Analysis of the coupled Navier-Stokes/Biot problem
Aycil Cesmelioglu































Oakland University, Department of Mathematics and Statistics, 146 Library Drive, Rochester, MI 48309


cesmelio@oakland.edu.






We analyze a weak formulation of the coupled  problem defining the interaction between a free fluid and a poroelastic structure. The problem is fully dynamic and 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. 




Navier-Stokes Darcy Biot poroelastic weak formulation existence





35Q30 35Q35









§ 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 equations. The steady-state case of this problem is analyzed mathematically in <cit.> and the time-dependent case in <cit.>. 

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 <cit.> and for the flow of a coupling of time-dependent 2D incompressible NSE to the linearly viscoelastic or the linearly elastic Koiter shell in <cit.>. The two layered structure version of this problem is discussed in <cit.>. 

In geosciences, aquifers and oil/gas reservoirs are porous and deformable affecting groundwater and oil/gas flow, respectively <cit.>. In biomedical sciences, blood flow is influenced by the porous and deformable nature of the arterial wall <cit.>. 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 <cit.> using a monolithic and a domain decomposition technique and the Stokes/Biot system is investigated in <cit.> using an operator splitting approach and in <cit.> using an optimization based decoupling strategy. In <cit.>, variational formulations for the the Stokes/Biot system are developed using semi-group methods. Also a two-layered version was studied in <cit.>.


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 <cit.>. 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 semi-discrete 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 well-known 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 paper. Section 6 contains the proof of the existence and uniqueness results and a priori estimates for the weak solution.





§ FLUID-POROELASTIC MODEL EQUATIONS



Let Ω⊂ℝ^d, d=2,3 be an open bounded domain with Lipschitz continuous boundary ∂Ω. The domain Ω is made up of two regions Ω_f, the fluid region, and
Ω_p, the poroelastic region, separated by a common interface
Γ_I = ∂Ω_f ∩∂Ω_p. Both Ω_1 and Ω_2 are assumed to be Lipschitz. See Figure  <ref>.
The first region Ω_f is occupied by a free fluid and has boundary Γ_f such that
Γ_f=Γ^in_f ∪Γ^out_f ∪Γ^ext_f
∪Γ_I, where Γ^in_f  and Γ^out_f represent the inlet and outlet boundary, respectively.
The second region Ω_p is occupied by a saturated poroelastic structure with boundary Γ_p such that Γ_p=Γ_p^s∪Γ^ext_p
∪Γ_I, where Γ^s_p∪Γ^ext_p represents the outer structure boundary.





Fluid flow is governed by the time-dependent incompressible Navier-Stokes equations:




    ρ_f  _f
     -2  μ_f ∇·(_f) + ρ_f_f·∇_f+ ∇
    p_f    = _f    Ω_f × (0,T) , 
    ∇· _f     = 0    Ω_f× (0,T)  .




Here _f denotes the
velocity vector of the fluid, p_f the pressure of the fluid, ρ_f the density
of the fluid, μ_f the constant fluid viscosity, and _f the body force acting on the fluid. We used the dot above a symbol to denote the time derivative.
The strain rate tensor (_f) is defined by:


    (_f) = 1/2( ∇_f + (∇_f)^T  ).


The Cauchy stress tensor is given by:


    _f =  2μ_f D(_f)  - p_f .


So (<ref>) can also be written as

    ρ_f  _f
     -∇·_f + ρ_f_f·∇_f = _f.

Equation (<ref>) represents the conservation of linear momentum, while equation (<ref>) is the incompressibility condition that represents the conservation of mass.

The poroelastic system is a fully dynamic coupled system of mixed hyperbolic-parabolic
type represented by the Biot model <cit.>:




    ρ_p  
     -2 μ_s  ∇·()  - λ_s ∇ (∇·)  + α∇ p_p    = _s   Ω_p× (0,T)  , 
    
    
    (s_0 _p+ α∇· ) -∇·∇ p_p
       =  f_p   Ω_p× (0,T)  ,


 

where  is the displacement of the structure, p_p is the pore pressure of the fluid, and _p is the fluid velocity in the pores. Here, f_p is the source/sink term and _s is the body force. The parameters ν_s and λ_s denote the Lamé constants for the solid skeleton.
The density of the saturated medium is denoted by ρ_p, and the hydraulic conductivity by . In the Biot model, the first equation, (<ref>), is the momentum equation for the balance of forces and the second equation, (<ref>), is the diffusion equation of fluid mass.

The total stress tensor for the poroelastic structure is given by:


    _p = _p^E - α  p_p ,

where _p^E is the elasticity stress tensor defined by _p^E = 2 μ_s () + λ_s (∇· ).
Therefore, (<ref>) can also be written as

    ρ_p  
    -∇·_p= _s.

The constrained specific storage coefficient is denoted by s_0 and the Biot-Willis 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 _f and _p denote the outward unit normal vectors of Ω_f and Ω_p, respectively and _Γ is the unit normal vector of the interface Γ_I pointing from Ω_f to Ω_p. Hence, _Γ_I=_f|_Γ_I=-_p|_Γ_I. Furthermore _Γ^l, l=1,,d-1 denotes an orthonormal set of unit vectors on the tangent plane to Γ_I.



  §.§.§ 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 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, 




      Γ_f^in×(0,T)    ,    _f _f   = - P_in(t)_f ,  

        Γ_f^out×(0,T)    ,    _f  _f =   ,  

       Γ_f^ext×(0,T)    ,    _f  =     ,  


      Γ_p^s×(0,T)   ,    ∇p_p·_p=0,   =   ,     


      Γ_p^ext×(0,T)    ,   
{ 
[ p_p=0,   _p·_p^E  _p =   ,;         _p^l·=0,  1≤l≤d-1. ]
 .         






  §.§.§ Initial conditions:
 

As initial conditions, we assume that everything is at rest in the beginning.


    At  t=0:   _f =  ,    p_p = 0,   =,   = .




  §.§.§ Interface conditions on Γ_I× (0,T):


The interface conditions are given by 



    _f ·_Γ    =  ( -∇ p_p) ·_Γ , 
    _f  _Γ    =   _p  _Γ , 
    _Γ·_f  _Γ    =  - p_p
      , 
    _Γ·_f  _Γ^l     =  - β (_f - )
     ·_Γ^l    , 1≤ l≤ d-1,




where β denotes the resistance parameter in the tangential direction.
Condition (<ref>) is the continuity of normal flux that satisfies mass conservation, condition (<ref>) is the balance of stresses, that is, the total stresses of the fluid and the poroelastic medium must match at the interface. Condition (<ref>) guarantees the balance of normal components of the stress in the fluid phase across the interface. Finally condition (<ref>) is the Beavers-Joseph-Saffman condition <cit.> 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 <cit.>. 





§ NOTATION AND USEFUL RESULTS.


Let Ω⊂^d, d=2, 3 be a bounded, open connected domain with a Lipschitz continuous boundary ∂Ω. Let (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:

    ^s(Ω):=[H^s(Ω)]^d,  ^s(Ω):=[H^s(Ω)]^d× d

and still write |·|_s,Ω and ·_s,Ω for the corresponding seminorms and norms.
When s=0, instead of H^0(Ω), ^0(Ω) and ^0(Ω) we write L^2(Ω), ^2(Ω) and ^2(Ω) and instead of |·|_0,Ω and ·_0,Ω we write |·|_Ω and ·_Ω. The scalar product of L^2(Ω) is denoted by (·, ·)_Ω.
The spaces (;Ω)={∈^2(Ω): ∇·∈ L^2(Ω)} and ^3/2(;Ω)={∈^2(Ω): ∇·∈ L^3/2(Ω)} are equipped with the graph norm.
If Γ⊂∂Ω, and v∈ H^1/2(Γ), we define
the extension ṽ of v as ṽ=v on Γ, ṽ=0 on ∂Ω\Γ and define the space of traces of all functions of H^1(Ω) that vanish on ∂Ω\Γ as follows:

    H_00^1/2(Γ)={v∈ L^2(Γ): ṽ∈ H^1/2(∂Ω)}.
   
We also define for any 1≤ r≤∞,

    L^r(a, b;X)={f : f_L^r(a,b;X)<∞}

equipped with the norm

    f_L^r(a,b;X)=(∫_a^bf(t)^r_X dt)^1/r

for 1≤ r<∞ and 

    f_L^∞(a,b;X)= esssup_t∈ [a,b]f(t)_X.

Furthermore, 
(0,T;X) denotes the set of all functions that are continuous into X and finally we define 

    H^1(a,b;X)={f∈ L^2(a,b;X): ḟ∈ L^2(a,b;X)}.




  §.§.§ Useful results.
 
