Topological asymptotic expansion for the full Navier

Abstract

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

Keywords

Asymptotic expansion topological sensitivity analysis Navier–Stokes equations nonlinear operator topological gradient fluid mechanics topology optimization

1. Introduction

In fluid mechanics, shape and topology optimization methods have wide fields of applications in aerodynamic, cardiology, biology and hydrodynamic problems. One of the first studies was developed by Pironneau in [36]. After that, many shape optimization approaches are performed to determine the design of minimum drag bodies [27,37], valves [31], diffuser [12], airfoils [15] and optimization of heat exchangers design [16]. The majority of the classical methods dealing with optimal design of flow domains are limited to determine the optimal shape of an existing boundary.

To remedy this drawback, various topology optimization based methods have been developed during the two last decades. In contrast to shape optimization, where only the boundary of the design domain is optimized and the topology of the structure is kept unchanged, topology optimization allows the creation of new geometric perturbations during the optimization process. For instance, one can cite the work of Borrvall and Petersson [11]. Based on the density approach, they performed a topology optimization method to minimize the power dissipated in a Stokes flow problem. The main idea of this method is based on the parametrization technique where the fluid flow domain has been identified by its characteristic function. Furthermore, based on this methodology, various numerical approaches have been carried out to solve engineering applications for fluidic devices in Stokes flow [11], Navier–Stokes flows [20], Darcy-Stokes flows [21], non-Newtonian flows [35] and Stokes-Brinkmann flows [39].

Recently, the topological sensitivity notion [1–4,7,9,10,19,23–25,34] gives new perspectives on shape and topology optimization. It provides an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The topology of the computational domain can change during the optimization process. The most simple way of modifying the topology consists in creating a small hole in the considered domain. In the case of structural topology optimization, creating a hole means simply removing some material. In the case of fluid mechanics where the domain represents the fluid, creating a hole means inserting an obstacle $O_{z, ε}$ of small size $ε > 0$ around a point $z \in Ω$ . The topological sensitivity tools allow to find the location where creating the geometric perturbation $O_{z, ε}$ will bring the best improvement of the shape function j. It leads to an asymptotic expansion of the form $\begin{matrix} j (Ω ∖ \overline{O_{z, ε}}) = j (Ω) + ρ (ε) δ j (z) + o (ρ (ε)) \end{matrix}$ where $ε \mapsto ρ (ε)$ is a positive scalar function going to zero with ε and $δ j$ is called the topological gradient or the topological sensitivity. In order to minimize the shape function j, the best location to insert a small perturbation in Ω is where $δ j$ is most negative. In fact, if $δ j (z) < 0$ , we have $j (Ω ∖ \overline{O_{z, ε}}) < j (Ω)$ .

In fluid mechanics, the topological sensitivity analysis approach has been the subject of several theoretical and numerical studies. Indeed, the expressions of the topological derivatives have been calculated for a variety of shape functions depending on the domain Ω via the velocity field of the fluid in the two and three dimensional cases. Various topological gradient-based algorithms have been investigated and applied for dealing with valuable applications in fluid mechanics. One can refer to the optimization of the injectors location in a lake for maximizing the oxygenation process [22,26], topology and shape optimization in Stokes flow [24], detecting of flaws location in moulds filling process [9], control of mechanical aeration process [4], geometric control problem in Stokes flow [3], concentration phenomena [41], etc..

In this work, we extend the topological sensitivity analysis notion to the non stationary Navier–Stokes problem. The aim is to derive a topological asymptotic expansion for the full nonlinear Navier–Stokes operator, which provide an interesting tool for topology optimization in fluid mechanics. From the mathematical point of view, the main difficulties arise from the non-linearity of the operator. For instance, the derivation of the energy estimate should be examined in a different way compared to the linear case since the used functional spaces depend on ε in a very non trivial way.

In this paper, we will present two main results. The first one concerns the velocity field variation caused by the presence of a small obstacle $O_{z, ε}$ inside the fluid flow domain. We will establish an estimate showing the behavior of the perturbed velocity with respect to the obstacle size ε. We will prove that the leading term of the velocity field variation admits an explicit expression involving the Green function of the steady state Stokes operator and the non perturbed Navier–Stokes problem solution.

The second main result provides a topological sensitivity analysis for the full non-linear Navier–Stokes operator with respect to the creation of a small geometric perturbation inside the fluid flow domain. We will derive a topological asymptotic expansion valid for a large class of shape functions and arbitrarily shaped perturbations. The proposed mathematical analysis is general and can be adapted for various non-linear partial differential equations (PDEs).

The rest of this paper is organized as follows. Section 2 is concerned with the formulation of the considered problem. In Section 2.1, we present the Navier–Stokes equations and we recall some existence and uniqueness results. The perturbed Navier–Stokes problem is described in Section 2.2. In Section 3, we examine the influence of the inserted obstacle $O_{z, ε}$ on the fluid flow. We derive an estimate showing the behavior of the velocity field variation with respect to the obstacle size ε. Section 4 is devoted to a topological sensitivity analysis for the nonlinear Navier–Stokes operator. We develop a topological asymptotic formula valid for a large class of shape functions.

2. Formulation of the problem

Let $T > 0$ and Ω be a bounded domain of $R^{2}$ with smooth boundary $Σ = \partial Ω$ (of class $C^{2}$ ), containing an incompressible Newtonian fluid. For simplicity and without any loss of generality, we assume that the system of units is chosen in such a way that the fluid density is equal to one. In addition, we assume that the fluid motion in the spatio-temporal domain $Ω \times (0, T)$ is governed by the non-stationary Navier–Stokes equations. Then, for a given source term $G \in L^{2} (0, T; L^{2} (Ω))$ , the velocity field w and the pressure field p satisfy the following system $\begin{matrix} (2.1) & \{\begin{matrix} \frac{\partial w}{\partial t} - ν Δ w + (w \cdot \nabla) w + \nabla p = G & in Ω \times (0, T), \\ div w = 0 & in Ω \times (0, T), \\ w = 0 & on Σ \times (0, T), \\ w (\cdot, 0) = w_{I} & in Ω . \end{matrix} \end{matrix}$ where $ν > 0$ denotes the kinematic viscosity coefficient of the fluid and $w_{I}$ is a given initial velocity.

In (2.1), we use the standards notation $(w \cdot \nabla) w = (w_{1} \partial_{1} + w_{2} \partial_{2}) w$ , where $w_{k}$ and $\partial_{k}$ are the kth component of the vector w and the partial derivative with respect the kth coordinate, respectively.

2.1. Weak formulation and well-posedness

In this section, we recall some results concerning the well-posedness of the full Navier–Stokes problem. To this end, we introduce the following required functional spaces $\begin{array}{rcl} V & = & {u \in H^{1} (Ω), div (u) = 0, u = 0 on Σ} \\ H & = & {u \in L^{2} (Ω), div (u) = 0, u \cdot n = 0 on Σ}, \end{array}$ where $H^{1} (Ω) = H^{1} (Ω) \times H^{1} (Ω)$ and $L^{2} (Ω) = L^{2} (Ω) \times = L^{2} (Ω)$ are the usual Hilbert spaces of vector-valued functions. Also, we denote by $T$ the trilinear form defined as $\begin{matrix} T (u, v, ξ) = \int_{Ω} (u \cdot \nabla) v \cdot ξ d x, \forall u, v, ξ \in H^{1} (Ω) . \end{matrix}$ In the literature, one may find many different definitions of the Navier–Stokes weak solutions. Following the construction procedure of weak solutions proposed by J. Leray (see [28,29] or [30]), R.Temam([40], see chapter III, Section 3) illustrated the variational formulation of the Navier–Stokes problem by the following system: $\begin{array}{l} (2.2) & ⊳ Given G \in L^{2} (0, T; V^{'}) and w_{I} \in H \\ (2.3) & ⊳ Find w \in L^{2} (0, T; V) such that \\ (2.4) & \frac{d}{d t} \int_{Ω} w \cdot ξ d x + ν \int_{Ω} \nabla w : \nabla ξ d x + T (w, w, ξ) = \int_{Ω} G \cdot ξ d x, \forall ξ \in V \\ (2.5) & w (\cdot, 0) = w_{I} . \end{array}$ This weak formulation has been also used by many researchers for the description of the Navier–Stokes weak solutions; see for example J.L. Lions [32] or G. Galdi [18]. Next, we briefly recall some basic results concerning the well-posedness of the full Navier–Stokes equations.

An existence result: In ([40], Theorem 3.1, p. 226), R. Temam proved that the problem (2.2)–(2.5) admits at least one solution. The presented proof is based on the construction of an approximate solution by the Galerkin method. The convergence of the constructed sequence is established with the help of a compactness theorem. The result is valid for each open Lipschitzian set $Ω \subset R^{d}$ , $d ⩽ 4$ .

A uniqueness result: The uniqueness result is presented for the two dimensional case in ([40], Theorem 3.2, p. 235), see also J.L. Lions and G. Prodi [33]. It is based on the following lemma

Lemma 2.1.
For any open set $Ω \subset R^{2}$ , we have $\begin{array}{rcl} (2.6) & ‖ u ‖_{L^{4} (Ω)} ⩽ 2^{1 / 4} ‖ u ‖_{L^{2} (Ω)}^{1 / 2} ‖ \nabla u ‖_{L^{2} (Ω)}^{1 / 2}, \forall u \in H_{0}^{1} (Ω) . \end{array}$

Remark 2.2.
As a consequence of (2.6), the unique solution of the Navier–Stokes equations (if exists) satisfies $\begin{matrix} w \in L^{4} (Ω \times (0, T)) . \end{matrix}$

Some regularity results: As a regularity result, it is proved in ([40], see Theorem 3.6, p. 242) that if Ω is a bounded set of class $C^{2}$ and the source term $G \in L^{\infty} (0, T; H)$ , then the unique solution w belongs to $L^{\infty} (0, T; H^{2} (Ω))$ .

In the literature, there are many others regularity results for the weak solution to the Navier–Stokes equations. For instance, we recall the following theorem, which has been proved in ([38], p. 49-51) in the two space dimensions.

Theorem 2.3.
Assume that $Ω \subset R^{2}$ is a bounded set of class $C^{2}$ , $w_{I} \in V$ and $G \in L^{2} (0, T; L^{2} (Ω))$ . Then, the weak solution $(w, p)$ to the Navier–Stokes equations ( 2.1 ) satisfies: $\begin{matrix} \nabla^{2} w \in L^{2} (0, T; L^{2} {(Ω)}^{2}), \frac{\partial w}{\partial t}, \nabla p \in L^{2} (0, T; L^{2} (Ω)) and \nabla w \in L^{\infty} (0, T; L^{2} {(Ω)}^{2}) . \end{matrix}$ Moreover $\begin{array}{l} {‖ \nabla^{2} w ‖}_{L^{2} (0, T; L^{2} {(Ω)}^{2})} + {‖ \frac{\partial w}{\partial t} ‖}_{L^{2} (0, T; L^{2} (Ω))} + ‖ \nabla p ‖_{L^{2} (0, T; L^{2} (Ω))} + ‖ \nabla w ‖_{L^{\infty} (0, T; L^{2} {(Ω)}^{2})} \\ ⩽ C (‖ G ‖_{L^{2} (0, T; L^{2} (Ω))}, ‖ w_{I} ‖_{H^{1} (Ω)}) . \end{array}$

Remark 2.4.
According to ([40], Remark 3.7, p. 243), it is important to signal that one can obtain a smooth solution to the full Navier–Stokes equations if the given data are regular. Particularly, if Ω is of class $C^{\infty}$ , $w_{I} \in C^{\infty} (Ω)$ and $G \in C^{\infty} (\overline{Ω \times (0, T)})$ , then the solution w belongs to $C^{\infty} (\overline{Ω \times (0, T)})$ .

