Beavers-Joseph-Saffman-Jones interface boundary condition and establishing their uniqueness property.Definition 1.For any T ∈ (0, +∞], we first define a function space X(0, T ) as in Ω × (0, T ), and fulfills the conditions (1.2)- (1.4).
Next, we present the main result of this paper.
Proposition 1.1.(Local well-posedness) Let µ 1 , µ 2 both be positive constants and assume that the initial data u 0 ∈ H 2 (Ω f ) is divergence free, and the compatibility condition holds as follows where u 0 t = u t | t=0 and P 0 1 = P 1 | t=0 .There exists a time T > 0 such that the 3-D coupled Navier-Stokes-Darcy systems (1.2)-(1.4)have a unique strong solution (u, P 2 ) ∈ X(0, T ).Theorem 1.2.(Global well-posedness) The permeability tensor Π satisfies (1.5) and the initial divergence free velocity field u 0 ∈ H 2 (Ω f ) satisfies the compatibility condition (1.6), then there exists a positive constant C depending only on µ 1 , Ω f , and λ, such that if the three-dimensional coupled Navier-Stokes-Darcy systems (1.2)-(1.4)have a unique global strong solution (u, P 2 ) ∈ X(0, +∞) as described in Definition 1.Moreover, the solution (u, P 2 ) has a decay rate where the positive constants are Remark 1.In divergence from the findings of [41,42], this paper makes two primary contributions.First, in terms of analytical techniques, the involvement of more intricate directional derivatives under the three-dimensional model renders estimations challenging and convoluted.Second, regarding outcomes, our research is centered on a strip domain, breaking away from the conventional assumption of periodicity.This marks a pioneering achievement as the first three-dimensional outcome in the rigorous examination of robust solutions for the Navier-Stokes-Darcy system.Notably, our results extend beyond, demonstrating applicability, even in the context of periodic domains.Furthermore, 0 in Theorem 1.2 will depend on ∇u 0 L 2 (Ω f ) .
Remark 2. The decay rate obtained in (1.7) indicates that after time t > 0, the solution (u, P 2 ) is smooth, and all its derivatives decay for any order.
The existence and uniqueness of local solutions to this problem can be obtained similarly to the approach in [41].Therefore, our subsequent focus will be on the a priori estimates of the global solution.

A priori estimates
As is well known, the global strong solution to the nonlinear partial differential equations can be obtained by combining local solutions with global a priori estimates.The local solution is proved similarly to that in [41] and is omitted here.Instead, we present the crucial a priori estimates pivotal to establishing global well-posedness below.Note that K = Π µ 2 and W = µ 1 α √ tr Π for convenience in the subsequent discussion.
then the following estimates hold: and where C depends only on µ 1 , Ω f , and λ.
The proof of Proposition 2.1 can be successfully summarized by the following Lemmas 2.1-2.4.
Lemma 2.1.Under the conditions of Proposition 2.1, it holds that where C depends on µ 1 and Ω f .Next, we come to estimate tells us to deduce the estimations on D(u x ) L 2 (Ω f ) minutely, we omit the details of the estimations on D(u y ) L 2 (Ω f ) due to the symmetry in the horizontal direction.We know from (2.8) and (2.23)-(2.28) that

≤C( D(u
(2.9) Next, taking the partial derivative of (1.3) 1 with respect to x, we get Multiplying (2.10) by u x and integrating the result inequality with respect to x over Ω m , then using Hölder's inequality, Gagliardo-Nirenberg inequality, Young's inequality, Korn's inequality, and (2.1), one can get where we have used Integrating (2.13) over (0, t), we can obtain (2.6) with Lemma 2.1, Young's inequality, and Thus, the proof of Lemma 2.2 is completed.
While combining Lemmas 2.1 and 2.2, we have Therefore, the proof of (2.2) is complete.
Lemma 2.3.Under the conditions of Proposition 2.1, it holds that where we have used Hölder's inequality, Young's inequality, Gagliardo-Nirenberg inequality, and Korn's inequality, then we can obtain by Grönwall's inequality that The proof is completed with the supports of Lemmas 2.1 and 2.2.
Lemma 2.4.Under the conditions of Proposition 2.1, it holds that where we have used Poincaré's inequality, Hölder's inequality, Young's inequality, Gagliardo-Nirenberg inequality, and Korn's inequality.Similarly, we can derive that Therefore, know from (2.9) and (2.19) that and it is easy to get that (see [42]) The proof of Proposition 2.1 has been finished.Next, we will proceed with the higher-order estimation involving the time weighting.

