Boundary Layer Analysis of the Navier-Stokes Equations with Generalized Navier Boundary Conditions

BOUNDARY LAYER ANALYSIS OF THE NAVIER-STOKES EQUATIONS WITH GENERALIZED NAVIER BOUNDARY CONDITIONS

GUNG-MIN GIE1 AND JAMES P. KELLIHER1

Abstract. We study the weak boundary layer phenomenon of the Navier-Stokes equations with generalized Navier friction boundary conditions, u n =0,[S(u)n]tan + u = 0, in a bounded domain in R3 when the viscosity, ε > 0, is· small. Here, S(u) isA the symmetric gradient of the velocity, u, and is a type (1, 1) tensor on the boundary. When = αI we obtain Navier boundary conditions,A and when is the shape operator weA obtain the conditions, u n = (curl u) n = 0. By constructingA an explicit corrector, we prove the convergence,· as ε tends× to zero, of the Navier-Stokes solutions to the Euler solution both in the natural energy norm and uniformly in time and space.

Contents 1. Introduction 2 2. Existence and Uniqueness Theorems 6 3. Channel domain 8 3.1. The corrector 9 3.2. Bounds on the corrector 10 3.3. Error analysis 11 3.4. Proof of convergence 14 4. Bounded domain 14 4.1. The corrector 17 4.2. The corrector in principal curvature coordinates 19 4.3. Bounds on the corrector 21 4.4. Error analysis 22 4.5. Proof of convergence 25 5. Uniform convergence 25 Appendix A. An anisotropic Agmon’s inequality 25 Appendix B. Special boundary conditions 27 Appendix C. Some lemmas 28 Acknowledgements 28 References 29

Date: 13 May 2011, updated 12 July 2011 and 28 May 2012 (compiled on June 8, 2012). 2000 Mathematics Subject Classiﬁcation. 35B25, 35C20, 76D05, 76D10. Key words and phrases. boundary layers, singular perturbations, Navier-Stokes equations, Euler equations, Navier friction boundary condition. 1 2 G. GIE AND J. KELLIHER

1. Introduction The flow of an incompressible, constant-density, constant-viscosity Newtonian fluid is described by the Navier-Stokes equations, ∂uε ε∆uε +(uε )uε + pε = f in Ω (0, T ), ∂t − · ∇ ∇ ×  div uε = 0 in Ω (0, T ), (1.1)  uε = u in Ω.× |t=0 0 R3 The fluid is contained in the bounded domain, Ω , with smooth boundary, Γ. The parameter, ε > 0, is the viscosity and T > 0⊂ is fixed (see Theorem 2.2). The equations are to be solved for the velocity of the fluid, uε, and pressure, pε, given the forcing function, f, and initial velocity, u0. The regularity of Γ, f, and u0 that we assume is specified in (1.7), but our emphasis is not on optimal regularity requirements. We also impose Navier boundary conditions on uε, described below, which include the impermeable condition, uε n = 0. When ε = 0, we formally· obtain the Euler equations, ∂u0 +(u0 )u0 + p0 = f in Ω (0, T ), ∂t · ∇ ∇ ×  div u0 = 0 in Ω (0, T ), (1.2)  u0 n =0 onΓ× (0, T ),  u0 · = u in Ω,× |t=0 0 n  where is the outer unit normal vector on Γ. The Euler equations, being first-order, need only the impermeable boundary condition, u0 n = 0, reflecting no entry or exit of fluid from the domain. No-slip boundary conditions,· uε = 0 on Γ, are those most often prescribed for the Navier-Stokes equations. This, of necessity, leads to a discrepancy between uε and u0 at the boundary, resulting in boundary layer effects. Prandtl [46] was the first to make real progress on analyzing these effects, and much of a pragmatic nature has been discovered, but to this day the mathematical understanding is woefully inadequate. (See [15, 9, 40, 17] for reviews of the mathematical literature. See [24], which builds on linear results of [18, 19, 25], for ill-posedness of Prandtl’s boundary layer equations; [19] gives a review of earlier ill-posedness results. See [31, 57, 60, 10, 33, 34, 49] for conditional results on convergence in the vanishing viscosity limit.) In part because of these difficulties with no-slip boundary conditions, and in part because of very real physical applications, researchers have turned to other boundary conditions. Of particular interest are boundary conditions variously called Navier friction, Navier slip, or simply Navier boundary conditions (other names have been used as well). These boundary conditions can be written as uε n =0, [S(uε)n + αuε] =0onΓ, (1.3) · tan where

1 ⊺ 1 ∂uj 1 ∂ui S(u) := u +( u) = + , for u =(u1,u2,u3). (1.4) 2 ∇ ∇ 2 ∂xi 2 ∂xj 1≤i,j≤3 Here (x1, x2, x3), (or (x, y, z) in Section 3), denotes the Cartesian coordinates of a point x R3, α is the (positive or negative) friction coeﬃcient, which is independent of ε. The∈ notation [ ] in (1.3) denotes the tangential components of a vector on Γ. · tan BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 3

In this paper, we use the generalization of (1.3), uε n =0onΓ, · (1.5) ( [S(uε)n] + uε =0onΓ, tan A of the Navier boundary conditions. Here, is a type (1, 1) tensor on the boundary having at least C2-regularity. In coordinatesA on the boundary, can be written in matrix form as = α . Note that uε lies in the tangent plane,A as does uε. A ij 1≤ij≤2 A It is easy to see that when = αI, the product of a function α on Γ and the identity tensor, the generalized NavierA boundary conditions, (1.5), reduce to the usual Navier friction boundary conditions, (1.3). In fact, the analysis using a general in place of αI is changed only slightly from using αI with α a constant (we say a bitA more on this in Remark 2.5). The primary motivation for generalizing Navier boundary conditions in this manner is that when is the shape operator (Weingarten map) on Γ, one obtains, as a special case, the boundaryA conditions, uε n = (curl uε) n =0, (1.6) · × as we show in Appendix B. (This fact is implicit in [3].) Such boundary conditions have been studied (in 3D) by several authors, including [2, 3, 61] (and see the references therein), [7, 6] for an inhomogeneous version of (1.6), and [4, 5] for related boundary conditions. In this special case, stronger convergence can be obtained (at least in a channel), in large part because vorticity can be controlled near the boundary. Hence, somewhat diﬀerent issues arise, and the bodies of literature studying boundary conditions (1.6) and (1.3) are somewhat disjoint. We introduce the Hilbert space, H = u L2(Ω)3 : div u = 0 and u n =0onΓ , ∈ · equipped with the usual L2 inner product. Then, letting T > 0 be an arbitrary time less than any T appearing in Theorems 2.2 and 2.4, we state our main result:

Theorem 1.1. Let Γa be the interior tubular neighborhood of Ω with width a > 0. Assume that u H Hm(Ω), f C∞ ([0, ); C∞(Ω)), Γ is Cm+2 for m 5. (1.7) 0 ∈ ∩ ∈ loc ∞ ≥ Then uε, a solution of the Navier-Stokes equations, (1.1), with Generalized Navier boundary conditions, (1.5), converges to u0, the solution of the Euler equations, (1.2), as the viscous parameter ε tends to zero, in the sense that

3 1 ε 0 4 ε 0 4 u u 2 κε , u u 2 1 κε , (1.8) − L∞(0,T ;L (Ω)) ≤ − L (0,T ;H (Ω)) ≤ for some T > 0 and for a constant κ = κ(T, α,u , f ), α = m . If m > 6 and 0 kAkC (Γ) f 0 then ≡ 3 − 3 3 9 ε 0 8 8(m 1) ε 0 4 − 8m u u κε − , u u κε , (1.9) − L∞([0,T ]×Γa) ≤ − L∞([0,T ]×Ω\Γa) ≤ where now κ = κ(T, α,m,a,u0, f). 4 G. GIE AND J. KELLIHER

Because we will only have existence of uε when 5 m 6 (Theorem 2.1), by uε we mean an arbitrary choice of the possibly multiple solutions≤ ≤when we consider the limit as ε 0. When m > 6 the solutions are unique as shown by Masmoudi and Rousset (see Theorem→ 2.4), and this arbitrary choice becomes unnecessary. Remark 1.2. Standard boundary layer analysis indicates that a linear corrector will 1/2 ∞ 1 3 be of order ε in L ([0, T ] Ω), so an exponent of 2 rather than 8 in (1.9) should formally be considered optimal× (for C∞ initial data). Navier boundary conditions go back to [44], in which Navier first proposed them, and to [42], in which Maxwell derived them from the kinetic theory of gases. There has been intermittent interest in them since, but revival of active interest in the mathematical community working on the vanishing viscosity limit started with the paper of Clopeau, Mikelić, and Robert [13], which gives a vanishing viscosity result in two dimensions. Also, the work of J-M Coron in [14] on the controllability of the 2D Navier-Stokes equations with Navier boundary conditions, which precedes [13], initiated interest in these boundary conditions in the PDE control theory community. By now there is a fairly substantial mathematical literature on the subject, but the three papers, [29, 30, 41], are of particular concern to us here. Both [29] and [41] give existence theorems for solutions to (1.1, 1.3), with uniqueness holding for stronger initial data. We quote these results in Theorems 2.1 and 2.4. Even with Navier boundary conditions there is a discrepancy between u0 and uε on the boundary, so we expect boundary layer effects to occur. As first shown (in 3D) by Iftimie and Planas in [29], however, this boundary layer effect is mild enough to allow convergence of uε to u0 in L∞(0, T ; L2(Ω)) without using any artificial function correcting the difference uε u0 on the boundary. (This result of [29] was for α 0, but the argument is easily modified− to allow α to be negative.) Thus, it makes sense≥ to refer to the boundary layer as weak. Specifically, Iftimie and Planas show in [29] that