2.2. Perturbed Navier–Stokes problem

Let $O_{z, ε}$ be a small obstacle inserted in the fluid flow domain Ω around an arbitrary point $z \in Ω$ . We assume that the employed obstacle has the form $\begin{matrix} O_{z, ε} = z + ε O, \end{matrix}$ with z is an arbitrary point inside the domain Ω, $ε > 0$ is a small parameter represents the obstacle size and $O$ is a given bounded domain of $R^{2}$ containing the origin whose boundary $\partial O$ is connected and of class $C^{2}$ . In this study, we assume that the fluid flow satisfies an homogeneous Dirichlet boundary condition on $\partial O_{z, ε}$ . Then, in the perturbed domain $Ω_{z, ε} = Ω ∖ \overline{O_{z, ε}}$ , the velocity field $w_{ε}$ and the pressure field $p_{ε}$ solves the following Navier–Stokes systems $\begin{matrix} (2.7) & \{\begin{matrix} \frac{\partial w_{ε}}{\partial t} - ν Δ w_{ε} + (w_{ε} \cdot \nabla) w_{ε} + \nabla p_{ε} = G & in Ω_{z, ε} \times (0, T), \\ div w_{ε} = 0 & in Ω_{z, ε} \times (0, T), \\ w_{ε} = 0 & on Σ \times (0, T), \\ w_{ε} = 0 & on \partial O_{z, ε} \times (0, T), \\ w_{ε} (\cdot, 0) = w_{I} & in Ω_{z, ε} . \end{matrix} \end{matrix}$

2.3. The purpose of this work

Let j be a shape function defined as: $\begin{matrix} (2.8) & j (Ω ∖ \overline{O_{z, ε}}) = \int_{0}^{T} Φ_{ε} (w_{ε} (\cdot, t)) d t, \end{matrix}$ with $Φ_{ε} : H^{1} (Ω ∖ \overline{O_{z, ε}}) ⟶ R$ is a given cost function and $w_{ε}$ is the velocity field, solution to the Navier–Stokes problem (2.7) in the perturbed fluid flow domain $Ω ∖ \overline{O_{z, ε}}$ . Note that if $ε = 0$ (absence of obstacles), we have $Ω_{0, z} = Ω$ and the shape function j is defined as $\begin{matrix} (2.9) & j (Ω) = \int_{0}^{T} Φ_{0} (w_{0} (\cdot, t)) d t, \end{matrix}$ where $w_{0}$ is the solution to the following unperturbed Navier–Stokes equations $\begin{matrix} (2.10) & \{\begin{matrix} \frac{\partial w_{0}}{\partial t} - ν Δ w_{0} + (w_{0} \cdot \nabla) w_{0} + \nabla p_{0} = G & in Ω \times (0, T), \\ div w_{0} = 0 & in Ω \times (0, T), \\ w_{0} = 0 & on Σ \times (0, T), \\ w_{0} (\cdot, 0) = w_{I} & in Ω . \end{matrix} \end{matrix}$

The purpose of this work is to derive a topological asymptotic expansion for the full non-linear Navier–Stokes operator, valid for a large class of shape functions j and arbitrarily shaped geometric perturbation $O_{z, ε}$ . As mentioned in the introduction, we will evaluate the behavior of j with respect to the perturbation size ε and establish an asymptotic formula on the form $\begin{matrix} j (Ω ∖ \overline{O_{z, ε}}) = j (Ω) + ρ (ε) δ j (z) + o (ρ (ε)) \end{matrix}$ In this expansion, the term $ρ (ε)$ describes the behavior of j with respect to ε. As it will be shown later, the expression of $ρ (ε)$ will be dictated by the behavior of the perturbed velocity field with respect to ε. The term $δ j$ (called the topological gradient) represents the leading term of the shape function variation $j (Ω ∖ \overline{O_{z, ε}}) - j (Ω)$ . Its expression depends on the considered model equation as well as the chosen cost function $Φ_{ε}$ .

In order to derive the expected asymptotic formula, we will start our analysis by evaluating the influence of the geometric perturbation $O_{z, ε}$ on the velocity field. The leading term of the variation $w_{ε} - w_{0}$ will be constructed with the help of the fundamental solution of the steady state Stokes operator. We will derive an estimate describing the behavior of the perturbed velocity with respect to ε.

In order to focus our efforts on the topological sensitivity analysis for the full nonlinear Navier–Stokes operator, which is the main objective of this paper, we introduce an additional assumption to overcome difficulties caused by the lack of regularity in data.

Assumptions 2.5.

The source term G and the initial velocity $w_{I}$ are sufficiently smooth in such a way that the solution $w_{0}$ of the Navier–Stokes belongs to $C^{1} (Ω \times [0, T])$ .

There exists $β > 0$ such that $‖ \nabla w_{0} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} ⩽ β < ν / κ (Ω)$ , where $κ (Ω) = \frac{1}{2} meas {(Ω_{z, ε})}^{1 / 2}$ .

Remark 2.6.

The assumption $(i)$ is not restrictive since the source term G represents in general the gravitational force which is a smooth function. For the initial condition, one can deal with $w_{I} = 0$ in Ω.

The assumption $(i i)$ have been used in [17] (see also [7]) as a sufficient uniqueness condition for the Navier–Stokes problem.

3. Estimate of the perturbed velocity field

In this section, we examine the influence of the inserted obstacle $O_{z, ε}$ on the fluid flow. We will derive an estimate showing the behavior of the velocity field variation $w_{ε} - w_{0}$ with respect to the parameter ε. To this end, we introduce the following functions defined in $Ω \times (0, T)$ by $\begin{matrix} (3.1) & W (x, t) = 4 π ν \sum_{j = 1}^{2} [E^{j} (x - z) w_{0} (z, t)] e_{j}, \end{matrix}$ where $z \in Ω$ represents the obstacle location, ${e_{j}}_{j = 1, 2}$ is the canonical basis of $R^{2}$ and $E^{j} (y) = E (y) e_{j}$ , with $(E, Π)$ is the fundamental solution of the steady-state Stokes system in $R^{2}$ $\begin{matrix} (3.2) & E (y) = \frac{1}{4 π ν} (- log (r) I + e_{r} e_{r}^{T}), Π (y) = \frac{y}{2 π r^{2}}, \forall y \in R^{2}, \end{matrix}$ where $I$ is the $2 \times 2$ identity matrix, $r = ‖ y ‖$ , $e_{r} = \frac{y}{r}$ and $e_{r}^{T}$ is the transposed vector of $e_{r}$ .

The following Theorem gives an estimate of the velocity field variation caused by the presence of a small obstacle $O_{z, ε}$ inside the fluid flow domain Ω.

Theorem 3.1.
Let $O_{z, ε}$ be a small obstacle, strictly embedded into the fluid flow domain Ω. Then, under the assumptions 2.5 , there exists a real number $c > 0$ independent on ε, such that $\begin{array}{rcl} {‖ w_{ε} - w_{0} - \frac{1}{log (ε)} W ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ w_{ε} - w_{0} - \frac{1}{log (ε)} W ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{array}$

In order to prove Theorem 3.1, we first introduce some notations and state the Navier–Stokes type problem that satisfied by $w_{ε} - w_{0} - \frac{1}{log (ε)} W$ . Indeed, let Q be the pressure field associated to the velocity field W (see (3.1)): $\begin{matrix} (3.3) & Q (x, t) = 4 π ν Π (x - z) \cdot w_{0} (z, t) = 4 π ν \sum_{j = 1}^{2} Π^{j} (x - z) w_{0}^{j} (z, t), \end{matrix}$ where $y \mapsto Π^{j} (y) = Π (y) \cdot e_{j}$ is the pressure field associated to the velocity field $E^{j}$ .

Setting: $\begin{matrix} (3.4) & z_{ε} = w_{ε} - w_{0} - \frac{1}{log (ε)} W and s_{ε} = p_{ε} - p_{0} - \frac{1}{log (ε)} Q . \end{matrix}$ From (2.7), (2.10) and (3.1), it follows that $(z_{ε}, s_{ε})$ is solution to $\begin{matrix} (3.5) & \{\begin{matrix} \frac{\partial z_{ε}}{\partial t} - ν Δ z_{ε} + (h_{ε} \cdot \nabla) z_{ε} + (z_{ε} \cdot \nabla) h_{ε} + (z_{ε} \cdot \nabla) z_{ε} + \nabla s_{ε} = G_{ε} & in Ω_{z, ε} \times (0, T), \\ div z_{ε} = 0 & in Ω_{z, ε} \times (0, T), \\ z_{ε} = - \frac{1}{log (ε)} W & on Σ \times (0, T), \\ z_{ε} = - w_{0} - \frac{1}{log (ε)} W & on \partial O_{z, ε} \times (0, T), \\ z_{ε} (\cdot, 0) = 0 & in Ω_{z, ε} . \end{matrix} \end{matrix}$ where $\begin{array}{rcl} (3.6) & h_{ε} & = & w_{0} + \frac{1}{log (ε)} W, \\ (3.7) & G_{ε} & = & \frac{1}{log (ε)} [- \frac{\partial W}{\partial t} + \nabla w_{0} \cdot W + \nabla W \cdot w_{0} + \frac{1}{log (ε)} \nabla W \cdot W] \end{array}$ Using the relation $E ((x - z) / ε) = E (x - z) + \frac{log (ε)}{4 π ν} I$ , the last boundary condition can be rewritten as $\begin{array}{l} z_{ε} = - w_{0} - \frac{1}{log (ε)} W = - w_{0} (x, t) + w_{0} (z, t) - \frac{4 π ν}{log (ε)} E ((x - z) / ε) w_{0} (z, t) \\ (3.8) & on \partial O_{z, ε} \times (0, T) . \end{array}$ Our aim is to prove that there exists a constant $c > 0$ (independent of ε) such that $\begin{matrix} ‖ z_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + ‖ z_{ε} ‖_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$ To this end, we decompose the solution $(z_{ε}, s_{ε})$ in three terms as $\begin{matrix} z_{ε} = z_{ε}^{1} + z_{ε}^{2} + z_{ε}^{3} and s_{ε} = s_{ε}^{1} + s_{ε}^{2} + s_{ε}^{3}, \end{matrix}$ and we will estimate each term separately.
The first term $(z_{ε}^{1}, s_{ε}^{1})$ is chosen as the solution to the following non-stationary Stokes system $\begin{matrix} (3.9) & \{\begin{matrix} \frac{\partial z_{ε}^{1}}{\partial t} - ν Δ z_{ε}^{1} + \nabla s_{ε}^{1} = 0 & in Ω_{z, ε} \times (0, T), \\ div z_{ε}^{1} = 0 & in Ω_{z, ε} \times (0, T), \\ z_{ε}^{1} = 0 & on Σ \times (0, T), \\ z_{ε}^{1} = - w_{0} (x, t) + w_{0} (z, t) & on \partial O_{z, ε} \times (0, T), \\ z_{ε}^{1} (\cdot, 0) = 0 & in Ω_{z, ε} . \end{matrix} \end{matrix}$

