Boundary layer analysis of nonlinear reaction-diffusion equations in a smooth domain

Chang-Yeol Jung; Eunhee Park; Roger Temam

doi:10.1515/anona-2015-0148

Artikel Open Access

Boundary layer analysis of nonlinear reaction-diffusion equations in a smooth domain

Chang-Yeol Jung , Eunhee Park und Roger Temam

Veröffentlicht/Copyright: 14. Januar 2016

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 6 Heft 3

Abstract

In this article, we consider a singularly perturbed nonlinear reaction-diffusion equation whose solutions display thin boundary layers near the boundary of the domain. We fully analyse the singular behaviours of the solutions at any given order with respect to the small parameter ε, with suitable asymptotic expansions consisting of the outer solutions and of the boundary layer correctors. The systematic treatment of the nonlinear reaction terms at any given order is novel along the singular perturbation analysis. We believe that the analysis can be suitably extended to other nonlinear problems.

Keywords: Boundary layers; singular perturbations; reaction-diffusion; boundary fitted coordinates

MSC 2010: 35B25; 35C20; 76D10; 35K05

1 Introduction

Nonlinear reaction-diffusion equations arise in many areas in systems consisting of interacting components. The equations describe, e.g., chemical reactions, pattern-formation, population dynamics, predator-prey equations, and competition dynamics in biological systems (see, e.g., [7, 11, 12, 31, 32, 33, 39]). One can consider a typical form of systems of reaction-diffusion equations in the form

(1.1)𝐮t=D⁢Δ⁢𝐮+𝐠⁢(𝐮),

where 𝐠=𝐠⁢(𝐮) describes a change or a local reaction of the state 𝐮 and D represents a diffusion coefficient matrix. It is also possible that the reaction 𝐠 may depend on the spatial domain variable 𝐱 and of a derivative of 𝐮, i.e., 𝐠=𝐠⁢(𝐱,𝐮,∇⁡𝐮).

In real applications like a fast reaction system, the magnitude of some coefficients in the diffusion matrix D is relatively small and hence the system can be singularly perturbed.

In this article, for the singular perturbation and boundary layer analysis aimed here, we consider the steady state system of (1.1) and study the following scalar nonlinear singularly perturbed problem which can serve as a guide for more general systems:

(1.2){-ε⁢Δ⁢uε+g⁢(uε)=fin Ω,uε=0at ∂⁡Ω.

Here, 0<ε≪1, Ω is a general smooth domain, f=f⁢(x,y) and g=g⁢(u) are given smooth functions with

(1.3)g⁢(0)=0,g′⁢(u)≥λ>0 for all ⁢u∈ℝ.

For example, g⁢(u)=u3+λ⁢u.

For small ε>0, the solutions to (1.2) display thin sharp transition layers called boundary layers which are formed due to the discrepancies between the limit solutions when ε=0 (see (2.3) below) and the boundary conditions in (1.2). The discrepancies are inevitable because the limit problem (see (2.2) below) loses high order derivatives and hence in general its solutions cannot meet the boundary conditions. Then, the small diffusion term -ε⁢Δ⁢uε smoothes out the discrepancies, which leads to sharp transition boundary layers.

Another motivation of studying boundary layers is the vanishing viscosity problem in fluid dynamics, see, e.g., [2, 5, 6, 24, 25, 8, 13, 28, 30, 23, 27, 35, 36, 37]. The typical question is on the behaviour of the Navier–Stokes flows at small viscosity, i.e., the limit behaviour or convergence to Euler flows as the viscosity tends to zero. The boundary layers play a crucial role for connecting the Navier–Stokes and Euler flows and they also do so for the singular perturbation analysis in the nonlinear reaction-diffusion equations considered here.

An additional motivation comes from the computational aspects in numerical simulations. Due to the thin boundary layers, the computational meshes are classically refined near the boundary ∂⁡Ω and this causes high cost in the simulations. Rather than refining meshes we propose to enrich with suitable boundary layer correctors the Galerkin or finite element space (or finite volume space). Then, we are able to use a coarse mesh and this reduces substantially the computational cost. See, e.g., [18, 17, 20, 21, 22, 38, 41] for the method of spaces enriched with boundary layer correctors. For singular perturbations analysis, see [15, 26, 42, 44] and also the recent review article [14]. See other perspectives in singular perturbations and boundary layers in [9, 16, 3, 19, 4, 10, 29, 43].

In what follows, we discuss the problems posed on a channel domain in Section 2 which is relatively easier to handle thanks to the simple geometry of the boundary. In Section 3, we cast the nonlinear reaction-diffusion equations in a general domain. We need to take into account the geometrical properties, like curvature, using the boundary fitted coordinates. Throughout this paper, we systematically handle the nonlinear term g along the singular perturbation analysis at any orders. This nonlinear treatment can apply to other nonlinear problems.

For the analysis below, we shall consider the Sobolev spaces Hs⁢(Ω) and we define the weighted energy norm,

∥u∥ε=(ε⁢∥∇⁡u∥L2⁢(Ω)2+∥u∥L2⁢(Ω)2)12.

An exponentially small term, denoted e.s.t., is a function whose norm in all Sobolev spaces Hs⁢(Ω) is exponentially small with, for each s, a bound of the form c1⁢e-c2/εγ, c1,c2,γ>0, with ci, γ depending possibly on s.

2 Channel domains

For general domains, which will be studied in Section 3, we consider the domains with smooth boundaries. Since the boundary layer correctors act locally in the inward direction normal to the boundaries, transforming the Cartesian coordinate into the so-called boundary fitted one, the boundary layers can be described in channel domains, which are relatively easy to analyse. We thus consider first the simpler case of channel domains, which possess boundary layers only on one side at a flat boundary.

Let us consider the problem in a channel domain as follows:

(2.1){-ε⁢Δ⁢uε+g⁢(uε)=fin Ω=(0,L1)×(0,L2),uε=0at x=0,L1,uε⁢(x,y)=uε⁢(x,y+L2)in Ω∞=(0,L1)×ℝ,

where f=f⁢(x,y) is smooth and L2- periodic in y. Then, the limit problem reads

(2.2)g⁢(u0)=f in Ω.

Since g is invertible, we write

(2.3)u0=g-1⁢(f).

To give an idea on how to construct the boundary layers, for now we assume

(2.4)f=0 at x=L1,

which, as we will see, reduces the boundary layer at x=L1, so that only the boundary layer at x=0 persists.

Thanks to (1.3) and (2.3), 0≤λ⁢(u0)2≤(g⁢(u0)-g⁢(0))⁢u0=f⁢u0, and hence

(2.5)u0=0 at x=L1 and u0⁢(x,y)=u0⁢(x,y+L2).

2.1 Boundary layer analysis at order ε0

We now construct a zeroth order corrector to account for the discrepancy between uε and u0 at x=0. Formally, substituting uε∼u0+θ0 in (2.1) and subtracting (2.2) from (2.1), we find that

-ε⁢Δ⁢(u0+θ0)+g⁢(u0+θ0)-g⁢(u0)=0.

Using the stretched variable x¯=x/ε and dropping non-stiff small terms, we find the zeroth order corrector equation for θ0:

-ε⁢θx⁢x0+g⁢(u0+θ0)-g⁢(u0)=0.

However, in general uε-u0 does not satisfy the boundary condition in (2.1), and hence at the boundary x=0, so we propose a boundary layer corrector θ0 satisfying

(2.6){-ε⁢θx⁢x0+g⁢(u0+θ0)-g⁢(u0)=0in Ω,θ0=-u0⁢(0,y)at x=0,θ0=0at x=L1.

Although θ0 is not known explicitly, unlike in many linear problems, we can derive pointwise estimates for θ0.

Lemma 2.1.

The corrector θ0 satisfies

(2.7)|θ0(x,y)|≤|u0(0,y)|exp(-λεx).

Proof.

Setting θ¯⁢θ0=|u0⁢(0,y)|⁢exp⁡(-λ⁢x/ε), writing θ~⁢θ0=θ0-θ¯⁢θ0 and then substituting in (2.6), we obtain -ε⁢θ~⁢θx⁢x0+g⁢(u0+θ0)-g⁢(u0)-λ⁢θ¯⁢θ0=0. Since g′⁢(η)-λ≥0 for all η∈ℝ and thanks to the mean value theorem, we find, for some η1 with |η1-u0|<|θ0|, that

(2.8)-ε⁢θ~⁢θx⁢x0+g′⁢(η1)⁢θ~⁢θ0=(-g′⁢(η1)+λ)⁢θ¯⁢θ0≤0.

Multiplying (2.8) by θ~⁢θ+0=max⁡{θ~⁢θ0,0}, integrating over (0,L1) and noting that θ~⁢θ+0=0 at x=0,L1, we obtain

ε⁢∫0L1((θ~⁢θ+0)x)2+λ⁢∫0L1(θ~⁢θ+0)2≤0.

This implies θ~⁢θ+0=0 and thus θ0-θ¯⁢θ0=θ~⁢θ0≤0. On the other hand, considering this time θ~⁢θ0=-θ0-θ¯⁢θ0 we find that -ε⁢θ~⁢θx⁢x0+g⁢(u0)-g⁢(u0+θ0)-λ⁢θ¯⁢θ0=0. We then similarly obtain (2.8) for this θ~⁢θ0, and hence we deduce the same conclusion, i.e., -θ0-θ¯⁢θ0=θ~⁢θ0≤0. This proves the lemma. ∎

We can also deduce some norm estimates.

Lemma 2.2.

There exists a constant c>0, independent of ε, such that

(2.9)∥θ0⁢(⋅,y)∥Hx1⁢(0,L1)≤c⁢ε-14,∥θ0⁢(⋅,y)∥Lx2⁢(0,L1)≤c⁢ε14.

Proof.

The second estimate of (2.9) directly follows from (2.7). To obtain the first estimate, we introduce θ¯⁢θ=-u0⁢(0,y)⁢e-x/ε⁢δ⁢(x), where δ⁢(x) is a smooth cut-off function with δ⁢(x)=1 for x∈[0,L1/4] and δ⁢(x)=0 for x∈[3⁢L1/4,∞). We observe that for a.e. y∈ℝ,

ε∫0L1|(θ0-θ¯θ)x|2dx=-ε∫0L1(θ0-θ¯θ)x⁢x(θ0-θ¯θ)dx=∫0L1((-g(u0+θ0)+g(u0))+εθ¯θx⁢x)(θ0-θ¯θ)dx≤cε12.

where the last inequality follows from the mean value theorem and the L2-estimate of θ¯⁢θx⁢x, θ¯⁢θ, and θ0. This implies the lemma. ∎

Theorem 2.3.

Assume that (2.4) holds. Then, there exists a constant c>0 such that

(2.10)∥uε-u0-θ0∥ε≤c⁢ε.

Proof.

