BOUNDARY LAYERS IN SMOOTH CURVILINEAR DOMAINS: PARABOLIC PROBLEMS

. The goal of this article is to study the boundary layer of the heat equation with thermal diﬀusivity in a general (curved), bounded and smooth domain in R d , d ≥ 2, when the diﬀusivity parameter ǫ is small. Using a curvilinear coordinate system ﬁtting the boundary, an asymptotic expansion, with respect to ǫ , of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ǫ and ǫ 1 / 2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.


1.
Introduction. In this article, we study the boundary layer of the heat equation with small thermal diffusivity in a general bounded and smooth domain; we aim to simplify and extend, in different ways, the results presented in [33] (see also [20] and [36]). We consider the heat equation in a bounded smooth domain Ω ⊂ R d , d ≥ 2, with boundary ∂Ω: , (x, t) ∈ Ω × (0, T ), u ǫ (x, t) = 0 on ∂Ω, where f and u 0 are given smooth functions, and ǫ is a small strictly positive parameter. We will specify the regularities of ∂Ω, f and u 0 below when we perform the error analysis although the emphasis in this article is not on optimal regularity requirements. Moreover, we impose a consistency condition on the data, namely: As we see below, due to the curvature of the boundary, the usual expansion in powers of ǫ will not give a suitable approximation. Indeed, as we show below, the usual expansion has to be adapted to the current situation by introducing terms of order ǫ j+1/2 in the expansion; these terms appear because, for a general curved domain, unlike a channel or a cube domain, the normal direction is changing along the boundary. Using the techniques of differential geometry and continuing the work in [12], we find explicit expressions of the correctors at all orders and obtain the optimal convergence rate as stated in Theorems 3.1 and 4.1 hereafter.
Using the techniques developed in this work, we intend in the future to study the linearized and non-linear Navier-Stokes equations in a general bounded and smooth domain; this will constitute a continuation of e.g. [32]- [35], [16], [17] and [19]. It is known that, at small viscosity, the solution of the time-dependent linearized Navier-Stokes equations (LNSE) behaves, to some (limited) extent, in parts of the domain under consideration, like the heat solution. This is especially the case when the boundary is characteristic, i.e. the homogeneous Dirichlet boundary conditions are imposed. In this direction, and in order to study the LNSE, Temam and Wang considered first a channel domain to avoid the geometric complexity of a general domain; see [32]. Later, they focused on the study of the boundary layer of the heat equation and the LNSE in a two-dimensional general domain. More precisely, they proposed in [33] the use of a curvilinear coordinate system adapted to the geometry of the boundary, which was also used in the convergence result for the fully non-linear Navier-Stokes problem; see [34]. We recall here that, when studying the boundary layers, the non-characteristic case for the Navier-Stokes problem does not involve the heat equation; the readers may consult [16] and [35].
The article is organized as follows: first, in Section 2 we propose a formal asymptotic expansion of u ǫ , solution of (1.1), and express the Laplace operator in terms of a curvilinear coordinate system adapted to the boundary. Then, in Section 3, we explicitly give the correctors at order ǫ 0 and ǫ 1/2 and perform the error estimates; in fact, we will see that, by adding the corrector at order ǫ 1/2 in the expansion of u ǫ , we recover the optimal convergence rate of the remainder. Finally, in Section 4, we expand our results to all orders ǫ N and ǫ N +1/2 , N ≥ 1.