The second term $(z_{ε}^{2}, s_{ε}^{2})$ is constructed as $\begin{matrix} (3.10) & \{\begin{matrix} z_{ε}^{2} (x, t) = - \frac{4 π ν}{log (ε)} Γ_{ε} (x) w_{0} (z, t) = - \frac{4 π ν}{log (ε)} \sum_{i = 1}^{2} Γ_{ε}^{j} (x) w_{0}^{j} (z, t), \\ s_{ε}^{2} (x, t) = - \frac{4 π ν}{log (ε)} θ_{ε} (x) \cdot w_{0} (z, t) = - \frac{4 π ν}{log (ε)} \sum_{i = 1}^{2} θ_{ε}^{j} (x) w_{0}^{j} (z, t), \end{matrix} \end{matrix}$ where $w_{0}^{j} (z, t)$ is the jth component of $w_{0} (z, t)$ and the matrix $Γ_{ε}$ and the vector $θ_{ε}$ are chosen such that the jth component $(Γ_{ε}^{j}, θ_{ε}^{j}) = (Γ_{ε} e_{j}, θ_{ε} . e_{j})$ is the solution to the following stationary Stokes problem $\begin{matrix} (3.11) & \{\begin{matrix} - ν Δ Γ_{ε}^{j} + \nabla θ_{ε}^{j} = 0 & in Ω_{z, ε}, \\ div Γ_{ε}^{j} = 0 & in Ω_{z, ε}, \\ Γ_{ε}^{j} = E^{j} (x - z) & on Σ, \\ Γ_{ε}^{j} = E^{j} ((x - z) / ε) & on \partial O_{z, ε}, \end{matrix} \end{matrix}$ where $E^{j} (y) = E (y) e_{j}$ is the jth column of the Stokes fundamental solution E (see (3.2)).

The third term $z_{ε}^{3}$ is chosen such that $z_{ε}^{3} = z_{ε} - (z_{ε}^{1} + z_{ε}^{2})$ . Using (3.5), (3.9) and (3.10) it follows that $(z_{ε}^{3}, s_{ε}^{3})$ is solution to $\begin{matrix} (3.12) & \{\begin{matrix} \frac{\partial z_{ε}^{3}}{\partial t} - ν Δ z_{ε}^{3} + (Ψ_{ε} \cdot \nabla) z_{ε}^{3} + (z_{ε}^{3} \cdot \nabla) Ψ_{ε} + (z_{ε}^{3} \cdot \nabla) z_{ε}^{3} + \nabla s_{ε}^{3} \\ = G_{ε}^{0} - \sum_{k = 1}^{4} G_{ε}^{k} & in Ω_{z, ε} \times (0, T), \\ div z_{ε}^{3} = 0 & in Ω_{z, ε} \times (0, T), \\ z_{ε}^{3} = 0 & on Σ \times (0, T), \\ z_{ε}^{3} = 0 & on \partial O_{z, ε} \times (0, T), \\ z_{ε}^{3} (\cdot, 0) = 0 & in Ω_{z, ε}, \end{matrix} \end{matrix}$ where the convective field $Ψ_{ε}$ is given by $\begin{array}{l} Ψ_{ε} (x, t) = w_{0} (x, t) + \frac{1}{log (ε)} W (x, t) + z_{ε}^{1} (x, t) + z_{ε}^{2} (x, t), (x, t) \in Ω_{z, ε} \times (0, T) . \end{array}$ The source terms are defined for almost every $(x, t) \in Ω_{z, ε} \times (0, T)$ as $\begin{array}{rcl} G_{ε}^{0} (x, t) & = & \frac{4 π ν}{log (ε)} [Γ_{ε} (x) - E (x - z)] \frac{\partial w_{0}}{\partial t} (z, t), \\ G_{ε}^{1} (x, t) & = & \frac{1}{log (ε)} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) W + \frac{1}{log (ε)} (W \cdot \nabla) w_{0}, \\ G_{ε}^{2} (x, t) & = & ([z_{ε}^{1} + z_{ε}^{2}] \cdot \nabla) [w_{0} + \frac{1}{log (ε)} W], \\ G_{ε}^{3} (x, t) & = & ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) [z_{ε}^{1} + z_{ε}^{2}], \\ G_{ε}^{4} (x, t) & = & ([z_{ε}^{1} + z_{ε}^{2}] \cdot \nabla) [z_{ε}^{1} + z_{ε}^{2}] . \end{array}$

3.1. Estimate of the term $z_{ε}^{1}$

In this paragraph, we examine the first part of $z_{ε}$ . We will prove that the term $z_{ε}^{1}$ satisfies the following estimate.

Proposition 3.2.
There exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} {‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$

To prove this result, we need some preliminary estimates related to the non-stationary Stokes equations.
3.1.1. Preliminary results for the non-stationary Stokes problem

In the following lemma, we recall a standard energy estimate for the non-stationary Stokes problem.

Lemma 3.3 (See [6], Theorems 8.2.3 and 8.2.9).

Let $T > 0$ be a final time. For a given source term $G \in L^{2} (0, T; L^{2} (Ω))$ and an initial data $ψ \in L^{2} (Ω)$ such that $div ψ = 0 in Ω$ , the time-dependent Stokes problem $\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} - ν Δ u + \nabla r = G & in Ω \times (0, T), \\ div u = 0 & in Ω \times (0, T), \\ u = 0 & on Σ \times (0, T), \\ u (\cdot, 0) = ψ & in Ω, \end{matrix} \end{matrix}$ has a unique solution $u \in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C ([0, T]; L^{2} (Ω))$ . Moreover, there exists a constant $c > 0$ (which only depends on Ω) such that: $\begin{matrix} ‖ u ‖_{L^{\infty} (0, T; L^{2} (Ω))} + ‖ u ‖_{L^{2} (0, T; H_{0}^{1} (Ω))} ⩽ c (‖ ψ ‖_{L^{2} (Ω)} + ‖ G ‖_{L^{2} (0, T; L^{2} (Ω))}) . \end{matrix}$

Proof.
For the proof and more details about this result, one can consult ([6], Chapter 8). More precisely, this estimate has been established for a general parabolic operator in Section 8.2.2 (see Theorem 8.2.3). The Stokes operator is discussed in Section 8.2.3 (see Theorem 8.2.9). □
Lemma 3.4.
Let $R > 0$ such that $O \subset B (0, R) = {x \in Ω, ‖ x ‖ < R}$ , where $‖ . ‖$ is the Euclidean norm. For $φ \in L^{2} (0, T; H^{3 / 2} (\partial O))$ , let $(U, ζ)$ be the solution to the following parabolic problem $\begin{matrix} \{\begin{matrix} \frac{\partial U}{\partial t} - ν Δ U + \nabla ζ = 0 & in (B (0, R) ∖ \overline{O}) \times (0, T), \\ div U = 0 & in (B (0, R) ∖ \overline{O}) \times (0, T), \\ U = 0 & on \partial B (0, R) \times (0, T), \\ U = φ & on \partial O \times (0, T), \\ U (\cdot, 0) = 0 & in B (0, R) ∖ \overline{O} . \end{matrix} \end{matrix}$ Then, there exists $c > 0$ , such that $\begin{matrix} (3.13) & ‖ U ‖_{H^{1} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ U ‖_{L^{2} (0, T; H^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ‖ φ ‖_{L^{2} (0, T; H^{3 / 2} (\partial O))} . \end{matrix}$
Proof.
For the proof and more details about this result, one can consult [14, Theorem 1.43]. More precisely, this estimate has been established for a general parabolic operator with the help of [14, Theorem 1.42(v)] and [8, Theorem II]. □

Applying the previous Lemma in the particular case $\begin{matrix} φ (y, t) = w_{0} (z + ε y, t) - w_{0} (z, t) on \partial O \times (0, T) . \end{matrix}$ We deduce the following preliminary estimate: Lemma 3.5.
Let $(U_{ε}, ζ_{ε})$ be the solution to the following non-stationary Stokes problem $\begin{matrix} \{\begin{matrix} \frac{\partial U_{ε}}{\partial t} - ν Δ U_{ε} + \nabla ζ_{ε} = 0 & in (B (0, R) ∖ \overline{O}) \times (0, T), \\ div U_{ε} = 0 & in (B (0, R) ∖ \overline{O}) \times (0, T), \\ U_{ε} = 0 & on \partial B (0, R) \times (0, T), \\ U_{ε} = w_{0} (z + ε y, t) - w_{0} (z, t) & on \partial O \times (0, T), \\ U_{ε} (\cdot, 0) = 0 & in B (0, R) ∖ \overline{O} . \end{matrix} \end{matrix}$ Then, there exists $c > 0$ , independent of ε, such that $\begin{array}{l} (3.14) & {‖ \frac{\partial U_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ε, \\ (3.15) & ‖ U_{ε} ‖_{L^{\infty} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ε . \end{array}$
Proof.
Using the fact that the function $(x, t) \mapsto w_{0} (x, t)$ is of class $C^{1}$ in the neighborhood of each point $z \in Ω$ and each time $t \in (0, T)$ , then by Taylor formula, one can write $\begin{matrix} (3.16) & w_{0} (z + ε y, t) - w_{0} (z, t) = ε D w_{0} (z + ε β (y), t) y, \end{matrix}$ where $D w_{0} (z + ε β (y), t)$ is the matrix defined by $\begin{matrix} {(D w_{0} (z + ε β (y), t))}_{i, j} = \frac{\partial w_{0}^{i}}{\partial x_{j}} (z + ε β_{i, j} (y), t), 1 ⩽ i, j ⩽ 2, \end{matrix}$ with $| β_{i, j} (y) | ⩽ | y | < R$ . Since the parameter ε is small, the points $z + ε β_{i, j} (y)$ are in the neighborhood of z. Then, it follows that the functions $(y, t) \mapsto {(D w_{0} (z + ε β (y), t))}_{i, j}$ are differentiable and their derivatives are uniformly bounded with respect to ε. Consequently, one can deduce that $\begin{matrix} (3.17) & {‖ w_{0} (z + ε y, t) - w_{0} (z, t) ‖}_{L^{2} (0, T; H^{3 / 2} (\partial O))} ⩽ c ε . \end{matrix}$ From Lemma 3.4, it follows $\begin{array}{l} (3.18) & {‖ \frac{\partial U_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ε . \end{array}$ Using Lemma 3.3 in a slightly different context (i.e. with a non homogeneous condition on $\partial O$ ), one can deduce the following standard estimate $\begin{array}{l} ‖ U_{ε} ‖_{L^{\infty} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} \\ ⩽ {‖ w_{0} (z + ε y, t) - w_{0} (z, t) ‖}_{L^{2} (0, T; H^{1 / 2} (\partial O))} ⩽ c ε . \end{array}$ □

3.1.2. Proof of Proposition 3.2

Recall that we aim to prove that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} {‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$ Let $V_{ε} (x, t) = U_{ε} (\frac{x - z}{ε}, t)$ , with $U_{ε}$ is the velocity field defined in Lemma 3.4. Since the support of the function $U_{ε}$ is included in the set $\overline{B (0, R) ∖ O} \times [0, T]$ (i.e. $supp (U_{ε}) \subset \overline{B (0, R) ∖ O} \times [0, T]$ ), we have $\begin{matrix} supp (V_{ε}) \subset (\overline{Ω_{z, ε}} \cap \overline{B (z, R ε)}) \times [0, T] . \end{matrix}$ Extending $V_{ε}$ by zero in $Ω ∖ B (z, R ε) \times [0, T]$ and setting $\begin{matrix} \tilde{w_{ε}} = z_{ε}^{1} - V_{ε} . \end{matrix}$ Then, one can check that the function $\tilde{w_{ε}}$ belongs to $L^{2} (0, T; H_{0}^{1} {(Ω_{z, ε})}^{2})$ . Therefore, from the weak formulation of the system (3.9) and we taking $\tilde{w_{ε}}$ as a test function, it follows $\begin{array}{l} \frac{1}{2} \int_{Ω_{z, ε}} {| z_{ε}^{1} (\cdot, t_{0}) |}^{2} d x + ν \int_{0}^{t_{0}} \int_{Ω_{z, ε}} {| \nabla z_{ε}^{1} |}^{2} d x d t \\ (3.19) & = \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \frac{\partial z_{ε}^{1}}{\partial t} V_{ε} d x d t + ν \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \nabla z_{ε}^{1} . \nabla V_{ε} d x d t, for almost t_{0} \in [0, T] . \end{array}$ The previous formula is obtained using the fact that $div z_{ε}^{1} = div V_{ε} = 0$ in $Ω_{z, ε}$ . Next, we will derive an estimate for each term on the right-hand-side of the equality (3.19).

