Boundary layers generated by singularities in the source function

Abstract

Our aim in this article is to study the Linearized Navier–Stokes (LNS) problem including an interior singularity of the source function. At small viscosity, in addition to the classical boundary layers, interior layers are then developed inside the domain due to the discontinuities appearing in the limit inviscid solution. Using boundary layer functions method, the asymptotic expansion of the viscous solution is constructed and the uniform validity of the approximate solution is then proved.

Keywords

interior layers correctors singular problems discontinuous and non-smooth solution Navier–Stokes equations

1. Introduction

The boundary-value problems for singularly perturbed partial differential equations are often used as mathematical models describing processes in biomechanics and physics (see e.g. [2,3,24] and [15]). In recent years, more attention was paid to the study of singular perturbation problems both in the linear and the nonlinear cases (see e.g. [5,9,13,16,17,21] and [22]). Most of these works are related to the classical boundary layers occurring at the boundary while few of them concern interior layers. In this article, we will discuss the interior layers for the Navier–Stokes problem in a 3D channel, for boundary layers problem in general domains see e.g. [4]. More precisely, we recall here the Navier–Stokes system: $\{\begin{matrix} \frac{\partial u^{ε}}{\partial t} - ε Δ u^{ε} + (u^{ε} \cdot \nabla) u^{ε} + \nabla p^{ε} = f in (0, T) \times Ω_{\infty}, \\ div u^{ε} = 0 in (0, T) \times Ω_{\infty}, \\ u^{ε} = 0 on (0, T) \times Γ_{\infty}, \\ u^{ε} is periodic in the x and y directions with periods L_{1} and L_{2}, \\ u_{|_{t = 0}}^{ε} = u_{0} . \end{matrix}$ (1.1) Here $Ω_{\infty} = R^{2} \times (0, 2 h)$ and $Γ_{\infty} = \partial Ω_{\infty} = R^{2} \times {0, 2 h}$ is its boundary. The function f is given by $f = H_{1} (t; x, y) H_{2} (z),$ (1.2) where $H_{1}$ is supposed to be as regular as necessary and it verifies $\int_{0}^{t} H_{1} (s, x, y) d s ≢ 0 for all (x, y) \in (0, L_{1}) \times (0, L_{2}),$ (H) whereas $H_{2}$ is the Heaviside function $H_{2} (z) = \{\begin{matrix} 1 & if z \in (h, 2 h], \\ 0 & if z \in [0, h) . \end{matrix}$ We will justify later on the utility of the hypothesis (H).

The initial data $u_{0}$ is supposed to be as regular as necessary. Because of the periodicity condition (1.1) $_{4}$ , we will consider in our calculations a portion of the channel $Ω_{\infty}$ that we denote by $Ω = (0, L_{1}) \times (0, L_{2}) \times (0, 2 h)$ , and its boundary $Γ = (0, L_{1}) \times (0, L_{2}) \times {0, 2 h}$ .

The limit system ( $ε = 0$ ) corresponding to (1.1) is the Euler problem. More precisely, the limit solution, denoted here by $u^{0}$ , is the solution of the following system: $\{\begin{matrix} \frac{\partial u^{0}}{\partial t} + (u^{0} \cdot \nabla) u^{0} + \nabla p^{0} = f in (0, T) \times Ω, \\ div u^{0} = 0 in (0, T) \times Ω, \\ u_{3}^{0} = 0 on (0, T) \times Γ, \\ u^{0} is periodic in the x and y directions with periods L_{1} and L_{2}, \\ u_{|_{t = 0}}^{0} = u_{0} . \end{matrix}$ (1.3)

Concerning the study of the classical boundary layer analysis related to the Navier–Stokes equations, we refer the reader to, e.g., [6,7,10,12,20] and [23]. In [11], the authors studied the same problem but with a smooth source function. The usual boundary layers develop at the boundary due to the difference in boundary values for the viscous and inviscid solutions. A natural question consists in studying this problem with singularities in the source function. This can be useful in studying the shear layers. In this article, we aim to study the asymptotic behavior of the Linearized Navier–Stokes (LNS), that is the system (1.1) without the nonlinear term $(u^{ε} \cdot \nabla) u^{ε}$ in the first equation of (1.1). In Section 2, we first treat the asymptotic behavior of the heat equation with a singularity in the source function. This problem will be useful, as we will see, in the study of the (LNS) problem which we consider in Section 3. We start by giving the formal limit solution which presents singularities inside the domain. The passage to the limit when $ε \to 0$ generates, in addition to the usual boundary layers at the boundary, another kind of boundary layers called “interior layers”. We actually construct the corrector which absorbs the singularities of the limit solution; this allows us to obtain an approximation of $u^{ε}$ when $ε \to 0$ . Then, we explicitly give the expression of the corrector and some estimates on its tangential derivatives. Finally, we state our main results and give a theorem for the error estimates.

2. The heat equation

In this section, we start by considering the linear heat problem in a 3D channel with a small parameter ε at the Laplacian term. More precisely, we are interested in the following system with f as in (1.2): $\{\begin{matrix} \frac{\partial u^{ε}}{\partial t} - ε Δ u^{ε} = f in (0, T) \times Ω, \\ u^{ε} = 0 on (0, T) \times \partial Ω, \\ u^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ u_{|_{t = 0}}^{ε} = u_{0} . \end{matrix}$ (2.1) We assume that we have the compatibility conditions by which the initial data $u_{0}$ satisfies the following boundary conditions: $u_{0} (x, y, 0) = u_{0} (x, y, 2 h) = 0,$ (2.2) and $u_{0}$ is periodic in the x and y directions with period $L_{1}$ and $L_{2}$ . For more details related to the compatibility conditions see for e.g., [19] and [8]. We propose to study the asymptotic behavior of $u^{ε}$ , the solution of (2.1), up to the boundary when ε tends to zero. This result will be useful later on in the study of the Navier–Stokes problem. By taking $ε = 0$ in (2.1), we formally obtain the limit problem which reduces to $\{\begin{matrix} \frac{\partial u^{0}}{\partial t} = f in (0, T) \times Ω, \\ u^{0} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ u_{|_{t = 0}}^{0} = u_{0} . \end{matrix}$ (2.3) The solution of (2.3) has the following expression: $u^{0} (t; x, y, z) = u_{0} (x, y, z) + \int_{0}^{t} f (s; x, y, z) d s,$ (2.4) or also, $u^{0} (t; x, y, z) = \{\begin{matrix} u_{0} (x, y, z) & if z \in [0, h), \\ u_{0} (x, y, z) + \int_{0}^{t} H_{1} (s; x, y) d s & if z \in (h, 2 h] . \end{matrix}$ (2.5) It is easy to deduce the regularity properties of $u^{ε}$ and $u^{0}$ ; if $f \in L^{2} (0, T; L^{2} (Ω))$ and $u_{0} \in L^{2} (Ω)$ , then $u^{ε} \in L^{\infty} (0, T; H^{1} (Ω))$ and $u^{0} \in L^{\infty} (0, T; L^{2} (Ω))$ . Concerning the convergence of $u^{ε}$ to $u^{0}$ , as ε tends to zero, we firstly see that $u^{ε}$ and $u^{0}$ do not have the same traces at $z = 2 h$ ; this produces boundary layers that will not be our primary aim here. Secondly, we notice that $u^{0}$ contains a discontinuity inside Ω, at $z = h$ , which is not present in $u^{ε}$ as we will see later on. Our primary goal is to derive corrector functions which will absorb the discrepancies between $u^{ε}$ and $u^{0}$ and the singularity developed by the later one near the interface $z = h$ . We propose the following formal approximation: $u^{ε} \sim u^{0} + θ^{ε} + R^{ε},$ where $θ^{ε}$ and $R^{ε}$ that we define later on, $θ^{ε}$ is the corrector and $R^{ε}$ is the rest which will confirm our choice for the correctors (to be specified later on in the asymptotic analysis).

2.1. Construction of the correctors

We first introduce a preliminary form of the zeroth order corrector $θ^{ε}$ . This corrector solves at the same time the classical and interior boundary layers, respectively, at $z = 0, 2 h$ and $z = h$ . More precisely, $θ^{ε}$ verifies: $\{\begin{matrix} \frac{\partial θ^{ε}}{\partial t} - ε Δ θ^{ε} = 0 in (0, T) \times Ω, \\ θ^{ε} = - u^{0} on (0, T) \times \partial Ω, \\ θ^{ε} = - u^{0} (h^{\pm}) at z = h^{\pm}, \\ θ^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{|_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.6) We set $Ω_{-} : = (0, L_{1}) \times (0, L_{2}) \times (0, h)$ and $Ω_{+} : = (0, L_{1}) \times (0, L_{2}) \times (h, 2 h)$ and then we decompose $θ^{ε}$ as follows: $θ^{ε} = θ_{1, h^{-}}^{ε} χ_{Ω_{-}} + (θ_{1, h^{+}}^{ε} + θ_{2 h}^{ε}) χ_{Ω_{+}},$ (2.7) where $θ_{2 h}^{ε}$ is the corrector solving the boundary layer generated near $z = 2 h$ : $\{\begin{matrix} \frac{\partial θ_{2 h}^{ε}}{\partial t} - ε Δ θ_{2 h}^{ε} = 0 in (0, T) \times Ω_{+}, \\ θ_{2 h}^{ε} = - u^{0} at z = 2 h, \\ θ_{2 h}^{ε} = 0 at z = h^{+}, \\ θ_{2 h}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{2 h |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.8) The boundary layers in the interface $z = h$ are described by the tow following correctors: $\{\begin{matrix} \frac{\partial θ_{1, h^{+}}^{ε}}{\partial t} - ε Δ θ_{1, h^{+}}^{ε} = 0 in (0, T) \times Ω_{+}, \\ θ_{1, h^{+}}^{ε} = - u^{0} (h^{+}) at z = h^{+}, \\ θ_{1, h^{+}}^{ε} = 0 at z = 2 h, \\ θ_{1, h^{+}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{1, h^{+} |_{t = 0}}^{ε} = 0 \end{matrix}$ (2.9) and $\{\begin{matrix} \frac{\partial θ_{1, h^{-}}^{ε}}{\partial t} - ε Δ θ_{1, h^{-}}^{ε} = 0 in (0, T) \times Ω_{-}, \\ θ_{1, h^{-}}^{ε} = - u^{0} (h^{-}) at z = h^{-}, \\ θ_{1, h^{-}}^{ε} = 0 at z = 0, \\ θ_{1, h^{-}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{1, h^{-} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.10) We note here that the corrector $θ^{ε}$ is a discontinuous function at $z = h$ as long as $u^{0}$ is.

2.2. Explicit form of the correctors and tangential derivative estimates

Our next goal is to derive explicit asymptotic formulas for the correctors. Starting with the function $θ_{2 h}^{ε}$ , we simplify the choice of the corrector by dropping the term $ε Δ_{x y} θ_{2 h}^{ε}$ in Eq. (2.8) $_{1}$ . Using the local variable $\tilde{z} = (2 h - z) / \sqrt{ε}$ , we define an approximation ${\tilde{θ}}_{2 h}^{ε}$ of $θ_{2 h}^{ε}$ which satisfies the following problem: $\{\begin{matrix} \frac{\partial {\tilde{θ}}_{2 h}^{ε}}{\partial t} - \frac{\partial^{2} {\tilde{θ}}_{2 h}^{ε}}{\partial {\tilde{z}}^{2}} = 0 in (0, T) \times Ω_{\infty}, \\ {\tilde{θ}}_{2 h}^{ε} = - u^{0} at \tilde{z} = 0, \\ {\tilde{θ}}_{2 h}^{ε} \to 0 as \tilde{z} \to \infty, \\ {\tilde{θ}}_{2 h}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{θ}}_{2 h |_{t = 0}}^{ε} = 0, \end{matrix}$ (2.11) with $Ω_{\infty} = (0, L_{1}) \times (0, L_{2}) \times (0, + \infty)$ .

