RECENT PROGRESSES IN BOUNDARY LAYER THEORY

Transcription

1 RECENT PROGRESSES IN BOUNDARY LAYER THEORY GUNG-MIN GIE 1, CHANG-YEOL JUNG, AND ROGER TEMAM Abstract. In this article, we review recent progresses in boundary layer analysis of some singular perturbation problems. Using the techniques of differential geometry, an asymptotic expansion of reaction-diffusion or heat equation in a domain with curved boundary is constructed and validated in some suitable functional spaces. In addition, we investigate the effect of curvature as well as that of an ill-prepared initial data. Concerning convection-diffusion equations, the asymptotic behavior of their solutions is difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, boundary layer analysis is performed on relatively simpler domains, typically intervals, rectangles, or circles. We consider also the interior transition layers at the turning point characteristics in an interval domain and classical ordinary), characteristic parabolic) and corner elliptic) boundary layers in a rectangular domain using the technique of correctors and the tools of functional analysis. The validity of our asymptotic expansions is also established in suitable spaces. Contents 1. Introduction. Boundary layers in a curved domain in R n, n =, 4.1. Elements of differential geometry Curvilinear coordinate system adapted to the boundary Examples of the curvilinear system with some special geometries 7.. Reaction-diffusion equations in a curved domain Boundary layer analysis at order ε 9... Boundary layer analysis at order ε 1/ : The effect of curvature 1... Asymptotic expansions at arbitrary orders ε n and ε n+1/, n 15.. Parabolic equations in a curved domain Boundary layer analysis at orders ε and ε 1/... Boundary layer analysis at arbitrary orders ε n and ε n+1/, n 1... Analysis of the initial layer: The case of ill-prepared initial data 5. Convection-diffusion equations in a bounded interval with a turning point 7.1. Interior layer analysis at arbitrary order ε n f, b compatible f, b noncompatible 4 4. Convection-diffusion equations in a domain with corners rectangle) Boundary layer analysis at order ε Boundary layer analysis at arbitrary order ε n, n Concluding Remarks 54 Acknowledgements 54 References 55 Date: September, 15. Mathematics Subject Classification. 5B5, 5C, 76D1, 5K5. Key words and phrases. boundary layers, singular perturbations, curvilinear coordinates, initial layers, turning points, corner layers. 1

2 G.-M. GIE, C. JUNG, AND R. TEMAM 1. Introduction We review in this article some recent developments in the study of singularly perturbed problems, that is, initial and) boundary value problems that contain a small parameter affecting the highest order derivatives. The engineering literature on boundary layers is very vast andincludes thetheory of Prandtl[114, 115] in fluidmechanics as well as works byvon Kármán [141], [11], and others, and the beautiful experimental book of Van Dyke [7]. On the theoretical side, the study of singular perturbations and boundary layers remains very challenging, and includes the notorious problem of turbulent boundary layer [11]. Nevertheless such problems have attracted the attention of mathematicians during the last 6 years or so, and some substantial insight has been gained in such problems; see, e.g., [, 91, 17, 11, 1, 19, 14]; some of the many other relevant references will be quoted below in the introduction or when we study the corresponding problems. One classical motivation of studying singularly perturbed problems is from the so-called vanishing viscosity limit in fluid mechanics, see, e.g., [1, 15, 4, 8, 8, 8, 84, 86, 96, 1, 19, 111, 11, 15, 11]. Beside the vanishing viscosity problem which underlies this article, one can mention many singular perturbation problems, in particular in geophysical fluid mechanics, in relation with rotating fluids and Ekman layers, see, e.g., [16, 46]. Returning to the vanishing viscosity limit, we recall that the motion of viscous and inviscid fluids is modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at small viscosity as a singular perturbation of the Euler equations, a major problem, still essentially open, is the asymptotic behavior of the Navier-Stokes solutions, i.e., to verify if the Navier- Stokes solutions converge, in some function spaces, to the Euler solutions as the viscosity tends to zero. This is an outstanding fundamental problem in analysis. Because an inviscid flow of the Euler type is free to slip along the boundary while any viscous flow of the Navier- Stokes type must adhere to the boundary, boundary layers must occur at small viscosity. Of course controlling the boundary layers of singularly perturbed problems is the main key to understanding the nature of vanishing viscosity limit. Another important motivation for singular perturbations is associated with the numerical computation of solutions to those problems. In approximating solutions to singularly perturbed boundary value problems, it is well-known that a very large discrete) gradient is created near the boundary. Hence, in most classical simulations of singularly perturbed equations, a drastic mesh refinement is usually required near the boundary to obtain an accurate approximation of the solution; a fine mesh of order ε 1/, where ε is the non-dimensional viscosity, is usually suggested [16]. Departing from massive mesh refinements, some new semi-analytic methods have been proposed and successfully applied under the names of enriched spaces, extended finite element method XFEM), and generalized finite element method GFEM). The common idea is to add to the Galerkin basis or its analogue in finite differences, finite elements, or finite volumes, some specific shape functions which carry the inherent singularity of the problem. For singular perturbations, the specific shape functions are related to the boundary layer correctors computed analytically; in crack theory, the shape functions embolden the singularity at the tip of a crack. See [, 4, 16] for the XFEM method for cracks; see [14, 15] for GFEM; and see e.g. [18, 19, 1] for singular perturbations as well as the articles [65, 66, 67, 71, 74, 76] which this article covers in part and generalizes. Such methods have proven to be highly efficient without any help of mesh refinement near the boundary. In this review article, we consider several classes of singular perturbation problems in some non-classical settings. Summarizing results in the recent works [5, 6, 8, 9, 4, 41, 4, 44, 7, 14], we study, in Section, the boundary layers in a smooth domain with curved boundary; see also [56, 57, 69, 1, 1] for the case of a flat boundary. Toward this end, we first recall some elements of differential geometry and give some concrete examples of the domains under

