Topological sensitivity analysis for the 3D nonlinear Navier

Abstract

This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.

Keywords

Topological asymptotic expansion 3D Navier–Stokes equations analysis of sensitivity topological gradient based methods fluid mechanics

1. Introduction

Let $T > 0$ and $D$ be a bounded domain of $R^{3}$ with smooth boundary $Γ = \partial D$ (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 $D \times (0, T)$ is governed by the non-stationary Navier–Stokes equations. Then, for a given source term F and an initial data $u_{I}$ , the velocity field u and the pressure field p satisfy the following system $\begin{matrix} (1.1) & \{\begin{matrix} \frac{\partial u}{\partial t} - ν Δ u + N (u) + \nabla p = F & in D \times (0, T), \\ div u = 0 & in D \times (0, T), \\ u = 0 & on Γ \times (0, T), \\ u (\cdot, 0) = u_{I} & in D \end{matrix} \end{matrix}$ where $ν > 0$ is the kinematic viscosity coefficient of the fluid and $N (u) = (u \cdot \nabla) u$ denotes the nonlinear term, which is defined as $\begin{matrix} N (u) = [\sum_{j = 1}^{3} u_{j} \partial / \partial x_{j}] u, \end{matrix}$ with $u_{j}$ is the $j^{th}$ component of the velocity field u and $\partial / \partial x_{j}$ is the partial derivative with respect the $j^{th}$ coordinate $x_{j}$ .

Theory and methods of shape optimization have wide range of applications in various areas of fluid mechanics such as in hydrodynamic, aerodynamic, biology and cardiology. Many shape optimization approaches have been achieved and applied for solving different engineering design problems including namely optimal design of minimum drag bodies [23,32], diffuser [9], heat exchangers [12], valves [27],and airfoils [11]. However, the majority of the classical methods dealing with optimal design of fluid flow domains are limited to determine the optimal shape of an existing boundary.

To overcome this drawback, different topology optimization based methods have been performed during the past few decades. Contrary to the classical shape optimization methods, where only the boundary of the design domain is optimized and the topology of the structure is kept unchanged, topology optimization based approaches permit the creation of new holes during the optimization procedure. In the context of fluid mechanics, various topology optimization algorithms have been carried out for solving engineering applications for fluidic devices in Stokes flow [8], Darcy–Stokes flows [17], Navier–Stokes flows [16], Stokes–Brinkmann flows [33] and non-Newtonian flows [31].

Throughout the two last decades, topological gradient-based methods [1,2,4,6,7,15,19–21,30] provide important advances in the field of shape and topology optimization techniques. Such kind of optimization methods are based on a topological sensitivity analysis of a given design function $S$ with respect to the creation of a small hole inside the computational 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 $z \in D$ . The topological gradient method leads to develop an asymptotic expansion of the form $\begin{matrix} S (D ∖ \overline{O_{z, ε}}) = S (D) + ψ (ε) δ S (z) + o (ψ (ε)) \end{matrix}$ where

$ψ (ε)$ is a positive scalar function going to zero with ε. It represents the asymptotic behavior of the variation $S (D ∖ \overline{O_{z, ε}}) - S (D)$ with respect to the geometric perturbation size.

$δ S$ is called the topological sensitivity function or the topological gradient. It represents the leading term of the design function variation with respect to the insertion of the obstacle $O_{z, ε}$ . It plays a crucial role in the topology optimization procedure. Indeed, in order to minimize the design function $O_{z, ε} \mapsto S (D ∖ \overline{O_{z, ε}})$ , the best location to insert a small geometric perturbation inside the domain $D$ is where $z \mapsto δ S (z)$ is most negative. This is due to the fact that if $δ S (z) < 0$ , we have $S (D ∖ \overline{O_{z, ε}}) < S (D)$ .

In fluid mechanics, topological gradient-based methods have been the subject of several theoretical and numerical investigations. The topological asymptotic expansion has been derived for different model problems and some design functions in the two and three dimensional cases. Various topological optimization algorithms have been constructed and exploited for solving valuable applications in fluid mechanics, such as the optimization of the injectors placement in a lake for maximizing the oxygenation process [18,22], the detection of flaws location in molds filling process [6], optimal shape design for Stokes flow in a cavity [20], control of mechanical aeration process [2], geometric control problem in Stokes flow [1], … etc.