Here we state inequalities and results to be used throughout the paper. More details can be found in <cit.>. 
We define the following spaces for the weak solution:


    _f   ={∈^1(Ω_f): =Γ_f^ext},
    _f   ={∈_f: ∇·=0 },
    _p   ={∈^1(Ω_p):  =Γ_p^s,  _p^l·=0,   1≤ l≤ d-1 Γ_p^ext},
    
    Q_f   =L^2(Ω_f),
    
    Q_p   ={r∈ H^1(Ω_p):   r=0 Γ_p^ext}.

On these spaces, we have the following trace inequalities: 


   ∀∈_f,   _Γ_I   ≤     T_1||_1,Ω_f   _Γ_f^in≤T_2||_1,Ω_f

   ∀q∈Q_p,  q_H^1/2(Γ_I)   ≤     T_3|q|_1,Ω_p,   q_Γ_p^s≤T_4|q|_1,Ω_p, 

   ∀∈_p,   _Γ_I   ≤     T_5||_1, Ω_p,


the Poincaré inequalities:


   ∀∈_f,   _Ω_f   ≤     P_1||_1, Ω_f, 

   ∀∈_p,   _Ω_p   ≤     P_2||_1, Ω_p,

   ∀q∈Q_p,  q_Ω_p   ≤     P_3|q|_1, Ω_p, 


a Sobolev inequality: 


    ∀∈_f,   _^4(Ω_f)≤ S_f||_^1(Ω_f),


and finally a Korn's inequality:


    ∀∈_f,    ||_1,Ω_f≤ K_f()_Ω_f,


where T_1-T_5, P_1-P_3, S_f and K_f are positive constants depending only on their corresponding domain.

<cit.>
Let


    ζ(t)≤ B+C∫_0^tζ(s)ds,  t ∈ (0,T)


where ζ is a continuous nonnegative function and B, C≥ 0 are constants.   Then

    ζ(t)≤ B e^Ct.



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.
<cit.>
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 L^p(0,T;X) where 1≤ p< ∞ and let the set {∂ f/∂ t}_f∈ F be bounded in L^1(0,T;Y). Then F is relatively compact in L^p(0,T;B).





§ 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 ∈^∞(Ω_p) is independent of time, uniformly bounded and positive definite. There exists K_min, K_max>0 such that


    ∀∈Ω_p,    K_min·≤·≤ K_max·.


Further, we assume that _f∈^2(0,T;^2(Ω_f)), _s∈^2(0,T;^2(Ω_p)), f_p∈ L^2(0,T;L^2(Ω_p)) and P_in∈ L^2(0,T;H^1/2(Γ_f^in)).

The weak formulation we propose for the problem is the following:

(WF1) Find _f∈ L^∞(0,T;^2(Ω_f))∩ L^2(0,T;_f), p_f∈ L^1(0,T;Q_f), ∈ W^1,∞(0,T;^2(Ω_p))× H^1(0,T;_p) and p_p∈ L^∞(0,T;L^2(Ω_p))∩ L^2(0,T;Q_p) where _f∈ L^1(0,T;^3/2(Ω_f)), 
such that for all ∈_f, q∈ Q_f, ∈_p and r∈ Q_p, 


    (ρ_f_f,)_Ω_f+(2μ_f(_f),())_Ω_f+ρ_f(_f·∇_f,)_Ω_f-(p_f,∇·)_Ω_f
       
    +(ρ_s,)_Ω_p+(2μ_s(),())_Ω_p+(λ_s∇·,∇·)_Ω_p-(α p_p,∇·)_Ω_p
       +(s_0ṗ_̇ṗ+α∇·,r)_Ω_p+(∇ p_p,∇ r)_Ω_p
       +⟨ p_p_Γ,-⟩_Γ_I+Σ_l=1^d-1⟨β(_f-)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨(-_f)·_Γ,r⟩_Γ_I
       =-⟨ P_in(t)_Γ,⟩_Γ_f^in+(_f,)_Ω_f+(_s,)_Ω_p+(f_p,r)_Ω_p,
       (∇·_f,q)_Ω_f=0,


a.e. in (0,T), and 

    _f(0)=Ω_f,   (0)=,  (0)=,    p_p(0)=0 Ω_p.

The reason for looking for a solution in L^∞(0,T;L^2(Ω_i)), i=f, p may not seem obvious at this point, since typically the solutions are sought in L^2(0,T;X) where X is an appropriate Sobolev space, but we will prove that such a solution exists.



 §.§ 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.

Let the data satisfy 
the assumptions listed in the previous section. Then each solution _f∈ L^∞(0,T;^2(Ω_f))∩ L^2(0,T;_f) such that _f∈ L^1(0,T;^3/2(Ω_f)), p_f∈ L^1(0,T;Q_f), ∈ W^1,∞(0,T;^2(Ω_p))∩ H^1(0,T;_p) 
 and p_p∈  L^∞(0,T;L^2(Ω_p))∩ L^2(0,T;Q_p) of the problem defined by (<ref>)-(<ref>), (<ref>)-(<ref>) and (<ref>)-(<ref>) is also a solution of the variational problem (WF1) and conversely.



We first show sufficiency. To simplify the presentation, we included in  <ref> the justification of using Green's formula in the following proof. Let (_f, p_f, , p_p) be a solution of the coupled problem defined by (<ref>)-(<ref>), (<ref>)-(<ref>), (<ref>)-(<ref>) satisfying the regularity stated in the proposition. We multiply (<ref>) by ∈_f. After integration by parts:


    (ρ_f  _f, )_Ω_f
    +(2  μ_f  D(_f),D())_Ω_f+ρ_f( _f·∇_f,)_Ω_f- (
    p_f,∇·)_Ω_f
    
     -⟨_f_f, ⟩_∂Ω_f = (_f, )_Ω_f.


Using (<ref>), (<ref>), = on Γ_f^ext and _f=_Γ on Γ_I, we have


    (ρ_f  _f, )_Ω_f
    +(2  μ_f  D(_f),D())_Ω_f+ρ_f( _f·∇_f,)_Ω_f- (
    p_f,∇·)_Ω_f
    -⟨_f_Γ, ⟩_Γ_I = (_f, )_Ω_f-⟨ P_in(t), ⟩_Γ_f^in.


Next we multiply (<ref>) by q∈ Q_f and integrate to get


    (∇·_f,q)_Ω_f=0.


Multiplying (<ref>) by ∈_p and integrating by parts yields:


    (ρ_p ,)_Ω_p
    +(2 μ_s    D(), D())_Ω_p + (λ_s (∇·), ∇·)_Ω_p  -(α  p_p,∇·)_Ω_p
    -⟨_p_p,⟩_∂Ω_p
    = (_s,)_Ω_p.


Observe that (<ref>) implies _p_p=_p^E_p=Σ_l=1^d-1(_p^l·_p^E_p)·_p^l on Γ_p^ext. Then since _p^l·=0 on Γ_p^ext,  =0 on Γ_p^s and _p=-_Γ on Γ_I, we have


    (ρ_p  ,)_Ω_p
    +(2 μ_s    D(), D())_Ω_p + (λ_s (∇·), ∇·)_Ω_p - (α  p_p,∇·)_Ω_p
    
    +⟨_p_Γ,⟩_Γ_I= (_s,)_Ω_p.


Multiplying (<ref>) by r∈ Q_p and integrating over Ω_p, we obtain


    (s_0 _p+ α∇·,r)_Ω_p +(∇ p_p, ∇ r)_Ω_p
    -⟨∇ p_p·_p, r⟩_∂Ω_p= (f_p,r)_Ω_p.


Using (<ref>), r=0 on Γ_p^ext, and _p=-_Γ on Γ_I, we have


    (s_0 _p+ α∇·,r)_Ω_p +(∇ p_p, ∇ r)_Ω_p
    +⟨∇ p_p·_Γ, r⟩_Γ_I= (f_p,r)_Ω_p.


Next, we rewrite the interface integrals using the interface conditions (<ref>)-(<ref>).
On Γ_I, by (<ref>) and (<ref>), we have


    _f_Γ=(_Γ·_f_Γ)_Γ+Σ_l=1^d-1(_Γ^l·_f_Γ)_Γ^l= -p_p_Γ-Σ_l=1^d-1(β(_f-)·_Γ^l)_Γ^l.


(Note here that since σ_f is symmetric, _Γ^l·_f_Γ=_Γ·_f_Γ^l.)
Adding (<ref>), (<ref>), (<ref>) and (<ref>) while using (<ref>) for ⟨_f_Γ,⟩_Γ_I, (<ref>) for ⟨∇ p_p·_Γ, r⟩_Γ_I and (<ref>) and (<ref>) for ⟨_p_Γ,⟩_Γ_I gives  the weak formulation (WF1).