2. Asymptotic expansions. In this article, (x 1 , . . . , x d ) denotes the Cartesian coordinates of a point x ∈ R d and ∆ is the Laplace operator with respect to the x variable. We assume that the domain Ω satisfies the following property: where each Γ i is a connected component of ∂Ω which is a smooth Jordan surface in R d−1 with Ω lying locally on one side of Γ i . We choose a curvilinear coordinate system ξ = (ξ ′ , ξ d ), ξ ′ = (ξ 1 , . . . , ξ d−1 ), adapted to ∂Ω, such that, more precisely: and Ω is located on the side ξ d > 0}. (2.2) For δ "small", we will consider the 3δ-neighborhood Ω 3δ of ∂Ω, Ω 3δ = {x ∈ Ω| dist(x, ∂Ω) < 3δ}, (2.3) and assume that this set is diffeomorphic to the following set in the space R d ξ : (2.4) where ω ′ is an open bounded set in R d−1 ξ ′ . We will also consider Ω 2δ = {x ∈ Ω| dist(x, ∂Ω) < 2δ}, (2.5) and In this article, we assume that ǫ << δ, which is reasonable since we aim to study the asymptotic behavior of the solution of (1.1) when the parameter ǫ tends to 0. Without loss of generality, we assume that our curvilinear coordinate system ξ is orthogonal and satisfies and we introduce the following classical geometrical notations for the coordinates ξ (see e.g. [5] and [37]): If we set h = √ g > 0, h ′ = ∂h/∂ξ d , then, using (2.8), we can write the Laplace operator in the ξ variable in the following form: (2.9) see [5] and [37] for more details.
To study the singularly perturbed problem (1.1), we will look for an asymptotic expansion of u ǫ in the form: where the u j correspond to the external expansion (outside of the boundary layer) and the correctors θ j , θ j+ 1 2 correspond to the inner expansion (inside the boundary layer); see e.g. [9], [18], [20] or [26] for general results on singular perturbation problems.
To obtain the external expansion of u ǫ , we formally set u ǫ ≃ ∞ j=0 ǫ j u j and insert this expression in (1.1) 1 and (1.1) 3 . By identifying all the terms of order ǫ j , for each j in each equation, we obtain the following set of equations and initial conditions: (2.11) Integrating (2.11) over (0, t), we recursively obtain the u j , j ≥ 0 in the form: (2.12) note that the u j are well-defined for all j ≥ 0 under the regularity assumption In the boundary layer, (2.14) as we will see, the terms ǫ j+ 1 2 θ j+ 1 2 appear because of the geometry; we will explain below the need to introduce the correctors θ j+ 1 2 which ensure optimal convergence results (see Remark 2.2 and Theorems 3.1 and 4.1). Now, we introduce the stretched variable ξ d of ξ d : and, using (2.15), we rewrite the Laplacian as follows: (2.16) By considering h(ξ ′ , ξ d ) = h(ξ ′ , ǫ 1/2 ξ d ), with ξ ′ , ξ d variables of order one in the boundary layer, a dependency of S and L on ǫ appears in (2.16). To address this dependency, we first introduce a notation: for any φ ∈ C ∞ (Ω 2δ ), we set and we write the Taylor expansion of where (h) j , (1/h) j , (h ′ ) j and (g γγ ) j can be made explicit by using the Faà di Bruno formula; see [6] and also [11], [12].
Thanks to (2.16) and (2.19), we now find the following expansions of S and L: , which are independent of ǫ and ξ d , are well-defined if ∂Ω is of class C j+2 . (2.21) It is noteworthy to observe that the S j/2 are tangential operators near ∂Ω and that the operators L j/2 are proportional to ǫ 1 2 ∂/∂ξ d . Of course, if we just want to study the asymptotic expansion of u ǫ at order ǫ 0 (this is what we will do in Section 3), we do not need the expansions (2.20). However they are needed for the higher-order cases starting from ǫ 1/2 .
Using (2.16) and (2.20), (2.14) yields which allows us to find the equations for the correctors θ j and θ j+ 1 2 , j ≥ 0; these equations will be presented in the following sections.
Remark 2.1. We will use the stretched variable ξ d to "weight" the different terms in the equation (2.22) and other similar equations. Otherwise we will generally revert to the initial variable ξ d .
Remark 2.2. If we study the problem (1.1) in a domain with a flat boundary, we notice that all the terms of order ǫ j+ 1 2 , j ≥ 0, disappear in the expansion (2.20); that is, in (2.16), h and g γγ , 1 ≤ γ ≤ 1 − d, are independent of ξ d . Hence we do not require the correctors θ j+ 1 2 , j ≥ 0, to obtain the optimal estimates in Theorems 3.1 and 4.1 below.