Integrating by parts in time, the first term on the right-hand-side of the equality (3.19) can be written as $\begin{array}{l} (3.20) & \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \frac{\partial z_{ε}^{1}}{\partial t} V_{ε} d x d t = \int_{Ω_{z, ε}} z_{ε}^{1} (\cdot, t_{0}) V_{ε} (\cdot, t_{0}) d x - \int_{0}^{t_{0}} \int_{Ω_{z, ε}} z_{ε}^{1} \frac{\partial V_{ε}}{\partial t} d x d t . \end{array}$ By Cauchy–Schwarz inequality and taking the supremum over $[0, T]$ , we deduce $\begin{array}{l} | \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \frac{\partial z_{ε}^{1}}{\partial t} V_{ε} d x d t | ⩽ & {‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} ‖ V_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} \\ + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} {‖ \frac{\partial V_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Similarly, the second term on the right-hand-side of the equality (3.19) can be estimated as $\begin{array}{l} | ν \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \nabla z_{ε}^{1} . \nabla V_{ε} d x d t | ⩽ ν {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ‖ \nabla V_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Using the change of variables $x = z + ε y$ and Lemma 3.5, we deduce that there exists $c > 0$ , independent of ε such that $\begin{matrix} ‖ V_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ \frac{\partial V_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} + ‖ \nabla V_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} ⩽ c ε, \end{matrix}$ which implies $\begin{array}{l} | \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \frac{\partial z_{ε}^{1}}{\partial t} V_{ε} d x d t + ν \int_{0}^{t_{0}} \int_{Ω_{z, ε}} \nabla z_{ε}^{1} . \nabla V_{ε} d x d t | \\ ⩽ c ε ({‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}), \end{array}$ for almost $t_{0} \in [0, T]$ .

On the other hand, from the previous estimate and (3.19), one can deduce $\begin{array}{l} (3.21) & {‖ \nabla z_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} ⩽ c ε ({‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}), \\ (3.22) & \frac{1}{2} \int_{Ω_{z, ε}} {| z_{ε}^{1} (\cdot, t_{0}) |}^{2} d x ⩽ c ε ({‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}) . \end{array}$ Taking the supremum for all $t_{0} \in [0, T]$ in (3.22), we obtain $\begin{array}{l} (3.23) & {‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))}^{2} ⩽ 2 c ε ({‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}) . \end{array}$ Integrating between 0 and T, from (3.22) we derive $\begin{array}{l} (3.24) & {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} ⩽ 2 c ε T ({‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}) . \end{array}$ Combining the estimates (3.23), (3.24) and (3.21) we deduce $\begin{matrix} (3.25) & {‖ z_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ c ε = o (\frac{- 1}{log (ε)}) . \end{matrix}$

3.2. Estimate of the term $z_{ε}^{2}$

The second part of $z_{ε}$ , defined by the formula (3.10), admits the following estimate.

Proposition 3.6.
There exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} {‖ z_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$

As one can observe here the behavior of the term $z_{ε}^{2}$ , with respect to ε, is dictated by its space component $Γ_{ε}$ (see (3.10)). Then, in order to prove this result, we start our analysis by recalling some preliminary results for the perturbed steady state Stokes problem.
3.2.1. Preliminary results for the perturbed Stokes problem

Lemma 3.7 (See [13], Lemma C.3).

Let $ε > 0$ . For $λ \in R^{2}$ , let $(h_{ε}, ξ_{ε}) \in H^{1} (Ω_{z, ε}) \times L_{0}^{2} (Ω_{z, ε})$ be the solution of the Stokes problem $\begin{matrix} \{\begin{matrix} - ν Δ h_{ε} + \nabla ξ_{ε} = 0 & in Ω_{z, ε}, \\ div h_{ε} = 0 & in Ω_{z, ε}, \\ h_{ε} = 0 & on Σ, \\ h_{ε} = λ & on \partial O_{z, ε} . \end{matrix} \end{matrix}$ Then, there exists a constant $c > 0$ (independent of ε) such that: $\begin{matrix} (3.26) & ‖ h_{ε} ‖_{H^{1} (Ω_{z, ε})} ⩽ \frac{c | λ |}{\sqrt{| log (ε) |}} . \end{matrix}$

Lemma 3.8 (See [13], Lemma 4.1).

Let $ε > 0$ and φ be a vectorial function verifying $y \mapsto φ (z + ε y) \in H^{1 / 2} {(\partial O)}^{2}$ . For each $g \in H^{1 / 2} (Σ)$ such that $\int_{Σ} g \cdot n d s + \int_{\partial O_{z, ε}} φ \cdot n d s = 0$ , we denote by $(u_{ε}, r_{ε}) \in H^{1} (Ω_{z, ε}) \times L_{0}^{2} (Ω_{z, ε})$ the solution of the problem $\begin{matrix} \{\begin{matrix} - ν Δ u_{ε} + \nabla r_{ε} = 0 & in Ω_{z, ε}, \\ div u_{ε} = 0 & in Ω_{z, ε}, \\ u_{ε} = g & on Σ, \\ u_{ε} = φ & on \partial O_{z, ε} . \end{matrix} \end{matrix}$ Then, there exists a constant $c > 0$ (independent of ε) such that: $\begin{matrix} ‖ u_{ε} ‖_{H^{1} (Ω_{z, ε})} ⩽ c (‖ g ‖_{H^{1 / 2} (Σ)} + {‖ φ (z + ε y) ‖}_{H^{1 / 2} (\partial O)}) . \end{matrix}$

Proof.
A detailed proof of this estimate is given in ([13], Appendix C, pages 360–363). □
3.2.2. Proof of Proposition 3.6

Due to Lemma 3.8, one can check that there exists a constant $c > 0$ (independent of ε) such that: $\begin{matrix} {‖ Γ_{ε}^{j} ‖}_{H^{1} (Ω_{z, ε})} ⩽ c ({‖ E^{j} (x - z) ‖}_{H^{1 / 2} (Σ)} + {‖ E^{j} (x) ‖}_{H^{1 / 2} (\partial O)}), j = 1, 2 . \end{matrix}$ Using (3.10) and the smoothness of $w_{0}$ (of class $C^{1}$ near the point z), one can deduce $\begin{array}{l} {‖ z_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} \\ (3.27) & ⩽ \frac{c}{| log (ε) |} ({‖ E (x - z) ‖}_{H^{1 / 2} (Σ)} + {‖ E (y) ‖}_{H^{1 / 2} (\partial O)}) . \end{array}$ Then, to obtain the desired estimate it suffices to show that the terms $‖ E (y) ‖_{H^{1 / 2} (\partial O)}$ and $‖ E (x - z) ‖_{H^{1 / 2} (Σ)}$ are uniformly bounded. For the term $‖ E (y) ‖_{H^{1 / 2} (\partial O)}$ , we use the fact that $O$ is an open bounded set containing the origin. More precisely, one can find $r_{1} > 0$ such that $\overline{B (0, r_{1})}$ is stricly included in $O$ . Setting $O_{r_{1}} = O ∖ \overline{B (0, r_{1})}$ . Since $0 \notin \overline{O_{r_{1}}}$ , the function $y \mapsto E (y)$ belongs to $C^{1} (\overline{O_{r_{1}}})$ . Using the trace theorem, one can deduce $\begin{array}{rcl} {‖ E (y) ‖}_{H^{1 / 2} (\partial O)} : = inf {{‖ Ψ (y) ‖}_{H^{1} (O_{r_{1}})}, Ψ = E on \partial O} ⩽ {‖ E (y) ‖}_{L^{2} (O_{r_{1}})} + {‖ \nabla E (y) ‖}_{L^{2} (O_{r_{1}})}, \end{array}$ which implies that the quantity $‖ E (y) ‖_{H^{1 / 2} (\partial O)}$ is uniformly bounded.

The term $‖ E (x - z) ‖_{H^{1 / 2} (Σ)}$ can be examined by the same idea. Let $r_{2} > 0$ be such that $\overline{B (z, r_{2})} \subset Ω$ . Since $z \notin Ω_{r_{2}} = Ω ∖ \overline{B (z, r_{2})}$ , the function $x \mapsto E (x - z)$ belongs to $C^{1} (\overline{Ω_{r_{2}}})$ and the norm $‖ E (x - z) ‖_{H^{1 / 2} (Σ)}$ is uniformly bounded. More precisely, we have $\begin{array}{rcl} {‖ E (x - z) ‖}_{H^{1 / 2} (Σ)} ⩽ {‖ E (x - z) ‖}_{L^{2} (O_{r_{2}})} + {‖ \nabla E (x - z) ‖}_{L^{2} (O_{r_{2}})} = O (1) . \end{array}$ Combining the above estimates, it follows that there exists $c > 0$ , independent of ε, such that $\begin{matrix} (3.28) & {‖ z_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$

3.3. Estimate of the term $z_{ε}^{3}$

This section is concerned with the third part of $z_{ε}$ . We will establish that the term $z_{ε}^{3}$ satisfies the following estimate.

Proposition 3.9.
There exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} (3.29) & {‖ z_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{3} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$

To prove this result, we need some preliminary results for treating the nonlinear and convection terms.
3.3.1. Preliminary results for the trilinear form

In the perturbed domain $Ω_{z, ε}$ , the trilinear form $T$ is defined as $\begin{matrix} T (u, v, w) = \int_{Ω_{z, ε}} (u \cdot \nabla) v \cdot w d x, \forall u, v, w \in H^{1} (Ω_{z, ε}) . \end{matrix}$ As one can observe if $u \in L^{p} (Ω_{z, ε})$ and $w \in L^{q} (Ω_{z, ε})$ such that $1 / p + 1 / q = 1 / 2$ , by the Hölder inequality we have $\begin{matrix} | T (u, v, w) | ⩽ ‖ u ‖_{L^{p} (Ω_{z, ε})} ‖ \nabla v ‖_{L^{2} (Ω_{z, ε})} ‖ u ‖_{L^{q} (Ω_{z, ε})} . \end{matrix}$ For instance, when $p = q = 4$ , the Sobolev embedding Theorem in the two-dimensional space (see Adam’s [5], Section V, p.97) implies that $H^{1} (Ω_{z, ε}) ↪ L^{4} (Ω_{z, ε})$ . Then, one can deduce that $\begin{array}{rcl} (3.30) & | T (u, v, w) | ⩽ ‖ u ‖_{L^{4} (Ω_{z, ε})} ‖ \nabla v ‖_{L^{2} (Ω_{z, ε})} ‖ u ‖_{L^{4} (Ω_{z, ε})}, \forall u, v, w \in H^{1} (Ω_{z, ε}) . \end{array}$ In the following Lemmas, we present some properties that will be extensively used in the proof of Proposition 3.9.