For the converse, let (_f, p_f, , p_p) be a solution of (WF1). 
We pick first ∈(Ω_f), r=0 and =, second ∈, r∈ D(Ω_p) and = and last =, r=0 and ∈(Ω_p). This gives (<ref>) on Ω_f and (<ref>) and (<ref>) on Ω_p in the sense of distributions. Next we multiply (<ref>) with ∈_f, (<ref>) with ∈_p and (<ref>) with r∈ Q_p and apply Green's formulas and add the outcomes to get


    (ρ_f_f   ,)_Ω_f+(2μ_f(_f),())_Ω_f+ρ_f(_f·∇_f,)_Ω_f-(p_f,∇·)_Ω_f
       +(ρ_s,)_Ω_p+(2μ_s(),())_Ω_p+(λ_s∇·,∇·)_Ω_p-(α p_p,∇·)_Ω_p
       +(s_0ṗ_̇ṗ+α∇·,r)_Ω_p+(∇ p_p,∇ r)_Ω_p
       -⟨_f
    _f,⟩_Γ_f^in∪Γ_f^out∪Γ_I-⟨_p
    _p, ⟩_Γ_p^ext∪Γ_I-⟨∇ p_p·_p,r ⟩_Γ_p^s∪Γ_I
       =(_f,)_Ω_f+(_s,)_Ω_p+(f_p,r)_Ω_p.


Comparing this with (WF1) gives


    ⟨ p_p_Γ,-⟩_Γ_I   +Σ_l=1^d-1⟨β(_f-)·_Γ^l,(-)·_Γ^l⟩_Γ_I
       +⟨(-_f)·_Γ,r⟩_Γ_I
    +⟨ P_in(t)_f,⟩_Γ_f^in
    
    =-   ⟨_f
    _f,⟩_Γ_f^in∪Γ_f^out∪Γ_I-⟨_p
    _p, ⟩_Γ_p^ext∪Γ_I-⟨∇ p_p·_p,r ⟩_Γ_p^s∪Γ_I


for all (, , r)∈_f×_p× Q_p.
If we let =, = in (<ref>), we get 


    -⟨∇ p_p·_p,r ⟩_Γ_p^s∪Γ_I=⟨(-_f)·_Γ,r⟩_Γ_I, ∀ r∈ Q_p.


The choice r∈ Q_p such that r|_Γ_I=0 yields

    ⟨∇ p_p·_p,r ⟩_Γ_p^s=0

which implies the first condition of (<ref>).
Using this  in (<ref>), we get 

    -⟨∇ p_p·_p,r⟩_Γ_I=⟨(-_f)·_Γ,r⟩_Γ_I, ∀ r∈ Q_p

 which yields (<ref>). This reduces 
(<ref>) to 


    ⟨ p_p_Γ,-⟩_Γ_I+∑_l=1^d-1⟨β(_f-)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨ P_in(t)_f,⟩_Γ_f^in
       =-⟨_f
    _f,⟩_Γ_f^in∪Γ_f^out∪Γ_I-⟨_p
    _p, ⟩_Γ_p^ext∪Γ_I, ∀∈_f, ∀∈_p.


Now we let = in (<ref>). Then

    -⟨ p_p_Γ,⟩_Γ_I-∑_l=1^d-1⟨β(_f-)·_Γ^l,·_Γ^l⟩_Γ_I
    
    =-⟨_p_p, ⟩_Γ_p^ext∪Γ_I,  ∀∈_p.

Since p_p=0 on Γ_p^ext, the choice ∈_p  
such that = on Γ_I implies

    0
    =⟨_p_p, ⟩_Γ_ext^p=⟨_p·_p_p, ·_p⟩_Γ_ext^p=⟨_p·_p^E_p, ·_p⟩_Γ_ext^p.

Therefore we recover 
in the sense of distributions the second condition in (<ref>). This also yields

    ⟨ -p_p_Γ-∑_l=1^d-1β((_f-)·_Γ^l)_Γ^l,⟩_Γ_I
    
    =-⟨_p_p, ⟩_Γ_I,  ∀∈_p.

Therefore 


    -p_p_Γ-∑_l=1^d-1β((_f-)·_Γ^l)_Γ^l=_p_Γ


holds in the sense of distributions.
This reduces (<ref>) to 


    ⟨ p_p_Γ,⟩_Γ_I+∑_l=1^d-1⟨β(_f-)·_Γ^l,·_Γ^l⟩_Γ_I+⟨ P_in(t)_f,⟩_Γ_f^in
    
    =-⟨_f_f,⟩_∂Ω_f, ∀∈_f.


Letting ∈_f such that = on Γ_I gives

    ⟨ P_in(t)_f,⟩_Γ_f^in=-⟨_f_f,⟩_Γ_f^in∪Γ_f^out.

Picking  such that = on Γ_f^out implies 
(<ref>). If we plug this in the above equation we get 0=-⟨_f_f,⟩_Γ_f^out concluding that _f_f= on Γ_f^out× (0,T) in the distributional sense. This gives (<ref>) and also implies that 

    ⟨ p_p_Γ,⟩_Γ_I+∑_l=1^d-1⟨β(_f-)·_Γ^l,·_Γ^l⟩_Γ_I=-⟨_f_f,⟩_Γ_I, ∀∈_f.

Therefore 

    p_p_Γ+∑_l=1^d-1⟨(β(_f-)·_Γ^l)_Γ^l=-_f_f.
 
This compared to (<ref>) implies (<ref>) and also gives (<ref>) and (<ref>) after dotted with _Γ and _Γ^l, 1≤ j≤ d-1. 




§ MAIN RESULTS.

This section summarizes the main results of this paper. First, for the sake of simplicity, we define the following functions of time:




    _1(t)   =(3 T_2^2K_f^24μ_fP_in(t)^2_Γ_f^in+ 3P_1^2K_f^24μ_f_f(t)^2_Ω_f+P_3^22K_minf_p(t)_Ω_p^2+12_s(t)_Ω_p^2)^1/2,
    _2(t)   =(3 T_2^2K_f^22μ_fṖ_in(t)_Γ_f^in^2+3 P_1^2K_f^22μ_f_̇ḟ(t)^2_Ω_f+P_3^22K_minḟ_̇ṗ(t)_Ω_p^2+12_s(t)^2_Ω_p)^1/2,

a.e. in (0,T),

and the following constant:


    _3=   (C_j^2ρ_fP_in(0)^2_H_00^1/2(Γ_f^in)+1ρ_f_f(0)_Ω_f^2
    +12s_0f_p(0)^2_Ω_p+12ρ_s_s(0)^2_Ω_p)^1/2,




where the constants T_2, K_f, P_1, P_3 are defined in Section <ref> and C_j is the continuity constant of the continuous lifting operator from H^1/2(∂Ω_f)→ H^1(Ω_f). 
Observe that _1, _2 and _3 depend only on the data of the  problem. 
We now present out main existence and uniqueness result.

Assume that _f∈ H^1(0,T;^2(Ω_f)), _s∈ H^1(0,T;^2(Ω_p)), _p∈ H^1(0,T;L^2(Ω_f)) and P_in∈ H^1(0,T;H^1/2(Γ_f^in)) and that the following small data condition holds:


    (1+Tρ_se^T/ρ_s)_2_L^2(0,T)^2+Tρ_s^2e^T/ρ_s_s(0)^2_Ω_p
    +(1+1ρ_se^T/ρ_s+Tρ_s^2e^T/ρ_s)_1_L^2(0,T)^2+_1^2_L^∞(0,T)<μ_f^39ρ_f^2S_f^4K_f^6.


Then, problem (WF1)has a unique solution (_f, p_f,, p_p) 
such that 


    ρ_f2_f_L^∞(0,T;^2(Ω_f))^2
    +ρ_s/2_L^∞(0,T;^2(Ω_p))^2+μ_s()^2_L^∞(0,T;^2(Ω_p))
       +s_0/2p_p_L^∞(0,T;L^2(Ω_f))^2
    +μ_f(_f)^2_L^2(0,T;^2(Ω_f))+12^1/2∇ p_p^2_L^2(0,T;^2(Ω_p))
       ≤(1+Tρ_se^T/ρ_s)_1^2_L^2(0,T).