The explicit expression of (2.11) is given by (see e.g. [1]): $\begin{array}{rclr} {\tilde{θ}}_{2 h}^{ε} (t; x, y, 2 h) & = & - u^{0} (0; x, y, 2 h) (1 - 2 erf (2 h - z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (2 h - z / \sqrt{2 ε (t - s)})] \frac{\partial u^{0}}{\partial s} (s; x, y, 2 h) d s . & (2.12) \end{array}$ To avoid the singularities that may be generated by the classical boundary layer at the boundary and which could appear in the equation satisfied by the global corrector, we introduce an other approximation ${\tilde{\tilde{θ}}}_{2 h}^{ε}$ of ${\tilde{θ}}_{2 h}^{ε}$ such as: ${\tilde{\tilde{θ}}}_{2 h}^{ε} = {\tilde{θ}}_{2 h}^{ε} - α^{ε} (t; x, y),$ where $\begin{array}{rclr} α^{ε} (t; x, y) & = & - u^{0} (0; x, y, 2 h) (1 - 2 erf (h / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (h / \sqrt{2 ε (t - s)})] \frac{\partial u^{0}}{\partial s} (s; x, y, 2 h) d s . & (2.13) \end{array}$ The explicit expression of ${\tilde{\tilde{θ}}}_{2 h}^{ε}$ is then given by: $\begin{array}{rclr} {\tilde{\tilde{θ}}}_{2 h}^{ε} & = & u^{0} (0; x, y, 2 h) (2 erf (h / \sqrt{2 ε t}) - 2 erf (2 h - z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [2 erf (h / \sqrt{2 ε (t - s)}) - 2 erf (2 h - z / \sqrt{2 ε (t - s)})] \frac{\partial u^{0}}{\partial s} (s; x, y, 2 h) d s, & (2.14) \end{array}$ which satisfies: $\{\begin{matrix} \frac{\partial {\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial t} - \frac{\partial^{2} {\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial {\tilde{z}}^{2}} = \frac{\partial α^{ε}}{\partial t} in (0, T) \times Ω_{\infty}, \\ {\tilde{\tilde{θ}}}_{2 h}^{ε} = - u^{0} - α^{ε} at \tilde{z} = 0, \\ {\tilde{\tilde{θ}}}_{2 h}^{ε} \to 0 as \tilde{z} \to \infty, \\ {\tilde{\tilde{θ}}}_{2 h}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{\tilde{θ}}}_{2 h |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.15) The corrector ${\tilde{\tilde{θ}}}_{2 h}^{ε}$ fulfills now the desired boundary condition: ${\tilde{\tilde{θ}}}_{2 h}^{ε} = 0 at z = h .$ (2.16) We notice that the study of the boundary layers generated near $z = 2 h$ and described by the corrector ${\tilde{\tilde{θ}}}_{2 h}^{ε}$ is not our primary aim here. But, in order to prove a global convergence result, it is necessary to treat at once all the boundary layers, both at the boundary and in the interior of the domain. We now study the boundary layer generated near $z = h^{+}$ . For that purpose, we introduce the local variable $\overset{ˇ}{z} = (z - h) / \sqrt{ε}$ . Then, by neglecting the tangential derivatives against the normal ones, we define an approximation ${\tilde{θ}}_{1, h^{+}}^{ε}$ that satisfies: $\{\begin{matrix} \frac{\partial {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial t} - \frac{\partial^{2} {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial {\tilde{z}}^{2}} = 0 in (0, T) \times Ω_{\infty}, \\ {\tilde{θ}}_{1, h^{+}}^{ε} = - u^{0} (h^{+}) at \overset{ˇ}{z} = 0, \\ {\tilde{θ}}_{1, h^{+}}^{ε} \to 0 as \overset{ˇ}{z} \to \infty, \\ {\tilde{θ}}_{1, h^{+}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{θ}}_{1, h^{+} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.17) After reintroducing the variable z, the explicit expression of (2.17) is given by: $\begin{array}{rclr} {\tilde{θ}}_{1, h^{+}}^{ε} (t; x, y, z) & = & - u^{0} (0; x, y, h^{+}) (1 - 2 erf (z - h / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (z - h / \sqrt{2 ε (t - s)})] \frac{\partial u^{0}}{\partial s} (s; x, y, h^{+}) d s . & (2.18) \end{array}$ Similarly, at $z = h^{-}$ , we introduce an approximation ${\bar{θ}}_{1, h^{-}}^{ε}$ of $θ_{1, h^{+}}^{ε}$ , having the same structure as ${\tilde{θ}}_{1, h^{+}}^{ε}$ with the role of $\overset{ˇ}{z}$ and $\bar{z} = (h - z) / \sqrt{ε}$ being exchanged. The corrector ${\bar{θ}}_{1, h^{-}}^{ε}$ verifies: $\{\begin{matrix} \frac{\partial {\bar{θ}}_{1, h^{-}}^{ε}}{\partial t} - \frac{\partial^{2} {\bar{θ}}_{1, h^{-}}^{ε}}{\partial {\bar{z}}^{2}} = 0 in (0, T) \times Ω_{\infty}, \\ {\bar{θ}}_{1, h^{-}}^{ε} = - u^{0} (h^{-}) at \bar{z} = 0, \\ {\bar{θ}}_{1, h^{-}}^{ε} \to 0 as \bar{z} \to \infty, \\ {\bar{θ}}_{1, h^{-}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{θ}}_{1, h^{-} |_{t = 0}}^{ε} = 0 \end{matrix}$ (2.19) and has the following expression: $\begin{array}{rclr} {\bar{θ}}_{1, h^{-}}^{ε} (t; x, y, z) & = & - u^{0} (0; x, y, h^{-}) (1 - 2 erf (h - z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (h - z / \sqrt{2 ε (t - s)})] \frac{\partial u^{0}}{\partial s} (s; x, y, h^{-}) d s . & (2.20) \end{array}$ The role of the correctors ${\bar{θ}}_{1, h^{-}}^{ε}$ and ${\tilde{θ}}_{1, h^{+}}^{ε}$ is to absorb the main part of the singularity generated by the discontinuity of $u^{0}$ at $z = h$ . At this stage, the function $u^{ε}$ is tentatively approximated by $(u^{0} + {\bar{θ}}_{1, h^{-}}^{ε} χ_{Ω_{-}} + {\tilde{θ}}_{1, h^{+}}^{ε} χ_{Ω_{+}})$ which is, in particular, continuous at $z = h$ . It is here convenient to recall that our aim is to construct an approximation of $u^{ε}$ , solution of problem (2.1) which is in $C^{1} (\bar{Ω})$ . Now, we will take care of the partial derivative in the z direction of the function $(u^{0} + {\bar{θ}}_{1, h^{-}}^{ε} χ_{(0, h)} + {\tilde{θ}}_{1, h^{+}}^{ε} χ_{(h, 2 h)})$ which is discontinuous at $z = h$ due to the difference of the values of the correctors at $z = h$ . We will handle these discrepancies by introducing another corrector denoted by $θ_{2, h}^{ε} = θ_{2, h^{-}}^{ε} χ_{(0, h)} + θ_{2, h^{+}}^{ε} χ_{(h, 2 h)}$ . We propose that $θ_{2, h^{+}}^{ε}$ verifies: $\{\begin{matrix} \frac{\partial θ_{2, h^{+}}^{ε}}{\partial t} - ε Δ θ_{2, h^{+}}^{ε} = 0 in (0, T) \times Ω_{+}, \\ θ_{2, h^{+}}^{ε} = 0 at z = 2 h, \\ \frac{\partial θ_{2, h^{+}}^{ε}}{\partial z} = - (\frac{\partial u^{0}}{\partial z} + \frac{\partial {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial z}) (t; x, y, h^{+}) at z = h^{+}, \\ θ_{2, h^{+}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{2, h^{+} |_{t = 0}}^{ε} = 0 \end{matrix}$ (2.21) and $θ_{2, h^{-}}^{ε}$ satisfies the following system: $\{\begin{matrix} \frac{\partial θ_{2, h^{-}}^{ε}}{\partial t} - ε Δ θ_{2, h^{-}}^{ε} = 0 in (0, T) \times Ω_{-}, \\ θ_{2, h^{-}}^{ε} = 0 at z = 0, \\ \frac{\partial θ_{2, h^{-}}^{ε}}{\partial z} = - (\frac{\partial u^{0}}{\partial z} + \frac{\partial θ_{1, h^{-}}^{ε}}{\partial z}) (t; x, y, h^{-}) at z = h^{-}, \\ θ_{2, h^{-}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{2, h^{-} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.22) We now give the explicit expression of the correctors $θ_{2, h^{+}}^{ε}$ and $θ_{2, h^{-}}^{ε}$ of which we respectively give an approximation. We start by the corrector $θ_{2, h^{+}}^{ε}$ . We use again the local variable $\overset{ˇ}{z} = (z - h) / \sqrt{ε}$ , and by neglecting the tangential derivatives against the normal ones, we define an approximation ${\tilde{θ}}_{2, h^{+}}^{ε}$ that verifies: $\{\begin{matrix} \frac{\partial {\tilde{θ}}_{2, h^{+}}^{ε}}{\partial t} - \frac{\partial^{2} {\tilde{θ}}_{2, h^{+}}^{ε}}{\partial {\overset{ˇ}{z}}^{2}} = 0 in (0, T) \times Ω_{\infty}, \\ \frac{\partial {\tilde{θ}}_{2, h^{+}}^{ε}}{\partial \overset{ˇ}{z}} = - \sqrt{ε} (\frac{\partial u^{0}}{\partial z} + \frac{\partial {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial z}) (t; x, y, h^{+}) at \overset{ˇ}{z} = 0, \\ {\tilde{θ}}_{h^{+}}^{ε} \to 0 as \overset{ˇ}{z} \to \infty, \\ {\tilde{θ}}_{2, h^{+}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{θ}}_{2, h^{+} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.23) The solution of (2.23) is given in [1] by: ${\tilde{θ}}_{2, h^{+}}^{ε} (t; x, y, z) = \sqrt{\frac{ε}{π}} \int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} e^{- {(z - h)}^{2} / 4 ε (t - τ)} [\frac{\partial u^{0}}{\partial z} + \frac{\partial {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial z}] (τ, x, y, h^{+}) d τ .$ (2.24) Hence, we define ${\tilde{θ}}_{h^{+}}^{ε}$ as being an approximation, in a sense to be specified later, of the corrector $θ_{h^{+}}^{ε}$ at $z = h^{+}$ : ${\tilde{θ}}_{h^{+}}^{ε} : = {\tilde{θ}}_{1, h^{+}}^{ε} + {\tilde{θ}}_{2, h^{+}}^{ε} .$ Similarly, for the corrector $θ_{2, h^{-}}^{ε}$ , using the local variable $\bar{z} = (h - z) / \sqrt{ε}$ , we define an approximation ${\bar{θ}}_{2, h^{-}}^{ε}$ of $θ_{2, h^{-}}^{ε}$ which is solution of: $\{\begin{matrix} \frac{\partial {\bar{θ}}_{2, h^{-}}^{ε}}{\partial t} - \frac{\partial^{2} {\bar{θ}}_{2, h^{-}}^{ε}}{\partial {\bar{z}}^{2}} = 0 in (0, T) \times Ω_{\infty}, \\ \frac{\partial {\bar{θ}}_{2, h^{-}}^{ε}}{\partial \bar{z}} = \sqrt{ε} (\frac{\partial u^{0}}{\partial z} + \frac{\partial {\bar{θ}}_{1, h^{-}}^{ε}}{\partial z}) (t; x, y, h^{-}) at \bar{z} = 0, \\ {\tilde{θ}}_{2, h^{-}}^{ε} \to 0 as \bar{\bar{z}} \to \infty, \\ {\bar{θ}}_{2, h^{-}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{θ}}_{2, h^{-} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.25) The explicit expression of the solution of system (2.25) is given by: ${\bar{θ}}_{2, h^{-}}^{ε} (t; x, y, z) = - \sqrt{\frac{ε}{π}} \int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} e^{- {(h - z)}^{2} / 4 ε (t - τ)} [\frac{\partial {\bar{θ}}_{1, h^{-}}^{ε}}{\partial z} + \frac{\partial u^{0}}{\partial z}] (τ, x, y, h^{-}) d τ .$ (2.26) Then, ${\bar{θ}}_{h^{-}}^{ε}$ is the corrector that we propose to solve the boundary layer problem at $z = h^{-}$ , and we decompose it as follows: ${\bar{θ}}_{h^{-}}^{ε} = {\bar{θ}}_{1, h^{-}}^{ε} + {\bar{θ}}_{2, h^{-}}^{ε} .$

