ON BLOCH WAVES FOR THE STOKES EQUATIONS

In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in Rd. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

1. Introduction and main results.The method of Bloch wave decomposition (or Floquet decomposition) is well known for reducing the problem of solving the Schrödinger equation in an infinite periodic medium to a family of simpler Schrödinger equations posed in a single periodicity cell and parametrized by the so-called Bloch frequency [9], [10], [17], [23].We extend this method to the Stokes equations of incompressible fluid mechanics or equivalently to the equations of linear incompressible elasticity.
where D(ξ) = (∇ + iξ), µ ∈ L ∞ # (Y ) is a given uniformly positive viscosity, and κ ∈ L ∞ # (Y ) is a given damping coefficient.We assume µ(y) ≥ µ 0 > 0 and κ(y) ≥ 1 a.e. in Y (without loss of generality since adding a constant to κ is equivalent to shifting the spectrum).All our results could be generalized to the case of κ being a symmetric matrix and µ a symmetric coercive fourth-order tensor.We also discuss the limit case κ(y) = +∞ in some subset T ⊂ Y which corresponds to a hole or obstacle T supporting a Dirichlet boundary condition.
We are mainly interested in the continuity and differentiability properties of the eigenvalues and eigenfunctions of (1).Indeed, it is well-known for the Schrödinger equation [7], [9], that the Hessian of the first eigenvalue at the origin ξ = 0 is the effective tensor for the corresponding homogenized problem.A first rigorous analysis of the homogenization process along these lines was given in [10].Their results have been generalized to some extent for systems of diffusion equation [4] when the first eigenvalue is simple, and for the elasticity system [13] using merely directional derivability.Therefore, the differentiability structure of ( 1) is a problem of paramount importance for homogenization which is our main motivation here.The difficulty, in the case of Stokes equations, is that (1) is not posed in a fixed functional space when ξ varies because the incompressibility constraint D(ξ) • φ = 0 precisely depends on ξ.
Our first result (Proposition 2.8) is that, contrary to the above examples, the eigenvalues and eigenfunctions of (1) are usually not even continuous at the origin.The main reason for this strange phenomenon is that the limit of the incompressibility constraint D(ξ) • φ = 0, as ξ goes to 0 in a fixed direction e, is not only D(0) • φ = 0 but it also includes the additional constraint Y e • φ dy = 0.A similar discontinuity of the Bloch eigenvalues at the origin was already obtained for a completely different model of fluid structure interaction in [5].
Our main result (Theorem 3.7) is to nevertheless prove that there exists a new family of spectral Stokes problem, featuring the additional constraint Y e•φ dy = 0, which are the limits of (1) when ξ goes to 0 in a fixed direction e.We prove that eigenvalues and eigenfunctions are thus directionally analytic.Then, as a final result (Lemma 4.2), we partially recover the usual homogenized effective tensor of Stokes equations, whose entries are linked to the second-order derivatives of the first eigenvalues.
The content of the present paper is the following.In Section 2 we first study the simpler non-homogeneous Stokes problem, i.e. we replace the right hand side of (1) by a fixed force term.Already for this simpler problem we obtain a discontinuity result in Proposition 2.8.Section 3 is devoted to the analyticity properties of the Bloch waves and contain our main result, Theorem 3.7.In Section 4 we compute the derivatives of the first Bloch eigenvalue at ξ = 0 and we show that we can partially recover the homogenized tensor of the Stokes problem (see Lemma 4.2).In Section 5 we present the Bloch decomposition of the space of divergence-free vector fields on R d .In Section 6, we study the Bloch decomposition and the regularity of the Bloch waves in the case of a periodically perforated domain, in a similar way as for nonperforated domains.Finally Section 7 contains some 2-d numerical computations which illustrate the discontinuity of the Bloch eigenvalues.