A priori estimates (II): Higher order estimates
Let σ(t) = min{1, t} and, from now on, the generic positive constant is defined by the right term of (2.3) as N C( u 0 2 We have the following third-order estimates.
Lemma 2.5.It holds that (2.22) Proof.It follows from the fact that First, we focus on u z,z .By (1.3) 1 , we have and taking divergence to (1.3) 1 , the elliptic problem can be obtained as follows: With the boundary condition obtained by (1.3) 2 and (1.4) 1 , we have It is clear to show by Lemma 2.5 [41] and the Trace theorem that The estimation of u • ∇u 2 L 2 (Ω f ) can be estimated by Hölder's inequality, Gagliardo-Nirenberg inequality, Young's inequality, and Korn's inequality as follows: Obviously, from (2.23) with Hölder's inequality, Young's inequality, Gagliardo-Nirenberg inequality, Poincáre's inequality, Sobolev inequality, and Korn's inequality we have , where we have used the estimation obtained based on (2.27) as follows: and, again, using Lemma 2.5 [41] together with the Trace theorem to get that Next, we will get the bound of each term at the righthand of (2.31) step by step.
Step 1. Next, we apply ∂ x ∂ y to (1.3) 1 and multiply u t,x,y before integrating the result equations over Ω f ; meanwhile, we apply ∂ x ∂ y to (1.2) and integrate it over Ω m after multiplying P 2,x,y on it.Finally, summing up the two resulting equations would come to (2.32), with the help of Hölder's inequality, Gagliardo-Nirenberg inequality, Young's inequality, and Korn's inequality, Multiplying (2.32) by σ(t) and integrating the result over (0, t), with integration by parts and Young's inequality, obtains that Similarly, we can get and Multiplying (2.36) by σ(t) and integrating the result over (0, t), with Lemma 2.3 and Young's inequality, obtains that (2.37) Step 3. Undoubtedly, from (2.25), (2.29), and (2.30), we get that which are derived by Hölder's inequality, Gagliardo-Nirenberg inequality, and Young's inequality.Thus, from (2.38), we could derive that which means that and, similarly, we have ) ds, and t 0 σ(s) D(u y,y,y ) 2 L 2 (Ω m ) ds can be obtained in a similar derivation.
Apply ∂ 2 x ∂ y to (1.3) 1 and multiply by σ(t)u x,x,y before integrating the resulting equations over Ω f .Meanwhile, apply ∂ 2 x ∂ y to (1.2) and multiply σ(t)∂ 2 x ∂ y P 2 on it after that, integrate over Ω m .Finally, summing up the above two resulting equations could come to the proof of which is reckoned from the Gagliardo-Nirenberg inequality and Young's inequality.It is easy to find that then, similarly, we have and it is more concisely to be derived that so that the proof of Lemma 2.5 is completed with Young's inequality.
Using a similar argument as that in the proof of Lemma 2.5, we can easily obtain the following fourth-order estimates.
Lemma 2.6.It holds that Proof.It follows from the fact that Obviously, from (2.23) with Hölder's inequality, Young's inequality, Gagliardo-Nirenberg inequality, Poincáre's inequality, Sobolev inequality, and Korn's inequality we have that where Additionally, we leverage Lemma 2.5 [41], coupled with the Trace theorem.The methodology employed here mirrors that of (2.26).We obtain such that one gets (2.45) Step 1.
We detail work on σ(t) x ∂ y P 2 , integrate the two equations with respect to x by parts, then add up the resulting formulas to get Step 2. For ∇ 2 u t L 2 (Ω f ) , we have (2.49) First, we apply ∂ x ∂ t to the Eq (1. 3) 1 and multiply by u t,t,x .Meanwhile, apply ∂ x ∂ t to (1.2) and multiply by ∂ x ∂ t P 2 , integrate the two equations with respect to x by parts, then we add up the resulting formulas to get where we have used Lemma 2.5.Next, it's time to deal with ∇∂ t P 1 L 2 (Ω f ) .We differentiate (2.24) and (2.25) with respect to t, and then using the standard L 2 -estimate for the elliptic system with (2.3), (2.51), and (2.52), we have (2.57) Step 3. We work on ∇u x,z,z L 2 (Ω f ) in this step.Applying the gradient operator ∇ to (2.38), then multiplying by σ(s) 2 and integrating over (0, t) yields We apply Lemma 2.5 in [41] together with the Trace theorem on (2.24) and (2.25), similarly as (2.39), to get that First of all, we know that the systems (1.3) and (1.4) have a unique local strong solution (u, P 2 ) on Ω × (0, T * ] for some T * > 0, so it's time to verify the continuity of the strong solution to extend it globally in time with counter-evidence. It follows from the fact that u 0 ∈ H 2 (Ω f ) and Proposition 1.1 that there exists a T 1 ∈ (0, T * ] such that (2.1) holds for T = T Finally, to finish the proof of Theorem 1.2, we have from (2.5) that 1 2 According to the Poincar inequality for three-dimensional cases, we have then substituting (3.6) into (3.5),we get

Conclusions
In conclusion, this study has made some progress in understanding the global well-posedness of a coupled Navier-Stokes-Darcy model with the Beavers-Joseph-Saffman-Jones interface boundary condition in three-dimensional Euclidean space.Through our investigation, we have achieved the establishment of a global strong solution for the system, marking a crucial advancement in the field.Moreover, we have demonstrated the exponential stability of this strong solution, further reinforcing its reliability.The implications of our findings extend to the analysis of subsurface flow problems, notably in the realm of karst aquifers, where such coupled systems play a pivotal role.By shedding light on the dynamics and behaviors of these systems, our research contributes to a deeper understanding of fluid flow phenomena in complex geological formations, offering valuable insights for both theoretical developments and practical applications in hydrogeology and related disciplines.

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) , first apply ∂ 2 t to (1.3) 1 and multiply by σ(t) 2 ∂ 2 t u.Meanwhile, apply ∂ 2 t to (1.2) and multiply by σ(t) 2 ∂ 2 t P 2 , then integrate it on Ω m .At last, integrate the two resulting equations with respect to x and t and sum them up to arrive at .59) Now, we respectively apply ∂ 2 x , ∂ x ∂ y , ∂ 2 y to (2.23) and multiply by σ(s) 2 to get 3. Proof of Theorem 1.2