Stability of oscillating boundary layers in rotating fluids

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(1)ISSN 0012-9593. ASENAH. quatrième série - tome 41. fascicule 6. novembre-décembre 2008. ANNALES SCIENTIFIQUES de L ÉCOLE NORMALE SUPÉRIEURE Nader MASMOUDI & Frédéric ROUSSET. Stability of oscillating boundary layers in rotating fluids. SOCIÉTÉ MATHÉMATIQUE DE FRANCE.

(2) Ann. Scient. Éc. Norm. Sup.. 4 e série, t. 41, 2008, p. 955 à 1002. STABILITY OF OSCILLATING BOUNDARY LAYERS IN ROTATING FLUIDS.  N MASMOUDI  F ROUSSET. A. – We prove the linear and non-linear stability of oscillating Ekman boundary layers for rotating fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to ε. This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated. R. – On prouve la stabilité linéaire et non-linéaire de couches limites oscillantes de type Ekman pour les fluides tournant dans le cas de données mal préparées sous une hypothèse spectrale. On s’intéresse au cas où la viscosité et le nombre de Rossby sont du même ordre ε. Cette étude généralise celle de [23] où une condition de petitesse était imposée et celle de [26] où les données bien préparées étaient traitées.. 1. Introduction We consider the following system describing the evolution of a rotating fluid in a rectangular domain  ε  ∂ uε + uε · ∇uε + e × u + ∇p − ε∆uε = 0, t ε ε (1)  ∇ · uε = 0 for x = (y, z) ∈ Ω = T2a × (0, 1) with the Dirichlet boundary condition (2). uε/∂Ω = 0. and the initial condition (3). uε/t=0 = uε,0 .. Here T2a is the periodic torus with periods a1 and a2 , namely, T2a = R2 /(a1 Z × a2 Z) and ε a1 , a2 > 0. Moreover, e = e3 is the vertical unit vector and e×u is the Coriolis force. ε. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 0012-9593/06/© 2008 Société Mathématique de France. Tous droits réservés.

(3) 956. N. MASMOUDI AND F. ROUSSET. This system describes the motion of a rotating fluid as the Ekman and Rossby numbers go to zero (see Pedlovsky [25], and Greenspan [14]). It can model the dynamics of the ocean or the atmosphere far from the equator or a rotating fluid in a container. Note that, here, we take the horizontal viscosity and the vertical viscosity to be equal. We point out that in many previous works the horizontal viscosity was supposed constant whereas the vertical viscosity ν goes to 0 (see for instance [17]) or in some other cases, the vertical viscosity was supposed much smaller than the horizontal viscosity. This anisotropy has the advantage of making the boundary layers more stable. In this paper, we look at the case where the vertical and the horizontal viscosities are equal. We study the convergence of solutions to (1) towards a solution of the limit system (9) defined below once the time oscillations are filtered out. We recall that this system and related ones were studied by several authors. In the “wellprepared” case in domains with boundary, like Ω, we refer to Colin, Fabrie [5], Grenier, Masmoudi [17], Masmoudi [22]. For general initial data, and for the periodic case, we refer to Grenier [15], Embid and Majda [9], Babin, Mahalov and Nicolaenko [2, 1], Gallagher [12] or in particular cases where there is no boundary layer, or where the boundary layer can be eliminated by symmetry (Beale and Bourgeois [3]). These results rely on the introduction of a group to filter the oscillations in time, a method which was previously used by Schochet [30] to investigate related problems in the torus concerning the compressible-incompressible limit. In [23], the “group method” was extended to the case of domains with boundary, by solving a superposition of an infinite number of boundary layers. These layers create an extra term in the limit equation. In [23], the stability of these boundary layers was proved in the case where the horizontal viscosity goes to zero slower than the Rossby number (or in the small data case). In this paper, we would like to give a spectral assumption (which we think is optimal) and which yields the stability of such boundary layers. In the well prepared case, a similar spectral assumption was used to prove the stability of the boundary layer [26]. This spectral assumption is optimal since the instability of the boundary layer was proved in [7] if the spectral assumption does not hold. In the following sections, we recall the main properties of the approximate solution of (1) constructed in [23] (see also [4]), in particular, we recall the properties of the limit system, of the boundary layers and the assumptions on the torus which are needed. Next, we shall give our main assumption on the spectral stability of the boundary layers and state our main result.. Acknowledgement N. Masmoudi was partially supported by an NSF grant DMS-0703145. This work was carried out during visits of N. Masmoudi to the university of Nice and visits of F. Rousset to the Courant Institute. The hospitality of both Institutes is highly acknowledged. 4 e SÉRIE – TOME 41 – 2008 – No 6.

(4) STABILITY OF BOUNDARY LAYERS. 957. 1.1. Properties of the approximate solution To state our main result, we first recall the main properties of the approximate solution uapp of (1) constructed in [23]. In particular, uapp describes the formal limit of (1) and the boundary layers. The details of the construction will be recalled later. The approximate solution is under the form Å ã Å ã t t z 1−z app int b , t, x + u , , , t, y + ur , x = (y, z) ∈ T2a × (0, 1) (4) u =u ε ε ε ε where the remainder term ur satisfies ur = O(ε) (a precise statement will be given later). The interior term uint can be expressed as uint (τ, t, x) = L(τ )wint (t, x) where L(τ ) = eτ L , Lu = −P(e × u) and P is the Leray projector on divergence-free vector 2π 2π 2π 2π 2 fields with zero normal component in Ω. We denote Z3a = 2π a1 Z× a2 Z× 2 Z, Za = a1 Z× a2 Z 2π 2π 3 3 2 and we denote elements of Za by k̄ = (k, k3 ) ∈ Za with k ∈ Za = a1 Z × a2 Z . We have an expansion X b(t, k̄)eik·y M k̄ (z), (5) wint (t, x) = k̄∈Z3a. so that (6). uint (τ, t, x) = L(τ )(wint (t, x)) =. X. b(t, k̄)eik·y M k̄ (z)eiλ(k̄)τ. k̄∈Z3a. and wint solves the limit system (9). Note that N k̄ = eik·y M k̄ is an eigenvector of L. We assume that the initial data is chosen such that b(0, (0, k3 )) = 0 for every k3 i.e. we exclude initial values with modes which depend only on z. We shall also assume that the torus is non resonant in the sense of [2] to insure that the condition b(t, (0, k3 )) = 0 for every k3 remains true for positive times (see below for a precise definition). We can express the dominant boundary layer term ub as ub (τ, Z, Z 0 , t, y) = ub,0 (τ, Z, t, y) + ub,1 (τ, Z 0 , t, y) where ub,σ (τ, Z, t, y) = −. 1X √ η k̄,+ Z √ η k̄,− Z − 1+i − 1−i b(t, k̄)eik·y+iλ(k̄)τ (−1)σk3 hk̄,+ e 2 + hk̄,− e 2 2 k̄. for σ = 0, 1, with η k̄,± =. » 1 ± λ(k̄),. hk̄,± = M k̄ (0) ∓ ie × M k̄ (0).. Note that since terms under the form (0, k3 ) are excluded in the above sum, we have η k̄,± > 0 and hence, we have a superposition of terms which are small far from the boundary. Nevertheless, the rate of decay, η k̄,± , goes to zero when |kk̄|3 tends to ±1. In view of the above definition of the boundary layers, we introduce the operators 1X √ η k̄,+ Z √ η k̄,− Z − 1−i − 1+i B σ (τ, Z)q = − qk̄ eiλ(k̄)τ (−1)σk3 hk̄,+ e 2 + hk̄,− e 2 2 k̄. for any sequence q = (qk̄ )k̄∈Z3a so that if q is taken under the form qk̄ = b(t, k̄)eik·y , we have B σ (τ, y, Z)q = ub,σ (τ, Z, t, y). ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(5) 958. N. MASMOUDI AND F. ROUSSET. In a similar way, we also define Lσ (τ )q =. X. qk̄ M k̄ (σ)eiλ(k̄)τ ,. σ = 0, 1.. k̄. Again, note that if q is taken such that qk̄ = b(t, k̄)eik·y , then we have Lσ (τ )q = w(t, y, σ). We shall allways assume that the initial data is sufficiently smooth and vanishes at a sufficient order at z = 0, z = 1 in order that b(t, k̄) decay to zero sufficiently fast. In particular, we assume that X (7) kw0 k2Vsym = |b(0, k̄)|2 |k̄|2s < ∞ for some s big enough. s k̄∈Z3a. This yields that w(t) ∈ for 0 < t < T ∗ where T ∗ is the life span of a smooth solution of the limit system (9). Hence, by using that s > 32 + 2, we have since 1 2 2|k̄|2 |k̄| = ≤ ≤ 2|k̄|2 2 k1 + k22 |k̄| − |k3 | η k̄,± s Vsym. that X k̄. 1 2 1 2 |b(t, k̄)| 1 + k̄,+ + k̄,− <∞ η η. to finally obtain the important property Z +∞ (8) sup ∂Z B σ (τ, Z)(w(t, y, σ)) (1 + |Z| + |Z 2 |) dZ < +∞, y. σ = 0, 1. 0. which insures that the boundary layers are sufficiently localized in the vicinity of the boundary. 1.2. The limit system P s We denote by wint = k̄∈Z3a b(t, k̄)N k̄ the solution in L∞ (0, T ∗ ; Vsym ) of the following system   ∂t wint + Q(wint , wint ) + S(wint ) = −∇p in Ω,       ∇ · wint = 0 in Ω, (9) int    w · n = ±w3 = 0 on ∂Ω,     wint (t = 0) = w0 , where T ∗ is the time of existence of the smooth solution wint of (9) and Q(wint , wint ), S(wint ) are respectively a bilinear and a linear operator of wint . The bilinear operator is given by X (10) Q(wint , wint ) = b(t, ¯l)b(t, m̄)αl̄m̄k̄ N k̄ (X). l̄,m̄,k̄ k̄∈A(l̄,m̄) λ(l̄)+λ(m̄)=λ(k̄). The numbers αl̄m̄k̄ are constants and the set A(¯l, m̄) = {¯l + m̄, S ¯l + m̄, ¯l + S m̄, S ¯l + S m̄} with the notation S(¯l1 , ¯l2 , ¯l3 ) = (¯l1 , ¯l2 , −l3 ) is the set of possible resonances. 4 e SÉRIE – TOME 41 – 2008 – No 6.

