A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

DOI:10.1051/cocv/2015045 www.esaim-cocv.org

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

Franc ¸ois Bouchon

, Gunther H. Peichl

, Mohamed Sayeh

and Rachid Touzani

Abstract. A free boundary problem for the Stokes equations governing a viscous flow with over- determined condition on the free boundary is investigated. This free boundary problem is transformed into a shape optimization one which consists in minimizing a Kohn–Vogelius energy cost functional.

Existence of the material derivatives of the states is proven and the corresponding variational problems are derived. Existence of the shape derivative of the cost functional is also proven and the analytic expression of the shape derivative is given in the Hadamard structure form.

Mathematics Subject Classification. 35R35, 49Q10, 35Q30, 76D07.

Received November 10, 2014. Revised August 26, 2015. Accepted September 17, 2015.

1. Introduction

This paper aims at studying a free boundary problem for fluid flows governed by the Stokes equations. This problem originates from numerous applications such as magnetic shaping processes, where the shape of the fluid is determined by the Lorentz force. For such configurations, the model is described by the fluid flow equations, here the Stokes equations, and a pressure balance equation on the unknown boundary in the case where surface tension effects are neglected. Depending on the application, two types of models can be considered: an interior problem where the fluid is confined in a mould and has an internal unknown boundary and an exterior problem where a part of the fluid boundary adheres to a solid and the remaining (unknown) part is free and is in contact with the ambient air. We shall focus our study on this last case for two-dimensional geometries.

Consider a bounded C^2,1 domainA⊂R² with boundaryΓ_f. The fluid is considered in levitation aroundA and occupies then the domainΩ=B\A, whereB is a bounded domain with boundaryΓ that containsA. Let uandpstand for the fluid velocity and pressure respectively. Letf ∈H_loc¹ (R²)² denote the density of a given body force andg∈H⁵²(Γ_f)² such that

Γf

g·ν = 0, whereν is the outer unit normal vector toΓ_f.

Keywords and phrases.Shape derivative, free boundary problems, Stokes Problem.

1 Clermont Universit´e, Universit´e Blaise-Pascal, Laboratoire de Math´ematiques, BP 10448, 63000 Clermont-Ferrand, France.

2 CNRS, UMR 6620, LM, 63171 Aubi`ere, France.francois.bouchon,rachid.touzani@univ-bpclermont.fr

3 University of Graz, Institute for Mathematics and Scientific Computing, NAWI Graz, Heinrichstr. 36, 8010 Graz, Austria.

gunther.peichl@uni-graz.at

4 Laboratoire de Mod´elisation Math´ematique et Num´erique dans les Sciences de l’Ing´enieur (LAMSIN), El Manar University, Tunis, Tunisia.Mohamed.Sayeh@ipein.rnu.tn

∗The work of this author was supported by the Austrian Science Fund (FWF) under grant SFB F32.

Article published by EDP Sciences c EDP Sciences, SMAI 2016

F. BOUCHONET AL.

For a given scalarλ we consider the free boundary problem of determining the domainΩ occupied by the fluid, the fluid velocityuand its pressurepsuch that:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

−2 divσ(u) +∇p=f in Ω,

divu= 0 in Ω,

u=g onΓ_f,

u·ν = 0 onΓ,

ν^Tσ(u)τ= 0 onΓ,

2ν^Tσ(u)ν−p=λ onΓ.

(1.1)

We start with the following observation concerning the scalarλ: if (Ω,u, p) is a solution of (1.1) corresponding to λ= 0, then (Ω,u, p−λ) is a solution of the same problem where 0 would be replaced byλ. For technical reasons, we will need to assume that

Ωp= 0 by adding a constant to the pressure, therefore we need to keep the scalarλin our analysis.

Above,ν andτ are the outer normal and tangent unit vectors to the boundaryΓ respectively and σ(u) = 1

2

∇u+∇u^T

is the symmetric deformation tensor. We note that the prescribed velocity g expresses the motion of the

‘mould’Γ_f.

Problem (1.1) can be viewed as an analog of the Bernoulli free boundary problem, where the Laplace operator is replaced by the Stokes operator. Assuming the existence of a solution to (1.1) we adopt the so-called Kohn–

Vogelius formulation, which applies for inverse problems (see [13,14,26] for instance) and which consists in matching aDirichletand aNeumannproblem. For an application of this technique to the Bernoulli free boundary problem we refer to [3,9].