3 RECENT PROGRESSES IN BOUNDARY LAYER THEORY consideration in Section.1. In Sections. and., boundary layers of the reaction diffusion and heat equations are respectively investigated. Here we construct an asymptotic expansion of the singularly perturbed reaction-diffusion solution or the heat solution at an arbitrary order with respect to the small diffusivity. The point of view that we systematically use in this article and others) is the utilization of correctors as proposed by J. L. Lions [91]. The corrector is in fact the solution of an equivalent of the Prandtl equation [114, 115, 19] for the problem and it accounts for the rapid variation of the functions and their normal derivatives in the boundary layer. It also corrects hence the name) the discrepancies between the boundary values of the viscous and inviscid solutions. This construction is, of course, closely related to the matching asymptotic method, see, e.g., [9,, 88]. Hence the analysis is at first informal and of a physical nature. Then the rigorous validity of our asymptotic expansions is confirmed globally in the whole domain by performing energy estimates on the difference of the diffusive solution and the proposed expansion. Our expansion at an arbitrary order with respect to the small diffusivity provides the complete structural information of the boundary layers. In Section, we discuss a class of singularly perturbed problems with a turning point in an interval domain. The literature about the analysis of singularly perturbed problems with a turningpointisnotverylarge; seehowever[7,,,68,18,17,144,145,146,147,148]. Here we follow[7, 79]. The cases where the limit problem is compatible and non-compatible with the given data are considered. With limited compatibility conditions on the data, the asymptotic expansions can be constructed only up to the order allowed by the level of compatibilities. However, using a smooth cut-off function compactly supported around the turning point, which localizes the singularities due to the non-compatible data, we obtain the asymptotic expansions up to any order. In Section 4, we consider domains with corners. Corners generate singularities which have to be corrected by boundary layers techniques, and/or they affect existing boundary layers due to the singularly perturbed nature of the equations. Of the first type, and without any small coefficient in the equation, are the singularities created at t = by the incompatible initial and boundary data see [14, 18, 17,, ]) or the singularities created by corners in a geometric domain see [51, 5] and [8, 58, 149], and the references therein). All these singularities develop already with regular differential operators, that is, differential operators with order one coefficients for the leading derivatives. Another type of problems on which we will focus, concerns the interaction of corner singularities with the boundary layers due to singularly perturbed differential operators, that is, differential operators with a small coefficient affecting the highest derivatives. The remarkable article [1] illustrates the variety and complexity of the boundary layers that occur for a singularly perturbed elliptic differential operator of the second order in a square. These problems have been studied in a variety of contexts in, e.g., [94, 95] and in, e.g., the articles [6, 4, 44, 75] on which this section is partly based. In this section, we study the asymptotic behavior of solutions to a convection-diffusion equation in a rectangular domain Ω. The Dirichlet boundary condition is then supplemented along the edges and at the corners. The elliptic corner layers are introduced to handle the interaction of the parabolic and classical boundary layers at the four corners. Our analysis simplifies that of [1] by minimizing the construction of boundary layers needed and extends the asymptotic expansions of [1] up to any order. In summary aiming to study the asymptotic behavior of the solutions to some singular perturbation problems, we construct the boundary layer or interior layer) correctors and obtain the full structural information of 1) the boundary layers of the reaction-diffusion equation in a smooth curved domain ) the boundary and initial layers of the heat equation in a smooth curved domain ) the interior layers of the convection-diffusion equation with turning points in an interval domain

4 4 G.-M. GIE, C. JUNG, AND R. TEMAM 4) the interaction of the boundary and corner layers for the convection-diffusion equation in a rectangular domain This article is not meant to beexhaustive in any way. Besides thetopics of this article, more subjects will be covered in a forthcoming book [7]. Not covered in this article is the case of convection-diffusion equations where the limit problem is hyperbolic. This project was studied from various point of views in, e.g., [5, 45, 85, 89, 16, 18]. Our contributions on the subject appear in [77, 78], in the review article on the subject)[7], andin [66] for some computational aspects. Other problems not discussed in this article include many problems of classical and geophysical fluidflows withsmallviscosity; see, e.g., [16,4, 7,4, 6,8, 4,41, 4, 44,49, 5, 54, 55, 56, 57, 59, 8, 81, 8, 9, 94, 95, 96, 1, 1, 1, 17, 116, 1, 1, 1, 1, 14, 15]; electromagnetism, acoustic, and the Helmholtz equation [, 9, 117, 118]; see also in e.g. [5] and in the references therein the issue of boundary layers for hyperbolic equations; the issue already mentioned of the numerical approximation of singularly perturbed problems, in particular in the context of enriched spaces [18, 19, 6, 61, 6, 64, 65, 66, 67, 71, 74, 76, 85, 14, 15, 16, 11, 119, 16, 17]; the important subject of vanishing viscosity for the Hamilton-Jacobi equations which is a whole subject by itself; see among many references, [1, 9]. See also various perspectives in singular perturbations and boundary layers in [1, 1, 1, 14, 5, 6, 1, 5, 47, 48, 9, 99, 11, 1, 17, 14, 14]. Let us mention also the article [97] and the subsequent book[98] which offer totally new perspectives in boundary layer separation. Finally let us mention that we considered convection-diffusion equations in a circular domain where two characteristic points appear. The singular behaviors may occur at these points depending on the behavior of the given data. However, this case is not covered here and the reader is referred to the review article [7]. Convection-diffusion equations in general curved domains will be studied elsewhere.. Boundary layers in a curved domain in R n, n =,.1. Elements of differential geometry. The domain Ω is assumed to be bounded and smooth in R. In this section, we construct an orthogonal curvilinear coordinate system adapted to the boundary Γ := Ω and write some differential operators with respect to the curvilinear system. Any smooth and bounded domain in R can be handled in a similar but easier) manner by suppressing the second tangential variable as in Section.1. below. More information about the geometry and construction of a special coordinate system is well-described in, e.g., [6,, 87] as well as [4, 4] Curvilinear coordinate system adapted to the boundary. We let x = x 1,x,x ) denote the Cartesian coordinates of a point in Ω R. To avoid some technical difficulties of geometry, we assume that the smooth boundary Γ is a D compact manifold in R having no umbilical points the two principal curvatures are different at each point on Γ). Concerning general classes of domains including isolated) umbilical points, the difficulties and some proper treatments are explained in, e.g., Section 4 in [4]. Then one can construct a curvilinear system globally on Γ in which the metric tensor is diagonal and the coordinate lines at each point are parallel to the principal directions. Such a coordinate system is called the principal curvature coordinate system. Inside of a tubular neighborhood Ω δ with a small, but fixed, width δ >, we extend the principal curvature coordinates on Γ in the direction of n where n is the outer unit normal on Γ. As a result, we obtain a triply orthogonal coordinate system ξ in R ξ, such that Ω δ is diffeomorphic to Ω δ,ξ := {ξ = ξ, ξ ) R ξ ξ = ξ 1, ξ ) ω ξ, < ξ < δ},.1)

5 RECENT PROGRESSES IN BOUNDARY LAYER THEORY 5 for some bounded set ω ξ in R ξ. The normal component ξ measures the distance from a point in Ω δ to Γ and hence we write the boundary Γ in the form, Γ = {ξ R ξ ξ = ξ 1,ξ ) ω ξ, ξ = }..) The need to introduce such tubular domains near the boundary comes from the fact that the boundary layer phenomena are local near the boundary in the direction orthogonal to the boundary but are otherwise nonlocal in the tangential directions. Using the covariant basis g i = x/ ξ i, 1 i, we write the metric tensor of ξ, g ij ) 1 i,j := g i g j )1 i,j [ 1 κ1 ξ ] )ξ g11 ξ ) = [ 1 κ ξ ] )ξ g ξ ), 1 where κ i ξ ), i = 1,, is the principal curvature on Γ, g i, i = 1,, are the covariant basis of the principal curvature coordinate system on Γ, and g ii = g i g i. By the choice of a small thickness δ >, we have.) gξ) := detg ij ) 1 i,j > for all ξ in the closure of Ω δ,ξ..4) We introduce the normalized covariant vectors, and set e i = g i g i, 1 i,.5) h i ξ) = g ii, i = 1,, hξ) = g..6) The function hξ) > is the magnitude of the Jacobian determinant for the transformation from x in Ω δ to ξ in Ω δ,ξ. Similarly the function hξ,) > is the magnitude of the Jacobian determinant for the transformation from x on Γ to ξ in ω ξ. For a smooth scalar function v, defined at least in Ω δ, we write the gradient of v in the ξ variable, v = The Laplacian of v is given in the form, where Sv = i=1, i=1 1 v e i + v e..7) h i ξ i ξ v = Sv +Lv + v ξ,.8) 1 h h ξ i h i v ), Lv = 1 h v..9) ξ i h ξ ξ A vector valued function v, defined at least in Ω δ, can be written in the curvilinear system, e 1, e, e as v = v i ξ)e i..1) i=1 One can classically express the divergence and curl operators acting on v in the ξ variable, divv = 1 h i=1 h v i )+ 1 hv ),.11) ξ i h i h ξ

