Biot problem

Accepted Manuscript Analysis of the coupled Navier–Stokes/Biot problem

Aycil Cesmelioglu

PII: DOI: Reference:

S0022-247X(17)30700-X http://dx.doi.org/10.1016/j.jmaa.2017.07.037 YJMAA 21568

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

28 February 2017

Please cite this article in press as: A. Cesmelioglu, Analysis of the coupled Navier–Stokes/Biot problem, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.07.037

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Analysis of the coupled Navier-Stokes/Biot problem Aycil Cesmelioglu Oakland University, Department of Mathematics and Statistics, 146 Library Drive, Rochester, MI 48309

Abstract We analyze a weak formulation of the coupled problem defining the interaction between a free fluid and a poroelastic structure. The problem is governed by the time-dependent incompressible Navier-Stokes equations and the Biot equations. Under a small data assumption, existence and uniqueness results are proved and a priori estimates are provided. Keywords: Navier-Stokes, Darcy, Biot, poroelastic, weak formulation, existence 2010 MSC: 35Q30, 35Q35 1. Introduction. We consider a fully dynamic model for the interaction of an incompressible Newtonian fluid with a poroelastic material where the boundary is assumed to be fixed. The fluid flow is governed by the time-dependent incompressible Navier-Stokes equations. For the poroelastic material we use the Biot system with appropriate flow and stress couplings on the interface between the fluid and the poroelastic regions. This problem is a fully dynamic coupled system of mixed hyperbolic-parabolic type and inherits all the difficulties mathematically and numerically involved in the standard fluid-structure interaction and Stokes/Navier-Stokes-Darcy coupling. The literature is rich in works on related coupled problems and here we provide only a partial list of relevant publications. One related problem deals with the interaction of an incompressible fluid with a porous material and modeled by the coupling of the Stokes/Navier-Stokes equations to the Darcy Email address: [email protected]. (Aycil Cesmelioglu) Preprint submitted to Elsevier

July 17, 2017

equations. The steady-state case of this problem is analyzed mathematically in [20, 15, 2] and the time-dependent case in [13, 10, 11]. Another related problem is that of the fluid-structure interaction. The analysis of a weak solution for the time dependent coupling of the Stokes and the linear elasticity equations is discussed in [16] and for the flow of a coupling of time-dependent 2D incompressible NSE to the linearly viscoelastic or the linearly elastic Koiter shell in [27]. The two layered structure version of this problem is discussed in [28, 29]. In geosciences, aquifers and oil/gas reservoirs are porous and deformable affecting groundwater and oil/gas flow, respectively [25, 23, 18]. In biomedical sciences, blood flow is influenced by the porous and deformable nature of the arterial wall [3, 7, 8]. Therefore, mathematical models that are used to simulate these flow problems must account for both the effects of porosity and elasticity. The Navier-Stokes/Biot system is investigated numerically in [3] using a monolithic and a domain decomposition technique and the Stokes/Biot system is investigated in [9] using an operator splitting approach and in [12] using an optimization based decoupling strategy. In [31], variational formulations for the the Stokes/Biot system are developed using semi-group methods. Also a two-layered version was studied in [8]. In this paper, we focus on the coupling of the fully dynamic incompressible Navier-Stokes equations (for the free fluid) with the Biot system (for the poroelastic structure completely saturated with fluid). This coupled problem is the nonlinear version of the problem presented in [9, 12]. We construct a weak formulation and show the existence and uniqueness (local) of its solution under small data assumption. We note that this small data assumption is not needed if the fluid is represented by the linear Stokes equations rather than the Navier-Stokes equations and the result would also be global. We assume that the boundaries and the interface between the fluid and the poroelastic material are fixed. The proof proceeds by constructing a semidiscrete Galerkin approximations, obtaining the necessary a priori estimates, and passing to the limit. To the author’s knowledge, there is no such analysis for this fully dynamic nonlinear coupled system. The outline of this paper is as follows: In Section 2, we introduce the equations governing the problem and appropriate interface, boundary and initial conditions. The next section is devoted to notation and some wellknown results that are used in the forthcoming sections. Section 4 sets the assumptions on data, presents the weak formulation and shows that it is equivalent to the problem. Section 5 summarizes the main result of the 2

paper. Section 6 contains the proof of the existence and uniqueness results and a priori estimates for the weak solution. 2. Fluid-poroelastic model equations Let Ω ⊂ Rd , d = 2, 3 be an open bounded domain with Lipschitz continuous boundary ∂Ω. The domain Ω is made up of two regions Ωf , the fluid region, and Ωp , the poroelastic region, separated by a common interface ΓI = ∂Ωf ∩ ∂Ωp . Both Ωf and Ωp are assumed to be Lipschitz. See Figure 1. The first region Ωf is occupied by a free fluid and has boundary Γf such out ext in out that Γf = Γin represent the inlet and f ∪ Γf ∪ Γf ∪ ΓI , where Γf and Γf outlet boundary, respectively. The second region Ωp is occupied by a saturated poroelastic structure with boundary Γp such that Γp = Γsp ∪ Γext p ∪ ΓI , where Γsp ∪ Γext represents the outer structure boundary. p

Figure 1: Fluid-poroelastic domain

Fluid flow is governed by the time-dependent incompressible NavierStokes equations: ρf u˙ f − 2μf ∇ · D(uf ) + ρf uf · ∇uf + ∇pf = ff

in Ωf × (0, T ) ,

∇ · uf = 0 in Ωf × (0, T ) .

(1a) (1b)

Here uf denotes the velocity vector of the fluid, pf the pressure of the fluid, ρf the density of the fluid, μf the constant fluid viscosity, and ff the body force acting on the fluid. We used the dot above a symbol to denote the  time 1 derivative. The strain rate tensor D(uf ) is defined by D(uf ) = 2 ∇uf + 3

 (∇uf )T and the Cauchy stress tensor is given by σ f = 2μf D(uf ) − pf I. So (1a) can also be written as ρf u˙ f − ∇ · σ f + ρf uf · ∇uf = ff . Equation (1a) represents the conservation of linear momentum, while equation (1b) is the incompressibility condition that represents the conservation of mass. The poroelastic system is a fully dynamic coupled system of mixed hyperbolicparabolic type represented by the Biot model [5, 6]: ¨ − 2μs ∇ · D(η) − λs ∇(∇ · η) + α∇pp = fs ρp η

in Ωp × (0, T ) ,

(2a)

˙ − ∇ · K∇pp = fp (s0 p˙p + α∇ · η)

in Ωp × (0, T ) ,

(2b)

where η is the displacement of the structure and pp is the pore pressure of the fluid. Here, fp is the source/sink term and fs is the body force. The parameters μs and λs denote the Lam´e constants for the solid skeleton. The density of the saturated medium is denoted by ρp , and the hydraulic conductivity by K. In the Biot model, the first equation, (2a), is the momentum equation for the balance of forces and the second equation, (2b), is the diffusion equation of fluid mass. The total stress tensor for the poroelastic structure E is defined as σ p = σ E p − α pp I, where σ p is the elasticity stress tensor defined by σ E p = 2μs D(η) + λs (∇ · η)I. Therefore, (2a) can also be written as ¨ − ∇ · σ p = fs . ρp η The constrained specific storage coefficient is denoted by s0 and the BiotWillis constant by α, the latter is usually close to unity. In the subsequent discussion, we assume that the motion of the structure is small enough so that the domain is fixed at its reference position. All the physical parameters are assumed to be constant in space and time. Next, we prescribe boundary, interface and initial conditions where nf and np denote the outward unit normal vectors of Ωf and Ωp , respectively and nΓ is the unit normal vector of the interface ΓI pointing from Ωf to Ωp . Hence, nΓ = nf |ΓI = −np |ΓI . Furthermore tlΓ , l = 1, . . . , d − 1 denotes an orthonormal set of unit vectors on the tangent plane to ΓI . Boundary conditions: Since the boundary conditions have no significant effect on the fluid poroelastic interaction, for simplicity they are chosen such that the normal fluid 4

stress is prescribed on the inlet and outlet boundaries, the poroelastic structure is assumed to be fixed at the inlet and outlet boundaries and have zero tangential displacement on the external structure boundary, that is, on Γin f × (0, T ) ,

σ f nf = −Pin (t)nf ,

(3a)

on Γout f × (0, T ),

σ f nf = 0,

(3b)

on Γext f × (0, T ),

uf = 0 ,

(3c)

on Γsp × (0, T ) ,

K∇pp · np = 0, η = 0,  pp = 0, np · σ E p np = 0, tlp · η = 0, 1 ≤ l ≤ d − 1.

(3d)

on Γext p × (0, T ),

(3e)

Initial conditions: As initial conditions, we assume that everything is at rest in the beginning, that is, At t = 0 :

uf = 0,

pp = 0,

η = 0,

η˙ = 0 .