Lemma 3.10 (see [17], Lemma IX.1.1, p. 588).

Let $Ω \subset R^{d}$ ( $d = 2, 3$ ) be a bounded and locally Lipschitz domain. Then, the trilinear form $T$ is continuous in the space $H_{0}^{1} (Ω_{z, ε}) \times H^{1} (Ω_{z, ε}) \times H_{0}^{1} (Ω_{z, ε})$ and we have $\begin{array}{rcl} (3.31) & | T (u, v, w) | = | \int_{Ω_{z, ε}} (u \cdot \nabla) v \cdot w d x | ⩽ κ (Ω_{z, ε}) ‖ \nabla u ‖_{L^{2} (Ω_{z, ε})} ‖ \nabla v ‖_{L^{2} (Ω_{z, ε})} ‖ \nabla w ‖_{L^{2} (Ω_{z, ε})} . \end{array}$ where $\begin{matrix} κ (Ω_{z, ε}) = \{\begin{matrix} \frac{2 \sqrt{2}}{3} meas {(Ω_{z, ε})}^{1 / 6} & if d = 3, \\ \frac{1}{2} meas {(Ω_{z, ε})}^{1 / 2} & if d = 2 . \end{matrix} \end{matrix}$

Remark 3.11.
Since $Ω_{z, ε} \subset Ω$ , we have $κ (Ω_{z, ε}) ⩽ κ (Ω)$ which implies $\begin{array}{rcl} (3.32) & | \int_{Ω_{z, ε}} (u \cdot \nabla) v \cdot w d x | ⩽ κ (Ω) ‖ \nabla u ‖_{L^{2} (Ω_{z, ε})} ‖ \nabla v ‖_{L^{2} (Ω_{z, ε})} ‖ \nabla w ‖_{L^{2} (Ω_{z, ε})}, \end{array}$ for all $(u, v, w) \in H_{0}^{1} (Ω_{z, ε}) \times H^{1} (Ω_{z, ε}) \times H_{0}^{1} (Ω_{z, ε})$ .
Lemma 3.12 (see [17], Lemma IX.2.1, p. 591).

Let $Ω \subset R^{d}$ ( $d = 2, 3$ ) be a bounded and locally Lipschitz domain. Consider $u \in H^{1} (Ω_{z, ε})$ such that $div u = 0$ in $Ω_{z, ε}$ . Then, for all $v \in H_{0}^{1} (Ω_{z, ε})$ we have $\begin{matrix} T (u, v, v) = \int_{Ω_{z, ε}} (u \cdot \nabla) v \cdot v d x = 0 . \end{matrix}$ Thus, in particular, $\begin{matrix} T (u, v, w) = - T (u, w, v), \forall v, w \in H_{0}^{1} (Ω_{z, ε}) . \end{matrix}$

3.3.2. Proof of Proposition 3.9

Aiming to derive the estimate (3.29), we start our proof by applying the weak formulation of the system (3.12). Then, one can get $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} \frac{\partial z_{ε}^{3}}{\partial t} z_{ε}^{3} d x d t + ν \int_{0}^{τ} \int_{Ω_{z, ε}} {| \nabla z_{ε}^{3} |}^{2} d x d t + \int_{0}^{τ} \int_{Ω_{z, ε}} (Ψ_{ε} \cdot \nabla) z_{ε}^{3} \cdot z_{ε}^{3} d x d t \\ + \int_{0}^{τ} \int_{Ω_{z, ε}} (z_{ε}^{3} \cdot \nabla) z_{ε}^{3} \cdot z_{ε}^{3} d x d t + \int_{0}^{τ} \int_{Ω_{z, ε}} (z_{ε}^{3} \cdot \nabla) Ψ_{ε} \cdot z_{ε}^{3} d x d t \\ = \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{0} \cdot z_{ε}^{3} d x d t - \sum_{k = 1}^{4} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{k} \cdot z_{ε}^{3} d x d t \end{array}$ Integrating by parts in time and using the fact that $ξ_{ε} (\cdot, 0) = 0$ in $Ω_{z, ε}$ , the first integral term can be rewritten as $\begin{array}{rcl} (3.33) & \int_{0}^{τ} \int_{Ω_{z, ε}} \frac{\partial z_{ε}^{3}}{\partial t} z_{ε}^{3} d x d t = \frac{1}{2} \int_{Ω_{z, ε}} {| z_{ε}^{3} (x, τ) |}^{2} d x . \end{array}$ According to Lemma 3.12, we have $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} (z_{ε}^{3} \cdot \nabla) z_{ε}^{3} \cdot z_{ε}^{3} d x d t = 0 and \\ \int_{0}^{τ} \int_{Ω_{z, ε}} (Ψ_{ε} \cdot \nabla) z_{ε}^{3} \cdot z_{ε}^{3} d x d t = 0 . \end{array}$ Then, it follows $\begin{array}{l} \frac{1}{2} \int_{Ω_{z, ε}} {| z_{ε}^{3} (x, τ) |}^{2} d x + ν \int_{0}^{τ} \int_{Ω_{z, ε}} {| \nabla z_{ε}^{3} |}^{2} d x d t + \int_{0}^{τ} \int_{Ω_{z, ε}} (z_{ε}^{3} \cdot \nabla) Ψ_{ε} \cdot z_{ε}^{3} d x d t \\ (3.34) & = \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{0} \cdot z_{ε}^{3} d x d t - \sum_{k = 1}^{4} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{k} \cdot z_{ε}^{3} d x d t \end{array}$ The rest of the proof will be divided into three steps:

Step 1: Estimates of the source terms :

We will examine the terms $\int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{k} \cdot z_{ε}^{3} d x d t$ , $0 ⩽ k ⩽ 4$ , separately.

Estimate of the term involving $G_{ε}^{0}$ : by Cauchy–Schwarz inequality one can get $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{0} \cdot z_{ε}^{3} d x d t | \\ = \frac{4 π ν}{| log (ε) |} | \int_{0}^{τ} \int_{Ω_{z, ε}} [Γ_{ε} (x) - E (x - z)] \frac{\partial w_{0}}{\partial t} (z, t) \cdot z_{ε}^{3} d x d t | \\ ⩽ \frac{4 π ν}{| log (ε) |} \int_{0}^{T} | \frac{\partial w_{0}}{\partial t} (z, t) | {‖ Γ_{ε} (x) - E (x - z) ‖}_{L^{2} (Ω_{z, ε})} {‖ z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} d t \\ ⩽ \frac{4 π ν}{| log (ε) |} {‖ \frac{\partial w_{0}}{\partial t} (z, t) ‖}_{L^{2} (0, T)} {‖ Γ_{ε} (x) - E (x - z) ‖}_{L^{2} (Ω_{z, ε})} {‖ z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Using (3.11) and Lemma 3.8 one can check $‖ Γ_{ε} ‖_{L^{2} (Ω_{z, ε})}$ is uniformly bounded with respect to ε i.e. $\begin{array}{l} (3.35) & ‖ Γ_{ε} ‖_{L^{2} (Ω_{z, ε})} = O (1) . \end{array}$ To estimate the term $‖ E (x - z) ‖_{L^{2} (Ω_{z, ε})}$ , we introduce the following notations:

From the fact that $O$ is an open domain containing the origin, there exists $r > 0$ such that: $\overline{B (0, r)} \subset O$ .

The domain Ω is bounded in such a way that there exists $R > 0$ such that: $\overline{Ω} \subset B (z, R)$ , $\forall z \in Ω$ .

Then, we have

\begin{array}{l} (3.36) & Ω_{z, ε} - z = {x - z, x \in Ω_{z, ε}} \subset C (0, r ε, R) : = {y \in R^{2}; r ε < | y | < R} . \end{array}

Using the polar coordinate system and an explicit calculation, one can prove that the quantity

‖ E (x - z) ‖_{L^{2} (Ω_{z, ε})}

is uniformly bounded with respect to ε i.e.

\begin{array}{l} (3.37) & {‖ E (x - z) ‖}_{L^{2} (Ω_{z, ε})} ⩽ c {‖ E (y) ‖}_{C (0, r ε, R)} = O (1) . \end{array}

Using the smoothness of

w_{0}

and the fact that

‖ Γ_{ε} (x) - E (x - z) ‖_{L^{2} (Ω_{z, ε})}

is uniformly bounded, it follows that there exists

c > 0

, independent of ε, such that

\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{0} \cdot z_{ε}^{3} d x d t | ⩽ \frac{c}{| log (ε) |} {‖ z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}

Estimate of the term involving $G_{ε}^{1}$ : we have $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{1} \cdot z_{ε}^{3} d x d t \\ = \frac{1}{log (ε)} [\int_{0}^{τ} \int_{Ω_{z, ε}} (W \cdot \nabla) w_{0} \cdot z_{ε}^{3} d x d t \\ (3.38) & + \int_{0}^{τ} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) W \cdot z_{ε}^{3} d x d t] . \end{array}$ Since $z_{ε}^{3} = 0$ on $\partial Ω \cup \partial O_{z, ε}$ and $div (w_{0}) = div (W) = 0$ in $Ω_{z, ε}$ , the last equality can be rewritten as $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{1} \cdot z_{ε}^{3} d x d t \\ = \frac{- 1}{log (ε)} [\int_{0}^{τ} \int_{Ω_{z, ε}} (W \cdot \nabla) z_{ε}^{3} \cdot w_{0} d x d t \\ + \int_{0}^{τ} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) z_{ε}^{3} \cdot W d x d t] . \end{array}$ By Holder inequality, one can derive the following estimate $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} (W \cdot \nabla) z_{ε}^{3} \cdot w_{0} d x d t | ⩽ \int_{0}^{T} ‖ W ‖_{L^{4} (Ω_{z, ε})} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} ‖ w_{0} ‖_{L^{4} (Ω_{z, ε})} d t . \end{array}$ Using the fact that $H^{1} (Ω) ↪ L^{4} (Ω)$ in the two dimensional case, one can check that there exists $c > 0$ , independent of ε, such that $\begin{matrix} ‖ w_{0} ‖_{L^{4} (Ω_{z, ε})} ⩽ ‖ w_{0} ‖_{L^{4} (Ω)} ⩽ c ‖ w_{0} ‖_{H^{1} (Ω)} . \end{matrix}$ Using the relation (3.1), one can deduce that the term $‖ W ‖_{L^{4} (Ω_{z, ε})}$ is uniformly bounded with respect to ε i.e. $\begin{array}{rcl} ‖ W ‖_{L^{4} (Ω_{z, ε})} & = & 4 π ν {‖ E (x - z) w_{0} (z, t) ‖}_{L^{4} (Ω_{z, ε})} \\ ⩽ & 4 π ν {‖ w_{0} (z, t) ‖}_{L^{\infty} (0, T)} {‖ E (x - z) ‖}_{L^{4} (Ω_{z, ε})} = O (1) . \end{array}$ Then, it follows $\begin{array}{l} (3.39) & | \int_{0}^{τ} \int_{Ω_{z, ε}} (W \cdot \nabla) z_{ε}^{3} \cdot w_{0} d x d t | ⩽ c ‖ w_{0} ‖_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Similarly $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) z_{ε}^{3} \cdot W d x d t | \\ ⩽ c \int_{0}^{T} {‖ w_{0} + \frac{1}{log (ε)} W ‖}_{L^{4} (Ω_{z, ε})} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} ‖ W ‖_{L^{4} (Ω_{z, ε})} d t . \end{array}$ Moreover, using the triangular inequality and the previous estimates one can derive $\begin{array}{l} {‖ w_{0} + \frac{1}{log (ε)} W ‖}_{L^{4} (Ω_{z, ε})} & ⩽ ‖ w_{0} ‖_{L^{4} (Ω_{z, ε})} + \frac{1}{| log (ε) |} ‖ W ‖_{L^{4} (Ω_{z, ε})} ⩽ c [‖ w_{0} ‖_{H^{1} (Ω)} + \frac{1}{| log (ε) |}] \end{array}$ which implies $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) z_{ε}^{3} \cdot W d x d t | \\ (3.40) & ⩽ c [‖ w_{0} ‖_{L^{2} (0, T; H^{1} (Ω))} + \frac{\sqrt{T}}{| log (ε) |}] {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Finally, from (3.38), (3.39) and (3.40) one can deduce that there exists $c > 0$ independent of ε, such that $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{1} \cdot z_{ε}^{3} d x d t | ⩽ \frac{c}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$