The goal of this work is to extend the mathematical framework of the topological sensitivity analysis method to the full nonlinear Navier–Stokes operator. The motivation of this study is to provide efficient tools for solving a wide variety of topology optimization problems 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.

Two main issues will be addressed in this study. The first one concerns the velocity field perturbation caused by the insertion of a small obstacle inside the fluid flow domain $D$ . We will derive an estimate describing the behavior of the velocity field variation with respect to the obstacle size ε. In this part, we will prove that the leading term of the velocity field variation admits an explicit formula involving the Green function of the steady state Stokes operator and the non perturbed Navier–Stokes problem solution.

The second one is devoted to the development of a topological sensitivity analysis for the full non-linear Navier–Stokes operator. We will derive an asymptotic formula modeling the design function variation with respect to the creation of a small geometric perturbation inside the fluid flow domain. The performed mathematical analysis is general and can be applied for a large class of design functions and arbitrarily shaped perturbations.

The remainder of this paper is outlined as follows. Section 2 is concerned with the problem statement. In Section 3, we present the weak formulation of the Navier–Stokes equations and we recall some existing results concerning existence, uniqueness and regularity of the weak solution. The main results of this study are illustrated in Section 4. We examine the behavior of the perturbed velocity field with respect to the obstacle size in Section 5. Section 6 is concerned with a topological asymptotic formula valid for a large class of shape functions.

2. Problem statement

Let $O_{z, ε}$ be a small obstacle inserted in the fluid flow domain Ω around an arbitrary point $z \in D$ . 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 $D$ , $ε > 0$ is a small parameter represents the obstacle size and $O$ is a given bounded domain of $R^{3}$ 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 $D_{z, ε} = D ∖ \overline{O_{z, ε}}$ , the velocity field $u_{ε}$ and the pressure field $p_{ε}$ solve the following Navier–Stokes system $\begin{matrix} (2.1) & \{\begin{matrix} \frac{\partial u_{ε}}{\partial t} - ν Δ u_{ε} + N (u_{ε}) + \nabla p_{ε} = F & in D_{z, ε} \times (0, T), \\ div u_{ε} = 0 & in D_{z, ε} \times (0, T), \\ u_{ε} = 0 & on Γ \times (0, T), \\ u_{ε} = 0 & on \partial O_{z, ε} \times (0, T), \\ u_{ε} (\cdot, 0) = u_{I} & in D_{z, ε} . \end{matrix} \end{matrix}$

Let $S$ be a shape function defined as: $\begin{matrix} (2.2) & S (D ∖ \overline{O_{z, ε}}) = \int_{0}^{T} F_{ε} (u_{ε} (\cdot, t)) d t, \end{matrix}$ with $F_{ε} : H^{1} (D ∖ \overline{O_{z, ε}}) ⟶ R$ is a given cost function and $u_{ε}$ is the velocity field, solution to the Navier–Stokes problem (2.1) in the perturbed fluid flow domain $D ∖ \overline{O_{z, ε}}$ . Note that if $ε = 0$ (absence of obstacles), we have $D_{0, z} = D$ and the shape function $S$ is defined as $\begin{matrix} (2.3) & S (D) = \int_{0}^{T} F_{0} (u_{0} (\cdot, t)) d t, \end{matrix}$ where $u_{0}$ is the solution to the following unperturbed Navier–Stokes equations $\begin{matrix} (2.4) & \{\begin{matrix} \frac{\partial u_{0}}{\partial t} - ν Δ u_{0} + N (u_{0}) + \nabla p_{0} = F & in D \times (0, T), \\ div u_{0} = 0 & in D \times (0, T), \\ u_{0} = 0 & on Γ \times (0, T), \\ u_{0} (\cdot, 0) = u_{I} & in D . \end{matrix} \end{matrix}$

