contentsgie/review_gie_jung_temam.pdfconvection-diﬀusion equations in a bounded interval with a...

RECENT PROGRESSES IN BOUNDARY LAYER THEORY

GUNG-MIN GIE1, CHANG-YEOL JUNG2, AND ROGER TEMAM3

Abstract. In this article, we review recent progresses in boundary layer analysis of somesingular perturbation problems. Using the techniques of differential geometry, an asymptoticexpansion of reaction-diffusion or heat equation in a domain with curved boundary is con-structed and validated in some suitable functional spaces. In addition, we investigate the effectof curvature as well as that of an ill-prepared initial data. Concerning convection-diffusionequations, the asymptotic behavior of their solutions is difficult and delicate to analyze be-cause it largely depends on the characteristics of the corresponding limit problems, which arefirst order hyperbolic differential equations. Thus, boundary layer analysis is performed onrelatively simpler domains, typically intervals, rectangles, or circles. We consider also the in-terior transition layers at the turning point characteristics in an interval domain and classical(ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangulardomain using the technique of correctors and the tools of functional analysis. The validity ofour asymptotic expansions is also established in suitable spaces.

Contents

1. Introduction 22. Boundary layers in a curved domain in R

n, n = 2, 3 42.1. Elements of differential geometry 42.1.1. Curvilinear coordinate system adapted to the boundary 42.1.2. Examples of the curvilinear system with some special geometries 72.2. Reaction-diffusion equations in a curved domain 82.2.1. Boundary layer analysis at order ε0 92.2.2. Boundary layer analysis at order ε1/2: The effect of curvature 122.2.3. Asymptotic expansions at arbitrary orders εn and εn+1/2, n ≥ 0 152.3. Parabolic equations in a curved domain 192.3.1. Boundary layer analysis at orders ε0 and ε1/2 202.3.2. Boundary layer analysis at arbitrary orders εn and εn+1/2, n ≥ 0 312.3.3. Analysis of the initial layer: The case of ill-prepared initial data 353. Convection-diffusion equations in a bounded interval with a turning point 373.1. Interior layer analysis at arbitrary order εn 383.1.1. f , b compatible 393.1.2. f , b noncompatible 424. Convection-diffusion equations in a domain with corners (rectangle) 444.1. Boundary layer analysis at order ε0 454.2. Boundary layer analysis at arbitrary order εn, n ≥ 0 495. Concluding Remarks 54Acknowledgements 54References 55

Date: September 23, 2015.2000 Mathematics Subject Classification. 35B25, 35C20, 76D10, 35K05.Key words and phrases. boundary layers, singular perturbations, curvilinear coordinates, initial layers, turn-

ing 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 perturbedproblems, that is, (initial and) boundary value problems that contain a small parameter affect-ing the highest order derivatives. The engineering literature on boundary layers is very vastand includes the theory of Prandtl [114, 115] in fluid mechanics as well as works by von Karman[141], [121], and others, and the beautiful experimental book of Van Dyke [27]. On the theo-retical side, the study of singular perturbations and boundary layers remains very challenging,and includes the notorious problem of turbulent boundary layer [121]. Nevertheless such prob-lems have attracted the attention of mathematicians during the last 60 years or so, and somesubstantial insight has been gained in such problems; see, e.g., [30, 91, 107, 110, 122, 139, 140];some of the many other relevant references will be quoted below in the introduction or whenwe study the corresponding problems.

One classical motivation of studying singularly perturbed problems is from the so-calledvanishing viscosity limit in fluid mechanics, see, e.g., [10, 15, 24, 28, 80, 82, 84, 86, 96, 100,109, 111, 112, 125, 131]. Beside the vanishing viscosity problem which underlies this article, onecan 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 vanishingviscosity limit, we recall that the motion of viscous and inviscid fluids is modeled respectivelyby the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at smallviscosity as a singular perturbation of the Euler equations, a major problem, still essentiallyopen, 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 viscositytends to zero. This is an outstanding fundamental problem in analysis. Because an inviscidflow 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. Ofcourse controlling the boundary layers of singularly perturbed problems is the main key tounderstanding the nature of vanishing viscosity limit.

Another important motivation for singular perturbations is associated with the numericalcomputation of solutions to those problems. In approximating solutions to singularly perturbedboundary value problems, it is well-known that a very large (discrete) gradient is created nearthe boundary. Hence, in most classical simulations of singularly perturbed equations, a drasticmesh refinement is usually required near the boundary to obtain an accurate approximationof the solution; a fine mesh of order ε1/2, where ε is the non-dimensional viscosity, is usuallysuggested [126]. Departing from massive mesh refinements, some new semi-analytic methodshave been proposed and successfully applied under the names of enriched spaces, extendedfinite element method (XFEM), and generalized finite element method (GFEM). The commonidea is to add to the Galerkin basis or its analogue in finite differences, finite elements, or finitevolumes, some specific shape functions which carry the inherent singularity of the problem. Forsingular perturbations, the specific shape functions are related to the boundary layer correctorscomputed analytically; in crack theory, the shape functions embolden the singularity at the tipof a crack. See [3, 4, 106] for the XFEM method for cracks; see [104, 105] for GFEM; and seee.g. [18, 19, 122] for singular perturbations as well as the articles [65, 66, 67, 71, 74, 76] whichthis article covers in part and generalizes. Such methods have proven to be highly efficientwithout any help of mesh refinement near the boundary.

In this review article, we consider several classes of singular perturbation problems in somenon-classical settings. Summarizing results in the recent works [35, 36, 38, 39, 40, 41, 43, 44, 72,134], we study, in Section 2, the boundary layers in a smooth domain with curved boundary;see also [56, 57, 69, 132, 133] for the case of a flat boundary. Toward this end, we first recallsome elements of differential geometry and give some concrete examples of the domains under

RECENT PROGRESSES IN BOUNDARY LAYER THEORY 3

consideration in Section 2.1. In Sections 2.2 and 2.3, boundary layers of the reaction diffusionand heat equations are respectively investigated. Here we construct an asymptotic expansionof the singularly perturbed reaction-diffusion solution or the heat solution at an arbitrary orderwith 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 isin fact the solution of an equivalent of the Prandtl equation [114, 115, 129] for the problemand it accounts for the rapid variation of the functions and their normal derivatives in theboundary layer. It also corrects (hence the name) the discrepancies between the boundaryvalues of the viscous and inviscid solutions. This construction is, of course, closely related tothe matching asymptotic method, see, e.g., [29, 30, 88]. Hence the analysis is at first informaland of a physical nature. Then the rigorous validity of our asymptotic expansions is confirmedglobally in the whole domain by performing energy estimates on the difference of the diffusivesolution and the proposed expansion. Our expansion at an arbitrary order with respect to thesmall diffusivity provides the complete structural information of the boundary layers.

In Section 3, we discuss a class of singularly perturbed problems with a turning point inan interval domain. The literature about the analysis of singularly perturbed problems with aturning point is not very large; see however [7, 22, 23, 68, 108, 127, 144, 145, 146, 147, 148]. Herewe follow [73, 79]. The cases where the limit problem is compatible and non-compatible with thegiven data are considered. With limited compatibility conditions on the data, the asymptoticexpansions 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 asymptoticexpansions up to any order.

In Section 4, we consider domains with corners. Corners generate singularities which haveto be corrected by boundary layers techniques, and/or they affect existing boundary layers dueto the singularly perturbed nature of the equations.

Of the first type, and without any “small coefficient” in the equation, are the singularitiescreated at t = 0 by the incompatible initial and boundary data (see [124, 128, 17, 32, 33]) orthe singularities created by corners in a geometric domain (see [51, 52] and [8, 58, 149], and thereferences therein). All these singularities develop already with regular differential operators,that is, differential operators with order one coefficients for the leading derivatives. Anothertype of problems on which we will focus, concerns the interaction of corner singularities withthe boundary layers due to singularly perturbed differential operators, that is, differentialoperators with a small coefficient affecting the highest derivatives. The remarkable article[122] illustrates the variety and complexity of the boundary layers that occur for a singularlyperturbed elliptic differential operator of the second order in a square. These problems havebeen studied in a variety of contexts in, e.g., [94, 95] and in, e.g., the articles [36, 42, 44, 75]on which this section is partly based. In this section, we study the asymptotic behavior ofsolutions to a convection-diffusion equation in a rectangular domain Ω. The Dirichlet boundarycondition is then supplemented along the edges and at the corners. The elliptic corner layersare introduced to handle the interaction of the parabolic and classical boundary layers at thefour corners. Our analysis simplifies that of [122] by minimizing the construction of boundarylayers needed and extends the asymptotic expansions of [122] up to any order.

In summary aiming to study the asymptotic behavior of the solutions to some singularperturbation problems, we construct the boundary layer (or interior layer) correctors andobtain the full structural information of

(1) the boundary layers of the reaction-diffusion equation in a smooth curved domain(2) the boundary and initial layers of the heat equation in a smooth curved domain(3) the interior layers of the convection-diffusion equation with turning points in an interval

domain


(4) the interaction of the boundary and corner layers for the convection-diffusion equationin a rectangular domain

This article is not meant to be exhaustive in any way. Besides the topics of this article, moresubjects will be covered in a forthcoming book [37]. Not covered in this article is the case ofconvection-diffusion equations where the limit problem is hyperbolic. This project was studiedfrom various point of views in, e.g., [5, 45, 85, 89, 136, 138]. Our contributions on the subjectappear in [77, 78], in the review article (on the subject) [70], and in [66] for some computationalaspects. Other problems not discussed in this article include many problems of classical andgeophysical fluid flows with small viscosity; see, e.g., [16, 24, 27, 34, 36, 38, 40, 41, 43, 44, 49, 50,54, 55, 56, 57, 59, 80, 81, 83, 93, 94, 95, 96, 100, 102, 103, 107, 116, 123, 130, 132, 133, 134, 135];electromagnetism, acoustic, and the Helmholtz equation [2, 9, 117, 118]; see also in e.g. [53] andin the references therein the issue of boundary layers for hyperbolic equations; the issue alreadymentioned of the numerical approximation of singularly perturbed problems, in particular inthe context of enriched spaces [18, 19, 60, 61, 62, 64, 65, 66, 67, 71, 74, 76, 85, 104, 105,106, 113, 119, 126, 127]; the important subject of vanishing viscosity for the Hamilton-Jacobiequations which is a whole subject by itself; see among many references, [21, 92]. See alsovarious perspectives in singular perturbations and boundary layers in [1, 12, 13, 14, 25, 26,31, 53, 47, 48, 90, 99, 101, 120, 137, 142, 143]. Let us mention also the article [97] and thesubsequent book [98] which offer totally new perspectives in boundary layer separation. Finallylet us mention that we considered convection-diffusion equations in a circular domain wheretwo characteristic points appear. The singular behaviors may occur at these points dependingon the behavior of the given data. However, this case is not covered here and the reader isreferred to the review article [70]. Convection-diffusion equations in general curved domainswill be studied elsewhere.

2. Boundary layers in a curved domain in Rn, n = 2, 3

2.1. Elements of differential geometry.The domain Ω is assumed to be bounded and smooth in R

3. In this section, we constructan orthogonal curvilinear coordinate system adapted to the boundary Γ := ∂Ω and write somedifferential operators with respect to the curvilinear system. Any smooth and bounded domainin R

2 can be handled in a similar (but easier) manner by suppressing the second tangentialvariable as in Section 2.1.2 below. More information about the geometry and construction ofa special coordinate system is well-described in, e.g., [6, 20, 87] as well as [40, 43].

2.1.1. Curvilinear coordinate system adapted to the boundary.We let x = (x1, x2, x3) denote the Cartesian coordinates of a point in Ω ⊂ R

3. To avoidsome technical difficulties of geometry, we assume that the smooth boundary Γ is a 2D compactmanifold in R

3 having no umbilical points (the two principal curvatures are different at eachpoint on Γ). Concerning general classes of domains including (isolated) umbilical points, thedifficulties and some proper treatments are explained in, e.g., Section 4 in [43]. Then onecan construct a curvilinear system globally on Γ in which the metric tensor is diagonal andthe coordinate lines at each point are parallel to the principal directions. Such a coordinatesystem is called the principal curvature coordinate system. Inside of a tubular neighborhoodΩδ with a small, but fixed, width δ > 0, we extend the principal curvature coordinates on Γin the direction of −n where n is the outer unit normal on Γ. As a result, we obtain a triplyorthogonal coordinate system ξ in R

3ξ, such that Ωδ is diffeomorphic to

Ωδ, ξ := ξ = (ξ′, ξ3) ∈ R3ξ| ξ′ = (ξ1, ξ2) ∈ ωξ′ , 0 < ξ3 < δ, (2.1)


for some bounded set ωξ′ in R2ξ′ . The normal component ξ3 measures the distance from a point

in Ωδ to Γ and hence we write the boundary Γ in the form,

Γ = ξ ∈ R3ξ| ξ′ = (ξ1, ξ2) ∈ ωξ′ , ξ3 = 0. (2.2)

The need to introduce such tubular domains near the boundary comes from the fact that theboundary layer phenomena are local near the boundary in the direction orthogonal to theboundary but are otherwise nonlocal in the tangential directions.

Using the covariant basis gi = ∂x/∂ξi, 1 ≤ i ≤ 3, we write the metric tensor of ξ,

(gij)1≤i,j≤3 := (gi · gj)1≤i,j≤3

=

[1− κ1(ξ

′)ξ3]2g11(ξ

′) 0 0

0[1− κ2(ξ

′)ξ3]2g22(ξ

′) 00 0 1

,

(2.3)

where κi(ξ′), i = 1, 2, is the principal curvature on Γ , gi, i = 1, 2, are the covariant basis of

the principal curvature coordinate system on Γ, and gii = gi · gi.By the choice of a small thickness δ > 0, we have

g(ξ) := det(gij)1≤i,j≤3 > 0 for all ξ in the closure of Ωδ, ξ. (2.4)

We introduce the normalized covariant vectors,

ei =gi

|gi|, 1 ≤ i ≤ 3, (2.5)

and set

hi(ξ) =√gii, i = 1, 2, h(ξ) =

√g. (2.6)

The function h(ξ) > 0 is the magnitude of the Jacobian determinant for the transformationfrom x in Ωδ to ξ in Ωδ, ξ. Similarly the function h(ξ′, 0) > 0 is the magnitude of the Jacobiandeterminant for the transformation from x on Γ to ξ′ in ωξ′ .

For a smooth scalar function v, defined at least in Ωδ, we write the gradient of v in the ξ

variable,

∇v =

2∑

i=1

1

hi

∂v

∂ξiei +

∂v

∂ξ3e3. (2.7)

The Laplacian of v is given in the form,

∆v = Sv + Lv +∂2v

∂ξ23, (2.8)

where

Sv =∑

i=1,2

1

h

∂

∂ξi

( h

hi2

∂v

∂ξi

), Lv =

1

h

∂h

∂ξ3

∂v

∂ξ3. (2.9)

A vector valued function v, defined at least in Ωδ, can be written in the curvilinear system,e1, e2, e3 as

v =

3∑

i=1

vi(ξ)ei. (2.10)

One can classically express the divergence and curl operators acting on v in the ξ variable,

divv =1

h

2∑

i=1

∂

∂ξi

( hhivi

)+

1

h

∂(hv3)

∂ξ3, (2.11)


and

curlv =h1h

∂v3∂ξ2

− ∂(h2v2)

∂ξ3

e1 +

h2h

∂(h1v1)∂ξ3

− ∂v3∂ξ1

e2

+1

h

∂(h2v2)∂ξ1

− ∂(h1v1)

∂ξ2

e3.

(2.12)

The Laplacian1 of v is given in the form,

∆v =3∑

i=1

(Siv + Lvi +

∂2vi∂ξ23

)ei, (2.13)

where

Siv =(

linear combination of tangential derivativesof vj , 1 ≤ j ≤ 3, in ξ′, up to order 2

),

Lvi =(proportional to

∂vi∂ξ3

).

(2.14)

Remark 2.1. The coefficients of Si, 1 ≤ i ≤ 3, and L are multiples of h , 1/h, hi , 1/hi,i = 1, 2 and their derivatives. Thanks to (2.4), all these quantities are well-defined at least inΩδ, ξ.

Considering smooth vector fields in Ωδ of the form,

v =

3∑

i=1

vi(ξ)ei, w =

3∑

i=1

wi(ξ)ei,

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,

∇v w =

3∑

i=1

Pi(v,w) + v3

∂wi

∂ξ3+Qi(v,w) +Ri(v,w)

ei, (2.15)

where

Pi(v,w) =2∑

j=1

1

hjvj∂wi

∂ξj, 1 ≤ i ≤ 3,

Qi(v,w) =

1

h1h2

( ∂hi∂ξ3−i

vi −∂h3−i

∂ξiv3−i

)w3−i, i = 1, 2,

−2∑

j=1

1

hj

∂hj∂ξ3

vjwj , i = 3,

Ri(v,w) =1

hi

∂hi∂ξ3

viw3, i = 1, 2, R3(v,w) = 0.

(2.16)

Remark 2.2. The Qi(v,w) and Ri(v,w), 1 ≤ i ≤ 3, are related to the Christoffel symbolsof the second kind that reflect the twisting effects of the curvilinear system.

1The Laplacian (Laplace-Beltrami operator) of a vector field is defined by the identity ∆v = ∇(divv) −curl(curlv); see, e.g., [20, 87, 6]. We know that other definitions of the Laplacian of a vector, which possessdifferent properties, are used in different contexts.


2.1.2. Examples of the curvilinear system with some special geometries.We present some examples of the curvilinear coordinates discussed in Section 2.1.1 when a

certain symmetry is imposed to the domain Ω. All the analysis below in Sections 2.2 and 2.3is valid (and can be made explicit) by using the Lame coefficients hi, 1 ≤ i ≤ 3 in Section 2.1.1replaced by the corresponding expressions in this section.

Polar coordinate systemWe consider the domain Ω in R

2 as a disk with radius R > 0,

Ω =x ∈ R

2∣∣x21 + x22 < R

. (2.17)

Using the polar coordinates, we construct a curvilinear system adapted to the boundary Γby setting,

x =((R − ξ3) cos ξ1, (R− ξ3) sin ξ1

), (2.18)

for ξ = (ξ1, ξ3) ∈ [0, 2π)× [0, R), i.e., all points x in Ω \ (0, 0). Here we suppressed the secondtangential variable ξ2 to use ξ3 as the normal variable as appearing in Section 2.1.1.

Differentiating x in (2.18) with respect to the variable ξ, we write the covariant basis,

g1 =(− (R− ξ3) sin ξ1, (R− ξ3) cos ξ1

),

g3 = (− cos ξ1, − sin ξ1).(2.19)

Using the orthogonality of gii=1,3, we write the metric tensor,

(gi,j)i,j=1,3 =: (gi · gj)i,j=1,3 =

((R− ξ3)

2 0

0 1

). (2.20)

Introducing the normalized covariant vectors, ei = gi/|gi|, i = 1, 3, we find the positiveLame coefficients,

h1(ξ3) = R− ξ3, h3 = 1. (2.21)

The function h(ξ1, ξ3) := h1(ξ3) is the magnitude of the Jacobian determinant for the trans-formation from x to ξ, and it is positive for all ξ in Ω \ (0, 0).

Cylindrical coordinate systemThe domain Ω is given as a cylinder in R

3,

Ω =x ∈ R

3∣∣x1 ∈ R, x22 + x23 < R

. (2.22)

Using the cylindrical coordinates, we construct a curvilinear system adapted to the bound-ary Γ by setting,

x =(ξ2, (R− ξ3) cos ξ1, (R− ξ3) sin ξ1

), (2.23)

for ξ = (ξ1, ξ2, ξ3) ∈ [0, 2π) × R× [0, R), i.e., all points x in Ω \ x1 = 0.Differentiating x in (2.23) with respect to ξ, we write the covariant basis,

g1 =(0, −(R− ξ3) sin ξ1, (R− ξ3) cos ξ1

),

g2 =(1, 0, 0

),

g3 = (0, − cos ξ1, − sin ξ1).