Estimate of the term involving $G_{ε}^{2}$ : Using the fact that $div (z_{ε}^{1} + z_{ε}^{2}) = 0$ in $Ω_{z, ε}$ and $z_{ε}^{3} = 0$ on $\partial Ω \cup \partial O_{z, ε}$ , one can write $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{2} \cdot z_{ε}^{3} d x d t & = \int_{0}^{τ} \int_{Ω_{z, ε}} ([z_{ε}^{1} + z_{ε}^{2}] \cdot \nabla) [w_{0} + \frac{1}{log (ε)} W] \cdot z_{ε}^{3} d x d t \\ = - \int_{0}^{τ} \int_{Ω_{z, ε}} ([z_{ε}^{1} + z_{ε}^{2}] \cdot \nabla) z_{ε}^{3} \cdot [w_{0} + \frac{1}{log (ε)} W] d x d t . \end{array}$ Then, we obtain $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{2} \cdot z_{ε}^{3} d x d t | \\ (3.41) & ⩽ \int_{0}^{T} {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{4} (Ω_{z, ε})} {‖ w_{0} + \frac{1}{log (ε)} W ‖}_{L^{4} (Ω_{z, ε})} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} d t . \end{array}$ Exploiting the smoothness of $w_{0}$ , we have $\begin{array}{l} {‖ w_{0} + \frac{1}{log (ε)} W ‖}_{L^{4} (Ω_{z, ε})} \\ ⩽ ‖ w_{0} ‖_{L^{4} (Ω)} + \frac{1}{| log (ε) |} {‖ E (x - z) w_{0} (z, t) ‖}_{L^{4} (Ω_{z, ε})} \\ ⩽ ‖ w_{0} ‖_{L^{\infty} (0, T; L^{4} (Ω))} + \frac{1}{| log (ε) |} {‖ w_{0} (z, t) ‖}_{L^{\infty} (0, T)} {‖ E (x - z) ‖}_{L^{4} (Ω_{z, ε})} \end{array}$ which implies that the term $‖ w_{0} + \frac{1}{log (ε)} W ‖_{L^{4} (Ω_{z, ε})}$ is uniformly bounded with respect to the time. Then, it follows $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{2} \cdot z_{ε}^{3} d x d t | & ⩽ c \int_{0}^{T} {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{4} (Ω_{z, ε})} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} d t \\ ⩽ c {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{2} (0, T; L^{4} (Ω_{z, ε}))} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} \end{array}$ Applying the embedding theorem, we may deduce $\begin{array}{l} {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{2} (0, T; L^{4} (Ω_{z, ε}))} ⩽ c [{‖ z_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} + {‖ z_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}] . \end{array}$ Finally, using the established results in Propositions 3.2 and 3.6, it follows that the term $G_{ε}^{2}$ verifies the following estimate $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{2} \cdot z_{ε}^{3} d x d t | & ⩽ \frac{c}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} \end{array}$

Estimate of the term involving $G_{ε}^{3}$ : Integrating by parts, the employed integral term in this case can be rewritten as $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{3} \cdot z_{ε}^{3} d x d t & = \int_{0}^{τ} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) [z_{ε}^{1} + z_{ε}^{2}] \cdot z_{ε}^{3} d x d t \\ = - \int_{0}^{τ} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) z_{ε}^{3} \cdot [z_{ε}^{1} + z_{ε}^{2}] d x d t . \end{array}$ In a similar way as in the previous case (see (3.41)), we obtain $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{3} \cdot z_{ε}^{3} d x d t | & ⩽ \int_{0}^{T} {‖ w_{0} + \frac{1}{log (ε)} W ‖}_{L^{4} (Ω_{z, ε})} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{4} (Ω_{z, ε})} d t \\ ⩽ \frac{c}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$

Estimate of the term involving $G_{ε}^{4}$ : We start by reformulating this term as $\begin{array}{l} \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{4} \cdot z_{ε}^{3} d x d t & = \int_{0}^{τ} \int_{Ω_{z, ε}} ([z_{ε}^{1} + z_{ε}^{2}] \cdot \nabla) [z_{ε}^{1} + z_{ε}^{2}] \cdot z_{ε}^{3} d x d t \\ = - \int_{0}^{τ} \int_{Ω_{z, ε}} ([z_{ε}^{1} + z_{ε}^{2}] \cdot \nabla) z_{ε}^{3} \cdot [z_{ε}^{1} + z_{ε}^{2}] d x d t . \end{array}$ By a Cauchy–Schwarz inequality, one can obtain $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{4} \cdot z_{ε}^{3} d x d t | & ⩽ \int_{0}^{T} {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{4} (Ω_{z, ε})}^{2} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})} d t \\ ⩽ {‖ z_{ε}^{1} + z_{ε}^{2} ‖}_{L^{4} (0, T; L^{4} (Ω_{z, ε}))}^{2} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Using the embedding theorem with the help of the established estimates in Propositions 3.2 and 3.6, one can check that there exists $c > 0$ , independent of ε, such that $\begin{array}{l} | \int_{0}^{τ} \int_{Ω_{z, ε}} G_{ε}^{4} \cdot z_{ε}^{3} d x d t | & ⩽ \frac{c}{| log (ε) |^{2}} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$

Step 2: Estimate of the trilinear term : Since $z_{ε}^{3} = 0$ on $\partial Ω$ , by Lemma 3.10 one can derive $\begin{array}{rcl} | \int_{0}^{τ} \int_{Ω_{z, ε}} (z_{ε}^{3} \cdot \nabla) Ψ_{ε} \cdot z_{ε}^{3} d x d t | & ⩽ & κ (Ω) \int_{0}^{T} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (Ω_{z, ε})}^{2} ‖ \nabla Ψ_{ε} ‖_{L^{2} (Ω_{z, ε})} d t \\ ⩽ & κ (Ω) {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} ‖ \nabla Ψ_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Furthermore, from the fact that $\begin{array}{rcl} ‖ \nabla W ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} = 4 π ν {‖ w_{0} (z, t) ‖}_{L^{\infty} (0, T)} {‖ \nabla E (x - z) ‖}_{L^{2} (Ω_{z, ε})} ⩽ c \sqrt{| log (ε) |} \end{array}$ one can deduce $\begin{array}{rcl} ‖ \nabla Ψ_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} & = & {‖ \nabla w_{0} + \frac{1}{log (ε)} \nabla W + \nabla z_{ε}^{1} + \nabla z_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} \\ ⩽ & ‖ \nabla w_{0} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + \frac{c}{\sqrt{| log (ε) |}} + \frac{c}{| log (ε) |} . \end{array}$ Consequently, from the Assumptions 2.5 (i.e. $‖ \nabla w_{0} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} ⩽ β < ν / κ (Ω)$ ) and the fact that $\frac{c}{\sqrt{| log (ε) |}} + \frac{c}{| log (ε) |}$ goes to zero when ε goes to zero, one can check that there exist $ε_{0} > 0$ and a constant $\tilde{β} > 0$ , such that $\begin{array}{rcl} ‖ \nabla Ψ_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} ⩽ \tilde{β} < ν / κ (Ω), \forall ε \in] 0, ε_{0} [. \end{array}$ Here, one can choose $\tilde{β} = β + δ β$ , with $δ β = \frac{ν / κ (Ω) - β}{2}$ . Therefore, the trilinear term admits the estimate $\begin{array}{rcl} (3.42) & | \int_{0}^{τ} \int_{Ω_{z, ε}} (z_{ε}^{3} \cdot \nabla) Ψ_{ε} \cdot z_{ε}^{3} d x d t | ⩽ \tilde{β} κ (Ω) {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} . \end{array}$

Step 3: Estimate of the term $z_{ε}^{3}$ . Inserting the above estimates into (3.34), we get $\begin{array}{l} \frac{1}{2} \int_{Ω_{z, ε}} {| z_{ε}^{3} (x, τ) |}^{2} d x + ν \int_{0}^{τ} \int_{Ω_{z, ε}} {| \nabla z_{ε}^{3} |}^{2} d x d t \\ ⩽ \tilde{β} κ (Ω) {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} + \frac{c}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}, \end{array}$ for almost all $τ \in [0, T]$ . Then, taking the supremum in τ over $[0, T]$ , we obtain $\begin{array}{l} (3.43) & {‖ z_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))}^{2} + 2 [ν - \tilde{β} κ (Ω)] {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} \\ ⩽ \frac{2 c}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Since $ν - \tilde{β} κ (Ω) > 0$ , one can write $\begin{array}{rcl} (3.44) & {‖ z_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))}^{2} ⩽ \frac{2 c}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}, \end{array}$ which implies $\begin{array}{rcl} (3.45) & {‖ z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}^{2} ⩽ T {‖ z_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))}^{2} ⩽ \frac{2 c T}{| log (ε) |} {‖ \nabla z_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$

Finally, combining the relations (3.43), (3.44) and (3.45) one can deduce that there exists

c > 0

(idependent of ε) such that

\begin{array}{rcl} {‖ z_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ z_{ε}^{3} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{array}

4. Sensitivity analysis framework

This section is concerned with the development of a general mathematical framework for the topological sensitivity analysis of the function j with respect to the inserted obstacle size ε. We will derive a topological asymptotic expansion valid for all function $Φ_{ε}$ verifying the following assumption:

Assumptions 4.1.

The function $Φ_{0}$ is differentiable on $H^{1} (Ω)$ and its derivative will be denoted by $D Φ_{0} (w)$ .

For each $ε ⩾ 0$ , the function $t \mapsto Φ_{ε} (w_{ε} (\cdot, t))$ belongs to $L^{1} (0, T)$ .