3. Analysis at lower orders ǫ 0 and ǫ To make (3.1) well-posed, we need to impose two boundary conditions which are not available (see (1.1) 2 and (2.2)). To overcome this difficulty, we replace (3.1) by the following equation (3.2) 1 and add the boundary and initial conditions: Here σ(·) is a cut-off function such that σ(ξ d ) ∈ [0, 1] and, 3) The existence and uniqueness of a solution for (3.2) is established in Section 3.4. For θ 1 2 , we collect all the terms of order ǫ 1 2 in (2.22) and find ∂θ where (see (2.19) and (2.20)) We modify (3.4) as for the corrector θ 0 , and impose the boundary and initial conditions; we then obtain the equation for θ  Note that, due to the presence of σ in (3.2) and (3.6), the problems (3.2) and (3.6) are well-posed with only one boundary condition 1 at ξ d = 0. Furthermore, we notice that, on Ω \ Ω 2δ , i.e. for (ξ ′ , ξ d ) ∈ Ω such that ξ d ≥ 2δ, To perform the error analysis at orders ǫ 0 and ǫ 1 2 , we define the remainders: (3.8) where Similarly, computing the difference between (1.1) and the sum of (2.11) 1 , (3.2) and ǫ 1 2 (3.6), we obtain the equation for w Multiplying (3.9) 1 by w 0 ǫ and integrating over Ω, we find Hence, using the Gronwall inequality, we obtain where κ T denotes a constant depending on T and the data, but independent of ǫ, and which may be different at different occurrences. Back to (3.13), we also notice that For w 1 2 ǫ , we multiply the equation (3.11) by w 1 2 ǫ , perform the same computation as for w 0 ǫ , and obtain (3.16) Note that, using (2.13) with j = 0, (3.14)-(3.16) are valid when and Due to the compatibility condition (1.2), the explicit expressions of θ 0 and θ 1 2 are given in the form (see [3]): and where f 1/2 ǫ = L 0 θ 0 has been defined in (3.19); see also (3.5). In (3.20), Using polar coordinates, one can verify the following useful inequality: for z > 0, and hence For the sake of convenience, we introduce a notation for a function φ in C ∞ ((0, T ) × Ω 2δ ): where ∂ k /∂τ k is any tangential operator of order k, Now we state and prove the following lemmas: (3.22), satisfies the following pointwise estimates: for m = 1, 2, and, for m ≥ 3, where κ m is a constant depending on m, but independent of ǫ, and which may be different at different occurrences.
Proof. For (3.27), using (3.22), we notice (3.29) hence (3.27) follows. Moreover, by differentiating (3.29) 2 (m − 2)-times with respect to ξ d , and using the Leibnitz formula for the m-th derivative of a product, we find, for m ≥ 3, where r 0,m is given in the form: for some strictly positive integers a i, (3.31) Using (3.31), we can bound the lower order term r 0,m , in (3.30), with respect to ǫ: and, from (3.30) and (3.32), we obtain (3.28) for m ≥ 3.
Lemma 3.2. For any p ≥ 0 and q ≥ 1, we have

33)
where κ T is a constant depending on T , but independent of ǫ. Proof. To verify (3.33), since the Jacobian determinant h is bounded on Ω 2δ , we write (3.34) (3.33) follows.
Assume that ∂Ω is of class C 2 , and that, for k ≥ 0, u 0 and f belong to H k (Ω) and L ∞ (0, T ; H k (Ω)) respectively. Then, the approximate corrector θ 0 given by (3.20) satisfies the following pointwise estimates: Moreover, for (t, ξ) ∈ (0, T ) × Ω 2δ,ξ , and for j = 0 and m ≥ 2, and j ≥ 1 and m ≥ 0, we have where κ T,k,m and κ T,j,k,m are constants depending on T , j, k, m, and the other data, but independent of ǫ.