Let w=uε-u0-θ0, then, thanks to (2.5), w=0 on ∂⁡Ω. Subtracting (2.2) and (3.14) from (2.1) we find that

(2.11){-ε⁢Δ⁢w+g⁢(uε)-g⁢(u0+θ0)=ε⁢Δ⁢u0+ε⁢θy⁢y0in Ω,w=0on ∂⁡Ω.

Multiplying by w and integrating over Ω we find that

ε∫Ω|∇w|2dxdy+∫Ω(g(uε)-g(u0+θ0))wdxdy≤λ2∫Ω|w|2dxdy+cε2.

Here, the L2-norm of θy⁢y0 is derived in Lemma 2.8 below. Thanks to the mean value theorem again and by observing that (g⁢(uε)-g⁢(u0+θ0))⁢w≥λ⁢|w|2, the theorem is proved. ∎

We can also obtain the lower bound of |θ0⁢(x,y)|.

Lemma 2.4.

The corrector θ0 satisfies

(2.12)|θ0⁢(x,y)|≥|u0⁢(0,y)|⁢exp⁡(-λ0ε⁢x)+e.s.t.,

where λ0=max|η-u0|≤|θ0|⁡g′⁢(η).

Proof.

From Lemma 2.1 and (2.3), we note that u0,θ0 are bounded and hence λ0>0 is too. We write θ~⁢θ0=θ0-θ¯⁢θ0, where θ¯⁢θ0=|u0⁢(0,y)|⁢(exp⁡(-λ0⁢x/ε)-L1-1⁢x⁢exp⁡(-λ0⁢L1/ε))≥0.

Fixing y, we first prove (2.12) for the case u0⁢(0,y)≤0. We note that θ~⁢θ0=0 at x=0,L1. Following the proof of Lemma 2.1, we similarly find that for some η1 with |η1-u0|<|θ0|,

-ε⁢θ~⁢θx⁢x0+g′⁢(η1)⁢θ~⁢θ0≥(-g′⁢(η1)+λ0)⁢θ¯⁢θ0≥0.

Multiplying by -θ~⁢θ-0=-max⁡{-θ~⁢θ0,0} and integrating over (0,L1), we obtain

ε⁢∫0L1((θ~⁢θ-0)x)2+λ⁢∫0L1(θ~⁢θ-0)2≤0.

This implies θ~⁢θ-0=0 and hence |θ0|≥θ0≥θ¯⁢θ0, which proves (2.12) for the case u0⁢(0,y)≤0. For the case u0⁢(0,y)>0, we write θ~⁢θ0=-θ0-θ¯⁢θ0. Then, we similarly deduce that θ~⁢θ-0=0 and hence |θ0|≥-θ0≥θ¯⁢θ0. This proves the lemma. ∎

Remark 2.5.

Thanks to the estimate for θ0 in L2, established in Lemma 2.2 with (2.4), Theorem 2.3 implies that

(2.13)∥uε-u0∥L2⁢(Ω)≤c⁢ε14.

Furthermore, for u0⁢(0,y)≠0 at some y∈(0,L2) the L2-norm in (2.13) has a lower bound, i.e., for some c0>0,

∥uε-u0∥L2⁢(Ω)≥c0⁢ε14.

Indeed, from Lemma 2.4 and Theorem 2.3, we find that

∥uε-u0∥L2≥∥θ0∥L2-∥uε-u0-θ0∥L2≥c2⁢ε14-c⁢ε≥c0⁢ε14.

2.2 Boundary layer analysis at arbitrary order εn, n≥0

Outer expansion. We now consider the higher order outer expansions uε∼∑j=0∞εj⁢uj. Substituting in (2.1) and using (2.2), we formally write

(2.14)-ε⁢Δ⁢(∑j=0∞εj⁢uj)+g⁢(∑j=0∞εj⁢uj)=f.

Dropping 𝒪⁢(εn+1) terms, we have

-ε⁢Δ⁢(∑j=0n-1εj⁢uj)+g⁢(∑j=0nεj⁢uj)≃f.

We identify at the order 𝒪⁢(εj), j=0,1,…,n, and find

(2.15){g⁢(u0)=f,-εj⁢Δ⁢uj-1+g⁢(∑k=0jεk⁢uk)-g⁢(∑k=0j-1εk⁢uk)=0,j≥1.

We then obtain, e.g.,

u0=g-1⁢(f),

u1=ε-1⁢g-1⁢(g⁢(u0)+ε⁢Δ⁢u0)-ε-1⁢u0.

More generally, we recursively obtain

(2.16)uj=ε-j⁢g-1⁢(g⁢(∑k=0j-1εk⁢uk)+εj⁢Δ⁢uj-1)-ε-j⁢∑k=0j-1εk⁢uk for ⁢j≥1.

To construct the higher order correctors, we assume, for simplicity, that f is infinitely flat at x=L1, i.e.,

(2.17)Dα⁢f=∂|α|⁡f∂⁡xα1⁢∂⁡yα2=0 at ⁢x=L1⁢, for all ⁢α≥0,

using the multi-index notation

(2.18)α=(α1,α2) with ⁢|α|=α1+α2.

This implies that the uj, j≥0, are infinitely flat at x=L1, that is

Dα⁢uj=0 at x=L1, for all j,α≥0.

Thus, we only have boundary layers at x=0 corresponding to uj. Correctors. We now proceed with the determination of the correctors. Substituting uε∼∑j=0∞εj⁢(uj+θj) in (2.1), we have formally

(2.19)-ε⁢Δ⁢(∑j=0∞εj⁢(uj+θj))+g⁢(∑j=0∞εj⁢(uj+θj))=f.

We subtract (2.14) from (2.19) to obtain

(2.20)-ε⁢Δ⁢(∑j=0∞εj⁢θj)+g⁢(∑j=0∞εj⁢(uj+θj))-g⁢(∑j=0∞εj⁢uj)=0.

We first need to handle the nonlinear term to identify the quantities of order εj and this is discussed below.

2.3 Treatment of the nonlinear term g⁢(u)

In this section, we formally write the nonlinear term g⁢(∑j=0∞εj⁢(uj+θj))-g⁢(∑j=0∞εj⁢uj) at each order εj. Thanks to the Taylor expansion of g about u0, we have

g⁢(∑j=0∞εj⁢uj)=g⁢(u0+∑j=1∞εj⁢uj)=∑k=0∞g(k)⁢(u0)k!⁢(∑j=1∞εj⁢uj)k.

Here, we formally consider ∑j=1∞εj⁢uj=𝒪⁢(ε). Similarly, expanding at u0+θ0, we write

g⁢(∑j=0∞εj⁢(uj+θj))=∑k=0∞gk⁢(u0+θ0)k!⁢(∑j=1∞εj⁢(uj+θj))k.

We first observe that

(∑j=1∞εj⁢uj)k=∑|α|=k(ka)⁢((ε1⁢u1)α1⁢⋯⁢(εl⁢ul)αl⁢⋯)

=∑|α|=k(ka)⁢((u1)α1⁢⋯⁢(ul)αl⁢⋯)⁢ε(α1+2⁢α2+⋯+l⁢αl+⋯)

=∑|α|=k(ka)⁢uα⁢ε(α1+2⁢α2+⋯+l⁢αl+⋯),

where

uα=(u1)α1⁢⋯⁢(ul)αl⁢⋯,

using the multi-index notation

(2.21)α=(α1,…,αl,…) with ⁢|α|=α1+⋯+αl+⋯,

and

(kα)=k!α1!⁢⋯⁢αl!⁢⋯.

We similarly find that

(∑j=1∞εj⁢(uj+θj))k=∑|α|=k(kα)⁢(u+θ)α⁢ε(α1+2⁢α2+⋯+l⁢αl+⋯),

where

(u+θ)α=(u1+θ1)α1⁢⋯⁢(ul+θl)αl⁢⋯.

Hence, we note that

g⁢(∑j=0∞εj⁢(uj+θj))-g⁢(∑j=0∞εj⁢uj)=∑k=0∞1k!⁢[g(k)⁢(u0+θ0)⁢(∑j=1∞εj⁢(uj+θj))k-g(k)⁢(u0)⁢(∑j=1∞εj⁢uj)k]

(2.22)=∑k=0∞∑|α|=k(kα)⁢1k!⁢[g(k)⁢(u0+θ0)⁢(u+θ)α-g(k)⁢(u0)⁢uα]⁢εα1+2⁢α2+⋯+l⁢αl+⋯,

using the notation (2.21). To arrange the terms at each order εj, we set α1+2⁢α2+⋯+l⁢αl+⋯=j. Since the multi-index α satisfies |α|=k, we easily note that k=|α|≤α1+2⁢α2+⋯+l⁢αl+⋯=j. If one of the αl with l≥j+1 is greater than or equal to 1, then α1+2⁢α2+⋯+l⁢αl+⋯≥j+1, and hence αj+1=αj+2=⋯=0. Thus, we may write the multi-index notations as

(2.23){α=(α1,…,αj),|α|=α1+⋯+αj,(u+θ)α=(u1+θ1)α1⁢⋯⁢(uj+θj)αj,uα=(u1)α1⁢⋯⁢(uj)αj.

Hence, using the multi-index notations in (2.23), we formally write

g⁢(∑j=0∞εj⁢(uj+θj))-g⁢(∑j=0∞εj⁢uj)

(2.24)=∑j=0∞{∑k=0j∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)1k![g(k)(u0+θ0)(u+θ)α-g(k)(u0)uα]}εj.

For the analysis below, we estimate the truncation error corresponding to the expansion (2.24).

Lemma 2.6.

There exists a constant C>0, independent of ε, such that

|g(∑j=0nεj(uj+θj))-g(∑j=0nεjuj)-Gn|≤Cεn+1,

where

Gn=∑j=0n{∑k=0j∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)1k![g(k)(u0+θ0)(u+θ)α-g(k)(u0)uα]}εj,

and the multi-index notations are given in (2.23).

Proof.

We first note that the Gn given above can be written as

Gn=∑k=0n1k!∑j=0n{∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)[g(k)(u0+θ0)(u+θ)α-g(k)(u0)uα]}εj.

Thanks to the multinomial theorem, we observe that

(2.25)Hn,k:=g(k)⁢(u0+θ0)⁢(∑j=1nεj⁢(uj+θj))k-g(k)⁢(u0)⁢(∑j=1nεj⁢uj)k

(2.26)=∑j=0n{∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)[g(k)(u0+θ0)(u+θ)α-g(k)(u0)uα]}εj+Jn,k,

where

(2.27)|Jn,k|≤C⁢εn+1.

On the other hand, we find from Taylor’s theorem that

g⁢(∑j=0nεj⁢(uj+θj))-g⁢(∑j=0nεj⁢uj)=∑k=0n1k!⁢Hn,k+R0=Gn+∑k=0n1k!⁢Jn,k+R0,

where

|R0|≤|g(n+1)(n+1)!⁢(ξ1)⁢(∑j=1nεj⁢(uj+θj))n+1|+|g(n+1)(n+1)!⁢(ξ2)⁢(∑j=1nεj⁢uj)n+1|.