Notations.Let us make precise the definition of D(ξ).If φ = (φ k ) 1≤k≤d is a vector-valued function, then D(ξ)φ = ∇φ + iφ ⊗ ξ is a matrix of entries (∂φ k /∂y l + iφ k ξ l ) 1≤k,l≤d .Let L 2 # (Y ) denote the space of functions in L 2 loc (R d ) which are Yperiodic.A similar definition holds for the Sobolev space H 1 # (Y ).Its dual is denoted by H −1 # (Y ).We denote by L 2 #,0 (Y ) the subspace of L 2 # (Y ) made of functions with zero-average on Y .All these spaces are made of complex-valued functions.
2. The non-homogeneous Stokes problem.We first consider the Stokes equations with a source term Note that in the case ξ = 0, the well-posedness theory of ( 2) is well-known: for each the pressure is defined up to an additive constant).As we shall see the existence theory in the case ξ = 0 is different but not more difficult.However, the surprising property that we shall establish is the lack of continuity of (2) as ξ goes to zero (see Proposition 2.8).
where the constant C > 0 does not depend on ξ and m is the averaging operator defined by (Note that, in this case, the pressure is uniquely defined.)As usual, for ξ = 0, there exists a unique solution Before proving Proposition 2.1 we need a series of lemma.We introduce the spaces of "generalized" divergence-free velocities which, upon introducing the Fourier series ϕ(y) and we deduce that ), and we conclude that that is, in terms of Fourier series, and since φ(k) satisfies and moreover and the representation is unique, which completes the proof.
On the other hand, we have the following regularity result for the pressure.
where m is the averaging operator defined by m(f Let us assume that m(g) = 0, therefore multiplying by −i(ξ + k) we obtain that for k = 0. Furthermore, if ξ = 0 we have p(0) = 0, while if ξ = 0 the value of p(0) is free.Thus, for ξ = 0, there exists a positive constant c, independent on k and ξ, such that, for every When ξ = 0, inequality (5) holds true if we choose p such that p(0) = 0. Therefore, for any value of ξ, summing up the squared inequalities (5) we conclude that p ∈ L 2 0,# (Y ) and Now, if we consider a general function and applying (6) we deduce that and p ∈ L 2 # (Y ).Proof of Proposition 2.1 For ξ = 0 the result is well known [18].For ξ = 0 we define the bilinear form on which is easily seen to be symmetric, continuous and coercive since there exists a positive constant C such that (see [9]) By the Lax-Milgram's Lemma, the problem which proves the existence and uniqueness of the solution of (2).Furthermore, since κ(y) ≥ 1, the bilinear form is uniformly coercive, namely there exists a positive constant C, independent of ξ, such that which is simply obtained by integrating D(ξ) • φ = 0 over Y .Therefore, in the limit as ξ goes to 0 in a fixed direction e, we formally obtain two limit constraints: ∇ • φ = 0 and e • Y φ dy = 0.This explains the introduction of a new Stokes problem ( 7) below.
To study the limit of the Stokes problem (2) when ξ goes to 0, we introduce a new family of Stokes problems, parametrized by a unit vector e ∈ R d with |e| = 1, Since there is an additional constraint on the velocity in (7), there is also an additional Lagrange multiplier q 0 e, where q 0 is a real constant.A possible physical interpretation is that adding the constraint e • Y u = 0 amounts to apply an affine pressure term q 0 e • y.
We introduce new spaces of "extended" divergence-free velocities: for each unit vector e ∈ R d , |e| = 1, we define which thus admits a unique solution u ∈ V e .To recover the pressure and the constant q 0 , we use Lemma 2.6 below.A priori the pressure q is defined up to an additive constant, but forcing its average to be zero (as is the case in L 2 #,0 (Y )) uniquely determines q.Lemma 2.6.Let e ∈ R d be a unit vector.We have We are now ready to prove that the family of Stokes equations (7) are the limits of (2) when ξ goes to 0. In other words the Stokes problem (2) is not continuous as ξ goes to 0. Proposition 2.8.For a given unit vector e ∈ R d , |e| = 1, we define ξ = εe.Then, as ε tends to 0, the solution (φ(ξ), p(ξ)) of (2) satisfies #,0 (Y ) iξm(p(ξ)) → q 0 (e)e, (8) where (u(e), q(e), q 0 (e)) is the unique solution of (7).