Finally, we gather all the previously introduced correctors and we denote by ${\tilde{θ}}^{ε}$ the global corrector: ${\tilde{θ}}^{ε} = {\bar{θ}}_{h^{-}}^{ε} χ_{Ω_{-}} + ({\tilde{θ}}_{h^{+}}^{ε} + {\tilde{\tilde{θ}}}_{2 h}^{ε}) χ_{Ω_{+}} .$ (2.27) We have the following lemma.

Lemma 2.1.

There exists a constant κ independent of ε such that, for all $i, j \in N$ , we have: $\begin{array}{rclr} {∥ \frac{\partial^{i + j} {\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial x^{i} \partial y^{j}} ∥}_{L^{\infty} ((0, T), L^{2} (Ω))} ⩽ κ ε^{1 / 4}, & (2.28) \\ {∥ \frac{\partial^{i + j} {\bar{θ}}_{1, h^{-}}^{ε}}{\partial x^{i} \partial y^{j}} ∥}_{L^{\infty} ((0, T), L^{2} (Ω))} ⩽ κ ε^{1 / 4}, & (2.29) \\ {∥ \frac{\partial^{i + j} {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial x^{i} \partial y^{j}} ∥}_{L^{\infty} ((0, T), L^{2} (Ω))} ⩽ κ ε^{1 / 4}, & (2.30) \\ {∥ \frac{\partial^{i + j} {\bar{θ}}_{2, h^{-}}^{ε}}{\partial x^{i} \partial y^{j}} ∥}_{L^{\infty} ((0, T), L^{2} (Ω))} ⩽ κ ε^{1 / 4}, & (2.31) \\ {∥ \frac{\partial^{i + j} {\tilde{θ}}_{2, h^{+}}^{ε}}{\partial x^{i} \partial y^{j}} ∥}_{L^{\infty} ((0, T), L^{2} (Ω))} ⩽ κ ε^{1 / 4} . & (2.32) \end{array}$

Proof.

To prove (2.28), (2.29) and (2.30), it is sufficient to estimate $v (t, z) = g (0, 0) (1 - 2 erf (z / \sqrt{2 t})) + \int_{0}^{t} [1 - 2 erf (z / \sqrt{2 (t - s)})] \frac{\partial g}{\partial s} (s, 0) d s,$ where g is a sufficiently smooth function. We first observe that, since $erf (\infty) = 1 / 2$ , we have $| I | = | 1 - 2 erf (z) | = \sqrt{\frac{2}{π}} \int_{z}^{\infty} e^{- s^{2} / 2} d x,$ and since $e^{- s^{2} / 2} ⩽ e^{- c^{2} / 2 - c s} for any c > 0,$ we can write $e^{- s^{2} / 2} ⩽ κ (c) e^{- c s} .$ Hence, we deduce that: $| g (0, 0) (1 - 2 erf (z / \sqrt{2 t})) | ⩽ κ ∥ g ∥_{L^{\infty} ((0, T) \times Ω)} e^{- z / \sqrt{2 T}}$ and $\begin{array}{rcl} | \int_{0}^{t} [1 - 2 erf (z / \sqrt{2 (t - s)})] \frac{\partial g}{\partial s} (s, 0) d s | & ⩽ & κ {∥ \frac{\partial g}{\partial s} ∥}_{L^{\infty} ((0, T) \times Ω)} \int_{0}^{t} e^{- z / \sqrt{2 (t - s)}} d s \\ ⩽ & κ T {∥ \frac{\partial g}{\partial s} ∥}_{L^{\infty} ((0, T) \times Ω)} e^{- z / \sqrt{2 T}} . \end{array}$ Since $\frac{\partial^{i + j} u^{0}}{\partial x^{i} \partial y^{j}} χ_{Ω_{-}}$ and $\frac{\partial^{i + j} u^{0}}{\partial x^{i} \partial y^{j}} χ_{Ω_{+}}$ are smooth, we have $\begin{array}{rcl} | \frac{\partial^{i + j} {\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial x^{i} \partial y^{j}} | & ⩽ & | \frac{\partial^{i + j} {\tilde{θ}}_{2 h}^{ε}}{\partial x^{i} \partial y^{j}} | + | \frac{\partial^{i + j} α^{ε}}{\partial x^{i} \partial y^{j}} | \\ ⩽ & κ [e^{- (2 h - z) / \sqrt{T ε}} + e^{- h / \sqrt{T ε}}] \forall (t; x, y, z) \in (0, T) \times (Ω_{-} \cup Ω_{+}) . \end{array}$ Then, we conclude that: ${∥ \frac{\partial^{i + j} {\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial x^{i} \partial y^{j}} ∥}_{L^{\infty} ((0, T), L^{2} (Ω))}^{2} ⩽ κ \int_{Ω} e^{- 2 (2 h - z) / \sqrt{T ε}} d Ω ⩽ κ ε^{1 / 2} .$ The estimates (2.29) and (2.30) can be similarly handled.

To prove (2.31) and (2.32), we first observe that: $\frac{\partial^{i + j}}{\partial x^{i} \partial y^{j}} {\bar{θ}}_{2, h^{-}}^{ε} (t; x, y, z) = \sqrt{ε} \int_{0}^{t} \frac{1}{\sqrt{π (t - τ)}} e^{- {(h - z)}^{2} / 4 ε (t - τ)} \frac{\partial^{1 + i + j}}{\partial z \partial x^{i} \partial y^{j}} {\bar{θ}}_{1, h^{-}}^{ε} (τ; x, y, h^{-}) d τ,$ with $\frac{\partial {\bar{θ}}_{1, h^{-}}^{ε}}{\partial z} (τ; x, y, h^{-}) = \frac{1}{\sqrt{π ε τ}} u^{0} (0, x, y, h^{-}) + \int_{0}^{τ} \frac{1}{\sqrt{π ε (τ - s)}} \frac{\partial u^{0}}{\partial s} (s, x, y, h^{-}) d s .$ (2.33) Hence, we have $\begin{array}{rclr} \frac{\partial^{i + j} {\bar{θ}}_{2, h^{-}}^{ε}}{\partial x^{i} \partial y^{j}} & = & \sqrt{ε} \int_{0}^{t} \frac{1}{\sqrt{π (t - τ)}} e^{- {(h - z)}^{2} / 4 ε (t - τ)} [\frac{1}{\sqrt{π ε τ}} \frac{\partial^{i + j} u^{0}}{\partial x^{i} \partial y^{j}} (0; x, y, h^{-})] d τ \\ + \sqrt{ε} \int_{0}^{t} \frac{1}{\sqrt{π (t - τ)}} e^{- {(h - z)}^{2} / 4 ε (t - τ)} \\ \times [\int_{0}^{τ} \frac{1}{\sqrt{π ε (τ - s)}} \frac{\partial^{1 + i + j} u^{0}}{\partial s \partial x^{i} \partial y^{j}} (s, x, y, h^{-}) d s] d τ . & (2.34) \end{array}$ Since for all $(t, x, y) \in (0, T) \times (0, L_{1}) \times (0, L_{2})$ , the function $\frac{\partial^{i + j} u^{0}}{\partial x^{i} \partial y^{j}} (t; x, y, h^{-})$ is smooth as long as $H_{1}$ is, and using the fact that: $\int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} \frac{1}{\sqrt{τ}} d τ = π,$ we obtain $\begin{array}{rcl} {∥ \frac{\partial^{i + j}}{\partial x^{i} \partial y^{j}} {\bar{θ}}_{2, h^{-}}^{ε} ∥}_{L^{\infty} (0, T), L^{2} (Ω)} & ⩽ & κ e^{- {(h - z)}^{2} / 4 ε T} [\int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} \frac{1}{\sqrt{τ}} d τ + \int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} \sqrt{τ} d τ] \\ ⩽ & κ e^{- (h - z) / \sqrt{ε T}} . \end{array}$ Similarly, we prove that: $| \frac{\partial^{i + j}}{\partial x^{i} \partial y^{j}} {\tilde{θ}}_{2, h^{+}}^{ε} | ⩽ κ e^{- (z - h) / \sqrt{ε T}} .$ (2.35) Finally, the estimates (2.31) and (2.32) are immediate consequences of the above estimates. □

Remark 1.

In this article, the positive real number κ is a generic constant depending on the data but not on ε; its values may change from one place to another, but we keep the same notation for it.