(4)

Interface conditions on ΓI × (0, T ): The interface conditions are given by uf · nΓ = (η˙ − K∇pp ) · nΓ , σ f nΓ = σ p nΓ ,

(5a) (5b)

nΓ · σ f nΓ = −pp ,

(5c)

˙ · tlΓ , 1 ≤ l ≤ d − 1, nΓ · σ f tlΓ = −β(uf − η)

(5d)

where β denotes the resistance parameter in the tangential direction. Condition (5a) is the continuity of normal flux that satisfies mass conservation, condition (5b) is the balance of stresses, that is, the total stresses of the fluid and the poroelastic medium must match at the interface. Condition (5c) guarantees the balance of normal components of the stress in the fluid phase across the interface. Finally condition (5d) is the Beavers-Joseph-Saffman condition [4, 30, 22] which assumes that the tangential stress of the fluid is proportional to the slip rate. More details on the interface conditions can be found in [26, 31]. 5

3. Notation and useful results. Let Ω ⊂ Rd , d = 2, 3 be a bounded, open connected domain with a Lipschitz continuous boundary ∂Ω. Let D(Ω) be the space of all infinitely differentiable functions with compact support in Ω. For s ∈ R, H s (Ω) denotes the standard Sobolev space of order s equipped with its standard seminorm | · |s,Ω and norm  · s,Ω . We denote the vector- and matrix-valued Sobolev spaces as follows: Hs (Ω) := [H s (Ω)]d×d

Hs (Ω) := [H s (Ω)]d ,

and still write | · |s,Ω and  · s,Ω for the corresponding seminorms and norms. When s = 0, instead of H 0 (Ω), H0 (Ω) and H0 (Ω) we write L2 (Ω), L2 (Ω) and L2 (Ω) and instead of | · |0,Ω and  · 0,Ω we write | · |Ω and  · Ω . The scalar product of L2 (Ω) is denoted by (·, ·)Ω . The spaces H(div; Ω) = {v ∈ L2 (Ω) : ∇ · v ∈ L2 (Ω)} and L3/2 (div; Ω) = {v ∈ L2 (Ω) : ∇ · v ∈ L3/2 (Ω)} are equipped with the graph norm. If Γ ⊂ ∂Ω, and v ∈ H 1/2 (Γ), we define the extension v˜ of v as v˜ = v on Γ, v˜ = 0 on ∂Ω\Γ and define the space of traces of all functions of H 1 (Ω) that vanish on ∂Ω\Γ as follows: 1/2

H00 (Γ) = {v ∈ L2 (Γ) : v˜ ∈ H 1/2 (∂Ω)}. We also define for any 1 ≤ r ≤ ∞, Lr (a, b; X) = {f measurable in : f Lr (a,b;X) < ∞}  1/r b equipped with the norm f Lr (a,b;X) = a f (t)rX dt for 1 ≤ r < ∞ and f L∞ (a,b;X) = ess supt∈[a,b] f (t)X . Furthermore, C(0, T ; X) denotes the set of all functions that are continuous into X and finally we define H 1 (a, b; X) = {f ∈ L2 (a, b; X) : f˙ ∈ L2 (a, b; X)}. 3.0.1. Useful results. Here we state inequalities and results to be used throughout the paper. More details can be found in [1, 17]. We define the following spaces for the weak solution: Xf = {v ∈ H1 (Ωf ) : v = 0 on Γext f }, Vf = {v ∈ Xf : ∇ · v = 0}, Xp = {ξ ∈ H1 (Ωp ) : ξ = 0 on Γsp , tlp · ξ = 0, 1 ≤ l ≤ d − 1 on Γext p }, Qf = L2 (Ωf ), Qp = {r ∈ H 1 (Ωp ) : r = 0 on Γext p }. 6

On these spaces, we have the following trace inequalities: ∀v ∈ Xf ,

vΓI

≤ T1 |v|1,Ωf

vΓin ≤ T2 |v|1,Ωf f

(6)

∀q ∈ Qp ,

qH 1/2 (ΓI ) ≤ T3 |q|1,Ωp ,

qΓsp ≤ T4 |q|1,Ωp ,

(7)

∀v ∈ Xp ,

ηΓI

≤ T5 |η|1,Ωp ,

(8)

the Poincar´e inequalities: ∀v ∈ Xf ,

vΩf ≤ P1 |v|1,Ωf ,

(9)

∀η ∈ Xp , ηΩp ≤ P2 |η|1,Ωp , ∀q ∈ Qp , qΩp ≤ P3 |q|1,Ωp ,

(10) (11)

a Sobolev inequality: ∀v ∈ Xf ,

vL4 (Ωf ) ≤ Sf |v|H1 (Ωf ) ,

(12)

|v|1,Ωf ≤ Kf D(v)Ωf ,

(13)

and finally a Korn’s inequality: ∀v ∈ Xf ,

where T1 − T5 , P1 − P3 , Sf and Kf are positive constants depending only on their corresponding domain. Theorem 3.1 (Gronwall’s inequality in integral form). Let  t ζ(s)ds, a.e. t ∈ (0, T ) ζ(t) ≤ B + C 0

where ζ is a continuous nonnegative function and B, C ≥ 0 are constants. Then ζ(t) ≤ BeCt . The next theorem is a compactness result that provides a strong convergence result which is used to pass to the limit in the nonlinear terms of the Galerkin solution. Theorem 3.2. [32, Corollary 4] Let X, B and Y be Banach spaces such that X ⊂ B ⊂ Y where the imbedding of X into B is compact. Let F be a bounded set in Lp (0, T ; X) where 1 ≤ p < ∞ and let the set { ∂f } be bounded in ∂t f ∈F 1 p L (0, T ; Y ). Then F is relatively compact in L (0, T ; B). 7

4. Weak Formulation. In this section we derive the weak formulation of the problem. But first, we introduce additional notation and present assumptions on the problem data. We assume that K ∈ L∞ (Ωp ) is independent of time, uniformly bounded and positive definite. There exists Kmin , Kmax > 0 such that ∀x ∈ Ωp ,

Kmin x · x ≤ Kx · x ≤ Kmax x · x.

(14)

Further, we assume that ff ∈ L2 (0, T ; L2 (Ωf )), fs ∈ L2 (0, T ; L2 (Ωp )), fp ∈ L2 (0, T ; L2 (Ωp )) and Pin ∈ L2 (0, T ; H 1/2 (Γin f )). The weak formulation we propose for the problem is the following: (WF1) Find uf ∈ L∞ (0, T ; L2 (Ωf )) ∩ L2 (0, T ; Xf ), pf ∈ L1 (0, T ; Qf ), η ∈ W 1,∞ (0, T ; L2 (Ωp )) × H 1 (0, T ; Xp ) and pp ∈ L∞ (0, T ; L2 (Ωp )) ∩ L2 (0, T ; Qp ) 3 where u˙ f ∈ L1 (0, T ; L 2 (Ωf )), such that for all v ∈ Xf , q ∈ Qf , ξ ∈ Xp and r ∈ Qp , (ρf u˙ f , v)Ωf + (2μf D(uf ), D(v))Ωf + ρf (uf · ∇uf , v)Ωf − (pf , ∇ · v)Ωf ¨ , ξ)Ωp + (2μs D(η), D(ξ))Ωp + (λs ∇ · η, ∇ · ξ)Ωp − (αpp , ∇ · ξ)Ωp + (ρs η ˙ r)Ωp + (K∇pp , ∇r)Ωp + (s0 p˙p + α∇ · η, ˙ · tlΓ , (v − ξ) · tlΓ ΓI + (η˙ − uf ) · nΓ , r ΓI + pp nΓ , v − ξ ΓI + d−1 l=1 β(uf − η) = − Pin (t)nΓ , v Γin + (ff , v)Ωf + (fs , ξ)Ωp + (fp , r)Ωp , f (∇ · uf , q)Ωf = 0, a.e. in (0, T ), and uf (0) = 0 a.e. in Ωf ,

η(0) = 0,

˙ η(0) = 0,

pp (0) = 0 a.e. in Ωp .

The reason for looking for a solution in L∞ (0, T ; L2 (Ωi )), i = f, p may not seem obvious at this point, since typically the solutions are sought in L2 (0, T ; X) where X is an appropriate Sobolev space, but we will prove that such a solution exists. 4.1. Equivalence of the weak formulation (WF1). The following proposition establishes the equivalence between the coupled problem and the weak formulation (WF1) proposed in the previous section.

8

Proposition 4.1. Let the data satisfy the assumptions listed in the previous section. Then each solution uf ∈ L∞ (0, T ; L2 (Ωf )) ∩ L2 (0, T ; Xf ) such 3 that u˙ f ∈ L1 (0, T ; L 2 (Ωf )), pf ∈ L1 (0, T ; Qf ), η ∈ W 1,∞ (0, T ; L2 (Ωp )) ∩ H 1 (0, T ; Xp ) and pp ∈ L∞ (0, T ; L2 (Ωp )) ∩ L2 (0, T ; Qp ) of the problem defined by (1a)-(1b), (2a)-(2b) and (3a)-(5d) is also a solution of the variational problem (WF1) and conversely. Proof. We first show sufficiency. To simplify the presentation, we included in Appendix A the justification of using Green’s formula in the following proof. Let (uf , pf , η, pp ) be a solution of the coupled problem defined by (1)-(5) satisfying the regularity stated in the proposition. We multiply (1a) by v ∈ Xf . After integration by parts and using (3a), (3b), v = 0 on Γext f and nf = nΓ on ΓI , we have (ρf u˙ f , v)Ωf + (2μf D(uf ), D(v))Ωf + ρf (uf · ∇uf , v)Ωf − (pf , ∇ · v)Ωf − σ f nΓ , v ΓI = (ff , v)Ωf − Pin (t), v Γin . (15) f Next we multiply (1b) by q ∈ Qf and integrate to get (∇ · uf , q)Ωf = 0.