(6) STABILITY OF BOUNDARY LAYERS. 959. The linear operator is defined by X S(wint ) = (D(k̄) + iI(k̄))b(t, k̄)N k̄ (X) k̄. where D(k̄) =. © √ ¶ 1 2 (1 − λ(k̄)2 ) 2 ,. I(k̄) =. © √ ¶ 1 2 λ(k̄)(1 − λ(k̄)2 ) 2 .. Note that S(wint ) is a damping term, since D(k̄) ≥ 0, that depends on the frequencies λ(k̄). 1.3. Non resonance assumption on the torus In the case of a non resonant torus (see [2] for the definition), the quadratic term Q(w, w) only includes trivial resonances, namely the resonances only take place with the 2d non oscillating geostrophic part: (11). {(k̄, ¯l, m̄) | k̄ ∈ A(¯l, m̄)} ⊂ {(k̄, ¯l, m̄) | k3 m3 l3 = 0}. which yields in particular the global existence of strong solutions to the limit system. We also know in that case that for k̄ ∈ A(¯l, m̄), and λ(¯l) + λ(m̄) − λ(k̄) 6= 0 1 (12) ≤ C(|¯l|d + |m̄|d ) ¯ λ(l) + λ(m̄) − λ(k̄) for some d > 4. We recall that for almost all choices of a1 and a2 , the torus T3a is non resonant (see [2]). Besides, if at t = 0, we have Z w0 dxdy = 0,. x,y. we see that this holds for any t. Indeed, in the non resonant case there are only trivial resonances, namely with the slow modes (the geostrophic modes) (k1 , k2 , 0). Notice then that the modes (k1 , k2 , 0), and (−k1 , −k2 , k3 ) do not create a resonance with (0, 0, k3 ), since (k1 , k2 ) 6= (0, 0), and then |λ(−k1 , −k2 , k3 )| < 1. Hence we get that for all t the modes such that λ(k̄) = ±1 are absent. This is a crucial fact in our analysis since the boundary layers for the modes λ(k̄) = ±1 behave like the boundary layers in the vanishing viscosity limit of the Navier-Stokes equation without the fast rotation (these layers are of Prandtl-type). For such a case, the stability of the boundary layer is known if the horizontal viscosity is much bigger than the vertical one (see [22]), in other cases instability is more expected [16] except in dimension 1 [27] or for analytic data [29]. 1.4. Stability assumption on the boundary layer profiles The main difficulty in the convergence proof is to get an estimate for (1) linearized about the approximate solution uapp : we study e×v = 0, x ∈ T2a × (0, 1) (13) ∂t v + uapp · ∇v + v · ∇uapp − ε∆v + ∇p + ε with the boundary condition (2) and the initial condition v/t=0 = v0 (x). We would like to prove an estimate like kv(T )k2 ≤ eγT kv0 k2 for some norm k · k with γ > 0 independent of ε. Even in the well-prepared case such an estimate is not always true, it depends on a spectral stability property of the boundary layer ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(7) 960. N. MASMOUDI AND F. ROUSSET. profiles. If the boundary layer profiles are spectrally stable, this estimate can be proven as well as nonlinear stability, [26]. Whereas if they are unstable, we can only get an estimate with γ of the order of ε−1 and in this case nonlinear instability can be proven [7]. The spectral stability depends on the amplitude of the boundary layer, numerically, one can prove that boundary layers with too large amplitude are unstable [21]. The aim of the next subsection is to formulate a stability assumption on the boundary layer profiles which generalize the spectral stability assumption of the well-prepared case formulated in [26]. We start by freezing the slow variables t = t0 and y = y 0 in the coefficients of the approx0 imate solution. Let us set q = q(t0 , y 0 ) = (b(t0 , k̄)eik·y )k̄∈Z3a . We want to study the stability property of the equations t z t t z ∇p e × v t + − ε∆v = 0. ∂t v + Lσ ( )q + B σ ( , )q · ∇v + v · ∇ Lσ ( )q + B σ ( , )q + ε ε ε ε ε ε ε ε Let us define for each sequence q = (qk̄ )k̄∈Z3a the oscillating boundary layer profile V (τ, Z, q) as V σ (τ, Z, q) = Lσ (τ )q + B σ (τ, Z)q, we can take the Fourier transform in y and set Z = z/ε and τ = t/ε to get the family of one-dimensional problems ! ! iεk iεk σ ∂τ w + V (τ, Z, q) · w+w· V σ (τ, Z, q) ∂Z ∂Z ! iεk + p + e × w + ε2 |k|2 w − ∂ZZ w = 0, ∂Z iεk · wh + ∂Z w3 = 0 which is now set for Z ∈ (0, +∞) with the boundary condition (14). w(τ, εk, 0) = 0.. We recall here that we use the notation k̄ = (k, k3 ) where k ∈ Z2a . Finally, we can set ξ = εk and use for ξ 6= 0 the Leray projection P+ (ξ) in the half-space which is recalled in section 19 to rewrite the equation as ∂τ w = P+ (ξ)Lσ+ (ξ, q)w,. (15). iξ · wh + ∂Z w3 = 0. where Lσ+ (τ, ξ, q) is defined as (16). Lσ+ (τ, ξ, q)w. σ. = −V (τ, Z, q)·. iξ ∂Z. ! w−w·. iξ ∂Z. ! V σ (τ, Z, q)−e × w−|ξ|2 w+∂ZZ w.. The non-autonomous operator P+ Lσ+ (τ, ξ, q) generates a strongly continuous family of evolution operators in the sense of [20], Chapter 7, Sσ+ (τ, τ 0 , ξ, q) on Hξ = {w ∈ L2 (0, +∞), iξ · wh + ∂Z w3 = 0}. As usual, the main property of Sσ+ (τ, τ 0 , ξ, q) is that τ 7→ Sσ+ (τ, τ 0 , ξ, q)w0 is the unique solution of (15) for τ > τ 0 with value w0 for τ = τ 0 . Let us fix s0 > 1 such that (8) holds when we replace w(t, y, σ) by Lσ (τ )q. A set K ⊂ P or more precisely in {q | k̄∈Z3a |k̄|2s0 |qk̄ |2 < ∞} (using the identification between. s0 Vsym (Ω). 4 e SÉRIE – TOME 41 – 2008 – No 6.