In what follows, we aim to derive the boundary layer profile and the equations satisfied by the corrector function ${\tilde{θ}}^{ε}$ . We first observe that ${\tilde{θ}}^{ε}$ is not continuous at $z = h$ , and then, it verifies a singular equation described as below: $\frac{\partial {\tilde{θ}}^{ε}}{\partial t} - ε \frac{\partial^{2} {\tilde{θ}}^{ε}}{\partial z^{2}} = - \frac{\partial α^{ε}}{\partial t} - ε λ^{ε} (t; x, y) δ^{'} - ε μ^{ε} (t; x, y) δ,$ (2.36) where $\begin{array}{rclr} λ^{ε} (t; x, y) & = & {\tilde{θ}}_{h^{+}}^{ε} (t; x, y, h^{+}) + {\tilde{\tilde{θ}}}_{2 h}^{ε} (t; x, y, h^{+}) - {\bar{θ}}_{h^{-}}^{ε} (t; x, y, h^{-}) \\ = & (Thanks to (2.18), (2.24), (2.20), (2.26) and (2.14)) \\ = & - \int_{0}^{t} H_{1} (s; x, y) d s & (2.37) \end{array}$ and $\begin{array}{rclr} μ^{ε} (t; x, y) & = & \frac{\partial}{\partial z} [{\tilde{θ}}_{h^{+}}^{ε} + {\tilde{\tilde{θ}}}_{2 h}^{ε}] (t; x, y, h^{+}) - \frac{\partial {\bar{θ}}_{h^{-}}^{ε}}{\partial z} (t; x, y, h^{-}) \\ = & (Thanks to (2.18), (2.24), (2.20), (2.26) and (2.16)) \\ = & \frac{{\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial z} (t; x, y, h^{+}) . & (2.38) \end{array}$ The boundary conditions for ${\tilde{θ}}^{ε}$ are given by: ${\tilde{θ}}^{ε} (z = 0) = {\bar{θ}}_{h^{-}}^{ε} (z = 0) and {\tilde{θ}}^{ε} (z = 2 h) = {\tilde{θ}}_{h^{+}}^{ε} (z = 2 h) - α^{ε} (t; x, y) .$ It is easy to see that: ${∥ γ_{i} {\tilde{θ}}^{ε} ∥}_{L^{\infty} (L_{x, y}^{2})} ⩽ κ e^{- h / \sqrt{2 ε T}} \forall i \in {0, 2 h},$ (2.39) where $γ_{i}$ denotes for the trace at $z = i$ . Remark 2.

The discontinuity of ${\tilde{θ}}^{ε}$ at $z = h$ appears in particular in the derivatives with respect to the normal z-direction and it is characterized by the presence of the Dirac delta function measure δ (in the z variable). Such singularities become intensified with higher order derivatives of $δ_{z = h}$ with respect to z.

2.3. Convergence result

We conclude this section by stating and proving the following theorem which provides the asymptotic approximation of the viscous solution $u^{ε}$ confirming thus the formal expansion introduced before.

Theorem 2.1.
For all $T > 0$ and under the hypothesis ( H ), the solution $u^{ε}$ of the problem ( 2.1 ), with $u_{0}$ supposed to be sufficiently smooth and verifying ( 2.2 ), satisfies the following estimates: $\begin{array}{rclr} {∥ u^{ε} - u^{0} - {\tilde{θ}}^{ε} ∥}_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ κ ε, & (2.40) \\ {∥ u^{ε} - u^{0} - {\tilde{θ}}^{ε} ∥}_{L^{2} (0, T; H^{1} (Ω))} ⩽ κ ε^{1 / 2}, & (2.41) \end{array}$ where κ is a positive constant independent of ε, and $u^{0}$ and ${\tilde{θ}}^{ε}$ are defined respectively by ( 2.3 ) and ( 2.27 ) .
Proof.
We set $R^{ε} = u^{ε} - u^{0} - {\tilde{θ}}^{ε}$ , i.e., $R^{ε} = \{\begin{matrix} u^{ε} - u_{0} - {\bar{θ}}_{h^{-}}^{ε} & if z \in [0, h), \\ u^{ε} - u_{0} - \int_{0}^{t} H_{1} (s; x, y) d s - {\tilde{\tilde{θ}}}_{2 h}^{ε} - {\tilde{θ}}_{h^{+}}^{ε} & if z \in (h, 2 h] . \end{matrix}$ First, since $u^{0}$ is not continuous at $z = h$ , we remark that: $Δ u^{0} = Δ_{x y} u_{0} + \frac{\partial^{2} u_{0}}{\partial z^{2}} + δ^{'} \int_{0}^{t} H_{1} (s; x, y) d s .$ (2.42) Thanks to our choice of the corrector, the singularity produced in the right-hand side of (2.42) is absorbed, in the equation of the rest $R^{ε}$ which will be stated later on, by ${\tilde{θ}}^{ε}$ (see (2.36) and (2.37)). The boundary conditions satisfied by $R^{ε}$ are: $\begin{array}{rcl} R^{ε} (z = 0) = - {\bar{θ}}_{h^{-}}^{ε} (t; x, y, 0), \\ R^{ε} (z = 2 h) = - {\tilde{θ}}_{h^{+}}^{ε} (t; x, y, 2 h) + α^{ε} (t; x, y) . \end{array}$ Since $R^{ε}$ has nonzero boundary values on Γ, we define ${\tilde{R}}^{ε} = R^{ε} - ψ^{ε},$ where $\begin{array}{rclr} ψ^{ε} (t; x, y, z) & = & - [\frac{{\tilde{θ}}_{h^{+}}^{ε} (t; x, y, 2 h) - α^{ε} (t; x, y) - {\bar{θ}}_{h^{-}}^{ε} (t; x, y, 0)}{2 h} (z - h) \\ + \frac{{\tilde{θ}}_{h^{+}}^{ε} (t; x, y, 2 h) - α^{ε} (t; x, y) + {\bar{θ}}_{h^{-}}^{ε} (t; x, y, 0)}{2}], & (2.43) \end{array}$ and now ${\tilde{R}}^{ε}$ vanishes on Γ (i.e., $z = 0, 2 h$ ).

The function $ψ^{ε}$ is small in all the spaces $H^{m} ((0, T) \times Ω)$ for all $m \in N$ . More precisely, $ψ^{ε}$ verifies the following estimate: ${∥ ψ^{ε} ∥}_{H^{m} ((0, T) \times Ω)} ⩽ κ e^{- m / 2} e^{- h / \sqrt{ε T}} .$ (2.44) We also have: $| \frac{\partial ψ^{ε}}{\partial t} (t; x, y, z) | ⩽ κ (| \frac{\partial {\bar{θ}}_{h^{-}}^{ε}}{\partial t} (t; x, y, 0) | + | \frac{\partial α^{ε}}{\partial t} (t; x, y) | + | \frac{\partial {\tilde{θ}}_{h^{+}}^{ε}}{\partial t} (t; x, y, 2 h) |),$ which yields $| \frac{\partial ψ}{\partial t} | ⩽ κ e^{- h / \sqrt{ε T}} \forall (t; x, y, z) \in (0, T) \times Ω .$ (2.45) Thanks to (2.1), (2.3), (2.27) and (2.43), ${\tilde{R}}^{ε}$ satisfies the following system: $\{\begin{matrix} \frac{\partial {\tilde{R}}^{ε}}{\partial t} - ε Δ {\tilde{R}}^{ε} = ε Δ u_{0} + ε Δ_{x y} {\tilde{θ}}^{ε} + ε Δ_{x y} ψ^{ε} - \frac{\partial ψ^{ε}}{\partial t} + \frac{\partial α^{ε}}{\partial t} + ε μ^{ε} δ in (0, T) \times Ω, \\ {\tilde{R}}^{ε} = 0 on (0, T) \times \partial Ω, \\ {\tilde{R}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{R}}_{|_{t = 0}}^{ε} = 0 . \end{matrix}$ (2.46)

We now multiply (2.46) by ${\tilde{R}}^{ε}$ , integrate over Ω and apply the Cauchy–Schwarz inequality in the right-hand side of the resulting energy equality. We conclude that $\begin{array}{rclr} \frac{1}{2} \frac{d}{d t} {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} + ε {∥ \nabla {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} & ⩽ & ε (∥ Δ u_{0} ∥_{L^{2} (Ω)} + {∥ Δ_{x y} {\tilde{θ}}^{ε} ∥}_{L^{2} (Ω)} + {∥ Δ_{x y} ψ^{ε} ∥}_{L^{2} (Ω)}) {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)} \\ + ({∥ \frac{\partial α^{ε}}{\partial t} ∥}_{L^{2} (Ω)} + {∥ \frac{\partial ψ}{\partial t} ∥}_{L^{2} (Ω)}) {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)} \\ + ε | ⟨ μ^{ε} (t; x, y) δ, {\tilde{R}}^{ε} ⟩ | . & (2.47) \end{array}$ From Lemma 2.1, we can easily bound $∥ Δ_{x y} {\tilde{θ}}^{ε} ∥_{L^{2} (Ω)}$ by a constant. For $ψ^{ε}$ , we use (2.44), which remains true for $ψ^{ε}$ replaced by any tangential or time derivative at any order, and we conclude that we can bound $∥ Δ_{x y} ψ^{ε} ∥_{L^{2} (Ω)}$ by a constant. We can also bound $∥ \partial ψ / \partial t ∥_{L^{2} (Ω)}$ and $∥ \partial α^{ε} / \partial t ∥_{L^{2} (Ω)}$ by any power of ε since $ψ^{ε}$ and $α^{ε}$ are exponentially small terms. Finally, we have $\begin{array}{rclr} | μ^{ε} (t, x, y) | & ⩽ & | \frac{{\tilde{\tilde{θ}}}_{2 h}^{ε}}{\partial z} (t; x, y, h) | \\ ⩽ & κ ε^{- 1 / 2} e^{- h / \sqrt{2 ε t}} \\ ⩽ & κ ε \forall (t, x, y) \in (0, T) \times (0, L_{1}) \times (0, L_{2}) . & (2.48) \end{array}$ Since, on the one hand, $μ^{ε}$ is an exponentially small term, and on the other hand ${\tilde{R}}^{ε} \in H_{0}^{1} (Ω)$ , using the Poincare inequality, we deduce that $\begin{array}{rclr} | ⟨ ε μ^{ε} (t, x, y) δ, {\tilde{R}}^{ε} ⟩ | & = & ε | μ^{ε} (t, x, y) {\tilde{R}}^{ε} (z = h) | \\ ⩽ & κ ε^{2} {∥ {\tilde{R}}^{ε} ∥}_{H_{0}^{1} (Ω)} \\ ⩽ & κ ε^{3} + \frac{ε}{4 c} {∥ {\tilde{R}}^{ε} ∥}_{H_{0}^{1} (Ω)}^{2} \\ ⩽ & κ ε^{3} + \frac{ε}{4} {∥ \nabla {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2}, & (2.49) \end{array}$ where the bracket $⟨ \cdot, \cdot ⟩$ denotes the duality product between the Sobolev spaces $H_{0}^{m}$ and $H^{- m}$ ( $m = 1$ here).

