Asymptotic analysis for elliptic equations with Robin boundary condition

Abstract

We investigate the boundary layers of a singularly perturbed reaction-diffusion equation in a 3D channel domain. The equation is supplemented with a Robin boundary condition especially when the smooth function on the boundary, appearing in the Robin boundary condition, depends on the perturbation parameter. By constructing an explicit function, called corrector, which describes behavior of the perturbed solution near the boundary, we obtain an asymptotic expansion of the perturbed solution as the sum of the corresponding limit solution and the corrector, and show the convergence in $L^{2}$ of the perturbed solution to the limit solution as the perturbation parameter tends to zero.

Keywords

Boundary layers singular perturbations Robin boundary condition

1. Introduction

We consider the reaction-diffusion equation in a periodic channel Ω with boundary $\partial Ω$ at $z = 0, 1$ , $\begin{matrix} (1.1) & \{\begin{matrix} - ε Δ u^{ε} + λ u^{ε} = f, in Ω = {(0, L)}^{2} \times (0, 1) \subset R^{3}, \\ u^{ε} is periodic in x and y with period L, \end{matrix} \end{matrix}$ where the smooth data $f = f (x, y, z)$ is assumed to be periodic in x and y with period L. ε is the small perturbation parameter for our problem and λ is a fixed positive constant (independent of ε) as specified in (1.14) below. Here and later in this article, we use the notations for differential operators, $\begin{matrix} (1.2) & \partial_{x} = \partial / \partial_{x}, \partial_{y} = \partial / \partial_{y}, \partial_{z} = \partial / \partial_{z}, Δ = \partial_{x}^{2} + \partial_{y}^{2} + \partial_{z}^{2}, \nabla = (\partial_{x}, \partial_{y}, \partial_{z}) . \end{matrix}$

The equation (1.1) is supplemented with the Robin boundary condition, $\begin{matrix} (1.3) & \nabla u^{ε} \cdot n + \tilde{γ} u^{ε} = 0, at z = 0, 1, \end{matrix}$ where $\tilde{γ} = \tilde{γ} (x, y)$ is a smooth function on the boundary $\partial Ω$ at $z = 0, 1$ , and the unit outer normal $n$ on $\partial Ω$ is equal to $(0, 0, - 1)$ at $z = 0$ , and $(0, 0, 1)$ at $z = 1$ . Hence we have $\begin{matrix} (1.4) & \{\begin{matrix} - \partial_{z} u^{ε} + \tilde{γ} u^{ε} = 0, & at z = 0, \\ \partial_{z} u^{ε} + \tilde{γ} u^{ε} = 0, & at z = 1 . \end{matrix} \end{matrix}$

The well-posedness of the problem (1.1)–(1.3) is classical. In fact, when $f \in L^{2} (Ω)$ , there exists in $L^{2} (Ω)$ a unique solution $u^{ε}$ (at each fixed $ε > 0$ ), provided that $\tilde{γ}$ is sufficiently regular. Higher regularity of $u^{ε}$ follows from the sufficient regularity assumptions on the data as well. Therefore, in this article, we aim to study the asymptotic behavior of solution $u^{ε}$ to (1.1)–(1.3) as the parameter ε tends to 0.

Setting $ε = 0$ in (1.1)–(1.3), we formally obtain the corresponding limit solution $u^{0}$ , $\begin{matrix} (1.5) & u^{0} = \frac{1}{λ} f . \end{matrix}$ The smooth data f may not satisfy the Robin boundary condition (1.3) at $z = 0, 1$ , and hence we expect boundary layers of (1.1)–(1.3) to occur as the parameter ε gets smaller.

Boundary layers of the reaction-diffusion type equations (and other more general equations in fluids) are well-studied in the literature; see, e.g., [2,4,5,8] as well as some related applications in numerical simulations, e.g., [1,9,11,12,14,15]. Concerning the reaction-diffusion equation, like our model (1.1), it is well-understood that the boundary layer created by Dirichlet boundary condition ( $u^{ε} = 0$ ) is of order $e^{- z / \sqrt{ε}}$ near the boundary at $z = 0$ , while the boundary layer created by Neumann boundary condition ( $\partial_{z} u^{ε} = 0$ ) is of order $\sqrt{ε} e^{- z / \sqrt{ε}}$ near the boundary at $z = 0$ . Hence we call the boundary layer created by Neumann boundary condition as a weak boundary layer as its profile is smaller by a factor of $\sqrt{ε}$ than the one from Dirichlet boundary condition.

In fluid mechanics, there is a generalization of Robin type boundary conditions (like (1.4)) called the Navier-friction boundary condition, and it is often imposed the fluid equations, including the Stokes, Navier–Stokes, and (hydrostatic) primitive equations for oceanic and atmospheric dynamics, where the function equivalent to $\tilde{γ}$ in (1.4) is called the friction coefficient. In fact, in, e.g., [3,6,7,10,13], the boundary layers of the Navier–Stokes equations supplemented with the Navier-friction boundary condition are investigated for the case when the smooth friction coefficient (equivalent to $\tilde{γ}$ in (1.4)) is assumed to be independent of the perturbation parameter ε (called the viscosity in the context of fluid equations). It is verified in [3,6,7] that, as long as the friction coefficient (equivalent to $\tilde{γ}$ in (1.4)) is independent of the viscosity (equivalent to ε in (1.1)), the boundary layer created by the Navier-friction boundary condition (equivalent to the Robin boundary condition (1.4)) can be treated as the one from the Neumann boundary condition. However, it has not been verified yet in the literature and it is not clear at all how to handle the boundary layer from the Navier-friction (or Robin) boundary condition when the friction coefficient (or $\tilde{γ}$ in (1.4)) depends on the perturbation parameter ε.