Here, ξ1 is between (u0+θ0) and ∑j=0nεj⁢(uj+θj) and ξ2 is between u0 and ∑j=0nεj⁢uj. The lemma follows by observing that |R0|≤C⁢εn+1. ∎

We now define the boundary layer correctors θj at order 𝒪⁢(εj). From (2.20) and (2.24), using the stretched variable x¯=x/ε at each order 𝒪⁢(εj), j=0,1,…, we identify

(2.28){-ε⁢θx⁢x0+g⁢(u0+θ0)-g⁢(u0)=0,-εj+1⁢θx⁢xj+{∑k=1j∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)⁢1k!⁢[g(k)⁢(u0+θ0)⁢(u+θ)α-g(k)⁢(u0)⁢uα]}⁢εj=εj⁢θy⁢yj-1,j≥1.

Dividing by εj, rearranging terms in the latter equation at each order 𝒪⁢(εj), and using the fact that

|α|=k=1 and (α1+2α2+⋯+jαj=j)⇔α1=⋯=αj-1=0,αj=1,

we rewrite (2.28) as

-ε⁢θx⁢x0+g⁢(u0+θ0)-g⁢(u0)=0,

-ε⁢θx⁢x1+g′⁢(u0+θ0)⁢θ1=-(g′⁢(u0+θ0)-g′⁢(u0))⁢u1+θy⁢y0,

and for j≥2,

-ε⁢θx⁢xj+g′⁢(u0+θ0)⁢θj=-(g′⁢(u0+θ0)-g′⁢(u0))⁢uj

(2.29)-∑k=2j∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)⁢1k!⁢[g(k)⁢(u0+θ0)⁢(u+θ)α-g(k)⁢(u0)⁢uα]+θy⁢yj-1.

We supplement the boundary condition on θj, for each j=0,1,…, by

(2.30){θj=-uj⁢(0,y)at ⁢x=0,θj=0at ⁢x=L1.

Remark 2.7.

We note that the corrector equations for θj, j≥1, are all linear and this allows us to directly apply the maximum principle. Differentiating the equations in y, the maximum principle also holds for ∂m⁡θj∂⁡ym⁢(x,y) for m≥1, j≥0.

Lemma 2.8.

The correctors θj, j≥0, satisfy

(2.31)|∂m⁡θj∂⁡ym⁢(x,y)|≤C⁢exp⁡(-34⁢λε⁢x),m≥0.

Proof.

We use the maximum principle to prove the lemma. Let ℒ be the linear operator given by

ℒ⁢u:=-ε⁢ux⁢x+g′⁢(u0+θ0)⁢u.

For j=0, we have

-ε⁢θx⁢x0+g⁢(u0+θ0)-g⁢(u0)=0.

We introduce a barrier function Ψ=C01⁢exp⁡(-34⁢λε⁢x), where C01 will be chosen later. We use the mathematical induction on m starting from the case j=0. By (2.7), we already have |θ0⁢(x,y)|≤c⁢exp⁡(-34⁢λε⁢x). We then see that

|ℒθy0|=|-εθy⁢x⁢x0+g′(u0+θ0)θy0|=|-g′(u0+θ0)uy0+g′(u0)uy0|≤|g′′(η)||θ0||uy0|,

by the mean value theorem, for some η between (u0+θ0) and θ0. We also find

ℒ⁢Ψ=(g′⁢(u0+θ0)-916⁢λ)⁢C01⁢exp⁡(-34⁢λε⁢x).

Since g′⁢(u0+θ0)≥λ and |g′′⁢(η)|⁢|θ0|⁢|uy0| is bounded on Ω¯, we can find a positive constant C01′ such that

|ℒ⁢θy0|≤C01′⁢exp⁡(-34⁢λε⁢x) in ⁢Ω.

By the boundary conditions of θ0, we obtain

|θy0|≤|u0⁢(0,y)|≤C01⁢exp⁡(-34⁢λε⁢x) on ⁢∂⁡Ω,

where C01=max⁡(|u0⁢(0,y)|,C01′). The maximum principle implies that

|θy0⁢(x,y)|≤Ψ in ⁢Ω¯.

We suppose by induction that for k≤(m-1), m≥1, there exists a positive constant C0⁢k satisfying

|∂k⁡θ0∂⁡yk⁢(x,y)|≤C0⁢k⁢exp⁡(-34⁢λε⁢x) in ⁢Ω¯.

We then find that

ℒ⁢(∂m∂⁡ym⁢θ0)=∂m-1∂⁡ym-1⁢(g′⁢(u0)⁢uy0-g′⁢(u0+θ0)⁢uy0)

=∑k=0m-1[h0k(g′(u0),…,g(m)(u0),g′(u0+θ0),…,g(m)(u0+θ0),uy0,…,∂m⁡u0∂⁡ym,

(2.32)g′′(η1),…,g(m+1)(ηm))Pk(θ0)],

where P0⁢(θ0)=θ0 and Pk⁢(θ0)=∑α1+⋯+(m-1)⁢αm-1=k∏i=1m-1(∂yi⁡θ0)αi for k≥1 with some multivariate polynomials h0k and ηk between (u0+θ0) and u0. Since g and u0 are smooth, there exists C0⁢m′ such that

ℒ⁢(∂m∂⁡ym⁢θ0)≤C0⁢m′⁢exp⁡(-34⁢λε⁢x) in ⁢Ω.

We infer from the boundary conditions for θ0 that

|∂m∂⁡ym⁢θ0|≤|∂m∂⁡ym⁢u0⁢(0,y)| on ⁢∂⁡Ω,

and using the maximum principle we obtain that

(2.33)|∂m∂⁡ym⁢θ0|≤C0⁢m⁢exp⁡(-34⁢λε⁢x) in ⁢Ω¯,

where C0⁢m=max⁡(|∂m∂⁡ym⁢u0⁢(0,y)|,C0⁢m′). Similarly, for the case when j=1, we see that for m≥0,

ℒ⁢(∂m∂⁡ym⁢θ1)=∂m∂⁡ym⁢(g′⁢(u0)⁢u1-g′⁢(u0+θ0)⁢u1+θy⁢y0)

=∑k=0m+2[h1k(g′(u0),…,g(m+1)(u0),g′(u0+θ0),…,g(m+1)(u0+θ0),uy0,…,∂m⁡u0∂⁡ym,

(2.34)u1,…,∂m⁡u1∂⁡ym,g′′(η1),…,gm+2(ηm+1))Pk(θ0)],

where P0⁢(θ0)=θ0 and Pk⁢(θ0)=∑α1+⋯+(m+2)⁢αm+2=k∏i=1m+2(∂yi⁡θ0)αi for k≥1 with for some multivariate polynomial h1k and ηk between (u0+θ0) and u0. Since g and uj are smooth and h1k are polynomials, we can find a positive constant C1⁢m′, by the result for θ0, such that

(2.35)|ℒ⁢(∂m∂⁡ym⁢θ1)|≤C1⁢m′⁢exp⁡(-34⁢λε⁢x) in ⁢Ω.

By the boundary condition on θ1, we also have

(2.36)|∂m∂⁡ym⁢θ1|≤|∂m∂⁡ym⁢u1⁢(0,y)| on ⁢∂⁡Ω.

We find from (2.35) and (2.36) that for m≥0,

|∂m∂⁡ym⁢θ1|≤C1⁢m⁢exp⁡(-34⁢λε⁢x) in ⁢Ω¯,

by the maximum principle where C1⁢m=max⁡(C1⁢m′,|∂m∂⁡ym⁢u1⁢(0,y)|). We now suppose by induction that for k≤(j-1), j≥1, there exists a positive constant Cj⁢m such that

(2.37)|∂m⁡θk∂⁡ym⁢(x,y)|≤Cj⁢m⁢exp⁡(-34⁢λε⁢x),m=0,1,….

To prove (2.31) at order k=j, differentiating (2.29) in y we note that the first and third terms of the right-hand side of (2.29) are similarly estimated as for the case θ1 by (2.37). We thus estimate the second term there. Observing that for k≥2,

(2.38)(|α|=k,α1+2α2+⋯+jαj=j)⇔(|α|=k,α1+2α2+⋯+(j-1)α(j-1)=j),

it suffices to show that for any m≥0,

(2.39)|∂m∂⁡ym⁢(∑k=2j∑|α|=k,α1+2⁢α2+⋯+(j-1)⁢α(j-1)=j[g(k)⁢(u0+θ0)⁢(u+θ)α-g(k)⁢(u0)⁢uα])|≤Cj⁢m⁢exp⁡(-34⁢λε⁢x).

To prove this, we note that

(2.40)g(k)⁢(u0+θ0)⁢(u+θ)α-g(k)⁢(u0)⁢uα=(g(k)⁢(u0+θ0)-g(k)⁢(u0))⁢(u+θ)α+g(k)⁢(u0)⁢((u+θ)α-uα),

and using the factorization an-bn=(a-b)⁢∑i=1nan-i⁢bi-1,

(u+θ)α-uα=∑l=1j-1[θl⁢(∑i=1αl(ul)αl-i⁢(θl)i-1)⁢∏n=1l-1(un)αn⁢∏n=l+1j-1(un+θn)αn],

where α=(α1,…,αj-1). Differentiating (2.40) in y, thanks to the mean value theorem, the left-hand side of (2.39) can be written as the sum of the products of θk and their derivatives in y for k≤(j-1). We then conclude, by assumption (2.37), that (2.39) holds true. ∎

We now estimate, for each n=0,1,…, the norm of wn, where wn=uε-∑j=0nεj⁢(uj+θj). Summing (2.15) for j=0 to j=n, we find

(2.41)-ε⁢Δ⁢(∑j=0n-1εj⁢uj)+g⁢(∑j=0nεj⁢uj)=f.

Summing (2.28) for j=0 to j=n, we find

-ε∑j=0nεjθx⁢xj+∑j=0n{∑k=0j∑|α|=k,α1+2⁢α2+⋯+j⁢αj=j(kα)1k![g(k)(u0+θ0)(u+θ)α-g(k)(u0)uα]}εj=∑j=0nεjθj-1y⁢y.

Thanks to Lemma 2.6, this can be written in the form

(2.42)-ε⁢Δ⁢(∑j=0nεj⁢θj)+g⁢(∑j=0nεj⁢(uj+θj))-g⁢(∑j=0nεj⁢uj)+R1=-εn+1⁢θy⁢yn

with

(2.43)|R1|≤c⁢εn+1.

Adding the two above equations (2.41) and (2.42), we find

(2.44)-ε⁢Δ⁢(∑j=0nεj⁢(uj+θj))+g⁢(∑j=0nεj⁢(uj+θj))=f-εn+1⁢Δ⁢un-εn+1⁢θy⁢yn-R1.