We then infer from (2.47) that $\frac{1}{2} \frac{d}{d t} {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} + \frac{3 ε}{4} {∥ \nabla {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} ⩽ {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} + κ ε^{2} .$ (2.50) We now apply the Gronwall inequality to (2.50) and obtain ${∥ {\tilde{R}}^{ε} ∥}_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ κ ε$ (2.51) and ${∥ {\tilde{R}}^{ε} ∥}_{L^{2} (0, T; H^{1} (Ω))} ⩽ κ ε^{1 / 2} .$ (2.52) This ends the proof of Theorem 2.1. □

3. The linearized Navier–Stokes problem

Our main purpose in this section is to study the asymptotic behavior of the following system $\{\begin{matrix} \frac{\partial u^{ε}}{\partial t} - ε Δ u^{ε} + \nabla p^{ε} = f in (0, T) \times Ω, \\ div u^{ε} = 0 in (0, T) \times Ω, \\ u^{ε} = 0 on (0, T) \times \partial Ω, \\ u^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ u_{|_{t = 0}}^{ε} = u_{0}, \end{matrix}$ (3.1) as ε approaches zero. The initial data $u_{0}$ is supposed to be as regular as necessary. The function f is given such that $f = H_{1} (t; x, y) H_{2} (z)$ where $H_{1}$ is supposed to be also a sufficiently regular vector field in $R^{3}$ that verifies $\int_{0}^{t} H_{1} (s, x, y) d s ≢ 0_{R^{3}} for all (x, y) \in (0, L_{1}) \times (0, L_{2})$ and $H_{2}$ is the Heaviside function $H_{2} (z) = \{\begin{matrix} 1 & if z \in (h, 2 h], \\ 0 & if z \in [0, h) . \end{matrix}$ We suppose that $f = (f_{1}, f_{2}, 0)$ , i.e. $H_{1, 3} \equiv 0$ , where $H_{1, 3}$ is the normal z-component of $H_{1}$ . The case when $f_{3} ≢ 0$ is more complicate and will be studied in a subsequent work. The corresponding inviscid limit problem is the following linearized Euler system: $\{\begin{matrix} \frac{\partial u^{0}}{\partial t} + \nabla p^{0} = f in (0, T) \times Ω, \\ div u^{0} = 0 in (0, T) \times Ω, \\ u_{3}^{0} = 0 on (0, T) \times \partial Ω, \\ u^{0} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ u_{|_{t = 0}}^{0} = u_{0} . \end{matrix}$ (3.2) An explicit expression of (3.2) can be written as $u^{0} (t; x, y, z) = u_{0} (x, y, z) + \int_{0}^{t} [f (s; x, y, z) - \nabla p^{0}] d s,$ (3.3) or also, $u^{0} (t; x, y, z) = \{\begin{matrix} u_{0} (x, y, z) + \int_{0}^{t} (- \nabla p^{0}) d s & if z \in [0, h), \\ u_{0} (x, y, z) + \int_{0}^{t} (H_{1} (s; x, y) - \nabla p^{0}) d s & if z \in (h, 2 h] . \end{matrix}$ (3.4) The existence and uniqueness of a regular solution for (3.1) and (3.2) and the weak convergence of $u^{ε}$ to $u^{0}$ in $L^{2} (Ω) = {(L^{2} (Ω))}^{3}$ are easy to derive. More precisely, using the standard regularity results for the Stokes problem and since $f \in L^{2} (0, T; L^{2} (Ω))$ , we have $u^{0} \in L^{2} (0, T; L^{2} (Ω))$ . However, the strong convergence of $u^{ε}$ to $u^{0}$ and the asymptotic behavior of $u^{ε}$ in space $H^{1}$ are not easy to obtain due, on the one hand, to the divergence free constraint, which couples the velocity and the pressure equations, and on the other hand to the presence of discontinuity in the limit solution $u^{0}$ at the interface $z = h$ . Hence, two types of boundary layers are developed: the classical boundary layers that occur at the boundaries $z = 0, 2 h$ ; these are not our primary aim, and the interior boundary layers generated by the singularity in the source function. We note that under the hypothesis $f_{3} = 0$ in Ω, the normal component $u_{3}^{0}$ of $u^{0}$ is continuous, and only the tangential component of $u^{0}$ contains a discontinuity at $z = h$ . Our objective is to introduce a correcting term to absorb these singularities and then to obtain an approximation of $u^{ε}$ , as $ε \to 0$ , up to the boundary. Therefore, we propose the following expansion: $u^{ε} \sim u^{0} + θ^{ε} + R^{ε},$ where $θ^{ε}$ is the corrector that we will define later and $R^{ε}$ is the rest which we will use to prove the error estimates. In the next section, we first derive an appropriate corrector for the problem, and then determine the explicit expression of this corrector using some results already obtained in Section 2.

3.1. Construction of the corrector and the explicit form

Following the computations carried out in Section 2 and similar arguments inherited from the boundary layer theory, we choose a corrector $θ^{ε}$ which verifies: $\{\begin{matrix} \frac{\partial θ^{ε}}{\partial t} - ε Δ θ^{ε} = 0 in (0, T) \times Ω, \\ div θ^{ε} = 0 in (0, T) \times Ω, \\ θ^{ε} = - u^{0} in (0, T) \times \partial Ω, \\ θ^{ε} = - u^{0} at z = h^{\pm}, \\ θ^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{|_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.5) We decompose $θ^{ε}$ as follows $θ^{ε} = (θ_{0}^{ε} + θ_{h^{-}}^{ε}) χ_{(0, h)} + (θ_{h^{+}}^{ε} + θ_{2 h}^{ε}) χ_{(h, 2 h)} .$ (3.6) In what follows, we will define the correctors introduced in (3.6).

First, the corrector $θ_{2 h}^{ε}$ is aimed to solve the boundary layer generated near $z = 2 h$ . More precisely, $θ_{2 h}^{ε}$ is the solution of the following system: $\{\begin{matrix} \frac{\partial θ_{2 h}^{ε}}{\partial t} - ε Δ θ_{2 h}^{ε} = 0 in (0, T) \times Ω_{+}, \\ div θ_{2 h}^{ε} = 0 in (0, T) \times Ω_{+}, \\ θ_{2 h}^{ε} = - u^{0} at z = 2 h, \\ θ_{2 h}^{ε} = 0 at z = h^{+}, \\ θ_{2 h}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{2 h |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.7) Similarly, $θ_{0}^{ε}$ is the corrector of the boundary layer generated near $z = 0$ and is the solution of $\{\begin{matrix} \frac{\partial θ_{0}^{ε}}{\partial t} - ε Δ θ_{0}^{ε} = 0 in (0, T) \times Ω_{-}, \\ div θ_{0}^{ε} = 0 in (0, T) \times Ω_{-}, \\ θ_{0}^{ε} = - u^{0} at z = 0, \\ θ_{0}^{ε} = 0 at z = h^{-}, \\ θ_{0}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{0 |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.8) Now, for the boundary layer which is generated in the neighborhood of the interface $z = h$ , we propose the tow following correctors: $θ_{h^{-}}^{ε}$ and $θ_{h^{+}}^{ε}$ that we decompose as follows: $θ_{h^{-}}^{ε} = θ_{1, h^{-}}^{ε} + θ_{2, h^{-}}^{ε}, θ_{h^{+}}^{ε} = θ_{1, h^{+}}^{ε} + θ_{2, h^{+}}^{ε},$ (3.9) where $θ_{1, h^{-}}^{ε}$ satisfies: $\{\begin{matrix} \frac{\partial θ_{1, h^{-}}^{ε}}{\partial t} - ε Δ θ_{1, h^{-}}^{ε} = 0 in (0, T) \times Ω_{-}, \\ div θ_{1, h^{-}}^{ε} = 0 in (0, T) \times Ω_{-}, \\ θ_{1, h^{-}}^{ε} = 0 at z = 0, \\ θ_{1, h^{-}}^{ε} = - u^{0} (t; x, y, h^{-}) at z = h^{-}, \\ θ_{1, h^{-}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{1, h^{-} |_{t = 0}}^{ε} = 0, \end{matrix}$ (3.10) and $θ_{2, h^{-}}^{ε}$ is solution of $\{\begin{matrix} \frac{\partial θ_{2, h^{-}}^{ε}}{\partial t} - ε Δ θ_{2, h^{-}}^{ε} = 0 in (0, T) \times Ω_{-}, \\ div θ_{2, h^{-}}^{ε} = 0 in (0, T) \times Ω_{-}, \\ θ_{2, h^{-}}^{ε} = 0 at z = 0, \\ \frac{\partial θ_{2, h^{-}}^{ε}}{\partial z} = - (\frac{\partial u_{τ}^{0}}{\partial z} + \frac{\partial θ_{1, h^{-}}^{ε}}{\partial z}) (t; x, y, h^{-}) at z = h^{-}, \\ θ_{2, h^{-}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{2, h^{-} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.11) In a similar way, we define $θ_{1, h^{+}}^{ε}$ as the solution of the following system: $\{\begin{matrix} \frac{\partial θ_{1, h^{+}}^{ε}}{\partial t} - ε Δ θ_{1, h^{+}}^{ε} = 0 in (0, T) \times Ω_{+}, \\ div θ_{1, h^{+}}^{ε} = 0 in (0, T) \times Ω_{+}, \\ θ_{1, h^{+}}^{ε} = - u^{0} (t; x, y, h^{+}) at z = h^{+}, \\ θ_{1, h^{+}}^{ε} = 0 at z = 2 h, \\ θ_{1, h^{+}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{1, h^{+} |_{t = 0}}^{ε} = 0, \end{matrix}$ (3.12) and then $θ_{2, h^{+}}^{ε}$ as the solution of the above equations: $\{\begin{matrix} \frac{\partial θ_{2, h^{+}}^{ε}}{\partial t} - ε Δ θ_{2, h^{+}}^{ε} = 0 in (0, T) \times Ω_{+}, \\ div θ_{2, h^{+}}^{ε} = 0 in (0, T) \times Ω_{+}, \\ \frac{\partial θ_{2, h^{+}}^{ε}}{\partial z} = - (\frac{\partial u_{τ}^{0}}{\partial z} + \frac{\partial {\tilde{θ}}_{1, h^{+}}^{ε}}{\partial z}) (t; x, y, h^{+}) at z = h^{+}, \\ θ_{2, h^{+}}^{ε} = 0 at z = 2 h, \\ θ_{2, h^{+}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ θ_{2, h^{+} |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.13)

In order to derive the final error analysis, we need further estimates on the spatial derivatives of the correctors. Thanks to our approach which makes the explicit construction possible, we may obtain the explicit expression of these correctors. We will instead use an explicit approximate corrector for $θ_{i}^{ε}$ , $\forall i \in {0, 2 h, h^{+}, h^{-}}$ . First, we decompose such corrector as $θ_{i}^{ε} = θ_{i, τ}^{ε} {\vec{e}}_{τ} + θ_{i, n}^{ε} {\vec{e}}_{n}$ , where ${\vec{e}}_{τ}$ denotes the $(x, y)$ -tangential directions and ${\vec{e}}_{n}$ the z-normal direction. We then derive the expression of the approximate tangential component $θ_{i, τ}^{ε}$ . The normal component $θ_{i, n}^{ε}$ is deduced by simply using the incompressibility condition and the boundary conditions. We start the analysis for $θ_{0}^{ε}$ , we define ${\bar{θ}}_{0}^{ε}$ an approximate corrector of $θ_{0}^{ε}$ , for which the tangential component ${\bar{θ}}_{0, τ}^{ε}$ verifies: $\{\begin{matrix} \frac{\partial {\bar{θ}}_{0, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\bar{θ}}_{0, τ}^{ε}}{\partial z^{2}} = 0 in (0, T) \times Ω_{-}, \\ {\bar{θ}}_{0, τ}^{ε} = - u^{0} at z = 0, \\ {\bar{θ}}_{0, τ}^{ε} = 0 at z = h, \\ {\bar{θ}}_{0, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{θ}}_{τ, 0 |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.14) The explicit expression of ${\bar{θ}}_{0, τ}^{ε}$ is given by: $\begin{array}{rclr} {\bar{θ}}_{0, τ}^{ε} & = & - u_{τ}^{0} (0; x, y, 0) (1 - 2 erf (z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (z / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, 0) d s . & (3.15) \end{array}$ Moreover, we choose ${\bar{θ}}_{0}^{ε}$ which fulfills the incompressibility condition, i.e., $div {\bar{θ}}_{0}^{ε} = 0 in (0, T) \times Ω_{-} .$ (3.16) We then integrate (3.16) in z, use the following boundary condition (inherited from (3.2) $_{3}$ ): ${\bar{θ}}_{0, n}^{ε} = 0 at z = h,$ (3.17) and we find: ${\bar{θ}}_{0, n}^{ε} (t; x, y, z) = - \int_{h}^{z} {div}_{τ} θ_{0, τ}^{ε} (t; x, y, s) d s .$ (3.18) A similar treatment applies at $z = 2 h$ . We deduce the explicit expression of ${\tilde{θ}}_{2 h}^{ε}$ , the corrector at $z = 2 h$ , as: $\begin{array}{rclr} {\tilde{θ}}_{2 h, τ}^{ε} & = & - u_{τ}^{0} (0; x, y, 2 h) (1 - 2 erf (2 h - z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (2 h - z / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, 2 h) d s . & (3.19) \end{array}$ We similarly obtain the normal component of ${\tilde{θ}}_{2 h}^{ε}$ using the incompressibility condition: ${\tilde{θ}}_{2 h, n}^{ε} = - \int_{h}^{z} {div}_{τ} {\tilde{θ}}_{2 h, τ}^{ε} (t; x, y, s) d s .$ (3.20)