There exists a real number $δ Φ$ (depending on the obstacle location z) such that, for small $ε ⩾ 0$ , we have $\begin{array}{rcl} \int_{0}^{T} [Φ_{ε} (w_{ε} (\cdot, t)) - Φ_{0} (w_{0} (\cdot, t))] d t & = & \int_{0}^{T} D Φ_{0} (w_{0} (\cdot, t)) (w_{ε} (\cdot, t) - w_{0} (\cdot, t)) d t \\ + \frac{1}{| log (ε) |} δ Φ (z) + o (\frac{1}{| log (ε) |}), \end{array}$ where $w_{ε}$ is extended by zero inside the domain $O_{z, ε}$ .

The associated adjoint problem: find $(v_{0}, q_{0})$ such that $\begin{matrix} (4.1) & \{\begin{matrix} - \frac{\partial v_{0}}{\partial t} - ν Δ v_{0} + (v_{0} \cdot \nabla) w_{0} - (w_{0} \cdot \nabla) v_{0} + \nabla q_{0} = - D Φ (w_{0}) & in Ω \times (0, T), \\ div v_{0} = 0 & in Ω \times (0, T), \\ v_{0} = 0 & on Σ \times (0, T), \\ v_{0} (\cdot, T) = 0 & in Ω, \end{matrix} \end{matrix}$ admits a unique solution.

In this study, we will examine the variation of the shape function j with respect to the location z and the size ε of the inserted obstacle $O_{z, ε}$ .

Theorem 4.2.

Let $Φ_{ε}$ be a given cost function verifying the assumption $(A)$ . Then, the variation of the associated shape function j, with respect to the insertion of a small obstacle $O_{z, ε} = z + ε O$ , admits the following asymptotic behavior $\begin{array}{l} j (Ω ∖ \overline{O_{z, ε}}) = j (Ω) + \frac{1}{| log (ε) |} {4 π ν \int_{0}^{T} w_{0} (z, t) \cdot v_{0} (z, t) d t + δ Φ (z)} + o (\frac{1}{| log (ε) |}), \\ z \in Ω . \end{array}$

Proof.

To prove this theorem, we proceed by establishing some preliminary estimates. Using the assumption $(A)$ , it follows $\begin{array}{rcl} j (Ω ∖ \overline{O_{z, ε}}) - j (Ω) & = & \int_{0}^{T} Φ_{ε} (w_{ε} (\cdot, t)) d t - \int_{0}^{T} Φ_{0} (w_{0} (\cdot, t)) d t \\ = & \int_{0}^{T} D Φ_{0} (w_{0} (\cdot, t)) (w_{ε} (\cdot, t) - w_{0} (\cdot, t)) d t \\ (4.2) & + \frac{1}{| log (ε) |} δ Φ (z) + o (\frac{1}{| log (ε) |}) . \end{array}$ The velocity field variation $(ξ, π_{ε}) = (w_{ε} - w_{0}, p_{ε} - p_{0})$ satisfies $\begin{matrix} (4.3) & \{\begin{matrix} \frac{\partial ξ_{ε}}{\partial t} - ν Δ ξ_{ε} + (ξ_{ε} \cdot \nabla) w_{0} + (w_{0} \cdot \nabla) ξ_{ε} + (ξ_{ε} \cdot \nabla) ξ_{ε} + \nabla π_{ε} = 0 & in Ω_{z, ε} \times (0, T), \\ div ξ_{ε} = 0 & in Ω_{z, ε} \times (0, T), \\ ξ_{ε} = 0 & on Σ \times (0, T), \\ ξ_{ε} = - w_{0} & on \partial O_{z, ε} \times (0, T), \\ ξ_{ε} (\cdot, 0) = 0 & in Ω_{z, ε}, \end{matrix} \end{matrix}$ Using the weak formulation of the adjoint problem and the fact that $w_{ε} = 0$ inside $O_{z, ε}$ , it follows $\begin{array}{l} \int_{0}^{T} D Φ_{0} (w_{0} (\cdot, t)) (w_{ε} (\cdot, t) - w_{0} (\cdot, t)) d t \\ = \int_{0}^{T} \int_{Ω_{z, ε}} \frac{\partial v_{0}}{\partial t} ξ_{ε} d x d t - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} w_{0} d x d t - ν \int_{0}^{T} \int_{Ω_{z, ε}} \nabla v_{0} \nabla ξ_{ε} d x d t \\ (4.4) & + ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla w_{0} d x d t - \int_{0}^{T} \int_{Ω} [(v_{0} \cdot \nabla) w_{0} - (w_{0} \cdot \nabla) v_{0}] ξ_{ε} d x d t . \end{array}$ Integrating in time and using the fact that $v (\cdot, T) = 0$ and $ξ_{ε} (\cdot, 0) = 0$ , one can derive $\begin{matrix} \int_{0}^{T} \int_{Ω_{z, ε}} \frac{\partial v_{0}}{\partial t} ξ_{ε} d x d t = - \int_{0}^{T} \int_{Ω_{z, ε}} \frac{\partial ξ_{ε}}{\partial t} v_{0} d x d t . \end{matrix}$ Then, from the variational formulation of (4.3), one can deduce $\begin{array}{l} \int_{0}^{T} \int_{Ω_{z, ε}} \frac{\partial v_{0}}{\partial t} ξ_{ε} d x d t - ν \int_{0}^{T} \int_{Ω_{z, ε}} \nabla v_{0} \nabla ξ_{ε} d x d t \\ = - \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) n v_{0} d s d t + \int_{0}^{T} \int_{Ω} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t \\ + \int_{0}^{T} \int_{Ω} [(ξ_{ε} \cdot \nabla) w_{0} + (w_{0} \cdot \nabla) ξ_{ε}] \cdot v_{0} d x d t \\ (4.5) & + \int_{0}^{T} \int_{O_{z, ε}} (w_{0} \cdot \nabla) w_{0} \cdot v_{0} d x d t . \end{array}$ On the other hand, using the fact that $v_{0} = ξ_{ε} = 0$ on $\partial Ω$ , one can check $\begin{array}{l} \int_{0}^{T} \int_{Ω} [(ξ_{ε} \cdot \nabla) w_{0} + (w_{0} \cdot \nabla) ξ_{ε}] \cdot v_{0} d x d t \\ (4.6) & = \int_{0}^{T} \int_{Ω} [(v_{0} \cdot \nabla) w_{0} - (w_{0} \cdot \nabla) v_{0}] \cdot ξ_{ε} d x d t . \end{array}$ Combining the relations (4.4), (4.5) and (4.6), one can deduce $\begin{array}{l} \int_{0}^{T} D Φ_{0} (w_{0} (\cdot, t)) (w_{ε} (\cdot, t) - w_{0} (\cdot, t)) d t \\ = ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla w_{0} d x d t + \int_{0}^{T} \int_{O_{z, ε}} (w_{0} \cdot \nabla) w_{0} \cdot v_{0} d x d t \\ - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} w_{0} d x d t + \int_{0}^{T} \int_{Ω} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t \\ (4.7) & - \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) n v_{0} d s d t . \end{array}$ Since $ξ_{ε} = z_{ε} + \frac{1}{log (ε)} W$ and $π_{ε} = s_{ε} + \frac{1}{log (ε)} Q$ , it follows that the last integral term in the previous equality can be decomposed as $\begin{array}{rcl} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) n v_{0} d s d t & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (z_{ε}, s_{ε}) n v_{0} d s d t \\ (4.8) & + \frac{1}{log (ε)} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (W, Q) n v_{0} d s d t . \end{array}$ Inserting (4.7) and (4.8) into (4.2), it follows that the variation of j satisfies the following preliminary estimate $\begin{array}{rcl} (4.9) & j (Ω ∖ \overline{O_{z, ε}}) - j (Ω) & = & \sum_{k = 1}^{5} T^{k} (ε) + \frac{1}{| log (ε) |} δ Φ (z) + o (\frac{1}{| log (ε) |}), \end{array}$ where the terms $T^{k} (ε)$ , $1 ⩽ k ⩽ 4$ are defined as $\begin{array}{rcl} T^{1} (ε) & = & ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla w_{0} d x d t + \int_{0}^{T} \int_{O_{z, ε}} (w_{0} \cdot \nabla) w_{0} \cdot v_{0} d x d t \\ - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} w_{0} d x d t + \int_{0}^{T} \int_{Ω} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t, \\ T^{2} (ε) & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) (x, t) n [v_{0} (z, t) - v_{0} (x, t)] d s d t, \\ T^{3} (ε) & = & - \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) (x, t) n v_{0} (z, t) d s d t, \\ T^{4} (ε) & = & \frac{- 1}{log (ε)} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (W, Q) (x, t) n v_{0} (x, t) d s d t . \end{array}$ Next, we will examine the behavior of the integral terms $T^{k} (ε)$ , $k = 1, \dots, 4$ with respect to ε.

Estimate of $T^{1} (ε)$ : Since $w_{0}$ and $v_{0}$ are smooth enough (of class $C^{1}$ ) near the point z, then for small values of ε, one can check that there exists a constant $c > 0$ (independent of ε) such that $\begin{array}{rcl} | ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla w_{0} d x d t + \int_{0}^{T} \int_{O_{z, ε}} (w_{0} \cdot \nabla) w_{0} \cdot v_{0} d x d t - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} w_{0} d x d t | ⩽ c | O_{z, ε} |, \end{array}$ with $| O_{z, ε} |$ is the Lebesque measure of the subdomain $O_{z, ε}$ . To examine the last integral term in $T^{1} (ε)$ , we start with an integration by parts. Using the fact that $div ξ_{ε} = 0$ in Ω and $v = 0$ on $\partial Ω$ , one gets $\begin{array}{rcl} \int_{0}^{T} \int_{Ω} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t = - \int_{0}^{T} \int_{Ω} (ξ_{ε} \cdot \nabla) v_{0} \cdot ξ_{ε} d x d t . \end{array}$ Then, it follows $\begin{array}{rcl} | \int_{0}^{T} \int_{Ω} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t | ⩽ ‖ \nabla v_{0} ‖_{L^{2} (0, T; L^{\infty} (Ω))} ‖ ξ_{ε} ‖_{L^{2} (0, T; L^{2} (Ω))}^{2} . \end{array}$ Using the fact that $ξ_{ε} = - w_{0}$ in $O_{z, ε}$ and $ξ_{ε} = z_{ε} + \frac{1}{| log (ε) |} W$ in $Ω_{z, ε}$ , we obtain $\begin{array}{rcl} ‖ ξ_{ε} ‖_{L^{2} (0, T; L^{2} (Ω))}^{2} & ⩽ & {‖ z_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} + \frac{1}{log (ε)} ‖ W ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))}}^{2} + ‖ w_{0} ‖_{L^{2} (0, T; L^{2} (O_{z, ε}))}^{2} \\ (4.10) & ⩽ & c {\frac{1}{| log (ε) |^{2}} + | O_{z, ε} |} . \end{array}$ Since $| O_{z, ε} | = O (ε^{2})$ , it follows that the term $T^{1} (ε)$ obeys the estimate $\begin{array}{rcl} T^{1} (ε) = o (\frac{1}{| log (ε) |}) . \end{array}$