(16)

d−1 l E l ext Observe that (3e) implies σ p np = σ E p np = l=1 (tp · σ p np ) · tp on Γp . Then s since tlp · ξ = 0 on Γext p , ξ = 0 on Γp and np = −nΓ on ΓI , multiplying (2a) by ξ ∈ Xp and integrating by parts yields ¨ , ξ)Ωp + (2μs D(η), D(ξ))Ωp + (λs (∇ · η), ∇ · ξ)Ωp − (αpp , ∇ · ξ)Ωp (ρp η + σ p nΓ , ξ ΓI = (fs , ξ)Ωp . (17) Multiplying (2b) by r ∈ Qp , integrating over Ωp and using (3d), r = 0 on Γext p , and np = −nΓ on ΓI , we obtain ˙ r)Ωp + (K∇pp , ∇r)Ωp + K∇pp · nΓ , r ΓI = (fp , r)Ωp . (18) (s0 p˙p + α∇ · η, Next, we rewrite the interface integrals using the interface conditions (5a)(5d). On ΓI , by (5c) and (5d), we have d−1 l l ˙ lΓ )tlΓ . σ f nΓ = (nΓ ·σ f nΓ )nΓ + d−1 l=1 (tΓ ·σ f nΓ )tΓ = −pp nΓ − l=1 (β(uf − η)·t (19) 9

Adding (15), (16), (17) and (18) while using (19) for σ f nΓ , v ΓI , (5a) for K∇pp ·nΓ , r ΓI and (5b) and (19) for σ p nΓ , ξ ΓI gives the weak formulation (WF1). For the converse, let (uf , pf , η, pp ) be a solution of (WF1). We pick first v ∈ D(Ωf ), r = 0 and ξ = 0, second v ∈ 0, r ∈ D(Ωp ) and ξ = 0 and last v = 0, r = 0 and ξ ∈ D(Ωp ). This gives (1a) on Ωf and (2a) and (2b) on Ωp in the sense of distributions. Next, we multiply (1a) with v ∈ Xf , (2a) with ξ ∈ Xp and (2b) with r ∈ Qp , apply Green’s formulas, add the outcomes and compare with (WF1). This gives ˙ · tlΓ , (v − ξ) · tlΓ ΓI pp nΓ , v − ξ ΓI + d−1 l=1 β(uf − η) + (η˙ − uf ) · nΓ , r ΓI + Pin (t)nf , v Γin (20) f out = − σ f nf , v Γin − σ p np , ξ Γext − K∇pp · np , r Γsp ∪ΓI p ∪ΓI f ∪Γf ∪ΓI

for all (v, ξ, r) ∈ Xf × Xp × Qp . If we let v = 0, ξ = 0 in (20), we get − K∇pp · np , r Γsp ∪ΓI = (η˙ − uf ) · nΓ , r ΓI , ∀r ∈ Qp .

(21)

The choice r ∈ Qp such that r|ΓI = 0 yields K∇pp · np , r Γsp = 0 which implies the first condition of (3d). Using this in (21), we get − K∇pp · np , r ΓI = (η˙ − uf ) · nΓ , r ΓI , ∀r ∈ Qp which yields (5a). This reduces (20) to pp nΓ , v − ξ ΓI +

d−1

˙ · tlΓ , (v − ξ) · tlΓ ΓI + Pin (t)nf , v Γin β(uf − η) (22) f

l=1 out − σ p np , ξ Γext , ∀v ∈ Xf , ∀ξ ∈ Xp . = − σ f nf , v Γin p ∪ΓI f ∪Γf ∪ΓI

Now we let v = 0 in (22). Then − pp nΓ , ξ ΓI −

d−1

˙ · tlΓ , ξ · tlΓ ΓI = − σ p np , ξ Γext β(uf − η) , ∀ξ ∈ Xp . p ∪ΓI

l=1

Since pp = 0 on Γext p , the choice ξ ∈ Xp such that ξ = 0 on ΓI implies p 0 = σ p np , ξ Γpext = np · σ p np , ξ · np Γpext = np · σ E p np , ξ · np Γext .

10

Therefore, we recover in the sense of distributions the second condition in (3e). This also yields −pp nΓ −

d−1

˙ · tlΓ )tlΓ , ξ ΓI = − σ p np , ξ ΓI , ∀ξ ∈ Xp . β((uf − η)

l=1

Therefore, −pp nΓ −

d−1

˙ · tlΓ )tlΓ = σ p nΓ β((uf − η)

(23)

l=1

holds in the sense of distributions. This reduces (22) to pp nΓ , v ΓI +

d−1

˙ · tlΓ , v · tlΓ ΓI + Pin (t)nf , v Γin β(uf − η) f

l=1

= − σ f nf , v ∂Ωf , ∀v ∈ Xf . Letting v ∈ Xf such that v = 0 on ΓI gives out . Pin (t)nf , v Γin = − σ f nf , v Γin f f ∪Γf

implies (3a). If we plug this in the above Picking v such that v = 0 on Γout f equation we get 0 = − σ f nf , v Γout concluding that σ f nf = 0 on Γout f ×(0, T ) f in the distributional sense. This gives (3b) and also implies that pp nΓ , v ΓI +

d−1

˙ · tlΓ , v · tlΓ ΓI = − σ f nf , v ΓI , ∀v ∈ Xf . β(uf − η)

l=1

Therefore, p p nΓ +

d−1

˙ · tlΓ )tlΓ = −σ f nf . (β(uf − η)

l=1

This compared to (23) implies (5b) and also gives (5c) and (5d) once dotted with nΓ and tlΓ , 1 ≤ j ≤ d − 1.

11

5. Main results. This section summarizes the main results of this paper. First, for the sake of simplicity, we define the following functions of time: C1 (t) =

C2 (t) =

 3T 2 K 2 2

f

4μf

f

 3T 2 K 2 2

2μf

f

1/2 3P12 Kf2 P32 1 ff (t)2Ωf + fp (t)2Ωp + fs (t)2Ωp , 4μf 2Kmin 2 (24a) 2 2 1/2 3P1 Kf P32 1 ˙ 2 2 2 ˙ ˙ + ff (t)Ωf + fp (t)Ωp + fs (t)Ωp , 2μf 2Kmin 2 (24b)

Pin (t)2Γin +

P˙ in (t)2Γin f

a.e. in (0, T ), and the following constant:  C3 =

1/2 Cj2 1 1 1 2 2 2 2 Pin (0)H 1/2 (Γin ) + ff (0)Ωf + fp (0)Ωp + fs (0)Ωp , ρf ρf 2s0 2ρs 00 f (24c)

where the constants T2 , Kf , P1 , P3 are defined in Section 3.0.1 and Cj is the continuity constant of the continuous lifting operator from H 1/2 (∂Ωf ) → H 1 (Ωf ). Observe that C1 , C2 and C3 depend only on the data of the problem. We now present out main existence and uniqueness result. Theorem 5.1. Assume that ff ∈ H 1 (0, T ; L2 (Ωf )), fs ∈ H 1 (0, T ; L2 (Ωp )), fp ∈ H 1 (0, T ; L2 (Ωf )) and Pin ∈ H 1 (0, T ; H 1/2 (Γin f )) and that the following small data condition holds: (1 +

T T /ρs T e )C2 2L2 (0,T ) + eT /ρs C32 ρs ρs 2μ3f 1 T + (1 + eT /ρs + 2 eT /ρs )C1 2L2 (0,T ) + C1 2L∞ (0,T ) < 2 4 6 . (25) ρs ρs 9ρf Sf Kf

Then, problem (WF1)has a unique solution (uf , pf , η, pp ) such that ρf ρs ˙ 2L∞ (0,T ;L2 (Ωp )) + μs D(η)2L∞ (0,T ;L2 (Ωp )) uf 2L∞ (0,T ;L2 (Ωf )) + η 2 2 s0 1 + pp 2L∞ (0,T ;L2 (Ωf )) + μf D(uf )2L2 (0,T ;L2 (Ωf )) + K1/2 ∇pp 2L2 (0,T ;L2 (Ωp )) 2 2  T ≤ 1 + eT /ρs C1 2L2 (0,T ) . (26) ρs 12

Furthermore, D(uf )L∞ (0,T ;L2 (Ωf )) <

μf , 3ρf Sf2 Kf3

(27)