6 6 G.-M. GIE, C. JUNG, AND R. TEMAM and where curlv = h { 1 v h v ) } e 1 + h { h1 v 1 ) v } e h ξ ξ h ξ ξ { h v ) h 1v 1 ) } e. h ξ 1 ξ The Laplacian 1 of v is given in the form, v = i=1 S i v +Lv i + v i ξ linear combination of tangential derivatives S i v = of v j, 1 j, in ξ, up to order Lv i = proportional to v ) i. ξ.1) ) e i,.1) ),.14) Remark.1. The coefficients of S i, 1 i, and L are multiples of h, 1/h, h i, 1/h i, i = 1, and their derivatives. Thanks to.4), all these quantities are well-defined at least in Ω δ,ξ. Considering smooth vector fields in Ω δ of the form, v = v i ξ)e i, w = i=1 w i ξ)e i, the covariant derivative of w in the direction v, which is denoted by v w and gives v w in the Cartesian coordinate system, can be written in the ξ variable, where v w = i=1 P i v,w) = i=1 { P i w } i v,w)+v +Q i v,w)+r i v,w) e i,.15) ξ j=1 Q i v,w) = 1 h j v j w i ξ j, 1 i, 1 hi v i h ) i v i w i, i = 1,, h 1 h ξ i ξ i j=1 1 h j h j ξ v j w j, i =,.16) R i v,w) = 1 h i h i ξ v i w, i = 1,, R v,w) =. Remark.. The Q i v,w) and R i v,w), 1 i, are related to the Christoffel symbols of the second kind that reflect the twisting effects of the curvilinear system. 1 The Laplacian Laplace-Beltrami operator) of a vector field is defined by the identity v = divv) curlcurlv); see, e.g., [, 87, 6]. We know that other definitions of the Laplacian of a vector, which possess different properties, are used in different contexts.

7 RECENT PROGRESSES IN BOUNDARY LAYER THEORY Examples of the curvilinear system with some special geometries. We present some examples of the curvilinear coordinates discussed in Section.1.1 when a certain symmetry is imposed to the domain Ω. All the analysis below in Sections. and. is valid and can be made explicit) by using the Lamé coefficients h i, 1 i in Section.1.1 replaced by the corresponding expressions in this section. Polar coordinate system We consider the domain Ω in R as a disk with radius R >, Ω = { x R x 1 +x < R }..17) Using the polar coordinates, we construct a curvilinear system adapted to the boundary Γ by setting, x = R ξ )cosξ 1, R ξ )sinξ 1 ),.18) for ξ = ξ 1,ξ ) [,π) [,R), i.e., all points x in Ω\,). Here we suppressed the second tangential variable ξ to use ξ as the normal variable as appearing in Section.1.1. Differentiating x in.18) with respect to the variable ξ, we write the covariant basis, { g1 = ) R ξ )sinξ 1, R ξ )cosξ 1,.19) g = cosξ 1, sinξ 1 ). Using the orthogonality of {g i } i=1,, we write the metric tensor, ) R ξ ) g i,j ) i,j=1, =: g i g j ) i,j=1, =..) 1 Introducing the normalized covariant vectors, e i = g i / g i, i = 1,, we find the positive Lamé coefficients, h 1 ξ ) = R ξ, h = 1..1) The function hξ 1,ξ ) := h 1 ξ ) is the magnitude of the Jacobian determinant for the transformation from x to ξ, and it is positive for all ξ in Ω\,). Cylindrical coordinate system The domain Ω is given as a cylinder in R, Ω = { x R x1 R, x +x < R }..) Using the cylindrical coordinates, we construct a curvilinear system adapted to the boundary Γ by setting, x = ξ, R ξ )cosξ 1, R ξ )sinξ 1 ),.) for ξ = ξ 1,ξ,ξ ) [,π) R [,R), i.e., all points x in Ω\{x 1 = }. Differentiating x in.) with respect to ξ, we write the covariant basis, g 1 = ), R ξ )sinξ 1, R ξ )cosξ 1, g = 1,, ), g =, cosξ 1, sinξ 1 ). and.4) The metric tensor and positive Lamé coefficients are given in the form, R ξ ) g i,j ) 1 i,j =: g i g j ) 1 i,j = 1,.5) 1 h 1 ξ ) = R ξ, h = h = 1..6)

8 8 G.-M. GIE, C. JUNG, AND R. TEMAM The function hξ 1,ξ ) := h 1 ξ ) is the magnitude of the Jacobian determinant for the transformation from x to ξ, and it is positive for all ξ in Ω\{x 1 = }. Toroidal coordinate system We consider the domain Ω enclosed by a toroidal surface Γ described by Γξ 1,ξ ) := a+bcosξ 1 )cosξ, a+bcosξ 1 )sinξ, bsinξ 1 ), ξ1,ξ ) [,π),.7) for fixed < b < a. Setting ξ as the distance from a point inside of Ω to the toroidal surface Γ in the direction of n on Γ), we construct a curvilinear system ξ = ξ 1,ξ,ξ ) via the mapping, x = a+b ξ )cosξ 1 )cosξ, a+b ξ )cosξ 1 )sinξ, b ξ )sinξ 1 ),.8) for any point x in the closure of Ω, except for those along the circle, C sing. = {x R x 1 +x = a and x = }..9) Differentiating x in.8) with respect to ξ, we find the covariant basis, g 1 = b ξ ) ) sinξ 1 cosξ, sinξ 1 sinξ, cosξ 1, g = ) a+bcosξ 1 ) cosξ 1 )ξ sinξ, cosξ, ), g =.) ) cosξ 1 cosξ, cosξ 1 sinξ, sinξ 1. Using the orthogonality of {g i } 1 i, we write the metric tensor and positive Lamé coefficients in the form, b ξ ) g i,j ) 1 i,j =: g i g j ) 1 i,j = ) a+b ξ )cosξ 1,.1) 1 and h 1 ξ ) = b ξ, h ξ 1,ξ ) = a+b ξ )cosξ 1, h = 1..) The function hξ 1,ξ ) := h 1 ξ )h ξ 1,ξ ) is the magnitude of the Jacobian determinant for the transformation from x to ξ, and it is positive for all ξ in Ω\C sing.. Remark.. Spherical coordinate system Considering a ball in R, the spherical coordinate system is one of the natural choices to locate a point on the sphere, limiting the ball, but it contains some coordinate singularities at the north and south poles of the sphere and this issue is well-known in atmospheric sciences. To resolve the issue of the pole singularities, one can construct smooth coordinate systems locally away from the north and south poles and glue them properly in the mid-latitude regions around the equator. We will not discuss this case in details... Reaction-diffusion equations in a curved domain. We consider a reaction-diffusion equation, { ε u ε +u ε = f, in Ω, u ε =, on Γ,.) where Ω is a bounded and smooth domain in R as discussed in Section.1 and ε is a small and strictly positive parameter. We expect that a boundary layer occurs near Γ as ε tends to because the equation.) 1 formally converges to u = f, in Ω,.4) but f may not vanish on Γ in general.