(8) STABILITY OF BOUNDARY LAYERS. 961. the function and its Fourier coefficients) will be called a uniform stability set if for every r, R, 0 < r < R, there exist C(r, R) and α(r, R) > 0 such that (17) ∀v ∈ Hξ ,. 0. |Sσ+ (τ, τ 0 , ξ, q)v|L2 (R+ ) ≤ Ce−α(τ −τ ) |v|L2 (R+ ) ,. ∀τ ≥ τ 0 ≥ 0, σ = 0, 1. for every q ∈ K and ξ such that r ≤ |ξ| ≤ R. We point out that there are uniform stability sets. Indeed, a vicinity of zero is a uniform stability set since we can prove as in [23, 6] that all weak amplitude boundary layers are stable when (8) is matched. P Let wint (t) = k̄∈Z3a b(t, k̄)N k̄ be the solution of (9). Our main stability assumption is: (H) We assume that the set K = {(qk (t, y) := b(t, k̄)eik·y )k̄∈Z3a , t ∈ [0, T ], y ∈ T2a }, is a uniform stability set. s0 R 1. – Note that we can define more generally the stability set K0 ⊂ Vsym (Ω) as the set of q having the property that for every 0 < r < R, there exist C(r, R, q) > 0 and α(r, R, q) > 0 such that (17) holds. By continuity of S+ with respect to q and compactness, we easily get that every compact set K ⊂ K0 is a uniform stability set. In particular, this s (Ω) with s > s0 , we have that if B is included yields that for every bounded set B ⊂ Vsym in K0 , then B is a uniform stability set. Consequently, the assumption (H) is matched if s (Ω) for s > s0 . K ⊂ K0 (where K is defined in the statement of (H) and K is bounded in Vsym. Since the operator Lσ+ has only quasi-periodic coefficients there is no easy characterization of the assumption (H), for example in term of the spectrum of L+ . When the initial data is prepared so that the coefficients of Lσ+ are periodic in time, we can use Floquet’s theory to replace in the assumption (H) the decay estimate (17) by an assumption on the spectrum of Sσ+ (T, 0, ξ, q) where T is the period: if the spectrum σ(Sσ+ (T, 0, ξ, q)) is contained in the open unit disk D, then the estimate (17) is verified. Finally, we note that (H) is the natural generalization of the assumption used in the well-prepared case. Indeed, in the well prepared case, since Lσ+ does not depend on τ the stability assumption was formulated in term of the spectrum of P+ (ξ)Lσ+ (ξ, q): it was basically assumed in [26] that the spectrum of P+ (ξ)Lσ+ (ξ, q) is contained in {Re λ < 0}. By the standard theory of analytic semi-groups, it is easy to prove that this assumption on the spectrum implies the estimate (17). This is proven in [13] in a close setting. 1.5. Notations We denote by k · k the norm of L2 (T2a × (0, 1)) and by (·, ·) the associated scalar product. We also define the weighted higher order norms: kvk21,ε = kvk2 + ε2 k∇vk2 , kvk22,ε = kvk2 + ε2 k∇vk2 + ε4 k∇2 vk2 . We will also use some anisotropic norms, namely X (18) kvk2m =. α. kZ vk2. α∈Z3 , |α|≤m m where Z 1 = ∂y1 , Z 2 = ∂y2 , and Z 3 = εz(1 − z)∂z and denote by Hanis the Hilbert space defined by this norm.. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(9) 962. N. MASMOUDI AND F. ROUSSET. 1.6. Main result Our main result is: s T 2. – We consider a torus T3a non-resonant in the sense of [2] and w0 ∈ Vsym for s sufficiently large such that in the expansion X w0 (x) = b0 (k̄)eik·y M k̄ (z) k̄∈Z3a. we have b0 (k̄) = 0 if k̄ = (0, 0, k3 ). Moreover, with the notation q 0 (y) = (qk̄0 (y))k̄ = (b0 (k̄)eik·y )k̄ , we assume that ( 0 α kuε,0 − w0 − B 0 (0, zε )(q 0 (y)) − B 1 (0, 1−z ε )(q (y))km ≤ cε (19) 0 α εk∇(uε,0 − w0 − B 0 (0, zε )(q 0 (y)) − B 1 (0, 1−z ε )(q (y)))km ≤ cε for some m ≥ 2, 3/4 < α ≤ 1 and some constant c > 0. Let wint (t) be the solution of (9), we assume that (H) holds. Then, there exists ε0 such that for all 0 < ε < ε0 , the system (1) has a unique weak solution uε ∈ L∞ (0, T ; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) with initial value uε,0 . Moreover, (20) Å ã Å ã Å ã t z t 1−z t int 0 1 ε u −L w −B , (q(t, y)) − B (q(t, y)) ≤ CT εα . ε ε ε ε ε ∞ 2 L (0,T ;L (Ω)) In addition, if α = 3/4 in (19), there exists a time T0 which depends on w0 and on c (but not on ε) such that (20) holds on (0, T0 ). Let us give a few remarks about this theorem. R 3. – 1) First, we note that when 3/4 < α ≤ 1, the uniform time of existence and convergence T is only limited by the stability assumption (b(t, k̄)eik·y )k̄∈Z3a ∈ K0 , in particular, it may be arbitrarily large even if the data is large. This is due to the regularity of the limit system which is better than the regularity of the 3D Navier-Stokes in the nonresonant case (see [2]). 2) We also point out that the assumption that the torus is non-resonant is used to ensure that b(t, k̄) = 0 for k̄ = (0, 0, k3 ) and that the result holds if we know that b(t, k̄) = 0 for k̄ = (0, 0, k3 ) as well as an estimate of the type (12). Indeed, for the modes k̄ = (0, 0, k3 ), the boundary layer we get is of Prandtl type (the rotation does not play any part). It was handled in [23] only with the crucial assumption that the vertical viscosity over the horizontal one also goes to zero. 3) We shall see in the proof that the result is actually more precise. A sufficient condition on the regularity of w0 is s > d + 5 where d is given in (12). The convergence will take place in a space with horizontal regularity m where 2 ≤ m ≤ s − d − 3. Moreover, we notice that the error estimate, namely CT εα sees the boundary layer and hence the boundary layer cannot be removed from the estimate (20). This is stronger than the estimates in [17, 23]. However, our result requires the use of an initial data which depends on ε and which is sufficiently close to the approximate solution at t = 0. The convergence is stated in L∞ (0, T ; L2 (Ω)) but as will be seen from the proof in section 5.1, we need to prove estimates in a stronger 4 e SÉRIE – TOME 41 – 2008 – No 6.

(10) STABILITY OF BOUNDARY LAYERS. 963. space, namely Ym . In particular, the solution we construct is a strong solution which yields the uniqueness of the weak solutions by the classical strong-weak uniqueness argument. 4) Finally, we note that the proof of Theorem 2 is completely different from the proof of the stability in the well-prepared case in [26] and the proof of the small data case in the illprepared case [23]. The paper is organized as follows: in the next section, we give some details about the construction of the approximate solution, then, in the next sections, we prove the linear stability of this approximate solution; finally section 5 is devoted to the proof of Theorem 2. Appendix A is devoted to the definition and the proof of some simple properties of semi-classical operator-valued pseudo-differential calculus. These properties are crucial in the study of the linear stability. Finally, the Appendix B gathers some useful properties of the Leray projection in the strip and in the half-space. 2. Construction of an approximate solution 2.1. Some definitions and notations We will use the notations of [23]. Let us denote by V 0 the subspace of L2 (Ω)3 consisting of divergence-free vectors (div u = 0), and tangent to ∂Ω (u3 (z = 0) = u3 (z = h) = 0 ) V 0 = {u ∈ L2 (Ω)3 , ∇ · u = 0, u3 (z = 0) = u3 (z = h) = 0} ; we also define, for m > 0, V m the space V m = H m (Ω)3 ∩ V 0 , where H m (Ω) is the classical Sobolev space W 2,m (Ω). Let T3a = T2a ×] − 1, 1[= T2a × R/2Z be the torus of periods respectively a1 , a2 and 2, and E the linear operator from L2 (Ω)3 into L2 (T3a )3 defined by Eu(z) = u(z) for 0 < z < h, and for −h < z < 0 (21). Eu(z) = S(u(−z)),. where S is the linear operator from R3 into R3 , defined by (22). SX1 = X1. SX2 = X2. SX3 = −X3. for all X ∈ R3 , which corresponds to a symmetry with respect to the plan X3 = 0. We also introduce V 0m = {u ∈ H m (T3a )3 , ∇ · u = 0}, Hence E 0 E u = IdV 0 where E 0 is m the restriction from L2 (T3 )3 onto L2 (Ω)3 . In the sequel we will work in the space Vsym , m Vsym = E 0 (V 0m ∩ E(V m )),. which consists of vectors u ∈ V m that satisfy extra boundary conditions on the vertical derivatives. m We also introduce a norm on Vsym. 1 |Eu|2V 0m , 2 m m , for every τ ∈ R. We will which is conserved by the group L, namely |L(τ )u|Vsym = |u|Vsym also use the following notations (see the appendix 2.2 of [23] for a precise construction): |u|2Vsym = m. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(11) 964. N. MASMOUDI AND F. ROUSSET. • X = (y1 , y2 , z) = (y, z); • N k̄ (X) = M k̄ (z)eik.y is an eigenvector of L associated to the eigenvalue iλ(k̄) = i k3 /|k|; s • for w ∈ L∞ (0, T, Vsym ), we set X (23) w(t, X) = b(t, k̄)N k̄ (X), k̄∈Z3a. then we have for all t, |w(t)|2Vsym = 2|Ω| s. P. k̄. |b(t, k̄)|2 |k̄|2s .. 2.2. Study of the group We study here the group L, in particular, we give the expression of the eigenvectors N k . Using the construction in the appendix of [23], we get for k̄ = (k, k3 ) ∈ Z3a and k 6= (0, 0) that N k̄ (X) = M k̄ (z)eik.y where Ü ê 2 cos(k3 z)n1 (k̄) M k̄ (z) =. (24). 2 cos(k3 z)n2 (k̄) 2i sin(k3 z)n3 (k̄). and. Å ã k2 k1 n3 (k̄) i − λ( k̄) 1 − λ(k̄)2 |k̄| |k̄| Å ã n3 (k̄) k1 k2 n2 (k̄) = −i − λ( k̄) 1 − λ(k̄)2 |k̄| |k̄| n1 (k̄) =. n3 (k̄) =. 1 − λ(k̄)2 . 2. s , and that we have LN k̄ = λ(k̄)N k̄ . Notice then that N k̄ (X) ∈ Vsym. 2.3. Approximate solution Here, we construct the approximate solution U app = (uapp , papp ). The aim is that U app satisfies (1) up to a small error and the boundary condition (2) exactly. To guess a good choice for U app , we expand the solution in the following form U 0 +εU 1 + 2 2 ε U + ... where (25). U 0 = U 0 (τ, t, x) + Ũ 0 (τ, t, y, Z) + Ŭ 0 (τ, t, y, Z 0 ),. where we recall that τ = t/ε, Z = z/ε and Z 0 = (1 − z)/ε. Even though it is not clear whether we can push this expansion to all order, this will allow us to guess the first terms. Hence, arguing as in [23], we get u0 (τ, t) = L(τ )u0 (τ = 0, t) = L(τ )wint (t) which is exactly the term uint in (4). We notice that u0 does not vanish on ∂Ω; we only have (u0 · n)/∂Ω , where n is the normal to the boundary. This requires the introduction of a boundary layer. For the boundary layer, we only construct Ũ 0 , near z = 0, since the construction of Ŭ 0 near z = 1 is similar. We recall that the modes (0, k3 ) are excluded due to the assumption on the initial data and 4 e SÉRIE – TOME 41 – 2008 – No 6.