(2.24)

The metric tensor and positive Lame coefficients are given in the form,

(gi,j)1≤i,j≤3 =: (gi · gj)1≤i,j≤3 =

(R− ξ3)2 0 0

0 1 0

0 0 1

, (2.25)

andh1(ξ3) = R− ξ3, h2 = h3 = 1. (2.26)


The function h(ξ1, ξ3) := h1(ξ3) is the magnitude of the Jacobian determinant for the trans-formation from x to ξ, and it is positive for all ξ in Ω \ x1 = 0.

Toroidal coordinate systemWe consider the domain Ω enclosed by a toroidal surface Γ described by

Γ(ξ1, ξ2) :=((a+ b cos ξ1) cos ξ2, (a+ b cos ξ1) sin ξ2, b sin ξ1

), (ξ1, ξ2) ∈ [0, 2π)2, (2.27)

for fixed 0 < b < a. Setting ξ3 as the distance from a point inside of Ω to the toroidal surfaceΓ (in the direction of −n on Γ), we construct a curvilinear system ξ = (ξ1, ξ2, ξ3) via themapping,

x =((a+ (b− ξ3) cos ξ1) cos ξ2, (a+ (b− ξ3) cos ξ1) sin ξ2, (b− ξ3) sin ξ1

), (2.28)

for any point x in the closure of Ω, except for those along the circle,

Csing. = x ∈ R3| x21 + x22 = a2 and x3 = 0. (2.29)

Differentiating x in (2.28) with respect to ξ, we find the covariant basis,

g1 = (b− ξ3)(− sin ξ1 cos ξ2, − sin ξ1 sin ξ2, cos ξ1

),

g2 =((a+ b cos ξ1)− (cos ξ1)ξ3

)(− sin ξ2, cos ξ2, 0

),

g3 =(cos ξ1 cos ξ2, cos ξ1 sin ξ2, sin ξ1

).

(2.30)

Using the orthogonality of gi1≤i≤3, we write the metric tensor and positive Lame coeffi-cients in the form,

(gi,j)1≤i,j≤3 =: (gi · gj)1≤i,j≤3 =

(b− ξ3)2 0 0

0(a+ (b− ξ3) cos ξ1

)20

0 0 1

, (2.31)

andh1(ξ3) = b− ξ3, h2(ξ1, ξ3) = a+ (b− ξ3) cos ξ1, h3 = 1. (2.32)

The function h(ξ1, ξ3) := h1(ξ3)h2(ξ1, ξ3) is the magnitude of the Jacobian determinant for thetransformation from x to ξ, and it is positive for all ξ in Ω \ Csing..

Remark 2.3. Spherical coordinate system Considering a ball in R3, the spherical coordinate

system is one of the natural choices to locate a point on the sphere, limiting the ball, but itcontains some coordinate singularities at the north and south poles of the sphere and this issueis well-known in atmospheric sciences. To resolve the issue of the pole singularities, one canconstruct smooth coordinate systems “locally” away from the north and south poles and gluethem properly in the “mid-latitude” regions around the equator. We will not discuss this casein details.

2.2. Reaction-diffusion equations in a curved domain.We consider a reaction-diffusion equation,

−ε∆uε + uε = f, in Ω,

uε = 0, on Γ,(2.33)

where Ω is a bounded and smooth domain in R3 as discussed in Section 2.1 and ε is a small

and strictly positive parameter.We expect that a boundary layer occurs near Γ as ε tends to 0 because the equation (2.33)1

formally converges tou0 = f, in Ω, (2.34)

but f may not vanish on Γ in general.


We aim to study the asymptotic behavior of a solution uε to (2.33) at a small ε especiallywhen the boundary Γ is curved. As we will see below in Theorems 2.1, 2.2, and 2.3, thetraditional asymptotic expansion in powers of the small parameter ε has to be modified byadding terms of order εj+1/2 in the expansion. In fact these new terms are necessitated by thegeometric effect (curvature) of a curved boundary.

2.2.1. Boundary layer analysis at order ε0.In this section, we construct an asymptotic expansion at order ε0 of uε, solution of (2.33),

in the form,uε ∼= u0 + θ0, (2.35)

where u0 = f given in (2.34) and θ0 is a corrector function that we will determine below.As we shall see below, the main role of θ0 is to balance the discrepancy of uε and u0 on theboundary Γ.

To define a corrector θ0, we formally insert θ0 ∼= uε−u0 into the difference of the equations(2.33)1 and (2.34). Introducing a stretched variable ξ3 = ξ3/ε

α, α > 0, and using (2.8) with(2.9), we perform the matching asymptotics for the difference equation with respect to a smallε. Then we find that a proper scaling for the stretched variable is

ξ3 =ξ3√ε, (2.36)

and that the asymptotic equation for the corrector θ0 is

− ∂2θ0

∂ξ23+ θ0 = 0, at least in Ωδ. (2.37)

To make the equation (2.37) above useful in all of Ω, we first define an exponentially decayingfunction θ0 in the half space, ξ3 ≥ 0, as a solution of

−ε∂2θ0

∂ξ23+ θ0 = 0, 0 < ξ3 <∞,

θ0 = −u0, at ξ3 = 0,

θ0 → 0, as ξ3 → ∞.

(2.38)

The explicit expression of θ0 is given by

θ0(ξ) = −u0(ξ′, 0) e−ξ3√ε . (2.39)

Using the θ0 in (2.39), we define a corrector θ0 in the form,

θ0(ξ) := θ0(ξ)σ(ξ3), (2.40)

where σ = σ(ξ3) is a cut-off function of class C∞ such that

σ(ξ3) =

1, 0 ≤ ξ3 ≤ δ/3,

0, ξ3 ≥ δ/2,(2.41)

where δ > 0 is the (small) fixed thickness defined in (2.1).The equation for θ0 reads

−ε∂2θ0

∂ξ23+ θ0 = −ε

(σ′′θ0 + 2σ′

∂θ0

∂ξ3

), in Ω,

θ0 = −u0, on Γ.

(2.42)

Using (2.39) and (2.40), we observe that

(right-hand side of (2.42)1) = e.s.t., (2.43)


where e.s.t. denotes a term that is exponentially small with respect to a small parameter ε inany of the usual norms in Ω, e.g., Cs(Ω) or Hs(Ω).

To derive some useful estimates on the corrector θ0, we recall an elementary lemma below:

Lemma 2.1. For any 1 ≤ p ≤ ∞ and q ≥ 0, we have∥∥∥( ξ3√

ε

)qe− ξ3√

ε

∥∥∥Lp(0,∞)

≤ κε1

2p . (2.44)

Using (2.39), (2.40), and Lemma 2.1, we find that∥∥∥( ξ3√

ε

)q ∂k+mθ0

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κε1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mθ0

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κε1

2p−m

2 , (2.45)

for i = 1 or 2, 1 ≤ p ≤ ∞, q ≥ 0, and k,m ≥ 0.

We define the difference between the diffusive solution uε and its asymptotic expansion(2.35) in the form,

wε,0 := uε − (u0 + θ0). (2.46)

In the theorem below, we state and prove the validity of the asymptotic expansion (2.35) aswell as the convergence of uε to u0:

Theorem 2.1. Assuming that the data f belongs to f ∈ H2(Ω), f |Γ ∈ W 2,∞(Γ), thedifference wε,0 between the diffusive solution uε and its asymptotic expansion of order ε0, (see(2.46)), vanishes as the diffusivity parameter ε tends to zero in the sense that

‖wε,0‖Hm(Ω) ≤ κε3

4−m

2 , m = 0, 1, (2.47)

for a constant κ depending on the data, but independent of ε. Moreover, as ε tends to zero, uε

converges to the limit solution u0 in the sense that

‖uε − u0‖L2(Ω) ≤ κε1

4 . (2.48)

Furthermore, we have

limε→0

(∂uε∂ξ3

, ϕ)L2(Ω)

=(∂u0∂ξ3

, ϕ)L2(Ω)

−(u0, ϕ

)L2(Γ)

, ∀ϕ ∈ C(Ω), (2.49)

which expresses the fact that

limε→0

∂uε

∂ξ3=∂u0

∂ξ3− u0(·, 0) δΓ2, (2.50)

in the sense of weak convergence of bounded measures on Ω.

Proof. Using (2.8), (2.33), (2.34), (2.42), and (2.43), we write the equation for wε,0,

−ε∆wε,0 + wε,0 = ε∆u0 +R0 + e.s.t., in Ω,

wε,0 = 0, on Γ,(2.51)

whereR0 = εSθ0 + εLθ0. (2.52)

Thanks to (2.9) and (2.45), we notice that

‖R0‖L2(Ω) ≤ κε2∑

i=1

∥∥∥∂2θ0

∂ξ2i

∥∥∥L2(Ω)

+ κε∥∥∥∂θ0

∂ξ3

∥∥∥L2(Ω)

≤ κε3

4 . (2.53)

2Here δΓ is used to denote the delta measure on Γ and it should not be confused with the (“small”) number δused at other places in the text.


Hence, multiplying (2.51) by wε,0, integrating over Ω, and integrating by parts, we find that

ε‖∇wε,0‖2L2(Ω) + ‖wε,0‖2L2(Ω) ≤[ε‖∆u0‖L2(Ω) + ‖R0‖L2(Ω) + e.s.t.

]‖wε,0‖L2(Ω)

≤ κε2‖∆u0‖2L2(Ω) + κ‖R0‖2L2(Ω) +1

2‖wε,0‖2L2(Ω)

≤ κε3

2 +1

2‖wε,0‖2L2(Ω).

(2.54)

Then we deduce that

‖wε,0‖L2(Ω) ≤ κε3

4 , ‖∇wε,0‖L2(Ω) ≤ κε1

4 , (2.55)

and this implies (2.47).

Thanks to (2.45), (2.48) follows from (2.47) with m = 0.

To prove (2.49), we infer from (2.47) that∣∣∣∣(∂uε∂ξ3

−(∂u0∂ξ3

+∂θ0

∂ξ3

), ϕ)L2(Ω)

∣∣∣∣ ≤ κε1

4 , ∀ϕ ∈ C(Ω). (2.56)

Using (2.39)-(2.41), we write

(∂θ0∂ξ3

, ϕ)L2(Ω)

=(∂θ0∂ξ3

σ, ϕ)L2(Ω)

+(θ0σ′, ϕ

)L2(Ω)

. (2.57)

We observe from (2.39) and (2.41) that the second term on the right-hand side of (2.57) is ane.s.t.. Hence we notice from (2.56) that

limε→0

(∂uε∂ξ3

, ϕ)

L2(Ω)=(∂u0∂ξ3

, ϕ)

L2(Ω)+ lim

ε→0

(∂θ0∂ξ3

σ, ϕ)

L2(Ω), ∀ϕ ∈ C(Ω), (2.58)

if the limit on the right-hand side exists.We introduce the following approximation of the δ-measure on R:

ηε(x) =1√εη( x√

ε

), where η(x) =

1

2e−|x|, (2.59)

so that ‖η‖L1(R) = ‖ηε‖L1(R) = 1 for all ε > 0. Then we write

(∂θ0∂ξ3

σ, ϕ)L2(Ω)

=

∫

ωξ′

(∫ ∞

0

∂θ0

∂ξ3σ ϕhdξ3

)dξ′

= −∫

ωξ′

u0(ξ′, 0)

(∫ ∞

0

1√εe− ξ3√

ε σ ϕhdξ3

)dξ′

= −∫

ωξ′

u0(ξ′, 0)

(∫

R

ηε(ξ3)σ(|ξ3|)ϕ(ξ′, |ξ3|)h(ξ′, |ξ3|) dξ3)dξ′.

(2.60)

Since ηε is an approximation of the δ-measure, the inner integral in ξ3 converges to σ ϕhevaluated at ξ3 = 0 as ε tends to 0. Using this fact and that σ(0) = 1, we deduce from (2.60)that

limε→0

(∂θ0∂ξ3

σ, ϕ)

L2(Ω)= −

∫

ωξ′

u0(ξ′, 0)ϕ(ξ′, 0)h(ξ′, 0) dξ′ = −(u0, ϕ

)L2(Γ)

, ∀ϕ ∈ C(Ω).

(2.61)Hence (2.49) follows from (2.58) and (2.61), and now the proof of Theorem 2.1 is complete.


Remark 2.4. The weak convergence result (2.49) in the space of Radon measures is borrowedfrom [44] in which the authors verified the so-called vorticity accumulation on the boundaryof the solutions to the Navier-Stokes at small viscosity. This interesting phenomena was es-tablished earlier in [95] by using a method different from that used in [44]. In a closely relatedarticle [83], the author proved the equivalence of the vanishing viscosity limit and the vorticityaccumulation in a weaker sense, that is, an analogue of (2.49) with the test functions of classC(Ω) ∩H1Ω.

2.2.2. Boundary layer analysis at order ε1/2: The effect of curvature.Comparing to the case of a domain with flat boundaries, the convergence results in (2.47)

are away from the optimal rates by a factor of ε1/4; see (2.63) below. To understand whatcauses this loss of accuracy, we first notice from (2.52) and (2.53) that the bounds in (2.47)are determined by the L2 norm of the term,

εLθ0, (2.62)

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 ξ3, theidentical copy of the surface coordinates on Γ. By doing so, the matrix tensor in (2.3) becomesindependent of the normal variable ξ3, and hence in the expression (2.8) of Laplacian, the termL vanishes thanks to ∂h/∂ξ3 = 0. Moreover, in this flat-boundary case, one can improve theconvergence result (2.47)1,2 to

‖uε − (u0 + θ0)‖Hm(Ω) ≤ κε1−m2 , m = 0, 1, (2.63)

because the term including ∂θ0/∂ξ3 vanishes and hence the bound in (2.53) becomes κε.

In this section, we will construct a corrector θ1/2 to resolve the effect of curvature, i.e., the(dominant) error in (2.62), so that we improve the convergence results in (2.47)1,2 to those as

in (2.63) by adding θ1/2 in the asymptotic expansion of uε.Noticing from (2.9) and (2.40) that

εLθ0 ∼=√ε c(ξ′) e

− ξ3√ε + e.s.t.,

we propose an asymptotic expansion of uε at the order ε1/2 in the form,

uε ∼= u0 + θ0 + ε1

2 θ1

2 . (2.64)

Here the second corrector θ1/2 will be constructed below as an approximate solution of

− ∂2θ1

2

∂ξ23+ θ

1

2 =√εLθ0, at least in Ωδ. (2.65)

The natural boundary condition for θ1/2 is

θ1

2 = 0, on Γ,

because the discrepancy of uε and u0 on Γ is already taken care of by introducing θ0 in theexpansion.

For any smooth function v, we define

L0v =1

h

∣∣∣ξ3=0

∂h

∂ξ3

∣∣∣ξ3=0

∂v

∂ξ3. (2.66)

Then, using the Taylor expansions of 1/h and ∂h/∂ξ3 in ξ3 at ξ3 = 0, we notice that, pointwise:

|Lv − L0v| =∣∣∣∣(1

h

∂h

∂ξ3− 1

h

∣∣∣ξ3=0

∂h

∂ξ3

∣∣∣ξ3=0

)∂v

∂ξ3

∣∣∣∣ ≤ κε1

2

∣∣∣∣ξ3√ε

∂v

∂ξ3

∣∣∣∣. (2.67)


Using (2.65) and (2.66), we define an exponentially decaying function θ1/2 in the half space,ξ3 ≥ 0, as a solution of

−ε∂2θ

1

2

∂ξ23+ θ

1

2 =√εL0θ

0, 0 < ξ3 <∞,

θ1

2 = 0, at ξ3 = 0,

θ1

2 → 0, as ξ3 → ∞.

(2.68)

Here using (2.39) and (2.66), we set

√εL0θ

0 = u0(ξ′, 0)1

h

∣∣∣ξ3=0

∂h

∂ξ3

∣∣∣ξ3=0

e− ξ3√

ε ; (2.69)

this is the leading order term in small ε of the right-hand side of (2.65).To find the explicit expression of the solution θ1/2 for (2.68), we recall an elementary lemma

below:

Lemma 2.2. A particular solution of

− α2 d2F

dx2(x) + F (x) = βj

(xα

)je−

xα , j ∈ N, α, βj ∈ R \ 0, (2.70)

is given by F = Fα(x) = αjF1(x/α) where, for x = x/α,

F1(x) =βj2j+2

[ j+1∑

k=1

j!

k!

(2x)k]e−x. (2.71)

More generally, if the right-hand side of (2.70) is of the form Pn(x)e−x where Pn(x) is a

polynomial in x of degree n, then (2.70) has a particular solution of the form Pn+1(x)e−x, with

Pn, Pn+1 independent of α.

Thanks to Lemma 2.2, we find the solution θ1/2 of (2.68),

θ1

2 (ξ) =1

2u0(ξ′, 0)

1

h

∣∣∣ξ3=0

∂h

∂ξ3

∣∣∣ξ3=0

ξ3√εe− ξ3√

ε . (2.72)

Using the cut-off function σ in (2.41), we define the corrector θ1/2 in the form,

θ1

2 (ξ) := θ1

2 (ξ)σ(ξ3). (2.73)

The equation for θ1/2 reads

−ε∂2θ

1

2

∂ξ23+ θ

1

2 =√εσL0θ

0 − ε

(σ′′θ

1

2 + 2σ′∂θ

1

2

∂ξ3

), in Ω,

θ1

2 = 0, on Γ.

(2.74)

Using (2.72), (2.73), and Lemma 2.1, we find that

∥∥∥( ξ3√

ε

)q ∂k+mθ1

2

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κε1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mθ1

2

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κε1

2p−m

2 , (2.75)

for i = 1 or 2, 1 ≤ p ≤ ∞, q ≥ 0, and k,m ≥ 0.

We define the difference between uε and the asymptotic expansion at order ε1/2 as

wε, 12

:= uε − (u0 + θ0 + ε1

2 θ1

2 ). (2.76)

Now we state and prove the validity of the asymptotic expansion (2.64) which improves (2.47)in Theorem 2.1:


Theorem 2.2. Assuming that the data f belongs to f ∈ H2(Ω), f |Γ ∈ W 2,∞(Γ), the

difference wε,1/2 between the diffusive solution uε and its asymptotic expansion at order ε1/2

vanishes (or is bounded) as the diffusivity parameter ε tends to zero in the sense that∥∥wε, 1

2

∥∥Hm(Ω)

≤ κε1−m2 , m = 0, 1, 2, (2.77)

for a constant κ depending on the data, but independent of ε.

Proof. We notice from (2.39), (2.40), (2.72), and (2.73) that

(right-hand side of (2.74)1) =√εL0θ

0 +√ε θ0L0σ − ε

(σ′′θ

1

2 + 2σ′∂θ

1

2

∂ξ3

)

=√εL0θ

0 + e.s.t..