ε 0 1 u u 2 Cε 2 . (1.10) k − kL∞(0,T ;L (Ω)) ≤ Iftimie and Sueur [30] use a corrector, v, to improve the convergence rate in (1.10) to ε3/4 in this energy norm. More precisely, they consider an asymptotic expansion of uε as 0 the sum of u and v, where v is a correctore whose tangential components are deﬁned as a solution of a linearized Prandtl-type system of coupled equations. Using the properties of v, they show thate e ε 0 ∞ 2 1 2 1 u (u + v) is order ε in L (0, T ; L (Ω)), order ε 2 in L (0, T ; H (Ω)). e − These bounds with estimates on the corrector v then give e 3 1 ε 0 4 ε 0 4 u u 2 Cε , u u 2 1 Cε . (1.11) − L∞(0,T ;L (Ω)) ≤ e − L (0,T ;H (Ω)) ≤ We, on the other hand, use an asymptotic expansion of uε in the form uε u0 + θε, ∼= where the main part of the explicitly deﬁned corrector, θε, exponentially decays from the boundary; see (3.8, 3.9) and (4.24, 4.25).1 Both correctors are linear and both can be

1 A B A B O εr r> Here, and in all that follows, ∼= is used to indicate that = + ( ) for some 0. We write this only to motivate the asymptotic expansions we make, but always show that such expansions are, in fact, valid in specific norms. BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 5 used to obtain order ε3/4 convergence in the vanishing viscosity limit, (1.8), but Iftimie and Sueur achieve an order of convergence of ε for the corrected difference, uε u0 v, while our corrected difference still gives order ε3/4. The tradeoff is simplicity− of− the corrector versus rate of convergence of the corrected velocity. e We wish to emphasize that the techniques we employ in this paper differ considerably from those of [30]. While the approach in both papers originates in the work of Prandtl [46], our approach adheres much more closely to a by now well-established approach to boundary layer analysis, which we adapt to treat Navier boundary conditions. In this regard our arguments will be more familiar to many researchers, and hence, ultimately, we believe, easier to incorporate into the existing understanding of boundary layers as they appear in a variety of physical problems. (For a description of the general theory of boundary layer analysis, see, for example, [15, 16, 20, 22, 28, 38, 45, 47, 58]. Concerning boundary layer analysis related to the Navier-Stokes equations, we refer readers to [21, 23, 26, 27, 30, 35, 51, 52, 55, 56, 57].) A key aspect of our corrector is that it is coordinate-independent. This not only gives it geometric meaning, it removes the need for a partition of unity to patch together the corrector defined in charts throughout the boundary layer. Nonetheless, the corrector has a particularly simple form in principal curvature coordinates, which we discuss in Section 4.2. (Such coordinates are used in much the same way, though for different purposes, in [3].) We also, in (1.9), obtain convergence uniformly in time and space of order ε3/8−δ near the boundary and ε3/4−δ′ in the interior, with δ, δ′ decreasing as the regularity of the initial velocity is increased, by employing an anisotropic embedding inequality developed in Appendix A. We take great advantage of the regularity result of Masmoudi and Rousset [41] (Theorem 2.4) to obtain this convergence. The authors of [41] themselves take a similar approach; however, the anisotropic inequality they use requires control on norms higher than those in (1.11), and this is only sufficient to obtain boundedness of the sequence of solutions to (1.1, 1.3). A compactness argument then gives convergence uniformly in time and space, though without a rate of convergence. The body of this paper is organized as follows: The existence and uniqueness results for solutions to the Navier-Stokes equations and Euler equations that we will need are given in Section 2. We give the proof of (1.8), the first part of Theorem 1.1, in Sections 3 and 4. To avoid the geometrical difficulties of a curved boundary, which obscure the key ingredients of the argument, we first prove (1.8) for the case of a three-dimensional periodic channel domain. We do this in Section 3. Then, in Section 4, as a generalization of Section 3, we treat the case of a bounded domain in R3 with smooth (curved) boundary. In Section 5, we present the (very short) proof of (1.9), the second part of Theorem 1.1, which relies on the anisotropic Agmon’s inequality, which we establish in Theorem A.2. In Appendix B, we prove that (1.5) reduces to (1.6) when is the shape operator. Finally, Appendix C contains some standard lemmas which weA state without proof.

Remark 1.3 (Notational conventions). Let I = [a, b] be a time interval and X a function space on Ω. By Lp(a, b; X), 1 p , we mean the space of all functions, f, for which ≤ ≤∞ 6 G. GIE AND J. KELLIHER f(t) lies in X for almost all t in I and for which its norm,

b 1/p f(t) p dt for 1 p< or ess sup f(t) for p = , k kX ≤ ∞ t∈(a,b) k kX ∞ Za k j is finite. The space, C (I; X), is the space of all functions, f, for which t ∂t f(t) is continuous as a function from I to X for all derivatives of order j k. Its norm7→ is ≤ k ess sup ∂jf(t) . t∈I k t kX j=0 X 0 Similarly, Cw(I; X) is the space of all functions weakly continuous from I to X. The abbreviation, e.s.t., stands for “exponentially small term.” More precisely, e.s.t. is a function (or a constant) whose norm in all Sobolev spaces, Hs, and thus Hölder s γ spaces, C , is exponentially small with a bound of the form, c1 exp( c2/ε ), c1,c2,γ > 0, for each s. − 2. Existence and Uniqueness Theorems Thanks to Lemma C.1, by applying the Galerkin method, one can construct solutions to (1.1) with (1.5) in the following sense, as shown in [30] (see Remark 2.5): 2 2 3 Theorem 2.1 (Iftimie, Sueur [30]). Assuming that u0 lies in H and f lies in L (0, T ; L (Ω) ), ε 0 2 1 3 there exists a weak solution u Cw([0, T ]; H) L (0, T ; H (Ω) ) of the Navier-Stokes equations, (1.1), with the generalized∈ Navier friction∩ boundary conditions, (1.5), in the sense that, for any v C∞([0, T ]; C∞(Ω) H) with v(T )=0, ∈ ∩ T ∂v T uε dx dt +2ε S(uε) S(v) dx dt − · ∂t · Z0 ZΩ Z0 ZΩ T T +2ε uε vdSdt + (uε )uε vdx dt = u v dx. A · · ∇ · 0 · |t=0 Z0 ZΓ Z0 ZΩ ZΩ We have the following well-posedness result for solutions to the Euler equations: Theorem 2.2. Suppose that u0 lies in H Hm(Ω) C1,µ(Ω), µ in (0, 1], m 3 is an ∞ ∞ ∩ ∩ m+2 ≥ integer, f lies in Cloc([0, ); C (Ω)), and Γ is of class C . Then for some time T > 0 ∞ 0 1 m there exists a unique solution, u ,to(1.2) lying in Cb ([0, T ] Ω) C([0, T ]; H (Ω)). The corresponding pressure, p0, lies in L∞(0, T ; Hm+1(Ω)) and× is unique∩ up to an additive function of time. Proof. Combining Theorem 1 and Theorem 2 part 3 of [37] gives the existence, uniqueness, and regularity of u when f 0 and the boundary is smooth. The proof is straightforward to adapt to smooth forcing,≡ and the strongest restriction on the smoothness of the boundary comes through the use of the Leray projector (Lemma 2 of [37]), where, however, Cm+2-regularity suffices. The regularity of the pressure (as well as the well- posedness in Sobolev spaces) is proved in [54, 50]. When (1.7) holds, by virtue of Theorem 2.2 and Sobolev embedding, for some T > 0, u0 C1([0, T ] Ω) C([0, T ]; Hm(Ω)), p0 L∞(0, T ; Hm+1(Ω)) for m 5. (2.1) ∈ b × ∩ ∈ ≥ The regularity in (2.1) of the solution is what we require; we do not claim that the assumptions in (1.7) are the minimal ones that guarantee such regularity, however. BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 7

In [41], Masmoudi and Rousset obtain the well-posedness result that we state in Theorem 2.4 for solutions to (1.1, 1.5) in the conormal Sobolev spaces of Definition 2.3 (see Remark 2.5). Definition 2.3. Let Ω be a d-dimensional manifold, d 1, with Ck-boundary, k 1. Viewing vector fields as derivations of C∞(Ω), we say that≥ a vector field, X, is tangent≥ N to ∂Ω if Xf =0 on ∂Ω whenever f is constant on Ω. Let (Zj)j=1 be a set of generators of vector fields tangent to ∂Ω. (Locally, only d such vector fields are needed, but for a global basis, N will be greater than d.) The following are examples of generators for the set of vector fields tangent to the boundary for two different domains: (i) For a channel domain, Ω = R2 (0, h), periodic in x and y, let ζ be a nonnegative × C∞(Ω)-function positive on Ω with ζ(x, y, z) = z near the lower boundary and ζ(x, y, z) = h z near the upper boundary. Then (∂1, ∂2, ζ∂3) defined globally on the channel− generate the set of vector fields tangent to the boundary. (ii) For the unit disk, D, let ζ be a nonnegative C∞(D)-function positive on D with ζ(r, θ)=1 r for all r > 1 . Let ϕ be a compactly supported cutoff function on − 2 Ω. Then ((1 ϕ)∂θ, (1 ϕ)ζ∂r, ϕ∂x, ϕ∂y) generate the set of vector fields tangent to the boundary.− − At the beginning of the proof of Theorem A.2 we construct a set of generators for vector fields tangential to the boundary in charts supported in tubular neighborhoods of the boundary of an arbitrary bounded domain in R3. For a multiindex, β, let Zβ = Zβ1 ZβN . Define 1 ··· N Hm(Ω) = f L2(Ω): Zβf L2(Ω) for all β m co ∈ ∈ | | ≤ with 2 β 2 f m = Z f . k kHco(Ω) L2(Ω) |β|≤m X m,∞ We say that f is in the space, Wco , if β f m, := Z u < k kWco ∞ k kL∞(Ω) ∞ |βX|≤m and we define the space Em by Em := u Hm(Ω) u Hm−1(Ω) { ∈ co | ∇ ∈ co } with the obvious norm. Theorem 2.4 (Masmoudi, Rousset [41]). Let m be an integer satisfying m> 6 and Ω be m+2 m 1,∞ a C domain. Consider u0 E H such that u0 Wco . Then there exists T > 0 such that for all sufficiently small∈ ε∩there exists a∇ unique∈ solution, uε C([0, T ], Em), ε ∈ to (1.1, 1.5) with f = 0 and such that u 1, bounded on [0, T ]. Moreover, there k∇ kWco∞ exists C = C(α) > 0, where α = m , such that kAkC (Γ) T ε ε ε 2 2 sup u (t) Hm(Ω) + u (t) m 1 + u (t) 1, + ε u(s) m 1 ds C. k k co k∇ kHco− (Ω) k∇ kWco∞ k∇ kHco− (Ω) ≤ [0,T ] Z0 (2.2) 8 G. GIE AND J. KELLIHER

Remark 2.5. Theorems 2.1, 2.2, and 2.4 were proved for a bounded domain, but each of the proofs extends easily to a 3D channel. Theorems 2.1 and 2.4 were also proved assuming that = αI, where α is a constant, but they easily extend to a general A A by using α = m in place of α in certain boundary terms, much as we do in kAkC (Γ) Sections 3.2 and 4.3.