ρf ρs ˙ 2L∞ (0,T ;L2 (Ωp )) u˙ f 2L∞ (0,T ;L2 (Ωf )) + ¨ η 2L∞ (0,T ;L2 (Ωp )) + μs D(η) 2 2 s0 1 + p˙p 2L∞ (0,T ;L2 (Ωf )) + μf D(u˙ f )2L2 (0,T ;L2 (Ωf )) + K1/2 ∇p˙p 2L2 (0,T ;L2 (Ωp )) 2 2 2T /ρ2s T Te 2 C3 , ≤ (1 + e2T /ρs )C2 2L2 (0,T ) + (28) ρs 2 and 1 ρf u˙ f L∞ (0,T ;L2 (Ωf )) + 2μf D(uf )L∞ (0,T ;L2 (Ωf )) κ + Sf2 uf 2L∞ (0,T ;H1 (Ωf )) + T1 T3 pp L∞ (0,T ;H 1 (Ωp )) + βT12 uf L∞ (0,T ;H1 (Ωf ))  ∞ (0,T ;L2 (Ω )) . ˙ L∞ (0,T ;H1 (Ωp )) + T2 Pin L∞ (0,T ;L2 (Γin + βT1 T5 η + f  f L )) f f (29)

pf L∞ (0,T ;L2 (Ωf )) ≤

The constants T1 − T5 , Kf , Sf , P1 , P3 used in the above estimates are defined in Section 3.0.1. 6. Proof of Theorem 5.1. The proof consists of multiple steps. The main idea is to use Galerkin’s method on the divergence-free version of the weak problem (WF1) in which the fluid pressure pf is eliminated. We will first present the divergence-free formulation (WF2) and introduce its Galerkin approximation (GF). Next we prove that there exists a unique maximal Galerkin solution by writing (GF) as a system of first order equations and applying the theory of ordinary differential equations. However this existence result holds only on a finite subiniterval of [0, T ]. Demonstrating a priori bounds for the Galerkin solution guarantees the validity of this existence result on the entire interval [0, T ] and also allows us to pass to the limit. At the end of this process, we obtain a solution uf , η and pp of the divergence-free weak formulation. We conclude the proof using an inf-sup condition to recover the fluid pressure pf that was eliminated from the weak formulation, proving the equivalence of (WF2) and (WF1). This last step also provides a priori estimates for pf . 13

6.1. A divergence-free weak formulation. For the analysis of the problem, we will focus on the following divergence free version of the formulation (WF1). (WF2) Find uf ∈ L∞ (0, T ; L2 (Ωf ))∩L2 (0, T ; Xf ), η ∈ W 1,∞ (0, T ; L2 (Ωp ))× H 1 (0, T ; Xp ) and pp ∈ L∞ (0, T ; L2 (Ωp )) ∩ L2 (0, T ; Qp ) with 3 u˙ f ∈ L1 (0, T ; L 2 (Ωf )) such that for all v ∈ Vf , ξ ∈ Xp and r ∈ Qp , (ρf u˙ f , v)Ωf + (2μf D(uf ), D(v))Ωf + ρf (uf · ∇uf , v)Ωf ¨ , ξ)Ωp + (2μs D(η), D(ξ))Ωp + (λs ∇ · η, ∇ · ξ)Ωp − (αpp , ∇ · ξ)Ωp + (ρs η ˙ r)Ωp + (K∇pp , ∇r)Ωp + (s0 p˙p + α∇ · η, ˙ · tlΓ , (v − ξ) · tlΓ ΓI + (η˙ − uf ) · nΓ , r ΓI + pp nΓ , v − ξ ΓI + d−1 l=1 β(uf − η) = − Pin (t)nΓ , v Γin + (ff , v)Ωf + (fs , ξ)Ωp + (fp , r)Ωp f a.e. in (0, T ), and uf (0) = 0 a.e. in Ωf ,

η(0) = 0,

˙ η(0) = 0,

pp (0) = 0 a.e. in Ωp .

Note that the unknown pressure pf is no longer in the weak formulation. Furthermore, it is obvious that any solution of (WF1) is a solution of (WF2). The converse will be proved in Section 6.6 using an inf-sup condition. 6.2. A semi-discrete Galerkin formulation of (WF2). The existence result is proved by constructing a sequence of approximate problems and then passing to the limit, that is, using the Galerkin method. Separability of Vf × Xp × Qp implies the existence of a basis {(vi , ξ i , ri )}i≥0 m consisting of smooth functions. We define Vfm , Xm p and Qp as the span of the first m vi , ξ i and ri ’s, respectively, and use the following Galerkin approximations for the unknowns uf , η and pp : um (x, t) =

m

j=1

αj (t)vj (x), η m (x, t) =

m

j=1

βj (t)ξj (x), pm (x, t) =

m

γj (t)rj (x).

j=1

(30) Then, we can write the Galerkin approximation of the problem (WF2) as follows:

14

1 m (GF) Find um ∈ C 1 (0, T ; Vfm ), η m ∈ C 2 (0, T ; Xm p ), pm ∈ C (0, T ; Qp ) such that

¨ m , ξ)Ωp (ρf u˙ m , v)Ωf + (2μf D(um ), D(v))Ωf + ρf (um · ∇um , v)Ωf + (ρs η + (2μs D(η m ), D(ξ))Ωp + (λs ∇ · η m , ∇ · ξ)Ωp − (αpm , ∇ · ξ)Ωp + (s0 p˙m + α∇ · η˙ m , r)Ωp + (K∇pm , ∇r)Ωp + pm nΓ , v − ξ ΓI ˙ m ) · tlΓ , (v − ξ) · tlΓ ΓI + (η˙ m − um ) · nΓ , r ΓI + d−1 l=1 β(um − η = − Pin (t)nΓ , v Γin + (ff , v)Ωf + (fs , ξ)Ωp + (fp , r)Ωp , f m for all (v, ξ, r) ∈ Vfm × Xm p × Qp , a.e. t ∈ (0, T ) and

um (0) = 0,

η m (0) = 0,

η˙ m (0) = 0,

pm (0) = 0.

(31)

Lemma 6.1. For each positive integer m, the formulation (GF) has a unique maximal solution (um , η m , pm ) ∈ C 1 (0, Tm ; Vfm ) × C 2 (0, Tm ; Xm p ) × 1 m C (0, Tm ; Qp ) for some time Tm where 0 < Tm ≤ T . Proof. Using the Galerkin expansions given in (30), the problem (GF) can be represented in matrix form. The following is a standard finite-dimensional argument which is basically defining the problem as a square first order system of ordinary differential equations (ODE) with an initial condition. For the integrals on the left hand side for 1 ≤ i, j ≤ m, we define Afij = ρf (vj , vi )Ωf , Bfij = 2μf (D(vj ), D(vi ))Ωf + dl=1 β vj · tlΓ , vi · tlΓ ΓI , Ni = (ρf (vj · ∇vk , vi )Ωf )1≤j,k≤m , Asij = ρs (ξ j , ξ i )Ωp ,

Bsij = 2μs (D(ξ j ), D(ξ i ))Ωp + λs (∇ · ξj , ∇ · ξi )Ωp ,

Apij = s0 (rj , ri )Ωp ,

Bpij = (K∇rj , ∇ri )Ωp ,

Cij = α(rj , ∇ · ξi )Ωp + rj nΓ , ξ i ΓI , Dij = rj nΓ , vi ΓI , Eij = dl=1 β ξ j · tlΓ , vi · tlΓ ΓI , Fij = dl=1 β ξ j · tlΓ , ξ i · tlΓ ΓI . And finally for the right hand side integrals we define ai = − Pin (t)nf , vi Ωf + (ff , vi )Ωf ,

bi = (fs , ξ i )Ωp ,

ci = (fp , ri )Ωp .

The unknowns are αi = αi (t), β i = βi (t) and γ i = γi (t), i = 1, . . . , m and we define w(t) = [α(t), β(t), γ(t), θ(t)]T and set (N (w))i = Ni α · α. With 15

these definitions, (GF) is equivalent to the following system of autonomous first order ODE in w(t): ˙ = M−1 d(w) − M−1 Nw =: g(w), where w(0) is w ⎡ f ⎤ ⎡ A 0 0 Bf 0 0 D ⎢ 0 I 0 ⎥ ⎢ 0 ⎥ 0 0 ⎢ 0 M=⎢ ⎣ 0 0 Ap 0 ⎦ , N = ⎣ −DT 0 Bp 0 0 0 As −ET Bs −C

given, ⎤ −E −I ⎥ ⎥ CT ⎦ 0

and d(w) = [a − N (w), 0, c, b] . Since ρf , ρs and s0 are positive, Af , Ap and As are symmetric positive definite implying that M is invertible. The matrices M, N are 4m×4m and the vectors d, w have length 4m. It is obvious that the function g is continuous in time and locally Lipschitz continuous in w. Then, it follows from the theory of ordinary differential equations [14] that there is a unique maximal solution w in the interval [0, Tm ] for some Tm such that 0 < Tm ≤ T such that each component of w, i.e., each component of α, β, γ and θ = β˙ belongs to C 1 (0, Tm ). We need a priori bounds on the Galerkin solution to conclude that Tm = T . We discuss this next in Section 6.3. Remark 6.2. Note that if we consider the Stokes problem for the fluid part, so if there is no nonlinearity, an existence and uniqueness result will be global on [0, T ]. 6.3. A priori estimates for the Galerkin solution. We begin by stating the main result of this section. Theorem 6.3. Suppose that ff ∈ H 1 (0, T ; L2 (Ωf )), fs ∈ H 1 (0, T ; L2 (Ωp )), 1/2 fp ∈ H 1 (0, T ; L2 (Ωp )) and Pin ∈ H 1 (0, T ; H00 (Γin f )). In addition, assume that the small data condition (25) holds. Then, problem (GF) has a unique solution (um , η m , pm ) in the interval [0, T ]. Furthermore, it satisfies the following bounds: ρf ρs λs s0 um 2Ωf + η˙ m 2Ωp + μs D(η m )2Ωp + ∇ · η m 2Ωp + pm 2Ωp 2 2 2 2 1 1/2 + K ∇pm 2L2 (0,T ;L2 (Ωp )) + μf D(um )2L2 (0,T ;L2 (Ωf )) 2  T T /ρs C1 2L2 (0,T ) (32) ≤ 1+ e ρs 16