(2.78)

Here we used the fact that the exponentially decaying function e−ξ3/√ε multiplied by (1 − σ)

or its derivative at any order is an e.s.t..Using (2.8), (2.33), (2.34), (2.42), (2.43), (2.74), and (2.78), we write the equation for wε,1/2,

−ε∆wε, 12

+ wε, 12

= ε∆u0 +R 1

2

+ e.s.t., in Ω,

wε, 12

= 0, on Γ,(2.79)

where

R 1

2

= εSθ0 + ε(L− L0)θ0 + ε

3

2Sθ1

2 + ε3

2Lθ1

2 . (2.80)

Thanks to (2.9), (2.45), (2.67), and (2.75), we notice that

‖R 1

2

‖L2(Ω) ≤ κε∥∥∥∂2θ0

∂ξ2i

∥∥∥L2(Ω)

+ κε3

2

∥∥∥ξ3√ε

∂θ0

∂ξ3

∥∥∥L2(Ω)

+ κε3

2

∥∥∥∂2θ

1

2

∂ξ2i

∥∥∥L2(Ω)

+ κε3

2

∥∥∥∂θ

1

2

∂ξ3

∥∥∥L2(Ω)

≤ κε5

4 ,(2.81)

where i = 1 or 2.Multiplying (2.79) by wε,1/2, integrating over Ω, and integrating by parts, we find that

ε‖∇wε, 12

‖2L2(Ω) + ‖wε, 12

‖2L2(Ω) ≤[ε‖∆u0‖L2(Ω) + ‖R 1

2

‖L2(Ω) + e.s.t.]‖wε, 1

2

‖L2(Ω)

≤ κε2‖∆u0‖2L2(Ω) + κ‖R 1

2

‖2L2(Ω) +1

2‖wε, 1

2

‖2L2(Ω)

≤ κε2 +1

2‖wε, 1

2

‖2L2(Ω).

(2.82)

Then we deduce that

‖wε, 12

‖L2(Ω) ≤ κε, ‖∇wε, 12

‖L2(Ω) ≤ κε1

2 , (2.83)

and this implies (2.77) with m = 0, 1.To verify (2.77) with m = 2, we infer from (2.79)1, (2.81), and (2.83) that

‖∆wε, 12

‖L2(Ω) ≤ ε−1‖wε, 12

‖L2(Ω) + ‖∆u0‖L2(Ω) + ε−1‖R 1

2

‖L2(Ω) ≤ κ. (2.84)

Thanks to the regularity theory of elliptic equations, (2.77) with m = 2 follows from (2.84)because of (2.79)2.


2.2.3. Asymptotic expansions at arbitrary orders εn and εn+1/2, n ≥ 0.To extend the convergence results of the diffusive solution uε to (2.33) in Theorems 2.1 and

2.2, we construct below asymptotic expansions uεn and uεn+1/2 of uε at arbitrary orders n and

n+ 1/2, n ≥ 0, in the form,

uεn =n∑

j=0

(εjuj + εjθj

)+

n−1∑

j=0

εj+1

2 θj+1

2 ,

uεn+ 1

2

=

n∑

j=0

(εjuj + εjθj + εj+

1

2 θj+1

2

).

(2.85)

Here the uj correspond to the external expansion (outside of the boundary layer) and the

correctors θj and θj+1/2 correspond to the inner expansion (inside the boundary layer).To obtain the external expansion of uε, we formally insert the external expansion uε ∼=∑∞j=0 ε

juj into (2.33) and write

n∑

j=0

(− εj+1∆uj + εjuj

) ∼= f. (2.86)

By matching the terms of the same order εj , we write each uj in terms of the data f :

u0 = f, uj = ∆uj−1 = ∆jf, 1 ≤ j ≤ n. (2.87)

Note that generally the uj ’s, 0 ≤ j ≤ n, do not necessarily vanish on the boundary Γ. In factthe discrepancy between uε and

∑nj=0 ε

juj on Γ creates the boundary layers near Γ.To balance the discrepancy between uε and the external expansion at order n, we introduce

the inner expansion near Γ in the form,

uε −n∑

j=0

εjuj ∼=n∑

j=0

(εjθj + εj+

1

2 θj+1

2

), at least near Γ. (2.88)

As we will see below, the corrector θj, 0 ≤ j ≤ n, balances the discrepancy on Γ between uε

and the proposed external expansion, which is caused by the term uj. Then, at each orderεj , 0 ≤ j ≤ n, an additional corrector θj+1/2 is introduced in the inner expansion to handlethe geometry of the curved boundary. As it appears in Theorems 2.2 and 2.3 below, addingthe corrector θj+1/2 in the expansion ensures the optimal convergence rate at each order εj ,0 ≤ j ≤ n.

By matching the terms of the same order εj on Γ, we deduce from (2.88) the boundarycondition for each θj:

θj = −uj, θj+1

2 = 0, on Γ, 0 ≤ j ≤ n. (2.89)

To find suitable equations for θj and θj+1/2, 0 ≤ j ≤ n, we use (2.33)1 and (2.87) as well as(2.88), and write,

− ε

n∑

j=0

(εj∆θj + εj+

1

2∆θj+1

2

)+

∞∑

j=0

(εjθj + εj+

1

2 θj+1

2

) ∼= 0, at least near Γ. (2.90)

Recalling that the size of the boundary layers for the problem (2.33) is of order ε1/2, weuse below the stretched variable ξ3 = ξ3/

√ε as in (2.36). To describe the dependency of

the Laplacian on the diffusivity parameter ε, we introduce the Taylor expansion of a smoothfunction in the normal variable ξ3,

φ(ξ′, ξ3) = φ(ξ′,√ε ξ3) ∼=

∞∑

j=0

εj

2 ξj3 φj, ∀φ ∈ C∞(ωξ′ × [0,∞)), (2.91)


where

φj :=1

j!

∂jφ

∂ξj3(ξ′, 0), j ≥ 0. (2.92)

Using this form of the Taylor expansion for h, 1/h, ∂h/∂ξ3, and 1/h2i , i = 1, 2, we write theoperators S and L in (2.9) as

S ∼=∞∑

j=0

εj

2 ξj3 S j

2

, L ∼=∞∑

j=0

εj

2 ξj3 L j

2

, (2.93)

where

S j

2

=∑

i=1,2

∑

j1+j2+j3=j,(j1,j2,j3)∈N3

( 1h

)

j1

∂

∂ξi

(h)j2

( 1

hi2

)

j3

∂

∂ξi

,

L j

2

=∑

j1+j2=j,(j1,j2)∈N2

(1h

)j1(h′)j2

∂

∂ξ3= ε−

1

2

∑

j1+j2=j,(j1,j2)∈N2

(1h

)j1(h′)j2

∂

∂ξ3.

(2.94)

The operators Sj/2 and Lj/2, j ≥ 0, are well-defined if

∂Ω is of class Cj+2. (2.95)

Each Sj/2, j ≥ 0, is a tangential differential operator near Γ and the Lj/2 are proportional to

∂/∂ξ3 = ε−1/2∂/∂ξ3. Hence the Sj/2 and Lj/2 at each j ≥ 0 are respectively of order ε0 and

ε−1/2 with respect to the small ε.

Remark 2.5. We use the stretched variable ξ3 to weight the different terms in the equation(2.90). Otherwise in the analysis (above and below) we generally revert to the initial variableξ3.

Remark 2.6. As explained in Section 2.2.2, if the boundary of a domain is flat, the operatorL in (2.9) is identically zero and, in this case, the correctors θj+1/2, 0 ≤ j ≤ n, are not requiredto derive the optimal estimates in Theorems 2.2 and 2.3 below.

Using (2.9), (2.93), and (2.94), we collect all terms of order εj in (2.90) and find the equation

of θj and θj+1/2, j ≥ 0:

− ε∂2θj+

d2

∂ξ23+ θj+

d2 = f

j+ d2

ε (θ), d = 0, 1, at least in Ωδ, (2.96)

where

f jε (θ) :=

2j−2∑

k=0

ε−k2 ξk3S k

2

θj−1− k2 +

2j−1∑

k=0

ε1

2− k

2 ξk3L k2

θj−1

2− k

2 ,

fj+ 1

2ε (θ) :=

2j−1∑

k=0

ε−k2 ξk3S k

2

θj−1

2− k

2 +

2j∑

k=0

ε1

2− k

2 ξk3L k2

θj−k2 .

(2.97)

The equations above with n = 0 are identical to those in (2.37) and (2.65).Modifying the equations (2.96) and (2.97), and using the boundary conditions (2.89), we

define the exponentially decaying functions θj and θj+1/2, j ≥ 0, as the solutions of

−ε∂2θj

∂ξ23+ θj = f jε (θ), 0 < ξ3 <∞,

θj = −uj, at ξ3 = 0,

θj → 0, as ξ3 → ∞,

(2.98)


and

−ε∂2θj+

1

2

∂ξ23+ θn+

1

2 = fj+ 1

2ε (θ), 0 < ξ3 <∞,

θj+1

2 = 0, at ξ3 = 0,

θj+1

2 → 0, as ξ3 → ∞,

(2.99)

where

fj+ d

2ε (θ) := (the right-hand side of (2.97) with θ replaced by θ), d = 0, 1. (2.100)

The equations above with j = 0 are identical to (2.38) and (2.68).Thanks to Lemma 2.2, we find the structure (and not the explicit expression) of the θj and

θj+1/2, j ≥ 0, and this is sufficient for us to perform the error analysis later on:

Lemma 2.3. The solutions θj and θj+1/2, j ≥ 0, of the equations (2.98) and (2.99) are of theform,

θj(ξ) = P2j

( ξ3√ε

)exp(− ξ3√

ε

), θj+

1

2 (ξ) = P2j+1

( ξ3√ε

)exp(− ξ3√

ε

), (2.101)

where Pk(ξ3/√ε) is a polynomial of order k in ξ3/

√ε whose coefficients are smooth function

of ξ′ independent of ε.

Proof. We proceed by induction on j.Thanks to (2.39) and (2.72), we first notice that (2.101) holds true for j = 0.Now, we assume that (2.101) holds for 0 ≤ j ≤ l − 1. Then, to prove that (2.101)1 is true

when j is equal to l, we consider the equation (2.98) with j replaced by l. With the inductiveassumption, we observe that

f lε(θ) =2l−2∑

k=0

ε−k2 ξk3S j

2

θl−1− k2 +

2l−1∑

k=0

ε1

2− k

2 ξk3L k2

θl−1

2− k

2

= P2l−1

( ξ3√ε

)exp(− ξ3√

ε

).

(2.102)

Then, using Lemma 2.2, we obtain a particular solution of (2.98) with j replaced by l:

θlp = P2l

( ξ3√ε

)exp(− ξ3√

ε

). (2.103)

Therefore (2.101)1 holds true for j = l, since the homogeneous solution of (2.98) reads

θlh = −∆lf(ξ′, 0)exp(− ξ3√

ε

).

Because (2.101)2 can be proved in the same way, the proof is now complete.

Using the cut-off function σ in (2.41), we now define the correctors θj and θj+1/2, 0 ≤ j ≤ n,in the form,

θj+d2 (ξ) := θj+

d2 (ξ)σ(ξ3), d = 0, 1. (2.104)

We deduce from Lemma 2.1, (2.101), and (2.104) that

∥∥∥( ξ3√

ε

)q ∂k+mθj+d2

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κε1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mθj+d2

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κε1

2p−m

2 , (2.105)

for d = 1 or 2, 0 ≤ j ≤ n, i = 1 or 2, 1 ≤ p ≤ ∞, q ≥ 0, and k,m ≥ 0.


Using (2.98), (2.99), (2.101) and (2.104), we write the equation for θj and θj+1/2 as

−ε∂2θj

∂ξ23+ θj = f jε (θ) + e.s.t., in Ω,

θj = −uj, on Γ,

(2.106)

and

−ε∂2θj+

1

2

∂ξ23+ θj+

1

2 = fj+ 1

2ε (θ) + e.s.t., in Ω,

θj+1

2 = 0, on Γ.

(2.107)

We introduce the remainders at order εn and εn+1/2, n ≥ 0, in the form,

wε,n+ d2

:= uε − uεn+ d

2

, d = 0, 1, (2.108)

where the asymptotic expansion uεn+d/2, d = 0, 1, of uε is given in (2.85).

Now we state and prove the validity of the asymptotic expansion as a generalization ofTheorems 2.1 and 2.2:

Theorem 2.3. Assuming that the data f belongs to f ∈ H2n+2(Ω), f |Γ ∈ W 2n+2,∞(Γ),the difference wε,n+d/2 between the diffusive solution uε and its asymptotic expansion of order

εn+d/2, d = 0, 1 and n ≥ 0, satisfies∥∥∥wε,n+ d

2

∥∥∥Hm(Ω)

≤ κεn+3+d4

−m2 , m = 0, 1, 2, (2.109)

for a constant κ depending on the data, but independent of ε.

Proof. Using (2.33), (2.87), and (2.108), we write the equations for wε,n+d/2, d = 0, 1, N ≥ 0,

−ε∆wε,n+ d2

+ wε,n+ d2

= εn+1∆un +Rn+ d2

, in Ω,

wε,n+ d2

= 0, on Γ,(2.110)

where

Rn+ d2

=2n+d∑

j=0

(ε∆θj

2 − θj

2 ), d = 0, 1. (2.111)

We multiply the equation (2.110)1 by wε,n+d/2, integrate over Ω, and integrate by parts tofind

ε∥∥∥∇wε,n+ d

2

∥∥∥2

L2(Ω)+

1

2

∥∥∥wε,n+ d2

∥∥∥2

L2(Ω)≤ ε2n+2‖∆un‖2L2(Ω) +

∥∥∥Rn+ d2

∥∥∥2

L2(Ω). (2.112)

Using the expression of the Laplacian in (2.8), (2.9), and (2.93), and the equations of thecorrectors in (2.106) and (2.107), we notice that

Rn =

2n−2∑

j=0

εj

2+1S −

2n−j−2∑

k=0

ξk3S k2

θ

j

2 + εn+1

2Sθn−1

2 + εn+1Sθn

+

2n−1∑

j=0

εj

2+1L−

2n−j−1∑

k=0

ξk3L k2

θ

j

2 + εn+1Lθn + e.s.t.,

(2.113)


and

Rn+ 1

2

=

2n−1∑

j=0

εj

2+1S −

2n−j−1∑

k=0

ξk3S k2

θ

j

2 + εn+1Sθn + εn+3

2Sθn+1

2

+

2n∑

j=0

εn2+1L−

2n−j∑

k=0

ξkdL k2

θ

j2 + εn+

3

2Lθn+1

2 .

(2.114)

We recall that S and the Sk/2 are tangential differential operators, and that L and the Lk/2

are proportional to ∂/∂ξ3. Hence, using (2.105), we find that

∥∥∥Rn

∥∥∥L2(Ω)

≤ κ

εn+

1

2

∥∥∥∥2n−2∑

k=0

(ε−

1

2 ξ3)2n−k−1

Sn− k2− 1

2

θk2

∥∥∥∥L2(Ω)

+ εn+1

2

∥∥∥Sθn−1

2

∥∥∥L2(Ω)

+ εn+1∥∥∥Sθn

∥∥∥L2(Ω)

+ εn+1

∥∥∥∥2n−1∑

k=0

(ε−

1

2 ξ3)2n−k

Ln− k2

θk2

∥∥∥∥L2(Ω)

+ εn+1∥∥∥Lθn

∥∥∥L2(Ω)

≤ κεn+3

4 ,

(2.115)

and∥∥∥Rn+ 1

2

∥∥∥L2(Ω)

≤ κ

εn+1

∥∥∥∥2n−1∑

k=0

(ε−

1

2 ξ3)2n−k

Sn− k2

θk2

∥∥∥∥L2(Ω)


∥∥∥L2(Ω)

+ εn+3

2

∥∥∥Sθn+1

2

∥∥∥L2(Ω)

+ εn+3

2

∥∥∥∥2n∑

k=0

(ε−

1

2 ξ3)2n−k+1

Ln− k2+ 1

2

θk2

∥∥∥∥L2(Ω)

+ εn+3

2

∥∥∥Lθn+1

2

∥∥∥L2(Ω)

≤ κεn+5

4 .(2.116)

Then (2.109) with m = 0, 1 follows from (2.112), (2.115), and (2.116).To verify (2.109) with m = 2, we infer from (2.109) with m = 0, 1, (2.110)1, (2.115), and

(2.116) that∥∥∆wε,n+ d

2

∥∥L2(Ω)

≤ ε−1∥∥wε,n+ d

2

∥∥L2(Ω)

+ εn‖∆un‖L2(Ω) + ε−1∥∥Rn+ d

2

∥∥L2(Ω)

≤ κεn+d−1

4 , (2.117)

for d = 0, 1. Thanks to the regularity theory of elliptic equations, (2.109) with m = 2 followsfrom (2.117) and (2.110)2.

2.3. Parabolic equations in a curved domain.We consider the heat equation in a bounded smooth domain Ω of R3,

∂uε

∂t− ε∆uε = f, in Ω× (0, T ),

uε = 0, on Γ× (0, T ),

uε∣∣t=0

= u0, in Ω,

(2.118)

where f and u0 are given smooth data, T > 0 is an arbitrary but fixed time, and ε is a smallstrictly positive diffusivity parameter.


It is well-known that the solution to (2.118) at small ε > 0 produces a large gradient nearthe boundary Γ when the data u0 and f do not vanish on the boundary; see the equation(2.122) below as for the formal limit of uε at ε = 0. In this section, we study the asymptoticbehavior of the solutions of (2.118) with respect to the small parameter ε > 0. Using themethodology introduced in Section 2.2, we construct below the asymptotic expansion for uε

as the sum of the inner and outer expansions, which gives a complete structural informationof uε in powers of ε.

To explain the basis of the boundary layer analysis for (2.118), we assume in Sections 2.3.1and 2.3.2 that the smooth initial data is well-prepared,

u0 = 0, on Γ, (2.119)

and construct an asymptotic expansion of uε at an arbitrary order εn and εn+1/2, n ≥ 0.When the initial data is ill-prepared, that is,

u0 6= 0, on Γ, (2.120)

it is well-known that the so-called initial layer is impulsively created at the initial time t = 0.This interesting phenomenon will be discussed separately in Section 2.3.3.

2.3.1. Boundary layer analysis at orders ε0 and ε1/2.In this section, we propose an asymptotic expansion of uε solution of (2.118) in the form,

uε ∼= u0 + θ0 + ε1

2 θ1

2 , (2.121)

where the formal limit u0 of uε at ε = 0 and the two corrector functions θ0 and θ1/2 will bedetermined below.

The limit u0 is defined as the solution of equation (2.118) with ε = 0:

∂u0

∂t= f, in Ω× (0, T ), u0

∣∣t=0

= u0, in Ω. (2.122)

Integrating (2.122) in time, we find

u0(x, t) = u0(x) +

∫ t

0f(x, s) ds; (2.123)

u0 belongs to Ck+1([0, T ];Hm(Ω)) for any T > 0 and k,m ≥ 0, provided

u0 ∈ Hm(Ω), f ∈ Ck([0, T ];Hm(Ω)).

Thanks to the consistency condition (2.119) on the initial data, we infer from (2.123) that

u0|t=0 = 0, on Γ. (2.124)

Hence the boundary values of uε and u0 agree as 0 at time t = 0. However, for any t > 0, weinfer from (2.118) and (2.123) that

uε − u0 = −u0 6= 0, on Γ (in general). (2.125)

In fact, this discrepancy of uε and u0 on the boundary creates the boundary layers near Γ, andit necessitates the first corrector θ0 in the expansion (2.121) that satisfies,

θ0 = −u0, on Γ× (0, T ). (2.126)

The main role of the corrector θ0 is to balance the difference uε−u0 on and near the boundary.Then, to manage the geometric effect of a curved boundary, we add the second corrector θ1/2

in the expansion (2.121) that satisfies the boundary condition,

θ1

2 = 0, on Γ× (0, T ). (2.127)


Since the initial data of uε and u0 are the same as u0, it is natural to impose the zero initialcondition for both θ0 and θ1/2:

θd2

∣∣t=0

= 0, d = 0, 1. (2.128)

To find an equation for θ0 (∼= uε − u0), we perform the matching asymptotics for the dif-ference of (2.118) and (2.122) with respect to a small diffusivity parameter ε > 0. Usingthe curvilinear coordinates ξ of Section 2.1.1, we find that a proper scaling for the stretchedvariable is ξ3/

√ε, and that an asymptotic equation for θ0 with respect to ε is

∂θ0

∂t− ∂2θ0

∂ξ23= 0, at least in Ωδ × (0, T ). (2.129)

This process is exactly the same as what we did for the reaction-diffusion equation to obtain(2.37).