We now return to our objective of respectively comparing θ 0 and θ 0 , and θ Thanks to Lemma 3.3 and the definition of σ in (3.3), we notice that, for j, k, m ≥ 0, where κ is a constant depending on the data, and on the indices j, k and m, but not on ǫ. Essential here is the fact that (σ − 1), σ ′ and the higher derivatives of σ vanish for 0 < ξ d < δ. Here we call e.s.t. a function (or a constant) whose norm in all Sobolev spaces H s (and thus spaces C s ) is exponentially small with a bound of the form c 1 exp(−c 2 /ǫ α ), c 1 , c 2 , α > 0, for each s. Hence, we deduce from (3.55) that F 0 ǫ satisfies the condition (A.2) for any fixed J, K and M , and, by applying Lemma A.1 for Φ = θ 0 − θ 0 , we obtain For the first term in the right-hand side of (3.57), as for (3.55), observing that (σ − 1) and the derivatives of σ vanish for 0 < ξ d < δ, we use Lemma 3.4, and find Combining this with (3.56), we conclude that F     Note that, in the error analysis at lower orders ǫ 0 and ǫ 1 2 below, we will need (3.59) for j = 0, 0 ≤ k, m ≤ 2, or equivalently, (3.56) for 0 ≤ j ≤ 1, 0 ≤ k ≤ 2 and 0 ≤ m ≤ 3, which are valid under the regularity assumptions (3.17).

3.3.
Convergence results at lower orders ǫ 0 and ǫ 1 2 . Thanks to the results in Section 3.2, we are now able to make the estimates (3.14)-(3.16) more useful.
Using (3.7) and (3.10), we write In (3.60) and below, we use θ and θ to denote the whole collections of θ j 2 and θ j 2 , and do not specify those used (needed) for the expressions under consideration. By the expression (3.10) of R ǫ 0 , we see that Observing that S is a tangential differential operator and that L is the product of a bounded function with ǫ 1/2 ∂/∂ξ d , we infer from (3.55) and Lemmas 3.2 and 3.3 is an e.s.t., the last bound is also valid for R 0 ǫ (θ) and thus: Using (3.7) and (3.12), we estimate R 1 2 ǫ (θ) similarly: Due to the expression (3.12) of R (3.65) Moreover, using (2.20), we notice that and, finally we obtain R Now, by inserting (3.63) and (3.68) into (3.14)-(3.16), we obtain the convergence results at orders ǫ 0 and ǫ 1 2 that we summarize in the following theorem: where κ T is a constant depending on T and the other data, but independent of ǫ.
Remark 3.2. Theorem 3.1 shows that where Θ is a function of u 0 , and c > 0 depends on the function space considered.

3.4.
Existence and uniqueness of a solution for (3.2). In this section, we briefly establish the existence and uniqueness of a solution for the system (3.2). Firstly, to make the boundary condition homogeneous, we reuse the same cut-off function σ(ξ d ) from (3.3), and consider the function The function ϕ satisfies the following system: (3.76) We set Note that, if v and √ σ∂v/∂ξ d belong to L 2 (Ω 2δ,ξ ) (that is ∂v/∂ξ d ∈ L 2 (Ω δ,ξ )), the trace of v, at ξ d = 0, is well-defined and belongs to L 2 (ω ′ ). Then ϕ is looked for as a function from (0, T ) into W satisfying the following weak form of (3.75), that is (3.78) The problem (3.78) is a standard linear evolution problem for which we easily obtain the existence and uniqueness of a solution ϕ such that see e.g. [23] and [31]. Further regularity properties of ϕ (and thus of θ 0 ) are obtained by inspection of (3.2) 1 in a routine way.

4.1.