For any ψ
and passing to the limit as ε tends to zero, we obtain which is precisely a variational formulation for (7).Thus (u(e), q(e), q 0 (e)) is the unique solution of ( 7) and the entire sequence converges.On the other hand, we have and since φ(ξ) → u(e) strongly in which implies the strong convergence of φ(ξ) in H 1 # (Y ) d .A similar strong convergence for the pressure is finally obtained from Lemma 2.3.

Bloch Waves and Analyticity
Properties.This section is devoted to the study of the Bloch spectral problem for the Stokes equation.In particular a detailed study of the Bloch waves as ξ tends to zero is performed since the Bloch waves are not smooth at the origin ξ = 0.
For ξ ∈ Y we consider the eigenvalue problem We begin by stating a classical result of existence and regularity of the Bloch waves away from the origin ξ = 0. Theorem 3.1.For all fixed ξ ∈ Y \{0}, problem (9) admits a countable sequence of real positive eigenvalues each of which is of finite multiplicity.As usual, we arrange them in increasing order repeating each value as many times as its multiplicity:

The corresponding eigenfunctions denoted by {φ n (ξ)} n≥1 forms a Hilbert basis in
are Lipschitz continuous functions of ξ.We will refer to {λ n (ξ)} n≥1 as Bloch eigenvalues and to {φ n (ξ)} n≥1 as Bloch eigenvectors or Bloch waves associated to the classical incompressible Stokes equations.
Remark 3.2.The continuity of the functions λ n : Y \ {0} → R cannot be derived by using minimax principle as in [10], this is due to the dependence on the parameter ξ ∈ Y \ {0} of the spaces V ξ .Thus the proof of Theorem 3.1 is similar to the proof of Theorem 3.7 (see Appendix) and the previous proofs in [20,21,22, 24] so we shall not repeat it here.The main idea is to consider, for any λ > 0 and ξ ∈ Y \ {0}, the map and to use the Lyapunov-Schmidt method, which is well known in bifurcation theory.The point is that if λ(ξ) is an eigenvalue of multiplicity h of (9) and φ 1 (ξ), . . ., φ h (ξ) are the associated velocities and p 1 (ξ), . . ., p h (ξ) the corresponding pressures, we have Ker(A) = span {(φ j (ξ), p j (ξ)) : j = 1, . . ., h} .Then, the problem is reduced to the existence of the roots of a polynomial of degree h of the form where the coefficients a j (ξ) have an analytic dependence on the parameter ξ.Thus, since the parameter belongs to R n we can conclude only the continuity of the roots α j (ξ) of the polynomial and the eigenvalues have the form λ(ξ) = λ + α(ξ).We note that in the case of the scalar perturbation, the Weierstrass Preparation Theorem gives us the regularity of the eigenvalues.
The continuity of the functions (φ n , p n ) is an easy consequence of the continuity of λ n and the smoothness of the corresponding Green's operator.
Remark 3.3.The study of the analyticity properties of λ n with respect to ξ requires special attention.A standard application of Rellich's theorem (see [16] p. 392) shows the existence of branches of eigenvalues of the Green's operator of problem (1) which are real analytic with respect to each individual variable ξ l .However, perturbation theory is inadequate to study the real-analyticity of these branches with respect to all the variables ξ (see [16] p. 177).In the sequel, we will come back to study analyticity properties of λ n , but following a completely different strategy which in fact merely allows to get a directional analyticity of these functions.Remark 3.4.From Rellich's theory, one can check that if the Bloch eigenvalue at a given frequency ξ 0 , ξ 0 = 0, is simple, the branch ξ → λ(ξ) is also simple in a neighborhood of ξ 0 and moreover, it is analytic in this same neighborhood.Hence the map ξ → φ(ξ) can also be proved to be analytic in this neighborhood.
In the general case, if we consider ξ = ξ 0 + εe, with ε ∈ R \ {0}, ξ 0 ∈ Y \ {0} and e ∈ R d , a unit vector, the classical results of Rellich [24] or Kato [16] show that the branches of eigenvalues and eigenvectors are analytic with respect to the real parameter ε in a neighborhood of 0. This allows to compute directional derivatives of the Bloch waves at ξ 0 in the direction e.