Subtracting (2.44) and from the first equation in (2.1), we find

(2.45)-ε⁢Δ⁢wn+g⁢(uε)-g⁢(∑j=0nεj⁢(uj+θj))=εn+1⁢Δ⁢un+εn+1⁢θy⁢yn+R1.

We multiply (2.45) by wn and since g⁢(u)-g⁢(v)=g′⁢(ξ)⁢(u-v) and g′⁢(ξ)≥λ>0, we obtain, by a priori estimate, that

ε⁢∥wn∥H1+∥wn∥L2≤c⁢εn+1.

We hence proved the following convergence theorem.

Theorem 2.9.

Assume that (2.17) holds. Let uε be the solution of (2.1) and uj and θj be given as in (2.16) and (2.29)–(2.30), respectively. Then, there exists a positive constant c>0, independent of ε, such that

∥uε-∑j=0nεj⁢(uj+θj)∥ε≤c⁢εn+1.

2.4 Without the assumption (2.17)

If we consider a general smooth function f, i.e., if we remove the assumptions (2.17), we also expect similar boundary layers at x=L1. Let us denote similar boundary layers θj at x=0 by θlj, j≥0, given in (2.29). Similarly, we define the boundary layers at x=L1, denoted by θrj, which satisfy equations (2.29) but with different boundary conditions, i.e.,

{θrj=0at ⁢x=0,θrj=-uj⁢(L1,y)at ⁢x=L1.

Then, we define the boundary layers θj, j≥0,

(2.46)θj=θlj+θrj.

Since the corrector equations for θlj,θrj, j≥1, are linear, we infer from equation (2.42) that

(2.47)-ε⁢Δ⁢(∑j=0nεj⁢θj)+g⁢(∑j=0nεj⁢(uj+θj))-g⁢(∑j=0nεj⁢uj)+R~1=-εn+1⁢θy⁢yn.

Here,

(2.48)R~1=R1+R2,

where R1 is given in (2.42)–(2.43) and

R2=g⁢(u0+θl0)+g⁢(u0+θr0)-g⁢(u0+θ0)-g⁢(u0).

We now note that R2 is exponentially small. Indeed, we find that, for all α≥0,

∥Dα⁢R2∥L2⁢(Ω)≤∥Dα⁢R2∥L2⁢((0,L1/2)×(0,L2))+∥Dα⁢R2∥L2⁢((L1/2,L1)×(0,L2))

≤∥Dα⁢(g⁢(u0+θl0)+g⁢(u0+e.s.t)-g⁢(u0+θl0+e.s.t)-g⁢(u0))∥L2⁢((0,L1/2)×(0,L2))

+∥Dα⁢(g⁢(u0+e.s.t.)+g⁢(u0+θr0)-g⁢(u0+e.s.t+θr0)-g⁢(u0))∥L2⁢((L1/2,L1)×(0,L2)),

which is an exponentially small term. Then, the convergence analysis similarly follows as in the above, from which we infer the following theorem.

Theorem 2.10.

Let f be a general smooth function periodic in y with period L2, let uε be the solution of (2.1), and let uj and θj be given as in (2.16) and (2.46), respectively. Then, there exists a positive constant c>0, independent of ε, such that

∥uε-∑j=0nεj⁢(uj+θj)∥ε≤c⁢εn+1.

3 General domains

We now return to the case of a general smooth domain, where equation (1.2) is posed:

(3.1){-ε⁢Δ⁢uε+g⁢(uε)=fin Ω,uε=0at ∂⁡Ω.

Here, Ω is a general smooth domain, f=f⁢(x,y) and g⁢(u) are given smooth functions with

(3.2)g⁢(0)=0,g′⁢(u)≥λ>0 for all ⁢u∈ℝ.

The outer solutions uj, j≥0, are the same as in (2.16). That is, in Ω, we have

(3.3){u0=g-1⁢(f),u1=ε-1⁢g-1⁢(g⁢(u0)+ε⁢Δ⁢u0)-ε-1⁢u0,uj=ε-j⁢g-1⁢(g⁢(∑k=0j-1εk⁢uk)+εj⁢Δ⁢uj-1)-ε-j⁢∑k=0j-1εk⁢uk for ⁢j≥1.

However, the boundary layers appear in the direction normal to the curved boundary. Thus, the boundary fitted coordinates, i.e., the normal and tangential components along the boundary, are necessary to devise the boundary layer correctors. Here, we consider smooth boundaries ∂⁡Ω which are parametrised by an arclength η and we assume that (X⁢(η),Y⁢(η))∈∂⁡Ω is a regular curve, i.e., the tangent vector T=(X′,Y′)≠0 for the arclength 0≤η<L0 is measured counterclockwise, where L0 is the length of the boundary ∂⁡Ω. Hence, we may assume that it has unit speed, i.e., (X′)2+(Y′)2=1 (see [34]).

We then define the boundary fitted coordinates:

(3.4)ΩBL={(x,y):x=X⁢(η)-ξ⁢Y′⁢(η),y=Y⁢(η)+ξ⁢X′⁢(η), 0≤η<L0, 0≤ξ<ξ0},

where ξ0 is the minimum radius of curvature of ∂⁡Ω, i.e., ξ0=1/max⁡κ⁢(η). Here we note that (-Y′⁢(η),X′⁢(η)) is the inward unit normal vector to the boundary ∂⁡Ω.

3.1 Boundary fitted coordinates

We introduce the local orthogonal coordinate basis 𝐠k, i=1,2, on the subdomain ΩBL by setting

𝐠1=∂⁡x∂⁡ξ⁢𝐞1+∂⁡y∂⁡ξ⁢𝐞2=-Y′⁢𝐞1+X′⁢𝐞2,

𝐠2=∂⁡x∂⁡η⁢𝐞1+∂⁡y∂⁡η⁢𝐞2=(X′-ξ⁢Y′′)⁢𝐞1+(Y′+ξ⁢X′′)⁢𝐞2,

where 𝐞1,𝐞2 is the standard basis in ℝ2 (see, e.g., [1, 40]). We can easily compute that

(3.5)𝐠1⋅𝐠1=(X′)2+(Y′)2=1,

(3.6)𝐠1⋅𝐠2=ξ⁢X′⁢X′′+ξ⁢Y′⁢Y′′=0,

𝐠2⋅𝐠2=1+2⁢ξ⁢(Y′⁢X′′-X′⁢Y′′)+ξ2⁢((X′′)2+(Y′′)2).

Here, we note that differentiating (3.5) implies (3.6). The curvature κ⁢(η) of ∂⁡Ω is derived

κ⁢(η)=X′⁢Y′′-Y′⁢X′′((X′)2+(Y′)2)32=X′⁢Y′′-Y′⁢X′′

and, by the Frenet–Serret relation, T′=(X′′,Y′′)=κ⁢(η)⁢N⁢(η) where N⁢(η) is the unit normal outward vector. Hence, we obtain that

𝐠2⋅𝐠2=(1-κ⁢(η)⁢ξ)2.

For the variables (ξ,η) in

Ωξ,η={(ξ,η):0≤η<L0, 0≤ξ<ξ0},

since 0≤ξ<ξ0=1/max⁡κ⁢(η), we find that 1-κ⁢(η)⁢ξ>0, and thus we have

h1:=𝐠1⋅𝐠1=1,h2:=𝐠2⋅𝐠2=1-κ⁢(η)⁢ξ.

The gradient and Laplacian operators are then defined as follows:

∇=∑i=12𝐠ihi2⁢∂∂⁡ζi,Δ=1h1⁢h2⁢∑i=12∂∂⁡ζi⁢(h1⁢h2hi2⁢∂∂⁡ζi),

where (ζ1,ζ2)=(ξ,η). Hence, we find that

∇=(-Y′⁢∂∂⁡ξ+(X′-ξ⁢Y′′)⁢σ2⁢(ξ,η)⁢∂∂⁡η,X′⁢∂∂⁡ξ+(Y′+ξ⁢X′′)⁢σ2⁢(ξ,η)⁢∂∂⁡η),

(3.7)and Δ=∂2∂⁡ξ2-κ⁢(η)⁢σ⁢(ξ,η)⁢∂∂⁡ξ+σ2⁢(ξ,η)⁢∂2∂⁡η2+ξ⁢κ′⁢(η)⁢σ3⁢(ξ,η)⁢∂∂⁡η,

where

(3.8)σ⁢(ξ,η)=11-κ⁢(η)⁢ξ.

Then, the model equation in (3.1), can be written as

(3.9)-ε⁢∂2⁡uε∂⁡ξ2+ε⁢κ⁢(η)⁢σ⁢(ξ,η)⁢∂⁡uε∂⁡ξ-ε⁢σ2⁢(ξ,η)⁢∂2⁡uε∂⁡η2-ε⁢ξ⁢κ′⁢(η)⁢σ3⁢(ξ,η)⁢∂⁡uε∂⁡η+g⁢(uε)=f.

Remark 3.1.

On the unit circle domain, using polar coordinates, we find X⁢(η)=cos⁡η, Y⁢(η)=sin⁡η and σ⁢(ξ)=(1-ξ)-1 (note κ⁢(η)=1). The operators ∇, Δ are simplified to

∇=(-cos⁡η⁢∂∂⁡ξ-sin⁡η1-ξ⁢∂∂⁡η,-sin⁡η⁢∂∂⁡ξ+cos⁡η1-ξ⁢∂∂⁡η),

and Δ=1(1-ξ)2⁢∂2∂⁡η2-11-ξ⁢∂∂⁡ξ+∂2∂⁡ξ2.

3.2 Boundary layer analysis at orders ε0 and ε12

Unlike the channel domain (2.1), we will need to introduce a corrector θ12 to settle the error from the curvature κ⁢(η).

In general, to find appropriate asymptotic expansions for the boundary layers, we preform the following expansions near the boundary ξ=0. Subtracting (2.41) from (3.9) and observing that ε is the thickness of the boundary layers, as indicated in previous sections, which suggests to use the stretched variable ξ¯⁢ξ=ξ/ε, we find that

(-∂2∂⁡ξ¯⁢ξ2+ε⁢κ⁢(η)⁢σ⁢(ξ,η)⁢∂∂⁡ξ¯⁢ξ-ε⁢σ2⁢(ξ,η)⁢∂2∂⁡η2-ε⁢ξ⁢κ′⁢(η)⁢σ3⁢(ξ,η)⁢∂∂⁡η)⁢(uε-∑j=0nεj⁢uj)

(3.10)+g⁢(uε)-g⁢(∑j=0nεj⁢uj)=εn+1⁢Δ⁢un.