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 and arbitrarily shaped geometric perturbation $O_{z, ε}$ . As mentioned in the introduction, we will evaluate the behavior of $S$ with respect to the perturbation size ε and establish an asymptotic formula on the form $\begin{matrix} S (D ∖ \overline{O_{z, ε}}) = S (D) + ψ (ε) δ S (z) + o (ψ (ε)) \end{matrix}$ In this expansion, the term $ψ (ε)$ describes the behavior of $S$ with respect to ε. As it will be shown later, the expression of $ψ (ε)$ will be dictated by the behavior of the perturbed velocity field with respect to ε. The term $δ S$ (called the topological gradient) represents the leading term of the shape function variation $S (D ∖ \overline{O_{z, ε}}) - S (D)$ . Its expression depends on the considered model equation as well as the chosen cost function $F_{ε}$ .

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 $u_{ε} - u_{0}$ will be constructed with the help of the fundamental solution of the steady state Stokes operator. The main results of this study will be presented in Section 4. The developed mathematical framework will be detailed in Section 5 and Section 6. Next, we will recall some existing results concerning the weak solutions of the full Navier–Stokes equations.

3. Weak solutions of the full Navier–Stokes equations

In the literature there are different definitions of the Navier–Stokes weak solutions. According to the construction procedure of weak solutions proposed by J. Leray (see [24,25] or [26]) and adapted by R. Temam ([34], see chapter III, Section 3), the variational formulation of the full Navier–Stokes problem can be written as follow: $\begin{array}{l} (3.1) & Given F \in L^{2} (0, T; V^{'}) and u_{I} \in H, find u \in L^{2} (0, T; V) solution to \\ (3.2) & \frac{d}{d t} \int_{D} u \cdot ξ d x + ν \int_{D} \nabla u : \nabla ξ d x + T (u, u, ξ) = \int_{D} F \cdot ξ d x, \forall ξ \in V \\ (3.3) & u (\cdot, 0) = u_{I} in D, \end{array}$ where

$T$ is the trilinear form defined by $\begin{matrix} T (u, v, ξ) = \int_{Ω} (u \cdot \nabla) v \cdot ξ d x, \forall u, v, ξ \in H^{1} (D) . \end{matrix}$

$V$ and $H$ are the functional spaces $\begin{array}{l} (3.4) & \begin{aligned} V = {v \in H^{1} (D), div (v) = 0, v = 0 on Γ} \\ H = {v \in L^{2} (D), div (v) = 0, v \cdot n = 0 on Γ}, \end{aligned} \end{array}$ with $H^{1} (D) = {(H^{1} (D))}^{3}$ and $L^{2} (D) = {(L^{2} (D))}^{3}$ are the usual Hilbert spaces of vector-valued functions.

This weak formulation has been exploited by several researchers for describing the Navier–Stokes weak solutions; one can see for example J.L. Lions [28] or G. Galdi [14]. Next, we briefly recall some basic results about existence, uniqueness and regularity of the full Navier–Stokes weak solutions.

Existence: It is proved by J.L. Lions [28] (see also R. Temam [34]) that the problem (3.1)–(3.3) admits at least one solution $u \in L^{\infty} (0, T; H)$ . According to R. Temam [34] (see Theorem 3.1, p. 226), the constructed solution is obtained using an approximate procedure defined with the help of the Galerkin method. The convergence of the constructed sequence is justified by a compactness theorem. The result is valid for each open Lipschitzian set $D \subset R^{d}$ , $d ⩽ 4$ .

About regularity and uniqueness: Unlike the two dimensional case which is discussed by Temam in [34] (see Theorem 3.2, p. 235) also by J.L. Lions and G. Prodi in [29]), the uniqueness result in the three dimensional case require more regularity on the weak solution u. For instance, one can cite the uniqueness result established by Temam [34] (see Theorem 3.4, p. 238–239) under the assumptions $u \in L^{2} (0, T; V) \cap L^{\infty} (0, T; H)$ and $u \in L^{8} (0, T; L^{4} (D))$ . The presented proof is based on the following lemma.

Lemma 3.1 ([34], p. 237).