With the purpose of absorbing the singularities produced, in the equation satisfied by the global corrector (these singularities are very small in some appropriate Sobolev spaces), by ${\tilde{θ}}_{2 h}^{ε}$ and ${\bar{θ}}_{0}^{ε}$ at $z = h$ , we first define a corrector ${\tilde{\tilde{θ}}}_{2 h}^{ε}$ as follows ${\tilde{\tilde{θ}}}_{2 h}^{ε} = {\tilde{θ}}_{2 h}^{ε} - α^{ε} (t; x, y),$ where $\begin{array}{rcl} α^{ε} (t; x, y) & = & - u_{τ}^{0} (0; x, y, 2 h) (1 - 2 erf (h / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (h / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, 2 h) d s . \end{array}$ It is easy to see that: ${∥ α^{ε} ∥}_{H_{x, y}^{m}} ⩽ κ e^{- h / \sqrt{2 ε T}} \forall (x, y) \in (0, L_{1}) \times (0, L_{2}) .$ (3.21) The estimate (3.21) remains true for all the time and space derivatives of $α^{ε}$ . The corrector ${\tilde{\tilde{θ}}}_{2 h}^{ε}$ can be expressed as: $\begin{array}{rclr} {\tilde{\tilde{θ}}}_{2 h, τ}^{ε} & = & - u_{τ}^{0} (0; x, y, 2 h) (2 erf (h / \sqrt{2 ε t}) - 2 erf (2 h - z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [2 erf (h / \sqrt{2 ε (t - s)}) - 2 erf (2 h - z / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, 2 h) d s & (3.22) \end{array}$ and ${\tilde{\tilde{θ}}}_{2 h, n}^{ε} (t; x, y, z) = - \int_{h}^{z} {div}_{τ} {\tilde{\tilde{θ}}}_{2 h, τ}^{ε} (t; x, y, s) d s .$ (3.23) This ensures that $div {\tilde{\tilde{θ}}}_{2 h}^{ε} = 0 in (0, T) \times Ω_{+} .$ The corrector ${\tilde{\tilde{θ}}}_{2 h, τ}^{ε}$ satisfies the following system: $\{\begin{matrix} \frac{\partial {\tilde{\tilde{θ}}}_{2 h, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\tilde{\tilde{θ}}}_{2 h, τ}^{ε}}{\partial z^{2}} = - \frac{\partial α^{ε}}{\partial t} in (0, T) \times Ω_{+}, \\ {\tilde{\tilde{θ}}}_{2 h, τ}^{ε} = - u_{τ}^{0} - α^{ε} at z = 2 h, \\ {\tilde{\tilde{θ}}}_{2 h, τ}^{ε} = 0 at z = h^{+}, \\ {\tilde{\tilde{θ}}}_{2 h, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{\tilde{θ}}}_{2 h, τ |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.24) In the same way, we define ${\bar{\bar{θ}}}_{0}^{ε}$ as follows ${\bar{\bar{θ}}}_{0}^{ε} = {\bar{θ}}_{0}^{ε} - β^{ε} (t; x, y),$ where $\begin{array}{rcl} β^{ε} (t; x, y) & = & - u_{τ}^{0} (0; x, y, 0) (1 - 2 erf (h / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (h / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, 0) d s . \end{array}$ The explicit expression of ${\bar{\bar{θ}}}_{0}^{ε}$ is given by: $\begin{array}{rclr} {\bar{\bar{θ}}}_{0, τ}^{ε} & = & - u_{τ}^{0} (0; x, y, 0) (2 erf (h / \sqrt{2 ε t}) - 2 erf (z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [2 erf (h / \sqrt{2 ε (t - s)}) - 2 erf (z / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, 0) d s & (3.25) \end{array}$ and ${\bar{\bar{θ}}}_{0, n}^{ε} (t; x, y, z) = - \int_{h}^{z} {div}_{τ} {\bar{\bar{θ}}}_{0, τ}^{ε} (t; x, y, s) d s .$ (3.26) Hence, we have the incompressibility condition for ${\bar{\bar{θ}}}_{0}^{ε}$ : $div {\bar{\bar{θ}}}_{0}^{ε} = 0 in (0, T) \times Ω_{-} .$ Now, the corrector ${\bar{\bar{θ}}}_{0, τ}^{ε}$ satisfies the following system: $\{\begin{matrix} \frac{\partial {\bar{\bar{θ}}}_{0, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\bar{\bar{θ}}}_{0, τ}^{ε}}{\partial z^{2}} = - \frac{\partial β^{ε}}{\partial t} in (0, T) \times Ω_{-}, \\ {\bar{\bar{θ}}}_{0, τ}^{ε} = - u_{τ}^{0} - β^{ε} at z = 0, \\ {\bar{\bar{θ}}}_{0, τ}^{ε} = 0 at z = h^{-}, \\ {\bar{\bar{θ}}}_{0, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{\bar{θ}}}_{0, τ |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.27)

Remark 3.

Note that the choice of correctors satisfying the incompressibility condition avoids us to carry out a heavy treatment of the pressure. This is fundamental in our approach.

We now want to derive the explicit expressions of the correctors corresponding to the interior layers at the interface $z = h$ . We start by the corrector $θ_{1, h^{-}}^{ε}$ for which we propose an approximation ${\bar{θ}}_{1, h^{-}}^{ε}$ such that ${\bar{θ}}_{1, h^{-}, τ}^{ε}$ verifies: $\{\begin{matrix} \frac{\partial {\bar{θ}}_{1, h^{-}, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\bar{θ}}_{1, h^{-}, τ}^{ε}}{\partial z^{2}} = 0 in (0, T) \times Ω_{-}, \\ {\bar{θ}}_{1, h^{-}, τ}^{ε} = 0 at z = 0, \\ {\bar{θ}}_{1, h^{-}, τ}^{ε} = - u_{τ}^{0} (t; x, y, h^{-}) at z = h^{-}, \\ {\bar{θ}}_{1, h^{-}, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{θ}}_{1, h^{-}, τ |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.28) The explicit expression of ${\bar{θ}}_{1, h^{-}, τ}^{ε}$ is given by: $\begin{array}{rclr} {\bar{θ}}_{1, h^{-}, τ}^{ε} (t; x, y, z) & = & - u_{τ}^{0} (0; x, y, h^{-}) (1 - 2 erf (h - z / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (h - z / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, h^{-}) d s . & (3.29) \end{array}$ Using the condition ${\bar{θ}}_{1, h^{-}, n}^{ε} = 0 at z = h^{-},$ (3.30) and the incompressibility condition, the explicit expression of ${\bar{θ}}_{1, h^{-}, n}^{ε}$ is deduced as follows: ${\bar{θ}}_{1, h^{-}, n}^{ε} = - \int_{h}^{z} {div}_{τ} {\bar{θ}}_{1, h^{-}, τ}^{ε} (t; x, y, s) d s .$ (3.31) We also define an approximation ${\bar{θ}}_{2, h^{-}}^{ε}$ of $θ_{2, h^{-}}^{ε}$ such that ${\bar{θ}}_{2, h^{-}, τ}^{ε}$ verifies: $\{\begin{matrix} \frac{\partial {\bar{θ}}_{2, h^{-}, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\bar{θ}}_{2, h^{-}, τ}^{ε}}{\partial z^{2}} = 0 in (0, T) \times Ω_{-}, \\ {\bar{θ}}_{2, h^{-}, τ}^{ε} = 0 at z = 0, \\ \frac{\partial {\bar{θ}}_{2, h^{-}, τ}^{ε}}{\partial z} = - (\frac{\partial u_{τ}^{0}}{\partial z} + \frac{\partial {\bar{θ}}_{1, h^{-}, τ}^{ε}}{\partial z}) at z = h^{-}, \\ {\bar{θ}}_{2, h^{-}, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{θ}}_{2, h^{-}, τ |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.32) The approximate solution ${\bar{θ}}_{2, h^{-}, τ}^{ε}$ can be expressed as: $\begin{array}{rclr} {\bar{θ}}_{2, h^{-}, τ}^{ε} (t; x, y, z) = - \sqrt{\frac{ε}{π}} \int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} e^{- {(h - z)}^{2} / 4 ε (t - τ)} (\frac{\partial u_{τ}^{0}}{\partial z} + \frac{\partial {\bar{θ}}_{1, h^{-}, τ}^{ε}}{\partial z}) (τ, x, y, h^{-}) d τ, & (3.33) \end{array}$ and the expression of the normal component ${\bar{θ}}_{2, h^{-}, n}^{ε}$ , which satisfies the following boundary condition ${\bar{θ}}_{2, h^{-}, n}^{ε} = 0 at z = h^{-},$ (3.34) is deduced from the incompressibility condition ${\bar{θ}}_{2, h^{-}, n}^{ε} = - \int_{h}^{z} {div}_{τ} {\bar{θ}}_{2, h^{-}, τ}^{ε} (t; x, y, s) d s .$ (3.35) In place of looking for the explicit expression of $θ_{1, h^{+}}^{ε}$ , we introduce a simple ${\tilde{θ}}_{1, h^{+}}^{ε}$ of $θ_{1, h^{+}}^{ε}$ for which the tangential component ${\tilde{θ}}_{1, h^{+}, τ}^{ε}$ satisfies the following system: $\{\begin{matrix} \frac{\partial {\tilde{θ}}_{1, h^{+}, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\tilde{θ}}_{1, h^{+}, τ}^{ε}}{\partial z^{2}} = 0 in (0, T) \times Ω_{+}, \\ {\tilde{θ}}_{1, h^{+}, τ}^{ε} = - u_{τ}^{0} (h^{+}) at z = h^{+}, \\ θ_{1, h^{+}, τ}^{ε} = 0 at z = 2 h, \\ {\tilde{θ}}_{1, h^{+}, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{θ}}_{1, h^{+}, τ |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.36) The explicit expression of ${\tilde{θ}}_{1, h^{+}, τ}^{ε}$ can be expressed as: $\begin{array}{rclr} {\tilde{θ}}_{1, h^{+}, τ}^{ε} (t; x, y, z) & = & - u_{τ}^{0} (0; x, y, h^{+}) (1 - 2 erf (z - h / \sqrt{2 ε t})) \\ - \int_{0}^{t} [1 - 2 erf (z - h / \sqrt{2 ε (t - s)})] \frac{\partial u_{τ}^{0}}{\partial s} (s; x, y, h^{+}) d s . & (3.37) \end{array}$ By our choice of divergence free corrector which verifies the boundary condition: ${\tilde{θ}}_{h^{+}, n}^{ε} = 0 at z = h^{+},$ (3.38) the expression of ${\tilde{θ}}_{n, h^{+}}^{ε}$ is simply given by: ${\tilde{θ}}_{n, 1, h^{+}}^{ε} (t; x, y, z) = - \int_{h}^{z} {div}_{τ} θ_{τ, 1, h^{+}}^{ε} (t; x, y, s) d s .$ (3.39) Finally, we define an approximation ${\bar{θ}}_{2, h^{+}}^{ε}$ of $θ_{2, h^{+}}^{ε}$ such that ${\bar{θ}}_{2, h^{+}, τ}^{ε}$ verifies $\{\begin{matrix} \frac{\partial {\bar{θ}}_{2, h^{+}, τ}^{ε}}{\partial t} - ε \frac{\partial^{2} {\bar{θ}}_{2, h^{+}, τ}^{ε}}{\partial z^{2}} = 0 in (0, T) \times Ω_{+}, \\ \frac{\partial {\bar{θ}}_{2, h^{+}, τ}^{ε}}{\partial z} = - (\frac{\partial u_{τ}^{0}}{\partial z} + \frac{\partial {\bar{θ}}_{1, h^{+}, τ}^{ε}}{\partial z}) at z = h^{+}, \\ {\bar{θ}}_{2, h^{+}, τ}^{ε} = 0 at z = 2 h, \\ {\bar{θ}}_{2, h^{+}, τ}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\bar{θ}}_{2, h^{+}, τ |_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.40) The approximate expression of ${\bar{θ}}_{τ, 1, h^{+}}^{ε}$ is given by: ${\bar{θ}}_{2, h^{+}, τ}^{ε} (t; x, y, z) = \sqrt{\frac{ε}{π}} \int_{0}^{t} \frac{1}{\sqrt{(t - τ)}} e^{- {(z - h)}^{2} / 4 ε (t - τ)} (\frac{\partial u_{τ}^{0}}{\partial z} + \frac{\partial {\bar{θ}}_{1, h^{+}, τ}^{ε}}{\partial z}) (τ, x, y, h^{+}) d τ,$ (3.41) and ${\bar{θ}}_{n, 2, h^{+}}^{ε}$ can be expressed, using the incompressibility and the boundary conditions ( ${\bar{θ}}_{2, h^{+}, n}^{ε} = 0$ at $z = h^{+}$ ), as follows: ${\bar{θ}}_{2, h^{+}, n}^{ε} = - \int_{h}^{z} {div}_{τ} {\bar{θ}}_{2, h^{+}, τ}^{ε} (t; x, y, s) d s .$ (3.42) Hence, the global corrector that we propose is given by: ${\tilde{θ}}^{ε} = ({\bar{\bar{θ}}}_{0}^{ε} + {\bar{θ}}_{h^{-}}^{ε}) χ_{(0, h)} + ({\tilde{θ}}_{h^{+}}^{ε} + {\tilde{\tilde{θ}}}_{2 h}^{ε}) χ_{(h, 2 h)} .$ (3.43)