for all t ∈ [0, T ], D(um )Ωf <

μf , 3ρf Sf2 Kf3

(33)

and ρf ρs λs s0 u˙ m 2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp + ∇ · η˙ m 2Ωp + p˙m 2Ωp 2 2 2 2 1 + μf D(u˙ m )2L2 (0,T ;L2 (Ωf )) + K1/2 ∇p˙m 2L2 (0,T ;L2 (Ωp )) 2 T /ρs T Te C3 . ≤ (1 + eT /ρs )C2 2L2 (0,T ) + (34) ρs 2 Here C1 , C2 and C3 are defined in (24) and (24c). Remark 6.4. These bounds imply that {um } is bounded in H 1 (0, T ; H1 (Ωf )), ¨ m is bounded in L∞ (0, T ; L2 (Ωp )) and {η m } is bounded in H 1 (0, T ; H1 (Ωp )), η {pm } is bounded in H 1 (0, T ; H 1 (Ωp )). In the next few sections, we verify the bounds (32), (33) and (34) in the interval [0, Tm ] which will then imply the global existence of the maximal solution (um , η m , pm ) in the interval [0, T ] as stated in the theorem. 6.3.1. Proof of (32). We let v = um , ξ = η˙ m , r = pm in the Galerkin formulation (GF). Then the Cauch-Schwarz inequality, inequalities (6), (9), (11), (12), (13) and assumption (14) on K imply ρf (u˙ m , um )Ωf + 2μf D(um )2Ωf + ρs (¨ η m , η˙ m )Ωp + 2μs (D(η m ), D(η˙ m ))Ωp + λs (∇ · η m , ∇ · η˙ m )Ωp + s0 (p˙m , pm )Ωp + K1/2 ∇pm 2Ωp ≤ ρf Sf2 Kf3 D(um )3Ωf + T2 Kf Pin (t)Γin D(um )Ωf f + P1 Kf ff Ωf D(um )Ωf + fs Ωp η˙ m Ωp +

P3 1/2 Kmin

(35) fp Ωp K1/2 ∇pm Ωp .

Here the only problematic terms on the right hand side are D(um )3Ωf and η˙ m Ωp . The rest can easily be hidden in the left hand side. Observe that since um (0) = 0 and um is continuous, there exists a time T m , 0 < T m ≤ Tm such that μf D(um )Ωf < ∀t ∈ [0, T m ]. (36) 3ρf Sf2 Kf3 17

In fact, this condition holds true on [0, Tm ]. For the sake of presentation, we postpone this proof to Section 6.3.2. Using this condition together with Young’s inequality with > 0, we obtain ρf (u˙ m , um )Ωf + 2μf D(um )2Ωf + ρs (¨ η m , η˙ m )Ωp + 2μs (D(η m ), D(η˙ m ))Ωp + λs (∇ · η m , ∇ · η˙ m )Ωp + s0 (p˙m , pm )Ωp + K1/2 ∇pm 2Ωp 3T22 Kf2 3P12 Kf2 1 ≤ μf D(um )2Ωf + K1/2 ∇pm 2Ωp + Pin (t)2Γin + ff 2Ωf f 2 4μf 4μf P32 1 1 1 + fp 2Ωp + fs 2Ωp + η˙ m 2Ωp = C12 (t) + η˙ m 2Ωp (37) 2Kmin 2 2 2 for all t ∈ [0, Tm ], where C1 is defined in (24). Integrating with respect to t, due to (31), we can deduce  2 1 t 2 2 η˙ m (t)Ωp ≤ C1 L2 (0,T ) + η˙ m (s)2Ωp ds. ρs ρs 0 ˙ m (t)2Ωp ∈ C 1 (0, T ) and applying TheTherefore, since η m ∈ C 2 (0, T ; Xm p ), η 1 2 orem 3.1 with ζ(t) = η˙ m (t)2Ωp , C = and B = C1 2L2 (0,T ) yields ρs ρs 2 η˙ m (t)2Ωp ≤ eT /ρs C1 2L2 (0,T ) . (38) ρs Plugging this in (37), we have ρf d d ρs d um 2Ωf + μf D(um )2Ωf + η˙ m 2Ωp + μs D(η m )2Ωp 2 dt 2 dt dt d λs d s 1 0 ∇ · η m 2Ωp + pm 2Ωp + K1/2 ∇pm 2Ωp + 2 dt 2 dt 2 1 ≤ C12 (t) + eT /ρs C1 2L2 (0,T ) ρs

(39)

for all t ∈ [0, Tm ]. After integrating with respect to t and using (31) implies the bound (32). 6.3.2. Proof of (33) and (34). Proof. Recalling (36), assume for a contradiction that there exists T ∗ ∈ (0, Tm ] such that μf μf , D(um )(T ∗ )Ωf = . ∀t ∈ [0, T ∗ ), D(um )(t)Ωf < 2 3 3ρf Sf Kf 3ρf Sf2 Kf3 (40) 18

Similar arguments leading to (39) and using Young’s inequality yield 2μf D(um )2Ωf ρf ρs λs s0 ≤ u˙ m 2Ωf + η˙ m 2Ωp + μs D(η m )2Ωp + ∇ · η m 2Ωp + pm 2Ωp 2 2 2 2 ρf ρ λ s0 s s η m 2Ωp + μs D(η˙ m )2Ωp + ∇ · η˙ m 2Ωp + p˙m 2Ωp + um 2Ωf + ¨ 2 2 2 2 1 + C12 (t) + eT /ρs C1 (t)2L2 (0,T ) (41) ρs for all t ∈ [0, T ∗ ]. To bound the first five terms we differentiate (GF) with respect to time. The specifics of this technique can be found in detail in [24]. ¨ m , v)Ωf + (2μf D(u˙ m ), D(v))Ωf + ρf (u˙ m · ∇um , v)Ωf + ρf (um · ∇u˙ m , v)Ωf (ρf u ... + (ρs η m , ξ)Ωp + (2μs D(η˙ m ), D(ξ))Ωp + (λs ∇ · η˙ m , ∇ · ξ)Ωp − (αp˙m , ∇ · ξ)Ωp ¨ m , r)Ωp + (K∇p˙m , ∇r)Ωp + p˙m nΓ , v − ξ ΓI + (s0 p¨m + α∇ · η d−1 ¨ m ) · tlΓ , (v − ξ) · tlΓ ΓI + (¨ + l=1 β(u˙ m − η η m − u˙ m ) · nΓ , r ΓI = − P˙in nΓ , v Γin + (f˙f , v)Ω + (f˙s , ξ)Ωp + (f˙p , r)Ωp . f

f

¨ m in the above, use the Cauch-Schwarz We let v = u˙ m , r = p˙m and ξ = η inequality, assumption (40), inequalities (6), (9), (11), (12), (13) and assumption (14) on K to get ρf d d ρs d u˙ m 2Ωf + 2μf D(u˙ m )2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp 2 dt 2 dt dt λs d s0 d 2 2 1/2 2 ∇ · η˙ m Ωp + p˙m Ωp + K ∇p˙m Ωp + 2 dt 2 dt

T22 Kf2 2μf 1 1 D(u˙ m )2Ωf + D(u˙ m )2Ωf + P˙ in 2Γin + D(u˙ m )2Ωf ≤ f 3 2

2 2

2 2

P1 Kf 1 P32 f˙f 2Ωf + f˙s Ωp ¨ + η m Ωp + K1/2 ∇p˙m 2Ωp + f˙p 2Ωp . 2 2 2Kmin

19

Picking =

3 , we obtain μf

ρf d d ρs d u˙ m 2Ωf + μf D(u˙ m )2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp 2 dt 2 dt dt d λs d s 1 0 ∇ · η˙ m 2Ωp + p˙m 2Ωp + K1/2 ∇p˙m 2Ωp + 2 dt 2 dt 2 2 2 3T22 Kf2 3P K 1 1 P32 1 f ˙ 2 η m 2Ωp + ≤ P˙ in 2Γin + ff Ωf + f˙s 2Ωp + ¨ f˙p 2Ωp f 2μf 2μf 2 2 2Kmin 1 η m 2Ωp , = C22 (t) + ¨ (42) 2 ¨ m . Since where C2 ∈ L2 (0, T ) is defined in (24). Now we need a bound for η ¨ m is continuous, we can use Gronwall’s inequality again. by assumption η Integrating from 0 to t for any t ∈ [0, T ∗ ] and recalling (31) we obtain ρf ρs λs s0 u˙ m 2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp + ∇ · η˙ m 2Ωp + p˙m 2Ωp 2 2 2 2 ρf ρs s0 2 2 2 η (0)Ωp + p˙m (0)Ωp ≤ u˙ m (0)Ωf + ¨ 2 2 m 2  1 t 2 + C2 L2 (0,T ) + ¨ η m (s)2Ωp ds. (43) 2 0 η m (0)2Ωp and p˙m (0)2Ωp . For that Now we need bounds for u˙ m (0)2Ωf , ¨ ¨ m (0) in (GF) at t = 0. Due to (31) purpose we let v = 0, r = 0 and ξ = η and the Cauchy-Schwarz inequality ¨ η m (0)Ωp ≤