In this article, we aim to study boundary layers of the reaction-diffusion equation (1.1), especially when the function $\tilde{γ}$ , in the Robin boundary condition (1.3), depends on the perturbation parameter ε. To study this interesting case, we first set $γ : = \tilde{γ} / (‖ \tilde{γ} ‖_{C^{0} (\overline{\partial Ω})} + 1)$ and choose a proper $α \in R$ so that $ε^{α} = 1 / (‖ \tilde{γ} ‖_{C^{0} (\overline{\partial Ω})} + 1)$ where $‖ \tilde{γ} ‖_{C^{0} (\overline{\partial Ω})} : = {sup}_{\overline{\partial Ω}} | \tilde{γ} (x, y) |$ . Then we write the boundary condition (1.3) in the form, $\begin{matrix} (1.6) & ε^{α} \nabla u^{ε} \cdot n + γ u^{ε} = 0, at z = 0, 1, where α \in R, ‖ γ ‖_{C^{0} (\overline{\partial Ω})} < 1 . \end{matrix}$

To study the asymptotic behavior of $u^{ε}$ , satisfying (1.1) and (1.6), we first recall briefly the energy estimate of $u^{ε}$ :

We multiply the equation (1.1) and integrate over Ω. Using the Schwarz and Young inequalities on the right-hand side, we find $\begin{matrix} (1.7) & - ε \int_{Ω} Δ u^{ε} u^{ε} d x + λ {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ \frac{1}{2 λ} ‖ f ‖_{L^{2} (Ω)}^{2} + \frac{λ}{2} {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2}, x = (x, y, z) . \end{matrix}$

We integrate by parts the first term on the left-hand side of (1.7) and write $\begin{matrix} (1.8) & - ε \int_{Ω} Δ u^{ε} u^{ε} d x = ε {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}^{2} + ε \int_{\partial Ω} \nabla u^{ε} \cdot n u^{ε} d x d y . \end{matrix}$ Then, using the boundary condition (1.6), we find $\begin{matrix} (1.9) & | ε \int_{\partial Ω} \nabla u^{ε} \cdot n u^{ε} d x d y | ⩽ ε^{1 - α} \int_{\partial Ω} | γ | {(u^{ε})}^{2} d x d y ⩽ ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (\partial Ω)}^{2} . \end{matrix}$ For the further analysis, we recall the trace theorem saying that, there exists a constant $C_{Ω} ⩾ 1$ , depending only on the domain Ω (hence independent of ε), such that $\begin{matrix} (1.10) & ‖ v ‖_{L^{2} (\partial Ω)} ⩽ C_{Ω} ‖ v ‖_{L^{2} (Ω)}^{\frac{1}{2}} ‖ v ‖_{H^{1} (Ω)}^{\frac{1}{2}}, \forall v \in H^{1} (Ω) . \end{matrix}$ Now, thanks to the trace theorem and using the Young inequality, we see that $\begin{array}{l} ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (\partial Ω)}^{2} & ⩽ C_{Ω}^{2} ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (Ω)} {({‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} + {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}^{2})}^{\frac{1}{2}} \\ ⩽ C_{Ω}^{2} ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (Ω)} ({‖ u^{ε} ‖}_{L^{2} (Ω)} + {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}) \\ ⩽ C_{Ω}^{2} ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} + C_{Ω}^{2} ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (Ω)} {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)} \\ (1.11) & ⩽ C_{Ω}^{2} ε^{1 - α} {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} + \frac{ε}{2} {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}^{2} + \frac{C_{Ω}^{4}}{2} ε^{1 - 2 α} {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} . \end{array}$

Combining the inequalities above, we obtain that $\begin{matrix} (1.12) & ε {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}^{2} + (λ - 2 C_{Ω}^{2} ε^{1 - α} - C_{Ω}^{4} ε^{1 - 2 α}) {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ \frac{1}{λ} ‖ f ‖_{L^{2} (Ω)}^{2} . \end{matrix}$

$(A)$ When $α < 1 / 2$ , for ε small enough, there exists a constant $C_{α 1} > 0$ , independent of ε, such that $C_{α 1} < λ - 2 C_{Ω}^{2} ε^{1 - α} - C_{Ω}^{4} ε^{1 - 2 α}$ , and hence $\begin{matrix} (1.13) & ε {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}^{2} + C_{α 1} {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ \frac{1}{λ} ‖ f ‖_{L^{2} (Ω)}^{2} . \end{matrix}$

$(B)$ When $α = 1 / 2$ , with a technical assumption on the constant λ, $\begin{matrix} (1.14) & λ > C_{Ω}^{4} (> 1), \end{matrix}$ there exists a constant $C_{α 2} > 0$ , independent of ε, such that $C_{α 2} < λ - 2 C_{Ω}^{2} ε^{1 / 2} - C_{Ω}^{4}$ , and hence $\begin{matrix} (1.15) & ε {‖ \nabla u^{ε} ‖}_{L^{2} (Ω)}^{2} + C_{α 2} {‖ u^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ \frac{1}{λ} ‖ f ‖_{L^{2} (Ω)}^{2} . \end{matrix}$

The brief energy estimates $(A)$ and $(B)$ above (which is valid for each fixed $ε > 0$ ) indicate that the $L^{2}$ -norm of $u^{ε}$ is uniformly bounded with respect to the small ε only when $α ⩽ 1 / 2$ (with a technical assumption (1.14) for $α = 1 / 2$ ), and it may blow up otherwise as ε tends to zero. Hence, hereafter, we consider the case when $α ⩽ 1 / 2$ only. More precisely, we study the asymptotic behavior of $u^{ε}$ satisfying (1.1) with $\begin{matrix} (1.16) & ε^{α} \nabla u^{ε} \cdot n + γ u^{ε} = 0, at z = 0, 1, where α ⩽ 1 / 2, ‖ γ ‖_{C^{0} (\overline{\partial Ω})} < 1, \end{matrix}$ which can be written in the form, $\begin{matrix} (1.17) & \{\begin{matrix} - ε^{α} \partial_{z} u^{ε} + γ u^{ε} = 0, & at z = 0, \\ ε^{α} \partial_{z} u^{ε} + γ u^{ε} = 0, & at z = 1 . \end{matrix} \end{matrix}$ In fact, as appearing below in Theorems 3.1 and 4.1, we construct an asymptotic expansion of $u^{ε}$ for $α ⩽ 1 / 2$ , and show the strong convergence in $L^{2} (Ω)$ of $u^{ε}$ to $u^{0}$ as the perturbation parameter ε tends to zero.