To address the terms at all the orders of ε in the boundary layers, we have to resolve the effect of curvature κ⁢(η). Here, κ⁢(η)=0 for the channel domain. We first observe that σ⁢(ξ,η)=(1-κ⁢(η))-1=∑l=0∞(κ⁢(η)⁢ξ¯⁢ξ)l⁢εl2 and the powers of σ⁢(ξ,η) can be similarly expressed (see also (3.38) below). We then take into account the mean value theorem for the first two terms in the second line of (3.10), i.e., g⁢(uε)-g⁢(∑j=0nεj⁢uj)=g′⁢(u)⁢(uε-∑j=0nεj⁢uj) for some u. In this way, we can balance the difference between uε and the outer expansion at order n by using the inner expansion near ∂⁡Ω in the form

(3.11)uε-∑j=0nεj⁢uj≃∑j=0n(ε⁢θj+εj+12⁢θj+12) near ⁢∂⁡Ω.

By comparing the terms of the same order εj on ∂⁡Ω, we deduce from (3.11) the following boundary conditions for j≥0:

(3.12){θj=-ujon ⁢∂⁡Ω,θj+12=0on ⁢∂⁡Ω.

We now find two leading order correctors θ0 and θ12 satisfying

(3.13)-ε⁢(θξ⁢ξ0+ε⁢θξ⁢ξ12)+ε⁢κ⁢(η)⁢(θξ0+ε⁢θξ12)+g⁢(u0+θ0+ε⁢θ12)-g⁢(u0)≃0.

By the Taylor expansion, dropping smaller terms and using the stretched variable ξ¯⁢ξ=ξ/ε, we obtain θ¯⁢θ0 at order 𝒪⁢(ε0), which is a solution of

(3.14)-θ¯⁢θξ¯⁢ξ⁢ξ¯⁢ξ0+g⁢(u0+θ¯⁢θ0)-g⁢(u0)=0,

and from (3.13) at order 𝒪⁢(ε12), we again find θ¯⁢θ12 such that

(3.15)-θ¯⁢θξ¯⁢ξ⁢ξ¯⁢ξ12+g′⁢(u0+θ¯⁢θ0)⁢θ¯⁢θ12=-κ⁢(η)⁢θ¯⁢θξ¯⁢ξ0.

These equations are supplemented with the respective boundary conditions, i.e.,

(3.16){θ¯⁢θ0=-u0|ξ=0=-u0⁢(X⁢(η),Y⁢(η))at ⁢ξ=0,θ¯⁢θ0=0at ⁢ξ=ξ0, {θ¯⁢θ12=0at ⁢ξ=0,θ¯⁢θ12=0at ⁢ξ=ξ0.

Then, we extend θ¯⁢θ0 and θ¯⁢θ12 by zero for ξ>ξ0 and these extensions are still denoted by the same notations. However, these extensions are not smooth. Thus, for the analysis below, let us define

θ0=θ¯⁢θ0⁢δ⁢(ξ),θ12=θ¯⁢θ12⁢δ⁢(ξ),

where δ⁢(ξ) is a smooth cut-off function given by

(3.17)δ⁢(ξ)={1,ξ∈[0,ξ0/4],0,ξ∈[3⁢ξ0/4,∞).

Remark 3.2.

We obtain θ0 by multiplying θ¯⁢θ0 by the cut-off function. This allows us to use the same estimates for θ0 as θ¯⁢θ0. We proceed similarly for θ12.

Lemma 3.3.

The following pointwise estimate holds for θ¯⁢θ0:

(3.18)|θ¯⁢θ0|≤|u0|ξ=0⁢exp⁡(-λε⁢ξ).

Furthermore, the derivatives of θ¯⁢θ0 satisfy pointwise, for l,m,n≥0, the following:

(3.19)|ξn∂l+m⁡θ¯⁢θ0∂⁡ξl⁢∂⁡ηm|≤cεn-l2exp(-12λεξ).

Proof.

We first set ψ=|u0|ξ=0⁢exp⁡(-λ⁢ξ¯⁢ξ). Writing θ~⁢θ=θ¯⁢θ0-ψ, we deduce from (3.14) that

-θ~⁢θξ¯⁢ξ⁢ξ¯⁢ξ+g⁢(u0+θ¯⁢θ0)-g⁢(u0)-λ⁢ψ=0,

which implies

(3.20)-θ~⁢θξ¯⁢ξ⁢ξ¯⁢ξ+g′⁢(η)⁢θ~⁢θ≤0.

Thanks to (3.16), multiplying (3.20) by θ~⁢θ+⁢(ξ¯⁢ξ,⋅)∈H01⁢(0,∞) and integrating over (0,∞), we find that

∫0∞|(θ~θ+)ξ¯⁢ξ|2dξ¯ξ+λ∫0∞|θ~θ+|2dξ¯ξ≤0.

This implies θ~⁢θ+=0 for all ξ¯⁢ξ≥0 and thus θ¯⁢θ0≤ψ. The other inequality -θ¯⁢θ0≤ψ similarly follows.

We now find the estimates for the derivatives as in (3.19). For l=0 and n=0, it immediately follows, from Lemma 2.8, that

|∂m⁡θ¯⁢θ0∂⁡ηm|≤c⁢exp⁡(-34⁢λε⁢ξ).

Multiplying by ξn, the estimate (3.19) follows for l=0 and n≥0.

Then, at higher orders l, we use the multi-index α=(l-2,m) with l≥2 and Dα=∂ξ¯⁢ξl-2⁡∂ηm. Applying the operator Dα to (3.14), we have

∂ξ¯⁢ξl⁡∂ηm⁡θ¯⁢θ0=(Dα⁢θ¯⁢θ0)ξ¯⁢ξ⁢ξ¯⁢ξ=Dα⁢(g⁢(u0+θ¯⁢θ0)-g⁢(u0)).

Thanks to the mean value theorem, we observe that

(3.21)Dα⁢(g⁢(u0+θ¯⁢θ0)-g⁢(u0))=finite sum of products of Dβ⁢θ¯⁢θ0 with β≤(l-2,m),

and we can thus inductively prove (3.19) for any l≥0 as long as the case for l=1 is proved.

For l=1, we let μ>0, which will be determined. We infer from (3.14) that

(θ¯⁢θξ¯⁢ξ0⁢eμ⁢ξ¯⁢ξ)ξ¯⁢ξ=(g⁢(u0+θ¯⁢θ0)-g⁢(u0))⁢eμ⁢ξ¯⁢ξ+μ⁢θ¯⁢θξ¯⁢ξ0⁢eμ⁢ξ¯⁢ξ=(g⁢(u0+θ¯⁢θ0)-g⁢(u0))⁢eμ⁢ξ¯⁢ξ+(μ⁢θ¯⁢θ0⁢eμ⁢ξ¯⁢ξ)ξ¯⁢ξ-μ2⁢θ¯⁢θ0⁢eμ⁢ξ¯⁢ξ.

Integrating over (ξ¯⁢ξ,∞), we find

C-θ¯⁢θξ¯⁢ξ0⁢eμ⁢ξ¯⁢ξ=I1⁢(ξ¯⁢ξ)-μ⁢θ¯⁢θ0⁢eμ⁢ξ¯⁢ξ,

where

I1⁢(ξ¯⁢ξ)=∫ξ¯⁢ξ∞(g⁢(u0+θ¯⁢θ0)-g⁢(u0)-μ2⁢θ¯⁢θ0)⁢(t,η)⁢eμ⁢t⁢dt.

Then,

(3.22)θ¯⁢θξ¯⁢ξ0=C⁢e-μ⁢ξ¯⁢ξ-I1⁢(ξ)⁢e-μ⁢ξ¯⁢ξ+μ⁢θ¯⁢θ0,

and integrating over (ξ¯⁢ξ,∞), we find that

(3.23)-θ¯⁢θ0=D+μ-1⁢C⁢e-μ⁢ξ¯⁢ξ-∫ξ¯⁢ξ∞I1⁢(s)⁢e-μ⁢s⁢ds+μ⁢∫ξ¯⁢ξ∞θ¯⁢θ0⁢(s,η)⁢ds.

Let μ=p⁢λ, where p, a constant independent of ε, is to be determined. For 0<p<1, since (3.18) yields |θ¯⁢θ0⁢(t,η)|≤c⁢exp⁡(-λ⁢t), we find that

|I1⁢(ξ¯⁢ξ)|≤c⁢∫ξ¯⁢ξ∞e-λ⁢(1-p)⁢t⁢dt≤c⁢e-λ⁢(1-p)⁢ξ¯⁢ξ,

and

|∫ξ¯⁢ξ∞I1(s)e-μ⁢sds|≤c∫∞ξ¯⁢ξe-λ⁢sds≤ce-λ⁢ξ¯⁢ξ.

Let us choose p=34 and thus μ=34⁢λ. Applying the boundary conditions to (3.23), i.e., θ¯⁢θ0→0 as ξ¯⁢ξ→∞ and θ¯⁢θ0=-u0|ξ=0 at ξ¯⁢ξ=0, we find that D=0 and that

-u0⁢(0,η)=θ¯⁢θ0⁢(0,η)=-μ-1⁢C+∫0∞I1⁢(s)⁢e-μ⁢s⁢ds-μ⁢∫0∞θ¯⁢θ0⁢(s,η)⁢ds.

Thus,

C=μ⁢[u0⁢(0,η)+∫0∞I1⁢(s)⁢e-μ⁢s⁢ds-μ⁢∫0∞θ¯⁢θ0⁢(s,η)⁢ds].

To estimate ∂ηm⁡θ¯⁢θξ¯⁢ξ0, we apply ∂ηm to (3.22) and find that

|∂ηm⁡θ¯⁢θξ¯⁢ξ0|≤(|∂ηm⁡C|+|∂ηm⁡I1⁢(ξ¯⁢ξ)|)⁢exp⁡(-34⁢λε⁢ξ)+c⁢|∂ηm⁡θ¯⁢θ0|

(3.24)≤c⁢exp⁡(-34⁢λε⁢ξ).

Here, from (3.21) with α=(0,m), i.e., l=2, and (3.19) with l=0, we used the fact that

|∂ηm⁡C|+|∂ηm⁡I1⁢(ξ¯⁢ξ)|≤c.∎

We similarly derive the pointwise estimate for θ¯⁢θ12 and the result appears in the following lemma.

Lemma 3.4.

The corrector θ¯⁢θ12 satisfies pointwise, for l,m,n≥0, the following:

(3.25)|ξn⁢∂l+m⁡θ¯⁢θ12∂⁡ξl⁢∂⁡ηm|≤c⁢εn-l2⁢exp⁡(-12⁢λε⁢ξ).

Proof.

Let ℒ be the linear operator given by

ℒ⁢u=-uξ¯⁢ξ⁢ξ¯⁢ξ+g′⁢(u0+θ¯⁢θ0)⁢u.