9 RECENT PROGRESSES IN BOUNDARY LAYER THEORY 9 We aim to study the asymptotic behavior of a solution u ε to.) at a small ε especially when the boundary Γ is curved. As we will see below in Theorems.1,., and., the traditional asymptotic expansion in powers of the small parameter ε has to be modified by adding terms of order ε j+1/ in the expansion. In fact these new terms are necessitated by the geometric effect curvature) of a curved boundary...1. Boundary layer analysis at order ε. In this section, we construct an asymptotic expansion at order ε of u ε, solution of.), in the form, u ε = u +θ,.5) where u = f given in.4) and θ is a corrector function that we will determine below. As we shall see below, the main role of θ is to balance the discrepancy of u ε and u on the boundary Γ. To define a corrector θ, we formally insert θ = u ε u into the difference of the equations.) 1 and.4). Introducing a stretched variable ξ = ξ /ε α, α >, and using.8) with.9), we perform the matching asymptotics for the difference equation with respect to a small ε. Then we find that a proper scaling for the stretched variable is and that the asymptotic equation for the corrector θ is ξ = ξ ε,.6) θ ξ +θ =, at least in Ω δ..7) Tomaketheequation.7)aboveusefulinallofΩ,wefirstdefineanexponentiallydecaying function θ in the half space, ξ, as a solution of ε θ ξ + θ =, < ξ <, θ = u.8), at ξ =, θ, as ξ. The explicit expression of θ is given by Using the θ in.9), we define a corrector θ in the form, where σ = σξ ) is a cut-off function of class C such that { 1, ξ δ/, σξ ) =, ξ δ/, where δ > is the small) fixed thickness defined in.1). The equation for θ reads ε θ ξ +θ = ε θ = u, on Γ. Using.9) and.4), we observe that θ ξ) = u ξ,)e ξ ε..9) θ ξ) := θ ξ)σξ ),.4) σ θ +σ θ ξ ), in Ω,.41).4) right-hand side of.4) 1 ) = e.s.t.,.4)

10 1 G.-M. GIE, C. JUNG, AND R. TEMAM where e.s.t. denotes a term that is exponentially small with respect to a small parameter ε in any of the usual norms in Ω, e.g., C s Ω) or H s Ω). To derive some useful estimates on the corrector θ, we recall an elementary lemma below: Lemma.1. For any 1 p and q, we have ξ ) qe ξ εl κε p. 1.44) ε p, ) Using.9),.4), and Lemma.1, we find that ) ξ q k+m θ ε ξi k κε 1 p m ), ξ q k+m θ ξm L p ω ξ R + ) ε ξi k κε 1 p m,.45) ξm L p Ω) for i = 1 or, 1 p, q, and k,m. We define the difference between the diffusive solution u ε and its asymptotic expansion.5) in the form, w ε, := u ε u +θ )..46) In the theorem below, we state and prove the validity of the asymptotic expansion.5) as well as the convergence of u ε to u : Theorem.1. Assuming that the data f belongs to {f H Ω), f Γ W, Γ)}, the difference w ε, between the diffusive solution u ε and its asymptotic expansion of order ε, see.46)), vanishes as the diffusivity parameter ε tends to zero in the sense that w ε, H m Ω) κε 4 m, m =,1,.47) for a constant κ depending on the data, but independent of ε. Moreover, as ε tends to zero, u ε converges to the limit solution u in the sense that Furthermore, we have lim ε which expresses the fact that u ε u, ϕ ξ )LΩ) =, ϕ ξ lim ε in the sense of weak convergence of bounded measures on Ω. u ε u L Ω) κε ) ) L Ω) u, ϕ ) L Γ), ϕ C Ω),.49) u ε ξ = u ξ u,)δ Γ,.5) Proof. Using.8),.),.4),.4), and.4), we write the equation for w ε,, { ε wε, +w ε, = ε u +R +e.s.t., in Ω, where w ε, =, on Γ,.51) R = εsθ +εlθ..5) Thanks to.9) and.45), we notice that R L Ω) κε θ +κε θ L L Ω) ξ 4. Ω) κε.5) i=1 ξ i Here δγ is used to denote the delta measure on Γ and it should not be confused with the small ) number δ used at other places in the text.

11 RECENT PROGRESSES IN BOUNDARY LAYER THEORY 11 Hence, multiplying.51) by w ε,, integrating over Ω, and integrating by parts, we find that ] ε w ε, L Ω) + w ε, L Ω) [ε u L Ω) + R L Ω) +e.s.t. w ε, L Ω) κε u L Ω) +κ R L Ω) + 1 w ε, L Ω).54) κε + 1 w ε, L Ω). Then we deduce that w ε, L Ω) κε 4, w ε, L Ω) κε 1 4,.55) and this implies.47). Thanks to.45),.48) follows from.47) with m =. To prove.49), we infer from.47) that u ε u + θ ) ), ϕ ξ ξ ξ L Ω) 4, κε1 ϕ C Ω)..56) Using.9)-.41), we write θ, ϕ ξ )LΩ) = θ ) σ, ϕ ξ L Ω) + θ σ, ϕ ) L Ω)..57) We observe from.9) and.41) that the second term on the right-hand side of.57) is an e.s.t.. Hence we notice from.56) that u ε u ) lim, ϕ ε ξ )LΩ) =, ϕ ξ + lim θ ) σ, ϕ L Ω) ε ξ, ϕ C Ω),.58) L Ω) if the limit on the right-hand side exists. We introduce the following approximation of the δ-measure on R: η ε x) = 1 x ε ) η, where ηx) = 1 ε e x,.59) so that η L 1 R) = η ε L 1 R) = 1 for all ε >. Then we write θ ) σ, ϕ ξ = θ ) σϕhdξ dξ L Ω) ξ ω ξ = = ω ξ ω ξ u ξ 1,) e ξ ε σϕhdξ )dξ ε u ξ,) η ε ξ )σ ξ )ϕξ, ξ )hξ, ξ )dξ )dξ. R.6) Since η ε is an approximation of the δ-measure, the inner integral in ξ converges to σϕh evaluated at ξ = as ε tends to. Using this fact and that σ) = 1, we deduce from.6) that θ ) lim σ, ϕ ε ξ = u ξ,)ϕξ,)hξ,)dξ = u, ϕ ), ϕ C Ω). L Ω) L ω Γ) ξ.61) Hence.49) follows from.58) and.61), and now the proof of Theorem.1 is complete.