Note that the technical assumption (1.14), added for the case when $α = 1 / 2$ , is somewhat natural and usually used to control the $L^{2}$ norm of reaction-diffusion equations. However, this assumption will be no longer needed for a generalization of our analysis to the time-dependent parabolic equations (which we intend to study in a subsequent article), thanks to the Gronwall inequality for time evolution equations.

2. Asymptotic expansion of $u^{ε}$

We aim to construct an asymptotic expansion of $u^{ε}$ (solution of (1.1) and (1.16)) in the form, $\begin{matrix} (2.1) & u^{ε} ≃ u^{0} + θ^{ε}, \end{matrix}$ where $u^{0} = f$ is the corresponding limit solution of $u^{ε}$ as introduced in (1.5), and $θ^{ε}$ is an artificial function (which we call corrector hereafter) that we will explicitly construct below.

Concerning $θ^{ε} ≃ u^{ε} - u^{0}$ , we consider the difference between the equations (1.1) and (1.5), and write $\begin{matrix} (2.2) & - ε Δ (u^{ε} - u^{0}) + (u^{ε} - u^{0}) = ε Δ u^{0}, in Ω . \end{matrix}$ Then, using the ansatz saying that the asymptotic order of $\partial_{x}$ and $\partial_{y}$ , with respect to the small parameter ε, is the same as the order of $\sqrt{ε} \partial_{z}$ , we collect the leading order terms (of order $ε^{0}$ ) in (2.2), and write, $\begin{matrix} (2.3) & - ε \partial_{z}^{2} θ^{ε} + λ θ^{ε} ≃ 0, in Ω; \end{matrix}$ this is the asymptotic equation for $θ^{ε}$ , which confirms that the size of boundary layers for the problem (1.1) is $\sqrt{ε}$ .

To derive a proper boundary condition for $θ^{ε}$ , we insert (2.1) into the boundary condition (1.17) and write, for $α ⩽ 1 / 2$ and $| γ | < 1$ , $\begin{matrix} (2.4) & \{\begin{matrix} - ε^{α} \partial_{z} (u^{0} + θ^{ε}) + γ (u^{0} + θ^{ε}) ≃ 0, & at z = 0, \\ ε^{α} \partial_{z} (u^{0} + θ^{ε}) + γ (u^{0} + θ^{ε}) ≃ 0, & at z = 1, \end{matrix} \end{matrix}$ which is equivalent to $\begin{matrix} (2.5) & \{\begin{matrix} \underset{: = I}{\underset{︸}{- ε^{α - \frac{1}{2}} \partial_{\overline{z}} θ^{ε}}} + \underset{: = II}{\underset{︸}{γ θ^{ε}}} ≃ ε^{α} \partial_{z} u^{0} - γ u^{0}, & at z = 0, \\ \overset{︷}{ε^{α - \frac{1}{2}} \partial_{\overline{z}} θ^{ε}} + \overset{︷}{γ θ^{ε}} ≃ - ε^{α} \partial_{z} u^{0} - γ u^{0}, & at z = 1, \end{matrix} \end{matrix}$ where $\overline{z} = z / \sqrt{ε}$ is the stretched variable of z so that $\partial_{\overline{z}}$ is of order $ε^{0}$ .

Depending on the value of $α ⩽ 1 / 2$ , we have two different cases:

When $α < 1 / 2$ , the term I is dominant over $II$ in (2.5), and hence we use the Neumann boundary condition, $\begin{matrix} (2.6) & \{\begin{matrix} \partial_{z} θ^{ε} = - \partial_{z} u^{0} + ε^{- α} γ u^{0}, & at z = 0, \\ \partial_{z} θ^{ε} = - \partial_{z} u^{0} - ε^{- α} γ u^{0}, & at z = 1 . \end{matrix} \end{matrix}$

When $α = 1 / 2$ , the asymptotic orders of the terms I and $II$ in (2.5) are balanced and hence we use the full Robin boundary condition as $\begin{matrix} (2.7) & \{\begin{matrix} - \sqrt{ε} \partial_{z} θ^{ε} + γ θ^{ε} = \sqrt{ε} \partial_{z} u^{0} - γ u^{0}, & at z = 0, \\ \sqrt{ε} \partial_{z} θ^{ε} + γ θ^{ε} = - \sqrt{ε} \partial_{z} u^{0} - γ u^{0}, & at z = 1 . \end{matrix} \end{matrix}$

In the following Sections 3 and 4 below, we construct the corrector $θ^{ε}$ for each Case 1 and Case 2 (as a solution of (2.3) supplemented with the boundary condition (2.6) or (2.7)), and perform the corresponding boundary layer analysis. As we will see below, more interesting case is the Case 2 when the terms I and $II$ in (2.5) are balanced with $α = 1 / 2$ .

3. Asymptotic analysis when $α < 1 / 2$

Here we aim to construct the asymptotic expansion (2.1) for Case 1 when $α < 1 / 2$ .

We recall the equation (2.3) with the boundary condition (2.6): $\begin{matrix} (3.1) & \{\begin{matrix} - ε \partial_{z}^{2} θ^{ε} + λ θ^{ε} ≃ 0, in Ω, \\ θ^{ε} is periodic in x and y with period L, \\ \partial_{z} θ^{ε} = - \partial_{z} u^{0} + ε^{- α} γ u^{0}, at z = 0, \\ \partial_{z} θ^{ε} = - \partial_{z} u^{0} - ε^{- α} γ u^{0}, at z = 1 . \end{matrix} \end{matrix}$