Theorem 3.1 has left apart the regularity of the spectrum of ( 9) at ξ = 0. Actually, the spectrum is not continuous at ξ = 0 because, as already mentioned in Remark 2.4, the incompressibility constraint D(ξ) • φ = 0 changes with ξ and its limit as ξ goes to 0 is not just D(0) • φ = 0. We shall establish below that the limits of ( 9) when ξ converges to 0 in the direction e are given by (10) To resolve this spectral problem, it is a classical technique to introduce the so-called Green's operator , where u ∈ V e is the unique weak solution of (7) given by Lemma 2.5.Applying next Rellich's compactness lemma it is straightforward to check that G e is a compact operator.Furthermore, it is also selfadjoint and it is easily verified that (ν(e), u(e)), ν(e) = 0 satisfies (10) iff u(e) and u(e) ≡ 0.
This means that ν(e) is a nonzero eigenvalue of problem (10) with corresponding eigenfunction u(e) iff 1 ν(e) is a nonzero eigenvalue of G e with eigenvector u(e).A standard application of the classical Hilbert-Schmidt theorem yields the following result about the spectrum of (10).Lemma 3.5.Problem (10) admits a countable sequence of real positive eigenvalues (ν n (e)) n≥1 converging to +∞ with n (repeated with their multiplicity) and an Hilbert basis of associated eigenfunctions (u n (e)) n≥1 .
To study the continuity of ( 9) when ξ converges to 0, we rewrite (9) in a slightly different form.For a given unit vector e ∈ R d , |e| = 1, we introduce a scalar parameter ε ∈ R and define The eigenvalue problem ( 9) is thus rewritten Introducing a new pressure variable we obtain that D(εe)p ε = D(εe)q ε + q ε 0 e with q ε 0 = iεm(p ε ).On the other hand, as already said in Remark 2.4, the condition D(εe) The spectral problem (12) has a structure similar to (10), so we can expect the former is an analytic perturbation of the latter.The study of analytic properties of the solution of (12) with respect to ε will follow from the general analytic perturbation theory for solutions of operators depending on one real parameter (see [24]).This is the starting point to obtain the analyticity of the spectrum in a neighborhood of ε = 0. Remark 3.6.As already noticed in Remark 2.7 the orthogonal of the space We are now ready to prove a result on the regularity of (λ We shall follow the generalization proposed in [20] of Rellich's method.We remark that the main difficulty for applying Rellich's method is that the functional spaces are varying with ε.Theorem 3.7.Assume that ν is an eigenvalue of multiplicity h of the Stokes system (10).Then there exist h analytic functions defined in a neighborhood of 2. for all ε small enough, λ ε j , φ ε j , q ε j , q ε 0,j is a solution of ( 12), 3. for all ε small enough, the set for each interval I ⊂ R such that I contains only the eigenvalue ν, and for all ε small enough, there are exactly h eigenvalues (counting the multiplicity) λ ε 1 , . . ., λ ε h of (11) contained in I.The proof of Theorem 3.7 is given in the Appendix.
Remark 3.8.In the above theorem, if ν correspond to the k + 1-th eigenvalue of (10) of multiplicity h, that is then there exist h regular functions ε → λ ε j , such that, λ ε j is an eigenvalue of (12) verifying λ ε j | ε=0 = ν and the branches of eigenvalues λ ε j correspond to the k + 1-th to k + h-th eigenvalues of (12), but not necessarily ordered in an increasing order.Remark 3.9.Since the Stokes problems (10) are the limits of the Bloch spectral problems (9) when ξ tends to 0, one can wonder if the purely periodic problem obtained by taking ξ = 0 in (9), has any connections to (10).It turns out that any eigenvalue and eigenvector of (13) is also an eigenvalue and eigenvector of (10) for a well-chosen vector e (orthogonal to the average of the eigenvector) and for q 0 = 0. Therefore, (13) gives no new contribution in the spectrum of the Bloch problems.