Using (2.8), (2.9), and (2.93), we notice that

∆(uε − u0 − θ0

) ∼= L0θ0 + l.o.t., (2.130)

with respect to ε. (The operator L0 is identical to that in (2.66)). Hence, following the

methodology in Section 2.2.2, we find an equation for√ε θ1/2 (∼= uε − u0 − θ0) as

∂θ1

2

∂t− ∂2θ

1

2

∂ξ23=

√εL0θ

0, at least in Ωδ × (0, T ). (2.131)

Now, using (2.126), (2.127), (2.128), (2.129), and (2.131), we define the approximate cor-

rectors θ0 and θ1

2 as solutions to the heat equations in the half-space, ξ3 ≥ 0,

∂θ0

∂t− ε

∂2θ0

∂ξ23= 0, ξ3, t > 0,

θ0 = −u0, at ξ3 = 0,

θ0 → 0, as ξ3 → ∞,

θ0∣∣t=0

= 0, ξ3 > 0,

(2.132)

and

∂θ1

2

∂t− ε

∂2θ1

2

∂ξ23=

√εL0θ

0, ξ3, t > 0,

θ1

2 = 0, at ξ3 = 0,

θ1

2 → 0, as ξ3 → ∞,

θ1

2

∣∣t=0

= 0, ξ3 > 0.

(2.133)

From, e.g., [11], we recall that the explicit expression of θ0 (when the initial data is well-prepared to satisfy (2.119)) is given by

θ0(ξ, t) = −2

∫ t

0

∂u0

∂t(ξ′, 0, s) erfc

( ξ3√2ε(t − s)

)ds

= −2

∫ t

0f(ξ′, 0, s) erfc

( ξ3√2ε(t− s)

)ds

= −2

∫ t

0f(ξ′, 0, t − s) erfc

( ξ3√2εs

)ds,

(2.134)


where the complementary error function erfc(·) on R+ is defined by

erfc(z) :=1√2π

∫ ∞

ze−y2/2 dy, (2.135)

so that

erfc(0) =1

2, erfc(∞) = 0. (2.136)

The approximate corrector θ1/2 is given in the form,

θ1

2 (ξ, t) = J+ − J−, (2.137)

where

J±(ξ, t) =∫ t

0

∫ ∞

0

∂

∂ξ3

erfc

( ξ3 ± η√2ε(t− s)

)√ε(L0θ

0)(ξ′, η, s) dηds. (2.138)

Using the cut-off function σ in (2.41) and the approximate correctors above, we define the

correctors θ0 and θ1/2 in the form,

θd2 (ξ, t) := θ

d2 (ξ, t)σ(ξ3), d = 0, 1, (2.139)

which are functions well-defined in Ω× [0, T ].

To derive some estimates on the correctors, we first state and prove the pointwise estimateson the complementary error function:

Lemma 2.4. For any (ξ3, t) in R2+, we have

∂m

∂ξm3

erfc

( ξ3√2εt

)≤

κ exp(− ξ23

4εt

), m = 0,

κ(εt)−m+ 1

2 ξm−13 exp

(− ξ23

4εt

), m = 1, 2,

κε−m+ 1

2 (1 + t−m+ 1

2 )(1 + ξm−13 ) exp

(− ξ23

4εt

), m ≥ 3.

(2.140)

for a constant κ independent of ε.

Proof. Using polar coordinates, we notice that

(erfc(z)

)2=

1

2π

∫ ∞

z

∫ ∞

ze−(y2

1+y2

2)/2dy1dy2 ≤

1

4

∫ ∞

√2ze−r2/2rdr ≤ 1

4e−z2 , (2.141)

and hence (2.140) follows for m = 0.Differentiating (2.135) with respect to ξ3, the left hand side of (2.140) is equal to

− 1

2√π(εt)−

1

2 exp(− ξ23

4εt

), for m = 1, and

1

4√π(εt)−

3

2 ξ3 exp(− ξ23

4εt

), for m = 2, (2.142)

and then (2.140) immediately follows for m = 1, 2.Thanks to the Leibnitz formula, we differentiate (2.142)2 (m− 2)-times (m ≥ 3) in ξ3 and

write

∂m

∂ξm3

erfc

( ξ3√2εt

)= − 1√

π(−2)−m+1(εt)−m+ 1

2 ξm−13 exp

(− ξ23

4εt

)+ r0,m(ξ3, t), (2.143)


where r0,m, m ≥ 3, is given in the form,

r0,2n−1(ξ3, t) =−1

2√π

n−2∑

i=0

ai,2n−1(−2)−2n+i+3(εt)−2n+i+ 5

2 ξ2n−2i−43 exp

(− ξ23

4εt

),

r0,2n(ξ3, t) =−1

2√π

n−2∑

i=0

ai,2n(−2)−2n+i+2(εt)−2n+i+ 3

2 ξ2n−2i−33 exp

(− ξ23

4εt

),

(2.144)

for some strictly positive integers ai,m, 0 ≤ i ≤ n− 2. Using (2.144), we bound the lower orderterm r0,m with respect to ε,

|r0,m(ξ3, t)| ≤ κε−m+ 3

2 (1 + t−m+ 3

2 )(1 + ξm−33 ) exp

(− ξ23

4εt

), (2.145)

and, from (2.143) and (2.145), we obtain (2.140) for m ≥ 3.

Now we state and prove some pointwise estimates on θd/2, d = 0, 1:

Lemma 2.5. Assuming that u0 satisfies the compatibility condition (2.119) and f |Γ belongsto C1([0, T ];W k,∞(Γ)), the approximate corrector θ0 satisfies the pointwise estimate,

∣∣∣∣∂ℓ+k+mθ0

∂tℓ∂ξki ∂ξm3

∣∣∣∣ ≤ κT ε−ℓ−m

2 exp(− ξ23

4εt

), (ξ, t) ∈ ωξ′ × R+ × (0, T ), (2.146)

for ℓ = 0, k ≥ 0, and 0 ≤ m ≤ 3, or ℓ = 1, k ≥ 0, and m = 0, 1, and∣∣∣∣∂ℓ+k+mθ0


∣∣∣∣ ≤ κT,δ ε−ℓ−m+ 1

2

∫ t

0(1 + s−2ℓ−m+ 1

2 ) exp(− ξ23

4εs

)ds, (ξ, t) ∈ ωξ′ × (0, δ) × (0, T ),

(2.147)for ℓ = 0, k ≥ 0, and m ≥ 4, ℓ = 1, k ≥ 0, and m ≥ 2, or ℓ = 2, and k,m ≥ 0. The constantκT (or κT,δ) depends on T (or T and δ) and the other data, but is independent of ε.

Proof. Using (2.134), we write, for i = 1, 2 and k,m ≥ 0,

∂k+mθ0

∂ξki ∂ξm3

= −2

∫ t

0

∂kf

∂ξki(ξ′, 0, t− s)

∂m

∂ξm3

erfc

( ξ3√2εs

). (2.148)

Then (2.146) with ℓ = 0, k ≥ 0, and m = 0, 1 follows from (2.140) because s−1/2 is integrableover (0, T ).

Using (2.132), we write

∂k+m+2θ0

∂ξki ∂ξm+23

= ε−1 ∂k+m+1θ0

∂t∂ξki ∂ξm3

= −2ε−1 ∂kf

∂ξki(ξ′, 0, 0)

∂m

∂ξm3

erfc

( ξ3√2εt

)

−2ε−1

∫ t

0

∂k+1f

∂t∂ξki(ξ′, 0, t− s)

∂m

∂ξm3

erfc

( ξ3√2εs

)ds.

(2.149)

Then (2.146) with ℓ = 0, k ≥ 0, and m = 2, 3 follows from (2.140), and hence we obtain (2.146)with ℓ = 1, k ≥ 0, and m = 0, 1 as well, using the heat equation (2.132).

We infer from (2.140) and (2.148) that∣∣∣∣∂k+mθ0

∂ξki ∂ξm3

∣∣∣∣ ≤ κT ε−m+ 1

2

∫ t

0

(1 + (s′)−m+ 1

2

)(1 + ξm−1

3 ) exp(− ξ23

4εs′

)ds′

≤ κT (1 + δm−1)ε−m+ 1

2

∫ t

0

(1 + (s′

)−m+ 1

2 ) exp(− ξ23

4εs′

)ds′, m ≥ 4,

(2.150)


and this implies (2.147) with ℓ = 0, k ≥ 0, and m ≥ 4.Using the heat equation (2.132), one can prove (2.147) with ℓ = 1, k ≥ 0, and m ≥ 2 or

ℓ ≥ 2 and k,m ≥ 0 by applying the same method as for (2.150).

Lemma 2.6. Assuming that u0 satisfies the compatibility condition (2.119) and f |Γ belongs

to C1([0, T ];W k,∞(Γ)), the approximate corrector θ1/2 satisfies the pointwise estimate,∣∣∣∣∂k+mθ

1

2

∂ξki ∂ξm3

∣∣∣∣ ≤ κT ε−m

2 exp(− ξ23

8(1 +m)εt

), (ξ, t) ∈ ωξ′ × R+ × (0, T ), (2.151)

for k ≥ 0 and m = 0, 1, and∣∣∣∣∂ℓ+k+mθ

1

2


∣∣∣∣ ≤ κT,δ ε−ℓ−m− 1

2

∫ t

0(1 + s−2ℓ−m− 1

2 ) exp(− ξ23

8εs

)ds, (ξ, t) ∈ ωξ′ × (0, δ) × (0, T ),

(2.152)for ℓ = 0, k ≥ 0, and m ≥ 2, or ℓ ≥ 1 and k,m ≥ 0. The constant κT (or κT,δ) depends on T(or T and δ) and the other data, but is independent of ε.

Proof. Using (2.137) and (2.138), we write

∂k+mθ1

2

∂ξki ∂ξm3

(ξ′, ξ3, t) =∂k+mJ+

∂ξki ∂ξm3

+∂k+mJ−∂ξki ∂ξ

m3

, i = 1, 2, k,m ≥ 0, (2.153)

where

∂k+mJ±∂ξki ∂ξ

m3

=

∫ t

0

∫ ∞

0

∂m+1

∂ξm+13

erfc

( ξ3 ± η√2ε(t− s)

)∂k(√εL0θ

0)

∂ξki(ξ′, η, s) dηds. (2.154)

Using (2.66) and (2.146), we observe that

∣∣∣∣∂k(√εL0θ

0)

∂ξki(ξ′, η, s)

∣∣∣∣ ≤ κk∑

r=0

∣∣∣∣∂r+1θ0

∂ξri ∂ξ3(ξ′, η, s)

∣∣∣∣ ≤ κT exp(− η2

4εs

), (2.155)

for each (ξ′, η, s) ∈ ωξ′ × R+ × (0, T ), and i = 1, 2.Now, concerning (2.151), we only show below the case when m = 1 because (2.151) with

m = 0 can be verified in a similar but easier way.To estimate ∂k+1J−/∂ξki ∂ξ3 pointwise, we write this term as the sum of two integrals

∂k+1J1−/∂ξ

ki ∂ξ3 on (0, t/2) and ∂k+1J2

−/∂ξki ∂ξ3 on (t/2, t), and estimate them below separately:

We first estimate the more problematic integral on (t/2, t) by using (2.140), (2.155), andthe Schwarz inequality,

∣∣∣∣∂k+1J2

−∂ξki ∂ξ3

∣∣∣∣ ≤ κT ε− 3

2

∫ t

t/2

∫ ∞

0

|ξ3 − η|(t− s)

3

2

exp(− (ξ3 − η)2

4ε(t− s)

)exp

(− η2

4εs

)dηds

≤ κT ε− 3

2

∫ t

t/2

(t− s)−1

[ ∫ ∞

0

|ξ3 − η|2t− s

exp(− (ξ3 − η)2

4ε(t − s)

)dη

] 1

2

[ ∫ ∞

0exp

(− (ξ3 − η)2

4ε(t− s)

)exp

(− η2

2εs

)dη

] 1

2ds.

(2.156)

Setting η′ = (η − ξ3)/√

2ε(t− s), we observe that∫ ∞

0

|ξ3 − η|2t− s

exp(− (ξ3 − η)2

4ε(t− s)

)dη ≤ (2ε)

3

2

√t− s

∫ ∞

−∞(η′)2e(η

′)2/2dη′

≤ κε3

2

√t− s.

(2.157)


Using the fact that t− s < s for t/2 < s < t and setting η′ = (η − ξ3/2)/√εs, we find

∫ ∞

0exp

(− (ξ3 − η)2

4ε(t− s)− η2

2εs

)dη ≤

∫ ∞

0exp

(− (ξ3 − η)2 + η2

4εs

)dη

≤ exp(− ξ23

8εs

) ∫ ∞

0exp

(− (η − ξ3/2)

2

2εs

)dη

≤ κε1

2

√s exp

(− ξ23

8εs

).

(2.158)

Combining (2.156)-(2.158), we see that

∣∣∣∣∂k+1J2

−∂ξki ∂ξ3

∣∣∣∣ ≤ κT ε− 1

2

∫ t

t/2(t− s)−

3

4 exp(− ξ23

16εs

)ds

≤ κT ε− 1

2 exp(− ξ23

16εt

) ∫ t

t/2(t− s)−

3

4 ds

≤ κT ε− 1

2 exp(− ξ23

16εt

).

(2.159)

The other integral ∂k+1J1−/∂ξ

ki ∂ξ3 on (0, t/2) satisfies the estimate (2.156) with the interval

(t/2, t) replaced by (0, t/2). Then, since (t − s)−3/2 is bounded from below and above by

(t/2)−3/2 and t−3/2 on (0, t/2), it is easy to see that |∂k+1J1−/∂ξ

ki ∂ξ3| is also bounded by the

right-hand side of (2.159), and we deduce that

∣∣∣∣∂k+1J−∂ξki ∂ξ3

∣∣∣∣ ≤ κT ε− 1

2 exp(− ξ23

16εt

). (2.160)

With the same (but easier) proof for the term ∂k+1J1+/∂ξ

ki ∂ξ3, we obtain (2.151) with m = 1.

To show (2.152) with ℓ = 0, k ≥ 0, and m ≥ 2, we use (2.4), (2.153), and (2.155), and findthat

∣∣∣∣∂k+mJ±∂ξki ∂ξ

m3

∣∣∣∣ ≤ κT ε−m− 1

2

∫ t

0

∫ ∞

0

[1 + (t− s)−m− 1

2

1 + |ξ3 ± η|m

exp(− (ξ3 ± η)2

4ε(t− s)− η2

4εs

)]dηds

≤ (using the analog of (2.158))

≤ κT,δ ε−m− 1

2

∫ t

0

1 + (t− s)−m− 1

2

exp

(− ξ23

8ε(t − s)

)ds.

(2.161)

Thus (2.152) with ℓ = 0, k ≥ 0, and m ≥ 2 follows.For (2.152) with ℓ ≥ 1 and k,m ≥ 0, we use (2.133) and write

∂k+m+1θ1

2

∂t∂ξki ∂ξm3

= ε∂k+m+2θ

1

2

∂ξki ∂ξm+23

+∂k+m(

√εL0θ

0)

∂ξki ∂ξm3

, (2.162)

and more generally,

∂ℓ+k+mθ1

2


= εℓ∂2ℓ+k+mθ

1

2

∂ξki ∂ξ2ℓ+m3

+ℓ−1∑

l=0

εl+1

2∂ℓ+k+m+l−1(L0θ

0)

∂tℓ−l−1∂ξki ∂ξ2l+m3

ℓ ≥ 1. (2.163)


Using (2.66) and (2.146), we find

∣∣∣∣ℓ−1∑

l=0

εl+1

2∂ℓ+k+m+l−1(L0θ

0)

∂tℓ−l−1∂ξki ∂ξ2l+m3

∣∣∣∣ ≤ κ

ℓ−1∑

l=0

εl+1

2

∣∣∣∣∂ℓ+k+m+lθ0

∂tℓ−l−1∂ξki ∂ξ2l+m+13

∣∣∣∣

≤ κT,δ ε−ℓ−m+1

∫ t

0(1 + s−2ℓ−m+ 3

2 ) exp(− ξ23

4εs

)ds.

(2.164)

Hence (2.152) with ℓ ≥ 1 and k,m ≥ 0 follows from (2.161), (2.163), and (2.164), and thelemma is now proved.

Using (2.139) and Lemmas 2.5 and 2.6, we notice that

∣∣∣∣∂ℓ+k+m(θ

d2 − θ

d2 )


∣∣∣∣ = e.s.t., ℓ, k,m ≥ 0, d = 0, 1, i = 1, 2, (2.165)

for (ξ′, ξ3, t) in ωξ′ × (0, δ) × (0, T ).Now we state and prove the following elementary lemma:

Lemma 2.7. For any 1 ≤ p ≤ ∞, q1 ≥ 0, and q2 ≥ 1, we have, for 0 ≤ t ≤ T ,∥∥∥∥(ε−

1

2 ξ3)q1 exp

(− ξ23

4q2εt

)∥∥∥∥Lp(R+)

≤ κT (εt)1

2p . (2.166)

Proof. Using the boundedness of the exponentially decaying function, we see that (2.166) withp = ∞ holds true.

Setting η = ξ3/√q2εt, we write

∥∥∥∥(ε−

1

2 ξ3)q1 exp

(− ξ23

4q2εt

)∥∥∥∥p

Lp(R+)

=

∫ ∞

0(ε−

1

2 ξ3)pq1 exp

(− p ξ23

4q2εt

)dξ3

≤ κT (εt)1

2

∫ ∞

0ηpq1e−pη2/4dη ≤ κT (εt)

1

2 .

(2.167)

Thus (2.166) follows for 1 ≤ p <∞.

Thanks to Lemmas 2.5, 2.6, and 2.7, and (2.165), we find the Lp estimates of θd/2 and θd/2,d = 0, 1,

∥∥∥( ξ3√

ε

)q ∂k+mθ0

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κT (εt)1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mθ0

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κ(εt)1

2p−m

2 ,

(2.168)for i = 1, 2, 1 ≤ p ≤ ∞, q ≥ 0, k ≥ 0, and 0 ≤ m ≤ 3, and

∥∥∥( ξ3√

ε

)q ∂k+mθ1

2

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κT (εt)1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mθ1

2

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κ(εt)1

2p−m

2 ,

(2.169)for i = 1, 2, 1 ≤ p ≤ ∞, q ≥ 0, k ≥ 0, and m = 0, 1.

Using (2.132), (2.133), (2.139), and (2.165), we write the equations for θd/2, d = 0, 1,

∂θ0

∂t− ε

∂2θ0

∂ξ23= e.s.t., in Ω× (0, T ),

θ0 = −u0, on Γ× (0, T ),

θ0∣∣t=0

= 0, in Ω,

(2.170)


and

∂θ1

2

∂t− ε

∂2θ1

2

∂ξ23=

√εL0θ

0 + e.s.t., in Ω× (0, T ),

θ1

2 = 0, on Γ× (0, T ),

θ1

2

∣∣t=0

= 0, in Ω.

(2.171)

We introduce the difference between the heat solution uε and the proposed asymptoticexpansions at order ε0 and ε1/2 in the form,

wε,0 := uε −(u0 + θ0

), wε, 1

2

:= uε −(u0 + θ0 + ε

1

2 θ1

2

). (2.172)

Now we state and prove the validity of the asymptotic expansion (2.121) as well as theconvergence of uε to u0 in the theorem below.