For any open set $D \subset R^{3}$ , we have $\begin{matrix} (3.5) & ‖ v ‖_{L^{4} (D)} ⩽ 2^{1 / 2} ‖ v ‖_{L^{2} (D)}^{1 / 4} ‖ \nabla v ‖_{L^{2} (D)}^{3 / 4}, \forall v \in H_{0}^{1} (D) . \end{matrix}$

Remark 3.2.
There are many similar uniqueness results which can be proved by assuming some other properties of regularity. For example, J.L. Lions proved that there is uniqueness in any dimension d if u belongs to $L^{s} (0, T; L^{r} (D))$ with $2 / s + d / r ⩽ 1$ if $D$ is bounded and $2 / s + d / r = 1$ if $D$ is unbounded.

Regularity results: It is proved in ([34], see Theorem 3.8, p. 246) that if $D$ is a bounded set of class $C^{2}$ , $F \in L^{\infty} (0, T; H)$ , $F^{'} \in L^{1} (0, T; H)$ and $u_{I} \in H^{2} (D) \cup V$ then the unique solution u satisfies $u \in L^{\infty} (0, T; H^{2} (D))$ .
Remark 3.3.
According to ([34], Remark 3.8, p. 247), it is important to signal that one can obtain a more regular solution to the full Navier–Stokes equations if the given data are smooth. For example, if $D$ is of class $C^{\infty}$ , $u_{I} \in C^{\infty} (D)$ and $F \in C^{\infty} (\overline{D \times (0, T)})$ , then the solution u belongs to $C^{\infty} (\overline{D \times (0, T)})$ .
In this study, aiming 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 additional assumptions to overcome difficulties caused by the lack of regularity in data. Assumptions 3.4.

The initial velocity is zero (i.e. $u_{I} = 0$ in $D$ ) and the source term F is sufficiently smooth in such a way that the solution u of the Navier–Stokes belongs to $C^{1} (D \times [0, T])$ .

There exists $δ > 0$ such that $‖ \nabla u_{0} ‖_{L^{\infty} (0, T; L^{2} (D))} ⩽ δ < ν / κ (D)$ , where $κ (D) = \frac{2 \sqrt{2}}{3} meas {(D)}^{1 / 6}$ .

Remark 3.5.

The assumption (i) is not restrictive since the source term F represents in general the gravitational force which is a smooth function.

The assumption (ii) have been used in [13] (see also [4]) as a sufficient uniqueness condition for the Navier–Stokes problem.

4. Main results

In this section, we present the main results of this study. Our aim is to extend the topological sensitivity analysis method to the full nonlinear Navier–Stokes operator. In the first part of this section, we will examine the obstacle influence and we calculate an estimate of the perturbed velocity field $u_{ε}$ . In Section 4.2, we will derive a topological asymptotic expansion describing the variation of the shape functional $S$ with respect to the insertion of a small obstacle inside the fluid flow domain. The proposed mathematical analysis is general and can be applied for others nonlinear problems.

4.1. Asymptotic behavior of the perturbed velocity

In this section, we evaluate the perturbation caused by the presence of a small obstacle $O_{z, ε} = z$ around an arbitrary point $z \in D$ . We will prove that the leading term $U_{z, ε}$ of the velocity field variation $u_{ε} - u_{0}$ is given as $\begin{matrix} (4.1) & U_{z, ε} (x, t) = \sum_{j = 1}^{3} u_{0}^{j} (z, t) Ψ^{j} (\frac{x - z}{ε}) \end{matrix}$ Its associated pressure field $P_{z, ε}$ is defined as $\begin{matrix} (4.2) & P_{z, ε} (x, t) = \frac{1}{ε} \sum_{j = 1}^{3} u_{0}^{j} (z, t) Π^{j} (\frac{x - z}{ε}), \end{matrix}$ where $(Ψ^{j}, Π^{j})$ is the solution to the following exterior Stokes problem $\begin{matrix} (4.3) & \{\begin{matrix} - ν Δ Ψ^{j} + \nabla Π^{j} = 0 & in R^{3} ∖ \overline{O}, \\ div Ψ^{j} = 0 & in R^{3} ∖ \overline{O}, \\ Ψ^{j} ⟶ 0 & at \infty, \\ Ψ^{j} = - e_{j} & on \partial O, \end{matrix} \end{matrix}$ with $e_{j}$ is the $j^{th}$ unit vector of the canonical basis of $R^{3}$ .