Furthermore,


    (_f)_L^∞(0,T;^2(Ω_f))   <μ_f3ρ_fS_f^2K_f^3,




    ρ_f2_f^2_L^∞(0,T;^2(Ω_f))
    +ρ_s2^2_L^∞(0,T;^2(Ω_p))+μ_s()^2_L^∞(0,T;^2(Ω_p))
       +s_02ṗ_p^2_L^∞(0,T;L^2(Ω_f))+μ_f(_f)^2_L^2(0,T;L^2(Ω_f))+12^1/2∇ṗ_p^2_L^2(0,T;^2(Ω_p))
       ≤ (1+Tρ_se^2T/ρ_s^2)_2^2_L^2(0,T)+Te^2T/ρ_s^22_3,


and 

    p_f_L^∞(0,T;L^2(Ω_f))≤1κ(ρ_f_f_L^∞(0,T;^2(Ω_f))+2μ_f(_f)_L^∞(0,T;^2(Ω_f))
       +S_f^2_f_L^∞(0,T;^1(Ω_f))^2
    +T_1T_3p_p_L^∞(0,T;H^1(Ω_p))+β T_1^2_f_L^∞(0,T;^1(Ω_f))
       +β T_1T_5_L^∞(0,T;^1(Ω_p))+T_2P_in_L^∞(0,T;L^2(Γ_f^in))+_f_L^∞(0,T;^2(Ω_f))).




The constants T_1-T_5, K_f, S_f, P_1, P_3 used in the above estimates are defined in Section <ref>.





§ PROOF OF THEOREM <REF>.

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 p_f 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 _f, and p_p of the divergence-free weak formulation. We conclude the proof using an inf-sup condition to recover the fluid pressure p_f that was eliminated from the weak formulation, proving the equivalence of (WF2) and (WF1). This last step also provides a priori estimates for p_f.




 §.§ 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 _f∈ L^∞(0,T;^2(Ω_f))∩ L^2(0,T;_f), ∈ W^1,∞(0,T;^2(Ω_p))× H^1(0,T;_p) and p_p∈ L^∞(0,T;L^2(Ω_p))∩ L^2(0,T;Q_p)  with 
_f∈ L^1(0,T;^3/2(Ω_f))  such that for all ∈_f, ∈_p and r∈ Q_p, 


    (ρ_f_f,)_Ω_f+(2μ_f(_f),())_Ω_f+ρ_f(_f·∇_f,)_Ω_f
       +(ρ_s,)_Ω_p+(2μ_s(),())_Ω_p+(λ_s∇·,∇·)_Ω_p-(α p_p,∇·)_Ω_p
       +(s_0ṗ_̇ṗ+α∇·,r)_Ω_p+(∇ p_p,∇ r)_Ω_p
       +⟨ p_p_Γ,-⟩_Γ_I+Σ_l=1^d-1⟨β(_f-)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨(-_f)·_Γ,r⟩_Γ_I
       =-⟨ P_in(t)_Γ,⟩_Γ_f^in+(_f,)_Ω_f+(_s,)_Ω_p+(f_p,r)_Ω_p


a.e. in (0,T), and 

    _f(0)=Ω_f,   (0)=,  (0)=,    p_p(0)=0 Ω_p.

Note that the unknown pressure p_f 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 <ref> using an inf-sup condition.


 §.§ 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 _f×_p× Q_p implies the existence of a basis {(_i, _i, r_i)}_i≥ 0 consisting of smooth functions. 
We define 


    _f^m   = {_i:i=1,,m}, 
    _p^m   ={_i:i=1,,m}, 
     Q_p^m   ={r_i:i=1,,m},


and use the following Galerkin approximations for the unknowns _f, and p_p:


    _m(,t)=∑_j=1^m α_j(t)_j(), _m(,t)=∑_j=1^m β_j(t)_j(), p_m(,t)=∑_j=1^m γ_j(t)r_j().


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

(GF)  Find _m∈^1(0,T;_f^m), _m∈^2(0,T;_p^m), p_m∈^1(0,T;Q_p^m) such that


    (ρ_f_m,)_Ω_f   +(2μ_f(_m),())_Ω_f+ρ_f(_m·∇_m,)_Ω_f+(ρ_s_m,)_Ω_p
       
    +(2μ_s(_m),())_Ω_p
    +(λ_s∇·_m,∇·)_Ω_p-(α p_m,∇·)_Ω_p
       +(s_0ṗ_m+α∇·_m,r)_Ω_p+(∇ p_m,∇ r)_Ω_p+⟨ p_m_Γ,-⟩_Γ_I
       +Σ_l=1^d-1⟨β(_m-_m)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨(_m-_m)·_Γ,r⟩_Γ_I
       =-⟨ P_in(t)_Γ,⟩_Γ_f^in+(_f,)_Ω_f+(_s,)_Ω_p+(f_p,r)_Ω_p,


for all (, , r)∈_f^m×_p^m× Q_p^m, a.e. t ∈ (0,T)
and 


    _m(0)=,   _m(0)=,   _m(0)=,    p_m(0)=0.



For each positive integer m, the formulation (GF)  has a unique maximal solution (_m,_m,p_m)∈^1(0,T_m;_f^m)×^2(0,T_m;_p^m)×^1(0,T_m;Q_p^m) for some time T_m where 0<T_m≤ T.



Using the Galerkin expansions given in (<ref>), 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 

    ^f_ij=ρ_f(_j,_i)_Ω_f,   _ij^f=2μ_f((_j),(_i))_Ω_f+Σ_l=1^dβ⟨_j·_Γ^l, _i·_Γ^l ⟩_Γ_I,


    _i=(ρ_f(_j·∇_k, _i)_Ω_f)_1≤ j,k≤ m,


    ^s_ij=ρ_s(_j,_i)_Ω_p,  _ij^s=2μ_s((_j),(_i))_Ω_p+λ_s(∇·_j,∇·_i)_Ω_p,


    ^p_ij=s_0(r_j,r_i)_Ω_p,  _ij^p=(∇ r_j,∇ r_i)_Ω_p,


    _ij=α(r_j,∇·_i)_Ω_p+⟨ r_j_Γ,_i⟩_Γ_I,  _ij=⟨ r_j_Γ,_i⟩_Γ_I,


    _ij=Σ_l=1^dβ⟨_j·_Γ^l, _i·_Γ^l ⟩_Γ_I, _ij=Σ_l=1^dβ⟨_j·_Γ^l, _i·_Γ^l ⟩_Γ_I.

And finally for the right hand side integrals we define

    _i=-⟨ P_in(t)_f,_i⟩_Ω_f+(f_f,_i)_Ω_f,  _i=(_s,_i)_Ω_p,   _i=(f_p,r_i)_Ω_p.

The unknowns are _i=α_i(t), _i=β_i(t) and _i=γ_i(t), i=1,, m and we define a vector that holds these unknowns and = as follows :

    (t)=[
    [ (t); (t); (t); (t) ]]

and set (())_i=_i·.
With these definitions, ( GF) is equivalent to finding , , such that


    ^f+^f+()+-=
    ^s+^s--^T+=
    ^p+^p+^T-^T=


where (0),  (0) and (0) are given.
We can rewrite this as a system of first order equations as follows:

    +=(),

where

    =[
    [ ^f         ;            ;       ^p   ;          ^s;    ]],  =[
    [  ^f           -;               -; -^T      ^p  ^T; -^T  ^s   -    ;     ]]

and 

    ()=[[ -();    ;    ;     ]].

Since ρ_f, ρ_s and s_0 are positive, ^f, ^p and ^s are symmetric positive definite implying that  is invertible. This defines an autonomous ODE in 
(t) such that 

    =^-1()-^-1=:g(),   (0)

The matrices , are 4m× 4m and the  vectors , have length 4m.
It is obvious that the function g is continuous in time and locally Lipschitz continuous in . Then, it follows from the theory of ordinary differential equations <cit.> that there is a unique maximal solution  in the interval [0,T_m] for some T_m such that 0<T_m≤ T such that each component of , i.e., each component of , , and = belongs to ^1(0,T_m). 


We need a priori bounds on the Galerkin solution to conclude that T_m=T. We discuss this next in Section <ref>. 

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].




 §.§ A priori estimates for the Galerkin solution.


We begin by stating the main result of this section. 


Suppose that _f∈ H^1(0,T;^2(Ω_f)), _s∈ H^1(0,T;^2(Ω_p)), _p∈ H^1(0,T;L^2(Ω_p)) and P_in∈ H^1(0,T;H^1/2_00(Γ_f^in)).  In addition, assume that the small data condition (<ref>) holds.


Then, problem (GF) has a unique solution (_m, _m, p_m)in the interval [0,T]. Furthermore, it satisfies the following bounds:


    ρ_f2_m_Ω_f^2
    +ρ_s/2_m_Ω_p^2+μ_s(_m)^2_Ω_p+λ_s/2∇·_m^2_Ω_p+s_0/2p_m_Ω_p^2
    
    +12^1/2∇ p_m^2_L^2(0,T;L^2(Ω_p))
    +μ_f(_m)^2_L^2(0,T;L^2(Ω_f))
    ≤(1+Tρ_se^T/ρ_s)_1^2_L^2(0,T)


for all t∈ [0,T],


    (_m)_Ω_f   <μ_f3ρ_fS_f^2K_f^3,


and 


    ρ_f2_m^2_Ω_f   +ρ_s2_m^2_Ω_p+μ_s(_m)^2_Ω_p+λ_s2∇·_m^2_Ω_p
    +s_02ṗ_m^2_Ω_p
       +μ_f(_m)^2_L^2(0,T;L^2(Ω_f))+12^1/2∇ṗ_m^2_L^2(0,T;L^2(Ω_p))
       ≤ (1+Tρ_se^2T/ρ_s^2)_2^2_L^2(0,T)+Te^2T/ρ_s^22_3.