Hence let (u_D, p_D), (u_N, p_N) denote for givenΩthe solution of the followingDirichlet, respectivelyNeumann

problem: ⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎩

−2 divσ(u_D) +∇p_D=f in Ω,

divu_D= 0 in Ω,

u_D=g onΓ_f,

ν^Tσ(u_D)τ = 0 onΓ, u_D·ν = 0 onΓ,

(1.2)

and ⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎩

−2 divσ(u_N) +∇p_N =f inΩ,

divu_N = 0 inΩ,

u_N =g onΓ_f,

ν^Tσ(u_N)τ = 0 onΓ, 2ν^Tσ(u_N)ν−p_N =λ onΓ.

(1.3)

Let us define the functionalJ on a suitable class of domainsΩby J(Ω) = 2

Ω|σ(u_D−u_N)|², where

|σ(u)|²:=

2 i,j=1

σ(u)²_ij.

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

We observe thatJ(Ω) = 0 if (u, Ω) is a solution of the free boundary problem (1.1) since thenu=u_D=u_N. Conversely, ifJ(Ω) = 0 Korn’s inequality implies u_D =u_N. Then (u, Ω) with u=u_D =u_N is a solution to the free boundary problem. Thus Problem (1.1) is equivalent to the shape optimization problem:

FindΩ such thatJ(Ω) = min

Ω˜ J( ˜Ω) = 0. (1.4)

Remark 1.1. Clearly the identityJ(Ω) = 0 is equivalent to the existence of a constantα∈Rsuch that (u_D, p_D) = (u_N, p_N+α).

In this paper we focus on a sensitivity analysis of the cost functional J. The computation of the shape gradient is complicated by the fact that applying the method of mapping to perturbations of Problem (1.2) neither preserves the divergence condition divu_D= 0 nor the boundary conditionu_D·ν = 0 onΓ. We address this issue and propose a novel strategy to overcome the ensuing difficulties in Section4.2. This technique only needs the weak form of the material derivative of the states.

The slip boundary condition u·ν = 0 on Γ allows tangential velocities on the free boundary, but inflow or outflow are not possible. This condition is appropriate for problems that involve free boundaries, flow past chemically reacting walls, and other examples where the usual no-slip condition u = 0 is not valid. There has been much discussion regarding the difference between the no-slip and slip boundary conditions. The no- slip condition is well established (see, e.g. Girault and Raviart [11] or Galdi [10]) for moderate pressures and velocities by direct observations and comparisons between numerical simulations and experimental results of a large class of complex flow problems. However, early experiments demonstrated that gases at low temperatures slip past solid surfaces. In particular, for sufficiently large Knudsen numbers velocity slip occurs at the wall surface. Such a wall slip has also been observed in the flow of nonlinear fluids such as lubricants, hydraulic fracturing fluids, biological fluids. A slip boundary condition is the appropriate physical model for a wide variety of applications including flow problems with free boundaries such as the coating flow (see [18,20,22,25]) and flows past chemically reacting walls (see [4]).

Free boundary problems and inverse problems with similar boundary conditions on the free boundary have already been investigated by several authors (see [2,16]). Solonnikov [24] considered the problem of filling a plane capillary. The problem of filling a circular capillary tube, coating flows and the steady flow in a capillary tube which is partially filled with liquid were considered. The latter free boundary problem was first considered by Sattinger [19] and investigated in more detail by Solonnikov [23]. As opposed to the problem discussed in this paper the free boundary however is in contact with a rigid wall.

As in most of the cited papers, we consider the simplified model where the boundary conditions on the free boundary do not take into account the surface tension terms. Thus, technical difficulties arising from the higher derivative terms are ignored.

The remainder of this paper is organized as follows: in Section 2 we recall some basic concepts, and results related to shape differentiation are given in Section 3. In Section 4 we prove the existence of the Eulerian derivatives of the state functionsu_Dandu_N and give the variational formulas verified by them. In Section 5 we prove the existence of the shape derivative of the functional J, and give its analytical representation together with the proof of this expression.

The results presented in Section 3 require theC^2,1 regularity of the boundaryΓ. Therefore, although many lemmas and theorems of other sections may required weaker assumptions, we considerΩ (and thenA andB) withC^2,1 boundaries. Both domainsAandB are assumed simply connected.

Remark 1.2. The results presented in this paper can be extended to the 3-dimensional case. This would need to rewrite some of the technical tools in Section 3, since we use the tangential vectorτ which is defined only in the 2-dimensional situation.

F. BOUCHONET AL.

2. Discussion of the equations for u

and u

In the followingL²(Ω) andH^k(Ω) will stand for classical Lebesgue and Sobolev spaces. Moreover, fork≥0, we shall denote byH^k(Ω) the space of vector valued functionsH^k(Ω)².