(12) 965. STABILITY OF BOUNDARY LAYERS. on the resonances of the torus. So, we only deal with the case −1 < λ(k̄) < 1. Using the P construction in section 4 of [23], we get ũ0 = k̄∈Z3a Rk̄ where ãò ï Å 1 λ(k)t (26) Rk̄ (z) = − b(t, k) exp i (k.y) + 2 ε ï ò (1 + i) z (1 − i) z × hk̄,+ exp(− √ η k̄,+ ) + hk̄,− exp(− √ η k̄,− ) . ε ε 2 2 P 0 k̄ k In a similar way, we get that ŭ = k̄∈Z3a T where T has the same formula as Rk . Hence, ub = ũ0 + ŭ0 . It remains to construct the remainder term ur . Of course, a good guess for ur is to take εU 1 . However, we would like to get an approximate solution uapp which is divergence-free and which vanishes on the boundary. The rest term is under the form X X ur = (R3k̄ + T3k̄ ) + rk̄ + εY + R3 k̄. k̄. and hence consists of four terms. We refer to [23] for the details on the construction of these four terms. The main property of this correction term is that for all |α| + |β| ≤ s − d − 5, (27). k(ϕ(z)∂z )β ∂yα ur kL∞ ≤ Cα ε,. k∂z (ϕ(z)∂z )β ∂yα ur kL∞ ≤ Cα ,. where ϕ(z) is a smooth bounded function which is equivalent to z and 1 − z in the vicinity of 0 and 1 respectively. • R3k̄ and T3k̄ which are introduced to insure that Rk̄ and T k̄ satisfy the divergence free condition. However, R3k̄ creates a trace at z = 0 and T3k̄ creates a trace at z = 1 of order ε. • rk̄ which is used to cancel the traces R3k̄ (z = 0) and T3k̄ (z = 1). Since, we have to take rk̄ which is divergence-free, we have to construct r1k̄ and r2k̄ which have a trace at z = 0 and z = 1 of order ε. This is actually easier to handle than the trace on the third component we started with. Besides, the term 1ε e × rk̄ is responsible for the Ekman damping in the limit equation. • εY is introduced to cancel the non resonant oscillating terms which do not yield a contribution in the limit equation. More precisely, we have Z τ Y(τ, t) = −L(τ ) L(−τ 0 )Q(w, w) − Q(w, w) (t)dτ 0 . 0. We also point out that due to the non resonance assumption, we know that kY(τ, t)kH s−d−1 ≤ C for a constant which does not depend on τ or t < T . • R3 which takes into account the boundary condition of r, Y. This was constructed in section 4.3 of [23]. 3. Linear stability We study the linearized system about the approximate solution: (28). T ε (v, p) = F,. ∇ · v = 0,. x ∈ T2a × (0, 1). where T ε (v, p) = ∂t v + uapp · ∇v + v · ∇uapp + ∇p + ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. e×v − ε∆v ε.

(13) 966. N. MASMOUDI AND F. ROUSSET. with initial data (29). v(0, x) = v0 (x). and the boundary condition (2). Let us define v HF by Fy v HF = 1ε|k|≥r Fy v, for some r > 0 and κs (εDy )v by Fy (κs v) = κs (εξ)Fy v(ξ) with κs (ξ) a smooth function which vanishes for |ξ| ≥ 2r. The main result of this section is T 4. – We assume that (H) holds. Then, there exist γ0 > 0, γ > 0 such that for every ε > 0, T > 0, with εeγ0 T ≤ 1, we have Z T (30) kv(T )k2 + ε2 k∇v HF (T )k2 + εk∇vk2 0. .e. γt. . 2. kv0 k +. kv0HF k21,ε. Z +. T s. Z. s. (κ F, κ v) + ε 0. T. kF k2 .. 0. Throughout the paper, . stands for ≤ C where C > 0 is independent of ε ∈ (0, 1), T if εeγ0 T ≤ 1. Note that by using the Cauchy Schwarz inequality and the Gronwall inequality, we can get from (30) the estimate Z T 2 2 HF 2 kv(T )k + ε k∇v (T )k + εk∇vk2 + ε−1 kv HF k21,ε 0. .e. γ̃t. . 2. kv0 k +. kv0HF k21,ε. Z +. T. (1 + ε)kF k2. . 0. for some γ̃ > γ which gives an estimate of v with respect to the source term F and the initial data only. Nevertheless, in order to handle the nonlinear stability, it is important to keep the term Z T. (κs F, κs v). 0. in the right-hand side since it will allow to use the structure of the nonlinear term of the Navier-Stokes equation and hence to get some better estimates. The aim of the remaining part of the section is to prove Theorem 4. Note that for the moment we have a control of the L∞ (0, T, H 1 ) norm only for the high frequency part of v. We will derive an estimate for all the frequencies in paragraph 3.6. 3.1. Proof of Theorem 4 We start with a localization in frequency of the equation similar to the well prepared case [26]. We will deal with large, medium and small frequencies in different ways. For a smooth bounded function κ, we apply the Fourier multiplier κ(εDy ) to the equation 13. We get T ε (κv, κp) = κF + C. (31). where the commutator C is defined as C = −[κ, uapp · ∇]v − [κ, Duapp ]v. By using the same argument as in [26], [28], we have the estimate (32) 4 e SÉRIE – TOME 41 – 2008 – No 6. kCk2 . ε2 k∇vk2 + kvk2 ..

(14) 967. STABILITY OF BOUNDARY LAYERS. Note that these commutator estimates are actually proven in a more general case in Lemma 16. We first deal with the case where κ = κL is supported in ε|k| ≥ R. We have P 5. – There exists R > 0 sufficiently large such that we have for every ε ∈ (0, 1) and every T > 0: Z T Z T L 2 −1 L 2 kκ v(T )k1,ε + ε kκ vk1,ε . kvk21,ε + εkF k2 + kκL v0 k21,ε . 0. 0. Note that for the proof of this lemma we do not need to use (H). Proof. – We use the same argument as in the well-prepared case treated in [26]. Using that uapp is divergence free, the standard energy estimate for (31) gives Z T Z T kκL v(T )k2 + εk∇κL vk2 . kκL v0 k2 + ε−1 kκL vk2 + kF k kκL vk + kCk2 0. 0. . kκL v0 k2 +. Z. T. . ε−1 kκL vk2 + εkF k2 + kCk2 .. 0. We notice that k∇κL vk2 ≥ R2 ε−1 kκL vk2 so that for R sufficiently large, the singular term ε−1 kκL vk2 in the right hand side of (33) can be absorbed by the left hand side. By using also (32), this yields Z T Z T L 2 −1 L 2 (33) kκ v(T )k + ε kκ vk1,ε . kvk21,ε + εkF k2 + kκL v0 k2 . 0. 0 2. L. To conclude, it suffices to estimate ε k∇κ v(T )k2 . This is an easy consequence of the following lemma: L 6. – Consider u a solution of e×u + ∇p = H, ∇ · u = 0, x ∈ Ω (34) ∂t u − ε∆u + ε with the initial condition u(0, x) = u0 (x) and the boundary condition (2). Then, we have the estimate Z T Z T 2 2 3 2 2 2 2 2 (35) ε k∇u(T )k + ε k∇ uk + εk∂t uk . ε k∇u0 k + ε−1 kuk2 + εkHk2 . 0. 0. We first end the proof of Proposition 5 by using Lemma 6. We can use Lemma 6 with u = κL v and H = −uapp · ∇κL v − (κL v) · ∇uapp + κL F + C. This yields 2. L. 2. 2. L. 2. Z. ε k∇κ vk . ε k∇κ v0 k +. T. . ε−1 kκL vk2 + εk∇κL vk2 + εkF k2 + kvk21,ε .. 0. To conclude, it suffices to add (33) times a sufficiently large constant and the last estimate. We now turn to the Proof of Lemma 6: ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(15) 968. N. MASMOUDI AND F. ROUSSET. Proof of Lemma 6. – We take the scalar product of (34) by ∂t u, since ∂t u is divergence free and verifies the boundary condition (2); we have d 1 (∇p, ∂t u) = 0, (−∆u, ∂t u) = k∇uk2 dt 2 and hence we get εk∇u(T )k2 +. T. Z. k∂t uk2 . εk∇u0 k2 +. Z. 0. T. . kHk + ε−1 kuk k∂t uk.. 0. By using the Young inequality, we find after multiplication by ε Z T Z T (36) ε2 k∇u(T )k2 + εk∂t uk2 . ε2 k∇u0 k2 + εkHk2 + ε−1 kuk2 . 0. 0. Next, we use the classical regularity result for the Stokes equation [11]. We consider (34) as −ε∆u + ∇p = H −. e×u − ∂t u, ε. ∇·u=0. and we find that ε2 k∇2 uk2 . kHk2 + ε−2 kuk2 + k∂t uk2 . To end the proof, it suffices to integrate in time, to multiply by ε the last estimate and to use (36). This ends the proof of Lemma 6. We now consider R as fixed. The next step is to consider the case where κ = κs is supported in ε|k| ≤ r. P 7. – There exists r > 0 sufficiently small such that we have for every ε ∈ (0, 1) and every T > 0 Z T Z T kκs v(T )k2 + εkκs ∇vk2 . kvk21,ε + |(κs F, κs v)| + kκs v0 k2 . 0. 0. Again, note that the assumption (H) is not used. Proof. – Again we use a direct energy estimate as in [26]. Since (e × κs v, κs v) = 0, we get Z T Z T s 2 s 2 (37) kκ v(T )k + εkκ ∇vk . kCk2 + kκs vk2 + |(κs F, κs v)| + S ε + kκs v0 k2 0. 0 ε. where the singular term S is defined by Z S ε = ε−1 |κs v3 | |∂Z ub | + |∂Z 0 ub | |κs v| dx. Ω. Again, we use (32) to estimate C and hence it remains to study S ε . To estimate this term, we shall use the inequality Z z Z 1 Z z 2 (38) |f (t, x)|2 ≤ |∂z f | ≤ z |∂z f |2 ≤ z |∂z f |2 if f/z=0 = 0. 0. 4 e SÉRIE – TOME 41 – 2008 – No 6. 0. 0.