3. Channel domain 3 In this section, we prove (1.8) for a periodic channel domain in R . We set Ω∞ := R2 (0, h), and consider solutions to (1.1, 1.5) in a channel domain Ω . That is, × ∞ ∂uε ε∆uε +(uε )uε + pε = f, in Ω (0, T ), ∂t − · ∇ ∇ ∞ ×  ε div u =0, in Ω∞ (0, T ),  × (3.1)  ε ε  u and p are periodic in x and y directions with periods L1 and L2, ε u t=0 = u0, in Ω∞.  |  Here,f and u0, satisfying (1.7), are assumed to be periodic in x and y directions with periods L1 and L2, respectively. For the sake of convenience, we set Ω := (0, L ) (0, L ) (0, h), 1 × 2 × and assume (to simplify the expressions in (3.6)) that ε< (h/8)2. Since n = (0, 0, 1) at z = 0 and n = (0, 0, 1) at z = h, we can write the Generalized Navier boundary condition,− appearing in (1.5) with (1.4), in the form ε u3 =0, at z =0, h, ε 2  ∂ui ε  2 αijuj =0, i =1, 2, at z =0,  ∂z −  j=1 (3.2)  X  ε 2 ∂ui ε +2 αiju =0, i =1, 2, at z = h.  ∂z j  j=1  X  The corresponding limit problem, (1.2), can be written as ∂u0 +(u0 )u0 + p0 = f, in Ω (0, T ), ∂t · ∇ ∇ ∞ ×  div u0 =0, in Ω (0, T ),  ∞ ×  0 0 (3.3)  u and p are periodic in x and y directions with periods L1 and L2,   0 u3 =0, at z =0, h,  0  u t=0 = u0, in Ω∞.  |  To study the boundary layer associated with the Navier-Stokes problem (3.1) with the Navier friction boundary conditions (3.2), we propose an asymptotic expansion of BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 9 uε with respect to small viscosity ε, ε 0 ε u ∼= u + θ , (3.4) where u0 is the solution of (3.3) and θε is a divergence-free corrector, which will be determined below. The main role of θε is to correct the tangential error related to the normal derivative of uε u0 on the boundary; see (3.5) below. − The corrector. ε ε ε ε ε 1/2 ε 3.1. To define a corrector, θ =(θ1, θ2, θ3), using the ansatz θ3 ∼= ε θi , i =1, 2, with respect to the order of ε in any Sobolev space, we first devote ourselves to ε find a suitable boundary condition for θi , i =1, 2. By inserting the expansion (3.4) into (3.2)2,3, we find that, for i =1, 2, ∂u0 2 ∂θε 2 i 2 α u0 + i 2 α θε = 0, at z =0, ∂z − ij j ∂z − ij j ∼  j=1 j=1  X X  0 2 ε 2  ∂ui ∂θi  +2 α u0 + +2 α θε = 0, at z = h. ∂z ij j ∂z ij j ∼ j=1 j=1  X X  ε For smooth αij, 1 i, j 2 on Γ, independent of ε, we expect that ∂θi /∂z 2 ε ≤ ≤ ε ≫ 2 j=1 αijθj , i =1, 2. Hence, we use the Neumann boundary condition for θi ,

ε 0 2 P ∂θi ∂ui 0 = ui,L := 2 αijuj , at z = 0 for i =1, 2,  ∂z − ∂z − j=1  X (3.5)  ε e 0 2  ∂θi ∂ui  = u := +2 α u0 , at z = h for i =1, 2. ∂z i,R − ∂z ij j j=1  X In the theory of boundary layer analysis, it is well known that the Neumann type  e boundary condition, (3.5), is useful when treating any weak boundary layer phenomenon. More precisely, to improve the convergence given in (1.10), it is sufficient to construct a corrector function that fixes the normal derivative of the difference uε u0 on the boundary, instead of the difference itself. − ∞ Toward this end, we first define cutoff functions, σL, σR, belonging to C ([0, h]), by 1, 0 z h/8, σL(z) := ≤ ≤ , σR(z) := σL(h z). (3.6) ( 0, h/4 z h. − ≤ ≤ ε ε ε ε ε Then we define the tangential component θi , i =1, 2, of the corrector θ =(θ1, θ2, θ3) as ε ε ε θi := θi,L + θi,R, i =1, 2, (3.7) where − z − z θε = √εu (x, y; t)σ (z)e √ε εu (x, y; t)σ′ (z) 1 e √ε i,L − i,L L − i,L L − ∂ z − √ε = εui,Le(x, y; t) σL(z) 1 ee , − ∂z − (3.8) n − h z o − h z θε = √εu (x, y; t)σ (z)e √−ε εu (x, y; t)σ′ (z) 1 e √−ε i,R ei,R R − i,R R − ∂ − h z = εu (x, y; t) σ (z) 1 e √−ε . − ei,R ∂z R − e n o e 10 G. GIE AND J. KELLIHER

ε ε ε To make θ divergence-free, we must define the normal component θ3 of θ as ε ε ε θ3 = θ3,L + θ3,R, where ∂u1,L ∂u2,L − z θε = ε + (x, y; t)σ (z) 1 e √ε , 3,L ∂x ∂y L − (3.9) ∂eu1,R ∂eu2,R − h z θε = ε + (x, y; t)σ (z) 1 e √−ε . 3,R ∂x ∂y R − (This form of the corrector ise as in [35e], adapted to Navier boundary conditions.) Thanks to (3.6, 3.8), by differentiating (3.7) with respect to the normal variable z, ε ε one can easily verify that the tangential components, θ1, θ2, satisfy the desired boundary condition (3.5). Moreover, from (3.9), we infer that ε θ3 =0, at z =0, h. (3.10) 3.2. Bounds on the corrector. We introduce the following convenient notation: ∂k any differential operator of order k := , k 0. ∂τ k with respect to the tangential variables x and y ≥ We also use the convention that κT = κT (T,u0, f) is a constant that depends on T , u0, and f, but is independent of ε and , and may vary from occurrence to occurrence. Letting A

α = m ,m> 6, (3.11) kAkC (Γ) (3.5) through (3.9) give ε ε ∂θi 1 ∂θi κ (1 + α)ε 2 , κ (1 + α), i =1, 2, (3.12) ∂τ ≤ T ∂z ≤ T L∞([0,T ]×Ω) L∞([0,T ]×Ω) ε ε ∂θ3 ∂θ 3 1 κ (1 + α)ε, κ (1 + α)ε 2 . (3.13) ∂τ ≤ T ∂z ≤ T L∞([0,T ]×Ω) L∞([0,T ]×Ω)

We have the following bounds on the corrector:

Lemma 3.1. Assume (1.7) holds and that k,l,n 0 are integers either l =1, k =0 or l =0, 0 k 2. Then the corrector, θε, defined≥ by (3.7) through (3.9), satisfies ≤ ≤ l+k+n ε ∂ θi 3 n 4 − 2 l k n C(1 + α)ε , i =1, 2, ∂t ∂τ ∂z 2 ≤  L∞(0,T ;L (Ω))  l+k ε l+k+n+1 ε  ∂ θ3 ∂ θ3 3 − n  4 2  l k C(1 + α)ε, l k n+1 C(1 + α)ε ∂t ∂τ 2 ≤ ∂t ∂τ ∂z 2 ≤ L∞(0,T ;L (Ω)) L∞(0,T ;L (Ω))   (3.14)  for a constant, C = C(T,l,k,n,u0, f). Proof. The assumptions (1.7) give the regularity of u0 in (2.1), and since m 5, this allows k to be at least as large as 2. To prove the lemma, using (3.6) through≥ (3.9), we ε ε first notice that it is sufficient to verify (3.14) with θi replaced by θi,L, 1 i 3. ε ≤ ≤ For (3.14)1 with θi,L, using (3.8)1, we write l+k ε l+k z l+k ∂ θi,L 1 ∂ ui,L 1 ′ − ∂ ui,L ′ = ε 2 σ (z) ε 2 σ (z) e √ε ε σ (z). (3.15) ∂tl∂τ k − ∂tl∂τ k L − L − ∂tl∂τ k L e e BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 11

Then, by diﬀerentiating (3.15) n times in the z variable, and using (3.5, 3.6), we ﬁnd l+k+n ε z ∂ θi,L 1 − n − C(1 + α)ε 2 2 e √ε + C(1 + α)ε + e.s.t.. (3.16) ∂tl∂τ k∂zn ≤

(See Remark 1.3 for the meaning of the abbreviation, e.s.t..) Hence, we ﬁnd

l+k+n ε 1 h z ∂ θi,L 1 − n − 2 2 C(1 + α)ε 2 2 e √ε dz + C(1 + α)ε ∂tl∂τ k∂zn ≤ 2 0 L∞(0,T ;L (Ω)) Z (setting z′ = z/√ε) (3.17) ∞ 1 3 − n −2z ′ 2 C(1 + α)ε 4 2 e ′ dz + C(1 + α)ε ≤ Z0 3 − n C(1 + α)ε 4 2 , for l,k,n 0. ≤ ≥ ε To prove (3.14)2 with θ3,L, using (3.9)1, we write l+k ε l+k l+k ∂ θ3,L ∂ ∂u1,L ∂u2,L ∂ ∂u1,L ∂u2,L − z = ε + σ (z) ε + σ (z)e √ε . ∂tl∂τ k ∂tl∂τ k ∂x ∂y L − ∂tl∂τ k ∂x ∂y L ε Hence, (3.14)2 with θ3,L followse bye applying exactly the samee computationse as (3.16, 3.17), and the proof of Lemma 3.1 is complete. We deﬁne a continuous piecewise linear function, ζ(z), by z, 0 z h/4, ζ(z) := h/4, h/≤4 ≤z 3h/4, (3.18)  h z, 3h/4≤ z≤ h.  − ≤ ≤ Then, using the analog of the proof of Lemma 3.1, one can verify that, i =1, 2,  ε ε ζ(z) ∂θi 1 ζ(z) ∂θ3 1 C(1 + α)ε 4 , C(1 + α)ε 2 . (3.19) √ε ∂z 2 ≤ √ε ∂z 2 ≤ L∞(0,T ;L (Ω)) L∞(0,T ;L (Ω))