12 1 G.-M. GIE, C. JUNG, AND R. TEMAM Remark.4. The weak convergence result.49) in the space of Radon measures is borrowed from [44] in which the authors verified the so-called vorticity accumulation on the boundary of the solutions to the Navier-Stokes at small viscosity. This interesting phenomena was established earlier in [95] by using a method different from that used in [44]. In a closely related article [8], the author proved the equivalence of the vanishing viscosity limit and the vorticity accumulation in a weaker sense, that is, an analogue of.49) with the test functions of class C Ω) H 1 Ω.... Boundary layer analysis at order ε 1/ : The effect of curvature. Comparing to the case of a domain with flat boundaries, the convergence results in.47) are away from the optimal rates by a factor of ε 1/4 ; see.6) below. To understand what causes this loss of accuracy, we first notice from.5) and.5) that the bounds in.47) are determined by the L norm of the term, εlθ,.6) which is created solely by the curvature of the boundary. In fact, when the boundary is flat, one can construct an orthogonal coordinates in Ω near Γ by taking, at each value of ξ, the identical copy of the surface coordinates on Γ. By doing so, the matrix tensor in.) becomes independent of the normal variable ξ, and hence in the expression.8) of Laplacian, the term L vanishes thanks to h/ ξ =. Moreover, in this flat-boundary case, one can improve the convergence result.47) 1, to u ε u +θ ) H m Ω) κε 1 m, m =,1,.6) because the term including θ / ξ vanishes and hence the bound in.5) becomes κε. In this section, we will construct a corrector θ 1/ to resolve the effect of curvature, i.e., the dominant) error in.6), so that we improve the convergence results in.47) 1, to those as in.6) by adding θ 1/ in the asymptotic expansion of u ε. Noticing from.9) and.4) that εlθ = εcξ )e ξ ε +e.s.t., we propose an asymptotic expansion of u ε at the order ε 1/ in the form, u ε = u +θ +ε 1 θ 1..64) Here the second corrector θ 1/ will be constructed below as an approximate solution of θ 1 ξ +θ 1 = εlθ, at least in Ω δ..65) The natural boundary condition for θ 1/ is θ 1 =, on Γ, because the discrepancy of u ε and u on Γ is already taken care of by introducing θ in the expansion. For any smooth function v, we define L v = 1 h ξ = h ξ v..66) ξ = ξ Then, usingthe Taylor expansions of 1/h and h/ ξ in ξ at ξ =, we notice that, pointwise: Lv L v = 1 h 1 h ) ξ v h ξ h ξ = ξ = ξ κε1 ξ v ε ξ..67)

13 RECENT PROGRESSES IN BOUNDARY LAYER THEORY 1 Using.65) and.66), we define an exponentially decaying function θ 1/ in the half space, ξ, as a solution of ε θ1 ξ + θ 1 = εl θ, < ξ <, θ 1 =, at ξ =, θ 1, as ξ..68) Here using.9) and.66), we set εl θ = u ξ,) h 1 h ξ ξ ξ = ξ = e ε ;.69) this is the leading order term in small ε of the right-hand side of.65). To find the explicit expression of the solution θ 1/ for.68), we recall an elementary lemma below: Lemma.. A particular solution of α d F x ) je j dx x)+fx) = β α, x α j N, α,β j R\{},.7) is given by F = F α x) = α j F 1 x/α) where, for x = x/α, [ j+1 F 1 x) = β j j+ k=1 ] j! ) k x e x..71) k! More generally, if the right-hand side of.7) is of the form P n x)e x where P n x) is a polynomial in x of degree n, then.7) has a particular solution of the form P n+1 x)e x, with P n, P n+1 independent of α. Thanks to Lemma., we find the solution θ 1/ of.68), θ ξ) 1 = 1 u ξ,) 1 h ξ ξ e ξ ε..7) h ξ = ξ = ε Using the cut-off function σ in.41), we define the corrector θ 1/ in the form, The equation for θ 1/ reads ε θ 1 ξ +θ 1 = εσl θ ε θ 1 =, on Γ. Using.7),.7), and Lemma.1, we find that ) ξ q k+m θ1 ε ξi k κε 1 p m, ξm L p ω ξ R + ) for i = 1 or, 1 p, q, and k,m. θ 1 ξ) := θ 1 ξ)σξ )..7) σ θ1 +σ θ 1 ξ ξ ε ) q k+m θ 1 ξ k i ξm ), in Ω,.74) L p Ω) κε 1 p m,.75) We define the difference between u ε and the asymptotic expansion at order ε 1/ as w ε, 1 := u ε u +θ +ε 1 θ 1 )..76) Now we state and prove the validity of the asymptotic expansion.64) which improves.47) in Theorem.1:

14 14 G.-M. GIE, C. JUNG, AND R. TEMAM Theorem.. Assuming that the data f belongs to {f H Ω), f Γ W, Γ)}, the difference w ε,1/ between the diffusive solution u ε and its asymptotic expansion at order ε 1/ vanishes or is bounded) as the diffusivity parameter ε tends to zero in the sense that wε, 1 H m Ω) κε1 m, m =,1,,.77) for a constant κ depending on the data, but independent of ε. Proof. We notice from.9),.4),.7), and.7) that right-hand side of.74) 1 ) = εl θ + ε θ L σ ε σ θ1 +σ θ 1 ) ξ = εl θ +e.s.t...78) Here we used the fact that the exponentially decaying function e ξ / ε multiplied by 1 σ) or its derivative at any order is an e.s.t.. Using.8),.),.4),.4),.4),.74), and.78), we writethe equation for w ε,1/, where ε w ε, 1 +w ε, 1 R1 w ε, 1 = ε u +R1 +e.s.t., in Ω, =, on Γ,.79) = εsθ +εl L )θ +ε Sθ 1 +ε Lθ 1..8) Thanks to.9),.45),.67), and.75), we notice that R1 L Ω) κε θ L Ω) +κε ξ θ L ε ξ Ω) +κε θ 1 L Ω) +κε θ1 L ξ Ω) ξ i κε 5 4,.81) where i = 1 or. Multiplying.79) by w ε,1/, integrating over Ω, and integrating by parts, we find that ] ε w ε, 1 L Ω) + w ε, 1 L Ω) [ε u L Ω) + R1 L Ω) +e.s.t. w ε, 1 L Ω) ξ i κε u L Ω) +κ R1 L Ω) + 1 w ε, 1 L Ω).8) κε + 1 w ε, 1 L Ω). Then we deduce that w ε, 1 L Ω) κε, w ε, 1 L Ω) κε 1,.8) and this implies.77) with m =,1. To verify.77) with m =, we infer from.79) 1,.81), and.8) that w ε, 1 L Ω) ε 1 w ε, 1 L Ω) + u L Ω) +ε 1 R1 L Ω) κ..84) Thanks to the regularity theory of elliptic equations,.77) with m = follows from.84) because of.79).