We repeat the same argument as in the proof of Lemma 3.3. For l=0, the estimates (3.25) hold true by the maximum principle applied to ℒ⁢(∂ηm⁡θ¯⁢θ12) with (3.15). For l=1, we infer from (3.15) that

(θ¯⁢θξ¯⁢ξ12⁢eμ⁢ξ¯⁢ξ)ξ¯⁢ξ=g′⁢(u0+θ¯⁢θ0)⁢θ¯⁢θ12⁢eμ⁢ξ¯⁢ξ+κ⁢(η)⁢θ¯⁢θξ¯⁢ξ0⁢eμ⁢ξ¯⁢ξ+(μ⁢θ¯⁢θ12⁢eμ⁢ξ¯⁢ξ)ξ¯⁢ξ-μ2⁢θ¯⁢θ12⁢eμ⁢ξ¯⁢ξ.

We integrate over (ξ¯⁢ξ,∞), to find that

C-θ¯⁢θξ¯⁢ξ12⁢eμ⁢ξ¯⁢ξ=I2⁢(ξ¯⁢ξ)-μ⁢θ¯⁢θ12⁢eμ⁢ξ¯⁢ξ,

where

I2⁢(ξ¯⁢ξ)=∫ξ¯⁢ξ∞(g′⁢(u0+θ¯⁢θ0)⁢θ¯⁢θ12+κ⁢(η)⁢θ¯⁢θξ¯⁢ξ0-μ2⁢θ¯⁢θ12)⁢(t,η)⁢eμ⁢t⁢dt.

Let μ=p⁢λ, where p is a constant independent of ε to be chosen later. From estimates (3.19), we note that |θ¯⁢θξ¯⁢ξ0⁢(t,η)|≤c⁢exp⁡(-λ⁢t/2). Using the same argument as in the proof of Lemma 3.3, we only need to show that for 0<p<1,

(3.26)|I2⁢(ξ¯⁢ξ)|≤c⁢∫ξ¯⁢ξ∞e-λ⁢(1-p)⁢t⁢dt≤c⁢e-λ⁢(1-p)⁢ξ¯⁢ξ.

Following then the same procedure as in the proof of Lemma 3.3 with the boundary conditions (3.16), we can obtain (3.24) for θ¯⁢θ12.

For l≥2, differentiating (3.15) in ξ¯⁢ξ and using estimates (3.19), the lemma is inductively proved. ∎

We now find the norm estimate in L2 and H1 for θ¯⁢θ0 and θ¯⁢θ12 in the next lemma.

Lemma 3.5.

Let l,m,n≥0. Then, there exists c>0 such that

(3.27)∥ξn⁢∂l+m⁡θ¯⁢θ0∂⁡ξl⁢∂⁡ηm∥Lξ2⁢(Ω)+∥ξn⁢∂l+m⁡θ¯⁢θ12∂⁡ξl⁢∂⁡ηm∥Lξ2⁢(Ω)≤c⁢εn-l2+14.

Proof.

We infer from Lemma 3.3 and Lemma 3.4 that (3.27) holds. ∎

We now introduce the analogue of Theorem 2.3.

Theorem 3.6.

Assume f is a general smooth function and Ω is a general smooth domain. Then, there exists a positive constant c>0 such that

∥uε-u0-θ0∥ε≤c⁢ε34.

Proof.

We set w=uε-u0-θ0. To avoid the singularity of σ⁢(ξ,η)=(1-κ⁢(η)⁢ξ)-1, we use the smooth cut-off function δ~⁢δ⁢(ξ) such that

(3.28)δ~⁢δ⁢(ξ)={1,ξ∈[0,ξ0/2],0,ξ∈[3⁢ξ0/4,∞).

Here, we recall that ξ0=1/max⁡κ⁢(η). Noting that θ0=0 for ξ≥ξ0/2 and denoting by (⋅,⋅) the scalar product in the space L2⁢(Ω), we can write

(ε⁢Δ⁢θ0-g⁢(u0+θ0)+g⁢(u0),w)=(ε⁢Δ⁢θ0-g⁢(u0+θ0)+g⁢(u0),δ~⁢δ⁢(ξ)⁢w)

=(G0+E0,δ~⁢δ⁢(ξ)⁢w)+(ε⁢Δ⁢θ¯⁢θ0-g⁢(u0+θ¯⁢θ0)+g⁢(u0),δ~⁢δ⁢(ξ)⁢w)

(3.29)=((G0+E0)⁢δ~⁢δ⁢(ξ),w)+(R0⁢δ~⁢δ⁢(ξ),w),

thanks to (3.14). Here,

E0=ε⁢Δ⁢(θ0-θ¯⁢θ0),

G0=g⁢(u0+θ¯⁢θ0)-g⁢(u0+θ0),

R0=-ε⁢κ⁢(η)⁢σ⁢(ξ,η)⁢∂⁡θ¯⁢θ0∂⁡ξ+ε⁢σ2⁢(ξ,η)⁢∂2⁡θ¯⁢θ0∂⁡η2+ε⁢ξ⁢κ′⁢(η)⁢σ3⁢(ξ,η)⁢∂⁡θ¯⁢θ0∂⁡η.

Since θ0-θ¯⁢θ0=θ¯⁢θ0⁢(1-δ⁢(ξ)), it is very easy to prove that G0 and E0 are e.s.t. We also find that

∥R0⁢δ~⁢δ⁢(ξ)∥L2⁢(Ω)≤c⁢ε34.

Taking the inner product of (3.3) and (3.1), respectively, with w, we write

(3.30)(-ε⁢Δ⁢uε+g⁢(uε),w)=(f,w),

(3.31)(g⁢(u0),w)=(f,w).

From (3.29), (3.30) and (3.31), we find that

(-ε⁢Δ⁢w+g⁢(uε)-g⁢(u0+θ0),w)

=(-ε⁢Δ⁢uε+g⁢(uε),w)-(g⁢(u0),w)+(ε⁢Δ⁢u0,w)+(ε⁢Δ⁢θ0-g⁢(u0+θ0)+g⁢(u0),w)

=(ε⁢Δ⁢u0,w)+(G0⁢δ~⁢δ⁢(ξ),w)+(R0⁢δ~⁢δ⁢(ξ),w)

≤c⁢ε34⁢∥w∥L2⁢(Ω).

Thanks to (3.2), this completes the proof of Theorem 3.6. ∎

Theorem 3.7.

Assume that f is a general smooth function and Ω is a general smooth domain. Then, there exists a positive constant c>0 such that

∥uε-u0-θ0-ε12⁢θ12∥ε≤c⁢ε.

Proof.

We define w12=uε-u0-θ0-ε⁢θ12, then we find

(ε⁢Δ⁢(θ0+ε⁢θ12)-g⁢(u0+θ0+ε⁢θ12)+g⁢(u0),w12)

=(ε⁢Δ⁢(θ0+ε⁢θ12)-g⁢(u0+θ0+ε⁢θ12)+g⁢(u0),δ~⁢δ⁢(ξ)⁢w12)

=(G12+E12,δ~⁢δ⁢(ξ)⁢w12)+(ε⁢Δ⁢(θ¯⁢θ0+ε⁢θ¯⁢θ12)-g⁢(u0+θ¯⁢θ0+ε⁢θ¯⁢θ12)+g⁢(u0),δ~⁢δ⁢(ξ)⁢w12)

(3.32)=((G12+E12)⁢δ~⁢δ⁢(ξ),w12)+(R12⁢δ~⁢δ⁢(ξ),w12),

by (3.14) and by (3.15), where

(3.33){E12=ε⁢Δ⁢(θ0+ε⁢θ12-θ¯⁢θ0-ε⁢θ¯⁢θ12),G12=g⁢(u0+θ¯⁢θ0+ε⁢θ¯⁢θ12)-g⁢(u0+θ0+ε⁢θ12),R12=-ε⁢κ⁢(η)⁢σ⁢(ξ,η)⁢∂∂⁡ξ⁢(θ¯⁢θ0+ε⁢θ¯⁢θ12)+ε⁢κ⁢(η)⁢∂∂⁡ξ⁢θ¯⁢θ0+ε⁢σ2⁢(ξ,η)⁢∂2∂⁡η2⁢(θ¯⁢θ0+ε⁢θ¯⁢θ12)+ε⁢ξ⁢κ′⁢(η)⁢σ3⁢(ξ,η)⁢∂∂⁡η⁢(θ¯⁢θ0+ε⁢θ¯⁢θ12)+g⁢(u0+θ¯⁢θ0)+g′⁢(u0+θ¯⁢θ0)⁢ε⁢θ¯⁢θ12-g⁢(u0+θ¯⁢θ0+ε⁢θ¯⁢θ12).

We note that thanks to the Taylor expansion, we have

g⁢(u0+θ¯⁢θ0+ε⁢θ¯⁢θ12)-g⁢(u0+θ¯⁢θ0)=g′⁢(u0+θ¯⁢θ0)⁢ε⁢θ¯⁢θ12+T12,

where

(3.34)∥T12∥L2⁢(Ω)≤c⁢∥ε⁢(θ¯⁢θ12)2∥L2⁢(Ω)≤c⁢ε54.

On the other hand,

∥-ε⁢κ⁢(η)⁢σ⁢(ξ,η)⁢δ~⁢δ⁢(ξ)⁢∂∂⁡ξ⁢(θ¯⁢θ0+ε⁢θ¯⁢θ12)+ε⁢κ⁢(η)⁢δ~⁢δ⁢(ξ)⁢∂∂⁡ξ⁢θ¯⁢θ0∥L2⁢(Ω)

=∥-ε⁢ε⁢κ⁢(η)⁢∂∂⁡ξ⁢θ¯⁢θ12⁢δ~⁢δ⁢(ξ)-ε⁢κ⁢(η)⁢∑j=1∞(κ⁢(η)⁢ε⁢ξ¯⁢ξ)j⁢δ~⁢δ⁢(ξ)⁢∂∂⁡ξ⁢(θ¯⁢θ0+ε⁢θ¯⁢θ12)∥L2⁢(Ω)

(3.35)≤c⁢ε54,

by Remark 3.2 and (3.27). We infer from Lemma 3.4, (3.34) and (3.35) that

∥R12⁢δ~⁢δ⁢(ξ)∥L2⁢(Ω)≤c⁢ε54,

while E12 and G12 are e.s.t. We now find from (3.32) that

(-ε⁢Δ⁢w12+g⁢(uε)-g⁢(u0+θ0+ε⁢θ12),w12)

=(-ε⁢Δ⁢uε+g⁢(uε)-g⁢(u0),w12)+(ε⁢Δ⁢u0,w12)+(ε⁢Δ⁢(θ0+ε⁢θ12)-g⁢(u0+θ0+ε⁢θ12)+g⁢(u0),δ~⁢δ⁢(ξ)⁢w12)

(3.36)=(ε⁢Δ⁢u0,w12)+((G12+E12)⁢δ~⁢δ⁢(ξ),w12)+(R12⁢δ~⁢δ⁢(ξ),w12).