(16) STABILITY OF BOUNDARY LAYERS. 969. Note that since ∂z κs v3 = −∂1 κs v1 − ∂2 κs v2 because of the incompressiblity condition, we can use (38) twice to get Z z Z z Z 1 |κs v3 |2 ≤ z |∂z κs v3 |2 . z |∇h κs vh |2 . z 3 |∂z ∇h κs vh |2 , 0. |κs v|2 . z. Z. 0. 0. 1. |∂z κs v|2 .. 0. This yields Z ε−1 |κs v3 | |∂Z ub,0 | |κs v| dx Ω. . ε2 k∂z ∇h κs vk k∂z κs vk sup y. Z. +∞. Z 2 |∂Z B 0 (τ, Z)wint (t, y, 0)| dZ. 0. . εrk∇κs vk2 . To get the last inequality, we have used that εk∇h κs f k ≤ rkκs f k by definition of κs and our regularity assumption which gives Z +∞ sup Z 2 |∂Z B 0 (τ, Z)wint (t, y, 0)| dZ < +∞ y. 0. following (8). The same argument in the vicinity of the boundary z = 1 shows that Z ε−1 |κs v3 | |∂Z 0 ub,1 | |κs v| dx . εrk∇κs vk2 . Ω. Consequently, we can choose r sufficiently small to absorb the singular term S ε in the left hand side of (37). This ends the proof of Proposition 7. Finally, it remains the most difficult case where κ(εk) = κl is supported in r/2 ≤ ε|k| ≤ 2R. Note that r and R are now fixed. We have the following estimate: P 8. – Under the assumptions of Theorem 4, we have for ε and T such that εe ≤1 Z T Z T kκl v(T )k21,ε + ε−1 kκl vk21,ε . eγ0 T kκl v0 k21,ε + εeγ0 T kF k2 + kvk21,ε . γ0 T. 0. 0 ik·y. The assumption (b(t, k̄)e. )k̄∈Z3a ∈ K is crucial in the proof of Proposition 8.. 3.2. Proof of Proposition 8 In this section, due to the oscillations in the boundary layers, we use an approach completely different from the one of the well-prepared case used in [26]. The proof of this Proposition is the most technical part and we split it into various steps. First, we rewrite (31) by using the Leray projection P(Dy ) which is recalled in section B.2. Let us set v l = κl v, we get the equation (39) ∂t v l = P(Dy )κl (εDy )Lε v l + P(Dy )κl κl F + κl C , ∇ · v l = 0, x ∈ Ω, where κl is compactly supported with a support slightly bigger than κl and takes the value 1 on the support of κl in order that κl κ = κ and Lε is defined by e×v Lε v = ε∆v − uapp · ∇v − v · ∇uapp − . ε ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(17) 970. N. MASMOUDI AND F. ROUSSET. Next, we shall estimate differently v l in the interior of the domain and in the vicinity of the boundary. We decompose v l as (40). z 1−z l v l = χb ( )v l + χint (z)v l + χb ( )v δ δ. where χb is compactly supported in [0, 2], and χint is compactly supported in [δ, 1 − δ]. Note that χint depends on δ though we forget this dependence in the notation. Multiplying (39) by χ for χ one of the truncation functions χint , χb,0 = χb (z/δ), χb,1 = χb (1 − z/δ), we get (41). ∂t (χv l ) = P(Dy )κl (εDy )Lε (χv l ) + H. (42). ∇ · (χv l ) = −∂z χ v3l. where (43). H = χP(Dy )κl κl F + κl C + C 1 + C 2. with the new commutators C 1 and C 2 defined by h i C 1 = χ, P(Dy )κl (εDy ) Lε v l , (44) h i (45) C 2 = P(Dy )κl (εDy ) χ, Lε v l . Thanks to (155) in Lemma 19, we get that (46). kC 1 k . kv l k + εk∇v l k + ε2 k∆v l k . kv l k2,ε .. Note that in the following . stands for ≤ C and that C may depend on δ. Besides, the explicit computation of C 2 and a new use of the commutator estimates (see the appendix of [26] or Lemma 16) gives that kC 2 k . kv l k + εk∇v l k . kv l k1,ε .. (47). l. l. Finally, note that if we choose κ such that the support of κ is again slightly larger than the one of κl then we have l. (48). H = κ (εDy )H.. 3.2.1. Interior estimates. – We start with the case χ = χint . The estimate of χint v l can be obtained by a direct energy estimate. We shall first establish the estimate: Z T Z T int l 2 −1 int l 2 int l 2 (49) kχ v (T )k + ε kχ v k1,ε . kχ v0 k + εkv l k22,ε + εkF k2 + kvk21,ε . 0. 0. The only difficulty is that we have to deal with the fact that χint v l is not divergence free. When we take the scalar product of (41) by χint v l , we can write P(Dy )κl (εDy )Lε (χint v l ), χint v l = Lε (χint v l ), P(Dy )κl (εDy )(χint v l ) = Lε (χint v l ), χint v l + Lε (χint v l ), [P(Dy )κl (εDy ), χint ]v l 4 e SÉRIE – TOME 41 – 2008 – No 6.

(18) 971. STABILITY OF BOUNDARY LAYERS. since v l is divergence free. Hence by using again the commutator estimate (155) of Lemma 19, we get P(Dy )κl (εDy )Lε (χint v l ), χint v l = Lε (χint v l ), χint v l + O(1) kv l k + εk∇v l k + ε2 k∆v l k kχint v l k. where O(1) is bounded by a constant which is independent of ε. The first term in the above equality can be handled by standard integration by parts as previously. This yields Z T int l 2 εk∇(χint v l )k2 (50) kχ v (T )k + . kχint v0l k2 +. Z. 0 T. . (kv l k2,ε + kHk) kχint v l k + S ε. . 0 ε. where the singular term S is given by S ε = ε−1 χint ∂Z ub v l , χint v l +ε−1 χint ∂Z 0 ub v l , χint v l := S1ε + S2ε . By using the localization of the support of χint we have η k,± δ X (51) kχint v l k2 |S1ε | . |b(t, k)||M k (0)|ε−1 exp − √ 2 ε k,± .. X |b(t, k)|M k (0) η k,±. k,±. kχint v l k2 . kχint v l k2. and hence this term is well controlled. The estimate of S2ε is similar. To conclude, we finally notice that since the Fourier transform of v l is localized in ε|k| ≥ r, we can write k∇(χint v l )k2 = k∇(χint κl (ε∂y )v l k2 = kκ∇(χint v l )k2 ≥ ε−2 kχint v l k2 and hence, we deduce (49) from (50) by using the Young inequality which gives for every η>0 (kv l k2,ε + kHk) kχint v l k ≤ C(η)ε(kHk2 + kv l k22,ε ) + η ε−1 kχint v l k2 and the estimates (32), (46), (47). 3.2.2. Estimates near the boundary. – We now explain how to estimate χb (z/δ)v l which will be denoted as χb v l in the following in order to simplify the notations. The estimate of χb,1 v l which can be obtained in a similar way will not be detailled. We shall establish Z T (52) kχb v l (T )k2 + ε−1 kχb v l k2 + εk∇(χb v l )k2 0. .e. γ0 T. kv. l. (0)k21,ε. + εe. γ0 T. Z. T. kF k2. 0. + ε eγ0 T. Z. T. kvk2 + ε2 k∇vk2 + ε4 k∆v l k2 + ε2 k∂t v l k2 .. 0 b,0. We study (41), (42) with χ = χ . Note that since χb v l is compactly supported in T2a ×[0, 2δ], we can use Lemma 20 and replace the Leray projection P(Dy ) by the Leray projection in the ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(19) 972. N. MASMOUDI AND F. ROUSSET. half-space P+ (Dy ) modulo a small remainder term. This means that we can study in T2a ×R+ the equation (53). ∂t (χb v l ) = P+ (Dy )κ(εDy )Lε (χb v l ) + H + E 0 ,. ∇ · (χb v l ) = −∂z χ v l. where E 0 = χb P(Dy ) − P+ (Dy ) κl (εDy )Lε (χb v l ) where χb is a smooth function with a support slightly bigger than χb . Thanks to Lemma 20, we have kE 0 k . kv l k + εk∇v l k + ε2 k∆v l k . kv l k2,ε .. (54). Again, note that E 0 verifies l. E 0 = κ (εDy )E 0 .. (55). We add to (53) the only boundary condition χb v l (t, y, 0) = 0.. (56). Since χb v l is not divergence free, we first lift the divergence to recover a problem with a divergence free constraint. We choose in a classical way d such that ∇ · d = −∂z χb v3l ,. (57). d(t, y, 0) = 0. and also in such a way that κl (εDy )d = d.. (58). This is possible (see [11]) since Z Z Z Z b l b l b ∂z χ v3 = − χ ∂z v3 = χ (z) ∇y · vhl = 0. z. y. Moreover, we can have kdkH s+1 . kv l kH s ,. (59). s ≥ 0.. Note that since d is chosen with the property (58), we can use that r . ε|k| on the support of κl and (59) to get that ε−1 kdk . kv l k.. (60). Moreover, by taking the time derivative of (57), we also get that ε−1 k∂t dk . k∂t v l k.. (61). Now, let us set w = χb v l − d, we deduce from (53), (57) that w solves (62). ∂t w = P+ (Dy )κ(εDy )Lb,ε w + H + E 0 + E 1 + E 2 ,. (63). ∇·w =0. for x = (y, z) ∈ T2a × R+ , with the boundary condition (64) 4 e SÉRIE – TOME 41 – 2008 – No 6. w(t, y, 0) = 0.