As mentioned in [22] (see also [20]), problem (4.3) can be treated with the help of a single layer potential and the solution $(Ψ^{j}, Π^{j})$ can be expressed as: $\begin{array}{l} Ψ^{j} (y) = \int_{\partial O} E (x - y) η^{j} (x) d s (x), \\ Π^{j} (y) = \int_{\partial O} Θ (x - y) \cdot η^{j} (x) d s (x), \end{array}$ with $η^{j} \in H^{- 1 / 2} (\partial O)$ solves the integral equation $\begin{matrix} (4.4) & \int_{\partial O} E (x - y) η^{j} (x) d s (x) = - e_{j}, y \in \partial O . \end{matrix}$ Here $(E, Θ)$ is the fundamental solution of the steady-state Stokes system in $R^{3}$ $\begin{matrix} (4.5) & E (y) = \frac{1}{8 π ν | y |} (I + e_{r} e_{r}^{T}), Θ (y) = \frac{e_{r}}{4 π | y |^{2}}, \forall y \in R^{3}, \end{matrix}$ where $| y |$ is the Euclidean norm of the vector y, $I$ is the $3 \times 3$ identity matrix, $r = | y |$ , $e_{r} = \frac{y}{r}$ and $e_{r}^{T}$ is the transposed vector of $e_{r}$ .

In the following Theorem, we present an estimate describing the behavior of the velocity field perturbation with respect to the obstacle size ε.

Theorem 4.1.
Let $O_{z, ε} = z + ε O$ be an obstacle of small size $0 < ε ≪ 1$ , immersed inside the fluid flow domain $D$ . Then, under the Assumptions 3.4 , there exists a constant $c > 0$ , independent of ε, such that $\begin{array}{rcl} ‖ u_{ε} - u_{0} - U_{z, ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + ‖ u_{ε} - u_{0} - U_{z, ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε . \end{array}$

Exploiting the systems (2.1), (2.4) and (4.3), one can easily show that the resulting fields $\begin{matrix} (4.6) & R_{ε} = u_{ε} - u_{0} - U_{z, ε} and s_{ε} = p_{ε} - p_{0} - P_{z, ε}, \end{matrix}$ solve the following Navier–Stokes type problem $\begin{matrix} (4.7) & \{\begin{matrix} \frac{\partial R_{ε}}{\partial t} - ν Δ R_{ε} + N (R_{ε}) + ⟨ D N (R_{ε}), h_{ε} ⟩ + \nabla s_{ε} = F_{ε} & in D_{z, ε} \times (0, T), \\ div R_{ε} = 0 & in D_{z, ε} \times (0, T), \\ R_{ε} = - U_{z, ε} & on Γ \times (0, T), \\ R_{ε} = - u_{0} + u_{0} (z, \cdot) & on \partial O_{z, ε} \times (0, T), \\ R_{ε} (\cdot, 0) = 0 & in D_{z, ε} \end{matrix} \end{matrix}$ where $D N (R_{ε})$ is the differential of the map $w \mapsto N (w)$ at the point $R_{ε}$ . It is applied to the field $h_{ε} = u_{0} + U_{z, ε}$ as $\begin{matrix} ⟨ D N (R_{ε}), h_{ε} ⟩ = [(u_{0} + U_{z, ε}) \cdot \nabla] R_{ε} + (R_{ε} \cdot \nabla) [u_{0} + U_{z, ε}] . \end{matrix}$ The source term $F_{ε}$ is defined as $\begin{array}{rcl} (4.8) & F_{ε} = - \frac{\partial U_{z, ε}}{\partial t} - [(u_{0} \cdot \nabla) U_{z, ε} + (U_{z, ε} \cdot \nabla) u_{0} + (U_{z, ε} \cdot \nabla) U_{z, ε}] . \end{array}$ Indeed, it is important to signal that the main difficulties in the proof of Theorem 4.1 arise from the terms $N (R_{ε})$ , $⟨ D N (R_{ε}), h_{ε} ⟩$ and $F_{ε}$ which depend on a nontrivial way on the parameter ε. Thus, to deal with such difficulties we will proceed by decomposing the solution $(R_{ε}, s_{ε})$ into three main parts and we will estimate each part separately. The detailed proof of Theorem 4.1 will be illustrated in Section 5.2 after presenting some preliminary lemmas in Section 5.1.
4.2. Topological asymptotic expansion