4. On the derivatives of the Bloch eigenpairs.This section focus on the computation of the derivatives of the Bloch eigenvalues and eigenvectors near the origin ξ = 0.It is well known, in the case of the Laplace operator, that the first Bloch eigenvalue is analytic in a neighborhood of ξ = 0, which is a consequence of the simplicity of the first eigenvalue, and its Hessian matrix is just the usual homogenized tensor [9].In the spirit of [12], we want to generalize this result for the Stokes equations.The main difficulty is that the first Bloch eigenvalue is not simple and not continuous.Fortunately we can compute directional derivatives due to the directional analyticity of the Bloch eigenvalues.
In view of the application that we have in mind (namely, the homogenization of the Stokes equations), we restrict ourselves to the special case For a unit vector e ∈ R d , we define ξ = εe, for ε ∈ R, and we study the directional derivatives of the first Bloch eigenvalues and eigenfunctions with respect to the real scalar parameter ε.Because of assumption ( 14), the first eigenvalue of ( 10) is The corresponding velocity eigenvectors are constant vectors orthogonal to e and the eigenpressures are 0. By Theorem 3.7 in the neighborhood of ε = 0, there exist d−1 (directionally) analytic branches of Bloch eigenvalues and Bloch eigenfunctions of ( 12) , q ε j | ε=0 = q j (e) = 0 and q ε 0,j | ε=0 = q 0,j (e) = 0, (15) where {u j (e)} 1≤j≤d−1 is an orthonormal family of vectors in R d orthogonal to the chosen direction e.Note that the labeling of the above eigenvalues is not the usual one of increasing order and depends on the direction e.As usual, we normalize the eigenvectors as follows We differentiate the eigenvalue problem (12) with respect to ε or, equivalently, we differentiate problem (9) in the direction e to obtain where φ j (ε), q j (ε), q 0,j (ε) and λ j (ε) are the derivatives at ε of φ ε j , q ε j , q ε 0,j and λ ε j respectively, with ( In order to identify the derivatives at ε = 0 in (17) we recall the so-called cell problems [7] which are useful for the classical homogenization of Stokes equations which admits a unique solution (w kl , π kl ) The homogenized tensor A * is then defined by ∇(w pq + y q e p ) dy.
We now compute the second order directional derivatives.Differentiating (17) with respect to ε yields where Lemma 4.2.The second order derivative of the eigenvalue at ε = 0 satisfies where δ jk is the Kronecker symbol and the homogenized tensor A * is written in the basis (18).
Proof.By the Fredholm alternative F (ε) must be orthogonal to φ ε k , which implies For ε = 0, we obtain Since multiplying equation ( 19) by w dk yields Y (µ(e ⊗ e j + ∇w dj )) : Remark 4.3.Formula ( 22) is similar to that obtained by Ganesh and Vanninathan in the elasticity case [13].Recalling our choice (18) of the basis of R d , it is equivalent to say that 1 2 λ j (0) is an eigenvalue and e j is an eigenvector of the symmetric matrix (A * didk ) i,k .There is however a notable change with the elasticity case in [13]: the indices j and k in (22) must be different from d, so the alluded matrix eigenvalue problem is of dimension d − 1 instead of d as in [13].In other words, the knowledge of only d − 1 branches of eigenvalues λ j yields less informations on the homogenized Stokes tensor A * than in the elasticity case.For example, among others, the homogenized coefficient A * dddd is not characterized by (22).In the elasticity case, Ganesh and Vanninathan were able to prove that the homogenized tensor A * is completely recovered from the knowledge of the d derivatives λ j (0) and d eigenvectors e j = u j (e).One can not expect such a result for the Stokes equation.Indeed, one can check that adding a multiple of I 2 ⊗ I 2 (where I 2 is the identity matrix of order d) to A * does not change formula (22) because it does not involve coefficients of the type A * jjdd .On the other hand, the underdeterminacy of A * up to the addition of I 2 ⊗I 2 does not matter in the homogenized Stokes equations since because u is a divergence-free vector field.Eventually we conjecture that the sole knowledge of the d − 1 derivatives λ j (0) and the d − 1 eigenvectors e j = u j (e) completely characterizes the homogenized Stokes tensor, up to the addition of an unimportant I 2 ⊗ I 2 term.However, we have been unable to prove such a result so far.