3.3. Error analysis. We set the remainder:

wε := uε u0 θε. (3.20) − − Then, using (3.1) through (3.3) with (3.5, 3.10, 3.20), the equations for wε read ∂wε ε∆wε + pε p0 = ε∆u0 + R (θε) J (uε,u0), in Ω (0, T ), ∂t − ∇ − ε − ε ∞ ×  ε  div w =0, in Ω∞ (0, T),  ×  wε is periodic in x and y directions with periods L and L ,  1 2   wε =0, at z =0, h,  3  2 2  ∂wε (3.21)  i 2 α wε =2 α θε, i =1, 2, at z =0,  ∂z − ij j ij j j=1 j=1 X X  ε 2 2  ∂wi ε ε  +2 αijwj = 2 αijθj , i =1, 2, at z = h,  ∂z −  j=1 j=1  X X  ε ε  w t=0 = θ t=0, in Ω∞,  | − |   12 G. GIE AND J. KELLIHER where ∂v R (v) := + ε∆v, for any smooth vector ﬁeld v, (3.22) ε − ∂t J (uε,u0):=(uε )uε (u0 )u0. (3.23) ε · ∇ − · ∇ ε We multiply (3.21)1 by w , integrate over Ω and then, integrate it by parts . As a result, after applying the Schwarz and Young inequalities as well, we ﬁnd:

d ε 2 ε 2 2 0 2 ε 2 ε 2 w 2 +2ε w 2 ε ∆u + R (θ ) 2 +2 w 2 dt k kL (Ω) k∇ kL (Ω) ≤ L2(Ω) k ε kL (Ω) k kL (Ω) (3.24)

+2ε wεn wε dS 2 J (uε,u0) wε dx. ∇ · − ε · Z{z=0,h} ZΩ Thanks to Lemma 3.1 and (3.22) with v replaced by θε, we ﬁnd that

2 3 ε 2 2 R (θ ) 2 κ (1 + α )ε . (3.25) k ε kL (Ω) ≤ T On the other hand, by remembering that n = (0, 0, 1) at z = 0 and n = (0, 0, 1) at − z = h, and using (3.21)5,6, we notice that ∂wε ∂wε 1 , 2 , at z =0, wεn = − ∂z ∂z ∇ tan  ∂w ε ∂wε  1 , 2 , at z = h,  ∂z ∂z (3.26) 2 2  = 2 α (wε + θε), α (wε + θε) , at z =0, h. − 1j j j 2j j j j=1 j=1 X X Then, using (3.26), we ﬁnd that

ε ε ε ε ε 2ε w n w dS κ αε [w + θ ] 2 [w ] 2 ∇ · ≤ T k tankL (Γ) k tankL (Γ) Z{z=0,h} ε 2 ε ε κ αε w 2 + κ αε [θ ] 2 w 2 ≤ T k kL (Γ) T k tankL (Γ) k kL (Γ) (using Lemma C.2, (3.8), and the Poincar´einequality for wε) (3.27) 3 ε ε 2 2 ε κ αε w 2 w 2 + κ (1 + α )ε w 2 ≤ T k kL (Ω) k∇ kL (Ω) T k∇ kL (Ω) ε 2 2 ε 2 4 2 ε w 2 + κ α ε w 2 + κ (1 + α )ε . ≤ k∇ kL (Ω) T k kL (Ω) T By applying (3.25, 3.27) to (3.24), we obtain

d ε 2 ε 2 w 2 + ε w 2 dt k kL (Ω) k∇ kL (Ω) (3.28) 3 2 4 2 2 ε ε 0 ε κ (1 + α )ε + κ (1 + α ) w 2 2 J (u ,u ) w dx. ≤ T T k kL (Ω) − ε · ZΩ To estimate the last term on the right-hand side of (3.28), using (3.20, 3.23), we ﬁrst notice that J (uε,u0)=(uε )wε +(wε )(uε wε)+(u0 )θε +(θε )u0 +(θε )θε. (3.29) ε · ∇ · ∇ − · ∇ · ∇ · ∇ BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 13

Then, we write: 5 J (uε,u0) wε dx := j, (3.30) ε · Jε Ω j=1 Z X where

1 = (uε )wε wε dx =0, 2 = (wε )(uε wε) wε dx, Jε · ∇ · Jε · ∇ − ·  ZΩ ZΩ   3 ε 0 ε 4 0 ε ε  ε = (θ )u w dx, ε = (u )θ w dx, (3.31)  J Ω · ∇ · J Ω · ∇ ·  Z Z  5 = (θε )θε wε dx.  ε  J Ω · ∇ ·  Z To bound 2, using (3.20), we ﬁrst write  Jε 2 = (wε )u0 wε dx + (wε )θε wε dx. Jε · ∇ · · ∇ · ZΩ ZΩ Then

ε 0 ε x 0 ε 2 ε 2 (w )u w d κT u L (Ω) w L2(Ω) κT w L2(Ω) Ω · ∇ · ≤ ∇ ∞ k k ≤ k k Z and, thanks to (3.12, 3.13),

ε ε ε ε ε 2 ε 2 (w )θ w dx κT θ w 2 κT (1 + α) w 2 . L∞(Ω) L (Ω) L (Ω) Ω · ∇ · ≤ k∇ k k k ≤ k k Z

Thus, 2 ε 2 κ (1 + α) w 2 . (3.32) Jε ≤ T k kL (Ω) Using Lemma 3.1, we bound 3 by Jε 3 0 ε ε ε ε ε u L (Ω) θ L2(Ω) w L2(Ω) κT θ L2(Ω) w L2(Ω) J ≤ ∇ ∞ k k k k ≤ k k k k (3.33) 3 3 2 4 ε 2 2 ε κ (1 + α)ε w 2 κ (1 + α )ε + w 2 . ≤ T k kL (Ω) ≤ T k kL (Ω) Since u0 vanishes at z =0 or h, using the regularity of u0, we bound 4 by 3 Jε 3 ∂θε ∂θε 3 ∂θε 4 u0 j + u0 j wε dx + u0 j wε dx Jε ≤ 1 ∂x 2 ∂y j 3 ∂z j j=1 ZΩ j=1 ZΩ X X 3 ε ε 0 ∂θj ∂θj ε u + wj 2 L∞(Ω) L (Ω) ≤ ∂x ∂y 2 j=1 L (Ω) X (3.34) 3 0 ε 1 u3 ζ(z) ∂θj ε + ε 2 w j L2(Ω) ζ(z) √ε ∂z 2 j=1 L∞(Ω) L (Ω) X (using (3.18, 3.19) and Lemma 3.1)

3 3 2 4 ε 2 2 ε κ (1 + α)ε w 2 κ (1 + α )ε + w 2 . ≤ T k kL (Ω) ≤ T k kL (Ω) 14 G. GIE AND J. KELLIHER

Thanks to (3.12, 3.13) and Lemma 3.1, was can bound 5 by Jε 3 ε ε 3 ε 5 ε ∂θj ε ∂θj ε ε ∂θj ε ε θ1 + θ2 wj dx + θ3 wj dx J ≤ Ω ∂x ∂y Ω ∂z j=1 Z j=1 Z X X 3 ∂θε ∂θε θε j wε + θε j wε 1 L∞(Ω) j L2(Ω) 2 L∞(Ω) j L2(Ω) ≤ k k ∂x 2 k k ∂y 2 j=1 L (Ω) L (Ω) (3.35) X n o 3 ∂θε + θε j wε 3 L∞(Ω) j L2(Ω) k k ∂z 2 j=1 L (Ω) X 5 5 2 2 4 ε 4 2 ε κ (1 + α )ε w 2 κ (1 + α )ε + w 2 . ≤ T k kL (Ω) ≤ T k kL (Ω) Then, using (3.32) through (3.35), (3.30) gives

3 2 ε 0 ε 4 2 ε Jε(u ,u ) w dx κT (1 + α )ε + κT (1 + α) w L2(Ω) . (3.36) Ω · ≤ k k Z Applying (3.36 ) to (3.28), we obtain

d 2 2 3 2 ε ε 4 2 2 ε w 2 + ε w 2 κ (1 + α )ε + κ (1 + α ) w 2 . dt k kL (Ω) k∇ kL (Ω) ≤ T T k kL (Ω) Moreover, using (3.7) through (3.9) and (3.21)7, we observe that z ε ε 1 − 3 w 2 = θ 2 κ (1 + α)ε 2 e √ε 2 + l.o.t. κ (1 + α)ε 4 . k |t=0kL (Ω) k |t=0kL (Ω) ≤ T k kL (Ω) ≤ T Thanks to the Gronwall inequality, we ﬁnally have the bounds on the remainder, wε, 3 1 ε 4 ε 4 w 2 κ(T, α,u0, f)ε , w 2 1 κ(T, α,u0, f)ε . (3.37) k kL∞(0,T ;L (Ω)) ≤ k kL (0,T ;H (Ω)) ≤ 3.4. Proof of convergence. Using (3.20), we ﬁrst notice that uε u0 wε + θε pointwise in Ω (0, T ). (3.38) | − |≤| | | | ∞ × Then, using (3.37) and Lemma 3.1, (1.8) follows from (3.38).