We first notice that exponentially decaying functions $θ_{L}^{ε}$ and $θ_{R}^{ε}$ , defined by, $\begin{matrix} (3.2) & θ_{L}^{ε} (x, y, z) = \sqrt{\frac{ε}{λ}} \partial_{z} u^{0} (x, y, 0) e^{- \sqrt{\frac{λ}{ε}} z} - \frac{ε^{\frac{1}{2} - α}}{\sqrt{λ}} γ (x, y) u^{0} (x, y, 0) e^{- \sqrt{\frac{λ}{ε}} z}, \end{matrix}$ and $\begin{matrix} (3.3) & θ_{R}^{ε} (x, y, z) = - \sqrt{\frac{ε}{λ}} \partial_{z} u^{0} (x, y, 1) e^{- \sqrt{\frac{λ}{ε}} (1 - z)} - \frac{ε^{\frac{1}{2} - α}}{\sqrt{λ}} γ (x, y) u^{0} (x, y, 1) e^{- \sqrt{\frac{λ}{ε}} (1 - z)}, \end{matrix}$ satisfy the equation (3.1)_{1, 2} with the zero right-hand side, and each of those $θ_{L}^{ε}$ and $θ_{R}^{ε}$ satisfies the boundary condition (3.1)_{3, 4} at $z = 0$ or $z = 1$ respectively. However, it is not a proper choice to define the corrector as the sum of $θ_{L}^{ε}$ and $θ_{R}^{ε}$ because the non-zero boundary value of $\partial_{z} θ_{L}^{ε}$ at $z = 1$ effects on the boundary value of $\partial_{z} θ_{R}^{ε}$ at $z = 1$ and vice versa, and hence $\partial_{z} (θ_{L}^{ε} + θ_{R}^{ε})$ does not satisfies the desired boundary condition (3.1)_{3, 4}.

To resolve this technical issue, we introduce a smooth cut-off functions $σ_{L}$ and $σ_{R}$ respectively near $z = 0$ and $z = 1$ such that $\begin{array}{l} (3.4) & σ_{L} (z) = \{\begin{matrix} 1, & 0 ⩽ z ⩽ 1 / 4, \\ 0, & 1 / 2 ⩽ z ⩽ 1, \end{matrix} σ_{R} (z) = \{\begin{matrix} 0, & 0 ⩽ z ⩽ 1 / 2, \\ 1, & 3 / 4 ⩽ z ⩽ 1 . \end{matrix} \end{array}$ Then, we define our corrector as $\begin{matrix} (3.5) & θ^{ε} (x, y, z) = σ_{L} (z) θ_{L}^{ε} (x, y, z) + σ_{R} (z) θ_{R}^{ε} (x, y, z), \end{matrix}$ which satisfies $\begin{matrix} (3.6) & \{\begin{matrix} - ε \partial_{z}^{2} θ^{ε} + λ θ^{ε} = R_{I}, in Ω, \\ θ^{ε} is periodic in x and y with period L, \\ \partial_{z} θ^{ε} = - \partial_{z} u^{0} + ε^{- α} γ u^{0}, at z = 0, \\ \partial_{z} θ^{ε} = - \partial_{z} u^{0} - ε^{- α} γ u^{0}, at z = 1 . \end{matrix} \end{matrix}$ Here $R_{I}$ on the right-hand side is given by $\begin{matrix} (3.7) & R_{I} = - ε (σ_{L}^{″} θ_{L} + 2 σ_{L}^{'} \partial_{z} θ_{L} + σ_{R}^{″} θ_{R} + 2 σ_{R}^{'} \partial_{z} θ_{R}) . \end{matrix}$

Recalling the fact that, for any $λ > 0$ , $\begin{matrix} (3.8) & {‖ z^{q} \partial_{z}^{m} e^{- \sqrt{λ} z / \sqrt{ε}} ‖}_{L^{p} (Ω)} ⩽ κ ε^{\frac{1}{2 p} - \frac{m}{2} + \frac{q}{2}}, m, q ⩾ 0, 1 ⩽ p ⩽ \infty, \end{matrix}$ one can easily verify that the corrector satisfies the following estimates: $\begin{matrix} (3.9) & {‖ z^{q} \partial_{x}^{k} \partial_{y}^{ℓ} \partial_{z}^{m} θ^{ε} ‖}_{L^{p} (Ω)} ⩽ κ ε^{\frac{1}{2} - α + \frac{1}{2 p} - \frac{m}{2} + \frac{q}{2}}, k, ℓ, m, q ⩾ 0, 1 ⩽ p ⩽ \infty . \end{matrix}$ Here and throughout this article, κ denotes a generic constant depending on the data, but independent from the parameter ε.

Moreover, since the derivatives of the cut-off functions vanish near $z = 0$ and $z = 1$ , i.e., $σ_{L}^{'} = σ_{L}^{″} = 0$ for $0 ⩽ z ⩽ 1 / 4$ and $σ_{R}^{'} = σ_{R}^{″} = 0$ for $3 / 4 ⩽ z ⩽ 1$ , we notice that $\begin{matrix} (3.10) & {‖ z^{q} \partial_{x}^{k} \partial_{y}^{ℓ} \partial_{z}^{m} R_{I} ‖}_{L^{p} (Ω)} = e . s . t . (exponentially small term), k, ℓ, m, q ⩾ 0, 1 ⩽ p ⩽ \infty . \end{matrix}$

Now, for the boundary layer analysis, we introduce the difference $w^{ε}$ between $u^{ε}$ and its asymptotic expansion $u^{0} + θ^{ε}$ as $\begin{matrix} (3.11) & w^{ε} = u^{ε} - (u^{0} + θ^{ε}) . \end{matrix}$

Using (1.1), (1.5), (1.16), and (3.6), we write the equation for $w^{ε}$ , $\begin{matrix} (3.12) & \{\begin{matrix} - ε Δ w^{ε} + λ w^{ε} = ε Δ u^{0} + ε \partial_{x}^{2} θ^{ε} + ε \partial_{y}^{2} θ^{ε} - R_{I}, in Ω, \\ w^{ε} is periodic in x and y with period L . \end{matrix} \end{matrix}$

Using the boundary conditions (1.17) and (3.6)₃, we write the boundary condition for $w^{ε}$ , $\begin{matrix} (3.13) & \{\begin{matrix} - ε^{α} \partial_{z} w^{ε} + γ w^{ε} = - γ θ^{ε}, & at z = 0, \\ ε^{α} \partial_{z} w^{ε} + γ w^{ε} = - γ θ^{ε}, & at z = 1 . \end{matrix} \end{matrix}$ Here $| γ | ⩽ 1$ on $\partial Ω$ and $α < 1 / 2$ as we are dealing with $Case 1$ .