To define the variational formulations we shall make use of the following Hilbert spaces:

V={v∈H¹(Ω) :v_|Γf = 0}, V₀={v∈V:v·ν= 0 onΓ}, V_0,Γ ={v∈H¹(Ω) :v·ν = 0 onΓ},

Q=L²(Ω), Q₀={q∈Q:

Ωq= 0}, and the bilinear formsa:V×V→R,b:V×Q→Rdefined by

a(u,v) = 2

Ωσ(u) :σ(v), b(u, q) =−

Ωqdivu.

The Dirichlet problem can then be cast into the variational formulation:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

Find (u_D, p_D)∈V_0,Γ×Q₀ such that:

a(u_D,v) +b(v, p_D) =

Ω

f·v ∀v∈V₀,

b(u_D, q) = 0 ∀q∈Q₀,

u_D=g onΓ_f.

(2.1)

We remark that the pressurep_D is determined by (1.2) only up to an additive constant.

Similarly one can derive a weak formulation of the Neumann problem (1.3):

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

Find (u_N, p_N)∈H¹(Ω)×Qsuch that:

a(u_N,v) +b(v, p_N) =

Ω

f·v+λ

Γ

v·ν ∀v∈V,

b(u_N, q) = 0 ∀q∈Q,

u_N =g onΓ_f.

(2.2)

Existence and uniqueness of the solutions (u_N, p_N) ∈ H¹(Ω)×Q and (u_D, p_D) ∈ V_0,Γ ×Q₀ to the mixed variational problems (2.1) and (2.2), follow from the following properties satisfied by the bilinear formsaandb, (i) The bilinear formais coercive, i.e. there is a constantC >0 such that

a(v,v)≥Cv²_H1(Ω) ∀ v∈V,

(ii) The bilinear formbsatisfies the inf-sup condition,i.e. there is a constantβ >0 such that

q∈Q,q=0inf sup

v∈V,v=0

b(v, q)

v_H¹_(Ω)q_L²_(Ω) > β.

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

For further reference we recall the following regularity result:

Theorem 2.1(see Thm. 4.1 in [8]).

Let Ω∈ Rⁿ be a bounded domain of class C^k, with 1 ≤k,0 ≤ ≤1. There exists a Lipschitz continuous function f :Rⁿ→Rⁿ such that:

Ω={y∈Rⁿ, f(y)>0}, ∂Ω={y∈Rⁿ, f(y) = 0}

and a neighborhood ω of ∂Ω such that:

f ∈C^k,(ω), ∇f = 0 onω andν=−∇f

|∇f| whereν is the outer unit normal to Ωon ∂Ω.

The verification of the inf-sup condition for (2.2) is based on the following Lemma which is adapted from [6].

Lemma 2.2. LetΩ⊂R²be a doubly connectedC^1,1-domain with boundaryΓ∪Γf. Then there exists a constant C >0 such that for allp∈L²(Ω)there is a functionv∈H¹(Ω)satisfying

divv=p inΩ, v= 0 onΓ_f, v_H¹ ≤Cp_L². Proof. For givenp∈L²(Ω), consider the problem

⎧⎪

⎨

⎪⎩

divv=p in Ω, v=0 onΓ_f, v=Kν onΓ whereK= _|Γ¹_|

Ωp.

The result follows from Lemma 3.3 in [1] observing that ν ∈ C^0,1(ω) ⊂W^1,∞(ω)⊂H¹(ω) ⊂H^1/2(Γ) by

Theorem2.1.

Usingv∈Vas specified by Lemma2.2one can show the inf-sup condition for problem (2.2) similarly as for problem (2.1). Hence problem (2.2) is also well-posed.

We conclude this section with an interpretation of the boundary condition in (2.2) which isa priorinot clear sincep_N ∈L²(Ω) only. We note that the first equation in (2.2) can be written as

2

Ωσ(u) :∇v−

Ωpdivv=

Ω

f·v+λ

Γ

v·ν, v∈V, (2.3)

which implies that the equation

−2 divσ(u) +∇p=f (2.4)

is satisfied in the distributional sense. Since

(−2 divσ(u) +∇p)_i= div(−2σ(u) +pI)e_i, i= 1,2,

wheree_iis theith vector of the canonical basis inR²and (−2σ(u) +pI)e_i∈L²(Ω) we conclude that (−2σ(u) + pI)e_i∈H¹(div, Ω),i= 1,2. In view of Theorem 2.5 in [11], the normal componente^T_i(−2σ(u) +pI)ν,i= 1,2, exists inH⁻¹²(Γ) and the following Green’s formula holds forϕ∈H¹(Ω):

((−2σ(u) +pI)e_i,∇ϕ) + (div(−2σ(u) +pI)e_i, ϕ) =

e^T_i (−2σ(u) +pI)ν, ϕ

(2.5)

F. BOUCHONET AL.

(·,·denotes the duality pairing betweenH¹²(Γ) andH⁻¹²(Γ)). Since forψ∈H¹(Ω) one can verify ((−2σ(u) +pI)e_i,∇ψ_i) =−2

Ωσ(u) :σ(ψ) +

Ωpdivψ, (div(−2σ(u) +pI)e_i, ψ_i) = (−2 divσ(u) +∇p, ψ)

we obtain the identity

Ω(−2σ(u) :σ(ψ) +pdivψ) +

Ω(−2 divσ(u) +∇p)·ψ=(−2σ(u) +pI)ν, ψ.