4. Bounded domain In this section we consider the Navier-Stokes equations, (1.1, 1.5), and the Euler equations, (1.2), in a bounded domain Ω in R3 with boundary Γ having regularity as in (1.7). To handle the geometric difficulties of a curved boundary, we must treat Ω as a manifold with boundary, first constructing charts on Γ = ∂Ω in a special way, as we describe below. We consider the boundary, Γ as a submanifold of R3. Then, since Γ is a compact and smooth surface in R3, we construct a system of finitely many charts where each chart is a Cm-map, ψ, from a domain, U, in R2 to a domain, V , in Γ. More precisely, we choose ′ an orthogonal curvilinear system (ξ )=(ξ1,ξ2) in U so that, for any point x on V Γ, we write e e e ⊂ ′ ′ x = ψ(ξ ), ξ =(ξ1,ξe2) U. e (4.1) ∈ e Differentiating (4.1) with respect to ξi, i =1, 2, variables, we obtain the covariant basis on U and the metric tensor: e e e

′ ∂x gi(ξ ) := , i =1, 2, (4.2) e ∂ξi e e BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 15 and g (ξ′) := g g = diag g g , g g . (4.3) ij 1≤i,j≤2 i · j 1≤i,j≤2 1 · 1 2 · 2 Moreover, we see that the determinant of the metric tensor is strictly positive; e ′ e e ′ e e e e g(ξ ) := det(gij) > 0, for all ξ in the closure of U. (4.4) For any smooth 2d compact manifold Γ in R3, one can construct a system of finitely many charts, whiche satisfy (4.3e, 4.4). Hence, the class of domainse under consideration in this paper covers all bounded domains in R3 having boundary regularity as in (1.7). Moreover, as we will see below in Section 4.2, the construction of the corrector is independent of our choice of charts. Hence, it is sufficient to restrict our attention to a single chart only, since any estimates we develop will apply equally to all of Ω. We define Γc to be the interior tubular neighborhood of Ω with width c for any sufficiently small c > 0, and let a > 0 be small enough that Γ3a is such a tubular neighborhood. We can globally define the coordinate ξ3 on Γ3a to be distance from the boundary, with positive distances directed inward. We fix the orientation of ξ′ variables on V so that

′ g1 g2 ′ n(ξ ) := e× (ψ(ξ )), (4.5) − g g | 1 × 2| e e where n(ξ′) is the unit outer normal vector on V . Then,e letting U = U (0, 3a), we define a chart ψ : U V (giving what are sometimese e called boundary normal× coordinates): → e e x = ψ(ξ)= ψ(ξ′) ξ n(ξ′), x =(x , x , x ) Γ . (4.6) − 3 1 2 3 ∈ 3a By differentiating ψ in ξ variables and using (4.2), we define the covariant basis of the e curvilinear system ξ:

′ ∂n ′ ′ gi(ξ)= gi(ξ ) ξ3 (ξ ), i =1, 2, g3(ξ)= n(ξ ); (4.7) − ∂ξi − hence, from (4.5, 4.7), wee see that the covariant basis satisﬁes the right-hand rule.

One important observation here is that the orthogonality, on V , of gi, i =1, 2, does not imply the orthogonality, in V , of gi, i =1, 2, 3. To see this, we ﬁrst notice that ∂n e e = (linear combination of g1 and g2), i =1, 2. (4.8) ∂ξi g g g g Thus, i 3 = 0 for i =1, 2, but 1 2 = 0 in general.e e Consequently, the metric tensor (g (ξ)) · := (g g satisﬁes:· 6 ij 1≤i,j≤3 i · j 1≤i,j≤3

gij =0 i = 3 and j =1, 2, or i =1, 2 and j =3, (4.9) ( g33 =1. Moreover, thanks to (4.4), by choosing the thickness 3a> 0 of the tubular neighborhood Γ3a small enough, we see that g(ξ) := det(g ) > 0 for all ξ in the closure of U = U (0, 3a); (4.10) ij 1≤i,j≤3 × The function, √g := g1/2, is the magnitude of the Jacobian determinant of the chart, ψ. e 16 G. GIE AND J. KELLIHER

The matrix of the contravariant metric components are deﬁned in the closure of U as well: g22 g12 0 ij −1 1 − (g )1≤i,j≤3 =(gij)1≤i,j≤3 = g12 g11 0 . (4.11) g  −0 01  We introduce the normalized covariant vectors:  gi ei = , 1 i 3. (4.12) g ≤ ≤ | i| Then, for a vector valued function F , deﬁned on U, in the form 3

F = Fiei, i=1 X one can classically express the divergence operator acting on F in the ξ variable (see [11] or [36]) as

2 1 ∂ √g 1 ∂(√gF3) div F = F + . (4.13) g ∂ξ g i g ∂ξ √ i=1 i √ ii √ 3 X We write the Laplacian of F as 3 ∂2F ∆F = iF + iF + i e , (4.14) S L i ∂ξ2 i i=1 3 X where i linear combination of tangential derivatives F = ′ , S of Fj, 1 j 3, in ξ , up to order 2  ≤ ≤ (4.15)  i ∂Fi  Fi = proportional to . L ∂ξ3  Remark 4.1.Note that the coeﬃcients of i and i, 1 i 3 in (4.15), are multiples S L ≤ ≤ of √g , 1/√g, √gii , 1/√gii, i =1, 2, g12, g21, and their derivatives. Thanks to (4.10), all these quantities are well-deﬁned because of the regularity assumed for Γ in (1.7). Remark 4.2. Thanks to (4.9), we notice that the tangential directions are perpendicular to the normal direction in the tubular neighborhood Γ3a. Indeed this property enables us to obtain the expression of Laplacian as (4.14, 4.15), which is essentially the same as for the case of orthogonal curvilinear system. The explicit expression of Laplacian in orthogonal system appears in, e.g., [21].

3 For smooth vector ﬁelds F, G : U R , we consider F G, the covariant derivative of G in the direction F , which gives F→ G in the Cartesian∇ coordinate system. More precisely, we consider the smooth functions· ∇ F and G in the form 3 3

F = Fiei, G = Giei. i=1 i=1 X X Then, one can write G in the ξ variable, ∇F BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 17

3 ∂G ∂G ∂G G = F , F : i , i + F i + (F : G) e , (4.16) ∇F Pi 1 2 ∂ξ ∂ξ 3 ∂ξ Qi i i=1 1 2 3 X n o where product of a linear combination of ∂Gi ∂Gi i F1, F2 : , = the tangential components F1, F2 of F , ,  P ∂ξ1 ∂ξ2   (4.17)  and sum of tangential derivatives of Gi   (F : G) = (linear combination of the products F G , 1 j, k 3 ). Qi j k ≤ ≤  Remark 4.3. i(F : G), 1 i 3, are related to the Christoﬀel symbols of the second kind, which comesQ from the≤ twisting≤ eﬀects of the curvilinear system ξ. For the case of an orthogonal system, the explicit expression of (4.16) is given in Appendix 2 of [1]. Using the expression of contravariant components of the strain rate tensor, and by remembering, from (4.3, 4.7, 4.9), that the covariant basis (and hence the normalized covariant basis) is triply orthogonal on Γ, we write the generalized Navier boundary 3 conditions, (1.5), for F = i=1 Fiei as

F3 =0, at ξ3 =0, P 2  1 ∂Fi ∂F3  + i Fi, + αijFj =0, in ei direction at ξ3 =0, i =1, 2,  −2 ∂ξ M ∂ξ  3 i j=1 X  (4.18) where

∂F3 linear combination of the tangential component Fi and i Fi, = . (4.19) M ∂ξi the derivative in ξi of the normal component F3 Remark 4.4. Thanks to (1.7, 4.10), the coefficients of (F ), i =1, 2, are well-defined. Mi Concerning an orthogonal system, the explicit expression of i(F ), i =1, 2, appears on p. 115 of [43]. M 4.1. The corrector. In defining the corrector, θε, we parallel the strategy we used in Section 3 for a periodic channel domain as closely as possible, employing an asymptotic expansion, uε = u0 + θε, asin (3.4), but adapting the corrector to the curved boundary. ∼ ε With the unit vectors, ei, defined as in (4.12), on U, θ can be written

3 ε ε θ := θi ei. (4.20) i=1 X ε The tangential components θi , i = 1, 2, will be constructed to correct the tangential discrepancy in the boundary conditions related to the normal derivative of uε u0 on ε − the boundary. Then the normal component θ3 will be deduced from the divergence-free condition on θε. To define the corrector θε appearing in (4.20), since u0 n = 0 on Γ, we first set · u =2 [S(u0)n] + u0 , tan A e 18 G. GIE AND J. KELLIHER defined on all of Γ, and write, in coordinates, 2 u(ξ′; t)= u (ξ′; t)e , u (ξ′; t) := u(ξ′; t) e . (4.21) i i|ξ3=0 i · i|ξ3=0 i=1 X ε 0 ε Then, we inserte the expansione u ∼= u + θ intoe the generalizede Navier boundary conditions, (1.5), and, thanks to (4.18)2, we find that, for i =1, 2, 1 1 ∂θε ∂θε 2 u (ξ′; t) i + θε, 3 + α θε = 0, at ξ =0. 2 i − 2 ∂ξ Mi i ∂ξ ij j ∼ 3 3 i j=1 X Using (4.19), wee expect that ∂θε/∂ξ (θε, ∂θε/∂ξ ) or 2 α θε, i = 1, 2, for i 3 ≫ Mi i 3 i j=1 ij j smooth αij, 1 i, j 2, independent of ε. Hence, we use the Neumann boundary ε≤ ≤ P condition for θi , ε ∂θi ′ = ui(ξ ; t), i =1, 2. (4.22) ∂ξ3 ξ3=0

We can now model the corrector after the flat-space version in (3.8). We define a e smooth cutoff function, σ(ξ3), with

1, 0 ξ3 a, σ(ξ3) := ≤ ≤ (4.23) ( 0, ξ3 2a. ≥ Letting

′ √g γi := γi(ξ )= , √gii ξ3=0

we define the tangential components of the corrector by √gii(ξ) ∂ ξ3 ε ′ − √ε θi (ξ; t) := ε (γiui)(ξ ; t) σ(ξ3) 1 e , i =1, 2. (4.24) − √g(ξ) ∂ξ3 − It follows from (4.13) that e 2 ε ∂ ∂ ξ3 ∂(√gθ ) ε ′ − √ε 3 √g div θ = ε (γiui)(ξ ; t) σ(ξ3) 1 e + − ∂ξi ∂ξ3 − ∂ξ3 i=1 X 2 ξ ε ∂ e∂ − 3 ∂(√gθ3) = ε (γ u )(ξ′; t)σ(ξ ) 1 e √ε + . − ∂ξ ∂ξ i i 3 − ∂ξ 3 i=1 i 3 n X o Thus, we can easily ensure that θε is divergence-freee by letting 2 ξ 1 ∂ − 3 θε(ξ; t) := ε (γ u )(ξ′; t) σ(ξ ) 1 e √ε . (4.25) 3 g(ξ) ∂ξ i i 3 − √ i=1 i n X o ε It is easy to see that θ3 vanishes at ξ3 =e 0, and by differentiating (4.24) and us- ε ing (4.23), we see that each tangential component θi , i = 1, 2, satisfies the boundary condition (4.22) to within order √ε: ε ∂θi ′ ′ = ui(ξ ; t) √εE(ξ ; t), (4.26) ∂ξ3 ξ3=0 −

e BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 19 where

′ ′ ∂ √gii ′ E(ξ ; t)= γi(ξ ) ui(ξ ; t). (4.27) ∂ξ3 √g ξ3=0

Due to the presence of σ in (4.24, 4.25), we also havee k ε ∂ θi k =0, 1 i, j 3, k 0. (4.28) ∂ξj ξ3≥2a ≤ ≤ ≥