1 fs (0)Ωp . ρs

(44)

Now let v = ξ = 0 and r = p˙ m (0) and t = 0 in (GF). Then by (31) we have p˙m (0)Ωp ≤

1 fp (0)Ωp . s0

(45)

Furthermore if we let η = 0, r = 0, v = u˙ m (0) and t = 0 in (GF), we get ρf u˙ m (0)2Ωf = − Pin (0)nΓin , u˙ m (0) Γin + (ff (0), u˙ m (0))Ωf . f f We need a good bound for Pin nΓ , u˙ m (0) Γin because otherwise u˙ m (0)Γin f f 1/2

cannot be hidden in the left hand side. Assuming Pin ∈ L2 (0, T ; H00 (Γin f )), 20

its extension P˜in by zero to ∂Ωf is in L2 (0, T ; H 1/2 (∂Ωf )). Then we can use the continuous lifting operator j : L2 (0, T ; H 1/2 (∂Ωf )) → L2 (0, T ; H 1 (Ωf )) with continuity constant Cj . Since ∇ · u˙ m = 0, Pin (0)nΓin , u˙ m (0) Γin = j(P˜in (0))n∂Ωf , u˙ m (0) ∂Ωf = (∇(j(P˜in (0))), u˙ m (0))Ωf f f ≤ j(P˜in (0))H 1 (Ωf ) u˙ m (0)Ωf ≤ Cj Pin (0)H 1/2 (Γin ) u˙ m (0)Ωf . 00

Then u˙ m (0)Ωf ≤

f

Cj 1 Pin (0)H 1/2 (Γin ) + ff (0)Ωf . 00 f ρf ρf

(46)

Plugging these in (43), ρs λs s0 ρf u˙ m 2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp + ∇ · η˙ m 2Ωp + p˙m 2Ωp 2 2 2 2  t 1 ≤ C32 + C2 2L2 (0,T ) + ¨ η m (s)2Ωp ds ρs 0 where C3 is defined in (24c). If we neglect all terms other than the second one on the left hand side of the above equation we get  2 2 2 1 t 2 2 ¨ η m Ωp ≤ C3 + C2 L2 (0,T ) + ¨ η m (s)2Ωp ds. ρs ρs ρs 0 Using Gronwall’s inequality with B = ζ(t) = ¨ η m (t)2Ωp , ¨ η m (t)2Ωp ≤eT /ρs

2 ρs

2 2 2 1 C3 + C2 2L2 (0,T ) , C = and ρs ρs ρs

C2 2L2 (0,T ) +

2 2 C . ρs 3

Therefore, putting this in (42) we have d ρs d ρf d u˙ m 2Ωf + μf D(u˙ m )2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp 2 dt 2 dt dt λs d s0 d 1 2 ∇ · η˙ m Ωp + p˙m 2Ωp + K1/2 ∇p˙m 2Ωp + 2 dt 2 dt 2  T /ρs  e ≤ C22 (t) + C2 2L2 (0,T ) + C32 . (47) ρs 21

Integrating from 0 to t in [0, T ∗ ] and recalling the bounds (44), (45) and (46) we obtain ρf ρs λs s0 u˙ m 2Ωf + ¨ η m 2Ωp + μs D(η˙ m )2Ωp + ∇ · η˙ m 2Ωp + p˙m 2Ωp 2 2 2  2  t t 1 1/2 K ∇p˙m 2Ωp dt + μf D(u˙ m )2Ωf dt + 0 0 2 T eT /ρs 2 T C3 , (48) ≤ (1 + eT /ρs )C2 2L2 (0,T ) + ρs ρs which implies (34). To find a bound for um Ωf in (41), we recall (35) and use assumption (40) to get (39) for all t ∈ [0, T ∗ ]. Neglecting the terms other than the ones with d/dt on the left hand side, we integrate (39) from 0 to t where t ∈ [0, T ∗ ]. Since the initial conditions are zero, we have ρf ρs λs s0 um 2Ωf + η˙ m 2Ωp + μs D(η m )2Ωp + ∇ · η m 2Ωp + pm 2Ωp 2 2 2 2 T T /ρs ≤ (1 + e )C1 2L2 (0,T ) . (49) ρs Therefore, using (48) and (49) in (41) we get   1 T 1  2 2 C1 (t) + 1 + eT /ρs + eT /ρs C1 2L2 (0,T ) D(um )Ωf ≤ 2μf ρs ρs   T T eT /ρs 2  + 1 + eT /ρs C2 2L2 (0,T ) + C3 ρs ρs for all t ∈ [0, T ∗ ]. So if the data satisfy   1  1 T C1 2L∞ (0,T ) + 1 + eT /ρs + eT /ρs C1 2L2 (0,T ) 2μf ρs ρs   μ2f T T eT /ρs 2  + 1 + eT /ρs C2 2L2 (0,T ) + C3 < 2 4 6 , ρs ρs 9ρf Sf Kf then we have D(um )Ωf <

μf 3ρf Sf2 Kf3

for all t ∈ [0, T ∗ ] which contradicts with the assumption. Remark 6.5. Note that if Pin (0) = 0, we immediately get u˙ m (0)Ωf ≤ 1 ff (0)Ωf rather than (46). In that case, it is not necessary to assume ρf 1/2 that Pin ∈ L2 (0, T ; H00 (Γin f )). 22

6.4. Passing to the limit. Since we established the boundedness of the Galerkin solution (um , η m , pm ) that we stated in Remark 6.4, we can pass to the limit in (GF) to obtain a solution to (WF2). Reflexivity of Vf , Xp and Qp and therefore of H 1 (0, T ; Vf ), H 1 (0, T ; Xp ) and H 1 (0, T ; Qp ) imply that there exists a subsequence of the Galerkin solution, still denoted by {(um , η m , pm )}m such that um ηm ¨m η pm

   

uf η ¨ η pp

in in in in

H 1 (0, T ; H1 (Ωf )), H 1 (0, T ; H1 (Ωp )), L2 (0, T ; L2 (Ωp )), H 1 (0, T ; H 1 (Ωp ))

where  stands for weak convergence. Note that due to the continuity of the trace operator we also have um  uf

in L2 (0, T ; H 1/2 (∂Ωf )),

η˙ m  η˙

in L2 (0, T ; H 1/2 (∂Ωp )),

pm  pp

in L2 (0, T ; H 1/2 (∂Ωp )).

We can pass to the limit in the linear terms with the convergence results above. However, the nonlinear term needs a stronger convergence result. To obtain such a result we recall that H1 (Ωf ) ⊂ L4 (Ωf ) ⊂ L2 (Ωf ) and that H1 (Ωf ) is compactly embedded in L4 (Ωf ) due to the Sobolev embedding theorem. Furthermore, Remark 6.4 implies that the subsequence {um }m is bounded in L4 (0, T ; H1 (Ωf )) and { ∂u∂tm }m is bounded in L1 (0, T ; L2 (Ωf )). Therefore, Theorem 3.2 implies that {um }m has a subsequence, still denoted the same, such that um → uf strongly in L4 (0, T ; L4 (Ωf )).

23

Fix an integer k ≥ 1. We multiply (GF) by ψ(t) ∈ L2 (0, T ) and integrate in time to obtain T T (ρf u˙ m , ψ(t)v)Ωf dt + 0 (2μf D(um ), ψ(t)D(v))Ωf dt 0 T T ¨ m , ψ(t)ξ)Ωp dt + 0 ρf (um · ∇um , ψ(t)v)Ωf dt + 0 (ρs η T T + 0 (2μs D(η m ), ψ(t)D(ξ))Ωp dt + 0 (λs ∇ · η m , ψ(t)∇ · ξ)Ωp dt T T − 0 (αpm , ψ(t)∇ · ξ)Ωp dt + 0 (s0 p˙m + α∇ · η˙ m , ψ(t)r)Ωp dt T T + 0 (K∇pm , ψ(t)∇r)Ωp dt + 0 pm nΓ , ψ(t)(v − ξ) ΓI dt T ˙ m ) · tlΓ , ψ(t)(v − ξ) · tlΓ ΓI dt + d−1 (50) l=1 0 β(um − η T T + 0 (η˙ m − um ) · nΓ , ψ(t)r ΓI dt = − 0 Pin (t)nΓ , ψ(t)v Γin dt f T T T + 0 (ff , ψ(t)v)Ωf dt + 0 (fs , ψ(t)ξ)Ωp dt + 0 (fp , ψ(t)r)Ωp dt, for all (v, ξ, r) ∈ Vfk × Xkp × Qkp , m ≥ k. Letting m → ∞ we obtain T 0