In view of (2.3) and (2.4) this implies

−λ

Γψ·ν =(−2σ(u) +pI)ν, ψ. Hence (u, p)∈H¹(Ω)×Qsatisfies the boundary condition

(2σ(u)−pI)·ν=λν

in H⁻¹²(Γ). If the solution turns out more regular,e.g. (u_N, p_N)∈H²(Ω)×H¹(Ω) this implies 2ν^Tσ(u_N)τ= 0 onΓ,

2ν^Tσ(u_N)ν−p=λ onΓ.

A similar discussion can be carried out for (u_D, p_D).

In order to derive the shape derivative ofJ we need (u_D, p_D)∈H³(Ω)×H²(Ω) which is guaranteed by ([17], Thm. 6.2):

Theorem 2.3. Assume that Γ_f andΓ are of classC³ andC⁴ respectively, f ∈H¹(Ω),g∈H⁵2(Γ_f), then (u_D, p_D)∈H³(Ω)×H²(Ω)

Moreover

u_D3+pD2≤C

f1+g⁵₂ .

We point out that under the assumptions of Theorem2.3we also have (u_N, p_N)∈H³(Ω)×H²(Ω). Results in [2]

indicate that this regularity could be obtained assumingΩ∈C^2,1 only (see also [5], Thm. IV.7.4). Given this regularity of the solution one only needsC^2,1-regularity ofΩto justify the derivation of the shape derivative of J. Therefore we shall assume for the rest of the paperΩ∈C^2,1and (u_D, p_D), (u_N, p_N)∈H³(Ω)×H²(Ω).

3. Some material on the shape derivative of J

In this section we briefly describe the idea of the shape derivative of J and recall some additional results which will be needed in the sequel. For a more thorough exposition we refer to [8,12,21]. In order to define the shape derivative of J we embed problems (1.2) and (1.3) into a family of perturbed problems which are defined on perturbations of aC^2,1reference domainΩconstructed by perturbing the identity. LetU be a convex bounded domain of classC^2,1such that ¯Ω⊂U and let

S={h∈C^2,1( ¯U ,R²) :h= 0on ∂U∪Γ_f andh·τ = 0on Γ} (3.1) be the space of feasible deformation fields endowed with the natural norm inC²( ¯U ,R²). This choice of feasible deformation fields anticipates the form of the shape gradient ofJ according to the Hadamard–Zol´esio structure theorem (see [7,15]). For a fixed fieldh∈ S define for everyt∈Rthe mappingF_t: ¯U →R² by

F_t=id+th.

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

Fortsufficiently smallF_tdefines a family ofC²-diffeomorphisms ofU onto itself. For suchtone sets Ω_t=F_t(Ω), Γ_t=F_t(Γ),

hence Ω₀ =Ω, Γ₀ =Γ, Γ_f = F_t(Γ_f) and ¯Ω_t ⊂ U. For eacht sufficiently small we consider problems (1.2)_t and (1.3)_t which correspond to problems (1.2) and (1.3) with (Ω, Γ_f, Γ) replaced by (Ω_t, Γ_f, Γ_t). In view of the previous discussion (1.2)_t and (1.3)_t have unique weak solutions (u_D,t, p_D,t) ∈ V_0,Γt ×Q_0,t, respectively (u_N,t, p_N,t)∈H¹(Ω_t)×Q_twhere

Q_t=L²(Ω_t), Q_0,t={q_t∈Q_t:

Ωt

q_t = 0}, Set

J(Ω_t) = 2

Ωt

|σ(u_D,t−u_N,t)|². Then the Eulerian derivative ofJ atΩ in the directionhis defined as

J(Ω;h) = lim

t→0

1

t(J(Ω_t)−J(Ω)).

The functional J is called shape differentiable atΩ if J(Ω;h) exists for all h ∈ S and defines a continuous linear functional onS.

In order to evaluateJ(Ω_t)−J(Ω) we transformJ(Ω_t) to an integral over the reference domainΩ. For this purpose we refer any functionϕ_t:Ω_t→R^k,k= 1,2, to the reference domain using the transformation

ϕ^t=ϕ_t◦F_t: Ω→R^k.