Correctors θ N and θ N + 1 2 . To derive an equation for θ N , N ≥ 0, we collect all terms of order ǫ N in (2.22) and find We modify (4.1) as for the lower orders and impose the proper boundary and initial conditions. Then, as a result, we obtain (propose) the following equation for θ N : For θ N + 1 2 , we collect all terms of order ǫ N + 1 2 in (2.22) and find: after we modify (4.4) and impose the boundary and initial conditions, we obtain the following proposed equation for θ N + 1 2 : As we have seen in the case when N = 0, the problems (4.3) and (4.6) are well-posed for any N ≥ 0. Moreover, on Ω \ Ω 2δ , (ξ d ≥ 2δ), we notice that To perform the error analysis at orders ǫ N and ǫ N + 1 2 , we consider the remainders:

4.2.
Approximations θ N and θ N + 1 2 of the correctors θ N and θ N + 1 2 , N ≥ 0. To make the estimates (4.12) more useful, we introduce the approximations θ N and θ N + 1 2 of the correctors θ N and θ N + 1 2 , defined as the solutions of the following heat equations on R + : where and Due to the linearity of the equation (4.14), we find the solution θ N , N ≥ 1 in the form: from (2.11), we see that the u j , j ≥ 1 satisfy the initial conditions: We also find the solution θ N + 1 2 , N ≥ 1 of (4.16) in the form: Now, we prove the following pointwise estimates for θ N and θ N + 1 2 : Lemma 4.1. Assume that, for k ≥ 0, ∂Ω is of class C 2N +2+k , and that u 0 and f belong to H 2N +k (Ω) and L ∞ (0, T ; H 2N +k (Ω)) respectively. Then, the approximate correctors θ N + r 2 , r = 0, 1, in (4.18) and (4.20), satisfy the following pointwise estimates: for (t, ξ) ∈ (0, T ) × ω ′ × R + , and for m = 0, 1, we have Moreover, for (t, ξ) ∈ (0, T ) × Ω 2δ,ξ , and for j = 0 and m ≥ 2, and j ≥ 1 and m ≥ 0, we have (4.22) where κ T,k,m and κ T,j,k,m are constants depending on T , j, k, m, and the other data, but independent of ǫ.
Proof. We proceed by induction on N . Thanks to Lemmas 3.3 and 3.4, we first see that (4.21) and (4.22) hold true when N is equal to 0. If we assume that (4.21) and (4.22) are valid when N ≤ k and r = 0, 1, then, using (4.15) and the inductive assumption, we notice that . As for the lower order cases, thanks to Lemma 4.1 and the fact that (σ − 1) and the derivatives of σ vanish for 0 < ξ d < δ, we find Then, using (3.56) to start induction on 0 ≤ n ≤ 2N + 1, and recursively using Lemma A.1 for Φ = θ Note that (4.26) is valid for all 0 ≤ n ≤ 2N + 1, under the regularity assumptions (4.13).

4.3.
Convergence results at general orders ǫ N and ǫ N + 1 2 . Due to the results in Section 4.2, we are now able to make the estimates (4.12) useful. Thanks to (4.7) and (4.26), we infer from the expressions (4.10) and (4.11) of R N +r/2 ǫ , r = 0, 1, that (4.28) Recalling that the S and S j/2 , j ≥ 0 are tangential differential operators, and that the L and L j/2 , j ≥ 0 are the products of a bounded function with ǫ 1/2 ∂/∂ξ d , we use (4.26) and Lemmas 3.2 and 4.1. As a result, we can bound |R N ǫ (θ)| in (4.28): Using exactly the same method as for R N ǫ (θ), we also estimate R N + 1 2 ǫ (θ) in (4.11): (4.30) Thanks to (4.27)-(4.30), we finally obtain Now, applying (4.31) to (4.12), we deduce the following convergence results at general orders ǫ N + r 2 , N ≥ 0, r = 0, 1:  32) and where κ T is a constant depending on T and the other data, but independent of   For simplicity, here we consider the Laplacian as the elliptic operator in the problem (1.1) and also use an orthogonal curvilinear system. One can generalize this work, using any boundary fitting (non-orthogonal) coordinates, to study the boundary layer of more general elliptic equations. We believe that many of the techniques used in this article (above and below in the Appendix) will help for such problems.