Here _1, _2 and _3 are defined in (<ref>).



  
These bounds imply that {_m} is bounded in H^1(0,T;^1(Ω_f)), {_m} is bounded in H^1(0,T;^1(Ω_p)), _m is bounded in L^∞(0,T;^2(Ω_p)) and
{p_m} is bounded in H^1(0,T;H^1(Ω_p)). 


In the next few sections, we verify the bounds (<ref>), (<ref>) and (<ref>) in the interval [0,T_m] which will then imply the global existence of the maximal solution (_m,_m,p_m) in the interval [0,T] as stated in the theorem.



  §.§.§ Proof of (<ref>).


We let =_m, =_m, r=p_m in the Galerkin formulation (GF). Then the Cauch-Schwarz inequality, inequalities (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and assumption (<ref>) on  imply (after neglecting the β term which is nonnegative) 


    ρ_f(_m,_m)_Ω_f+2μ_f(_m)^2_Ω_f
    +ρ_s(_m, _m)_Ω_p+2μ_s((_m), (_m))_Ω_p
       +λ_s(∇·_m, ∇·_m)_Ω_p+s_0(ṗ_m,p_m)_Ω_p+^1/2∇ p_m^2_Ω_p
    
    
       ≤ρ_fS_f^2K_f^3(_m)_Ω_f^3+T_2K_fP_in(t)_Γ_f^in(_m)_Ω_f
       +P_1K_f_f_Ω_f(_m)_Ω_f+_s_Ω_p_m_Ω_p+P_3K_min^1/2f_p_Ω_p^1/2∇ p_m_Ω_p.


Here the only problematic terms on the right hand side are (_m)_Ω_f^3 and _m_Ω_p. The rest can easily be hidden in the left hand side. Observe that since _m(0)= and _m is continuous, there exists a time T_m, 0<T_m≤ T_m such that 


    (_m)_Ω_f<μ_f3ρ_fS_f^2K_f^3  ∀ t∈ [0,T_m].
 

In fact, this condition holds true on [0,T_m]. For the sake of presentation, we postpone this proof to Section <ref>. Using this condition together with Young's inequality with ϵ>0, we obtain


    ρ_f(_m,_m)_Ω_f+2μ_f(_m)^2_Ω_f
    +ρ_s(_m, _m)_Ω_p+2μ_s((_m), (_m))_Ω_p
       +λ_s(∇·_m, ∇·_m)_Ω_p+s_0(ṗ_m,p_m)_Ω_p+^1/2∇ p_m^2_Ω_p
    
       ≤μ_f3(_m)_Ω_f^2+ϵ T_2^2K_f^22P_in(t)^2_Γ_f^in+12ϵ(_m)^2_Ω_f+ϵ P_1^2K_f^22_f^2_Ω_f
       +12ϵ(_m)^2_Ω_f+12_s^2_Ω_p+12_m^2_Ω_p+P_3^22K_minf_p_Ω_p^2+12^1/2∇ p_m_Ω_p^2


for all t∈ [0,T_m].
Picking ϵ=32μ_f, we get


    ρ_f(_m,_m)_Ω_f+2μ_f(_m)^2_Ω_f
    +ρ_s(_m, _m)_Ω_p+2μ_s((_m), (_m))_Ω_p
       +λ_s(∇·_m, ∇·_m)_Ω_p+s_0(ṗ_m,p_m)_Ω_p+^1/2∇ p_m^2_Ω_p
    
       ≤μ_f(_m)_Ω_f^2+12^1/2∇ p_m_Ω_p^2+3 T_2^2K_f^24μ_fP_in(t)^2_Γ_f^in+ 3P_1^2K_f^24μ_f_f^2_Ω_f
       +P_3^22K_minf_p_Ω_p^2+12_s^2_Ω_p+12_m^2_Ω_p=_1^2(t)+12_m^2_Ω_p



where _1 is defined in (<ref>).
Integrating with respect to t, due to (<ref>), we get:


    ρ_f/2_m_Ω_f^2
    +ρ_s/2_m_Ω_p^2+μ_s(_m)^2_Ω_p+λ_s/2∇·_m^2_Ω_p+s_0/2p_m_Ω_p^2
    
    +μ_f∫_0^t(_m)^2_Ω_fdt+12∫_0^t^1/2∇ p_m^2_Ω_pdt
    ≤_1^2_L^2(0,T)+12∫_0^t_m(s)^2_Ω_pds.


So 


    _m(t)_Ω_p^2
    ≤2ρ_s_1^2_L^2(0,T)+1ρ_s∫_0^t_m(s)^2_Ω_pds.


Therefore, since _m∈𝒞^2(0,T;_p^m), _m(t)_Ω_p^2∈^1(0,T) and applying Gronwall's inequality (<ref>) with ζ(t)=_m(t)_Ω_p^2, C=1ρ_s and B=2ρ_s_1^2_L^2(0,T) yields


    _m(t)_Ω_p^2≤2ρ_se^T/ρ_s_1_L^2(0,T)^2.


Plugging this in (<ref>), we have


    ρ_f(_m,_m)_Ω_f+μ_f(_m)^2_Ω_f
    +ρ_s(_m, _m)_Ω_p+2μ_s((_m), (_m))_Ω_p
       +λ_s(∇·_m, ∇·_m)_Ω_p+s_0(ṗ_m,p_m)_Ω_p+12^1/2∇ p_m^2_Ω_p
       ≤_1^2(t)+1ρ_se^T/ρ_s_1^2_L^2(0,T)


or


    ρ_f2ddt_m_Ω_f^2+μ_f(_m)^2_Ω_f
    +ρ_s/2d/dt_m_Ω_p^2+μ_sd/dt(_m)^2_Ω_p
    
    +λ_s/2d/dt∇·_m^2_Ω_p+s_0/2d/dtp_m_Ω_p^2+12^1/2∇ p_m^2_Ω_p
    ≤_1^2(t)+1ρ_se^T/ρ_s_1^2_L^2(0,T)


for all t∈ [0,T_m].
After integrating with respect to t and using (<ref>) implies the bound (<ref>).



  §.§.§ Proof of (<ref>) and (<ref>).



Recalling (<ref>), assume for a contradiction that there exists T^*∈ (0,T_m] such that 


    ∀ t∈ [0,T^*),   (_m)(t)_Ω_f<μ_f3ρ_fS_f^2K_f^3,  (_m)(T^*)_Ω_f=μ_f3ρ_fS_f^2K_f^3.
                                                                                                                                  

Similar arguments leading to (<ref>) and using Young's inequality yield 
 
    2μ_f   (_m)^2_Ω_f
    ≤   ρ_f2_m_Ω_f^2+ρ_s2_m^2_Ω_p+μ_s(_m)^2_Ω_p+
    λ_s2∇·_m^2_Ω_p+s_02p_m_Ω_p^2
       +ρ_f2_m_Ω_f^2+ρ_s2_m_Ω_p^2+μ_s(_m)^2_Ω_p+λ_s2∇·_m^2_Ω_p+s_02ṗ_m^2_Ω_p
       +_1^2(t)+1ρ_s e^T/ρ_s_1(t)_L^2(0,T)


 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 <cit.>.


    (ρ_f_m,)_Ω_f+(2μ_f(_m),())_Ω_f+ρ_f(_m·∇_m,)_Ω_f+ρ_f(_m·∇_m,)_Ω_f
       +(ρ_s⃛_m,)_Ω_p+(2μ_s(_m),())_Ω_p+(λ_s∇·_m,∇·)_Ω_p-(αṗ_m,∇·)_Ω_p
       +(s_0p̈_m+α∇·_m,r)_Ω_p+(∇ṗ_m,∇ r)_Ω_p
    +⟨ṗ_m_Γ,-⟩_Γ_I
       +Σ_l=1^d-1⟨β(_m-_m)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨(_m-_m)·_Γ,r⟩_Γ_I
       =-⟨Ṗ_̇i̇ṅ_Γ,⟩_Γ_f^in+(_f,)_Ω_f+(_s,)_Ω_p+(ḟ_p,r)_Ω_p.