T (ρf u˙ f ,ψ(t)v)Ωf dt + 0 (2μf D(uf ), ψ(t)D(v))Ωf dt T T ¨ , ψ(t)ξ)Ωp dt + 0 ρf (uf · ∇uf , ψ(t)v)Ωf dt + 0 (ρs η T T + 0 (2μs D(η), ψ(t)D(ξ))Ωp dt + 0 (λs ∇ · η, ψ(t)∇ · ξ)Ωp dt T T ˙ ψ(t)r)Ωp dt − 0 (αpp , ψ(t)∇ · ξ)Ωp dt + 0 (s0 p˙p + α∇ · η, T T + 0 (K∇pp , ψ(t)∇r)Ωp dt + 0 pp nΓ , ψ(t)(v − ξ) ΓI dt T ˙ · tlΓ , ψ(t)(v − ξ) · tlΓ ΓI dt + d−1 (51) l=1 0 β(uf − η) T T + 0 (η˙ − uf ) · nΓ , ψ(t)r ΓI dt = − 0 Pin (t)nΓ , ψ(t)v Γin dt f T T T + 0 (ff , ψ(t)v)Ωf dt + 0 (fs , ψ(t)ξ)Ωp dt + 0 (fp , ψ(t)r)Ωp dt,

for all (v, ξ, r) ∈ Vfk × Xkp × Qkp . Since any element of V, Xp , Qp can be approximated by elements of Vfk , Xkp , Qkp and ψ ∈ L2 (0, T ) is arbitrary, this also holds for all (v, ξ, r) ∈ Vf ×Xp ×Qp a.e. in (0, T ). Therefore, we recover the equation in (WF2). Last, we check whether the initial conditions are satisfied. In (51) we ˙ ) = 0 and integrate the first let ψ ∈ C 2 ([0, T ]) such that ψ(T ) = 0 and ψ(T and the eighth terms once and the fourth term twice with respect to time. Then we do the same in (50) and also take the limit as m → ∞ using the convergence properties established in the beginning of the proof and using 24

(31). Comparing the resulting equations yield: ˙ ˙ − (ρf u(0), ψ(0)v)Ωf − (ρs η(0), ψ(0)ξ)Ωp + (ρs η(0), ψ(0)ξ) Ωp − (s0 p(0), ψ(0)r)Ωp = 0, ˙ for all v ∈ V, ξ ∈ Xp , r ∈ Qp . Since ψ(0) and ψ(0) are arbitrary, ˙ −(ρf u(0), v)Ωf − (ρs η(0), ξ)Ωp + (ρs η(0), ξ)Ωp − (s0 p(0), r)Ωp = 0, for all v ∈ V, ξ ∈ Xp , r ∈ Qp . This yields the initial conditions stated in (WF2). Finally passing to the limit in (32), (33) and (34), we obtain (26), (27) and (28). 6.5. Uniqueness. For the Stokes flow, there is no issue of uniqueness. For the Navier-Stokes problem, we can only prove uniqueness for restricted solution. Theorem 6.6 (Local uniqueness). Let (uf , η, pp ) be a solution of (WF2). Then if μf D(uf )Ωf ≤ 2 3 , (52) Sf Kf then (uf , η, pp ) is unique. Proof. Let (u1 , η 1 , p1 ) and (u2 , η 2 , p2 ) be two solutions of (WF2) such that (52) holds. Then w = u1 − u2 , θ = η 1 − η 2 and φ = p1 − p2 satisfy: ˙ v)Ωf + (2μf D(w), D(v))Ωf + ρf (u1 · ∇u1 − u2 · ∇u2 , v)Ωf − (φ, ∇ · v)Ωf (ρf w, ¨ ξ)Ωp + (2μs D(θ), D(ξ))Ωp + (λs ∇ · θ, ∇ · ξ)Ωp − (αφ, ∇ · ξ)Ωp + (ρs θ, ˙ r)Ωp + (K∇φ, ∇r)Ωp + φnΓ , v − ξ Γ + (s0 φ˙ + α∇ · θ, I d−1 l l ˙ ˙ + l=1 β(w − θ) · tΓ , (v − ξ) · tΓ ΓI + (θ − w) · nΓ , r ΓI = 0 ˙ r = φ where for all v ∈ Vf , ξ ∈ Xp , r ∈ Qp . Letting v = w, ξ = θ, ˙ w(0) = 0, θ(0) = 0, θ(0) = 0, φ(0) = 0 and using (12), (13) and (52), ρf d d ρs d ˙ 2 w2Ωf + 2μf D(w)2Ωf + θΩp + μs D(θ)2Ωp 2 dt 2 dt dt λs d s0 d 2 2 ∇ · θΩp + φΩp ≤ −ρf (w · ∇u1 + u2 · ∇w, w)Ωf + 2 dt 2 dt ≤ Sf2 Kf3 D(w)2Ωf D(u1 )Ωf + Sf2 Kf3 D(u2 )Ωf D(w)2Ωf ≤ 2μf D(w)2Ωf . 25

Therefore, d ρf d ρs d ˙ 2 λs d s0 d w2Ωf + θΩp + μs D(θ)2Ωp + ∇ · θ2Ωp + φ2Ωp ≤ 0 2 dt 2 dt dt 2 dt 2 dt which implies using the initial conditions that ρf ρs ˙ 2 λs s0 2 w2Ωf + θ ∇ · θ2Ωp + φ2Ωp ≤ 0 Ωp + μs D(θ)Ωp + 2 2 2 2 Therefore w = 0, θ = 0, φ = 0. 6.6. Existence of a Navier-Stokes pressure. Next we prove that from any solution of (WF2) we can recover a solution of (WF1). In fact, an inf-sup condition is sufficient to show the existence of a pf which was eliminated when we restricted the search space L2 (0, T ; Xf ) to L2 (0, T ; Vf ). The following inf-sup condition holds [20, 11]: There exists a constant κ > 0 such that inf sup

q∈Qf v∈Xf

(∇ · v, q)Ωf ≥ κ. vH 1 (Ωf ) qL2 (Ωf )

Note that we can replace the supremum above with the supremum over all (v, ξ, r) ∈ Xf × Xp × Qp . In fact, the supremum is attained when ξ = 0 and r = 0. Lemma 6.7. If (uf , η, pp ) is a solution of problem (WF2), then there exists a unique pf ∈ L∞ (0, T ; Qf ) such that (uf , pf , η, pp ) is a solution of problem (WF1). Proof. Let’s define a mapping F such that for all (v, ξ, r) ∈ Xf × Xp × Qp F(v, ξ, r) = (ρf u˙ f , v)Ωf + (2μf D(uf ), D(v))Ωf + ρf (uf · ∇uf , v)Ωf ¨ , ξ)Ωp + (2μs D(η), D(ξ))Ωp + (λs ∇ · η, ∇ · ξ)Ωp − (αpp (t), ∇ · ξ)Ωp + (ρs η ˙ r)Ωp + (K∇pp , ∇r)Ωp + (s0 p˙p + α∇ · η, ˙ · tlΓ , (v − ξ) · tlΓ ΓI + (η˙ − uf ) · nΓ , r ΓI + pp nΓ , v − ξ ΓI + d−1 l=1 β(uf − η) + Pin (t)nf , v Γin − (ff , v)Ωf − (fs , ξ)Ωp − (fp , r)Ωp , a.e. in (0, T ). f It is straightforward to see that F is linear and continuous on Xf ×Xp ×Qp for a.e. t ∈ (0, T ). Furthermore, F(v, ξ, r) = 0 for any (v, ξ, r) ∈ Vf × Xf × Qp 26

for a.e. t ∈ (0, T ). Therefore, the theory of Babuska-Brezzi imply for a.e. t ∈ (0, T ) that there exists a unique function pf ∈ Qf such that (pf , ∇ · v)Ωf = F(v, ξ, r), ∀(v, ξ, r) ∈ Xf × Xf × Qp ,

(53)

that is, there is a unique pf ∈ L∞ (0, T ; Qf ) such that (uf , pf , η, pp ) is a solution of problem (WF1). Furthermore, letting r = 0, ξ = 0 in (53) and using the inf-sup condition again we have 1 pf Ωf ≤ ρf u˙ f Ωf + 2μf D(uf )Ωf + Sf2 uf 21,Ωf + T1 T3 pp 1,Ωp κ  ˙ 1,Ωp + T2 Pin Γin +βT12 uf 1,Ωf + βT1 T5 η + f  f Ωf , f a.e. in (0, T ) which implies the bound (29) on pf in Theorem 5.1. This concludes the proof of Theorem 5.1. Appendix A. The interface conditions. In this section we prove that the interface conditions are meaningful for a solution of (1)-(5) and therefore justify the derivation of the weak formulation in the proof of Proposition 4.1. Let us consider a solution of (1)-(5) such that uf ∈ L∞ (0, T ; L2 (Ωf )) ∩ L2 (0, T ; Xf ), η ∈ W 1,∞ (0, T ; L2 (Ωp ))×H 1 (0, T ; Xp ) and pp ∈ L∞ (0, T ; L2 (Ωp ))∩ 3 L2 (0, T ; Qp ) with u˙ f ∈ L1 (0, T ; L 2 (Ωf )). Interface conditions (5c) and (5d). Since uf ∈ L2 (0, T ; Xf ) and H1 (Ωf ) is embedded continuously in L6 (Ωf ), for d = 2, 3, uf ∈ L2 (0, T ; L6 (Ωf )). Then H¨older’s inequality implies uf · ∇uf L1 (0,T ;L3/2 (Ωf ) ≤ uf L2 (0,T ;L6 (Ωf )) uf L2 (0,T ;Xf ) . Therefore, uf · ∇uf ∈ L1 (0, T ; L3/2 (Ωf )). Then from (1a) ∇ · σ f = ff − ρf u˙ f − uf · ∇uf ∈ L1 (0, T ; L3/2 (Ωf )). Also

σ f = 2μf D(uf ) − pf I ∈ L1 (0, T ; L2 (Ωf )).