We now want to derive the equations satisfied by ${\tilde{θ}}^{ε} = {\tilde{θ}}_{τ}^{ε} {\vec{e}}_{τ} + {\tilde{θ}}_{n}^{ε} {\vec{e}}_{n}$ . Since the problem considered here is linear and thanks to the incompressibility condition preserved by our choice of correctors, we deduce that the normal components ${\bar{\bar{θ}}}_{0, n}^{ε}$ , ${\bar{θ}}_{h^{-}, n}^{ε}$ , ${\tilde{θ}}_{h^{+}, n}^{ε}$ and ${\tilde{\tilde{θ}}}_{2 h, n}^{ε}$ satisfy the same equations as the tangential components. It is important to notice that ${\tilde{θ}}_{τ}^{ε}$ is not continuous at $z = h$ . Then, the corrector ${\tilde{θ}}^{ε}$ verifies an equation with singular source term: $\frac{\partial {\tilde{θ}}^{ε}}{\partial t} - ε \frac{\partial^{2} {\tilde{θ}}^{ε}}{\partial z^{2}} = - \frac{\partial α^{ε}}{\partial t} - \frac{\partial β^{ε}}{\partial t} - ε λ (t; x, y) δ^{'} - ε μ (t; x, y) δ,$ (3.44) where $\begin{array}{rclr} λ^{ε} (t; x, y) & = & {\tilde{θ}}_{τ, h^{+}}^{ε} (t; x, y, h) + {\tilde{\tilde{θ}}}_{τ, 2 h}^{ε} (t; x, y, h) - ({\bar{\bar{θ}}}_{τ, 0}^{ε} (t; x, y, h) - {\bar{θ}}_{τ, h^{-}}^{ε} (t; x, y, h)) \\ = & - \int_{0}^{t} H_{1} (s; x, y) d s & (3.45) \end{array}$ and $\begin{array}{rclr} μ^{ε} (t; x, y) & = & \frac{\partial ({\tilde{θ}}_{τ, h^{+}}^{ε} + {\tilde{\tilde{θ}}}_{τ, 2 h}^{ε})}{\partial z} (t; x, y, h) - \frac{\partial ({\bar{\bar{θ}}}_{τ, 0}^{ε} + {\bar{θ}}_{τ, h^{-}}^{ε})}{\partial z} (t; x, y, h) \\ = & \frac{\partial {\tilde{\tilde{θ}}}_{τ, 2 h}^{ε}}{\partial z} (t; x, y, h) - \frac{\partial {\bar{\bar{θ}}}_{τ, 0}^{ε}}{\partial z} (t; x, y, h) . & (3.46) \end{array}$ We recall that ${\tilde{θ}}^{ε}$ satisfies the incompressibility condition $div {\tilde{θ}}^{ε} = 0 in (0, T) \times Ω .$ The boundary conditions for ${\tilde{θ}}^{ε}$ are given in the following.

At the boundary $z = 0$ , we have ${\tilde{θ}}_{τ}^{ε} = - u_{τ}^{0} (t, x, y, 0) + {\bar{θ}}_{h^{-}, τ}^{ε} (t; x, y, 0) - β^{ε},$ with $| {\bar{θ}}_{h^{-}, τ}^{ε} (t; x, y, 0) - β^{ε} (t, x, y) | ⩽ κ e^{- h / \sqrt{2 ε T}}$ and ${\bar{θ}}_{n}^{ε} = {\bar{θ}}_{0, n}^{ε} (t, x, y, 0) + {\bar{θ}}_{h, n^{-}}^{ε} (t; x, y, 0); | {\bar{θ}}_{n}^{ε} | ⩽ κ ε^{1 / 2} .$ At the boundary $z = 2 h$ , ${\tilde{θ}}_{τ}^{ε} = - u_{τ}^{0} (t, x, y, 2 h) + {\tilde{θ}}_{h^{+}, τ}^{ε} - α^{ε} (t; x, y),$ with $| {\tilde{θ}}_{h^{+}, τ}^{ε} - α^{ε} | ⩽ κ e^{- h / \sqrt{2 ε T}}$ and ${\tilde{θ}}_{n}^{ε} = {\tilde{θ}}_{2 h, n}^{ε} + {\tilde{θ}}_{h^{+}, n}^{ε} - α^{ε}; and | {\tilde{θ}}_{n}^{ε} | ⩽ κ ε^{1 / 2} .$

3.2. Convergence result

In this section, we state and prove our main results and give the rate convergence order with respect to the viscosity ε. We have the following theorem.

Theorem 3.1.

Under the hypothesis ( H ) and for all $T > 0$ , the solution $u^{ε}$ of the problem ( 3.1 ) with $u_{0}$ supposed to be sufficiently smooth, satisfies the following estimates: $\begin{array}{rclr} {∥ u^{ε} - u^{0} - {\tilde{θ}}^{ε} ∥}_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ κ ε^{3 / 4}, & (3.47) \\ {∥ u^{ε} - u^{0} - {\tilde{θ}}^{ε} ∥}_{L^{2} (0, T; H^{1} (Ω))} ⩽ κ ε^{1 / 4}, & (3.48) \end{array}$ where κ is a positive constant independent of ε, and $u^{0}$ and ${\tilde{θ}}^{ε}$ are defined respectively by ( 3.2 ) and ( 3.43 ) .

Before we prove Theorem 3.1, we need an auxiliary result subject of the following lemma which will be used later in the proof of the theorem.

Lemma 3.1.

Let $(u, π)$ be the solution of the steady Stokes system given as below: $\{\begin{matrix} - Δ u + \nabla π = 0 in Ω, \\ div u = 0 in Ω, \\ u = g on Γ = Γ_{-} \cup Γ_{+}, \\ u is periodic in the directions x and y with periods L_{1}, L_{2} . \end{matrix}$ (3.49) Here we should have of course $\int_{Γ} g \cdot n d Γ = 0$ for the well-posedness of the system ( 3.49 ) .

Then, there exists a constant c depending only on Ω such that: $∥ u ∥_{L^{2} (Ω)} ⩽ c ∥ g ∥_{H^{- 1 / 2} (Γ)} .$ (3.50)

Proof.

The idea of the proof is inspired from the transposition method introduced for general elliptic problems in [14].

Let $(v, q)$ be the solution of $\{\begin{matrix} - Δ v + \nabla q = u in Ω, \\ div v = 0 in Ω, \\ v = 0 on Γ = Γ_{-} \cup Γ_{+}, \\ v is periodic in the directions x and y with periods L_{1}, L_{2} . \end{matrix}$ (3.51) Hence, from the classical theory of the Stokes problems, one can easily obtain that (at least) $v \in H^{2} (Ω)$ and $q \in H^{1} (Ω)$ (see e.g. [2,3] and [18]). Consequently, we deduce that $\begin{array}{rclr} \int_{Ω} u^{2} d Ω & = & \int_{Ω} u (- Δ v + \nabla q) d Ω \\ = & \int_{Γ} [g \cdot n q - g \frac{\partial v}{\partial n}] d Γ \\ ⩽ & ∥ g ∥_{H^{- 1 / 2} (Γ)} {∥ \frac{\partial v}{\partial n} ∥}_{H^{1 / 2} (Γ)} + ∥ g \cdot n ∥_{H^{- 1 / 2} (Γ)} ∥ γ q ∥_{H^{1 / 2} (Γ)} \\ ⩽ & c ∥ g ∥_{H^{- 1 / 2} (Γ)} (∥ v ∥_{H^{2} (Ω)} + ∥ q ∥_{H^{1} (Ω)}) \\ ⩽ & c ∥ g ∥_{H^{- 1 / 2} (Γ)} ∥ u ∥_{L^{2} (Ω)} . & (3.52) \end{array}$ This ends the proof of the lemma. □

Now, we return to the proof of the main results stated in Theorem 3.1.

Proof of Theorem 3.1.