(20) STABILITY OF BOUNDARY LAYERS. 973. where Lb,ε is the operator Lb,ε w = ε∆w − uint ( εt , t, y, 0) + ub,0 ( εt , zε , t, y) + χ( zδ )(uint ( εt , t, y, z) − uint ( εt , t, y, 0)) · ∇w − w · ∇ uint ( εt , t, y, 0) + ub,0 ( εt , zε , t, y) + χ( zδ )(uint ( εt , t, y, z) − uint ( εt , t, y, 0)) −. e×w ε .. In this operator, we have introduced χ which is again a smooth compactly supported function such that χχb = χb . Moreover, we recall that the notation ub,0 refers to the main boundary layer in the vicinity of z = 0. The main interest of the introduction of χ is that the term wδ = χ( zδ ) uint ( εt , t, y, z) − uint ( εt , t, y, 0) verifies (65). kwδ kL∞ . δ,. k∇wδ kL∞ . 1.. In the above estimates, . is independent of δ for 0 < δ ≤ 1. In the right-hand side of (62), E 1 is the error term coming from d, i.e E 1 = ∂t d − P+ (Dy )κ(εDy )Lb,ε d. Hence we have kE 1 k . k∂t dk + k∇dk + kdk + εk∆dk + ε−1 kdk and hence, thanks to (59), (60), (61), we get (66). kE 1 k . εk∂t v l k + kv l k + εk∇v l k . k∂t v l k + kv l k1,ε .. The other error term E 2 is defined as r E 2 = −P+ (∂y )κ(ε∂y ) ub,1 ( εt , 1−z , y, 0) + u · ∇(χb v l ) ε r − χb v l · ∇ ub,1 ( εt , 1−z ε , t, y) + u so that by using (27) and the same trick as in (51) with the regularity assumption on the coefficients, we get (67). kE 2 k . εk∇v l k + kv l k . kv l k1,ε .. Finally, note that E 1 and E 2 verify (68). l. κ E1 = E1,. l. κ E2 = E2.. To estimate the solution of (62), (63), we can use the following general principle: L 9. – Consider a linear equation of parabolic type in a domain Ω (69). ∂t w = Aε w + F. with the boundary condition w/∂Ω = 0 and the initial condition w(0, x) = w0 (x). Consider two weighted norms NTε and k · kT,ε . Assume that there exists an approximate solver Gapp such that if we define wapp (t) = Gapp F, F = (F, w0 ) ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(21) 974. N. MASMOUDI AND F. ROUSSET. then wapp satisfies the boundary condition and the initial condition and moreover, there exists CT,ε > 0 such that NTε (wapp ) ≤ CT,ε kFkT,ε. (70). and if we define the rest operator Rapp as Rapp F = ∂t wapp − Aε wapp − F, 1 then, there exists CT,ε > 0 such that 1 k(Rapp F, 0)kT,ε ≤ CT,ε kFkT,ε .. (71). Moreover, for every ε > 0 and T > 0 such that 1 CT,ε < 1,. (72) there exists C > 0 (C =. 1 k k≥0 (CT,ε ) ). P. (73). such that the exact solution of (69) satisfies. NTε (w). ≤ CCT,ε kFkT,ε .. The proof of this lemma which only relies on a simple iteration scheme is postponed to the end of the section. We shall first explain how we can use Lemma 9 to estimate the solution of (62), (63). In other words, we need to find an approximate solver Gapp . A similar idea was used in [18]; nevertheless, here our approximate solver will be completely different. We define the dilation operators √ Mε f (Z) = εf (εZ), Mε f (τ, Z) = εf (ετ, εZ). Note that Mε is an isometry on L2 (R+ ) and Mε from L2 ([0, T ]×R+ ) to L2 ([0, T /ε]×R+ ). We notice that thanks to (65), we can rewrite the operator as 1 Lb,ε w = Mε−1 Lb q(t, y), εDy Mε w + O(1) δ|∇w| + |w| ε P+ (Dy ) = Mε−1 P+ (εDy )Mε where q(t, y) = (qk̄ (t, y)) with qk̄ (t, y) = b(t, k̄)eik·y . In the above equality, O(1) is bounded by a number independent of δ if ε/δ ≤ 1. The rescaled operator Lb is defined by ! iξ b 2 0 0 L (q, ξ)w = (∂ZZ − |ξ| )w − L q + B (τ, Z)q · w ∂z ! iξ 0 −w · L q + B 0 (τ, Z)q − e × w. ∂z Next, we use a frozen time approximation, we rewrite (62) as 1 −1 M P+ (εDy )κ(εDy )T (q(0, y), εDy )Mε w + Rε w = H 1 (74) ε ε where the symbol T (q, ξ) is the differential operator acting only on the τ and Z variables defined by T (q, ξ)w = ∂τ w − L0+ (q, ξ)w, 4 e SÉRIE – TOME 41 – 2008 – No 6.

(22) 975. STABILITY OF BOUNDARY LAYERS. where L0+ is defined in (16) and Rε w is defined by 1 Rε w = M−1 ε P+ (εDy )κ(εDy ) T (q(0, y), εDy ) − T (q(t, y), εDy ) Mε w ε +P+ (Dy )O(1) δ|∇w| + |w| and hence satisfies the estimate Z T Z (75) kRε wk2 . (T 2 + δ 2 ) 0. T. k∇wk2 + ε−2 (δ 2 + T 2 ). Z. 0. T. kwk2 .. 0. The source term H 1 in the right-hand side of (74) is defined by H 1 = H + E0 + E1 + E2.. (76). Thanks to our assumption (H), it is natural to define our approximate solver Gapp and our approximate solution wapp as (77). l. wapp = Gapp F = M−1 ε Opg κ (εDy )Mε Dε F,. where F=. H1 w0. ! ,. 1. Dε F = (εH 1 , ε 2 w0 ),. and with a slight abuse of notation, we define Mε F as Mε F = (Mε H 1 , Mε w0 ). To define the operator-valued symbol g, we first define for data F(τ, Z) = (F (τ, Z), u0 (Z)), the operator G(q, ξ) (acting on functions depending on τ and Z) such that G(q, ξ)F(τ, Z) := u(τ, Z) is the solution of (78). ∂τ u = P+ (ξ)L0+ (ξ)u + F,. iξ · uh + ∂Z u3 = 0. such that u(0, Z) = u0 (Z), u(τ, 0) = 0. By using the operator G(q, ξ), we can define a symbol g such that (79). g(y, ξ) = G(q(0, y), ξ). and then a semi-classical operator-valued pseudo-differential operator Opg as in the Appendix A. In particular for F(τ, y, Z), we have the definition X (80) Opg F = eiky G(q(0, y), εk)F̂(τ, k, Z). k. With a slight abuse of notation, we shall sometimes use the notation G(q(0, y), εDy ) in place of Opg . Note that the operator Mε acts only on the τ and Z variables. Consequently, we can write (81). Gapp F = Opgapp ,. g app (y, ξ) = M−1 ε g(y, ξ)Mε Dε. and hence Gapp is itself an operator valued semi-classical pseudo-differential operator with symbol g app . Note that because of (48), (55), (68), since we want to solve (74) only for data such that (82). l. κ (εDy )H 1 = H 1 ,. l. κ w0 = w0 ,. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(23) 976. N. MASMOUDI AND F. ROUSSET l. the introduction of κ in the definition (77) is justified. To prove that wapp = Gapp F is a good approximate solution, we want to use the semi-classical pseudo-differential calculus of section A seeing the symbol g(y, ξ) as an operator on L2 ([0, T ]×R+ ) or L∞ ([0, T ], L2 (R+ )). The aim of the following lemma is to study the dependence of G(q, ξ) in q. In particular, we prove that it is smooth in q and we estimate the derivatives. This will imply that g is smooth in y and hence we will be able to use Lemmas 16 and 17 of Appendix A. We introduce the following notations: |v|Θ,2 = kvkL2 ((0,Θ)×R+ ) ,. |v|Θ,∞ = kvkL∞ ((0,Θ),L2 (R+ )) ,. kvkΘ,2 = kvkL2 ((0,Θ)×T2a ×R+ ) ,. kvkΘ,∞ = kvkL∞ ((0,Θ),L2 (T2a ×R+ )) . l. L 10. – Thanks to (H), for every ξ in the support of κ , for every q in K and for every m, there exist αm > 0 and Cm > 0 such that for every Θ > 0, for every F = (F (t, ·), u0 (·)), F (t, ·), u0 (·) ∈ Hξ (83). |Dqm G(q, ξ)F|Θ,∞ + |Dqm G(q, ξ)F|Θ,2 + |∂Z Dqm G(q, ξ)F|Θ,2 ≤ Cm |F |Θ,2 + |u0 |L2 (R+ ). (84). |∂Z Dqm G(q, ξ)F|Θ,∞ + |∂τ Dqm G(q, ξ)F|Θ,2 + |∂ZZ Dqm G(q, ξ)F|Θ,2 ≤ Cm |F |Θ,2 + |u0 |H 1 (R+ ) .. We also postpone the proof of this Lemma to the end of the section. Thanks to Lemma 10 and Lemma 15, we get that l kwapp kT,2 = kGapp FkT,2 = Opg κ (ε∂y )Mε Dε F(τ 0 , Z) dτ 0 T /ε,2 √ √ ε 1 . εkM H kT /ε,2 + εkMε w0 k . εkH 1 kT,2 + εkw0 k. By the same method, we get l. 1. kwapp kT,∞ = kGapp FkT,∞ = ε− 2 Opg κ (ε∂y )Mε Dε F(τ 0 , Z) dτ 0 1 2. ε. 1 2. 1. T /ε,∞ 1. . ε kM H kT /ε,2 + kMε w0 k . ε kH kT,2 + kw0 k. In a similar way, since −1 ∂t M−1 M−1 ε =ε ε ∂τ ,. −1 ∂z M−1 M−1 ε =ε ε ∂Z ,. ∂Z Mε = εMε ∂z ,. we find 1. k∂z (Gapp F)kT,2 . kH 1 kT,2 + ε− 2 kw0 k, 1. 1. k∂t (Gapp F)kT,2 . kH 1 kT,2 + ε− 2 kw0 k + ε 2 k∇w0 k 3. 1. k∂zz (Gapp F)kT,2 . ε−1 kH 1 kT,2 + ε− 2 kw0 k + ε− 2 k∇w0 k, 1. k∂z (Gapp F)kT,∞ . ε− 2 kH 1 kT,2 + ε−1 kw0 k + k∇w0 k. Consequently, we define the weighted norm NTε (u) by 1. 1. NTε (u) = ε−1 kukT,2 + k∇ukT,2 + k∂t ukT,2 + εk∇2 ukT,2 + ε− 2 kukT,∞ + ε 2 k∇ukT,∞ . 4 e SÉRIE – TOME 41 – 2008 – No 6.