Theorem 2.4. Assuming that u0 belongs to H2(Ω) and satisfies the compatibility condition(2.119), and that f belongs to f ∈ C1([0, T ];H2(Ω)), f |Γ ∈ C1([0, T ];W 2,∞(Γ)), the differ-

ence wε,d/2 between the heat solution uε and its asymptotic expansion of order εd/2, d = 0, 1,vanishes as the diffusivity parameter ε tends to zero in the sense that

∥∥∥∥∂mwε,0

∂ξm3

∥∥∥∥L∞(0,T ;L2(Ω))

≤ κT ε3

4−m

2 ,

∥∥∥∥∂mwε,0

∂ξm3

∥∥∥∥L2(0,T ;H1(Ω))

≤ κT ε1

4−m

2 , (2.173)

for m = 0, 1, and that∥∥wε, 1

2

∥∥L∞(0,T ;L2(Ω))

≤ κT ε,∥∥wε, 1

2

∥∥L2(0,T ;H1(Ω))

≤ κT ε1

2 , (2.174)

for a constant κT depending on T and the other data, but independent of ε. Moreover, as εtends to zero, uε converges to the limit solution u0 in the sense that

‖uε − u0‖L∞(0,T ;L2(Ω)) ≤ κε1

4 . (2.175)

We also have

limε→0

(∂uε∂ξ3

, ϕ)L2(Ω)

=(∂u0∂ξ3

, ϕ)L2(Ω)

−(u0, ϕ

)L2(Γ)

, ∀ϕ ∈ C(Ω), (2.176)

uniformly in time, t ∈ [0, T ], which expresses the fact that

limε→0

∂uε

∂ξ3=∂u0

∂ξ3− u0(·, 0, ·) δΓ3, (2.177)

uniformly in time, t ∈ [0, T ], in the sense of weak convergence of bounded measures on Ω.

Proof. Using (2.8), (2.118), (2.122), (2.170), and (2.171), we write the equation for wε,d/2,d = 0, 1, as

∂wε, d2

∂t− ε∆wε, d

2

= ε∆u0 +R d2

+ e.s.t., in Ω× (0, T ),

wε, d2

= 0, on Γ× (0, T ),

wε, d2

∣∣t=0

= 0, in Ω,

(2.178)

where

R0 = εSθ0 + εLθ0,

R 1

2

= εSθ0 + ε(L− L0)θ0 + ε

3

2Sθ1

2 + ε3

2Lθ1

2 .(2.179)

3Here again δΓ denotes the delta measure supported on the boundary and is not related to the “small” coefficientδ.


Using (2.9), (2.168), (2.67), and (2.169), we notice that

‖R0‖L2(Ω) ≤ κε

∥∥∥∥∂2θ0

∂ξ2i

∥∥∥∥L2(Ω)

+ κε

∥∥∥∥∂θ0

∂ξ3

∥∥∥∥L2(Ω)

≤ κT ε3

4 , i = 1, 2, (2.180)

and that

‖R 1

2

‖L2(Ω) ≤ κε

∥∥∥∥∂2θ0

∂ξ2i

∥∥∥∥L2(Ω)

+ κε3

2

∥∥∥∥ξ3√ε

∂θ0

∂ξ3

∥∥∥∥L2(Ω)

+ κε3

2

∥∥∥∥∂2θ

1

2

∂ξ2i

∥∥∥∥L2(Ω)

+ κε3

2

∥∥∥∥∂θ

1

2

∂ξ3

∥∥∥∥L2(Ω)

≤ κT ε5

4 , i = 1, 2.(2.181)

Using these estimates, we multiply (2.178)1 by wε,d/2, integrate over Ω, and integrate by parts.Applying the Schwarz inequality as well, we find that

1

2

d

dt

∥∥wε, d2

∥∥2L2(Ω)

+ ε∥∥∇wε, d

2

∥∥2L2(Ω)

≤∥∥wε, d

2

∥∥2L2(Ω)

+ κε2∥∥∆u0

∥∥2L2(Ω)

+ κd∑

n=0

∥∥Rn2

∥∥2L2(Ω)

≤∥∥wε, d

2

∥∥2L2(Ω)

+ κT ε3

2 , d = 0, 1.

(2.182)Hence we obtain (2.173) with m = 0 and (2.174) using the Gronwall lemma.

The convergence result (2.175) follows from (2.173) with m = 0 and (2.168), thanks to thetriangle inequality.

To verify (2.173) with m = 1, we differentiate (2.178) with m = 0 in the normal variable ξ3and find the equation and initial condition for ∂wε,0/∂ξ3,

∂

∂t

(∂wε,0

∂ξ3

)− ε∆

(∂wε,0

∂ξ3

)= ε

∂∆u0

∂ξ3+∂R0

∂ξ3+ e.s.t., in Ω× (0, T ),

∂wε,0

∂ξ3

∣∣∣∣t=0

= 0, in Ω.

(2.183)

Using (2.8), we restrict (2.178)1 with m = 0 on Γ, (ξ3 = 0) and find that

− εLwε,0 − ε∂2wε,0

∂ξ23= ε(∆u0 + Lθ0

), on Γ× (0, T ). (2.184)

Here we used the fact that the e.s.t. on the right-hand side of (2.178)1 vanishes on Γ. Using(2.8) and (2.184), we now obtain the Robin boundary condition for ∂wε,0/∂ξ3 in the form,

∂2wε,0

∂ξ23+( 1h

∂h

∂ξ3

)∣∣∣∣ξ3=0

∂wε,0

∂ξ3= −∆u0 −

( 1h

∂h

∂ξ3

)∣∣∣∣ξ3=0

∂θ0

∂ξ3, on Γ× (0, T ). (2.185)

We multiply (2.183)1 by ∂wε,0/∂ξ3, integrate over Ω, and integrate by parts. After applyingthe Schwarz inequality, we find that

1

2

d

dt

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+ ε

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

≤ ε

∫

Γ

(∇∂wε,0

∂ξ3

)· n ∂wε,0

∂ξ3dx

+

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+ κε2∥∥∥∥∂∆u0

∂ξ3

∥∥∥∥2

L2(Ω)

+ κ

∥∥∥∥∂R0

∂ξ3

∥∥∥∥2

L2(Ω)

.

(2.186)


Using (2.185) and the fact that e3 = −n on Γ, we estimate the first term on the right-handside of (2.186),∣∣∣∣ε∫

Γ

(∇∂wε,0

∂ξ3

)· n ∂wε,0

∂ξ3dx

∣∣∣∣ ≤ ε

∣∣∣∣∫

Γ

∂2wε,0

∂ξ23

∂wε,0

∂ξ3dx

∣∣∣∣

≤ κε

∥∥∥∥∂2wε,0

∂ξ23

∥∥∥∥L2(Γ)

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Γ)

≤ κε

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Γ)

+ κε

∥∥∥∥1 +∂θ0

∂ξ3

∥∥∥∥L2(Γ)

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Γ)

.

(2.187)

Thanks to the trace theorem, we notice that

κε

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Γ)

≤ κε

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Ω)

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥H1(Ω)

≤ κε

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+ κε

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Ω)

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥L2(Ω)

≤ κε

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+1

4ε

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

.

(2.188)

Using (2.146), (2.165), and the continuity of θ0, we infer that∣∣∣∣∂θ0

∂ξ3

∣∣∣Γ

∣∣∣∣ ≤ κT ε− 1

2 . (2.189)

Hence, using the trace theorem as well, we estimate the second term on the right-hand side of(2.187),

κε

∥∥∥∥1 +∂θ0

∂ξ3

∥∥∥∥L2(Γ)

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Γ)

≤ κT ε1

2

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥1

2

L2(Ω)

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥1

2

H1(Ω)

≤ κT ε1

2

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Ω)

+ κT ε1

2

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥1

2

L2(Ω)

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥1

2

L2(Ω)

≤ κT ε1

4

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥L2(Ω)

+ κT ε3

4

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥L2(Ω)

≤ κT ε1

2 + κT

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+1

4ε

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

.

(2.190)We deduce from (2.187), (2.188), and (2.190) that

∣∣∣∣ε∫

Γ

(∇∂wε,0

∂ξ3

)· n ∂wε,0

∂ξ3dx

∣∣∣∣ ≤ κT ε1

2 + κT

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+1

2ε

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

. (2.191)

Using (2.9), (2.168), and (2.179), we observe that∥∥∥∥∂R0

∂ξ3

∥∥∥∥L2(Ω)

≤ κε

∥∥∥∥∂3θ0

∂ξ3∂ξ2i

∥∥∥∥L2(Ω)

+ κε

∥∥∥∥∂2θ0

∂ξ23

∥∥∥∥L2(Ω)

≤ κT ε1

4 , i = 1, 2. (2.192)


Combining (2.186), (2.191), and (2.192), we see that

d

dt

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

+ ε

∥∥∥∥∇∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

≤ κT ε1

2 + κT

∥∥∥∥∂wε,0

∂ξ3

∥∥∥∥2

L2(Ω)

, (2.193)

and (2.173) follows from (2.193) for m = 1.Now to show the weak convergence result in (2.176), we infer from (2.173) with m = 1 that

∣∣∣∣(∂uε∂ξ3

−(∂u0∂ξ3

+∂θ0

∂ξ3

), ϕ)L2(Ω)

∣∣∣∣ ≤ κT ε1

4 , ∀ϕ ∈ C(Ω), (2.194)

uniformly in 0 < t < T .Using (2.139) and (2.146), we notice that

(∂θ0∂ξ3

, ϕ)L2(Ω)

=(∂θ0∂ξ3

σ, ϕ)L2(Ω)

+ e.s.t.. (2.195)

Hence we deduce from (2.194) that

limε→0

(∂uε∂ξ3

, ϕ)L2(Ω)

=(∂u0∂ξ3

, ϕ)L2(Ω)

+ limε→0

(∂θ0∂ξ3

σ, ϕ)L2(Ω)

, ∀ϕ ∈ C(Ω), (2.196)

uniformly in 0 < t < T , if the limit on the right-hand side exists.We observe from (2.168) that the L∞(0, T ;L1(Ω)) norm of ∂θ0/∂ξ3 is bounded indepen-

dently of ε. Therefore the limit of ∂θ0/∂ξ3 at ε = 0 must exist in the space of Radon measuresin Ω uniformly in time 0 < t < T .

To find the explicit form of the limit for ∂θ0/∂ξ3 as ε tends to zero, we fix 0 < t < T andintroduce the following approximation of the δ-measure in R,

ηε(x, t) =∂

∂x

erfc

( x√2εt

)=

1

2√πεt

e−x2/(4εt), (2.197)

so that ‖ηε‖L1(R) = 1 for all ε, t > 0. Then, using (2.134) and (2.197), we write

(∂θ0∂ξ3

σ, ϕ)L2(Ω)

= −2

∫ t

0

∫

ωξ′

∂u0

∂t(ξ′, 0, t− s)

(∫ ∞

0ηε(ξ3, s)σ ϕhdξ3

)dξ′ds

= −∫ t

0

∫

ωξ′

∂u0

∂t(ξ′, 0, t− s)

(∫

R

ηε(ξ3)σ(|ξ3|)ϕ(ξ′, |ξ3|)h(ξ′, |ξ3|) dξ3)dξ′ds.

(2.198)

Since ηε is an approximation of the δ-measure, the most inner integral with respect to ξ3converges to σ ϕh evaluated at ξ3 = 0 as ε tends to 0. Using this fact, (2.124), and σ(0) = 1,we deduce from (2.198) that

limε→0

(∂θ0∂ξ3

σ, ϕ)L2(Ω)

= −∫

ωξ′

∫ t

0

∂u0

∂t(ξ′, 0, t − s)ϕ(ξ′, 0)h(ξ′, 0) dξ′

= −∫

ωξ′

u0(ξ′, 0, t)ϕ(ξ′, 0)h(ξ′, 0) dξ′

= −(u0, ϕ

)L2(Γ)

, ∀ϕ ∈ C(Ω).

(2.199)

Hence (2.176) follows from (2.196) and (2.199), and now the proof of Theorem 2.4 is complete.

Remark 2.7. The weak convergence result (2.176) in the space of Radon measures is borrowedfrom [44].


2.3.2. Boundary layer analysis at arbitrary orders εn and εn+1/2, n ≥ 0.To extend the convergence results of the heat solution uε of (2.118) in Theorem 2.4, we

construct the asymptotic expansions uεn and uεn+1/2 of uε at arbitrary orders n and n + 1/2,

n ≥ 0, in the form,

uεn =

n∑

j=0

(εjuj + εjθj

)+

n−1∑

j=0

εj+1

2 θj+1

2 ,

uεn+ 1

2

=

n∑

j=0

(εjuj + εjθj + εj+

1

2 θj+1

2

),

(2.200)

where the uj correspond to the external expansion and the correctors θj and θj+1/2 correspondto the inner expansion.

To obtain the external expansion of uε, we formally insert the external expansion uε ∼=∑∞j=0 ε

juj into (2.118) and write

n∑

j=0

(εj∂uj

∂t− εj+1∆uj

)∼= f. (2.201)

By matching the terms of the same order εj , we write the equation for each uj ,

∂uj

∂t= f, for j = 0, and ∆uj−1, for 1 ≤ j ≤ n. (2.202)

Each equation is supplemented with the initial condition,

uj∣∣t=0

= u0, for j = 0, and 0, for 1 ≤ j ≤ n. (2.203)

By sequentially solving the initial value problem (2.202)-(2.203), we find that

uj(x, t) =

(right-hand side of (2.123)), j = 0,

1

j!tj∆ju0(x) +

∫ t

0

1

j!(t− s)j∆jf(x, s) ds, 1 ≤ j ≤ n.

(2.204)

Each uj, 0 ≤ j ≤ n, belongs to Ck+j+1([0, T ];Hm(Ω)) for any T > 0 and k,m ≥ 0, providedthat

u0 ∈ Hm+2j(Ω), f ∈ Ck([0, T ];Hm+2j(Ω)).

Note that the uj, 0 ≤ j ≤ n, do not necessarily vanish on the boundary Γ, and in fact thediscrepancy between uε and

∑nj=0 ε

juj on Γ creates boundary layers near Γ.To balance the discrepancy of uε and the external expansion, we follow the methodology

introduced in Section 2.2.3 and propose an inner expansion near Γ in the form,

uε −n∑

j=0

εjuj ∼=n∑

j=0

(εjθj + εj+

1

2 θj+1

2

), at least near Γ. (2.205)

The main role of θj, 0 ≤ j ≤ n, is to balance the discrepancy between uε and the proposedexternal expansion at order εj while θj+1/2, 0 ≤ j ≤ n, is necessary to be added in theinner expansion to handle the error caused by the curvature of the boundary. As it appearsin Theorems 2.5 below, adding the corrector θj+1/2 in the expansion ensures the optimalconvergence rate at each order of εj , 0 ≤ j ≤ n.

By matching the terms of the same order εj on Γ, we find the boundary and initial conditionsfor each θj from (2.205):

θj = −uj and θj+1

2 = 0, on Γ× (0, T ), 0 ≤ j ≤ n, (2.206)


and

θj∣∣t=0

= θj+1

2

∣∣t=0

= 0, in Ω, 0 ≤ j ≤ n. (2.207)

To find the proper equations for θj and θj+1/2, 0 ≤ j ≤ n, we use (2.118)1, (2.204), and(2.205) to write,

∞∑

j=0

(εj∂θj

∂t+ εj+

1

2∂θj+

1

2

∂t

)− ε

n∑

j=0

(εj∆θj + εj+

1

2∆θj+1

2

) ∼= 0, at least near Γ. (2.208)

Following the procedure in Section 2.2.3, we use (2.9), (2.93), and (2.94), and collect all terms

of order εj in (2.208) to find the equation of θj and θj+1/2,

∂θj+d2

∂t− ε

∂2θj+d2

∂ξ23= f

j+ d2

ε (θ), d = 0, 1, at least in Ωδ, (2.209)

where

f jε (θ) :=

2j−2∑

k=0

ε−k2 ξk3S k

2

θj−1− k2 +

2j−1∑

k=0

ε1

2− k

2 ξk3L k2

θj−1

2− k

2 ,

fj+ 1

2ε (θ) :=

2j−1∑

k=0

ε−k2 ξk3S k

2

θj−1

2− k

2 +

2j∑

k=0

ε1

2− k

2 ξk3L k2

θj−k2 .

(2.210)

The equations above with j = 0 are identical to those in (2.129) and (2.131).Modifying the equations (2.209) and (2.210), and using the boundary and initial conditions

(2.206) and (2.207), we define θj and θj+1/2, 0 ≤ j ≤ n, as solutions of

∂θj

∂t− ε

∂2θj

∂ξ23= f jε (θ), ξ3, t > 0,

θj = −uj, at ξ3 = 0,

θj → 0, as ξ3 → ∞,

θj∣∣t=0

= 0, ξ3 > 0,

(2.211)

and

∂θj+1

2

∂t− ε

∂2θj+1

2

∂ξ23= f

j+ 1

2ε (θ), ξ3, t > 0,

θj+1

2 = 0, at ξ3 = 0,

θj+1

2 → 0, as ξ3 → ∞,

θj+1

2

∣∣t=0

= 0, ξ3 > 0,

(2.212)

where

fj+ d

2ε (θ) := (the right-hand side of (2.210) with θ replaced by θ), d = 0, 1. (2.213)

The equations above with n = 0 are identical to (2.132) and (2.133) with

f0ε (θ) = 0, f1

2ε (θ) =

√εL0θ

0. (2.214)

Thanks to (2.203) and the linearity of the equation (2.211), we infer that the solution θj,j ≥ 1, is given in the form,

θj = θjh + θjp, (2.215)


where θjh =

((2.134) with u0 replaced by uj

),

θjp =((2.137) with

√εL0θ

0 replaced by f jε (θ)).

(2.216)

In the same manner, we find that the solution θj+1/2, j ≥ 1, of (2.212) is given in the form,

θj+1

2 =((2.137) with

√εL0θ

0 replaced by fj+ 1

2ε (θ)

). (2.217)

Now, we prove the following pointwise estimates for θj and θj+1/2.

Lemma 2.8. Assuming that u0 satisfies the compatibility condition (2.119) and f |Γ belongs

to C1([0, T ];W 2n+k,∞(Γ)), the approximate corrector θj+d/2, 0 ≤ j ≤ n, d = 0, 1, satisfies thepointwise estimates,

∣∣∣∣∂k+mθj+

d2

∂ξki ∂ξm3

∣∣∣∣ ≤ κT ε−m

2 exp(− ξ23

2m42j+d+1εt

), (ξ, t) ∈ ωξ′ × R+ × (0, T ), (2.218)

for k ≥ 0 and m = 0, 1, and

∣∣∣∣∂ℓ+k+mθj+

d2

∂tℓ∂ξki ξm3

∣∣∣∣

≤ κT,δ ε−ℓ−m− 1

2

∫ t

0(1 + s−2ℓ−m− 1

2 ) exp(− ξ23

42j+d+1εs

)ds, (ξ, t) ∈ ωξ′ × (0, δ) × (0, T ),

(2.219)for ℓ = 0, k ≥ 0, and m ≥ 2, or ℓ ≥ 1 and k,m ≥ 0. The constant κT (or κT,δ) depends on T(or T and δ) and the other data, but is independent of ε.

Proof. We proceed by induction on j.Thanks to Lemmas 2.5 and 2.6, we immediately see that (2.218) and (2.219) hold true when

j is equal to 0.If we assume that (2.218) and (2.219) are valid when j ≤ k and d = 0, 1, then, using (2.213)

with d = 0 and the inductive assumption, we notice that∣∣∣∣∂kfk+1

ε (θ)

∂ξki

∣∣∣∣ ≤ κT1 + (ε−

1

2 ξ3)2k+1

exp

(− 4−(2k+2) ξ

23

2εt

). (2.220)

Then, using the same arguments as in the proofs of Lemmas 2.5 and 2.6, one can verify that(2.218) and (2.219) hold true when j = k + 1 and d = 0.