15 RECENT PROGRESSES IN BOUNDARY LAYER THEORY Asymptotic expansions at arbitrary orders ε n and ε n+1/, n. To extend the convergence results of the diffusive solution u ε to.) in Theorems.1 and., we construct below asymptotic expansions u ε n and u ε n+1/ of uε at arbitrary orders n and n+1/, n, in the form, n u ε n = ε j u j +ε j θ j) n 1 + ε j+1 θ j+1, u ε n+ 1 = n ) ε j u j +ε j θ j +ε j+1 θ j+1..85) Here the u j correspond to the external expansion outside of the boundary layer) and the correctors θ j and θ j+1/ correspond to the inner expansion inside the boundary layer). To obtain the external expansion of u ε, we formally insert the external expansion u ε = εj u j into.) and write n ε j+1 u j +ε j u j) = f..86) By matching the terms of the same order ε j, we write each u j in terms of the data f: u = f, u j = u j 1 = j f, 1 j n..87) Note that generally the u j s, j n, do not necessarily vanish on the boundary Γ. In fact the discrepancy between u ε and n εj u j on Γ creates the boundary layers near Γ. To balance the discrepancy between u ε and the external expansion at order n, we introduce the inner expansion near Γ in the form, n n u ε ε j u j = ε j θ j ) +ε j+1 θ j+1, at least near Γ..88) As we will see below, the corrector θ j, j n, balances the discrepancy on Γ between u ε and the proposed external expansion, which is caused by the term u j. Then, at each order ε j, j n, an additional corrector θ j+1/ is introduced in the inner expansion to handle the geometry of the curved boundary. As it appears in Theorems. and. below, adding the corrector θ j+1/ in the expansion ensures the optimal convergence rate at each order ε j, j n. By matching the terms of the same order ε j on Γ, we deduce from.88) the boundary condition for each θ j : θ j = u j, θ j+1 =, on Γ, j n..89) To find suitable equations for θ j and θ j+1/, j n, we use.) 1 and.87) as well as.88), and write, n ε ε j θ j +ε j+1 θ j+ ) 1 + ε j θ j ) +ε j+1 θ j+ 1 =, at least near Γ..9) Recalling that the size of the boundary layers for the problem.) is of order ε 1/, we use below the stretched variable ξ = ξ / ε as in.6). To describe the dependency of the Laplacian on the diffusivity parameter ε, we introduce the Taylor expansion of a smooth function in the normal variable ξ, φξ,ξ ) = φξ, ε ξ ) = ε j ξj φ j, φ C ω ξ [, ) ),.91)

16 16 G.-M. GIE, C. JUNG, AND R. TEMAM where φ j := 1 j φ ξ,), j..9) j! ξ j Using this form of the Taylor expansion for h, 1/h, h/ ξ, and 1/h i, i = 1,, we write the operators S and L in.9) as where Sj Lj = i=1, = S = j 1 +j =j, j 1,j ) N j 1 +j +j =j, j 1,j,j ) N ε j ξj Sj, L = 1 h) { 1 h) j j 1 ξ i 1 h )j 1 h ) j = ε 1 ξ The operators S j/ and L j/, j, are well-defined if ε j ξj Lj,.9) h i j 1 +j =j, j 1,j ) N ) }, j ξ i 1 h )j 1 h ) j. ξ.94) Ω is of class C j+..95) Each S j/, j, is a tangential differential operator near Γ and the L j/ are proportional to / ξ = ε 1/ / ξ. Hence the S j/ and L j/ at each j are respectively of order ε and ε 1/ with respect to the small ε. Remark.5. We use the stretched variable ξ to weight the different terms in the equation.9). Otherwise in the analysis above and below) we generally revert to the initial variable ξ. Remark.6. As explained in Section.., if the boundary of a domain is flat, the operator L in.9) is identically zero and, in this case, the correctors θ j+1/, j n, are not required to derive the optimal estimates in Theorems. and. below. Using.9),.9), and.94), wecollect all terms of order ε j in.9) and findtheequation of θ j and θ j+1/, j : where ε θ j+d ξ j fε j θ) := +θ j+d = f j+d ε θ), d =,1, at least in Ω δ,.96) k= j 1 ε k ξ k Sk θ j 1 k + k= j 1 f j+1 ε θ) := ε k ξ k Sk θ j 1 k + k= ε 1 k ξ k Lk θ j 1 k, j ε 1 k ξ k Lk θ j k. k=.97) The equations above with n = are identical to those in.7) and.65). Modifying the equations.96) and.97), and using the boundary conditions.89), we define the exponentially decaying functions θ j and θ j+1/, j, as the solutions of ε θj ξ + θ j = fε θ), j < ξ <, θ j = u j.98), at ξ =, θ j, as ξ,

17 RECENT PROGRESSES IN BOUNDARY LAYER THEORY 17 and where ε θj+ 1 ξ + θ n+1 = f j+1 ε θ), < ξ <, θ j+1 =, at ξ =, θ j+1, as ξ,.99) f j+d ε θ) := the right-hand side of.97) with θ replaced by θ), d =,1..1) The equations above with j = are identical to.8) and.68). Thanks to Lemma., we find the structure and not the explicit expression) of the θ j and θ j+1/, j, and this is sufficient for us to perform the error analysis later on: Lemma.. The solutions θ j and θ j+1/, j, of the equations.98) and.99) are of the form, θ j ξ ξ) = P j )exp ξ ), θj+ 1 ξ ξ) = P j+1 )exp ξ ),.11) ε ε ε ε where P k ξ / ε) is a polynomial of order k in ξ / ε whose coefficients are smooth function of ξ independent of ε. Proof. We proceed by induction on j. Thanks to.9) and.7), we first notice that.11) holds true for j =. Now, we assume that.11) holds for j l 1. Then, to prove that.11) 1 is true when j is equal to l, we consider the equation.98) with j replaced by l. With the inductive assumption, we observe that f l ε θ) = l k= l 1 ε k ξ k Sj θ l 1 k + = P l 1 ξ ε )exp k= ξ ε ). ε 1 k ξ k Lk θ l 1 k.1) Then, using Lemma., we obtain a particular solution of.98) with j replaced by l: θp l ξ = P l )exp ξ )..1) ε ε Therefore.11) 1 holds true for j = l, since the homogeneous solution of.98) reads θh l = l fξ,)exp ξ ). ε Because.11) can be proved in the same way, the proof is now complete. Using the cut-off function σ in.41), we now definethe correctors θ j and θ j+1/, j n, in the form, θ j+d ξ) := θ j+d ξ)σξ ), d =,1..14) We deduce from Lemma.1,.11), and.14) that ) ξ q k+m θj+ d ε ξi k κε 1 p m ), ξ q k+m θ j+d ξm L p ω ξ R + ) ε ξi k κε 1 p m,.15) ξm L p Ω) for d = 1 or, j n, i = 1 or, 1 p, q, and k,m.