We multiply (3.12) by $w^{ε}$ , integrate over Ω, and write, $\begin{array}{l} \underset{: = A_{1}}{\underset{︸}{- ε \int_{Ω} Δ w^{ε} w^{ε} d x}} + λ {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} \\ (3.14) & = \underset{: = A_{2}}{\underset{︸}{ε \int_{Ω} Δ u^{0} w^{ε} d x}} + \underset{: = A_{3}}{\underset{︸}{ε \int_{Ω} (\partial_{x}^{2} θ^{ε} + \partial_{y}^{2} θ^{ε}) w^{ε} d x}} - \underset{: = A_{4}}{\underset{︸}{ε \int_{Ω} R_{I} w^{ε} d x}} . \end{array}$

For the term $A 1$ , we integrate it by parts and write, $\begin{matrix} (3.15) & A_{1} = ε {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} - ε \int_{\partial Ω} (\nabla w^{ε} \cdot n) w^{ε} d x d y . \end{matrix}$

Using (3.13) and the Schwarz inequality, we estimate the second term on the right-hand side of (3.15) as $\begin{array}{l} | - ε \int_{\partial Ω} (\nabla w^{ε} \cdot n) w^{ε} d x d y | & ⩽ ε \int_{z = 0, 1} | \partial_{z} w^{ε} | | w^{ε} | d x d y \\ ⩽ ε^{1 - α} \int_{z = 0, 1} | γ | | w^{ε} + θ^{ε} | | w^{ε} | d x d y \\ (3.16) & ⩽ ε^{1 - α} [{‖ w^{ε} ‖}_{L^{2} (\partial Ω)}^{2} + {‖ θ^{ε} ‖}_{L^{2} (\partial Ω)} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)}] . \end{array}$ We notice from (3.2) and (3.3) that $\begin{matrix} (3.17) & {‖ θ^{ε} ‖}_{L^{2} (\partial Ω)} ⩽ κ {‖ θ^{ε} ‖}_{L^{\infty} (\partial Ω)} ⩽ κ ε^{\frac{1}{2} - α}, \end{matrix}$ and hence using Young inequality as well, we further estimate, $\begin{array}{l} R.H.S of (3.16) & ⩽ ε^{1 - α} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)}^{2} + κ ε^{1 - α} ε^{\frac{1}{2} - α} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)} \\ (3.18) & ⩽ ε^{1 - α} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)}^{2} + κ ε^{2 - 3 α}, \end{array}$ for a constant $κ > 0$ , depending on the data, but independent of ε.

Using the trace theorem, we find that $\begin{array}{l} ε^{1 - α} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)}^{2} & ⩽ κ ε^{1 - α} {‖ w^{ε} ‖}_{L^{2} (Ω)} ({‖ w^{ε} ‖}_{L^{2} (Ω)} + {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}) \\ ⩽ κ ε^{1 - α} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} + κ ε^{1 - α} {‖ w^{ε} ‖}_{L^{2} (Ω)} {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)} \\ (3.19) & ⩽ κ (ε^{1 - α} + ε^{1 - 2 α}) {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} + \frac{ε}{2} {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} . \end{array}$ Then we infer from (3.16)–(3.19) that $\begin{matrix} (3.20) & | - ε \int_{\partial Ω} (\nabla w^{ε} \cdot n) w^{ε} d x d y | ⩽ κ (ε^{1 - α} + ε^{1 - 2 α}) {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} + \frac{ε}{2} {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} + κ ε^{2 - 3 α} . \end{matrix}$

Thanks to (3.9) and (3.10), using the Schwarz and Young inequalities, it is easy to estimate the terms $A_{2}$ , $A_{3}$ , and $A_{4}$ in (3.14): $\begin{array}{l} (3.21) & | A_{2} | ⩽ & ε {‖ Δ u^{0} ‖}_{L^{2} (Ω)} {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ κ ε {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ κ ε^{2} + \frac{λ}{6} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2}, \\ | A_{3} | ⩽ & ε ({‖ \partial_{x}^{2} θ^{ε} ‖}_{L^{2} (Ω)} + {‖ \partial_{y}^{2} θ^{ε} ‖}_{L^{2} (Ω)}) {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ κ ε^{\frac{7}{4} - α} {‖ w^{ε} ‖}_{L^{2} (Ω)} \\ (3.22) & ⩽ & κ ε^{\frac{7}{2} - 2 α} + \frac{λ}{6} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2}, \end{array}$ and $\begin{matrix} (3.23) & | A_{4} | ⩽ ε ‖ R_{I} ‖_{L^{2} (Ω)} {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ e . s . t . + \frac{λ}{6} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} . \end{matrix}$

Combining the inequalities (3.14)–(3.23) above, we obtain $\begin{matrix} (3.24) & ε {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} + (\frac{λ}{2} - κ ε^{1 - α} - κ ε^{1 - 2 α}) {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ κ ε^{2 - 3 α} . \end{matrix}$ Because $α < 1 / 2$ , there exists a constant $c_{1} > 0$ (independent of ε) such that $λ / 2 - κ ε^{1 - α} - κ ε^{1 - 2 α} > c_{1}$ for sufficiently small $ε > 0$ , and hence we deduce that $\begin{matrix} (3.25) & ε {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} + c_{1} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ κ ε^{2 - 3 α} . \end{matrix}$

Noticing that the bound on the right-hand side of (3.25) decays to zero faster than $κ ε^{1 / 2}$ as $α < 1 / 2$ , the validity of our asymptotic expansion follows from (3.25). Moreover, using the estimate (3.9) of the corrector and the triangle inequality, we conclude the convergence of $u^{ε}$ to $u^{0}$ in the sense that $\begin{matrix} (3.26) & {‖ u^{ε} - u^{0} ‖}_{L^{2} (Ω)} ⩽ κ min (ε^{1 - \frac{3 α}{2}}, ε^{\frac{3}{4} - α}) . \end{matrix}$ Note that the estimate (3.26) above is valid for any fixed $λ > 0$ , and hence we do not need to impose the technical assumption (1.14) for this current case when $α < 1 / 2$ .