Using (2.218) and (2.219) for θk+1 and the inductive assumption, we repeat the proof of

Lemma 2.6 and verify that (2.218) and (2.219) hold true for θk+3/2. By the induction, theproof is now complete.

Using the cut-off function σ in (2.41) and the approximate correctors θj+d/2 above, wedefine the actual correctors in the form,

θj+d2 (ξ′, ξ3, t) := θj+

d2 (ξ′, ξ3, t)σ(ξ3), 0 ≤ j ≤ n, d = 0, 1, (2.221)

which are functions well-defined in Ω× [0, T ]. Then, using Lemma 2.8, we notice that

∣∣∣∣∂ℓ+k+m(θj+

d2 − θj+

d2 )


∣∣∣∣ = e.s.t., ℓ, k,m ≥ 0, 0 ≤ j ≤ n, d = 0, 1, i = 1, 2, (2.222)

for (ξ′, ξ3, t) in ωξ′ × (0, δ) × (0, T ).


Thanks to Lemmas 2.7 and 2.8, and (2.222), we find the Lp estimates of θj+d/2 and θj+d/2,0 ≤ j ≤ n, d = 0, 1,

∥∥∥( ξ3√

ε

)q ∂k+mθj+d2

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κT ε1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mθj+d2

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κε1

2p−m

2 ,

(2.223)for i = 1, 2, 1 ≤ p ≤ ∞, q ≥ 0, k ≥ 0, and m = 0, 1.

Using (2.211), (2.212), (2.221), and (2.222), we write the equations for θj+d/2, 0 ≤ j ≤ n,d = 0, 1,

∂θj

∂t− ε

∂2θj

∂ξ23= f jε (θ) + e.s.t., in Ω× (0, T ),

θj = −uj, on Γ× (0, T ),

θj∣∣t=0

= 0, in Ω,

(2.224)

and

∂θj+1

2

∂t− ε

∂2θj+1

2

∂ξ23= f

j+ 1

2ε (θ) + e.s.t., in Ω× (0, T ),

θj+1

2 = 0, on Γ× (0, T ),

θj+1

2

∣∣t=0

= 0, in Ω.

(2.225)

We introduce the remainders at order εn and εn+1/2, n ≥ 0, in the form,

wε,n+ d2

:= uε − uεn+ d

2

, d = 0, 1, (2.226)

where the asymptotic expansion uεn+d/2, d = 0, 1, of uε is given in (2.200).

Now we state and prove the validity of the asymptotic expansion as a generalization ofTheorems 2.4:

Theorem 2.5. Assuming that u0 belongs to H2n+2(Ω) and satisfies the compatibility condition(2.119), and that f belongs to f ∈ C1([0, T ];H2n+2(Ω)), f |Γ ∈ C1([0, T ];W 2n+2,∞(Γ)), thedifference wε,n+d/2 between the heat solution uε and its asymptotic expansion of order εn+d/2,d = 0, 1, n ≥ 0, vanishes as the diffusivity parameter ε tends to zero in the sense that

∥∥wε,n+ d2

∥∥L∞(0,T ;L2(Ω))

≤ κT εn+ 3+d

4 ,∥∥wε,n+ d

2

∥∥L2(0,T ;H1(Ω))

≤ κT εn+ 1+d

4 , (2.227)

for a constant κT depending on T and the other data, but independent of ε.

Proof. Using (2.118), (2.202), (2.203), and (2.226), we write the equations for wε,n+d/2, d = 0, 1,n ≥ 0,

∂wε,n+ d2

∂t− ε∆wε,n+ d

2

= εn+1∆un +Rn+ d2

, in Ω× (0, T ),

wε,n+ d2

= 0, on Γ× (0, T ),

wε,n+ d2

∣∣t=0

= 0, in Ω,

(2.228)

where

Rn+ d2

=

2n+d∑

n=0

(ε∆θn2 − θ

n2 ), d = 0, 1. (2.229)


We multiply the equation (2.228)1 by wε,n+d/2, integrate over Ω, and integrate by parts tofind

d

dt

∥∥∥wε,n+ d2

∥∥∥2

L2(Ω)+ 2ε

∥∥∥∇wε,n+ d2

∥∥∥2

L2(Ω)≤ 2∥∥∥wε,n+ d

2

∥∥∥2

L2(Ω)+ ε2N+2‖∆un‖2L2(Ω) +

∥∥∥Rn+ d2

∥∥∥2

L2(Ω).

(2.230)Using the expression of the Laplacian in (2.8), (2.9), and (2.93), and the equations of the

correctors in (2.224) and (2.225), we notice that

Rn =2n−2∑

j=0

εj

2+1S −

2n−j−2∑

k=0

ξk3S k2

θ

j

2 + εn+1

2Sθn−1

2 + εn+1Sθn

+

2n−1∑

j=0

εj

2+1L−

2n−j−1∑

k=0

ξk3L k2

θ

j

2 + εn+1Lθn + e.s.t.,

(2.231)

and

Rn+ 1

2

=

2n−1∑

j=0

εj

2+1S −

2n−j−1∑

k=0

ξk3S k2

θ

j

2 + εn+1Sθn + εn+3

2Sθn+1

2

+2n∑

j=0

εj

2+1L−

2n−j∑

k=0

ξkdL k2

θ

j

2 + εn+3

2Lθn+1

2 .

(2.232)

We recall that S and the Sk/2 are tangential differential operators, and that L and the Lk/2

are proportional to ∂/∂ξ3. Hence, using (2.223), we find that

∥∥∥Rn

∥∥∥L2(Ω)

≤ κ

εn+

1

2

∥∥∥∥2n−2∑

k=0

(ε−

1

2 ξ3)2n−k−1

Sn− k2− 1

2

θk2

∥∥∥∥L2(Ω)

+ εn+1

2

∥∥∥Sθn−1

2

∥∥∥L2(Ω)


∥∥∥L2(Ω)

+ εn+1

∥∥∥∥2n−1∑

k=0

(ε−

1

2 ξ3)2n−k

Ln− k2

θk2

∥∥∥∥L2(Ω)

+ εn+1∥∥∥Lθn

∥∥∥L2(Ω)

≤ κεn+3

4 ,

(2.233)

and

∥∥∥Rn+ 1

2

∥∥∥L2(Ω)

≤ κ

εn+1

∥∥∥∥2n−1∑

k=0

(ε−

1

2 ξ3)2n−k

Sn− k2

θk2

∥∥∥∥L2(Ω)


∥∥∥L2(Ω)

+ εn+3

2

∥∥∥Sθn+1

2

∥∥∥L2(Ω)

+ εn+3

2

∥∥∥∥2n∑

k=0

(ε−

1

2 ξ3)2n−k+1

Ln− k2+ 1

2

θk2

∥∥∥∥L2(Ω)

+ εn+3

2

∥∥∥Lθn+1

2

∥∥∥L2(Ω)

≤ κεn+5

4 .(2.234)

Thanks to the Gronwall inequality, (2.227) follows from (2.230), (2.233), and (2.234).

2.3.3. Analysis of the initial layer: The case of ill-prepared initial data.In this section, we study the asymptotic behavior of the heat solution uε of (2.118) when

the initial data is ill-prepared as appearing in (2.120). As we shall see below, in this interesting


case, the so-called initial layer is created at t = 0 in addition to the (regular) boundary layersthat we analyzed in the previous Sections 2.3.1 and 2.3.2. The initial layer can be understoodphysically as an impulsively started motion at t = 0 near the boundary.

To analyze the initial layer, it is required to introduce an additional corrector (ψσ below in(2.238)) in the asymptotic expansion uε ∼= u0 + θ0 (and hence in the expansions at all orders).In fact, in this case, a solution of the equation (2.132) is given in the form,

Θ0(ξ, t) = ψ + θ0, (2.235)

where

ψ(ξ, t) = −2u0(ξ′, 0) erfc

( ξ3√2εt

),

θ0(ξ, t) = (2.134).

(2.236)

Here the initial layer function ψ corresponds to the ill-prepared initial data u0 on Γ, and ψ = 0if u0 is well-prepared as in (2.119).

Thanks to Lemma 2.4, we obtain the following pointwise estimates for ψ:

Lemma 2.9. Assuming that u0|Γ belongs to W k,∞(Γ), then ψ satisfies the pointwise estimates,

∣∣∣∣∂k+mψ

∂ξki ∂ξm3

∣∣∣∣ ≤

κT exp(− ξ23

4εt

), k ≥ 0, m = 0,

κ(εt)−m+ 1

2 ξm−13 exp

(− ξ23

4εt

), k ≥ 0, m = 1, 2,

κε−m+ 1

2 (1 + t−m+ 1

2 )(1 + ξm−13 ) exp

(− ξ23

4εt

), k ≥ 0, m ≥ 3,

(2.237)

for (ξ, t) ∈ ωξ′ × R+ × (0, T ). The constant κT depends on T and the other data, but isindependent of ε.

Using the cut-off function σ in (2.41), we define

Θ0(ξ, t) := Θ0(ξ, t)σ(ξ3) = ψ(ξ, t)σ(ξ3) + θ0(ξ, t), (2.238)

with θ0 as in (2.139). Then Θ0 is well-defined in Ω× [0, T ].Using (2.165), (2.237), and the fact that ψ is a solution of the homogeneous heat equation

in R+ × (0, T ), we observe that∣∣∣∣∂ℓ+k+m(Θ0 −Θ

0)


∣∣∣∣ = e.s.t., ℓ, k,m ≥ 0, d = 0, 1, i = 1, 2, (2.239)

for (ξ′, ξ3, t) in ωξ′ × (0, δ) × (0, T ).

Using (2.239) and the fact that Θ0is a heat solution, we find the equation of Θ0,

∂Θ0

∂t− ε

∂2Θ0

∂ξ23= e.s.t., in Ω× (0, T ),

Θ0 = −u0, on Γ× (0, T ),

Θ0∣∣t=0

= 0, in Ω.

(2.240)

Thanks to Lemmas 2.5, 2.7, and 2.9, and (2.239), we find the Lp estimates of Θ0and Θ0,

∥∥∥( ξ3√

ε

)q ∂k+mΘ0

∂ξki ∂ξm3

∥∥∥Lp(ωξ′×R+)

≤ κT (εt)1

2p−m

2 ,∥∥∥( ξ3√

ε

)q ∂k+mΘ0

∂ξki ∂ξm3

∥∥∥Lp(Ω)

≤ κ(εt)1

2p−m

2 ,

(2.241)for i = 1, 2, 1 ≤ p ≤ ∞, q ≥ 0, k ≥ 0, and 0 ≤ m ≤ 2.


Remark 2.8. Comparing (2.241) with the Lp estimates of θ0 in (2.168), we see that handling

ψ (and hence ψ, Θ0, and Θ0) is more problematic than θ0 (and θ0), because its derivatives in

ξ3 or t are more singular at t = 0 than the derivatives of θ0. However the time singularity in(2.241) is manageable in the analysis of Theorem 2.6 below because it is integrable in time t.

Now we set

wε,n+ d2

:= uε − uε,n+ d2

, n ≥ 0, d = 0, 1, (2.242)

where the asymptotic expansion uεn+d/2 of uε is given by

uε,n+ d2

= (right-hand side of (2.200) with θ0 replaced by Θ0; see (2.139) and (2.238)),

(2.243)with n ≥ 0, d = 0, 1. Remembering that the time singularity in (2.241) is integrable, one canreproduce the convergence results in Theorems 2.4 and 2.5:

Theorem 2.6. Assuming that the initial data u0 is ill-prepared as appearing in (2.120), theconvergence results in Theorems 2.4 and 2.5 hold true with wε,n+d/2 replaced by wε,n+d/2, n ≥ 0,d = 0, 1, provided that the same regularity assumptions are imposed to the data.

3. Convection-diffusion equations in a bounded interval with a turning point

In this section we consider a one-dimensional singularly perturbed problem which has asingle turning point on Ω = (−1, 1),

−ε2d2uε

dx2− b

duε

dx= f in Ω,

uε(−1) = uε(1) = 0,

(3.1)

where 0 < ε << 1, b = b(x), f = f(x) are smooth on [−1, 1], and

b < 0 for x < 0, b = 0 for x = 0, b > 0 for x > 0, (3.2a)

db

dx≥ c0 > 0, ∀x ∈ [−1, 1]. (3.2b)

We start with the formal outer expansions uε ∼=∑∞

j=0 εjujl in x < 0 and uε ∼=

∑∞j=0 ε

jujrin x > 0. Substituting these expansions in Eq. (3.1)1 we find that, by identification at eachpower of ε,

O(1) : − bdu0ldx

= f in [−1, 0), −bdu0r

dx= f in (0, 1], (3.3a)

O(ε) : − bdu1ldx

= 0 in [−1, 0), −bdu1r

dx= 0 in (0, 1], (3.3b)

O(εj) : − bdujldx

=d2uj−2

l

dx2, in [−1, 0), −bdu

jr

dx=d2uj−2

r

dx2, in (0, 1], for j ≥ 2. (3.3c)

We firstly construct the outer expansions ujl , ujr as in (3.3). Here we impose the inflow

boundary conditions,

ujl (−1) = ujr(1) = 0, j ≥ 0, (3.4)

which will be justified below. We then notice with (3.3) that ujl = ujr = 0 for all odd j’s,1 ≤ j ≤ n. Furthermore, we are able to obtain the following explicit expressions:

u0l = −∫ x

−1b(s)−1f(s)ds, u0r =

∫ 1

xb(s)−1f(s)ds, (3.5a)


and for all j = 2k, k ≥ 1,

u2kl = −∫ x

−1b(s)−1 du

2(k−1)l

dx2(s)ds, u2kr =

∫ 1

xb(s)−1du

2(k−1)r

dx2(s)ds. (3.5b)

We follow the asymptotic analysis given in [73].

3.1. Interior layer analysis at arbitrary order εn. For later use, we first consider thehomogeneous boundary value problem with f = 0, with non-zero boundary conditions,

−ε2 d2uε

dx2− b

duε

dx= 0 in Ω,

uε(−1) = α, uε(1) = β.

(3.6)

To enforce the boundary values α, β at x = ±1 we incorporate the so-called interior layer θj.

Interior layers θj

To resolve the discrepancies between α and β if these numbers are different, we introduceas follows the so-called ordinary interior layers which are defined by the inner expansionsuε ∼=

∑∞j=0 ε

jθj with a stretched variable x = x/ε, θj = θj(x), x ∈ (−∞,∞). Using the formal

Taylor expansion for b = b(x) at x = 0 we obtain the asymptotic expansion for b:

b(x) =

∞∑

j=1

bjxj =

∞∑

j=1

bjεj xj; (3.7)

note that b0 = b(0) = 0 and b1 = bx(0) ≥ 1 by (3.2). Substituting (3.7) and the innerexpansions (

∑∞j=0 ε

jθj) for b and uε, respectively, in Eq. (3.1)1, we then obtain (with b0 = 0)the following formal expansion:

∞∑

j=0

−εj d

2θj

dx2− εj

[j∑

k=0

bj−k+1xj−k+1dθ

k

dx

]= 0. (3.8)

By identification at each power of ε, we find

O(1) : −d2θ0

dx2− b1x

dθ0

dx= 0, (3.9a)

O(ε) : −d2θ1

dx2− b1x

dθ1

dx= b2x

2 dθ0

dx(3.9b)

O(εj) : −d2θj

dx2− b1x

dθj

dx=

j−1∑

k=0

bj−k+1xj−k+1dθ

k

dx. (3.9c)

We impose the boundary conditions,

θ0(x = −1) = α, θ0(x = 1) = β, and θj(x = −1) = θj(x = 1) = 0, 1 ≤ j ≤ n. (3.10)

But for the purpose of the analysis below it is convenient to consider the approximate formof θj, namely θj satisfying equations (3.9) on all of R (for the variable x) with the followingboundary conditions:

θ0 → α as x→ −∞, θ0 → β as x→ ∞, (3.11a)

θj → 0 as x→ ±∞, j ≥ 1. (3.11b)


We show that θj and θj differ by an exponentially small term (see [73]), and

θ0 = c−10

[α

∫ ∞

xexp

(−b1s

2

2

)ds+ β

∫ x

−∞exp

(−b1s

2

2

)ds

], (3.12a)

θ1 = (α− β)b23−1c−1

0

∫ x

−∞s3 exp

(−b1s

2

2

)ds, (3.12b)

where c0 =∫∞−∞ exp(−b1s2/2)ds =

√2π/b1. The following pointwise and norm estimates for

θj, j ≥ 0, can then be derived ([73]).

Lemma 3.1. There exist positive constants κjm and c such that the following pointwise esti-mates hold

∣∣∣∣dmθj

dxm

∣∣∣∣ ≤ κjm

1 for j = 0 and m = 0,

ε−m exp

(−c |x|

ε

)for j ≥ 1 or m ≥ 1,

(3.13)

and for m ≥ 0,∥∥θj∥∥Hm(−1,1)

≤ κjm

(1 + ε−m+ 1

2

). (3.14)

We consider the compatible case which we now present.

3.1.1. f , b compatible. In this section, we consider the problem (3.1) with f arbitrary satisfyingthe compatibility conditions (3.15) below. If f(0) 6= 0, since b(0) = 0, the limit problem−bdu0/dx = f has an inconsistency at x = 0. That is its solution cannot be smooth. Toavoid the inconsistency between b and f , we assume in this section the following compatibilityconditions:

dif

dxi(0) = 0, i = 0, 1, · · · , 2n+ 1. (3.15)

If (3.15) does not hold, we will see that the solution uε of (3.1) possesses logarithmic sin-gularities at x = 0 as already indicated in the Introduction. Thanks to the compatibility

conditions (3.15), recalling the condition for b(x) as in (3.2b) the values of ujl (0−) and ujr(0+),

of dujl /dx(0−) and dujr/dx(0+) or of higher order derivatives, are finite if we take n sufficiently

large. More precisely, we have (see [73] for details):

Lemma 3.2. Let m ≥ 1 and k ≥ 0. Assume that the compatibility conditions (3.15) hold.Then, for m,k such that 0 ≤ m+ 2k ≤ 2n+ 2, there exists a positive constant κkm such that

∣∣∣∣dmu2kldxm

(0−)

∣∣∣∣ ,∣∣∣∣dmu2krdxm

(0+)

∣∣∣∣ ≤ κkm. (3.16)

Interior layers θjr, θjl , ζ

j

Assuming enough compatibility conditions, we can guarantee, as in Lemma 3.2, that |ujl (0−)|,|ujr(0+)|, |dujl /dx(0−)|, |dujr/dx(0+)| ≤ κj . In general, ujl (0

−) 6= ujr(0+). To resolve thesediscrepancies at x = 0, using the stretched variable x = x/ε, we introduce the functions

θjl (x), and θjr(x) which are defined as the solutions of the same equations (3.9) respectively on

(−∞, 0), and (0,∞) with the following boundary conditions.

θjr(x) = ujl (0−),

dθjrdx

(x) = εdθjrdx

= εdujldx

(0−) at x = 0, (3.17a)

θjl (x) = ujr(0+),

dθjldx

(x) = εdθjldx

= εdujrdx

(0+) at x = 0, (3.17b)


which allows us to determine the θjl , θjr explicitly. Notice that ujr = ujl = 0 for j odd. In

particular, for j = 0, 1, we find

θ0r = εdu0ldx

(0−)∫ x

0exp

(−b1s

2

2

)ds+ u0l (0

−), (3.18a)

θ1r = −εdu0l

dx(0−)b23

−1

∫ x

0s3 exp

(−b1s

2

2

)ds. (3.18b)

Here we note that as x→ ∞,

θ0r → εdu0ldx

(0−)cr,0 + u0l (0−) =: c0r,∞(ε), (3.19a)

θ1r → −εdu0l

dx(0−)b23

−1cr,1 =: c1r,∞(ε), (3.19b)

where cr,0 =∫∞0 exp

(−b1s2/2

)ds, cr,1 =

∫∞0 s3 exp

(−b1s2/2

)ds.