Bloch wave decomposition of H(R d
).Let V(R d ) be the space (without topology) We associate to every v ∈ H(R d ) and to every ξ the following function It is easily seen that v(•, ξ) belongs to L 2 # (Y ).The right hand-side series converges and define an element of H ξ as a function of x, satisfying the following properties: (i) We recover v from v by the following formula: (ii) The norms of v and v(•, ξ) are related by (i) For m ∈ N, the m th Bloch coefficient is defined by where the limit is taken in L 2 (Y ).(ii) Then the following inverse formula holds: where the limit is taken in the space H(R d ).(iii) In particular, we have the following Parseval identity: (iv) More generally, the following Plancherel Identity is also valid: The proof of the above result is analogous to the one in [7] so we omit it.
6. Bloch waves in a perforated domain.In this section we briefly explain how the previous results can be extended to the case of perforated domains.This is an important issue since many models of flows in porous media, like Darcy's law, are obtained by homogenization of the Stokes equations in a periodically perforated domain.At least formally, the case of Stokes equations in a domain with holes, supporting a Dirichlet boundary condition, can be recovered by letting the damping coefficient κ goes to +∞ inside the hole in equation ( 1).Nevertheless, in full mathematical rigor, some previous technical results need a specific proof for perforated domains that we now give.
Let us recall that Y = [0, 2π[ d and let T ⊂ Y be a smooth closed open subset.We assume that the hole T is isolated in the unit cell Y , i.e. the two holes of two adjacent cells do not touch (we denote this assumption by T ⊂⊂ Y ).We can thus define: We introduce the spaces of "generalized" divergence-free velocities In order to obtain uniform a priori estimate for the pressure in (25) we first need to generalize a result of Tartar [28] about a restriction operator on vector fields.Lemma 6.1.For each ξ ∈ Y there exists an operator where C does not depend on ξ and it satisfies: Proof.From Lemma 4 pp 373 in [28], there exists an operator The properties (ii) and (iii) of R ξ follow directly from the properties (b) and (c) of R. Property (i) follows from property (a) and the equality . By the existence of constant C 1 for R, we can deduce the existence of constant C for R ξ .
We establish an adequate version of De Rham's Theorem with a proof which is different from those of Lemmas 2.2 and 2.3.We denote by H 1 0,# (Y * ) the subspace of H 1 # (Y ) made of functions vanishing on T , and by Since F satisfies (26) we get that D(ξ)p = f since R ξ (ψ) = ψ.From (26) we easily obtain , the other estimates are then a consequence of Lemma 2.3.
where the constant C > 0 does not depend on ξ and P is the extension of p given in Lemma 6.2.As usual, for ξ = 0, there exists a unique solution (24).
Proof.For ξ = 0 the result is well known [18].For ξ = 0 we define the bilinear form on which is easily seen to be symmetric, continuous and coercive.Furthermore, since a Poincaré inequality is satisfied by the functions belonging to H 1 0,# (Y * ) d , the bilinear form is uniformly coercive.In the same manner as in Proposition 2.1, we can show that there exist ϕ which proves the existence and uniqueness of the solution of (24).Furthermore, since the bilinear form is uniformly coercive, we prove that 2 then yields the estimates on the pressure.
For ξ ∈ Y we consider the eigenvalue problem To study the continuity of (29) when ξ converges to 0, we rewrite (29) in a slightly different form.For a given unit vector e ∈ R d , |e| = 1, we introduce a scalar parameter ε ∈ R and define ξ = εe.
(31) We shall establish that the limit spectral problem of (31) is u(e) = 0.
(32) Theorem 6.5.Assume that ν is an eigenvalue of multiplicity h of the Stokes system (32).Then there exist h analytic functions defined in a neighborhood of ε = 0 with values in R, ε → λ ε j , and h analytic functions ε → (φ ε j , q ε j , q ε 0,j ), with values in 2. for all ε small enough, λ ε j , φ ε j , q ε j , q ε 0,j is a solution of (31), 3. for all ε small enough the set {φ ε for each interval I ⊂ R such that I contains only the eigenvalue ν, and for all ε small enough, there are exactly h eigenvalues (counting the multiplicity) λ ε 1 , . . ., λ ε h of (30) contained in I.