Let us define the material and shape derivatives of a function depending on the domain.

Definition 3.1. Letu(Ω) be a function defined and depending on the domainΩ. The material derivative ofu in the directionhatΩ is the limit

˙

u(h) := lim

t→0

u(Ω_t)◦F_t−u(Ω)

t ,

when it exists. The local derivative also called shape derivative in the directionhatΩis the limit u(h) := lim

t→0

u(Ω_t)−u(Ω)

t ·

The general strategy to derive shape derivatives is to first transfer the problem back to the original boundary before computing the limit, which results in the need to compute the material derivative. The chain rule combines both by the identity

˙

u(h) =u(h) +∇u·h.

Thus, ifudoes not depend on the geometry, one has

u(h) = 0 and u(h) =˙ ∇u·h.

The following technical results (see [8], Chap. 9) will be useful in deriving equations for the transported solutions (u^t_D, p^t_D) and (u^t_N, p^t_N).

F. BOUCHONET AL.

Lemma 3.2. Let ϕ_t∈H¹(Ω_t)andv_t∈H¹(Ω_t). Then ϕ^t∈H¹(Ω),v^t∈H¹(Ω)and we have (∇ϕ_t)◦F_t= (∇ϕ_t)^t=M_t∇ϕ^t,

(divv_t)◦F_t= (divv_t)^t=M_t^T :∇v^t, where

M_t= (DF_t)^−T = (∇F_t)⁻¹. Here,Ddenotes the Jacobian matrix and∇=D^T.

For further use we recall the definition of the tangential divergence of a vector field ϕ and the tangential gradient of a sufficiently smooth functionf:

div_Γϕ= divϕ_|Γ − ∇ϕν·ν,

∇_Γf =∇f_|Γ−∂f

∂νν.

Next we briefly discuss the variation of the normal fieldν_t. Letb_tdenote the signed distance function to Γ_t such thatν_t=∇b_t. By ([8], Thm. 5.4.3) we haveb_t∈C²in a neighborhood ofΓ_t. Define

b^t=b_t◦F_t, ν^t=ν_t◦F_t.

Since Γ is the zero level set of b^t we conclude that ∇b^t is collinear to ν =ν₀ and ∇b^t=αν for some α >0.

This implies

ν^t=∇bt◦F_t=M_t∇b^t=αM_tν.

In view of|ν^t|= 1 one eventually obtains

ν^t= M_tν

|M_tν|· (3.2)

Since ∇b provides an unitary C^1,1-extension of ν to a neighborhood of Γ we deduce from (3.2) the following properties of the normal and tangent vector fields:

Lemma 3.3. Let Ω be aC^2,1 domain. We have:

(∇ν)ν = (∇τ)τ = 0, (3.3)

(∇ν)τ+ (∇τ)ν= 0, (3.4)

(∇ν)τ =−(∇τ)ν =κτ. (3.5)

whereκdenotes the curvature.

Proof. (3.3) and (3.4) follow by differentiating the identities |ν|² =|τ|² = 1, respectively ν·τ = 0. (3.5) is a consequence of the following identities:

(∇ν)τ·ν = 0, (∇ν)τ·τ = div_Γν=κ,

and (3.4).

The relations (3.3)–(3.5) even hold on a neighborhood ofΓ sinceb is defined and regular on a neighborhood ofΓ.

Finally we report some properties of the deformation fields h∈ S.

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

Lemma 3.4. Let h∈ S. Then the following relations hold onΓ:

∇(h·ν) = (∇h)ν, (3.6)

div_Γh=κh·ν. (3.7)

Proof. Combiningh= (h·ν)ν onΓ and (3.3) we obtain

∇(h·ν) = (∇h)ν+ (∇ν)h= (∇h)ν+ (h·ν)(∇ν)ν = (∇h)ν.

For the proof of (3.7) we observe that div_Γhdoes not depend on the extension ˜hofhinto a neighborhood of Γ. Since h= (h·ν)ν onΓ one can use the identity

h˜ = (h· ∇b)∇b= (h·ν)ν for the computation of div_Γ h. Thus we obtain

div_Γh= div_Γh˜ = div ˜h−(∇h)˜ ν·ν.

Since∇h˜=∇(h·ν)ν^T+ (h·ν)∇ν, (3.3) leads to

(∇h)˜ ν·ν=∇(h·ν)ν^Tν·ν+ (h·ν)(∇ν)ν·ν =∇(h·ν)·ν.

Using divν =κwe therefore find

div_Γh= div((h·ν)ν)− ∇(h·ν)·ν

=∇(h·ν)·ν+ (h·ν) divν− ∇(h·ν)·ν =κ(h·ν).