We denote by ϕ ∪ ψ the function on (−1, 1) equal to (the restriction of) ϕ on (−1, 0) and

to (the restriction of) ψ on (0, 1) and consider the functions ujl ∪ θjr and θjl ∪ u

jr. Note that

due to (3.17), these functions belong to C1([−1, 1]) and to H2(−1, 1). In general, the followingpointwise and norm estimates can be derived.

Lemma 3.3. Assume that the compatibility conditions (3.15) hold. Then there exist positiveconstants κ and c such that for x ∈ [0, 1], 0 ≤ j ≤ 2n+ 2,

∣∣∣∣∣dmθjldxm

∣∣∣∣∣ ,∣∣∣∣∣dmθjrdxm

∣∣∣∣∣ ≤ κ

1 for m = 0,

ε−m+1 exp(−cx

ε

)for m ≥ 1,

(3.20)

and for m ≥ 0,∥∥∥θjl∥∥∥Hm(0,1)

,∥∥θjr∥∥Hm(0,1)

≤ κ(1 + ε−m+ 3

2

). (3.21)

By our construction, we then notice that the functions gj := −(ujl ∪ θjr) − (θjl ∪ u

jr) attain

the values −θjl = −cjl,∞(ε) + e.s.t at x = −1 and −θjr = −cjr,∞(ε) + e.s.t at x = 1. To

remedy these discrepancies between gj and uε at the boundaries x = −1, 1 (we recall thatuε(−1) = uε(1) = 0), we introduce interior layers ζj similar to θj but we use different boundaryconditions: the ζj = ζj(x) satisfy (3.9) and

ζj = −θjl , at x = −1, ζj = −θjr, at x = 1 for j ≥ 0. (3.22)

The same estimates as in Lemma 3.1 hold for ζj with j = 0 and m = 0, and j ≥ 1 or m ≥ 1being replaced respectively by m = 0, and m ≥ 1.

We let

wεn = uε − ξεn − ηεn − ζεn, (3.23)

where

ξεn =2n∑

j=0

εj(ujl ∪ θjr), ηεn =2n∑

j=0

εj(θjl ∪ ujr), ζεn =2n∑

j=0

εjζj. (3.24)

From the outer expansions (3.3) and the interior layers θjr, θjl , ζ

j, after some elementarycalculations, we find that

Lεwεn = R1n +R2

n +R3n + e.s.t. in Ω, (3.25a)

wεn(−1) = wεn(1) = 0, (3.25b)


where

R1n = ε2n+2

(d2u2nldx2

∪ d2u2nrdx2

), (3.25c)

R2n =

2n∑

j=0

εj(dθjldx

∪ dθjrdx

)Rj,2n(b), (3.25d)

R3n =

2n∑

j=0

εjdζj

dxRj,2n(b), (3.25e)

with

Rj,2n(b) = b(x)−2n+1−j∑

k=1

bkxk. (3.25f)

Here we used the fact that, by permuting the summations:

2n∑

j=0

εj

j∑

k=0

bj−k+1xj−k+1dθ

k

dx

=

2n∑

j=0

j∑

k=0

bj−k+1xj−k+1εk

dθk

dx

=2n∑

j=0

εjdθj

dx

2n+1−j∑

k=1

bkxk

=

2n∑

j=0

εjdθj

dx

(b(x)−Rj,2n(b)

).

(3.26)

We now estimate the L2- norm of Rn1 . We first notice that, by Taylor’s expansion,

|Rj,2n(b)| =∣∣∣∣∣b(x)−

2n+1−j∑

k=1

bkxk

∣∣∣∣∣ ≤ κ|x|2n+2−j ≤ κε2n+2−j |x|2n+2−j . (3.27)

We then estimate the L2- norms of R1n, R

2n and R3

n. From (3.16) with ujl = ujr = 0, j odd,

we note that |d2ujl /dx2(0−)|, |d2ujr/dx2(0+)| ≤ κn, 0 ≤ j ≤ 2n+ 1. Hence, we easily find that

‖R1n‖L2(Ω) ≤ κnε

2n+2. (3.28)

Using (3.27) and the pointwise estimates (3.20), we find

|R2n| ≤ κnε

2n+22n∑

j=0

|x|2n+2−j

(∣∣∣dθjrdx

∣∣∣χ(0,1] +∣∣∣dθjldx

∣∣∣χ[−1,0)

)

≤ κnε2n+2 exp

(−c|x|

2ε

),

(3.29)

where χA(x) is the characteristic function of the set A, and

‖R2n‖L2(Ω) ≤ κε2n+

5

2 , ‖R3n‖L2(Ω) ≤ κε2n+

3

2 . (3.30)

We define the weighted energy norm:

‖u‖ε = ‖u‖L2(Ω) +√ε‖u‖H1(Ω), (3.31)

and conclude that we have proved the following:

Theorem 3.1. Assume that the compatibility conditions (3.15) hold. Let uε be the solution of(3.1). Then there exists a constant κ > 0 independent of ε such that

‖uε − ξεn − ηεn − ζεn‖ε ≤ κε2n+3

2 , (3.32)

‖uε − ξεn − ηεn − ζεn‖H2(Ω) ≤ κε2n−3

2 , (3.33)

where ξεn, ηεn and ζεn are as in (3.24).


We now consider the case where f does not satisfy the compatibility conditions (3.15).

3.1.2. f , b noncompatible. We now want to remove the compatibility conditions. For thatpurpose, we decompose f , as explained below:

f = f +

2n+1∑

k=0

γkBk(x), (3.34a)

where

B0 = bx(x), B1 = b(x), (3.34b)

Bk+2 = b(x)

∫ x

0Bk(s)ds, k ≥ 0. (3.34c)

Note that since diBk/dxi(0) = 0 for i < k and diBk/dx

i(0) 6= 0 for i = k (recall b(0) = 0 andbx(0) ≥ c0 > 0), we can recursively find all the γk, k ≥ 0 so that the compatibility conditions

(3.15) for f = f holds for 0 ≤ i ≤ 2n + 1. For f = Bk(x), k odd, it turns out that the outersolutions are bounded in the neighborhood of x = 0 (see [73]) and this enables us to performthe same asymptotic analysis as before.

Hence, Theorem 3.1 holds for f = f , where

f = f −n∑

k=0

γ2kB2k(x). (3.35)

Thus, thanks to the superposition of solutions, it suffices to consider the special cases forf = B2J :

−ε2 d2uε

dx2− b

duε

dx= B2J in Ω,

uε(−1) = uε(1) = 0.

(3.36)

Following the analysis for f = B2J in [79] we need the following correctors,

u∗lε,m =

m∑

j=0

εj(u∗jl (x) ∪ u∗jl (0−)), (3.37a)

u∗rε,m =

m∑

j=0

εj(u∗jr (0+) ∪ u∗jr (x)), (3.37b)

ζ∗ε,m =m∑

j=0

εjζ∗j , (3.37c)

ϕ∗ε,p =

p∑

j=2J

δjϕj , for p ≥ 2J, (3.37d)

ς∗ε,q =q∑

j=0

εjςj , for p ≥ 2J, (3.37e)

with a parameter δ > 0, to be determined below. Here, the functions in (3.37) are defined asfollows:

• For u∗jl , u∗jr , for j′ ≥ 0, u∗,2j

′+1l = u∗,2j

′+1r = 0,

u∗0l = −∫ x

−1b(s)−1ρ(s)B2J (s)ds, u

∗0r =

∫ 1

xb(s)−1ρ(s)B2J (s)ds, (3.38)


where

ρ(x) =

eψ1

(xδ

)if |x| < δ,

1 if δ ≤ |x| <∞,(3.39)

with

ψ1(x) = exp(−(1− exp(1− (1− x2)−1)

)−1), (3.40)

and for j′ ≥ 1,

u∗2j′

l = −∫ x

−1b(s)−1 d

2u∗2(j′−1)l

dx2(s)ds, u∗2j

′

r =

∫ 1

xb(s)−1 d

2u∗2(j′−1)r

dx2(s)ds. (3.41)

• For ζ∗j, using the stretched variable x = x/ε we write

b(x) =∞∑

j=1

bjxj =

∞∑

j=1

bjεj xj, (3.42)

where bj = b(j)(0)/j! and we obtain the interior layer equations,

− d2ζ∗0

dx2− b1x

dζ∗0

dx= 0, (3.43a)

− d2ζ∗j

dx2− b1x

dζ∗j

dx=

j−1∑

k=0

bj−k+1xj−k+1dζ

∗k

dx, j ≥ 1. (3.43b)

supplemented with the boundary conditions, ζ∗j → −u∗jr (0+) as x→ −∞, ζ∗j →−u∗jl (0−) as x→ ∞.

• For ϕj , using the stretched variable x = x/δ we write

b(x) =

∞∑

j=1

bjxj =

∞∑

j=1

bjδj xj, (3.44)

and we obtain an outer expansion,

− ε2δ−2 d2ϕ2J

dx2− b1x

dϕ2J

dx= (1− ρ(δx))δ−2JB2J(δx), (3.45a)

− ε2δ−2 d2ϕj

dx2− b1x

dϕj

dx=

j−1∑

k=2J

bj+1−kxj+1−k dϕ

k

dx, j ≥ 2J + 1, (3.45b)

supplemented with the boundary conditions, ϕj(0) = 0, and ϕjx → 0 as x→ ±∞.

• For ςj , using the stretched variable x = x/ε, we obtain the interior layer equations,

− d2ς0

dx2− b1x

dς0

dx= 0, (3.46a)

− d2ςj

dx2− b1x

dςj

dx=

j−1∑

k=0

bj−k+1xj−k+1dς

k

dx, j ≥ 1, (3.46b)

supplemented with the boundary conditions, ς0 → −ϕ∗εp(−1) as x→ ∞, ς0 → −ϕ∗

εp(1)

as x→ −∞, ςj → 0 as x→ ±∞, j ≥ 1.

Along with the analysis presented above for f = f as in (3.35) we conclude for f noncom-patible:


Theorem 3.2. Let uε be the solution of (3.1). For n,N > 0 integers, define

ξεn,N =

2n∑

j=0

εj(ujl ∪ θjr) +n∑

k=0

γ2kξ∗2kε,6N , (3.47)

ηεn,N =

2n∑

j=0

εj(θjl ∪ ujr) +n∑

k=0

γ2kη∗2kε,6N , (3.48)

ζεn,N =

2n∑

j=0

εj ζj +

n∑

k=0

γ2kζ∗2kε,6N , (3.49)

ϕεn,N =

minN,n∑

k=0

γ2kϕ∗2kε,2N , (3.50)

ςεn,N =

minN,n∑

k=0

γ2kς∗2kε,N−k, (3.51)

where the γk’s are determined above, ujl = ujl , ujr = ujr, θ

jl = θjl , θ

jr = θjr and ζj = ζj are as in

(3.24) corresponding to f = f as in (3.35), and ξ∗2kε,6N = ξ∗ε,6N , η∗2kε,6N = η∗ε,6N , ζ∗2kε,6N = ζ∗ε,6N ,

ϕ∗2kε,2N = ϕ∗

ε,2N , ς∗2kε,N−k = ς∗ε,N−k are as in (3.37) with J = k and δ = ε5

6 .Then there exists κ = κn,N > 0, independent of ε, such that

‖uε − ξεn,N − ηεn,N − ζεn,N − ϕε2N − ςεn,N‖ε ≤ κ(ε2n+3

2 + εn+3

4 ), (3.52)

‖uε − ξεn,N − ηεn,N − ζεn,N − ϕε2N − ςεn,N‖H2(Ω) ≤ κε−3(ε2n+3

2 + εn+3

4 ). (3.53)

4. Convection-diffusion equations in a domain with corners (rectangle)

We consider a singularly perturbed convection-diffusion equation in a rectangular domainΩ = (0, 1) × (0, 1):

−ε∆uε − ∂uε

∂x= f in Ω,

uε = 0 on ∂Ω.(4.1)

Here ε is a small but strictly positive diffusivity parameter, and f = f(x, y) is a given smoothfunction with ‖Dαf‖L2(Ω) ≤ κα, independent of ε, for some α’s as needed in the analysis below.

In a related earlier work, [122], the following problem similar to (4.1) was discussed. More

precisely, in a rectangular domain Ω = (0, l0)× (0, l1), the authors considered

−ε′∆uε′ − b∂uε

′

∂x+ cuε

′= f in Ω,

uε′= g on ∂Ω,

(4.2)

where b > 0, c ≥ 0 are constants, f is a given smooth function in Ω. The function g = g(x, y)

is assumed to be continuous on ∂Ω and smooth on each edge of ∂Ω. Using a simple changeof variables which maps Ω onto Ω, and setting uε

′= uεeλx with a suitable λ, our analysis of

(4.1) in this article is applicable to (4.2) as well (see [42]).We will use a smooth cut-off function σ = σ(r), independent of ε, such that

σ(r) =

1 for 0 ≤ r ≤ 1/2,

0 for r ≥ 3/4,(4.3)


To study the asymptotic behavior of uε, solution of (4.1), we propose an asymptotic expansionof the following type:

uε ∼=∞∑

j=0

εj(uj +Θj

). (4.4)

Here, at each order of εj , j ≥ 0, uj corresponds to the outer expansion (outside of the boundarylayer) of uε as explained below. To balance the discrepancy on the boundary ∂Ω of uε and ofthe uj , 0 ≤ j ≤ n, we introduce the correctors Θj, 0 ≤ j ≤ n, which will contribute mainlyinside of the boundary layer: Θj will be itself the sum of several boundary layer functions aswe shall see.

To determine the asymptotic expansion (4.4), we start with the outer expansion for uε,uε ∼=

∑∞j=0 ε

juj. Inserting formally this expansion into (4.1), we find the equations for the uj :

−∂u0

∂x= f, −∂u

j

∂x= ∆uj−1, j ≥ 1. (4.5)

We supplement these equations with the inflow zero boundary conditions. Integrating theequations in x, we find the smooth outer solutions uj in the form,

u0 =

∫ 1

xf(x1, y) dx1, uj =

∫ 1

x∆uj−1(x1, y) dx1, j ≥ 1. (4.6)

For simplicity in the analysis below, we first decompose

f = f1 + f2, with f1 = σ(y)f , f2 = (1− σ(y))f . (4.7)

Thanks to the symmetry and by superposition, it suffices to consider Eq. (4.1) with f = f1.Thus, we may assume that f is infinitely flat at y = 1, i.e. ∂αy f = 0 at y = 1 for all α ≥ 0.

Hence, we notice that the uj ’s satisfy the boundary condition (4.1)2 along the edges x = 1 andy = 1, and not on the two other sides of ∂Ω, y = 0, and x = 0. Hence we expect boundarylayers to occur near those edges, and they are constructed as

Θj = ϕj + ξj + θj + ζj, j ≥ 0, (4.8)

where

ϕj is the parabolic boundary layer near y = 0,

ξj is the elliptic boundary layer resolving the compatibility issues,in the construction of ϕj , at the corners (1, 0),

θj is the ordinary boundary layer near x = 0,

ζj is the ordinary corner layer which manages the effectof ϕj along x = 0.

4.1. Boundary layer analysis at order ε0. Following the construction of the boundarylayers suggested in [42], we list them briefly below.

Parabolic boundary layers (PBL)At the bottom boundary, i.e., at y = 0, in general uε−u0 = −u0 6= 0. To resolve this difficulty,we construct below the parabolic boundary layer corrector ϕ0. Using the stretched variabley = y/

√ε, we find the dominating differential operators which lead to the parabolic boundary


layer corrector. For ϕj = ϕj(x, y), 0 ≤ j ≤ n,

−ε∂2ϕ0

∂y2− ∂ϕ0

∂x= 0 for (y, x) ∈ R

+ × (0, 1),

ϕ0 = −u0(x, 0) at y = 0,

ϕ0 → 0 as y → ∞,

ϕ0 = 0 at x = 1.

(4.9)

From [11], [71], [39] or [122], we recall the explicit expressions of ϕ0 = ϕ0(x, y),

ϕ0 = −√

2

π

∫ ∞

y/√

2(1−x)exp

(− y21

2

)u0(x+

y2

2y21, 0)dy1, (4.10)

Since u0(1, 0) = 0, some pointwise and Lp estimates for the ϕ0 can be deduced (see theAppendix in [42]).

Lemma 4.1. For each i,m, 0 ≤ i+m ≤ 1 and s ≥ 0, we have the estimates:∣∣∣∣y

s ∂i+mϕ0

∂xi∂ym

∣∣∣∣ ≤ κεs−m

2 exp(− c

y√ε

), p.t.w., (4.11)

and∥∥∥∥y

s ∂i+mϕ0

∂xi∂ym

∥∥∥∥Lp(Ω)

≤ κεs−m

2+ 1

2p , (4.12)

for a generic constant c > 0 independent of x, y and ε.

Ordinary boundary layers (OBL)We note that in general uε − u0 − ϕ0 = −u0 − ϕ0 6= 0 at x = 0. To resolve this difficulty, wefirst handle the value −u0 at x = 0 by the so-called ordinary boundary layer θ0 introduced inthis section. Using the stretched variable x = x/ε, we find the equations of θ0 = θ0(x, y):

−ε∂2θ0

∂x2− ∂θ0

∂x= 0 in Ω,

θ0 = −u0(0, y) at x = 0,

θ0 → 0 as x→ ∞.

(4.13)

The explicit expression of θ0 is readily available,

θ0 = −u0(0, y) exp(−xε

), (4.14)

and we easily deduce the following estimates.

Lemma 4.2. We have the estimates:∣∣∣∣x

r ∂i+mθ0

∂xi∂ym

∣∣∣∣ ≤ κεr−i exp(−cx

ε

), r, i,m ≥ 0, p.t.w., (4.15)

and∥∥∥∥x

r ∂i+mθ0

∂xi∂ym

∥∥∥∥Lp(Ω)

≤ κεr−i+ 1

p , r, i,m ≥ 0. (4.16)

for a constant c independent of x, y and ε.


Ordinary corner layers (OCL)We next handle the value −ϕ0 at x = 0 by introducing the ordinary corner layer correctorsζj. Using the stretched variables x = x/ε and y = y/

√ε near (0, 0), we find the equations for

ζ0 = ζ0(x, y):

−ε∂2ζ0

∂x2− ∂ζ0

∂x= 0 in Ω,

ζ0 = −ϕ0(0, y√

ε

)at x = 0,

ζ0 → 0 as x→ ∞.

(4.17)

The explicit solution ζ0 is readily available,

ζ0 = −ϕ0(0,

y√ε

)exp

(−xε

). (4.18)

Using Lemma 4.1 and (4.18), we obtain the estimates below.

Lemma 4.3. For 0 ≤ i+m ≤ 1, and r, s ≥ 0, we have the estimates:∣∣∣∣x

rys∂i+mζ0

∂xi∂ym

∣∣∣∣ ≤ κεr−i+ s−m2 exp

(− c(xε+

y√ε

)), p.t.w., (4.19)

and∥∥∥∥x

rys∂i+mζ0

∂xi∂ym

∥∥∥∥Lp(Ω)

≤ κεr−i+ s−m2

+ 3

2p , (4.20)


Now, we finally obtain that

uε − (u0 + ϕ0 + θ0 + ζ0)

=

0 at x = 0,

θ0 + ζ0 = e.s.t. at x = 1,

θ0 + ζ0 = (−u0 − ϕ0)(0, 0) exp(−xε

)= 0 at y = 0,

ϕ0 = e.s.t. at y = 1.

(4.21)

Hence, with the boundary layers ϕ0, θ0, ζ0, we resolve all the discrepancies between u0 and thezero boundary condition except for the e.s.t. terms.