27

Hence, each row of σ f belongs to L1 (0, T ; L3/2 (div; Ωf )). Since H1 (Ωf ) is dense in L3/2 (div; Ωf ), the following Green’s formula hold: ∀v ∈ L3/2 (div; Ωf ), ∀φ ∈ H 1 (Ωf ), v · n∂Ωf , φ ∂Ωf = (∇ · v, φ)Ωf + (v, ∇φ)Ωf . This allows σ f nΓ to be defined in a weak sense on ΓI . See also [11, p.541]. Next since u ∈ L2 (0, T ; Xf ), η˙ ∈ L 2 (0, T ; Xp ) and pp ∈ L2 (0, T ; Qp ) we  ˙ · tlΓ )tlΓ ) ∈ L2 (0, T ; L4 (ΓI )) where t1Γ , t2Γ are have (−pp nΓ − d−1 l=1 (β(uf − η) ΓI

the unit tangential vectors on ΓI . With these the interface conditions (5c) and (5d) make sense. For the rest of the interface conditions we use a global space-time argument in Rd+1 , d = 2, 3 as done in [21]. We define new variables in ˜ = (t, x1 , . . . , xd ), d = 2, 3 and also define the cylindrical region Rd+1 by x ˜ Ωp = [0, T ) × Ωp . Interface condition (5a). Defining Θ = (−(s0 pp + α∇ · η), K∇pp ), the equation (2b) can be written as −∇x˜ · Θ = fp d where ∇x˜ = ( , ∇·)T . dt ˜ p ) and Θ ∈ L2 (Ω ˜ p ), we have Θ ∈ H(div; Ω ˜ p ). So Since fp ∈ L2 (Ω ˜ p ). Therefore, the following Green’s Θ · nΩ˜ p |∂ Ω˜ p is well-defined in H −1/2 (∂ Ω formula holds [19]: ˜ ∈ H1 ( Ω ˜ p ), ∀φ

˜ ˜ = −(Θ, ∇x˜ φ) ˜ ˜ ˜ ˜ + (Θ · n ˜ , φ (∇x˜ · Θ, φ) ∂ Ωp Ωp Ωp Ωp

˜ p. where nΩ˜ p is the outward unit normal to Ω This allows the well-definition of K∇pp · np in the weak sense on ΓI . Interface condition (5b). From the above discussion it is enough to check whether σ p nΓ is mean˙ σ p ), ingful and we use a similar space-time argument. Defining Σ = (−ρs η, the equation (2a) can be written as −∇x˜ · Σ = fs . ˜ and Σ ∈ L2 (Ω ˜ p ) we have Σ ∈ H(div; Ω ˜ p ). Therefore, the Since fs ∈ L2 (Ω) following Green’s formula holds [19]: ˜ ∈ H 1 (Ω ˜ p ), ∀φ

˜ ˜ = −(Σ, ∇x˜ φ) ˜ ˜ . ˜ ˜ + Σ · n ˜ , φ (∇x˜ · Σ, φ) ∂ Ωp Ωp Ωp Ωp 28

This allows the well-definition of σ p np on ΓI and η˙ at t = 0, T in the weak sense. Acknowledgement This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Adams, R. A., 1978. Sobolev spaces. Pure and applied mathematics. Academic Press, New York. [2] Badea, L., Discacciati, M., Quarteroni, A., 2010. Numerical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 115 (2), 195–227. [3] Badia, S., Quaini, A., Quarteroni, A., 2009. Coupling Biot and NavierStokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228 (21), 7986–8014. [4] Beavers, G., Joseph, D., 1967. Boundary conditions at a naturally impermeable wall. J. Fluid. Mech 30, 197–207. [5] Biot, M. A., 1941. General theory of three dimensional consolidation. Journal of Applied Physics 12 (2), 155–164. [6] Biot, M. A., 1955. Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185. [7] Bodn´ar, T., Galdi, G. P., Neˇcasov´a, v. (Eds.), 2014. Fluid-structure interaction and biomedical applications. Advances in Mathematical Fluid Mechanics. Birkh¨auser/Springer, Basel. [8] Bukaˇc, M., Yotov, I., Zunino, P., 2015. An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure. Numer. Methods Partial Differential Equations 31 (4), 1054–1100. [9] Bukaˇc, M., Yotov, I., Zakerzadeh, R., Zunino, P., 2015. Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Computer Methods in Applied Mechanics and Engineering 292, 138 – 170, special Issue on Advances 29

in Simulations of Subsurface Flow and Transport (Honoring Professor Mary F. Wheeler). [10] Cao, Y., Gunzburger, M., Hua, F., Wang, X., 2010. Coupled StokesDarcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. 8 (1), 1–25. [11] Cesmelioglu, A., Girault, V., Rivi`ere, B., 2013. Time-dependent coupling of Navier-Stokes and Darcy flows. ESAIM Math. Model. Numer. Anal. 47 (2), 539–554. [12] Cesmelioglu, A., Lee, H., Quaini, A., Wang K., Yi, S.-Y., 2016. Optimization-based decoupling algorithms for a fluid-poroelastic system. in press. [13] C ¸ e¸smelio˘glu, A., Rivi`ere, B., 2008. Analysis of time-dependent NavierStokes flow coupled with Darcy flow. J. Numer. Math. 16 (4), 249–280. [14] Coddington, E. A., Levinson, N., 1955. Theory of differential equations. McGraw-Hill, New-York. [15] Discacciati, M., Quarteroni, A., 2009. Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22 (2), 315–426. [16] Du, Q., Gunzburger, M. D., Hou, L. S., Lee, J., 2003. Analysis of a linear fluid-structure interaction problem. Discrete and Continuous Dynamical Systems 9 (3), 633–650. [17] Ern, A., Guermond, J.-L., 2004. Theory and practice of finite elements. Applied mathematical sciences. Springer, New York. [18] Ganis, B., Mear, M. E., Sakhaee-Pour, A., Wheeler, M. F., Wick, T., 2014. Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method. Comput. Geosci. 18 (5), 613– 624. [19] Girault, V., Raviart, P.-A., 1986. Finite element methods for NavierStokes equations : theory and algorithms. Springer series in computational mathematics. Springer-Verlag, Berlin, New York, extended version of : Finite element approximation of the Navier-Stokes equations. 30

[20] Girault, V., Rivi´ere, B., 2009. DG approximation of coupled NavierStokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM Journal on Numerical Analysis 47 (3), 2052–2089. [21] Girault, V., Wheeler, M. F., Ganis, B., Mear, M. E., 2015. A lubrication fracture model in a poro-elastic medium. Mathematical Models and Methods in Applied Sciences 25 (04), 587–645. [22] J¨ager, W., Mikeli´c, A., 2000. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127. [23] Lesinigo, M., D’Angelo, C., Quarteroni, A., 2011. A multiscale DarcyBrinkman model for fluid flow in fractured porous media. Numer. Math. 117 (4), 717–752. [24] Lions, J. L., 1961. Equations differentielles operationnelles. SpringerVerlag, Berlin, Heidelb erg, New York. [25] Martin, V., Jaffr´e, J., Roberts, J. E., 2005. Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (5), 1667–1691 (electronic). [26] Mikeli´c, A., Wheeler, M. F., 2012. On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Mathematical Models and Methods in Applied Sciences 22 (11), 1250031. ˇ c, S., 2013. Existence of a weak solution to a nonlinear [27] Muha, B., Cani´ fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls. Arch. Ration. Mech. Anal. 207 (3), 919–968. ˇ c, S., 2014. Existence of a solution to a fluid-multi[28] Muha, B., Cani´ layered-structure interaction problem. J. Differential Equations 256 (2), 658–706. ˇ c, S., 2016. Existence of a weak solution to a fluid-elastic [29] Muha, B., Cani´ structure interaction problem with the Navier slip boundary condition. Journal of Differential Equations.

31

[30] Saffman, P., 1971. On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 292–315. [31] Showalter, R. E., 2005. Poroelastic filtration coupled to Stokes flow. In: Control theory of partial differential equations. Vol. 242 of Lect. Notes Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL, pp. 229–241. [32] Simon, J., 1987. Compact sets in the space Lp (0, T ; B). Ann. Mat. Pura Appl. (4) 146, 65–96.

32