We set $R^{ε} = u^{ε} - u^{0} - {\tilde{θ}}^{ε}$ , i.e. $R^{ε} = \{\begin{matrix} u^{ε} - u^{0} - {\bar{θ}}_{0}^{ε} - {\bar{θ}}_{h^{-}}^{ε} & if z \in [0, h), \\ u^{ε} - u^{0} - {\tilde{\tilde{θ}}}_{2 h}^{ε} - {\tilde{θ}}_{h^{+}}^{ε} & if z \in (h, 2 h] . \end{matrix}$ (3.53) As mentioned in the previous section, our primary goal is to absorb the singularity generated by the discontinuity of $u^{0}$ near $z = h$ . In fact, we have $Δ u^{0} = Δ_{x y} u_{0} + \frac{\partial^{2} u_{0}}{\partial z^{2}} + δ^{'} \int_{0}^{t} H_{1} (s; x, y) d s .$ (3.54) By our choice of corrector, the singularity produced in the right-hand side of (3.54) is absorbed by ${\tilde{θ}}^{ε}$ (see (3.44) and (3.45)). Let us examine $R^{ε} = (R_{τ}^{ε}, R_{n}^{ε})$ along the boundary of Ω.

At $z = 0$ , $R_{τ}^{ε} = - {\bar{θ}}_{τ, h^{-}}^{ε} (t; x, y, h^{-} / \sqrt{ε}) and R_{n}^{ε} = - {\bar{θ}}_{n, 0}^{ε} - {\bar{θ}}_{n, h^{-}}^{ε} .$ At $z = 2 h$ , $R_{τ}^{ε} = - {\tilde{θ}}_{τ, h^{+}}^{ε} (t; x, y, h^{+} / \sqrt{ε}) + α^{ε} (t; x, y) and R_{n}^{ε} = - {\tilde{θ}}_{n, h^{+}}^{ε} - {\tilde{\tilde{θ}}}_{n, 2 h}^{ε} .$ We note that $R^{ε}$ does not satisfy the desired boundary conditions ( $R^{ε} = 0 \in \partial Ω$ ). Hence, we introduce a supplementary corrector denoted by $ψ^{ε} = \sqrt{ε} {\tilde{ψ}}^{ε}$ where ${\tilde{ψ}}^{ε}$ is the solution of the following Stokes problem $\{\begin{matrix} - Δ {\tilde{ψ}}^{ε} + \nabla {\tilde{π}}^{ε} = 0 in (0, + \infty) \times Ω, \\ div {\tilde{ψ}}^{ε} = 0 in (0, + \infty) \times Ω, \\ {\tilde{ψ}}_{τ}^{ε} = - \frac{1}{\sqrt{ε}} γ (u_{τ}^{0} + {\tilde{θ}}_{τ}^{ε}) at z = 0, 2 h, \\ {\tilde{ψ}}^{ε} \cdot n = - \frac{1}{\sqrt{ε}} γ ({\tilde{θ}}_{n}^{ε}) at z = 0, 2 h, \\ {\tilde{ψ}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \end{matrix}$ (3.55) and $π = ε^{3 / 2} {\tilde{π}}^{ε}$ . Note that ${\tilde{ψ}}^{ε} = 0$ at $t = 0$ . Moreover, we observe that $γ ({\tilde{ψ}}_{τ}^{ε}) = {\tilde{ψ}}_{τ}^{ε} |_{z = 0, 2 h} = - \frac{1}{\sqrt{ε}} γ (u_{τ}^{0} + {\tilde{θ}}_{τ}^{ε})$ is an exponentially small term (e.s.t.). Also $| γ {\tilde{ψ}}_{n}^{ε} |$ is bounded in $L^{\infty} (0, T; L^{\infty} (\partial Ω))$ by a constant independent of ε. Then from the theory of steady Stokes problems, we observe that the solution ${\tilde{ψ}}^{ε}$ verifies for all $0 ⩽ t ⩽ T$ : $\begin{array}{rclr} {∥ \nabla {\tilde{ψ}}^{ε} ∥}_{L^{2} (Ω)} & ⩽ & \frac{κ}{\sqrt{ε}} {∥ γ ({\tilde{θ}}_{n}^{ε}) ∥}_{H^{1 / 2} (Γ)} + e . s . t . \\ ⩽ & \frac{κ}{\sqrt{ε}} {∥ {\tilde{θ}}_{n}^{ε} ∥}_{H^{1} (Ω)} + e . s . t . ⩽ c ε^{- 1 / 4} . & (3.56) \end{array}$ Hence, we have ${∥ ψ^{ε} ∥}_{L^{\infty} (0, T; H^{1} (Ω))} ⩽ c ε^{1 / 4} .$ (3.57) From Lemma 3.1 and the fact that $\int {\tilde{ψ}}^{ε} \cdot n d Γ = 0$ , we can estimate ${\tilde{ψ}}^{ε}$ in $L^{\infty} (0, T; L^{2} (Ω))$ as follows: $\begin{array}{rclr} {∥ {\tilde{ψ}}^{ε} ∥}_{L^{2} (Ω)} & ⩽ & \frac{κ}{\sqrt{ε}} {∥ γ ({\tilde{θ}}_{n}^{ε}) ∥}_{H^{- 1 / 2} (Γ)} + e . s . t . \\ ⩽ & \frac{κ}{\sqrt{ε}} {∥ {\tilde{θ}}_{n}^{ε} ∥}_{L^{2} (Ω)} + e . s . t . \\ ⩽ & κ ε^{1 / 4} \forall 0 ⩽ t ⩽ T . & (3.58) \end{array}$ Consequently, we obtain that ${∥ ψ^{ε} ∥}_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ κ ε^{3 / 4} .$ (3.59) The estimate (3.58) also holds for the time derivatives of ${\tilde{ψ}}^{ε}$ which is solution of the same problem (3.55) with ${\tilde{θ}}^{ε}$ and $u^{0}$ replaced by $\partial {\tilde{θ}}^{ε} / \partial t$ and $\partial u^{0} / \partial t$ . We thus deduce that: ${∥ \frac{\partial ψ^{ε}}{\partial t} ∥}_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ κ ε^{3 / 4} .$ (3.60) We now define ${\tilde{R}}^{ε} = R^{ε} - ψ^{ε},$ which satisfies: $\{\begin{matrix} \frac{\partial {\tilde{R}}^{ε}}{\partial t} - ε Δ {\tilde{R}}^{ε} + \nabla (p^{ε} - p^{0} - π^{ε}) \\ = ε Δ u^{0} + ε Δ_{x y} {\tilde{θ}}^{ε} + \frac{\partial ψ^{ε}}{\partial t} + \frac{\partial α^{ε}}{\partial t} + ε μ^{ε} δ in (0, T) \times Ω, \\ div {\tilde{R}}^{ε} = 0 in (0, T) \times Ω, \\ {\tilde{R}}^{ε} = 0 on (0, T) \times \partial Ω, \\ {\tilde{R}}^{ε} is periodic in the x and y directions with periods L_{1}, L_{2}, \\ {\tilde{R}}_{|_{t = 0}}^{ε} = 0 . \end{matrix}$ (3.61) We multiply (3.61) by ${\tilde{R}}^{ε}$ , integrate over Ω and apply the Cauchy–Schwarz inequality in the right-hand side of the resulting energy equality. We finally obtain $\begin{array}{rclr} \frac{1}{2} \frac{d}{d t} {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} + ε {∥ \nabla {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} & = & ε (∥ Δ u_{0} ∥_{L^{2} (Ω)} + {∥ Δ_{x y} {\tilde{θ}}^{ε} ∥}_{L^{2} (Ω)}) {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)} \\ + ({∥ \frac{\partial α^{ε}}{\partial t} ∥}_{L^{2} (Ω)} + {∥ \frac{\partial β^{ε}}{\partial t} ∥}_{L^{2} (Ω)} + {∥ \frac{\partial ψ}{\partial t} ∥}_{L^{2} (Ω)}) {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)} \\ + ε ⟨ μ^{ε} (t; x, y) δ, {\tilde{R}}^{ε} ⟩ . & (3.62) \end{array}$ We recall that $Δ u_{0}$ is bounded in $L^{2} (Ω)$ since $u_{0}$ is regular, and from Lemma 2.1, we can easily bound $∥ Δ_{x y} {\tilde{θ}}^{ε} ∥_{L^{2} (Ω)}$ by a constant. The functions $\partial α^{ε} / \partial t$ and $\partial β^{ε} / \partial t$ are exponentially small terms in all Sobolev spaces, so using (3.60) we have: $\begin{array}{rclr} | ⟨ \frac{\partial ψ^{ε}}{\partial t} + \frac{\partial α^{ε}}{\partial t} + \frac{\partial β^{ε}}{\partial t}, {\tilde{R}}^{ε} ⟩ | & ⩽ & κ {∥ \frac{\partial ψ^{ε}}{\partial t} ∥}_{L^{2} (Ω)}^{2} + \frac{1}{4} {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} \\ ⩽ & κ ε^{3 / 2} + \frac{1}{4} {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2}, & (3.63) \end{array}$ where the bracket $⟨ \cdot, \cdot ⟩$ denotes here the duality product of $H_{0}^{1} (Ω)$ with its dual space $H^{- 1} (Ω)$ .

Moreover, we have: $\begin{array}{rclr} | μ^{ε} (t, x, y) | & ⩽ & | \frac{{\tilde{\tilde{θ}}}_{τ, 2 h}^{ε}}{\partial z} (t; x, y, h) + \frac{{\bar{\bar{θ}}}_{τ, 0}^{ε}}{\partial z} (t; x, y, h) | \\ ⩽ & κ ε^{- 1 / 2} e^{- h / \sqrt{ε T}} \\ ⩽ & κ ε . & (3.64) \end{array}$ Since $μ^{ε}$ is an exponentially small term, using the fact that ${\tilde{R}}^{ε} \in H_{0}^{1} (Ω)$ and the Poincaré-type inequality, we deduce that $\begin{array}{rclr} | ⟨ ε μ^{ε} (t, x, y) δ, {\tilde{R}}^{ε} ⟩ | & = & ε μ^{ε} (t, x, y) {\tilde{R}}^{ε} (z = h) \\ ⩽ & κ ε^{2} {∥ {\tilde{R}}^{ε} ∥}_{H^{1} (Ω)} \\ ⩽ & κ ε^{3} + \frac{ε}{4} {∥ {\tilde{R}}^{ε} ∥}_{H^{1} (Ω)}^{2} \\ ⩽ & κ ε^{3} + \frac{ε}{4} {∥ \nabla {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} . & (3.65) \end{array}$ We then infer from (3.62) that $\frac{1}{2} \frac{d}{d t} {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} + \frac{3 ε}{4} {∥ \nabla {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} ⩽ {∥ {\tilde{R}}^{ε} ∥}_{L^{2} (Ω)}^{2} + κ ε^{2} .$ (3.66) We now apply the Gronwall inequality to (3.66) and we obtain ${∥ {\tilde{R}}^{ε} ∥}_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ κ ε^{3 / 4}$ (3.67) and ${∥ {\tilde{R}}^{ε} ∥}_{L^{2} (0, T; H^{1} (Ω))} ⩽ κ ε^{1 / 4} .$ (3.68) This achieves the proof of Theorem 3.1. □

Footnotes

Acknowledgement

The authors would like to thank Professor Roger Temam for suggesting this problem and for his advice.

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