The proof of Theorem 6.5 follows as the proof of Theorem 3.7.
The above results allow us to compute directional derivatives of the first Bloch eigenvalue.
Figure 2. Two eigenvectors for the first (double) Bloch eigenvalue for a circular hole with θ = 0 (periodic case) 7. Numerical results.To confirm our analysis and to check in which cases the Bloch eigenvalues are discontinuous at the origin, we perform numerical computations in dimension d = 2 with the finite element software FreeFem++ [14].We compute the first and second eigenvalues and eigenvectors of (1) in a unit cell (0, 1) 2 perforated by a hole T which is either a disk of radius 0.15, or an ellipsoid of principal axes aligned with the cell axes and of half sizes 0.1, 0.2.The holes support a Dirichlet boundary condition.The viscosity is uniform, µ(y) ≡ 1, and there is no zero-order term, κ(y) ≡ 0. By using a rescaled dual variable ξ = 2πθ we still have Y = (0, 1) 2 as the dual cell for θ.The incompressibility constraint is obtained by penalization, i.e. instead of solving (1) we solve with ν = 10 5 and D(θ) = (∇ + 2iπθ).We use P 2 finite elements and a triangular mesh with 7767 nodes for the circular hole and 7919 nodes for the ellipsoidal hole.We checked that our results are converged both with respect to mesh refinement and incompressibility penalization.The main difference between these two geometries is that, in the periodic case θ = 0, the first eigenvalue is double for the circular hole (see Figure 2) and simple for the ellipsoidal hole (see Figure 3).We plot the functions θ → λ 1 (θ), λ 2 (θ) on the dual cell (−0.5, +0.5) 2 with a zoom in the neighborhood (−0.05, +0.05) 2 of the origin (see Figures 4 and 5).In both cases we clearly see that the second eigenvalue is discontinuous at θ = 0.However, only in the case of the ellipsoidal hole is the first eigenvalue discontinuous: we see on Figure 5 that λ 1 (θ) is continuous in the x direction but discontinuous in the y direction.Actually, the limit of λ 1 (θ) in the y direction is precisely equal to λ 2 (0).We checked that higher eigenvalues (typically the 3rd and 4th) are also discontinuous at the origin for both geometries.In view of these numerical simulations it seems that, for symmetric obstacles in a Stokes flow, the first eigenvalue is "smooth".More precisely, we conjecture that the first eigenvalue λ 1 ( , e) of the spectral problem ( 12) is differentiable (again for symmetric holes).Recall that ( 12) is equivalent to (33) for θ = 0, with 2πθ = e with ∈ R and e a unit vector in R d .If this were true, such a result would pave the way for the homogenization of the unsteady Stokes equations in a porous media by using methods proposed in [4].Finally, let us mention that this discontinuity phenomenon for the Bloch eigenvalues at the origin was already numerically observed for a different model of fluid-structure interaction in [1].8. Appendix.In this appendix we give the proof of Theorem 3.7, i.e. the regularity in each direction e.To prove this regularity of the eigenvalues and eigenfunctions we use the Lyapunov-Schmidt method (see [30], [8, pp. 30]) Lemma 8.1.[8, Lemma 4.1, pp.31] Suppose that X and Z are Hilbert spaces and A : X −→ Z is a continuous linear operator.Let U : X −→ N (A), E : Z −→ R(A) be the orthogonal projection from X and Z on the kernel and range of A respectively.
Then, there exists a bounded linear operator K : R(A) −→ N (A) ⊥ called the right inverse of A such that Let Λ be a closed subset of a Banach space, such that IntΛ = ∅.