We let =_m, r=ṗ_m and =_m in the above, use the Cauch-Schwarz inequality, assumption (<ref>), inequalities (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and assumption (<ref>) on  to get (neglecting the nonnegative β term)


    ρ_f2ddt_m^2_Ω_f+2μ_f(_m)^2_Ω_f
    +ρ_s2ddt_m^2_Ω_p+μ_sddt(_m)^2_Ω_p
       +λ_s2ddt∇·_m^2_Ω_p+s_02ddtṗ_m^2_Ω_p+^1/2∇ṗ_m^2_Ω_p
    
       ≤ -ρ_f(_m·∇_m,_m)_Ω_f-ρ_f(_m·∇_m,_m)_Ω_f
       -⟨Ṗ_̇i̇ṅ_Γ,_m⟩_Γ_f^in+(_f,_m)_Ω_f+(_s,_m)_Ω_p+(ḟ_p,ṗ_m)_Ω_p


    ≤   2ρ_fS_f^2K_f^3(_m)^2_Ω_f(_m)_Ω_f+T_2K_fṖ_in_Γ_f^in(_m)_Ω_f
       +P_1K_f_̇ḟ_Ω_f(_m)_Ω_f+_s_Ω_p_m_Ω_p+P_3K_min^1/2ḟ_̇ṗ_Ω_p^1/2∇ṗ_m_Ω_p
    ≤   2μ_f3(_m)^2_Ω_f+12ϵ(_m)^2_Ω_f+ϵ T_2^2K_f^22Ṗ_in_Γ_f^in^2+12ϵ(_m)^2_Ω_f
       +ϵ P_1^2K_f^22_̇ḟ^2_Ω_f+_s_Ω_p_m_Ω_p+12^1/2∇ṗ_m_Ω_p^2+P_3^22K_minḟ_̇ṗ_Ω_p^2.


Picking ϵ=3μ_f, we obtain


    ρ_f2ddt_m^2_Ω_f+μ_f(_m)^2_Ω_f
    +ρ_s2ddt_m^2_Ω_p+μ_sddt(_m)^2_Ω_p
       +λ_s2ddt∇·_m^2_Ω_p+s_02ddtṗ_m^2_Ω_p+12^1/2∇ṗ_m^2_Ω_p
       ≤3 T_2^2K_f^22μ_fṖ_in_Γ_f^in^2+3 P_1^2K_f^22μ_f_̇ḟ^2_Ω_f+12_s^2_Ω_p+12_m^2_Ω_p+P_3^22K_minḟ_̇ṗ_Ω_p^2
       =_2^2(t)+12_m^2_Ω_p,


where _2∈ L^2(0,T) is defined in (<ref>).
Now we need a bound for _m. Since by assumption  is continuous, we can use Gronwall's inequality (<ref>) again. 
Integrating from 0 to t for any t∈ [0,T^*] and recalling (<ref>) we obtain


    ρ_f2_m^2_Ω_f
    +ρ_s2_m^2_Ω_p+μ_s(_m)^2_Ω_p
    +λ_s2∇·_m^2_Ω_p+s_02ṗ_m^2_Ω_p
    
    
       ≤ρ_f2_m(0)^2_Ω_f
    +ρ_s2_m(0)^2_Ω_p+s_02ṗ_m(0)^2_Ω_p
       +_2^2_L^2(0,T)+1ρ_s∫_0^t_m(s)^2_Ω_pds.


Now we need bounds for _m(0)^2_Ω_f, _m(0)^2_Ω_p and ṗ_m(0)^2_Ω_p. For that purpose we let =, r=0 and =_m(0) in (GF) at t=0. Due to (<ref>) this yields:


    ρ_s_m(0)_Ω_p^2=(_s(0), _m(0))_Ω_p≤_s(0)_Ω_p_m(0)_Ω_p.


Therefore, 


    _m(0)_Ω_p≤1ρ_s_s(0)_Ω_p.


Now let == and r=ṗ_m(0) and t=0 in (GF). Then by (<ref>) we have 


    ṗ_m(0)_Ω_p≤1s_0f_p(0)_Ω_p.


Furthermore if we let =, r=0, =_m(0) and t=0 in (GF), we get 

    ρ_f_m(0)^2_Ω_f=-⟨ P_in(0)_Γ_f^in, _m(0)⟩_Γ_f^in+(_f(0), _m(0))_Ω_f.

We need a good bound for ⟨ P_in_Γ, _m(0)⟩_Γ_f^in because otherwise _m(0)_Γ_f^in cannot be hidden in the left hand side. 
Assuming P_in∈ L^2(0,T;H_00^1/2(Γ_f^in)), its extension  P̃_in by zero  to ∂Ω_f is in L^2(0,T;H^1/2(∂Ω_f)). Then we can use the continuous lifting operator j:L^2(0,T;H^1/2(∂Ω_f))→ L^2(0,T;H^1( Ω_f)) with continuity constant C_j. Since ∇·_m=0


    ⟨ P_in(0)_Γ_f^in   , _m(0)⟩_Γ_f^in=⟨ j(P̃_in(0))_∂Ω_f, _m(0)⟩_∂Ω_f=(∇ (j(P̃_in(0))), _m(0) )_Ω_f
       ≤∇ (j(P̃_in(0))_Ω_f_m(0) _Ω_f≤j(P̃_in(0))_H^1(Ω_f)_m(0) _Ω_f
       ≤ C_jP̃_in(0)_H^1/2(∂Ω_f)_m(0) _Ω_f=C_jP_in(0)_H_00^1/2(Γ_f^in)_m(0) _Ω_f.


Then 


    _m(0)_Ω_f≤C_jρ_fP_in(0)_H_00^1/2(Γ_f^in)+1ρ_f_f(0)_Ω_f.


 Plugging these in (<ref>), 


    ρ_f2_m^2_Ω_f
    +ρ_s2_m^2_Ω_p+μ_s(_m)^2_Ω_p
    +λ_s2∇·_m^2_Ω_p+s_02ṗ_m^2_Ω_p
    
       ≤C_f^2ρ_fP_in(0)^2_H_00^1/2(Γ_f^in)+1ρ_f_f(0)_Ω_f^2
    +12ρ_s_s(0)^2_Ω_p+12s_0f_p(0)^2_Ω_p
       +_2^2_L^2(0,T)
    +1ρ_s∫_0^t_m(s)^2_Ω_pds
       = _3+_2^2_L^2(0,T)
    +1ρ_s∫_0^t_m(s)^2_Ω_pds


where 
_3 is defined in (<ref>).

If we neglect all terms other than the second one on the left hand side of the above equation we get


    _m^2_Ω_p≤2ρ_s_3+2ρ_s_2^2_L^2(0,T)
    +2ρ_s^2∫_0^t_m(s)^2_Ω_pds.


 Using Gronwall's inequality (<ref>) with
B= 2ρ_s_3+2ρ_s_2^2_L^2(0,T) , C=2ρ_s^2 and ζ(t)=_m(t)^2_Ω_p,


    _m(t)^2_Ω_p≤    e^2T/ρ_s^2(2ρ_s_2^2_L^2(0,T)+2ρ_s_3).


Therefore, putting this in (<ref>) we have


    ρ_f2ddt_m^2_Ω_f+μ_f(_m)^2_Ω_f
    +ρ_s2ddt_m^2_Ω_p+μ_sddt(_m)^2_Ω_p
       +λ_s2ddt∇·_m^2_Ω_p
    +s_02ddtṗ_m^2_Ω_p+12^1/2∇ṗ_m^2_Ω_p
       ≤_2^2(t)+e^2T/ρ_s^2ρ_s(_2^2_L^2(0,T)+_3).


Integrating from 0 to t in [0,T^*] and recalling the bounds (<ref>), (<ref>) and (<ref>) we obtain


    ρ_f2_m^2_Ω_f   +ρ_s2_m^2_Ω_p+μ_s(_m)^2_Ω_p+λ_s2∇·_m^2_Ω_p
    +s_02ṗ_m^2_Ω_p
       +∫_0^tμ_f(_m)^2_Ω_fdt+∫_0^t12^1/2∇ṗ_m^2_Ω_pdt
       ≤_2^2_L^2(0,T)+Tρ_se^2T/ρ_s^2_2^2_L^2(0,T)+Te^2T/ρ_s^22_3
       =(1+Tρ_se^2T/ρ_s^2)_2^2_L^2(0,T)+Te^2T/ρ_s^22_3,


which implies
(<ref>). Neglecting all terms on the left hand side other than the first we find



    _m^2_Ω_f≤2ρ_f((1+Tρ_se^2T/ρ_s^2)_2^2_L^2(0,T)+Te^2T/ρ_s^22_3).


To find a bound for _m_Ω_f in (<ref>),
we recall (<ref>), use assumption (<ref>) to get (<ref>) for all t∈ [0,T^*]. Neglecting the terms other than the ones with d/dt on the left hand side, we integrate (<ref>) from 0 to t where t∈[0,T^*]. Since the initial conditions are zero, we have


    ρ_f2_m_Ω_f^2
    +ρ_s/2_m_Ω_p^2+μ_s(_m)^2_Ω_p
    +λ_s/2∇·_m^2_Ω_p+s_0/2p_m_Ω_p^2
    ≤ (1+Tρ_se^T/ρ_s)_1^2_L^2(0,T).