Remark 4.5. For the case of a flat boundary Γ, one can choose curvilinear coordinates with the metric tensor (gij)1≤i,j≤3, defined in (4.9), as the identity matrix I3×3, and hence √g, √gii and γi, i = 1, 2, appearing in (4.24, 4.25, 4.27), are equal to 1. This implies that the expression of the corrector defined by (4.24, 4.25) are identical to (3.8, 3.9) in a channel domain where the error E in (4.27) is now equal to 0. Remark 4.6. The form of our corrector (4.24, 4.25) is similar to the background flow in Lemma 1 of [59]. 4.2. The corrector in principal curvature coordinates. In this section, we express the corrector in particularly convenient and geometrically meaningful coordinates called principal curvature coordinates. We define an umbilical point of Γ to be a point at which the principal curvatures, κ1 and κ2, are equal (this also includes what some authors refer to as a planar point, where both curvatures vanish). By Lemma 3.6.6 of [36], in some neighborhood of any non-umbilical point there exists a chart in which the metric tensor of (4.2) is diagonal (as is the second fundamental form) and the coordinate lines are parallel to the principal directions at each point. Such a chart is also called a principal curvature coordinate system. For now, we assume that we are working in such a chart, ψ : U V Γ, p p → ⊆ x = ψ (η′), η′ =(η , η ) U . p 1 2 ∈ p e e e The corresponding covariant basis and metric tensor are e e e ′ ∂ψp ′ qi(η )= , i =1, 2, (qij(η ))1≤i,j≤2 =(qi qj)1≤i,j≤2 = diag(q1 q1, q2 q2). (4.29) ∂ηi · · · e ′ ′ eUsing (4.6) with ξ andeψ replaced by ηe ande ψp, we definee a charte eψp efrom Up = Up (0, 3a) into Γ3a by × e e x = ψ (η)= ψ (η′) ξ n(η′), η =(η′,ξ ) U . e p p − 3 3 ∈ p As before, ξ3 is the distance from the boundary, which we note does not depend upon the choice of the boundary chart.e In the principal curvature coordinate system on Up, the unit outer normal vector n satisfies n ∂ ′ e = κi(η )qi, i =1, 2. ∂ηi Hence, differentiating ψp in the η variables gives the covariant basis of the coordinate system η, e q (η)=(1 κ (η′)ξ )q (η′), i =1, 2, q (η)= n(η′). (4.30) i − i 3 i 3 − e 20 G. GIE AND J. KELLIHER

Using (4.29, 4.30), the metric tensor (qij)1≤i,j≤3 is written in the form ′ 2 (1 κ1(η )ξ3) q11 0 0 (q ) = − 0 (1 κ (η′)ξ )2 q 0 (4.31) ij 1≤i,j≤3  − 2 3 22  0e 01 with its determinant, q(η), bounded away from zero. This ise guaranteed by simply choosing the thickness, 3a> 0, of the tubular neighborhood small enough. It is easy to see that the coordinate system, η, derived from the principal curvature coordinate system, satisﬁes (4.4, 4.9, 4.10). Hence we use the expression of the corrector θε, (4.20, 4.24, 4.25), in η coordinates and write ε ε ε θ = θτ + θ3e3, where 2 ξ ∂ − 3 √qii √q ε √ε ′ e e θτ = ε σ(ξ3) 1 e u(ξ ; t) i ξ =0 i,  − ∂ξ3 − √q √qii · 3 i=1 ξ3=0  X n o (4.32)  2  ξ3 1 ∂ √q e b b  ε − √ε ′  θ = εσ(ξ3) 1 e u(ξ ; t) ei , 3 ξ3=0 − q ∂ξ q ξ3=0 · √ i=1 i √ ii  n X o e q q e n for i = i/ i , i =1, 2, and 3 = . e b Using (4.31| ),| it is easy to see that−

b √qii √q 1 = ′ for i =1, 2 and j =3 i. (4.33) √q √qii 1 κj(η )ξ3 − ξ3=0 − Then, combining (4.32, 4.33), we ﬁnd

∂ ξ3 ε − √ε ′ θτ (η)= ε σ(ξ3) 1 e M(η)u(η ; t), (4.34) − ∂ξ3 − M where is a smooth type (1, 1) tensor deﬁned, in our coordinates,e by 2 1 2 M(η)F := F e , F = F e . (4.35) 1 κ (η′)ξ i i i i i=1 3−i 3 i=1 X − X On the boundary, thee divergence operator is e b e e b 1 2 ∂ q divτ = F . ∂η q i q i=1 i s ii ! X e Then by (4.31), p e e e √q = (1 κ (η′)ξ )(1 κ (η′)ξ ) q, − 1 3 − 2 3 ε so we can write θ3 in (4.32) as p e ξ3 − √ε ε 1 e θ = εσ(ξ ) − divτ u. (4.36) 3 3 (1 κ (η′)ξ )(1 κ (η′)ξ ) − 1 3 − 2 3 Although we assumed in its derivation that we were near a non-e umbilical point so that we could construct a principal curvature coordinate system, our expression for θε BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 21 is perfectly valid at an umbilical point, where we simply have κ1 = κ2 thanks to the smoothness of the curvatures in the tangential variables. Finally, a straightforward but lengthy calculation, which we omit, shows that (4.24, 4.25) transforms to (4.34, 4.36) under the change of variables from ψ to ψp, showing that our corrector in the form (4.24, 4.25) is covariant with respect to the changes of charts. (This is perhaps not immediately obvious, because (4.24, 4.25) involve the metrics both on the boundary and in the tubular neighborhood.)

Remark 4.7. For most smooth, bounded domains in R3, principal curvature coordinate systems can be constructed in the neighborhood of all but at most isolated points; in fact, having only isolated umbilical points is (in some sense) generic. And, for instance, a sphere, while it consists only of umbilical points, can be covered by two charts, both of which use principal curvature coordinates (essentially, spherical coordinates). When such coordinates suit the boundary of a domain, the expression for the tangential corrector in (4.34, 4.35) is both simpler to calculate and simpler to interpret than the expression in (4.24). In such coordinates, the expressions for the diﬀerential operators, such as div, curl, ∆, can also be written more simply. Though we used principal curvature coordinates in this section to prove that our corrector is independent of the choice of charts, we cannot restrict ourselves to such coordinates in the rest of our analysis, as that would put constraints on the geometry of the domains that we would be able to treat.

Remark 4.8. As a special example of a smooth bounded domain in R3, consider a solid torus, which has no umbilical points on its boundary. Hence, we need only one principal curvature coordinate system, if we allow it to be periodic in the tangential variables. In this sense, the solid torus is the simplest smooth bounded domain to work with in R3.

4.3. Bounds on the corrector. We follow the convention described at the beginning of Section 3.2, though now the tangential variables are ξ1 and ξ2, and as in (3.11), we let α = m ,m> 6. Then, from (4.24, 4.25), we infer that kAkC (Γ)

ε ε ∂θi 1 ∂θi κT (1 + α)ε 2 , κT (1 + α), i =1, 2, (4.37) ∂τ ≤ ∂ξ3 ≤ L∞([0,T ]×U) L∞([0,T ]×U)

and

ε ε ∂θ3 ∂θ3 1 κT (1 + α)ε, κT (1 + α)ε 2 . (4.38) ∂τ ≤ ∂ξ3 ≤ L∞([0,T ]×U) L∞([0,T ]×U)

We now state the estimates on the corrector in the lemma below, which we omit the proof as it is essentially the same as that of Lemma 3.1 because of (4.10). 22 G. GIE AND J. KELLIHER

Lemma 4.9. Assume (1.7) holds and that k,l,n 0 are integers either l =1, k =0 or l =0, 0 k 2. Then the corrector, θε, defined≥ by (4.24, 4.25), satisfies ≤ ≤ l+k+n ε ∂ θi 1 3 n 2 4 − 2 l k n C V (1 + α)ε , i =1, 2, ∂t ∂τ ∂ξ3 2 ≤ | |  L∞(0,T ;L (U))  l+k ε  ∂ θ3 1  C V 2 (1 + α)ε,  l k  ∂t ∂τ 2 ≤ | |  L∞(0,T ;L (U))  l+k+n +1 ε ∂ θ3 1 3 − n C V 2 (1 + α)ε 4 2 ,  l k n+1  ∂t ∂τ ∂ξ3 2 ≤ | |  L∞(0,T ;L (U))  for C = C(T,l,k,n,u , f) > 0, independent of ε, and the measure V of V = ψ(U).  0 A | | In addition to the estimates in Lemma 4.9, as for the case of the channel domain, one can easily verify that the corrector θε satisfies that, i =1, 2, ε ε ξ3 ∂θi 1 1 ξ3 ∂θ3 1 1 C V 2 (1 + α)ε 4 , C V 2 (1 + α)ε 2 . √ε ∂ξ3 2 ≤ | | √ε ∂ξ3 2 ≤ | | L∞(0,T ;L (U)) L∞(0,T ;L (U))

(4.39)