We then find from (3.36) and the mean value theorem that

(-ε⁢Δ⁢w12+g′⁢(ζ)⁢w12,w12)=(ε⁢Δ⁢u0,w12)+((G12+E12)⁢δ~⁢δ⁢(ξ),w12)+(R12⁢δ~⁢δ⁢(ξ),w12)

for some ζ between uε and (u0+θ0+ε⁢θ12). We then conclude

(-ε⁢Δ⁢w12+g′⁢(ζ)⁢w12,w12)≤c⁢ε⁢∥w12∥L2⁢(Ω),

which implies

ε⁢∥w12∥H1⁢(Ω)2+λ⁢∥w12∥L2⁢(Ω)2≤c⁢ε2+λ2⁢∥w12∥L2⁢(Ω)2.∎

3.3 Boundary layer analysis at arbitrary orders εn and εn+12, n≥0

Similarly as in (2.20), we formally write

(3.37)-ε⁢Δ⁢(∑j=0∞εj⁢(θj+ε⁢θj+12))+g⁢(∑j=0∞εj⁢(uj+θj+ε⁢θj+12))-g⁢(∑j=0∞εj⁢uj)=0.

Because of the one-dimensional nature of these boundary layers near the boundary in the direction normal to the boundary, we now introduce the boundary fitted coordinates. We transform the Laplacian Δ as in (3.7) and (3.8).

Using the geometric series expansions

(1-r)-1=∑l=0∞rl,(1-r)-2=∑l=0∞(l+1)⁢rl and (1-r)-3=12⁢∑l=0∞(l+1)⁢(l+2)⁢rl,

we formally write, e.g.,

(3.38)σ3⁢(ξ,η)=(1-κ⁢(η)⁢ξ)-3=12⁢∑l=0∞(l+1)⁢(l+2)⁢(κ⁢(η)⁢ξ¯⁢ξ)l⁢εl2.

Then,

ε⁢Δ⁢(∑j=0∞(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))=∂2∂⁡ξ¯⁢ξ2⁢(∑j=0∞(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))-κ⁢(η)⁢(∑l=0∞(κ⁢(η)⁢ξ¯⁢ξ)l⁢εl2)⁢∂∂⁡ξ¯⁢ξ⁢(∑j=0∞(εj+12⁢θ¯⁢θj+εj+1⁢θ¯⁢θj+12))

+(∑l=0∞(l+1)⁢(κ⁢(η)⁢ξ¯⁢ξ)l⁢εl2)⁢∂2∂⁡η2⁢(∑j=0∞(εj+1⁢θ¯⁢θj+εj+32⁢θ¯⁢θj+12))

+ξ¯⁢ξ2⁢κ′⁢(η)⁢(∑l=0∞(l+1)⁢(l+2)⁢(κ⁢(η)⁢ξ¯⁢ξ)l⁢εl2)⁢∂∂⁡η⁢(∑j=0∞(εj+32⁢θ¯⁢θj+εj+2⁢θ¯⁢θj+12))

(3.39) =∂2∂⁡ξ¯⁢ξ2⁢(∑j=0∞(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))+I1∞+I2∞+I3∞.

Using

∑l=0∞(κ⁢(η)⁢ξ¯⁢ξ)l⁢εl2=∑m=0∞(κ⁢(η)⁢ξ¯⁢ξ)2⁢m⁢εm+∑m=0∞(κ⁢(η)⁢ξ¯⁢ξ)2⁢m+1⁢εm+12,

we rearrange the terms according to the order of ε and then we find the last expression in (3.39). The Ik are defined and expanded as follows

(3.40){I1n=-κ⁢(η)⁢∑j=0n(εj⁢J1j⁢(θ¯⁢θ)+εj+12⁢J1j+12⁢(θ¯⁢θ)),I2n=∑j=0n(εj⁢J2j⁢(θ¯⁢θ)+εj+12⁢J2j+12⁢(θ¯⁢θ)),I3n=ξ¯⁢ξ2⁢κ′⁢(η)⁢∑j=0n(εj⁢J3j⁢(θ¯⁢θ)+εj+12⁢J3j+12⁢(θ¯⁢θ)),

where

(3.41){J1j⁢(θ¯⁢θ)=∑k=0j-1(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-1⁢∂⁡θ¯⁢θk∂⁡ξ¯⁢ξ+∑k=0j-1(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-2⁢∂⁡θ¯⁢θk+12∂⁡ξ¯⁢ξ,J1j+12⁢(θ¯⁢θ)=∑k=0j(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)⁢∂⁡θ¯⁢θk∂⁡ξ¯⁢ξ+∑k=0j-1(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-1⁢∂⁡θ¯⁢θk+12∂⁡ξ¯⁢ξ,

(3.42){J2j⁢(θ¯⁢θ)=∑k=0j-1(2⁢(j-k)-1)⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-2⁢∂2⁡θ¯⁢θk∂⁡η2+∑k=0j-2(2⁢(j-k)-2)⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-3⁢∂2⁡θ¯⁢θk+12∂⁡η2,J2j+12⁢(θ¯⁢θ)=∑k=0j-1(2⁢(j-k))⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-1⁢∂2⁡θ¯⁢θk∂⁡η2+∑k=0j-1(2⁢(j-k)-1)⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-2⁢∂2⁡θ¯⁢θk+12∂⁡η2,

(3.43){J3j⁢(θ¯⁢θ)=∑k=0j-2(2⁢(j-k)-2)⁢(2⁢(j-k)-1)⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-3⁢∂⁡θ¯⁢θk∂⁡η+∑k=0j-2(2⁢(j-k)-3)⁢(2⁢(j-k)-2)⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-4⁢∂⁡θ¯⁢θk+12∂⁡η,J3j+12⁢(θ¯⁢θ)=∑k=0j-1(2⁢(j-k)-1)⁢(2⁢(j-k))⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-2⁢∂⁡θ¯⁢θk∂⁡η+∑k=0j-2(2⁢(j-k)-2)⁢(2⁢(j-k)-1)⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-3⁢∂⁡θ¯⁢θk+12∂⁡η.

We note that when the upper limit and exponents in the expressions of Jkj and Jkj+12 are negative these terms are set to be zero. To handle the nonlinear term g, we derive the analogue of Lemma 2.6. Replacing ε by ε, n by 2⁢n+d (d=0,1), and then renaming u2⁢j, θ2⁢j, u2⁢j+1, θ2⁢j+1, respectively, as uj, θj, uj+12, θj+12 and setting uj+12=0, we obtain the following lemma.

Lemma 3.8.

There exists a constant C>0, independent of ε, such that

(3.44){|g(∑j=0n(εjuj+εjθj)+∑j=0n-1εj+12θj+12)-g(∑j=0nεjuj)-Gn|≤C⁢ε2⁢n+1,|g⁢(∑j=0n(εj⁢uj+εj⁢θj+εj+12⁢θj+12))-g⁢(∑j=0nεj⁢uj)-Gn+12|≤C⁢ε2⁢n+2,

where, for d=0,1,

(3.45)Gn+d2=∑m=02⁢n+d{∑k=0m∑|α|=k,α1+2⁢α2+⋯+m⁢αm=m(kα)⁢1k!⁢[g(k)⁢(u0+θ0)⁢(u+θ)α-g(k)⁢(u0)⁢uα]}⁢εm,

and the multi-index notations are defined, for d=0,1, as follows

(3.46){α=(α1,…,α2⁢j+d),|α|=α1+⋯+α2⁢j+d,(u+θ)α=(u12+θ12)α1⁢…⁢(uj+d2+θj+d2)α2⁢j+d,uα=(u12)α1⁢…⁢(uj+d2)α2⁢j+d,

with uj+12=(uj+12)α2⁢j+1=0 if α2⁢j+1≥1 and (uj+12)0=1 for j=0,1,2,….

We now construct high order boundary layers. From the formal sum (3.37) and Lemma 3.8 with n=∞, we can write

-ε⁢Δ⁢(∑j=0∞εj⁢(θj+ε⁢θj+12))

(3.47)+∑m=0∞{∑k=0m∑|α|=k,α1+2⁢α2+⋯+m⁢αm=m(kα)1k![g(k)(u0+θ0)(u+θ)α-g(k)(u0)uα]}εm=0.

We then observe that at order 𝒪⁢(ε)

-θ¯θξ¯⁢ξ⁢ξ¯⁢ξ1+g′(u0+θ¯θ0)θ¯θ1=(g′(u0)-g′(u0+θ¯θ0))u1-g′′⁢(u0+θ¯⁢θ0)2(θ¯θ12)2-κ(η)((κ(η)ξ¯ξθ¯θξ¯⁢ξ0+θ¯θξ¯⁢ξ)12+θ¯θη⁢η0.

Incorporating (3.39) and (3.40), and from (3.47) at m=2⁢j, we obtain at order 𝒪⁢(εj) for j≥1 that

-θ¯⁢θξ¯⁢ξ⁢ξ¯⁢ξj+g′⁢(u0+θ¯⁢θ0)⁢θ¯⁢θ⁢j=-(g′⁢(u0+θ¯⁢θ0)-g′⁢(u0))⁢uj

-{∑k=22⁢j∑|α|=k,α1+2⁢α2+⋯+2⁢j⁢α2⁢j=2⁢j(kα)⁢1k!⁢[g(k)⁢(u0+θ¯⁢θ0)⁢(u+θ¯⁢θ)α-g(k)⁢(u0)⁢uα]}

(3.48)-κ⁢(η)⁢J1j⁢(θ¯⁢θ)+J2j⁢(θ¯⁢θ)+ξ¯⁢ξ2⁢κ′⁢(η)⁢J3j⁢(θ¯⁢θ),

where the multi-index notation is described in (3.46). On the other hand, at order 𝒪⁢(εj+12) for j≥1, from (3.47) at m=2⁢j+1 we similarly find that

-θ¯⁢θξ¯⁢ξ⁢ξ¯⁢ξj+12+g′⁢(u0+θ¯⁢θ0)⁢θ¯⁢θj+12=-(g′⁢(u0+θ¯⁢θ0)-g′⁢(u0))⁢uj+12

-{∑k=22⁢j+1∑|α|=k,α1+2⁢α2+⋯+(2⁢j+1)⁢α2⁢j+1=2⁢j+1(kα)⁢1k!⁢[g(k)⁢(u0+θ¯⁢θ0)⁢(u+θ¯⁢θ)α-g(k)⁢(u0)⁢uα]}

(3.49)-κ⁢(η)⁢J1j+12⁢(θ¯⁢θ)+J2j+12⁢(θ¯⁢θ)+ξ¯⁢ξ2⁢κ′⁢(η)⁢J3j+12⁢(θ¯⁢θ).