Therefore using (<ref>) and (<ref>) in (<ref>) we get


    (_m)_Ω_f^2≤12μ_f(_1^2(t)
    +(1+1ρ_s e^T/ρ_s+Tρ_se^T/ρ_s)_1^2_L^2(0,T)
    
    +(1+Tρ_se^2T/ρ_s^2)_2^2_L^2(0,T)+Te^2T/ρ_s^22_3)


for all t∈ [0,T^*].

So if the data satisfy


    12μ_f(_1^2_L^∞(0,T)
    +(1+1ρ_s e^T/ρ_s+Tρ_se^T/ρ_s)_1^2_L^2(0,T)
    
    +(1+Tρ_se^2T/ρ_s^2)_2^2_L^2(0,T)+Te^2T/ρ_s^22_3)<μ_f^39ρ_f^2S_f^4K_f^6,
 

then we have

    (_m)_Ω_f<μ_f^23ρ_fS_f^2K_f^3

for all t∈ [0,T^*]
which contradicts with the assumption.



Note that if P_in(0)=0,  we immediately get _m(0)_Ω_f≤1ρ_f_f(0)_Ω_f rather than (<ref>). In that case, it is not necessary to assume that P_in∈ L^2(0,T;H_00^1/2(Γ_f^in)). 




 §.§ Passing to the limit.


Since we established the boundedness of the Galerkin solution (_m, _m, p_m) that we stated in Remark <ref>, we can pass to the limit in (GF) to obtain a solution to (WF2).
Reflexivity of _f, _p and Q_p and therefore of H^1(0,T;_f), H^1(0,T;_p) and H^1(0,T;Q_p) imply that there exists a subsequence of the Galerkin solution, still denoted by {(_m, _m, p_m)}_m
such that


_m   ⇀      _f     H^1(0,T;^1(Ω_f)), 




_m   ⇀        H^1(0,T;^1(Ω_p)),


_m   ⇀         L^2(0,T;^2(Ω_p)), 


p_m   ⇀   p_p       H^1(0,T;H^1(Ω_p))

where ⇀ stands for weak convergence.

Note that due to the continuity of the trace operator we also have

_m   ⇀      _f     L^2(0,T;H^1/2(∂Ω_f)), 


_m   ⇀        L^2(0,T;H^1/2(∂Ω_p)),

p_m   ⇀   p_p       L^2(0,T;H^1/2(∂Ω_p)).

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 ^1(Ω_f)⊂^4(Ω_f)⊂^2(Ω_f) and that ^1(Ω_f) is compactly embedded in ^4(Ω_f) due to the Sobolev embedding theorem. Furthermore, Remark <ref> implies that the subsequence {_m}_m is bounded in L^4(0,T;^1(Ω_f)) and {∂_m/∂ t}_m is 
bounded in L^1(0,T;^2(Ω_f)). Therefore, Theorem <ref> implies that {_m}_m has a subsequence, still denoted the same, such that 

    _m→_f L^4(0,T;^4(Ω_f)).
 
Fix an integer k≥ 1. We multiply (GF) by ψ(t)∈ L^2(0,T) 
 and integrate in time to obtain


    ∫_0^T(ρ_f_m   ,ψ(t))_Ω_fdt+∫_0^T(2μ_f(_m),ψ(t)())_Ω_fdt
       +∫_0^Tρ_f(_m·∇_m,ψ(t))_Ω_fdt
    +∫_0^T(ρ_s_m,ψ(t))_Ω_pdt
       +∫_0^T(2μ_s(_m),ψ(t)())_Ω_pdt
    +∫_0^T(λ_s∇·_m,ψ(t)∇·)_Ω_pdt
       -∫_0^T(α p_m,ψ(t)∇·)_Ω_pdt+∫_0^T(s_0ṗ_m+α∇·_m,ψ(t)r)_Ω_pdt
       +∫_0^T(∇ p_m,ψ(t)∇ r)_Ω_pdt
    +∫_0^T⟨ p_m_Γ,ψ(t)(-)⟩_Γ_Idt
       +Σ_l=1^d-1∫_0^T⟨β(_m-_m)·_Γ^l,ψ(t)(-)·_Γ^l⟩_Γ_Idt
       +∫_0^T⟨(_m-_m)·_Γ,ψ(t)r⟩_Γ_Idt=-∫_0^T⟨ P_in(t)_Γ,ψ(t)⟩_Γ_f^indt
       +∫_0^T(_f,ψ(t))_Ω_fdt+∫_0^T(_s,ψ(t))_Ω_pdt+∫_0^T(f_p,ψ(t)r)_Ω_pdt,


for all (, , r)∈_f^k×_p^k× Q_p^k, m≥ k. Letting m→∞ we obtain


    ∫_0^T(ρ_f_f,   ψ(t))_Ω_fdt+∫_0^T(2μ_f(_f),ψ(t)())_Ω_fdt
       +∫_0^Tρ_f(_f·∇_f,ψ(t))_Ω_fdt
    +∫_0^T(ρ_s,ψ(t))_Ω_pdt
       +∫_0^T(2μ_s(),ψ(t)())_Ω_pdt
    +∫_0^T(λ_s∇·,ψ(t)∇·)_Ω_pdt
       -∫_0^T(α p_p,ψ(t)∇·)_Ω_pdt+∫_0^T(s_0ṗ_p+α∇·,ψ(t)r)_Ω_pdt
       +∫_0^T(∇ p_p,ψ(t)∇ r)_Ω_pdt
    +∫_0^T⟨ p_p_Γ,ψ(t)(-)⟩_Γ_Idt
       +Σ_l=1^d-1∫_0^T⟨β(_f-)·_Γ^l,ψ(t)(-)·_Γ^l⟩_Γ_Idt
       +∫_0^T⟨(-_f)·_Γ,ψ(t)r⟩_Γ_Idt=-∫_0^T⟨ P_in(t)_Γ,ψ(t)⟩_Γ_f^indt
       +∫_0^T(_f,ψ(t))_Ω_fdt+∫_0^T(_s,ψ(t))_Ω_pdt+∫_0^T(f_p,ψ(t)r)_Ω_pdt,


for all (, , r)∈_f^k×_p^k× Q_p^k. Since any element of , _p, Q_p can be approximated by elements of _f^k, _p^k, Q_p^k and ψ∈ L^2(0,T) is arbitrary, this also holds for all (, , r)∈_f×_p× Q_p a.e. in (0,T). Therefore we recover the equation in (WF2).

Last, we check 
whether the initial conditions are satisfied. 
In (<ref>) we let ψ∈^2([0,T]) such that ψ(T)=0 and ψ̇(T)=0 and integrate the first and the eighth terms once and the fourth term twice with respect to time. Then we do the same in (<ref>) and also take the limit as m→∞ using the convergence properties established in the beginning of the proof and using (<ref>).  
Comparing the resulting equations yield:


    -(ρ_f(0),ψ(0))_Ω_f-(ρ_s(0),ψ(0))_Ω_p+(ρ_s(0),ψ̇(0))_Ω_p
    
    -(s_0 p(0),ψ(0)r)_Ω_p=0,


for all ∈, ∈_p, r∈ Q_p. Since ψ(0) and ψ̇(0) are arbitrary, 


    -(ρ_f(0),)_Ω_f-(ρ_s(0),)_Ω_p+(ρ_s(0),)_Ω_p
    -(s_0 p(0),r)_Ω_p=0,


for all ∈, ∈_p, r∈ Q_p.
This yields the initial conditions stated in (WF2).

Finally passing to the limit in (<ref>), (<ref>) and (<ref>), we obtain (<ref>), (<ref>) and (<ref>).


 §.§ Uniqueness.

For the Stokes flow, there is no issue of uniqueness. For the Navier-Stokes problem, we can only prove  uniqueness for restricted solution.


Let (_f, , p_p) be a solution of (WF2).  Then if 

    ()_Ω_f≤μS_f^2K_f^3,

then (_f, , p_p) is unique.



Let (_1, _1, p_1) and (_2, _2, p_2) be two solutions of (WF2). Then =_1-_2, =_1-_2 and ϕ=p_1-p_2 satisfy:


    (ρ_f,)_Ω_f+(2μ_f(),())_Ω_f+ρ_f(_1·∇_1-_2·∇_2,)_Ω_f-(ϕ,∇·)_Ω_f
       
    +(ρ_s,)_Ω_p+(2μ_s(),())_Ω_p+(λ_s∇·,∇·)_Ω_p-(αϕ,∇·)_Ω_p
       +(s_0ϕ̇+α∇·,r)_Ω_p+(∇ϕ,∇ r)_Ω_p
       +⟨ϕ_Γ,-⟩_Γ_I+Σ_l=1^d-1⟨β(-)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨(-)·_Γ,r⟩_Γ_I
       =0


for all , , r. Let =, =, r=ϕ where (0)=, (0)=, (0)=, ϕ(0)=0.