In this paragraph, we derive a topological asymptotic expansion describing the variation of a shape functional $S$ with respect to the presence of a small obstacle $O_{z, ε}$ inside the fluid flow domain. The developed mathematical framework is general and can be applied for a large class of cost functions and arbitrary-shaped obstacles. More precisely, the derived topological sensitivity analysis is valid for all functions $F_{ε}$ verifying the following assumptions:

Assumptions 4.2.

The cost function $F_{0}$ is differentiable on $H^{1} (D)$ , its derivative will be denoted by $D F_{0} (w)$ .

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

There exists a scalar function $δ F : D ⟶ R$ defined at each point $z \in D$ , such that for all small $ε ⩾ 0$ , we have $\begin{matrix} \int_{0}^{T} [F_{ε} (u_{ε} (\cdot, t)) - F_{0} (u_{0} (\cdot, t))] d t = \int_{0}^{T} D F_{0} (u_{0} (\cdot, t)) (u_{ε} (\cdot, t) - u_{0} (\cdot, t)) d t + ε δ F (z) + o (ε), \end{matrix}$ where $u_{ε}$ is extended by zero inside the domain $O_{z, ε}$ .

Posing $G (u) = {(\partial u_{i} / \partial x_{j})}_{1 ⩽ i, j ⩽ 3}$ . The adjoint problem: find $(v_{0}, q_{0})$ solution to $\begin{matrix} (4.9) & \{\begin{matrix} - \frac{\partial v_{0}}{\partial t} - ν Δ v_{0} + G (u_{0}) v_{0} -^{t} G (v_{0}) u_{0} + \nabla q_{0} = - D F_{0} (u_{0}) & in D \times (0, T), \\ div v_{0} = 0 & in D \times (0, T), \\ v_{0} = 0 & on Γ \times (0, T), \\ v_{0} (\cdot, T) = 0 & in D, \end{matrix} \end{matrix}$ admits a unique solution.

In order to determine the leading term of the shape function variation $S (D ∖ \overline{O_{z, ε}}) - S (D)$ with respect to the obstacle size, we introduce the tensor $T_{O}$ , which depends on the obstacle shape $O$ as follows $\begin{matrix} {T_{O}}_{i j} = - \int_{\partial O} η_{i}^{j} (y) d s (y), 1 ⩽ i, j ⩽ 3, \end{matrix}$ where $η^{j} = {(η_{i}^{j})}_{1 ⩽ i ⩽ 3}$ is a solution to the boundary integral equation (4.4).

The obtained asymptotic expansion, describing the variation of the shape function $S$ , with respect to the location z and the size ε of the inserted obstacle $O_{z, ε}$ , is illustrated by the following theorem.
Theorem 4.3.
Let $O_{z, ε} = z + ε O$ be an obstacle of small size $0 < ε ≪ 1$ inserted inside the fluid flow domain $D$ and let $S$ be a shape functional of the form $S (D ∖ \overline{O_{z, ε}}) = \int_{0}^{T} F_{ε} (u_{ε} (\cdot, t)) d t$ .

If the cost function $F_{ε}$ verifies the Assumptions 4.2 , then the shape functional $S$ admits the following asymptotic expansion $\begin{array}{rcl} (4.10) & S (D ∖ \overline{O_{z, ε}}) = S (D) + ε δ S (z) + o (ε), \end{array}$ where $δ S$ is the topological gradient, defined at any point $x \in D$ as $\begin{matrix} δ S (x) = \int_{0}^{T} u_{0} (x, t) \cdot T_{O} v_{0} (x, t) d t + δ F (x) . \end{matrix}$
Remark 4.4.
In the previous formula, the term $δ F (x)$ depends on the analytic expression of the considered cost function $F_{ε}$ .