We summarize the convergence results as a theorem below:

Theorem 3.1 (Case 1 when $α < 1 / 2$ ).

Under a sufficient regularity assumption on the data, e.g., $f \in H^{3} (Ω)$ and $γ \in L^{\infty} (\partial Ω)$ , the difference between the perturbed solution $u^{ε}$ and its asymptotic expansion vanishes as the perturbation parameter tends to zero in the sense that $\begin{matrix} (3.27) & {‖ u^{ε} - (u^{0} + θ^{ε}) ‖}_{L^{2} (Ω)} + ε^{\frac{1}{2}} {‖ \nabla u^{ε} - (u^{0} + θ^{ε}) ‖}_{L^{2} (Ω)} ⩽ κ ε^{1 - \frac{3 α}{2}}, \end{matrix}$ for a generic constant $κ = κ (Ω, f, λ) > 0$ depending on the indicated data, but independent of ε. Moreover the perturbed solution $u^{ε}$ converges, as $ε \to 0$ , to the corresponding limit solution $u^{0}$ in the sense that $\begin{matrix} (3.28) & {‖ u^{ε} - u^{0} ‖}_{L^{2} (Ω)} ⩽ κ min (ε^{1 - \frac{3 α}{2}}, ε^{\frac{3}{4} - α}) . \end{matrix}$

Remark 3.2.
From the construction of $θ^{ε}$ in (3.5) and the convergence result (3.27), we observe that, as long as $α < 1 / 2$ , the boundary layer created by the proposed Robin boundary condition can be treated as like the one from the Neumann boundary condition.

4. Asymptotic analysis when $α = 1 / 2$

In this section, concerning the case when $α = 1 / 2$ , we aim to construct the asymptotic expansion of $u^{ε}$ as in (2.1).

We consider the equation (2.3) with the boundary condition (2.7): $\begin{matrix} (4.1) & \{\begin{matrix} - ε \partial_{z}^{2} θ^{ε} + λ θ^{ε} ≃ 0, in Ω, \\ θ^{ε} is periodic in x and y with period L, \\ - \sqrt{ε} \partial_{z} θ^{ε} + γ θ^{ε} = \sqrt{ε} \partial_{z} u^{0} - γ u^{0}, at z = 0, \\ \sqrt{ε} \partial_{z} θ^{ε} + γ θ^{ε} = - \sqrt{ε} \partial_{z} u^{0} - γ u^{0}, at z = 1 . \end{matrix} \end{matrix}$

The main idea of constructing a solution to (4.1) is forming the linear combination of two solutions to the equation (4.1)_{1, 2} that respectively satisfy the boundary conditions, $\begin{matrix} (4.2) & \partial_{z} θ^{ε} = - \partial_{z} u^{0}, at z = 0, 1, \end{matrix}$ and $\begin{matrix} (4.3) & θ^{ε} = - u^{0}, at z = 0, 1 . \end{matrix}$ Then the resulting combination automatically satisfies the original boundary condition given in (4.1)_{3, 4}.

Following the process explained above, and thanks to the technical assumption (1.14) and the fact that $‖ γ ‖_{C^{0} (\overline{\partial Ω})} < 1$ , one can construct the exponentially decaying functions $θ_{L}^{ε}$ and $θ_{R}^{ε}$ , defined by $\begin{matrix} (4.4) & θ_{L}^{ε} (x, y, z) = \frac{\sqrt{ε}}{γ (x, y) + \sqrt{λ}} \partial_{z} u^{0} (x, y, 0) e^{- \sqrt{\frac{λ}{ε}} z} - \frac{γ (x, y)}{γ (x, y) + \sqrt{λ}} u^{0} (x, y, 0) e^{- \sqrt{\frac{λ}{ε}} z}, \end{matrix}$ and $\begin{array}{l} θ_{R}^{ε} (x, y, z) \\ (4.5) & = - \frac{\sqrt{ε}}{γ (x, y) + \sqrt{λ}} \partial_{z} u^{0} (x, y, 1) e^{- \sqrt{\frac{λ}{ε}} (1 - z)} - \frac{γ (x, y)}{γ (x, y) + \sqrt{λ}} u^{0} (x, y, 1) e^{- \sqrt{\frac{λ}{ε}} (1 - z)} . \end{array}$ The $θ_{L}^{ε}$ and $θ_{R}^{ε}$ above satisfy the equation (4.1)_{1, 2} with the zero right-hand side, and they satisfy respectively the boundary conditions (4.1)_{3, 4} at $z = 0$ and $z = 1$ .

Using $θ_{L}^{ε}$ and $θ_{R}^{ε}$ , and the cut-off functions in (3.4), we now define our corrector as $\begin{matrix} (4.6) & θ^{ε} (x, y, z) = σ_{L} (z) θ_{L}^{ε} (x, y, z) + σ_{R} (z) θ_{R}^{ε} (x, y, z), \end{matrix}$ that satisfies $\begin{matrix} (4.7) & \{\begin{matrix} - ε \partial_{z}^{2} θ^{ε} + λ θ^{ε} = R_{II}, in Ω, \\ θ^{ε} is periodic in x and y with period L, \\ - \sqrt{ε} \partial_{z} θ^{ε} + γ θ^{ε} = \sqrt{ε} \partial_{z} u^{0} - γ u^{0}, at z = 0, \\ \sqrt{ε} \partial_{z} θ^{ε} + γ θ^{ε} = - \sqrt{ε} \partial_{z} u^{0} - γ u^{0}, at z = 1 . \end{matrix} \end{matrix}$ Here $R_{II}$ on the right-hand side is given by $\begin{matrix} (4.8) & R_{II} = - ε (σ_{L}^{″} θ_{L} + 2 σ_{L}^{'} \partial_{z} θ_{L} + σ_{R}^{″} θ_{R} + 2 σ_{R}^{'} \partial_{z} θ_{R}) . \end{matrix}$