Applying the Implicit Function Theorem to (35), we deduce the existence of a neighborhood V ⊂ N (A) × Λ of (0, 0) and a function z * : V −→ N (A) ⊥ with the same regularity of N providing the solution of (35).Therefore, if {y 1 , . . ., y h } is an orthonormal basis of N (A), the solution x(λ) of (35) satisfies for suitable coefficients c 1 , . . ., c h .Then, (x, λ) ∈ V satisfy (34) iff which is a finite dimensional system of equations on the constants c 1 , . . ., c h .Let us define the operator S as Clearly, we have the following result.
Lemma 8.2.The map S is analytic in a neighborhood of ε = 0 with values in Now, we define two mappings T and A by By definition A is self-adjoint and R(A) = N (A) ⊥ where N (A) is the eigenspace associated to the eigenvalue ν of S(0).
In order to prove Theorem 3.7 we first prove that, if ν is an eigenvalue of multiplicity h, we can find a first set of eigenvalue ε → λ ε ∈ R, and eigenfunctions Y )×C for (12).To find the other h−1 branches of eigenvalues and eigenfunctions, we shall apply later an iterative method.Proposition 8.3.Assume that ν is an eigenvalue of multiplicity h of the problem (10).Then there exists at least one function ε ) is a solution of (12) for ξ = εe, associated to the eigenvalue λ ε .
Proof of Proposition 8.3.Let ν be an eigenvalue of multiplicity h of the problem (10) and let (u j , q j , q 0,j ), j = 1, . . ., h, be the associated eigenfunctions in . By definition of the operators S(ε) and T , (φ ε , q ε , q ε 0 ) is an eigenfunction associated to the eigenvalue From Lemma 8.1 we know that the map A = S(0) − νT has a right inverse operator K. Thus, in view of (36) we obtain that On the other hand, from (36 q 0,l ), (u j , q j , q 0,j ) , for all j = 1, ..., h which is a linear system of equations on the unknowns c l (ε).This system has a non trivial solution if and only if det Q ε [I − KQ ε ] −1 (u l , q l , q 0,l ), (u j , q j , q 0,j ) = 0.
Remark 8.4.Proposition 8.3 yields the existence of one branch of eigenpairs associated to the root α(ε) of (44).We do not use the eigenpairs associated to the other roots α j by now since, so far, we do not know whether they coincide or not with the eigenpair associated to α 1 (ε).
In other words, ν is an eigenvalue of multiplicity h−1 of the operator B = B(0) with eigenfunctions u 2 , . . ., u h .There are no other linearly independent eigenfunctions of B associated to ν.Indeed, if u is another eigenfunction of B associated to the eigenvalue ν such that u, u j = 0, j = 2, . . ., h, then u, u 1 = 0 (because u 1 is an eigenfunction associated to the eigenvalue ν − 1) and Bu = νu.Then that is, u is an eigenfunction of P associated to ν and thus, ν is an eigenvalue of multiplicity h + 1, which is impossible because the multiplicity of ν is h.
It is not difficult to see that B(ε) satisfies the same conditions of the Stokes problem (12) to apply the Lyapunov-Schmidt Method used in the proof of Proposition 8.3.Applying this method in an iterative form we obtain h − 1 analytic functions in a neighborhood of ε = 0, ε −→ λ ε j and ε −→ φ ε j , q ε j , q ε 0,j , with j = 2, . . ., h such that B(ε)φ ε j = λ ε j φ ε j .Moreover, the functions φ ε 2 , . . ., φ ε h form an orthonormal set in H εe .This shows us the existence of the h branches of eigenpairs.
We now prove the last part of the theorem.Since the eigenvalues ε → λ ε j are analytic in a neighborhood of ε = 0, there exist constants c i such that Therefore λ ε n−1 ∈ I and λ ε n+h ∈ I, that is, (11) has at most h eigenvalues contained in I counting multiplicity.This completes the proof of Theorem 3.7.

Figure 1 .
Figure 1.Periodic perforated cell At first, we consider the Stokes equations with a source term f ∈ L 2 # (Y * ), namely    

Figure 4 .Figure 5 .
Figure 4. First and second Bloch eigenvalue in Y for a circular hole: global picture (left), zoom around the origin (right)
We first prove an adequate version of the De Rham's Theorem.
Y ) by De Rham's Theorem.A similar proof applies to H e .