4.4. Error analysis. We set the remainder: wε := uε u0 θε. (4.40) − − ε ε Then, using (1.1, 1.2, 1.5, 4.40) and the fact that θ3 = 0 on Γ, the equations for w read ∂wε ε∆wε + pε p0 = ε∆u0 + R (θε) J (uε,u0), in Ω (0, T ), ∂t − ∇ − ε − ε ×  ε  div w =0, in Ω (0, T ),  ×  wε n =0, on Γ, (4.41)   · S(wε)n + wε = S(u0 + θε)n (u0 + θε), on Γ, tan A − tan −A  wε = θε , in Ω,  t=0 t=0  | − | where Rε( ) and Jε( , ) are deﬁned by (3.22) and (3.23).  · · · ε We multiply (4.41)1 by w , integrate it over Ω and then use Lemma C.1. After applying the Schwarz and Young inequalities to the right-hand side of the resulting equation, we ﬁnd:

d ε 2 ε 2 w 2 +4ε S(w ) 2 dt k kL (Ω) k kL (Ω) 2 0 2 ε 2 ε 2 ε ∆u + R (θ ) 2 +2 w 2 ≤ L2(Ω) k ε kL (Ω) k kL (Ω) (4.42) ε 0 ε 0 ε ε 4ε w + S(u + θ )n + (u + θ ) w dS − A tan A · ZΓ 2 J (uε,u0) wε dx. − ε · ZΩ To go further, using what is sometime called Korn’s second inequality, which requires no boundary conditions (see [12] for proofs of this inequality and some comments on its history), we have ε 2 ε 2 ε 2 κS w 2 S(w ) 2 + w 2 (4.43) k∇ kL (Ω) ≤k kL (Ω) k kL (Ω) BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 23 for a positive constant, κS, depending on the domain, but independent of ε and α. Restricted to the range, V , of any chart, ψ, we ﬁnd, using (3.22) with v replaced by θε and (4.14, 4.15) for each θε, that

3 ε 2 2 ε 2 ε 2 ∂θ i ε i ε ∂ θi Rε(θ ) L2(V ) + ε θ + θi + 2 k k ≤ ∂t L2(U) S L ∂ξ L2(U) i=1 3 X 2 3 κ T (1 + α )ε 2 , ≤ where we also used Remark 4.1 and Lemma 4.9. Since we have a ﬁnite number of charts ε on Γ3a, and since θ is supported in Γ3a by (4.28), the same estimate holds on Ω; namely,

2 3 ε 2 2 R (θ ) 2 κ (1 + α )ε . (4.44) k ε kL (Ω) ≤ T To estimate the fourth term in the right-hand side of (4.42), we write

ε S 0 ε n 0 ε ε 4ε w + (u + θ ) tan + (u + θ ) w dS Γ A A · Z (4.45) 2 0 ε 0 ε κT εα w L2(Γ) + κT ε S(u + θ )n + (u + θ ) w L2(Γ). ≤ k k tan A L2(Γ)k k

On each V Γ, the range of the boundary chart, ψ, using (4.18 , 4.19, 4.21, 4.26), we have ⊂ e 0 ε 0 ε e S(u + θ )n + (u + θ ) e tan A V 2 1 1 ∂θε ∂θ ε 2 = u i + θε, 3 + α θε e 2 i − 2 ∂ξ Mi i ∂ξ ij j i|ξ3=0 i=1 " 3 i j=1 # X X ξ3=0 2 e ∂θε 2 1 = θε, 3 + α θε + √εE(ξ ,ξ ; t) e , Mi i ∂ξ ij j 2 1 2 i|ξ3=0 i=1 " i j=1 # X X ξ3=0 where ui and E are deﬁned by (4.21, 4.27). Since this bound holds for all charts, using Remark 4.4 and (4.19, 4.37, 4.38), we ﬁnd that

e 0 ε 0 ε 2 3 ε S(u + θ )n + (u + θ ) κT (1 + α )ε 2 . (4.46) tan A L2(Γ) ≤

Thanks to (4.43, 4.44, 4.45, 4.46), applying Lemma C.2 and the Poincar´einequality, (4.42) yields that

d ε 2 ε 2 w 2 +2κSε w 2 dt k kL (Ω) k∇ kL (Ω) (4.47) 3 2 4 2 2 ε ε 0 ε κ (1 + α )ε + κ (1 + α ) w 2 2 J (u ,u ) w dx. ≤ T T k kL (Ω) − ε · ZΩ To estimate the last term of (4.47), using (3.29), we write

5 J (uε,u0) wε dx := j, ε · Jε Ω j=1 Z X 24 G. GIE AND J. KELLIHER

j where ε , 1 j 5, are given by (3.31). Due to (4.28, 4.37, 4.38) and Lemma 4.9, J ≤ ≤ j one can easily veriﬁes that ε , j =2, 3, satisﬁes the same estimate, appearing in (3.32, 3.33), as for the case of a channelJ domain. That is,

3 3 2 j 2 2 ε κ (1 + α )ε + κ (1 + α) w 2 . Jε ≤ T T k kL (Ω) j=1 X 4 5 To bound ε and ε , it is suﬃcient to work in a single chart, ψ : U V , as there a ﬁnite number,JN, of them,J which just introduces the constant, N. → For 4, using (3.31 , 4.28), we write Jε 4 4 0 ε ε (u )θ w 2 Jε ≤ · ∇ L2(V ) k kL (Ω) (using (4.16, 4.17 ) with F , G replaced by u0, θε, respectively)

3 2 ε 0 ε ∂θi ε κT u θi 2 + w 2 L∞(Ω) L (U) L (Ω) ≤ k k ∂ξ 2 k k i=1 j=1 j L (U) X n X o 0 3 ε u e3 ξ3 ∂θ i ε + κT √ε · w L2(Ω) ξ3 √ε ∂ξ3 2 k k L∞(Γ2a) i=1 L (U) X (using (4.39 ), Lemma 4.9 and the regularity of u0 with (u0 e ) = 0) · 3 |ξ3=0 3 3 2 4 ε 2 2 ε κ (1 + α)ε w 2 κ (1 + α )ε + w 2 . ≤ T k kL (Ω) ≤ T k kL (Ω) For 5, using (4.16, 4.17) with G and F replaced by θε, and using (4.28), we ﬁnd Jε 5 ε ε ε (θ )θ 2 w 2 Jε ≤k · ∇ kL (V ) k kL (Ω) 3 2 ε ε ε ∂θi ε κT θ θ 2 + w 2 L∞(U) i L (U) L (Ω) ≤ k k k k ∂ξ 2 k k i=1 j=1 j L (U) X n X o 3 ε ε ∂θi ε + κT θ w 2 3 L∞(U) L (Ω) k k ∂ξ 2 k k i=1 3 L (U) X (using (4.37, 4.38) and Lemma 4.9)

5 5 2 2 4 ε 4 2 ε κ (1 + α )ε w 2 κ (1 + α )ε + w 2 . ≤ T k kL (Ω) ≤ T k kL (Ω) Using these bounds on i, 1 i 5, (4.47) becomes Jε ≤ ≤ d 2 3 2 ε ε 2 4 2 2 ε w 2 +2κSε w 2 κ (1 + α )ε + κ (1 + α ) w 2 . dt k kL (Ω) |∇ |L (Ω) ≤ T T k kL (Ω)

Moreover, using (4.24, 4.25, 4.415), we see that

ξ 1 − 3 3 ε ε 2 √ε 4 w t=0 L2(Ω) = θ t=0 2 κT (1 + α)ε e L2(Γ ) + l.o.t. κT (1 + α)ε . k | k k | kL (Γ2a) ≤ k k 2a ≤ Thanks to the Gronwall inequality, we ﬁnally have the estimates,

3 1 ε 4 ε 4 w 2 κ(T, α,u0, f)ε , w 2 1 κ(T, α, u0, f)ε . (4.48) k kL∞(0,T ;L (Ω)) ≤ k kL (0,T ;H (Ω)) ≤ BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 25

4.5. Proof of convergence. Using (4.40), we ﬁrst notice that uε u0 wε + θε , (pointwise in Ω (0, T )). | − |≤| | | | × Then, using (4.48) and Lemma 4.9, (1.8) follows.

5. Uniform convergence With the estimates we now have, the proof of (1.9) is quite simple. m Because we assume that m > 6 in (1.7), u0 E H and (by Sobolev embedding) 1,∞ ∈ ∩ 2 u0 Wco . Hence, both (2.1) and the hypotheses for Theorem 2.4 hold , so we can use∇ (2.1∈ , 2.2) to conclude that ε 0 ε 0 u u m , (u u ) m 1 C. − L∞(0,T ;Hco(Ω)) ∇ − L∞(0,T ;Hco− (Ω)) ≤

Then using (1.8 ), Theorem A.2, and Remark A.3, (1.9 ) follows. Remark 5.1. Since also uε u0 lies in L∞([0, T ]; W 1,∞) by Theorem 2.4, we could use the Gagliardo-Nirenberg interpolation− inequality (see, for instance, p. 314 of [8]),

2 3 ε 0 ε 0 5 ε 0 5 u u C u u 2 u u 1, , − L∞ ≤ − L − W ∞ to give uε u0 Cε3/10. This is the same rate that is obtained for m =6in(1.9); L∞ since, however,k − wek require≤ that m> 6, (1.9) always gives a better rate than this.