Thanks to the estimate (3.8), one can easily verify that the corrector satisfies the following estimates: $\begin{matrix} (4.9) & {‖ z^{q} \partial_{x}^{k} \partial_{y}^{ℓ} \partial_{z}^{m} θ^{ε} ‖}_{L^{p} (Ω)} ⩽ κ ε^{\frac{1}{2 p} - \frac{m}{2} + \frac{q}{2}}, k, ℓ, m, q ⩾ 0, 1 ⩽ p ⩽ \infty . \end{matrix}$ Moreover, since the derivatives of the cut-off functions vanish near $z = 0$ and $z = 1$ , i.e., $σ_{L}^{'} = σ_{L}^{″} = 0$ for $0 ⩽ z ⩽ 1 / 4$ and $σ_{R}^{'} = σ_{R}^{″} = 0$ for $3 / 4 ⩽ z ⩽ 1$ , we notice that $\begin{matrix} (4.10) & {‖ z^{q} \partial_{x}^{k} \partial_{y}^{ℓ} \partial_{z}^{m} R_{II} ‖}_{L^{p} (Ω)} = e . s . t . (exponentially small term), k, ℓ, m, q ⩾ 0, 1 ⩽ p ⩽ \infty . \end{matrix}$

For the boundary layer analysis, we introduce the difference $w^{ε}$ between $u^{ε}$ and its asymptotic expansion $u^{0} + θ^{ε}$ as $\begin{matrix} (4.11) & w^{ε} = u^{ε} - (u^{0} + θ^{ε}) . \end{matrix}$ Using (1.1), (1.5), (1.16), and (4.7), we write the equation for $w^{ε}$ , $\begin{matrix} (4.12) & \{\begin{matrix} - ε Δ w^{ε} + λ w^{ε} = ε Δ u^{0} + ε \partial_{x}^{2} θ^{ε} + ε \partial_{y}^{2} θ^{ε} - R_{II}, in Ω, \\ w^{ε} is periodic in x and y with period L, \\ - \sqrt{ε} \partial_{z} w^{ε} + γ w^{ε} = 0, at z = 0, \\ \sqrt{ε} \partial_{z} w^{ε} + γ w^{ε} = 0, at z = 1 . \end{matrix} \end{matrix}$ We multiply (4.12)₁ by $w^{ε}$ , integrate over Ω, and write, $\begin{array}{l} \underset{: = B_{1}}{\underset{︸}{- ε \int_{Ω} Δ w^{ε} w^{ε} d x}} + λ {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} \\ (4.13) & = \underset{: = B_{2}}{\underset{︸}{ε \int_{Ω} Δ u^{0} w^{ε} d x}} + \underset{: = B_{3}}{\underset{︸}{ε \int_{Ω} (\partial_{x}^{2} θ^{ε} + \partial_{y}^{2} θ^{ε}) w^{ε} d x}} - \underset{: = B_{4}}{\underset{︸}{ε \int_{Ω} R_{II} w^{ε} d x}} . \end{array}$

For the term $B 1$ , we integrate it by parts and write, $\begin{matrix} (4.14) & B_{1} = ε {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} - ε \int_{\partial Ω} (\nabla w^{ε} \cdot n) w^{ε} d x d y . \end{matrix}$

Using the boundary condition of $w^{ε}$ in (4.12)_{3, 4} and using the Schwarz inequality, we estimate the second term on the right-hand side of (4.14) as $\begin{array}{l} (4.15) & | - ε \int_{\partial Ω} (\nabla w^{ε} \cdot n) w^{ε} d x d y | ⩽ ε \int_{z = 0, 1} | \partial_{z} w^{ε} | | w^{ε} | d x d y ⩽ ε^{\frac{1}{2}} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)}^{2} . \end{array}$ Using the trace theorem as in (1.10), we repeat exactly the same computations as in (1.11) with $u^{ε}$ and α respectively replaced by $w^{ε}$ and $1 / 2$ . Then we find that $\begin{matrix} (4.16) & ε^{\frac{1}{2}} {‖ w^{ε} ‖}_{L^{2} (\partial Ω)}^{2} ⩽ C_{Ω}^{2} ε^{\frac{1}{2}} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} + \frac{ε}{2} {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} + \frac{C_{Ω}^{4}}{2} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} . \end{matrix}$

Thanks to (4.9) and (4.10), using the Schwarz and Young inequalities, it is easy to estimate the terms $B_{2}$ , $B_{3}$ , and $B_{4}$ in (4.13): $\begin{array}{l} (4.17) & | B_{2} | ⩽ & ε {‖ Δ u^{0} ‖}_{L^{2} (Ω)} {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ κ ε {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ κ ε^{2} + \frac{λ}{6} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2}, \\ | B_{3} | ⩽ & ε ({‖ \partial_{x}^{2} θ^{ε} ‖}_{L^{2} (Ω)} + {‖ \partial_{y}^{2} θ^{ε} ‖}_{L^{2} (Ω)}) {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ κ ε^{\frac{5}{4}} {‖ w^{ε} ‖}_{L^{2} (Ω)} \\ (4.18) & ⩽ & κ ε^{\frac{5}{2}} + \frac{λ}{6} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2}, \end{array}$ and $\begin{matrix} (4.19) & | B_{4} | ⩽ ε ‖ R_{II} ‖_{L^{2} (Ω)} {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ e . s . t . + \frac{λ}{6} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} . \end{matrix}$

Combining the inequalities (4.13)–(4.19) above, we obtain $\begin{matrix} (4.20) & ε {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} + (λ - 2 C_{Ω}^{2} ε^{\frac{1}{2}} - C_{Ω}^{4}) {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ κ ε^{2} . \end{matrix}$ Thanks to the technical assumption (1.14), there exists a constant $c_{1} > 0$ , independent of ε, such that $λ - 2 C_{Ω}^{2} ε^{\frac{1}{2}} - C_{Ω}^{4} > c_{1}$ for sufficiently small $ε > 0$ . Thus we deduce that $\begin{matrix} (4.21) & ε {‖ \nabla w^{ε} ‖}_{L^{2} (Ω)}^{2} + c_{1} {‖ w^{ε} ‖}_{L^{2} (Ω)}^{2} ⩽ κ ε^{2}, \end{matrix}$ which implies the validity of our asymptotic expansion with the optimal convergence rate of ε. Furthermore, using the estimate (4.9) of the corrector, we conclude the convergence of $u^{ε}$ to $u^{0}$ in the sense that $\begin{matrix} (4.22) & {‖ u^{ε} - u^{0} ‖}_{L^{2} (Ω)} ⩽ κ ε^{\frac{1}{4}} . \end{matrix}$

The convergence results (4.21) and (4.22) are summarized below in Theorem 4.1.