It is important to signal here that the tensor $T_{O}$ can be calculated explicitly in the case of spherical-shaped obstacles. More precisely, when the fixed domain $O$ coincides with the unit ball $B (0, 1)$ , the density $η_{j}$ , $j = 1, 3$ admits the explicit expression $\begin{matrix} η_{j} (y) = - \frac{3 ν}{2} e_{j}, \forall y \in \partial O, \end{matrix}$ which implies, in this particular case, that the tensor $T_{O}$ is given as $\begin{matrix} {(T_{O})}_{i, j} = 6 π ν δ_{i, j}, 1 ⩽ i, j ⩽ 3, \end{matrix}$ where $δ_{i, j}$ denotes the usual Kronecker symbol.
Corollary 4.5.
Let $O_{z, ε} = z + ε B (0, 1)$ be a spherical-shaped obstacle of small size $0 < ε ≪ 1$ inserted in the fluid flow domain $D$ . Then, under the same hypotheses of Theorem 4.3 , the shape functional $S$ satisfies the following explicit asymptotic expansion $\begin{matrix} S (D ∖ \overline{O_{z, ε}}) = S (D) + ε [6 π ν \int_{0}^{T} u_{0} (z, t) \cdot v_{0} (z, t) d t + δ F (z)] + o (ε) . \end{matrix}$

The detailed proof of Theorem 4.3 will be presented in Section 6.2 after establishing some preliminary results.
5. Estimate of the perturbed velocity

This section is devoted to the proof of Theorem 4.1, which describes the asymptotic behavior of the perturbed velocity field with respect to the insertion of a small obstacle. To this end, we start our analysis by recalling some preliminary lemmas. The proof of Theorem 4.1 will be presented in Section 5.2.

5.1. Preliminary lemmas

We begin this paragraph by recalling some preliminary estimates related to the perturbed steady-state Stokes problem.

Lemma 5.1.
See [ 22 ] or [ 20 ]. Let $ε > 0$ and $φ \in H^{1 / 2} (\partial D)$ such that $\int_{\partial D} φ \cdot n d s = 0$ . Consider $(w_{ε}, ξ_{ε})$ be the solution to $\begin{matrix} (5.1) & \{\begin{matrix} - ν Δ w_{ε} + \nabla ξ_{ε} = 0 & in D_{z, ε} \\ div w_{ε} = 0 & in D_{z, ε} \\ w_{ε} = φ & on \partial D \\ w_{ε} = 0 & on \partial O_{z, ε} . \end{matrix} \end{matrix}$ Then, there exist $c > 0$ (independent of φ and ε) and $ε_{1} > 0$ such that for all $0 < ε < ε_{1}$ , we have $\begin{matrix} ‖ w_{ε} ‖_{1, D_{z, ε}} ⩽ c ‖ φ ‖_{1 / 2, \partial D} . \end{matrix}$
Lemma 5.2.
See [ 22 ] or [ 20 ]. Let $ϕ \in H^{1 / 2} (\partial O)$ such that $\int_{\partial O} ϕ \cdot n d s = 0$ . Let $(w, ξ) \in W^{1} (R^{3} ∖ \overline{O} \times L^{2} (R^{3} ∖ \overline{O}))$ be the solution to the exterior Stokes problem $\begin{matrix} \{\begin{matrix} - ν Δ w + \nabla ξ = 0 & in R^{3} ∖ \overline{O}, \\ div w = 0 & in R^{3} ∖ \overline{O}, \\ w ⟶ 0 & at \infty, \\ w = ϕ & on \partial O . \end{matrix} \end{matrix}$ Then, there exists $c > 0$ , independent of ε, such that $\begin{array}{l} ‖ w ‖_{L^{2} ((D_{z, ε} - z) / ε)} ⩽ c ε^{- 1 / 2} ‖ ϕ ‖_{1 / 2, \partial O}, \\ ‖ w ‖_{L^{4} ((D_{z, ε} - z) / ε)} ⩽ c ‖ ϕ ‖_{1 / 2, \partial O}, \\ ‖ \nabla w ‖_{L^{2} ((D_{z, ε} - z) / ε)} ⩽ c ‖ ϕ ‖_{1 / 2, \partial O}, \\ ‖ \nabla w ‖_{L^{2} ((D_{z, R} - z) / ε)} ⩽ c ε^{1 / 2} ‖ ϕ ‖_{1 / 2, \partial O}, \end{array}$ where $D_{z, ε} = D ∖ \overline{O_{z, ε}}$ and $D_{z, R} = D ∖ \overline{B (z, R)}$ , with $R > 0$ such that $\overline{B (z, R)} \subset D$ .
Lemma 5.3.
Let $R > 0$ such that $\overline{O}$ is strictly included in $B (0, R)$ . For $φ \in L^{2} (0, T; H^{3 / 2} (\partial O))$ , $(U, ζ)$ denotes the solution to the following evolutionary Stokes 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 $\begin{matrix} (5.2) & ‖ U ‖_{H^{1} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ U ‖_{L^{2} (0, T; H^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ‖ φ ‖_{L^{2} (0, T; H^{3 / 2} (\partial O))} . \end{matrix}$
Proof.
For the proof and more details about this result, one can consult [10, Theorem 1.43]. More precisely, this estimate has been established for a general parabolic operator with the help of [10, Theorem 1.42(v)] and [5, Theorem II]. □