The two equations (3.48) and (3.49) are, respectively, supplemented with the boundary conditions

{θ¯⁢θj=-uj|ξ=0=-uj⁢(X⁢(η),Y⁢(η))at ⁢ξ=0,θ¯⁢θj=0at ⁢ξ=ξ0, {θ¯⁢θj+12=0at ⁢ξ=0,θ¯⁢θj+12=0at ⁢ξ=ξ0.

As before, we extend θ¯⁢θj and θ¯⁢θj+12 by zero for ξ>ξ0 and these extensions are still denoted by the same notations.

Multiplying, respectively, (3.14) by ε0, (3.15) by ε12, (3.48) by εj and (3.49) by εj+12, from j=1 to j=n, and adding these resulting equations we find that

-∑j=0nεj⁢(θ¯⁢θξ¯⁢ξ⁢ξ¯⁢ξj+ε⁢θ¯⁢θξ¯⁢ξ⁢ξ¯⁢ξj+12)

(3.50)=-∑m=02⁢n+1{∑k=0m∑|α|=k,α1+2⁢α2+⋯+m⁢αm=m(kα)⁢1k!⁢[g(k)⁢(u0+θ¯⁢θ0)⁢(u+θ¯⁢θ)α-g(k)⁢(u0)⁢uα]}⁢εm+I1n+I2n+I3n,

where Ikn are described as in (3.40).

Lemma 3.9.

For l,m,n≥0 and j=0,1,2,…, there exist c>0 such that, pointwise,

(3.51)|ξn⁢∂l+m⁡θ¯⁢θj∂⁡ξl⁢∂⁡ηm|+|ξn⁢∂l+m⁡θ¯⁢θj+12∂⁡ξl⁢∂⁡ηm|≤c⁢εn-l2⁢exp⁡(-12⁢λε⁢ξ).

Moreover, θ¯⁢θj and θ¯⁢θj+12 satisfy, for j=0,1,2,…,

(3.52)∥ξn∂l+m⁡θ¯⁢θj∂⁡ξl⁢∂⁡ηm∥Lξ2⁢(Ω)+∥ξn∂l+m⁡θ¯⁢θj+12∂⁡ξl⁢∂⁡ηm∥Lξ2⁢(Ω)≤cεn-l2+14.

Proof.

Using Lemma 3.3 and Lemma 3.4 and the induction on j, from (3.48) and (3.49), we obtain the estimates (3.19) and (3.25) for θ¯⁢θj and θ¯⁢θj+12, j=0,1,2,…. Here, as indicated in (2.38), for k≥2 we used the fact that the right-hand side of (3.48) and (3.49) involve only the preceding boundary layer correctors. From the pointwise estimates (3.51), we readily obtain the L2- norm estimates (3.52). ∎

Lemma 3.10.

There exists c>0 such that

∥(ε⁢Δ⁢(∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))-In)⁢δ~⁢δ⁢(ξ)∥L2⁢(Ω)≤c⁢εn+54,

where δ~⁢δ⁢(ξ) is given in (3.28), and

In=∑j=0nεj⁢(θ¯⁢θj+ε⁢θ¯⁢θj+12)ξ¯⁢ξ⁢ξ¯⁢ξ+I1n+I2n+I3n

with Iin given by (3.40).

Proof.

Using the Laplacian (3.7) in terms of ξ,η, we write

(3.53)ε⁢Δ⁢(∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))-ε⁢∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12)ξ⁢ξ=K1n+K2n+K3n,

where

K1n=-ε⁢κ⁢(η)⁢σ⁢(ξ,η)⁢∂∂⁡ξ⁢(∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12)),K2n=ε⁢σ2⁢(ξ,η)⁢∂2∂⁡η2⁢(∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12)),K3n=ε⁢ξ⁢κ′⁢(η)⁢σ3⁢(ξ,η)⁢∂∂⁡η⁢(∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12)).

We now only have to prove that

(3.54)∥(Kin-Iin)⁢δ~⁢δ⁢(ξ)∥L2≤κ⁢εn+54,i=1,2,3.

Noting that ∑j=0n∑k=0j=∑k=0n∑j=kn and ∑j=0n∑k=0j-1=∑k=0n∑j=k+1n, from (3.40) and (3.41) we write

I1n=-κ⁢(η)⁢∑k=0n(∑j=k+1nεj⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-1+∑j=knεj+12⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k))⁢∂⁡θ¯⁢θk∂⁡ξ¯⁢ξ

-κ⁢(η)⁢∑k=0n(∑j=k+1nεj⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-2+∑j=k+1nεj+12⁢(κ⁢(η)⁢ξ¯⁢ξ)2⁢(j-k)-1)⁢∂⁡θ¯⁢θk+12∂⁡ξ¯⁢ξ

=-κ⁢(η)⁢∑k=0n(εk+1⁢∑m=02⁢(n-k)(κ⁢(η)⁢ξ)m)⁢∂⁡θ¯⁢θk∂⁡ξ-κ⁢(η)⁢∑k=0n(εk+32⁢∑m=02⁢(n-k)-1(κ⁢(η)⁢ξ)m)⁢∂⁡θ¯⁢θk+12∂⁡ξ.

Then, we find

∥(K1n-I1n)⁢δ~⁢δ⁢(ξ)∥L2≤∥ε⁢κ⁢(η)⁢(∑j=0nεj⁢(σ⁢(ξ,η)-∑m=02⁢(n-j)(κ⁢(η)⁢ξ)m)⁢∂⁡θ¯⁢θj∂⁡ξ)⁢δ~⁢δ⁢(ξ)∥L2

+∥εκ(η)(∑j=0nεj+12(σ(ξ,η)-∑m=02⁢(n-j)-1(κ(η)ξ)m)∂⁡θ¯⁢θj+12∂⁡ξ)δ~δ(ξ)∥L2

≤cε∑j=0n(∥εj(κ(η)ξ)2⁢(n-j)+1δ~δ(ξ)∂⁡θ¯⁢θj∂⁡ξ∥L2+∥εj+12(κ(η)ξ)2⁢(n-j)δ~δ(ξ)∂⁡θ¯⁢θj+12∂⁡ξ∥L2)

≤c⁢εn+54,

where in the last inequality we used Lemma 3.9. Note that in (3.42) and (3.43) the sum ∑k=0j-2 can be replaced by ∑k=0j-1 because the terms there for k=j-1 do not contribute. Then, permuting summations as above, we can also prove (3.54) for i=2,3. The lemma thus follows. ∎

We use Lemmas 3.8 and 3.10, and equation (3.50), to find that

(3.55)-ε⁢Δ⁢(∑j=0n(εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))+g⁢(∑j=0n(εj⁢uj+εj⁢θ¯⁢θj+εj+12⁢θ¯⁢θj+12))-g⁢(∑j=0nεj⁢uj)=Rn+R,

where

∥Rn⁢δ~⁢δ⁢(ξ)∥L2≤c⁢εn+54,|R|≤c⁢ε2⁢n+2.

As before, we define now θj=θ¯⁢θj⁢δ⁢(ξ) and θj+12=θ¯⁢θj+12⁢δ⁢(ξ) for each j≥0, where δ is defined as in (3.17). Note that θj,θj+12 satisfy the boundary conditions as in (3.12).

Theorem 3.11.

Assume that f is a smooth function, Ω is a general smooth domain and uε is solution of (2.1). Let uj and θj satisfy (3.3) and (3.48), respectively. Then, for every n≥0, there exists a constant c>0, independent of ε, such that

(3.56)∥uε-∑j=0nεj⁢(uj+θj+ε⁢θj+12)∥ε≤c⁢εn+1,

(3.57)∥uε-∑j=0nεj⁢(uj+θj)-∑j=0n-1εj+12⁢θj+12∥ε≤c⁢εn+34.

Proof.

We set

wn+12=uε-∑j=0nεj⁢(uj+θj+ε⁢θj+12).

We use the smooth cut-off function δ~⁢δ⁢(ξ) to eliminate the singularity of σ⁢(ξ,η) where δ~⁢δ⁢(ξ) is given by (3.28). Then, by a similar argument as before, we obtain that

(ε⁢Δ⁢∑j=0nεj⁢(θj+ε⁢θj+12)-g⁢(∑j=0nεj⁢(uj+θj+ε⁢θj+12))+g⁢(∑j=0nεj⁢uj),wn+12)

=((Gn+12+En+12)⁢δ~⁢δ⁢(ξ),wn+12)+(Rn+12⁢δ~⁢δ⁢(ξ),wn+12),

where

En+12=ε⁢Δ⁢(∑j=0nεj⁢(θj+ε⁢θj+12)-∑j=0nεj⁢(θ¯⁢θj+ε⁢θ¯⁢θj+12)),

Gn+12=g⁢(∑j=0nεj⁢(uj+θ¯⁢θj+ε⁢θ¯⁢θj+12))-g⁢(∑j=0nεj⁢(uj+θj+ε⁢θj+12)),

Rn+12=Rn+R,

where Rn and R are given in (3.55). We note that En+12 and Gn+12 are e.s.t., and ∥Rn+12⁢δ~⁢δ⁢(ξ)∥L2⁢(Ω)≤εn+1. We now find

(-ε⁢Δ⁢wn+12+g⁢(uε)-g⁢(∑j=0nεj⁢(uj+θj+ε⁢θj+12)),wn+12)

=(εn+1⁢Δ⁢un,wn+12)+((Gn+12+En+12)⁢δ~⁢δ⁢(ξ),wn+12)+(Rn+12⁢δ~⁢δ⁢(ξ),wn+12).

Here, we used (2.41) which is obtained from summing (3.3).

We finally obtain from the mean value theorem that, for some ζ between uε and ∑j=0nεj⁢(uj+θj+ε⁢θj+12),

(-ε⁢Δ⁢wn+12+g′⁢(ζ)⁢wn+12,wn+12)≤c⁢εn+1⁢∥wn+12∥L2⁢(Ω).

This proves (3.56). From Lemma 3.9, we note that ∥εn+12⁢θn+12∥ε≤c⁢εn+34, and the estimate (3.57) follows from (3.56). ∎

Funding source: National Science Foundation

Award Identifier / Grant number: DMS 1206438

Award Identifier / Grant number: DMS 1510249

Funding source: National Research Foundation of Korea

Award Identifier / Grant number: 2015K2A1A2070543

Award Identifier / Grant number: 2015R1D1A1A01059837

Funding statement: This work was supported in part by NSF grants DMS 1206438 and 1510249, by a research fund of Indiana University. Chang-Yeol Jung was supported under the framework of international cooperation program managed by the National Research Foundation of Korea (grant no. 2015K2A1A2070543) and supported by the National Research Foundation of Korea grant funded by the Ministry of Education (grant no. 2015R1D1A1A01059837).

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Received: 2015-11-1

Revised: 2015-11-29

Accepted: 2015-12-7

Published Online: 2016-1-14

Published in Print: 2017-8-1

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https://doi.org/10.1515/anona-2015-0148

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Boundary layers; singular perturbations; reaction-diffusion; boundary fitted coordinates

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