Theorem 4.1 (Case 1 when $α = 1 / 2$ ).

Under a sufficient regularity assumption on the data, e.g., $f \in H^{3} (Ω)$ and $γ \in L^{\infty} (\partial Ω)$ , and under the technical assumption ( 1.14 ), the difference between the perturbed solution $u^{ε}$ and its asymptotic expansion vanishes as the perturbation parameter tends to zero in the sense that $\begin{matrix} (4.23) & {‖ u^{ε} - (u^{0} + θ^{ε}) ‖}_{L^{2} (Ω)} + ε^{\frac{1}{2}} {‖ \nabla u^{ε} - \nabla (u^{0} + θ^{ε}) ‖}_{L^{2} (Ω)} ⩽ κ ε^{}, \end{matrix}$ for a generic constant $κ = κ (Ω, f, λ) > 0$ depending on the indicated data, but independent of ε. Moreover the perturbed solution $u^{ε}$ converges, as $ε \to 0$ , to the corresponding limit solution $u^{0}$ in the sense that $\begin{matrix} (4.24) & {‖ u^{ε} - u^{0} ‖}_{L^{2} (Ω)} ⩽ κ ε^{\frac{1}{4}} . \end{matrix}$

Remark 4.2.

In subsequent articles, it is planned to generalize our analysis in this article to more interesting and general cases when the domain is involved in a curved boundary, and/or for time-evolution equations.

References

Cheng and

Temam, Numerical approximation of one-dimensional stationary diffusion equations with boundary layers, Comput. & Fluids 31(4–7) (2002), 453–466, Dedicated to Professor Roger Peyret on the occasion of his 65th birthday (Marseille, 1999). doi:10.1016/S0045-7930(01)00060-3.

G.-M.

Gie,

Hamouda,

C.-Y.

Jung and

Temam, Singular Perturbations and Boundary Layers, Applied Mathematical Sciences., Vol. 200, Springer Nature, Switzerland, 2018. doi:10.1007/978-3-030-00638-9.

G.-M.

Gie and

C.-Y.

Jung, Vorticity layers of the 2D Navier–Stokes equations with a slip type boundary condition, Asymptotic Analysis 84(1) (2013), 17–33. doi:10.3233/ASY-131164.

G.-M.

Gie,

C.-Y.

Jung and

Temam, Recent progresses in boundary layer theory, Discrete and Continuous Dynamical Systems – Series A 36(5) (2016), 2521–2583. doi:10.3934/dcds.2016.36.2521.

G.-M.

Gie,

Kelliher,

Lopes Filho,

Mazzucato and

Nussenzveig Lopes, Vanishing viscosity limit of some symmetric flows, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire (2018). doi:10.1016/j.anihpc.2018.11.006.

G.-M.

Gie and

J.P.

Kelliher, Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions, Journal of Differential Equations 253(6) (2012), 1862–1892. doi:10.1016/j.jde.2012.06.008.

G.-M.

Gie and

J.P.

Whitehead, Boundary layer analysis for Navier–Slip Rayleigh–Bénard convection: The non-existence of an ultimate state, Journal of Mathematical Fluid Mechanics (2019). doi:10.1007/s00021-018-0404-3.

Han and

R.B.

Kellogg, Differentiability properties of solutions of the equation

- ϵ^{2} Δ u + r u = f (x, y)

in a square, SIAM J. Math. Anal. 21(2) (1990), 394–408. doi:10.1137/0521022.

P.W.

Hemker, A Numerical Study of Stiff Two-Point Boundary Problems, Mathematical Centre Tracts, Vol. 80, Mathematisch Centrum, Amsterdam, 1977.

10.

Iftimie and

Sueur, Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal. 199(1) (2011), 145–175. doi:10.1007/s00205-010-0320-z.

11.

C.-Y.

Jung and

Temam, Finite volume approximation of one-dimensional stiff convection-diffusion equations, J. Sci. Comput. 41(3) (2009), 384–410. doi:10.1007/s10915-009-9304-2.

12.

C.-Y.

Jung and

Temam, Finite volume approximation of two-dimensional stiff problems, Int. J. Numer. Anal. Model. 7(3) (2010), 462–476.

13.

Masmoudi and

Rousset, Uniform regularity for the Navier–Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal. 203(2) (2012), 529–575. doi:10.1007/s00205-011-0456-5.

14.

H.-G.

Roos,

Stynes and

Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, Vol. 24, Springer-Verlag, Berlin, 1996.

15.

Stynes, Steady-state convection-diffusion problems, Acta Numer. 14 (2005), 445–508. doi:10.1017/S0962492904000261.

Asymptotic analysis for elliptic equations with Robin boundary condition

Abstract

Keywords

1. Introduction

2. Asymptotic expansion of u ε

3. Asymptotic analysis when α < 1 / 2

Theorem 3.1 (Case 1 when α < 1 / 2 ).

Theorem 4.1 (Case 1 when α = 1 / 2 ).

References

2. Asymptotic expansion of $u^{ε}$

3. Asymptotic analysis when $α < 1 / 2$

Theorem 3.1 (Case 1 when $α < 1 / 2$ ).

Theorem 4.1 (Case 1 when $α = 1 / 2$ ).