Appendix A. An anisotropic Agmon’s inequality In this appendix we develop a version of Agmon’s inequality in d dimensions, d =2 or 3, that is suitable for applying to anisotropic problems in which there is more control over tangential (horizontal) derivatives than over normal (vertical) derivatives. We use the notation A B to mean that A CB for some constant, C, which may depend upon the geometry≪ of an underlying domain≤ but not upon anything else. If C depends on some parameter, m, then we write A m B. Our starting point is the following simple lemma:≪ Lemma A.1. Let U be a bounded domain in Rd, d = 1, 2, 3. For any f in Hk(U), k d, ≥ 1 1 1− 2k 2k f k f 2 f k if d =1, k kL∞(U) ≪ k kL (U) k kH (U) 1 1 1− k k f k f 2 f k if d =2, k kL∞(U) ≪ k kL (U) k kH (U) 3 3 1− 2k 2k f k f 2 f k if d =3. k kL∞(U) ≪ k kL (U) k kH (U) 1/2 1/2 Proof. Combine the 1D Agmon’s inequality, f f 2 f 1 , 2D Ag- L∞(U) L (U) H (U) 1/2 1/2 k k ≪ k k k k mon’s inequality, f f 2 f 2 , or 3D Agmon’s inequality, f L∞(U) L (U) H (U) L∞(U) 1/4 3/4 k k ≪k k k k k1−kj/k j/k≪ f f with the Sobolev interpolation inequality, f j f f , k kL2(U) k kH2(U) k kH (U) ≪k k kL2(U) k kHk(U) 0 j k. ≤ ≤ 2The hypotheses in Theorem 1.1 are not the minimal ones insuring this. 26 G. GIE AND J. KELLIHER

Theorem A.2. Let Ω be a bounded domain in R3 with Cm+1-boundary, m 3, and let ≥ Γa be the tubular neighborhood of fixed width a > 0 interior to Ω. Suppose that f and m f lie in the space Hco (Ω) of Definition 2.3. Then ∇ 1 1 1 1 2 2 − 2m 2m f m,a f 2 f m f 2 + f m , k kL∞(Γa) ≪ k kL (Ω) k kHco(Ω) k kL (Ω) k∇ kHco(Ω) 3 3 1− 2m 2m h i f m,a f 2 f m . k kL∞(Ω\Γa) ≪ k kL (Ω) k kHco(Ω) Proof. We define the chart, ψ, as in the beginning of Section 4. In this chart, we can −1 define a local conormal basis, (X1,X2,X3), by Xif(x)= ∂i(f ψ)(ψ (x)), i =1, 2 and ψ 1 ◦ X f(x)= 3− ∂ (f ψ)(ψ−1(x)). We will have need, however, only for X and X . 3 1+ψ 1 3 1 2 3− ◦ It suffices to assume that f lies in C∞(Ω). Restricting ourselves to the one chart, ψ, by Lemma A.1, we have, for any ξ =(ξ1,ξ2,ξ3) in U, 1 1 1− m m f ψ(ξ) m f ψ( , ,ξ3) 2 f ψ( , ,ξ3) m . ◦ ≪ k ◦ · · kL (U0) k ◦ · · kH (U0) Applying Lemma A.1 again, this time in 1D with k = 1, gives

2 ′ ′ 2 ′ ′ f ψ( , ,ξ3) 2 = f ψ(ξ ,ξ ,ξ3) dξ dξ k ◦ · · kL (U0) ◦ 1 2 1 2 ZU0 ′ ′ ′ ′ ′ ′ f ψ(ξ1,ξ2, ) L2(0,a) f ψ(ξ1,ξ2, ) H1(0,a) dξ1 dξ2 ≪ U0 k ◦ · k k ◦ · k Z 1 1 2 2 ′ ′ 2 ′ ′ ′ ′ 2 ′ ′ f ψ(ξ ,ξ , ) 2 dξ dξ f ψ(ξ ,ξ , ) 1 dξ dξ ≤ k ◦ 1 2 · kL (0,a) 1 2 k ◦ 1 2 · kH (0,a) 1 2 U0 U0 Z 1 Z 1 2 2 2 2 2 = f ψ(ξ) dξ f ψ(ξ) + f(ψ(ξ)) ∂3ψ(ξ) dξ U | ◦ | U | ◦ | |∇ · | Z Z 1 1 2 2 f ψ(ξ) 2 J(ξ) dξ f ψ(ξ) 2 + f(ψ(ξ)) 2 J(ξ) dξ ≪a | ◦ | | | | ◦ | |∇ | | | ZU ZU = f 2 f 1 . k kL (V ) k kH (V ) The second followed because the magnitude of the Jacobian determinant, J, is ≪ bounded away from zero and ∂3ψ is bounded above. Because there are a ﬁnite number of charts on Γa, the bounds are uniform over Γa. Similarly, for all multiindices, α = (α1, α2, 0) with α m, applying Lemma A.1 with k = 1 gives | | ≤ α 2 D (f ψ)( , ,ξ3) 2 k ◦ · · kL (U0) 1 1 2 2 α 2 α 2 α 2 m,a D (f ψ) D (f ψ) + ∂3D (f ψ) ≪ U | ◦ | U | ◦ | | ◦ | Z 1 Z 1 2 2 α 2 α 2 α 2 = X f X f + D ∂3(f ψ) V | | V | | U | ◦ | Z 1 Z Z 1 2 2 Xαf 2 Xαf 2 + Xα f 2 ≪ V | | V | | V | ∇ | Zα αZ Z = X f 2 X f 1 . k kL (V ) k kH (V ) BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 27

But this is true for all α m, so | | ≤ 1/2 1/2 f ψ( , ,ξ3) m f m f 2 + f m , k ◦ · · kH (U0) ≤k kHco(V ) k kL (Ω) k∇ kHco(V ) h i where we can use the conormal Sobolev space since the only derivative in the normal m+1 m+1 direction occurs in f itself. Also, because Γ is C , ψ can be chosen to be C (Γa) and hence f ψ has∇ suﬃcient smoothness. Combining◦ these bounds we have,

1 1 1 1 1 1 2m 2 − 2m 2 − 2m 2m f m,a f 2 f 1 f m f 2 + f m k kL∞(V ) ≪ k kL (V ) k kH (V ) k kHco(V ) k kL (Ω) k∇ kHco(V ) 1 1 1 1 h 2 i 2 − 2m 2m f 2 f m f 2 + f m . ≤k kL (V ) k kHco(V ) k kL (Ω) k∇ kHco(V ) h i Summing over all the V gives

1 1 1 1 2 2 − 2m 2m f m,a f 2 f m f 2 + f m . k kL∞(Γa) ≪ k kL (Ω) k kHco(Ω) k kL (Ω) k∇ kHco(V ) h i With W = Ω Γ , Lemma A.1 gives \ a 3 3 3 3 1− 2m 2m 1− 2m 2m f m f 2 f m a f 2 f m . k kL∞(W ) ≪ k kL (W ) k kH (W ) ≪ k kL (Ω) k kHco(Ω) Combining these last two inequalities completes the proof. Remark A.3. It is easy to see that Theorem A.2 holds as well for a channel domain. It is worth comparing the inequality in Theorem A.2 with the 3D Agmon’s anisotropic inequality in Proposition 2.2 of [53], which can be written, for any f in H2(Ω), as

3 1 1 4 4 f f 2 f 2 + ∂jf 2 + ∂j∂jf 2 k kL∞(Ω) ≪k kL (Ω) k kL (Ω) k kL (Ω) k kL (Ω) j=1 (A.1) Y 3 1 4 4 f L2(Ω) f L2(Ω) + f L2(Ω) + ∆f . ≪k k k k k∇ k k k L2(Ω) In [53], the authors have some control of the Laplacian but not (directly) of the full H2 2 norm. This inequality would not work for us, however, as it includes ∂3 f. Both Theorem A.2 and the inequality in (A.1) are descendants in spirit of Solonnikov’s Theorem 4 of [48]. The proof in (A.1) uses, in part, Solonnikov’s approach. The approach we have taken is, however, more elementary and direct than that of [48]. Another type of anisotropic inequality that is not a descendant of Solonnikov’s theorem (and is not of Agmon type) is the anisotropic embedding inequality of [35] (Corollary 1 7.3), which originated in Remark 4.2 of [51], which states that for all f in H0 (Ω),

1 1 1 1 1 1 2 2 2 2 2 2 f f 2 ∂3f 2 + ∂1f 2 ∂3f 2 + f 2 ∂1∂3f 2 . k kL∞(Ω) ≪k kL (Ω) k kL (Ω) k kL (Ω) k kL (Ω) k kL (Ω) k kL (Ω) Its proof, however, is entirely diﬀerent from that of Theorem A.2 or the inequalities in [48, 53] described above. (A 3D version of it can, however, be obtained using an argument somewhat along the lines of the proof of Theorem A.2.) 28 G. GIE AND J. KELLIHER

Appendix B. Special boundary conditions For the Navier-Stokes equations in 2D, when α = κ, the Navier boundary conditions reduce to the conditions, u n = ω(u)=0([13, 39, 32]).3 Here, κ is the curvature of the boundary of a planar bounded· domain and ω(u) is the scalar curl of u. The natural extension of this observation to 3D is Lemma B.1, which involves the shape operator,4 ∂n v := = n, A ∂v ∇v the directional derivative of n in the direction, v, for any vector, v, in the tangent plane. Lemma B.1. The boundary conditions in (1.5) reduce to those in (1.6) when is the shape operator. A Proof. Let be the shape operator and let τ be any unit tangent vector. Then since the shape operatorA is symmetric, we can write (1.5) as S(uε)n τ + τ uε = 0. But as in [2], 2S(uε)n τ = (curl uε n) τ 2uε ∂n , so (1.5) becomes· A · · × · − · ∂τ ∂n (curl uε n) τ =2 τ uε =0. × · ∂τ −A ·

Appendix C. Some lemmas We assume that Ω is a bounded domain in R3 having a Lipschitz boundary, Γ. Lemma C.1. Let f be a divergence-free vector field in H2(Ω) that satisfies S(f)n =ΦonΓ in the sense of a trace, where Φ lies in H3/2(Γ). Then, for any vector field, g, in H2, we have ∆f gdx =2 S(f) S(g) dx 2 Φ g dS, − · · − · ZΩ ZΩ ZΓ where A B = a b for matrices A =(a ) and B =(b ) . · 1≤i,j≤3 ij ij ij 1≤i,j≤3 ij 1≤i,j≤3 We recall theP following classical lemma: Lemma C.2. Let u be a divergence-free vector field, of class H1(Ω)3, in a bounded domain, Ω R3, with a C2-boundary, Γ. Then, if the normal component of u vanishes on Γ, we have⊂ 1 1 2 2 u 2 κ u u , | |L (Γ) ≤ Ω| |L2(Ω)|∇ |L2(Ω) for a constant κΩ depending on the domain. Acknowledgements The authors were supported in part by NSF Grants DMS-0842408 and DMS-1009545. The authors would like to thank Frederick Wilhelm for helpful discussions on the geometry of surfaces, and Drago¸sIftimie for pointing out the issue raised in Remark 5.1.

3In these references, the relation is written α = 2κ, since 2 (u) rather than (u) is used in the (2D version of) (1.3). S S 4When an inward unit normal convention is used, the expression for contains a negative sign. A BOUNDARY LAYER WITH GENERALIZED NAVIER BOUNDARY CONDITIONS 29

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1 Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521, U.S.A. E-mail address: [email protected] E-mail address: [email protected]