(24) STABILITY OF BOUNDARY LAYERS. 977. Note that the norm NTε (w) involves ∇w in our definition. Nevertheless, since we use only this norm for functions whose Fourier transform in y is supported in r ≤ ε|ξ| ≤ R, the terms involving ∇ in the norm actually give a useful, non redondant piece of information for ∂z only. We also define the weighted norm kFkT,ε as 1. 1. kFkT,ε = kH 1 kT,2 + ε− 2 kw0 k + ε 2 k∇w0 k; we have actually proven that NTε (Gapp F) . kFkT,ε .. (85). Moreover, by using again Lemma 10 and Lemma 15, we get by the same method that (86) NTε OpDyα gapp F . Cα kFkT,ε . We can now check that wapp is a suitable approximate solution. To have clear notations in the following computation, we use the notation OpT = T (q(0, y), εDy ). To check that Gapp is a good approximate solution, we write 1 −1 M P+ (εDy )κl (εDy )Op T Mε Op gapp F + Rε Opgapp F ε ε 1 = M−1 P+ (εDy )K l (εDy )Op T Mε Op gapp F + Rε Opgapp F + C r ε ε where K l (ξ) is a smooth compactly supported function with a support slightly bigger than the one of κ and such that K l κ = κ.. (87) The commutator C r is defined by. 1 −1 M P+ (εDy )K l [κ, OpT ]Opgapp F ε ε and hence is very similar to C. In particular, thanks to (85), we have Cr =. kC r kT,2 . εkFkT,ε .. (88) Next, we write. 1 −1 M P+ (εDy )K l (εDy )Op T Mε Op gapp F + Rε Opgapp F + C r ε ε 1 = M−1 Op P+ K l OpT Mε gapp F + Rε Opgapp F + C r + R1 F ε ε 1 = M−1 Op P+ K l T g Mε Dε F + Rε Opgapp F + C r + R1 F + R2 F. ε ε Since by definition, the symbol g, is chosen such that P+ (ξ)K l (ξ)T (q(0, y), ξ)g(y, ξ) = K l (ξ)Id, ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(25) 978. N. MASMOUDI AND F. ROUSSET. we get thanks to (82) and (87) that 1 −1 M P+ (εDy )κ(εDy )Op T Mε Op gapp F + Rε Opgapp F ε ε = K l (εDy )H 1 + Rε Opgapp F + C r + R1 F + R2 F = H 1 + C r + R1 F + R2 F. The remainder R1 F is defined by R1 F =. 1 −1 M P+ (εDy )κ(εDy )Opr1 F ε ε. with the symbol r1 given by r1 (y, ξ) = V (t/ε, q(t, y), Z) · εDy + ε2 Dy2 g app (y, ξ). Consequently, thanks to (86), we have kR1 FkT,2 . εkFkT,ε .. (89). In a similar way, thanks to Lemma 16 and (86), we have for m > 7 X (90) kR2 FkT,2 . ε N ε Op Dyα gapp F . εkFkT,ε . |α|≤2. Finally, thanks to (75), we also have (91). kRε Opgapp FkT,2 . (δ + T )kFkT,ε .. To use the result of Lemma 9, we set Rapp F = Rε Gapp (y, εDy )H 1 + C r + R1 F + R2 F and we see thanks to (88), (89), (90), (91) that (92). kRapp FkT,2 . (δ + T + ε)kFkT,ε .. Consequently, thanks to Lemma 9, for ε and δ sufficiently small, there exist T 0 and C 0 > 0 such that NTε 0 (w) ≤ C 0 kFkT 0 ,ε . To get an estimate on a longer interval of time, we can reiterate the process as long as q(t, y) ∈ K. Indeed, since q(T 0 , y) ∈ K, then we can use the same method as previously for T ≤ T 0 ; we rewrite the analogous of (74) but we replace T (wint,0 (0, y), εDy ) by T (wint,0 (T 0 , y), εDy ). The same argument as previously allows to get an estimate on [T 0 , 2T 0 ]. The iteration of the argument finally allows to get for some C > 0, γ0 > 0 independent of T , that (93). NTε (w) ≤ Ceγ0 T kFkT,ε. for every T > 0 such that wint (t, y, 0) ∈ K, for t ∈ [0, T ]. 4 e SÉRIE – TOME 41 – 2008 – No 6.

(26) 979. STABILITY OF BOUNDARY LAYERS 1. Finally, we can multiply (93) by ε 2 , use that w = χb v l − d and the estimates (43), (46), (47), (66), (67), (59), (60) to get in particular that T. Z. kχb v l (T )k2 +. (94). ε−1 kχb v l k2 + εk∇(χb v l )k2. . 0. . e. γ0 T. kv0l k21,ε. + εe. γ0 T. Z. T. kF k2. 0 T. Z. + ε eγ0 T. kvk2 + ε2 k∇vk2 + ε4 k∆v l k2 + ε2 k∂t v l k2 .. 0. This ends the proof of (52).. We can now end the proof of Proposition 8. At this point, we shall restrict ε and T such that εeγ0 T . 1; this will allow to absorb the terms in the right hand side still depending on v l which involve higher order derivatives by another estimates. At first, we can use the decompositions (40) and (52), (49) (we recall that we get an estimate near the upper boundary z = 1 completely analogous to (52)), to get kv l (T )k2 +. (95). Z. T. ε−1 kv l k2 + εk∇v l k2. . 0. . e. γ0 T. kv. l. (0)k21,ε. γ0 T. Z. T. kF k2. + εe. 0. +(1 + εeγ0 T ). (96). T. Z. . kvk2 + ε2 k∇vk2 + ε4 k∆v l k2 + ε2 k∂t v l k2 .. 0. To conclude, we use Lemma 6 to estimate higher order derivatives. We get 2. l. Z. 2. (97) ε k∇v (T )k +. T. . εk∂t v l k2 + ε3 k∇2 vk2. . 0. . ε2 k∇v0 k2 +. T. Z. ε−1 kv l k2 + εk∇v l k2 + εkvk2 + ε2 k∇vk2 + εkF k2 .. 0. Finally we can add (97) and (95) times a sufficiently large constant to get l. 2. 2. l 2. Z. T. . kv (T )k + ε k∇v k +. ε−1 kv l k2 + εk∇v l k2 + εk∂t v l k2 + ε3 k∇2 v l k2. . 0. . eγ0 T kv0 k21,ε + εeγ0 T. Z. T. kF k2. 0. + (1 + ε eγ0 T ). Z. T. . kvk2 + ε2 k∇vk2 + ε4 k∆v l k2 + ε2 k∂t v l k2. . 0. and hence, for ε sufficiently small, and εeγ0 T ≤ 1, we finally get the result of Proposition 8. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(27) 980. N. MASMOUDI AND F. ROUSSET. 3.3. End of the proof of Theorem 4 To get (30), we collect the estimates of Propositions 5, 7, 8. We get for ε, T such that εe ≤ 1: Z T kv(T )k2 + k∇v HF (T )k21,ε + εk∇vk2 + ε−1 kv HF k2 γ0 T. 0. . eγ0 T (kv0 k2 + kv0HF k21,ε ) +. T. Z. . εeγ0 T kF k2 + kvk2 + ε2 k∇vk2 + |(κs F, κs v)| .. 0. For ε sufficiently small, this gives (98). kv(T )k2 + k∇v HF k21,ε +. Z. T. . εk∇vk2 + ε−1 kv HF k2. . 0. . eγ0 T (kv0 k2 + kv0HF k21,ε ) +. Z. T. . εeγ0 T kF k2 + kvk2 + |(κs F, κs v)|. . 0. and hence Theorem 4 follows by using the Gronwall inequality. 3.4. Proof of Lemma 9 We represent the exact solution w of (69) as X w= wk k≥0. where w0 = Gapp (F, w0 ),. R0 = Rapp (F, w0 ). and for k ≥ 1 we define recursively wk and Rk as wk = −Gapp (Rk−1 , 0),. Rk = Rapp (Rk−1 , 0).. Thanks to (70), (71), we easily get by induction that 1 NTε (wk ) ≤ CT,ε (CT,ε )k k(F, w0 )kT,ε ,. 1 kRk kT,ε ≤ (CT,ε )k+1 k(F, w0 )kT,ε. and hence, thanks to (72), we get that X 1 NTε (w) ≤ CT,ε (CT,ε )k ) k(F, w0 )kT,ε ≤ CT,ε C k(F, w0 )kT,ε . k≥0. 3.5. Proof of Lemma 10 We start with the proof for m = 0. In this section, . means ≤ C where C is independent of Θ. We can write the solution of (78) under the form Z τ l l S+ (τ, τ 0 , q, ξ)κ (ξ)F (τ 0 ) dτ 0 G(q, ξ)F(τ ) = S+ (τ, 0, q, ξ)κ (ξ)w0 + 0. and hence, thanks to (H), we get that |G(q, ξ)F(τ )| . e−ατ |w0 | +. Z. τ. 0. e−α(τ −τ ) |F (τ 0 )| dτ 0 .. 0. This yields by standard results on convolutions that kG(q, ξ)F(τ )kΘ,∞ . |w0 | + kF kΘ,2 4 e SÉRIE – TOME 41 – 2008 – No 6.