As a consequence of (3.2) and Lemma 3.4 we obtain the following expression for the material derivative of the normal field (see also [12]).

Lemma 3.5. For each t sufficiently small ν^t is C¹ in a neighborhood of Γ andt →ν^t(x) is differentiable at t= 0uniformly in x∈Γ and its derivative at 0 is given by the following expression:

˙

ν =−∇Γ(h·ν). (3.8)

A similar statement holds for the transported tangent vectorτ^t. This follows from the representation τ^t= 1

|M_tν|(−(Mtν·τ)ν+ (M_tν·ν)τ).

Lemma 3.6. Forf, g∈H²(Ω)the following integration by parts formula holds

∂Ωf(∇g·τ) =−

∂Ωg(∇f·τ).

In particular we have forh∈ S

Γ

f∇hν·τ=−

Γ

(∇f·τ)h·ν.

Proof. Recall that the following integration by parts formula holds for domains of classC² andϕ∈H²(Ω) and ψ∈C¹( ¯Ω) such thatψ·ν= 0 on∂Ω

∂Ω∇Γϕ·ψ=−

∂Ωϕdiv_Γ ψ.

F. BOUCHONET AL.

Chooseϕ=gandψ=f_nτwhere (f_n)⊂C¹( ¯Ω) approximatesf inH²(Ω) andτis a continuously differentiable extension of the tangent field toΩwhich is unitary in a neighborhood of∂Ω. Note that in such a neighborhood of∂Ω we haveτ= (−b_y, b_x) and hence divτ= 0. This entails

div_Γ(f_nτ) =∇f_n·τ−D(f_nτ)ν·ν.

Then (3.5) implies

D(f_nτ)ν·ν=Df_nν τ·ν+f_nDτ ν·ν= 0.

Since∇g·τ=∇Γg·τ we obtain

∂Ωf_n(∇g·τ) =−

∂Ωg(∇f_n·τ)

which implies the result. The second statement follows choosing g = h·ν using (3.6). Note that g vanishes

onΓ_f.

We conclude this section with some well known formula:

dM_t

dt |t=0=−∇h and then dM_t^−T

dt |t=0=∇h^T (3.9)

and

dδ_t

dt |t=0 = divh (3.10)

where we have setδ_t= Det(DF_t) = det(I+tDh).

4. Differentiability of the states

For eacht∈I we define the bilinear formsa^t,a:˙ V×V→Randb^t,b˙:V×Q→Rby a^t(u,v) =

Ωδ_tM_t^T(M_t∇u+ (M_t∇u)^T) :∇v,

˙

a(u,v) = 2

Ω(divh)σ(u) :∇v−2

Ωσ(u) :∇h∇v−2

Ω∇h∇u:σ(v), b^t(u, q) =−

Ωδ_tq(M_t^T :∇u), b(u, q) =˙ −

Ωqdivhdivu+

Ωq∇h^T :∇u.

4.1. Differentiability of t → (u

, p

)

It was shown in Section2that the family of perturbed Neumann problems (1.3)_tis well posed inH¹(Ω_t)×Qt

fort in a sufficiently small neighborhoodI of 0. We observe that the transported solutions (u^t_N, p^t_N) belong to V×Q. One can verify that (u^t_N, p^t_N) satisfies the following system.

Theorem 4.1. For each t∈I the pair(u^t_N, p^t_N)∈H¹(Ω)×Qis the unique solution of the problem

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

a^t(u^t_N, ϕ) +b^t(ϕ, p^t_N) =

Ωδ_tf^t·ϕ+λ

Γδ_tϕ·M_tν ∀v∈V,

b^t(u^t_N, χ) = 0 ∀χ∈Q,

u^t_N =g onΓ_f.

(4.1)

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

Proof. We just indicate the transformation of the boundary integral in (2.2)_tdescribing the Neumann condition.

The transformation formula for boundary integrals implies

Γt

ϕ_t·ν_t=

Γ ϕ^t·ν^tδ_t|M_tν|=

Γδ_tϕ^t·M_tν

where we used (3.2) in the last step (see also [2], Lem. 3.1).

Now we can state the main result of this section.

Theorem 4.2. The mapping t → (u^t_N, p^t_N) is continuously strongly differentiable on a neighborhood of zero.

Furthermore,

( ˙u_N,p˙_N) := ∂

∂t(u^t_N, p^t_N)|t=0, is the unique solution of the following variational problem:

⎧⎪

⎨

⎪⎩

Find ( ˙u_N,p˙_N)∈V×Qsuch that:

a( ˙u_N, ϕ) +b(ϕ,p˙_N) =L_N(ϕ) ∀ ϕ∈V, b( ˙u_N, χ) =−b(u˙ _N, χ) ∀ χ∈Q,

(4.2)

whereL_N(ϕ) is given by

L_N(ϕ) :=−a(u˙ _N, ϕ)−b(ϕ, p˙ _N) +_N(ϕ), and

_N(ϕ) =

Ω

(divh f·ϕ+∇f^Th·ϕ) +λ

Γ

(divhϕ·ν−ϕ·(∇h)ν).