To handle the e.s.t. on ∂Ω, here and after as indicated in (4.21), using the smooth cut-offfunctions σ(x), σ(y) as in (4.3), we define

Θ0 = ϕ0 + θ0 + ζ0, Θ0= ϕ0 + θ0 + ζ0, (4.22)

where ϕ0 = σ(y)ϕ0, θ0 = σ(x)θ0, and ζ0 = σ(x)ζ0, ζ0 as in (4.17) but with ϕ0(0, y/√ε) being

replaced by ϕ0(0, y/√ε). We then set

wε,0 = uε −(u0 +Θ0

), wε,0 = uε −

(u0 +Θ

0). (4.23)

Let Lεu = −ε∆u− ∂u/∂x. Noting that wε,0 = 0 on ∂Ω, the equation of wε,0 now reads:

Lεwε,0 = ε

(∆u0 +

∂2ϕ0

∂x2+∂2θ0

∂y2+∂2ζ0

∂y2

)+ Lε(wε,0 − wε,0) in Ω,

wε,0 = 0 on ∂Ω.

(4.24)


Observing that

‖wε,0 − wε,0‖H1 ≤ κ exp

(− c√

ε

), c > 0, (4.25)

we thus have∣∣∣∣∫

ΩLε(wε,0 − wε,0)wε,0dxdy

∣∣∣∣ ≤ κ exp

(− c√

ε

)‖wε,0‖H1 , c > 0. (4.26)

Hence, this quantity will be absorbed by other norms and we will omit it.We now multiply (4.59) by exwε,0 and integrate over Ω. Integrating by parts and using

Lemmas 4.1, 4.2 and 4.3, we find that

ε‖wε,0‖2H1 +1− ε

2‖wε,0‖2L2

≤ ε

∫

Ω

∣∣∆u0∣∣∣∣wε,0

∣∣dxdy

+ ε

∫

Ω

∣∣∣∣∂ϕ0

∂x+∂θ0

∂y+∂ζ0

∂y

∣∣∣∣∣∣∇wε,0

∣∣dxdy

≤ κε3

2 +1

4‖wε,0‖2L2 +

ε

2‖wε,0‖2H1 .

(4.27)

If we impose one compatibility condition at the inflow, i.e. f(1, 0) = 0, the error estimateabove is improved. If f(1, 0) = 0, we note that u0(1, 0) = (∂u0/∂x)(1, 0) = 0. Then, Lemmas4.1 and 4.3 hold for 0 ≤ i+m ≤ 2 (see Lemmas 4.4 and 4.6 below and see also the Appendixin[42]). Using the Hardy inequality (see, e.g., [63]) we then find that

ε‖wε,0‖2H1 +1− ε

2‖wε,0‖2L2

≤ κε

∣∣∣∣∫

Ω

(∆u0 +

∂2ϕ0

∂x2+∂2θ0

∂y2

)wε,0dxdy

∣∣∣∣

+ κε

∫

Ωx

∣∣∣∣∂2ζ0

∂y2

∣∣∣∣∣∣∣∣wε,0

x

∣∣∣∣dxdy

≤ κε2 +1

4‖wε,0‖2L2 +

ε

2‖wε,0‖2H1 .

(4.28)

The H2 estimates follow from the elliptic regularity theory.The above errors were derived when f was infinitely flat at y = 1, and hence we have

constructed boundary and corner layers ϕ0 = ϕ0B , ζ

0 = ζ0B only at the bottom edge alongy = 0. For a general f , along the top edge, y = 1, the correctors are similarly constructed anddenoted respectively by ϕ0

T , and ζ0T (see Figure 1).

Thanks to the superposition of solutions (see (4.7) above), we conclude the following errorestimates.

Theorem 4.1. Let uε be the solution of Eq. (4.1). Without any compatibility conditions onf , we have

∥∥∥uε − (u0 + ϕ0B + ϕ0

T + θ0 + ζ0B + ζ0T )∥∥∥ε≤ κε

3

4 . (4.29)

With the compatibility conditions,

f(1, 0) = f(1, 1) = 0,


we have ∥∥∥uε − (u0 + ϕ0B + ϕ0

T + θ0 + ζ0B + ζ0T )∥∥∥ε≤ κε, (4.30)

∥∥∥uε − (u0 + ϕ0B + ϕ0

T + θ0 + ζ0B + ζ0T )∥∥∥H2(Ω)

≤ κε−1

2 . (4.31)

Remark 4.1. Without any compatibility condition, we will need the elliptic boundary layersξ0 = ξ0B + ξ0T as in Theorem 4.2 for n = 0 below.

ujθj

ϕjT

ϕjB

ξjT

ξjBζ

jB

ζjT

0

1

y

1 x

Figure 1. Location of the outer solution uj and the boundary layer correctors.

4.2. Boundary layer analysis at arbitrary order εn, n ≥ 0. For simplicity in the analysis,we assume that f is infinitely flat at y = 1. We now construct high order boundary layers Θj

so that uε −∑nj=0 ε

j(uj +Θj

)is small.

Parabolic boundary layers (PBL)At the bottom boundary, i.e., at y = 0, in general −uj 6= 0. To resolve this difficulty, wefirst construct the parabolic boundary layer correctors ϕj . We formally insert the asymptoticexpansion uε ∼=

∑∞j=0 ε

jϕj into (4.1). Using the stretched variable y = y/√ε and collecting the

terms of the same order εj , we find the equations of the parabolic boundary layer correctors:

−ε ∂2ϕj

∂y2− ∂ϕj

∂x=∂2ϕj−1

∂x2in Ω, (4.32)

where we set ϕ−1 = 0.To resolve the compatibility issues for the parabolic boundary layers at the inflow corner

(1, 0), using the smooth cut-off function σ(r) as in (4.3), we set

γj(x) = γj(x)σ(1 − x), (4.33)

where

γj(x)= −2n+1−2j∑

i=1

(x− 1)i

i!

∂iuj

∂xi(1, 0), 0 ≤ j ≤ n. (4.34)

The cut-off suppresses additional discrepancies at the outflow which can lead to the so-calledelliptic corner layers (see [122]). Then, introducing the stretched variable y = y/

√ε, the


equation (4.32) is written as well as a compatible boundary condition as follows. For ϕj =ϕj(x, y), 0 ≤ j ≤ n,

−ε∂2ϕj

∂y2− ∂ϕj

∂x=∂2ϕj−1

∂x2for (y, x) ∈ R+ × (0, 1),

ϕj = gj(x) := −uj(x, 0) − γj(x) at y = 0,

ϕj → 0 as y → ∞,

ϕj = 0 at x = 1.

(4.35)

Note that the smooth boundary conditions gj(x), 0 ≤ j ≤ n, along y = 0 are compatible withthe other zero boundary conditions at x = 1 in the sense that, for each 0 ≤ j ≤ n,

∂igj

dxi(1) = 0, 0 ≤ i ≤ 2n+ 1− 2j.

From [11], [71], [39] or [122], we recall the explicit expressions of ϕj = ϕj(x, y), 0 ≤ j ≤ n,

ϕ0 =

√2

π

∫ ∞

y/√

2(1−x)exp

(− y21

2

)g0(x+

y2

2y21

)dy1, (4.36)

ϕj =

√2

π

∫ ∞

y/√

2(1−x)exp

(− y21

2

)gj(x+

y2

2y21

)dy1

+1

2√π

∫ 1−x

0

∫ ∞

0

1√x1

exp

(− (y − y1)

2

4x1

)− exp

(− (y + y1)

2

4x1

)

× ∂2ϕj−1

∂x2(x+ x1, y1) dy1dx1.

(4.37)

One can verify that the expression (4.36) or (4.37) of the heat solution is identical to thatin (2.134) or (2.215) with (2.216) by using a change of variable and integrating by parts.

Some pointwise and Lp estimates for the ϕj , 0 ≤ j ≤ n can be deduced (see the Appendixin [42]):

Lemma 4.4. For each 0 ≤ j ≤ n, 0 ≤ i+m ≤ 2n+ 2− 2j and s ≥ 0, we have the estimates:∣∣∣∣y

s ∂i+mϕj

∂xi∂ym

∣∣∣∣ ≤ κεs−m

2 exp(− c

y√ε

), p.t.w., (4.38)

and ∥∥∥∥ys ∂

i+mϕj

∂xi∂ym

∥∥∥∥Lp(Ω)

≤ κεs−m

2+ 1

2p , (4.39)


Elliptic boundary layers (EBL)In Section 4.2, to construct the consistent parabolic boundary layer correctors ϕj , we consideredthe γj above. To cancel γj along y = 0, we introduce the elliptic boundary layers ξj :

−ε ∂2ξj

∂x2− ε

∂2ξj

∂y2− ∂ξj

∂x= 0 in Ω,

ξj = γj(x) at y = 0,

ξj = 0 at x = 0, 1, or y = 1.

(4.40)

Performing the energy estimate on (4.40), we find that

‖ξj‖ε ≤ κε1

4 , 0 ≤ j ≤ n. (4.41)


Ordinary boundary and corner layers (OBL, OCL)Finally, since in general −uj 6= 0 at x = 0, we resolve by introducing the so-called ordinaryboundary layer θj. We insert a formal expansion uε ∼=

∑∞j=0 ε

j θj into the diffusive equation

(4.1). Then, using the stretched variable x = x/ε, we find the equations of θj = θj(x, y):

−ε∂2θj

∂x2− ∂θj

∂x= ε−1∂

2θj−2

∂y2in Ω, (4.42)

where we set θ−1 = θ−2 = 0. Hence we supplement the equations (4.42) with the boundaryconditions,

θj = −uj(0, y) at x = 0, θj → 0 as x→ ∞. (4.43)

To account for the discrepancy of −ϕj at x = 0, we need to introduce the ordinary cornerlayers ζj. We insert a formal expansion uε ∼=

∑∞j=0 ε

j ζj into (4.1). Then, using the stretched

variables x = x/ε and y = y/√ε near (0, 0), we collect the terms of order εj . As a result, we

find the equations for ζj = ζj(x, y):

− ε∂2ζj

∂x2− ∂ζj

∂x=ζj−1

∂y2in Ω. (4.44)

Here, we set ζ−1 = 0. At each order of εj , to cancel the error −ϕj near the corner (0, 0), wesupplement the equations (4.44) with the boundary conditions,

ζj = −ϕj(0,

y√ε

)at x = 0, ζj → 0 as x→ ∞. (4.45)

However, in general −θj − ζj 6= 0 at y = 0 and hence we expect further discrepancies whichlead to the so-called elliptic corner layers near (0, 0) (see [122]). To avoid the the use of theseelliptic corner layers, we consider at once the discrepancies between −uj(0, y) − ϕj

(0, y/

√ε)

at x = 0 and the zero boundary condition. Combining the equations (4.42) - (4.45) and usingthe formal expansion uε ∼=

∑∞j=0 ε

jϑj with the stretched variable x, we define the ϑj:

−ε∂2ϑj

∂x2− ∂ϑj

∂x=∂2ϑj−1

∂y2in Ω,

ϑj = −uj(0, y)− ϕj(0, y√

ε

)at x = 0,

ϑj → 0 as x→ ∞,

(4.46)

where we set ϑ−1 = 0. The explicit solutions ϑj of (4.46) are inductively found to be of theform,

ϑj = P j(xε, y,

y√ε

)exp

(−xε

), (4.47)

where P j = P j(xε , y, y/

√ε)

=∑j

k=0Ak(y)xk is a polynomial in x/ε of degree j whose

coefficients are of the form

Ak(y) =

j∑

s=0

λs,kεs∂

2saj−s

∂y2s(y), k = 0, 1, · · · , j, (4.48)

where ak(y) = −uk(0, y) − ϕk(0, y/√ε), λs,k independent of ε, and P j(0, y, y/

√ε) = aj(y) =

−uj(0, y) − ϕj(0, y/√ε). Indeed, this expression is obvious for j = 0. At order j, we assume

that the coefficients are of the same form as in (4.48). At order j + 1, from (4.46)1 using thestretched variable x we find that

−∂2ϑj+1

∂x2− ∂ϑj+1

∂x= ε

∂2ϑj

∂y2=

j∑

k=0

[j∑

s=0

λs,kεs+1∂

2s+2aj−s(y)

∂y2s+2

]xk exp(−x). (4.49)


Imposing the boundary condition ϑj+1 = aj+1(y) = −uj+1(0, y)− ϕj+1(0, y/

√ε)at x = 0 and

solving for ϑj in the equation (4.49), we find that the coefficients are of the form (4.48) withj replaced by j + 1.

We now write ϑj = θj + ζj where

θj = P j1

(xε, y)exp

(−xε

), (4.50)

ζj = P j2

(xε,y√ε

)exp

(−xε

), (4.51)

where P j1 (x/ε, y) is a polynomial in x/ε of degree j whose coefficients are

j∑

s=0

λsεs(∂2suj−s/∂y2s)(0, y),

λs is independent of ε, and Pj1 (0, y) = −uj(0, y), and P j

2

(x/ε, y/

√ε)is a polynomial in x/ε of

degree j whose coefficients are

j∑

s=0

λsεs(∂2sϕj−s/∂y2s)(0, y/

√ε),

λs is independent of ε, and P j2 (0, y) = −ϕj(0, y/

√ε).

Based on (4.50), one can prove the following estimates (see [42]).

Lemma 4.5. For each 0 ≤ j ≤ n, we have the estimates:

∣∣∣∣xr ∂

i+mθj

∂xi∂ym

∣∣∣∣ ≤ κεr−i exp(−cx

ε

), r, i,m ≥ 0, p.t.w., (4.52)

and

∥∥∥∥xr ∂

i+mθj

∂xi∂ym

∥∥∥∥Lp(Ω)

≤ κεr−i+ 1

p , r, i,m ≥ 0. (4.53)

for a constant c independent of x, y and ε.

Using Lemma 4.4 and (4.51), we obtain in [42] the estimates below.

Lemma 4.6. For each 0 ≤ j ≤ n, 0 ≤ i+m ≤ 2n+2−2j, and r, s ≥ 0, we have the estimates:

∣∣∣∣xrys

∂i+mζj

∂xi∂ym

∣∣∣∣ ≤ κεr−i+ s−m2 exp

(− c(xε+

y√ε

)), p.t.w., (4.54)

and

∥∥∥∥xrys

∂i+mζj

∂xi∂ym

∥∥∥∥Lp(Ω)

≤ κεr−i+ s−m2

+ 3

2p , (4.55)



Now, we finally obtain that

uε −n∑

j=0

εj(uj + ϕj + ξj + θj + ζj)

=

0 at x = 0,

n∑

j=0

εj(θj + ζj) = e.s.t. at x = 1,

n∑

j=0

εj(−uj − ϕj)(0, 0) exp(−xε

)= 0 at y = 0,

n∑

j=0

εjϕj = e.s.t. at y = 1.

(4.56)

Hence, with the boundary layers, we resolve all the discrepancies between the uj and the zeroboundary value except for the e.s.t. terms.

As before, using the smooth cut-off functions σ(x), σ(y), we define

Θj = ϕj + ξj + θj + ζj, Θj= ϕj + ξj + θj + ζj (4.57)

where ϕj = σ(y)ϕj , θj = σ(x)θj, θj as in (4.50), and ζj = σ(x)ζj , ζj as in (4.51) but withϕj(0, y/

√ε) being replaced by ϕj(0, y/

√ε). Then we set

wεn = uε −n∑

j=0

εj(uj +Θj

), wεn = uε −

n∑

j=0

εj(uj +Θ

j). (4.58)

We note that wεn = 0 on ∂Ω. As explained in (4.25)-(4.26), we similarly drop the termLε(wεn − wεn). From (4.1), (4.5), (4.35) and (4.46), the equation of wεn then reads:

Lεwεn = εn+1

(∆un +

∂2ϕn

∂x2+∂2θn

∂y2+∂2ζn

∂y2

)in Ω,

wεn = 0 on ∂Ω.

(4.59)

We multiply (4.59) by exwεn and integrate over Ω. Then, using Lemmas 4.4, 4.5, 4.6 andthe Hardy inequality, we find that

ε‖wεn‖2H1 +1− ε

2‖wεn‖2L2

≤ κεn+1

∣∣∣∣∫

Ω

(∆un +

∂2ϕn

∂x2+∂2θn

∂y2

)wεndxdy

∣∣∣∣

+ κεn+1

∫

Ωx

∣∣∣∣∂2ζn

∂y2

∣∣∣∣∣∣∣∣wεn

x

∣∣∣∣dxdy

≤ κε2n+2 +1

4‖wεn‖2L2 +

ε

2‖wεn‖2H1 .

(4.60)

We have constructed boundary and corner layers at bottom edge along y = 0 respectively

denoted by ϕj = ϕjB , ξ

j = ξjB and ζj = ζjB. For a general f , as explained before, we similarly

construct boundary and corner layers along the top edge, y = 1, denoted respectively by ϕjT ,

ξjT and ζjT (see Figure 1).By linearity and superposition of solutions, we conclude the following theorem.


Theorem 4.2. Let uε be the solutions of Eq. (4.1). Without any compatibility conditions onf , we have for n ≥ 0,

∥∥∥uε −n∑

j=0

εj(uj + ϕj + ξj + θj + ζj

)∥∥∥ε≤ κεn+1, (4.61)

∥∥∥uε −n∑

j=0

εj(uj + ϕj + ξj + θj + ζj

)∥∥∥H2(Ω)

≤ κεn−1

2 , (4.62)

where ϕj = ϕjB + ϕj

T , ξj = ξjB + ξjT and ζj = ζjB + ζjT .

Remark 4.2. When f satisfies certain compatibility conditions as indicated in (4.34) so that

∂iuj

∂xi(1, 0) =

∂iuj

∂xi(1, 1) = 0, 1 ≤ i ≤ 2n+ 1− 2j, 0 ≤ j ≤ n, (4.63)

the elliptic boundary layers ξj are not needed in Theorem 4.2.

5. Concluding Remarks

Some recent progresses on the boundary layer theory are reviewed in this article. As wesaid in the Introduction, aiming to study the asymptotic behavior of the solutions to somesingular perturbation problems, we construct the boundary layer (or interior layer) correctorsand obtain the full structural information of

(1) the boundary layers of the reaction-diffusion equation in a smooth curved domain(2) the boundary and initial layers of the heat equation in a smooth curved domain(3) the interior layers of the convection-diffusion equation with turning points in an interval

domain(4) the interaction of the boundary and corner layers for the convection-diffusion equation

in a rectangular domain

Depending on, e.g., the characteristics of the corresponding limit problems and the geometryof the domain, various types of boundary layer correctors are introduced; the interior layercorrectors near the turning point, the ordinary, parabolic, and elliptic boundary layer correctorsas well as the initial layer correctors. These correctors successfully balance the discrepanciesbetween the diffusive (or viscous) solution and its limit solution at zero diffusivity (or viscosity)mainly inside of the boundary layers.

Ongoing projects along the same lines as those in this article, we have been generalizingthe methodologies introduced here for some equations in fluid mechanics, e.g., the stationaryor evolutionary Stokes, (linearized) Navier-Stokes, and (linearized) primitive equations. Thesecurrent projects will confirm that our method of correctors can be made useful in the analysisand numerical computations of many singular perturbation problems.

Acknowledgements

This work was supported in part by the Research Fund of Indiana University, the NSFGrants DMS 1206438, DMS 1212141 and DMS 1510249, and the National Research Foundationof Korea(NRF) grant funded by the Korea government(MSIP) (2012R1A1B3001167).


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1 Department of Mathematics, University of Louisville, 328 Natural Sciences Building, Louisville,KY 40292, U.S.A.

2 Department of Mathematical Sciences, School of Natural Science, Ulsan National Instituteof Science and Technology, UNIST-gil 50, Ulsan 689-798, Republic of Korea

3 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831East Third Street, Bloomington, IN 47405, U.S.A.

E-mail address: [email protected] address: [email protected] address: [email protected]

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