Estimate of $T^{2} (ε)$ : Let $(y_{ε}^{1}, r_{ε}^{1})$ be solution to the following system $\begin{matrix} (4.11) & \{\begin{matrix} \frac{\partial y_{ε}^{1}}{\partial t} - ν Δ y_{ε}^{1} + \nabla r_{ε}^{1} = 0 & in Ω_{z, ε} \times (0, T), \\ div y_{ε}^{1} = 0 & in Ω_{z, ε} \times (0, T), \\ y_{ε}^{1} = 0 & on Σ \times (0, T), \\ y_{ε}^{1} = - v_{0} (x, t) + v_{0} (z, t) & on \partial O_{z, ε} \times (0, T), \\ y_{ε}^{1} (\cdot, 0) = 0 & in Ω_{z, ε} . \end{matrix} \end{matrix}$ Then, using the weak formulation of (3.5), the term $T^{2} (ε)$ can be rewritten as $\begin{array}{rcl} T^{2} (ε) & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (z_{ε}, s_{ε}) (x, t) n y_{ε}^{1} d s d t \\ = & \int_{0}^{T} \int_{Ω_{z, ε}} \frac{\partial z_{ε}}{\partial t} y_{ε}^{1} d s d t + ν \int_{0}^{T} \int_{Ω_{z, ε}} \nabla z_{ε} \nabla y_{ε}^{1} d x d t \\ + \int_{0}^{T} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) z_{ε} \cdot y_{ε}^{1} d x d t \\ + \int_{0}^{T} \int_{Ω_{z, ε}} (z_{ε} \cdot \nabla) [w_{0} + \frac{1}{log (ε)} W] \cdot y_{ε}^{1} d x d t \\ + \int_{0}^{T} \int_{Ω_{z, ε}} (z_{ε} \cdot \nabla) z_{ε} \cdot y_{ε}^{1} d x d t - \int_{0}^{T} \int_{Ω_{z, ε}} G_{ε} \cdot y_{ε}^{1} d x d t . \end{array}$ By Cauchy–Schwarz inequality, one can deduce $\begin{array}{rcl} | T^{2} (ε) | & ⩽ & {‖ \frac{\partial z_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} {‖ y_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} + ‖ \nabla z_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} {‖ \nabla y_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} \\ + 2 c (Ω) {‖ w_{0} + \frac{1}{log (ε)} W ‖}_{L^{\infty} (0, T; H^{1} (Ω_{z, ε}))} ‖ z_{ε} ‖_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} {‖ y_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} \\ + c (Ω) ‖ z_{ε} ‖_{L^{2} (0, T; H^{1} (Ω_{z, ε}))}^{2} {‖ y_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} + ‖ G_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} {‖ y_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} . \end{array}$ Using the smoothness of $w_{0}$ and the fact that $‖ E (x - z) ‖_{L^{2} (Ω_{z, ε})}$ is uniformly bounded, one can derive $\begin{matrix} ‖ G_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} = O (\frac{1}{| log (ε) |}) . \end{matrix}$ The term $y_{ε}^{1}$ can be estimate in a similar way as $z_{ε}^{1}$ . Then, one can check that there exists $c > 0$ such that (see Proposition 3.2) $\begin{matrix} {‖ y_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} + {‖ y_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ \frac{c}{| log (ε) |} . \end{matrix}$ In other hand, we have (see Theorem 3.1) $\begin{matrix} ‖ z_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + ‖ z_{ε} ‖_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} = O (\frac{1}{| log (ε) |}) . \end{matrix}$ Therefore, one can deduce that $\begin{matrix} | T^{2} (ε) | ⩽ \frac{c}{| log (ε) |^{2}} = o (\frac{1}{| log (ε) |}) . \end{matrix}$

Estimate of $T^{3} (ε)$ : Denoting by $(ϑ_{ε}^{j}, γ_{ε}^{j})$ the solution to the following Stokes type equations $\begin{array}{rcl} \{\begin{matrix} - ν Δ ϑ_{ε}^{j} + \nabla γ_{ε}^{j} = 0 & in Ω_{z, ε}, \\ div ϑ_{ε}^{j} = 0 & in Ω_{z, ε}, \\ ϑ_{ε}^{j} = 0 & on Γ, \\ ϑ_{ε}^{j} = - e_{j} & on \partial O_{z, ε} . \end{matrix} \end{array}$ Setting $y_{ε}^{2} (x, t) = \sum_{j = 1}^{2} ϑ_{ε}^{j} (x) v_{0}^{j} (z, t)$ where $v_{0}^{j}$ is the jth component of $v_{0}$ . Then, the term $T^{3} (ε)$ reads $\begin{array}{rcl} T^{3} (ε) & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (z_{ε}, s_{ε}) (x, t) n y_{ε}^{2} d s d t \\ = & \int_{0}^{T} \int_{Ω_{z, ε}} \frac{\partial z_{ε}}{\partial t} y_{ε}^{2} d s d t + ν \int_{0}^{T} \int_{Ω_{z, ε}} \nabla z_{ε} \nabla y_{ε}^{2} d x d t \\ + \int_{0}^{T} \int_{Ω_{z, ε}} ([w_{0} + \frac{1}{log (ε)} W] \cdot \nabla) z_{ε} \cdot y_{ε}^{2} d x d t \\ + \int_{0}^{T} \int_{Ω_{z, ε}} (z_{ε} \cdot \nabla) [w_{0} + \frac{1}{log (ε)} W] \cdot y_{ε}^{2} d x d t \\ + \int_{0}^{T} \int_{Ω_{z, ε}} (z_{ε} \cdot \nabla) z_{ε} \cdot y_{ε}^{2} d x d t - \int_{0}^{T} \int_{Ω_{z, ε}} G_{ε} \cdot y_{ε}^{2} d x d t . \end{array}$ Since $v_{0}$ is a smooth function, by adapting the proof of Proposition 3.6, one can derive $\begin{matrix} {‖ y_{ε}^{2} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} + {‖ y_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} = O (\frac{1}{| log (ε) |}) . \end{matrix}$ Similarly to the term $T^{2} (ε)$ , one can deduce $\begin{matrix} | T^{3} (ε) | = o (\frac{1}{| log (ε) |}) . \end{matrix}$

Estimate of $T^{4} (ε)$ : From the definition (3.1) and the fact that $y_{ε}^{1} = - v_{0} (x, t) + v_{0} (z, t)$ on $\partial O_{z, ε}$ , one can deduce $\begin{array}{rcl} T^{4} (ε) & = & \frac{1}{log (ε)} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (W, Q) (x, t) n [v_{0} (z, t) - v (x, t)] d s d t \\ - \frac{1}{log (ε)} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (W, Q) (x, t) n v_{0} (z, t) d s d t \\ = & \frac{4 π ν}{log (ε)} \sum_{j = 1}^{2} [\int_{0}^{T} w_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n y_{ε}^{1} d s d t \\ - \int_{0}^{T} w_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n v_{0} (z, t) d s d t] . \end{array}$ Since $z \notin Ω_{z, ε}$ , $(E^{j}, Π^{j})$ satisfies $\begin{matrix} (4.12) & - ν Δ_{x} E^{j} (x - z) + \nabla Π_{x}^{j} (x - z) = 0, \forall x \in Ω_{z, ε} . \end{matrix}$ Hence, using the fact that $y_{ε}^{1} = 0$ on $Σ \times (0, T)$ , one can deduce $\begin{array}{l} \int_{0}^{T} w_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n y_{ε}^{1} d s d t \\ (4.13) & = ν \int_{0}^{T} w_{0}^{j} (z, t) \int_{Ω_{z, ε}} \nabla_{x} E^{j} (x - z) . \nabla y_{ε}^{1} (x, t) d x d t . \end{array}$ From the definition of $E^{j}$ , one can check that $\begin{matrix} | \nabla E^{j} (y) | = O (1 / r) . \end{matrix}$ Then, exploiting the relation (3.36) and the polar coordinate system, it follows $\begin{array}{rcl} {‖ \nabla E^{j} (x - z) ‖}_{L^{2} (Ω_{z, ε})} ⩽ c {‖ \nabla E^{j} (y) ‖}_{C (0, r ε, R)} = O (\sqrt{| log (ε) |}) . \end{array}$ Recall that $\begin{matrix} {‖ \nabla y_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} = O (\frac{1}{| log (ε) |}) . \end{matrix}$ Then, one can check that there exists $c > 0$ (independent of ε) such that $\begin{array}{l} | \int_{0}^{T} w_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n y_{ε}^{1} d s d t | \\ ⩽ {‖ w_{0}^{j} (z, t) ‖}_{L^{2} (0, T)} {‖ \nabla E^{j} (x - z) ‖}_{L^{2} (Ω_{z, ε})} {‖ \nabla y_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} \\ (4.14) & ⩽ \frac{c}{\sqrt{| log (ε) |}} \end{array}$ Consequently, the first part of $T^{4} (ε)$ admits the estimate $\begin{matrix} \frac{4 π ν}{log (ε)} \sum_{j = 1}^{2} \int_{0}^{T} w_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n y_{ε}^{1} d s d t = o (\frac{1}{| log (ε) |}), \end{matrix}$ which implies $\begin{array}{rcl} T^{4} (ε) & = & - \frac{4 π ν}{log (ε)} \sum_{j = 1}^{2} \int_{0}^{T} w_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n v_{0} (z, t) d s d t + o (\frac{1}{| log (ε) |}) . \end{array}$ To examine the last integral term, we use the fact that $div (σ (E^{j}, Π^{j}) (x - z)) = δ_{z} e_{j}$ in $O_{z, ε}$ , which leads to $\begin{array}{rcl} \int_{\partial O_{z, ε}} σ (E^{j}, Π^{j}) (x, t) n v_{0} (z, t) d s = v_{0}^{j} (z, t), j = 1, 2 . \end{array}$

Then, the last estimate becomes $\begin{matrix} (4.15) & T^{4} (ε) = - \frac{4 π ν}{log (ε)} \int_{0}^{T} w_{0} (z, t) \cdot v_{0} (z, t) d t + o (\frac{1}{| log (ε) |}) . \end{matrix}$

Consequently, all shape functions j satisfying the assumption $(A)$ admit the asymptotic expansion $\begin{matrix} j (Ω ∖ \overline{O_{z, ε}}) = j (Ω) + \frac{1}{| log (ε) |} [4 π ν \int_{0}^{T} w_{0} (z, t) \cdot v_{0} (z, t) d t + δ Φ (z)] + o (\frac{1}{| log (ε) |}), \end{matrix}$ for all small $ε > 0$ . □

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Topological asymptotic expansion for the full Navier–Stokes equations

Abstract

Keywords

1. Introduction

2. Formulation of the problem

2.1. Weak formulation and well-posedness

2.3. The purpose of this work

Lemma 3.3 (See [6], Theorems 8.2.3 and 8.2.9).

3.2. Estimate of the term z ε 2

Lemma 3.7 (See [13], Lemma C.3).

Lemma 3.8 (See [13], Lemma 4.1).

Proof. A detailed proof of this estimate is given in ([13], Appendix C, pages 360–363). □ 3.2.2. Proof of Proposition 3.6

3.3. Estimate of the term z ε 3

Lemma 3.10 (see [17], Lemma IX.1.1, p. 588).

3.3.2. Proof of Proposition 3.9

4. Sensitivity analysis framework

References

3.2. Estimate of the term $z_{ε}^{2}$

Proof.
A detailed proof of this estimate is given in ([13], Appendix C, pages 360–363). □
3.2.2. Proof of Proposition 3.6

3.3. Estimate of the term $z_{ε}^{3}$