Forϕ∈V∩H²(Ω)one can represent_N as _N(ϕ) =

Ω(divh f·ϕ+∇f^Th·ϕ) +λ

Γ(∇ϕτ·τ)h·ν.

Proof. Let us consider the mapping

H = (H₁, H₂) :I×V×Q→V^∗×Q^∗ defined by

H1(t,w, θ), ϕ=a^t(w+u_N, ϕ) +b^t(ϕ, θ)

−λ

Γδ_tϕ·M_tν−

Ωδ_tf^t·ϕ, ϕ∈V H₂(t,w, θ), χ=b^t(w+u_N, χ), χ∈Q.

(4.3)

Then (u^t_N −u_N, p^t_N) is the unique element ofV×Qwhich satisfies H(t,u^t_N −u_N, p^t_N) = 0 for eacht∈I. Since

D_(w,θ)H(0,0, p_N)(δw, δθ),(ϕ, χ)= (a(δw, ϕ) +b(ϕ, δθ), b(δw, χ)) (4.4)

F. BOUCHONET AL.

it follows from the discussion in Section2thatD_(w,θ)H(0,0, p_N) is an isomorphism fromV×QontoV^∗×Q^∗. Furthermore, one can verify thatt→H(t,w, θ) is weakly differentiable for every (w, θ)∈V×Qand

∂

∂tH₁(t,w, θ), ϕ= ∂

∂ta^t(w+u_N, ϕ) + ∂

∂tb^t(ϕ, θ)

−

Ω

δ˙_tf^t·ϕ−

Ωδ_t∇f^T◦F_th·ϕ

−λ

Γ

δ˙_tϕ·M_tν−λ

Γδ_tϕ·M˙_tν

∂

∂tH₂(t,w, θ), χ= ˙b^t(w+u_N, χ).

Since the derivatives ofδ_tandM_twith respect totexist inL^∞(Ω), respectivelyL^∞(Ω,R^2×2) one can show that the limits defining the above weak derivatives can be taken uniformly with respect toϕ∈V, respectivelyχ∈Q which implies thatt→H(t,w, θ) is strongly continuously differentiable. ThereforeH satisfies the assumptions of the implicit function theorem which ensures the strong differentiability oft →(u^t_N, p^t_N) in a neighborhood of 0. Its derivative at 0, ( ˙u_N,p˙_N), is the unique solution of the system

D_(w,θ)H(0,0, p_N)( ˙u_N,p˙_N),(ϕ, χ)=− ∂

∂tH(0,0, p_N),(ϕ, χ)

which reads

D_(w,θ)H₁(0,0, p_N)( ˙u_N,p˙_N), ϕ,D_(w,θ)H₂(0,0, p_N)( ˙u_N,p˙_N), χ

=−∂

∂t(H₁(0,0, p_N), ϕ,H₂(0,0, p_N), χ) which by (4.4) is equivalent to

a( ˙u_N, ϕ) +b(ϕ,p˙_N) =−∂

∂tH₁(0,0, p_N), ϕ b( ˙u_N, χ) =−∂

∂tH2(0,0, p_N), χ.

One can verify

∂

∂tH2(0,0, p_N), χ= ˙b(u_N, χ), similarly

∂

∂tH₁(0,0, p_N), ϕ= ˙a(u_N, ϕ) + ˙b(ϕ, p_N)−

Ω(divh f·ϕ+ (∇f^T)h·ϕ)

−λ

Γ(divhϕ·ν−ϕ·(∇h)ν).

A FREE BOUNDARY PROBLEM FOR THE STOKES EQUATIONS

Forϕ∈V∩H²(Ω) one can use Lemma3.6and (3.7), (3.3) and (3.5) to simplify the last term:

Γ((divh)ϕ·ν−ϕ· ∇hν) =

Γ((divh− ∇hν·ν)ϕ·ν−(ϕ·τ)∇hν·τ)

=

Γ(ϕ·ν div_Γh+

Γ(∇(ϕ·τ)·τ)h·ν

=

Γ(κϕ·ν+∇ϕτ·τ+∇τ ϕ·τ)h·ν

=

Γ

κϕ·ν+∇ϕτ·τ+ (∇τ ν·τ)(ϕ·ν)

h·ν

=

Γ

∇ϕτ·τ h·ν

which completes the proof of the theorem.