(28) 981. STABILITY OF BOUNDARY LAYERS. and that kG(q, ξ)F(τ )kΘ,2 . |w0 | + kF kΘ,2 .. (99). Next, we can reintroduce the pressure and rewrite the equation (78) as ! ! ! iξ iξ iξ ∂τ u + V (τ, Z, q) · (100) u+u· V (τ, Z, q) + p ∂Z ∂Z ∂Z +e × u + |ξ|2 u − ∂ZZ u = 0 with the divergence free condition iξ · uh + ∂Z u3 = 0. Consequently, the standard energy estimate gives Z τ Z |u(τ )|2 + |∂Z u|2 . |u0 |2 + 0. τ. |F |2 +. 0. Z. τ. |u|2. 0. and since the right-hand side is already estimated thanks to (99), we also get k∂Z G(q, ξ)F(τ )kΘ,2 . |w0 | + kF kΘ,2 .. (101). To estimate higher order derivatives, we use again Lemma 6 (with ε = 1); we get k∂Z uk2Θ,∞ + k∂τ uk2Θ,2 + k∂ZZ uk2Θ,2 . |∂Z u0 |2 + kF k2Θ,2 + kuk2Θ,2 and since the right-hand side is again already bounded thanks to (99), we get that k∂Z G(q, ξ)FkΘ,∞ + k∂τ G(q, ξ)FkΘ,2 + k∂ZZ G(q, ξ)FkΘ,2 . |w0 | + kF kΘ,2 . This ends the proof of (83), (84) for m = 0. The general case follows by induction, we shall just explain how to handle the case m = 1. The regularity of the solution of (78) with respect to q follows from standard regularity results for solutions of parabolic equations whose coefficients smoothly depend on a parameter [10]. Taking the differential of (78) with respect to q in the direction h, we find that ∂τ − P+ (ξ)L+ (τ, q, ξ) Dq u · h = R1 with 1. . R = − (Dq V · h) ·. iξ ∂Z. ! u+u·. iξ. !. ∂Z. Dq V · h. and Dq u · h/t=0 = 0. Consequently, we have Dq u · h = G(q, ξ)(R1 , 0). By using (83), (84) for m = 0, we get |Dq u · h|Θ,∞ + |Dq u · h|Θ,2 + |∂Z Dq u · h|Θ,2 +|∂Z Dq u · h|Θ,∞ + |∂τ Dq u · h|Θ,2 + |∂ZZ Dq u · h|Θ,2 . |R1 |Θ,2 . |h| |u|Θ,2 + |∂Z u|Θ,2 . Consequently, we can use again (83), (84) for m = 0 to get the result for m = 1. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(29) 982. N. MASMOUDI AND F. ROUSSET. 3.6. Estimates of the gradient The aim of this section is to estimate k∇v(T )k2 , we first give a crude estimate: T 11. – Under the assumptions of Theorem 4, we have Z T 3 2 (102) ε k∇v(T )k + ε2 k∂t vk2 + ε4 k∇2 vk2 + ε2 k∇pk2 0. Z . ε3 k∇v0 k2 + eγT kv0 k2 + kv HF k21,ε +. T. |(κs F, κs v)| + ε. 0. Z. T. kF k2. . 0. Note that this estimate is relatively crude since we only have a control of ε3 k∇v(T )k2 whereas, because of the size of the boundary layers, we would expect a control of ε2 k∇v(T )k2 as we had for the large frequency part of the solution. Nevertheless, this estimate will be useful in Section 4. The reason is that in the proof we do not use in an optimal way the structure of the singular term ε−1 e × v. Proof of Theorem 11 To get (102), it suffices to use Lemma 6, then multiply the estimate (35) by ε and finally use (30). To get better estimates of some components of ∇v, we shall rewrite the equation (28) under an equivalent form which is classically used in fluid mechanics. We define η = ∂1 v2 −∂2 v1 and w = v3 . Note that η is the third component of the curl of v. Taking the curl of (28) and using that v is divergence-free, we easily get that the equation for η is given by (103). ∂z w − ε∆η ∂t η − ε app app = ∂1 uapp · ∇v2 + v · ∇uapp + F − ∂ u · ∇v + v · ∇u + F . 2 2 1 1 2 1. For the equation on w, we first derive the equation for the pressure. We take the divergence of (28) to get η (104) ∆p = + ∇ · F − uapp · ∇v − v · ∇uapp ε and next, we take the Laplacian in the third component of (28) and we use (104) to get ∂z η (105) ∂t ∆w + − ε∆2 w = ∆ ◦ π3 − ∂z ∇ · − uapp · ∇v − v · ∇uapp + F ε where π3 stands for the projection on the third component i.e. π3 (v) = v3 . Next, thanks to (2) and the fact that ∇·w = 0, we notice that the boundary condition for (103), (105) is given by (106). η/∂Ω = w/∂Ω = ∂z w/∂Ω = 0.. For the system (103), (105), we can prove: 4 e SÉRIE – TOME 41 – 2008 – No 6.

(30) 983. STABILITY OF BOUNDARY LAYERS. T 12. – Under the same assumptions as in Theorem 4, we have Z T Z (107) ε2 (kη(T )k2 + k∇w(T )k2 ) . eγT kv0 k21,ε + |(κs F, κs v| + ε 0. T. kF k2. . 0. and also (108). Z ε2 (k∇h vh (T )k2 + k∇w(T )k2 ) . eγT kv0 k21,ε +. T. |(κs F, κs v)| + ε. 0. Z. T. kF k2. . 0. for some γ > 0. Proof. – To estimate the solution of (103), (105), we multiply (103) by η and (105) by −w and we add the two equations. We use (106) to get thanks to integration by parts that d 1 kηk2 + k∇wk2 + ε(k∇ηk2 + k∆wk2 ) − ε−1 (∂z w, η) + (∂z η, w) dt 2 . k∇vk + ε−1 kv ∂Z ub k + kF k k∇ηk + k∆wk . The crucial fact in the above identity is that (∂z w, η) + (∂z η, w) = 0 so that the singular term vanishes. Consequently, we can use the Young inequality, the estimate (38) and multiply by ε2 to get Z T 2 2 2 3 ε (kη(T )k + k∇w(T )k ) + ε (k∇ηk2 + k∆wk2 ) 0. . ε2 (kη0 k2 + k∇w0 k2 ) +. Z. T. . εk∇vk2 + kvk2 + εkF k2 .. 0. Since the right hand side of the above estimate was already estimated in (30), we get (107). To get (108), we use that ∇ · v = 0, to get ∆h v2 = ∂1 η − ∂2 ∂z w,. ∆h v1 = −∂2 η − ∂1 ∂z w,. ∆h = ∂12 + ∂22 .. This immediately yields that k∇h vi k . kηk + k∂z wk,. i = 1, 2. and hence (108) follows from (107).. 4. Higher order conormal derivatives The estimates of Theorem 4 are sufficient to get a nonlinear stability result when it is possible to construct a very accurate approximate solution. Indeed, we can very easily estimate weighted derivatives under the form ε|α| ∂yα . It suffices to apply the operator ε|α| ∂yα to (28), to rewrite the obtained equation as T ε (ε|α| ∂yα v, ε|α| ∂yα p) = ε|α| ∂yα F + C where C is a well-controlled commutator and then to apply Theorem 4. The drawback of this approach is that this yields by Sobolev embedding a bad control of the L∞ norm which is needed to prove the nonlinear stability by a fixed point argument. Since here we have been ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.