    ρ_f2ddt^2_Ω_f+2μ_f()^2_Ω_f
    +ρ_s2ddt^2_Ω_p+μ_sddt()^2_Ω_p
       +λ_s2ddt∇·^2_Ω_p+s_02ddtϕ^2_Ω_p+^1/2∇ϕ^2_Ω_p
    +Σ_l=1^d-1β-)·_Γ^l^2_Γ_I
    
       =-ρ_f(·∇_1+_2·∇,)_Ω_f
    
       ≤ S_f^2K_f^3()^2_Ω_f(_1)_Ω_f+S_f^2K_f^3(_2)_Ω_f()_Ω_f^2
    ≤ 2μ_f()^2_Ω_f.


Therefore, 

    ρ_f2ddt^2_Ω_f
    +ρ_s2ddt^2_Ω_p+μ_sddt()^2_Ω_p
    
    +λ_s2ddt∇·^2_Ω_p+s_02ddtϕ^2_Ω_p≤ 0

which implies using the initial conditions:

    ρ_f2(t)^2_Ω_f
    +ρ_s2(t)^2_Ω_p+μ_s((t))^2_Ω_p
    +λ_s2∇·^2_Ω_p+s_02ϕ^2_Ω_p≤ 0

This implies

    =, =, ϕ=0.





 §.§ 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 p_f which was eliminated when we restricted the search space L^2(0,T;_f) to L^2(0,T;_f).
The following inf-sup condition holds <cit.>: There exists a constant κ>0 such that 

    inf_q∈ Q_fsup_∈_f(∇·,q)_Ω_f_H^1(Ω_f)q_L^2(Ω_f)≥κ.

Note that we can replace the supremum above with the supremum over all (,,r)∈_f×_p× Q_p. In fact, the supremum is attained when = and r=0.


If (_f, , p_p) is a solution of problem (WF2), then there exists a unique p_f∈ L^∞(0,T;Q_f) such that (_f,p_f,, p_p) is a solution of problem (WF1).



Let's define a mapping  such that for all (,,r)∈_f×_p× Q_p 


    (, , r)=(ρ_f_f,)_Ω_f+(2μ_f(_f),())_Ω_f+ρ_f(_f·∇_f,)_Ω_f
       +(ρ_s,)_Ω_p+(2μ_s(),())_Ω_p+(λ_s∇·,∇·)_Ω_p-(α p_p(t),∇·)_Ω_p
       +(s_0ṗ_̇ṗ+α∇·,r)_Ω_p+(∇ p_p,∇ r)_Ω_p
       +⟨ p_p_Γ,-⟩_Γ_I+Σ_l=1^d-1⟨β(_f-)·_Γ^l,(-)·_Γ^l⟩_Γ_I+⟨(-_f)·_Γ,r⟩_Γ_I
       +⟨ P_in(t)_f,⟩_Γ_f^in-(_f,)_Ω_f-(_s,)_Ω_p-(f_p,r)_Ω_p, (0,T).


It is straightforward to see that  is linear and continuous on _f×_p× Q_p for a.e. t∈ (0,T). Furthermore, (,,r)=0 for any (,,r)∈_f×_f× Q_p 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 p_f∈ Q_f such that 


    (p_f,∇·)_Ω_f=(, , r), ∀ (,,r)∈_f×_f× Q_p,


that is, there is a unique p_f∈ L^∞(0,T;Q_f) such that (_f,p_f,, p_p) is a solution of problem (WF1). Furthermore, letting r=0, = in (<ref>) and using the inf-sup condition again 
we have 


    p_f_Ω_f≤1κ(ρ_f_f_Ω_f+2μ_f(_f)_Ω_f+S_f^2_f_1, Ω_f^2+T_1T_3p_p_1,Ω_p
    +β T_1^2_f_1,Ω_f+β T_1T_5_1,Ω_p+T_2P_in_Γ_f^in+_f_Ω_f),


a.e. in (0,T) which implies the bound (<ref>) on p_f in Theorem <ref>.


This concludes the proof of Theorem <ref>.




§ THE INTERFACE CONDITIONS.
 

In this section we prove that the interface conditions are meaningful for a solution of (<ref>), (<ref>), (<ref>) and (<ref>) 
and therefore justify the derivation of the weak formulation  in the proof of Proposition <ref>.

Let us consider a solution of (<ref>), (<ref>), (<ref>) and (<ref>) such that _f∈ L^2(0,T;_f), p_f∈ L^1(0,T;Q_f), ∈ H^1(0,T;_p) and  p_p ∈ L^2(0,T;Q_p) with _f∈ L^1(0,T;^3/2(Ω_f)). 



  §.§.§ Interface conditions (<ref>) and (<ref>).


Since _f∈ L^2(0,T;_f) and ^1(Ω_f) is  embedded continuously in ^6(Ω_f), for d=2,3, _f∈ L^2(0,T;^6(Ω_f)).
Then Hölder's inequality implies 


    _f·∇_f_L^1(0,T;^3/2(Ω_f)
    
    
    
    
       ≤_f_L^2(0,T;^6(Ω_f))_f_L^2(0,T;_f).


Therefore, _f·∇_f∈ L^1(0,T;^3/2(Ω_f)). 
Then from (<ref>)

    ∇·_f=_f-ρ_f_f-_f·∇_f∈ L^1(0,T;^3/2(Ω_f)).

Also 

    _f = 2μ_fD(_f)-p_fI ∈ L^1(0,T;^2(Ω_f)).

Hence, each row of _f belongs to L^1(0,T;^3/2(;Ω_f)). 
Since ^1(Ω_f) is dense in ^3/2(;Ω_f), the following Green's formula hold:

    ∀∈^3/2(;Ω_f), ∀ϕ∈ H^1(Ω_f), ⟨·_∂Ω_f, ϕ⟩_∂Ω_f=(∇·, ϕ)_Ω_f+(, ∇ϕ)_Ω_f.

This allows _f_Γ to be defined in a weak sense on Γ_I. See also <cit.>.
 
Next  since ∈ L^2(0,T;_f), ∈ L^2(0,T;_p) and p_p∈ L^2(0,T;Q_p) we have (-p_p_Γ-∑_l=1^d-1(β(_f-)·_Γ^l)_Γ^l)|_Γ_I∈ L^2(0,T;^4(Γ_I))
where _Γ^1, _Γ^2 are the unit tangential vectors on Γ_I. With these the interface conditions (<ref>) and (<ref>) make sense.


For the rest of the interface conditions we use a global space-time argument in ℝ^d+1, d=2, 3. We define new variables in ℝ^d+1 by =(t, x_1, , x_d), d=2, 3 and also define the cylindrical region Ω̃_p=[0,T]×Ω_p. 



  §.§.§ Interface condition (<ref>).
 

Defining Θ=(-(s_0p_p+α∇·), ∇ p_p),  the equation (<ref>) can be written as

    -∇_·Θ=f_p

where ∇_=(ddt,∇·)^T.

Since f_p∈ L^2(Ω̃_p) and Θ∈^2(Ω̃_p), we have Θ∈(;Ω̃_p). So Θ·_Ω̃_p|_∂Ω̃_p is well-defined in H^-1/2(∂Ω̃_p). Therefore the following Green's formula holds <cit.>:

    ∀∈^1(Ω̃_p),    (∇_·Θ, )_Ω̃_p=-(Θ,∇_)_Ω̃_p+⟨ (Θ·_Ω̃_p, ⟩_∂Ω̃_p

where _Ω̃_p outward unit normal to Ω̃_p. 

This allows the well-definition of ∇ p_p·_∂Ω_p in the weak sense 
 on Γ_I.



  §.§.§ Interface condition (<ref>).


From the above discussion it is enough to check whether _p_Γ is meaningful and we use a similar space-time argument. Defining Σ=(-ρ_s , _p), the equation (<ref>) can be written as

    -∇_·Σ=_s.

Since _s∈ L^2(Ω̃) and ∈^2(Ω̃_p) we have Σ∈(;Ω̃_p). Therefore the following Green's formula holds <cit.>:

    ∀∈^1(Ω̃_p),    (∇_·Σ, )_Ω̃_p=-(Σ,∇_)_Ω̃_p+⟨Σ·_Ω̃_p, ⟩_∂Ω̃_p.

This allows the well-definition of _p_Γ in the weak sense 
 on Γ_I.




§ REFERENCES



elsarticle-harv 