4.2. Differentiability of t → (u

, p

)

The analysis of the differentiability of t → (u^t_D, p^t_D) is more difficult since the family of perturbed prob- lems (1.2)_t is well posed only in the spaceV_0,Γt ×Q_0,t. However, the transported solutions (u^t_D, p^t_D) do not belong to the subspaceV_0,Γ ×Q₀. This is a significant difference to the discussion in the preceding section. At first we note that the condition

Ωtp_D,t= 0 leads to

Ωδ_tp^t_D= 0. Secondly we note that by (3.2) the condition u_D,t·ν_t= 0 onΓ_t is equivalent to

(u_D,t·ν_t)◦F_t=u^t_D·ν^t=u^t_D· M_tν

|Mtν| = 0 onΓ.

Hence u^t_D satisfies onΓ the boundary condition

M_t^Tu^t_D·ν = 0 onΓ

and therefore does not belong toV_0,Γ. Transforming (1.2)_tto the reference domain we obtain Lemma 4.3. For each t∈I the pair(u^t_D, p^t_D)is the unique solution of the following problem:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Find (u^t_D, p^t_D)∈H¹(Ω)×Q^t₀, such that:

a^t(u^t_D, ϕ^t) +b^t(ϕ^t, p^t_D) =

Ωδ_tf^t·ϕ^t ∀ ϕ^t∈V^t₀, b^t(u^t_D, χ^t) = 0, ∀ χ^t∈Q^t₀,

u^t_D=g onΓ_f,

u^t_D·M_tν = 0 onΓ,

(4.5)

where

V^t₀={ϕ∈V:M_t^Tϕ·ν= 0 onΓ}, Q^t₀={χ∈Q:

Ωδ_tχ= 0}.

The boundary condition inV^t₀ suggests to consider

w^t=M_t^T(u^t_D−u_D),

F. BOUCHONET AL.

where (u_D, p_D) is the solution of the following problem

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−divσ(u_D) +∇p_D= 0 inΩ,

divu_D= 0 inΩ,

u_D=g onΓ_f,

u_D= 0 onΓ.

(4.6)

Observe that w^t ∈ V₀ holds for all t ∈ I. This motivates to introduce the following family of auxiliary problems inV₀×Q₀: For eacht∈I

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Find (w^t,pˆ^t_D)∈V₀×Q₀ such that:

a^t

M_t^−Tw^t+u_D, M_t^−Tϕ +b^t

M_t^−Tϕ,pˆ^t_D

=

Ωδ_tf^t·M_t^−Tϕ ∀ϕ∈V₀, b^t

M_t^−Tw^t+u_D,χ δ_t

= 0, ∀χ∈Q₀.

(4.7)

Problems (4.5) and (4.7) are related as follows:

Lemma 4.4. Let (w^t,pˆ^t_D)∈V₀×Q₀ be a solution of (4.7)and define (u^t_D, p^t_D) =

M_t^−Tw^t+u_D,pˆ^t_D+c(t) with

c(t) =−

Ωδ_tpˆ^t_D

Ωδ_t

then (u^t_D, p^t_D) ∈ H¹(Ω)×Q^t₀ is the solution of (4.5). Conversely if (u^t_D, p^t_D) ∈ H¹(Ω)×Q^t₀ is the unique solution of (4.5)then

(w^t,pˆ^t_D) =

M_t^T(u^t_D−u_D), p^t_D− 1

|Ω|

Ωp^t_D

is a solution of (4.7)

As a consequence we conclude that (4.7) admits a unique solution for everyt∈I.

Lemma 4.5. The mappingt→(w^t,pˆ^t_D)is continuously differentiable in a neighborhood of0and we have that ( ˙w,p˙ˆ_D) =_∂t^∂(w^t,pˆ^t_D)|t=0∈V₀×Q₀ is the solution of the following variational problem:

⎧⎪

⎨

⎪⎩

Find ( ˙w,p˙ˆ_D)∈V₀×Q₀ such that:

a( ˙w, ϕ) +b(ϕ,p˙ˆ_D) = ˆL_D(ϕ) ∀ ϕ∈V₀, b( ˙w, χ) =−b(∇h^T(u_D−u_D), χ)−b(u˙ _D, χ) ∀ χ∈Q₀

(4.8)

where

Lˆ_D(ϕ) =−a(∇h^T(u_D−u_D), ϕ)−a(u_D,∇h^Tϕ)−a(u˙ _D, ϕ)

−b(∇h^Tϕ, p_D)−b(ϕ, p˙ _D) +

Ωdivhf·ϕ +

Ω∇f^Th·ϕ+

Ω

f· ∇h^Tϕ.