18 18 G.-M. GIE, C. JUNG, AND R. TEMAM Using.98),.99),.11) and.14), we write the equation for θ j and θ j+1/ as ε θ j ξ +θ j = fεθ)+e.s.t., j in Ω,.16) θ j = u j, on Γ, and ε θ j+1 ξ +θ j+1 = f j+1 ε θ) + e.s.t., in Ω, θ j+1 =, on Γ..17) We introduce the remainders at order ε n and ε n+1/, n, in the form, w ε,n+ d := u ε u ε, d =,1,.18) n+ d where the asymptotic expansion u ε n+d/, d =,1, of uε is given in.85). Now we state and prove the validity of the asymptotic expansion as a generalization of Theorems.1 and.: Theorem.. Assuming that the data f belongs to {f H n+ Ω), f Γ W n+, Γ)}, the difference w ε,n+d/ between the diffusive solution u ε and its asymptotic expansion of order ε n+d/, d =,1 and n, satisfies H m Ω) κεn++d 4 m, m =,1,,.19) w ε,n+ d for a constant κ depending on the data, but independent of ε. Proof. Using.),.87), and.18), we write the equations for w ε,n+d/, d =,1, N, where ε w ε,n+ d +w ε,n+ d R n+ d = w ε,n+ d n+d = ε n+1 u n +R n+ d, in Ω, =, on Γ,.11) ε θ j θ), j d =,1..111) We multiply the equation.11) 1 by w ε,n+d/, integrate over Ω, and integrate by parts to find ε w ε,n+ d + 1 w L Ω) ε,n+ d L Ω) εn+ u n L Ω) + Rn+ d..11) L Ω) Using the expression of the Laplacian in.8),.9), and.9), and the equations of the correctors in.16) and.17), we notice that R n = n ε j +1{ n j S n 1 + k= ε j +1{ n j 1 L k= ξ k } j Sk θ +ε n+1 Sθ n 1 +ε n+1 Sθ n ξ k } j Lk θ +ε n+1 Lθ n +e.s.t.,.11)

19 and R n+ 1 = RECENT PROGRESSES IN BOUNDARY LAYER THEORY 19 n 1 + n ε j +1{ n j 1 S k= ε n +1{ n j L k= ξ k Sk ξ k d Lk } θ j +ε n+1 Sθ n +ε n+ Sθ n+1 } θ j +ε n+ Lθ n ) We recall that S and the S k/ are tangential differential operators, and that L and the L k/ are proportional to / ξ. Hence, using.15), we find that L R n {ε κ n+1 Ω) n k= ε 1 ξ ) n k 1Sn k 1 +ε n+1 Sθ n n 1 ) L Ω) +εn+1 ε 1 n kln ξ k k= } +ε n+1 Lθ n L Ω) θ k +ε n+1 Sθ n 1 L Ω) L Ω) θ k L Ω).115) and R n+ 1 κε n+ 4, { n 1 κ ε n+1 L Ω) k= ε 1 ξ ) n ksn k +ε n+ Sθ n+1 L Ω) +εn+ +ε n+ Lθ n+1 L Ω) } θ k L Ω) n k= +ε n+1 Sθ n L Ω) ε 1 ξ ) n k+1ln k +1 θ k L Ω) κε n ) Then.19) with m =,1 follows from.11),.115), and.116). To verify.19) with m =, we infer from.19) with m =,1,.11) 1,.115), and.116) that w ε,n+ d L Ω) ε 1 w ε,n+ d L Ω) +εn u n L Ω)+ε 1 R n+ d L Ω) κεn+d 1 4,.117) for d =,1. Thanks to the regularity theory of elliptic equations,.19) with m = follows from.117) and.11)... Parabolic equations in a curved domain. We consider the heat equation in a bounded smooth domain Ω of R, u ε t ε uε = f, in Ω,T), u ε =, on Γ,T), u ε t= = u, in Ω,.118) where f and u are given smooth data, T > is an arbitrary but fixed time, and ε is a small strictly positive diffusivity parameter.

20 G.-M. GIE, C. JUNG, AND R. TEMAM It is well-known that the solution to.118) at small ε > produces a large gradient near the boundary Γ when the data u and f do not vanish on the boundary; see the equation.1) below as for the formal limit of u ε at ε =. In this section, we study the asymptotic behavior of the solutions of.118) with respect to the small parameter ε >. Using the methodology introduced in Section., we construct below the asymptotic expansion for u ε as the sum of the inner and outer expansions, which gives a complete structural information of u ε in powers of ε. To explain the basis of the boundary layer analysis for.118), we assume in Sections..1 and.. that the smooth initial data is well-prepared, u =, on Γ,.119) and construct an asymptotic expansion of u ε at an arbitrary order ε n and ε n+1/, n. When the initial data is ill-prepared, that is, u, on Γ,.1) it is well-known that the so-called initial layer is impulsively created at the initial time t =. This interesting phenomenon will be discussed separately in Section Boundary layer analysis at orders ε and ε 1/. In this section, we propose an asymptotic expansion of u ε solution of.118) in the form, u ε = u +θ +ε 1 θ 1,.11) where the formal limit u of u ε at ε = and the two corrector functions θ and θ 1/ will be determined below. The limit u is defined as the solution of equation.118) with ε = : u t = f, in Ω,T), u t= = u, in Ω..1) Integrating.1) in time, we find u x,t) = u x)+ u belongs to C k+1 [,T];H m Ω)) for any T > and k,m, provided u H m Ω), t fx, s) ds;.1) f C k [,T];H m Ω)). Thanks to the consistency condition.119) on the initial data, we infer from.1) that u t= =, on Γ..14) Hence the boundary values of u ε and u agree as at time t =. However, for any t >, we infer from.118) and.1) that u ε u = u, on Γ in general)..15) In fact, this discrepancy of u ε and u on the boundary creates the boundary layers near Γ, and it necessitates the first corrector θ in the expansion.11) that satisfies, θ = u, on Γ,T)..16) The main role of the corrector θ is to balance the difference u ε u on and near the boundary. Then, to manage the geometric effect of a curved boundary, we add the second corrector θ 1/ in the expansion.11) that satisfies the boundary condition, θ 1 =, on Γ,T)..17)

21 RECENT PROGRESSES IN BOUNDARY LAYER THEORY 1 Since the initial data of u ε and u are the same as u, it is natural to impose the zero initial condition for both θ and θ 1/ : θ d t= =, d =,1..18) To find an equation for θ = u ε u ), we perform the matching asymptotics for the difference of.118) and.1) with respect to a small diffusivity parameter ε >. Using the curvilinear coordinates ξ of Section.1.1, we find that a proper scaling for the stretched variable is ξ / ε, and that an asymptotic equation for θ with respect to ε is θ t θ ξ =, at least in Ω δ,t)..19) This process is exactly the same as what we did for the reaction-diffusion equation to obtain.7). Using.8),.9), and.9), we notice that u ε u θ ) = L θ +l.o.t.,.1) with respect to ε. The operator L is identical to that in.66)). Hence, following the methodology in Section.., we find an equation for εθ 1/ = u ε u θ ) as t 1 θ ξ θ 1 = εl θ, at least in Ω δ,t)..11) Now, using.16),.17),.18),.19), and.11), we define the approximate correctors θ and θ 1 as solutions to the heat equations in the half-space, ξ, θ ε θ t ξ =, ξ, t >, θ = u, at ξ =,.1) θ, as ξ, θ t= =, ξ >, and θ 1 ε θ 1 = εl t θ, ξ, t >, ξ θ 1 =, at ξ =,.1) θ 1 θ 1, as ξ, t= =, ξ >. From, e.g., [11], we recall that the explicit expression of θ when the initial data is wellprepared to satisfy.119)) is given by θ ξ,t) = = = t t t u ξ t ξ,,s)erfc )ds εt s) fξ ξ,,s)erfc )ds εt s) fξ ξ,,t s)erfc )ds, εs.14)

22 G.-M. GIE, C. JUNG, AND R. TEMAM where the complementary error function erfc ) on R + is defined by so that where The approximate corrector θ 1/ is given in the form, J ± ξ,t) = t erfcz) := 1 e y / dy,.15) π z erfc) = 1, erfc ) =..16) θ 1 ξ,t) = J + J,.17) { ξ ±η ) } erfc ) ε L θ ξ,η,s)dηds..18) ξ εt s) Using the cut-off function σ in.41) and the approximate correctors above, we define the correctors θ and θ 1/ in the form, which are functions well-defined in Ω [,T]. θ d ξ,t) := θ d ξ,t)σξ ), d =,1,.19) To derive some estimates on the correctors, we first state and prove the pointwise estimates on the complementary error function: Lemma.4. For any ξ,t) in R +, we have m ξ m { ξ ) } erfc εt for a constant κ independent of ε. κexp ξ ), m =, 4εt κεt) m+1 ξ m 1 exp ξ ), m = 1,, 4εt κε m+1 1+t m+1 )1+ξ m 1 )exp ξ ), m. 4εt.14) Proof. Using polar coordinates, we notice that erfcz) ) = 1 π z z e y 1 +y )/ dy 1 dy 1 4 z e r / rdr 1 4 e z,.141) and hence.14) follows for m =. Differentiating.15) with respect to ξ, the left hand side of.14) is equal to 1 π εt) 1 exp ξ ), for m = 1, and 4εt 1 4 ξ π εt) exp ξ ), for m =,.14) 4εt and then.14) immediately follows for m = 1,. Thanks to the Leibnitz formula, we differentiate.14) m )-times m ) in ξ and write m { ξ ) } erfc = 1 ) m+1 εt) m+1 ξ m 1 εt π exp ξ ) +r,m ξ,t),.14) 4εt ξ m

23 RECENT PROGRESSES IN BOUNDARY LAYER THEORY where r,m, m, is given in the form, r,n 1 ξ,t) = 1 n a i,n 1 ) n+i+ εt) n+i+5 ξ n i 4 exp ξ ), π 4εt i= r,n ξ,t) = 1 n π i= a i,n ) n+i+ εt) n+i+ ξ n i exp ξ ), 4εt.144) for some strictly positive integers a i,m, i n. Using.144), we bound the lower order term r,m with respect to ε, r,m ξ,t) κε m+ 1+t m+ )1+ξ m )exp ξ ),.145) 4εt and, from.14) and.145), we obtain.14) for m. Now we state and prove some pointwise estimates on θ d/, d =,1: Lemma.5. Assuming that u satisfies the compatibility condition.119) and f Γ belongs to C 1 [,T];W k, Γ)), the approximate corrector θ satisfies the pointwise estimate, l+k+m θ t l ξi k ξm κ T ε l m exp ξ ), ξ,t) ω ξ R +,T),.146) 4εt for l =, k, and m, or l = 1, k, and m =,1, and l+k+m θ t t l ξi k ξm κ T,δε l m+1 1+s l m+1 )exp ξ ) ds, ξ,t) ω ξ,δ),t), 4εs.147) for l =, k, and m 4, l = 1, k, and m, or l =, and k,m. The constant κ T or κ T,δ ) depends on T or T and δ) and the other data, but is independent of ε. Proof. Using.14), we write, for i = 1, and k,m, k+m θ ξ k i ξm = t k f ξ k i ξ,,t s) m ξ m { ξ ) } erfc..148) εs Then.146) with l =, k, and m =,1 follows from.14) because s 1/ is integrable over,t). Using.1), we write k+m+ θ ξ k i ξm+ = ε 1 k+m+1 θ t ξ k i ξm = ε 1 k f ξi k ξ,,) m ξ m ε 1 t { ξ ) } erfc εt k+1 f t ξi k ξ,,t s) m ξ m { ξ ) } erfc ds. εs.149) Then.146) with l =, k, and m =, follows from.14), and hence we obtain.146) with l = 1, k, and m =,1 as well, using the heat equation.1). We infer from.14) and.148) that k+m θ ξi k ξm κ T ε m+1 t κ T 1+δ m 1 )ε m+1 1+s ) m+1 ) 1+ξ m 1 )exp t ξ 4εs ) ds 1+s ) m+ 1 )exp ξ 4εs ) ds, m 4,.15)

24 4 G.-M. GIE, C. JUNG, AND R. TEMAM and this implies.147) with l =, k, and m 4. Using the heat equation.1), one can prove.147) with l = 1, k, and m or l and k,m by applying the same method as for.15). Lemma.6. Assuming that u satisfies the compatibility condition.119) and f Γ belongs to C 1 [,T];W k, Γ)), the approximate corrector θ 1/ satisfies the pointwise estimate, k+m θ1 ξi k ξm κ ξ T ε m exp ), ξ,t) ω ξ R +,T),.151) 81+m)εt for k and m =,1, and l+k+m θ1 t l ξi k ξm κ T,δε l m 1 t 1+s l m 1 )exp ξ ) ds, ξ,t) ω ξ,δ),t), 8εs.15) for l =, k, and m, or l 1 and k,m. The constant κ T or κ T,δ ) depends on T or T and δ) and the other data, but is independent of ε. Proof. Using.17) and.18), we write where k+m J ± ξ k i ξm k+m θ1 ξi k ξ,ξ,t) = k+m J + ξm ξi k ξm = ξ k i t m+1 { ξ m+1 erfc + k+m J ξi k, i = 1,, k,m,.15) ξm ξ ±η ) } k εl ) θ ξ,η,s)dηds..154) εt s) Using.66) and.146), we observe that k εl ) θ ξ k,η,s) κ r+1 θ ξi r ξ ξ,η,s) κ T exp η ),.155) 4εs r= for each ξ,η,s) ω ξ R +,T), and i = 1,. Now, concerning.151), we only show below the case when m = 1 because.151) with m = can be verified in a similar but easier way. To estimate k+1 J / ξi k ξ pointwise, we write this term as the sum of two integrals k+1 J / ξ 1 i k ξ on,t/) and k+1 J / ξ i k ξ ont/,t), andestimate thembelowseparately: We first estimate the more problematic integral on t/, t) by using.14),.155), and the Schwarz inequality, k+1 J t ξi k ξ ξ κ T ε η exp ξ η) ) exp η ) dηds t/ t s) 4εt s) 4εs κ T ε t t/ {t s) 1 [ ξ η t s ξ k i exp ξ η) 4εt s) ]1 ) dη [ exp ξ η) ]1 ) exp η ) } dη ds. 4εt s) εs.156) Setting η = η ξ )/ εt s), we observe that ξ η exp ξ η) ) dη ε) t s η ) e η ) / dη t s 4εt s).157) κε t s.