Let $ψ : D \times (0, T) ⟶ R$ be a function of class $C^{1}$ , with respect to the space variable, in a neighborhood of the point $(z, t)$ for each $t \in (0, T)$ . Applying Lemma 5.3 in the particular case where $\begin{matrix} φ (y, t) = ψ (z + ε y, t) - ψ (z, t) on \partial O \times (0, T), \end{matrix}$ one can deduce the following result.
Lemma 5.4.
Consider $R > 0$ such that $\overline{O}$ is strictly included in $B (0, R)$ . Let $(U_{ε}, ζ_{ε})$ be the solution to the following evolutionary Stokes problem $\begin{matrix} (5.3) & \{\begin{matrix} \frac{\partial U_{ε}}{\partial t} - ν Δ U_{ε} + \nabla ζ_{ε} = 0 & in (B (0, R) ∖ \overline{O}) \times (0, T), \\ div U_{ε} = 0 & in (B (0, R) ∖ \overline{O}) \times (0, T), \\ U_{ε} = 0 & on \partial B (0, R) \times (0, T), \\ U_{ε} = ψ (z + ε y, t) - ψ (z, t) & on \partial O \times (0, T), \\ U_{ε} (\cdot, 0) = 0 & in B (0, R) ∖ \overline{O} . \end{matrix} \end{matrix}$ Then, there exists $c > 0$ , independent of ε, such that $\begin{array}{l} (5.4) & {‖ \frac{\partial U_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ε, \\ (5.5) & ‖ U_{ε} ‖_{L^{\infty} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ε . \end{array}$
Proof.
Since $(x, t) \mapsto ψ (x, t)$ is of class $C^{1}$ (with respect to x) in a neighborhood of $(z, t)$ , then by a Taylor formula, one can obtain $\begin{matrix} (5.6) & ψ (z + ε y, t) - ψ (z, t) = ε D ψ (z + ε β (y), t) y, \end{matrix}$ where $D ψ (z + ε β (y), t)$ is the matrix defined by $\begin{matrix} {(D ψ (z + ε β (y), t))}_{i, j} = \frac{\partial ψ^{i}}{\partial x_{j}} (z + ε β_{i, j} (y), t), 1 ⩽ i, j ⩽ 3, \end{matrix}$ with $| β_{i, j} (y) | ⩽ | y | < R$ . Since ε is very small $0 < ε ≪ 1$ , the points $z + ε β_{i, j} (y)$ are in a neighborhood of z. Then, it follows that the maps $(y, t) \mapsto {(D w_{0} (z + ε β (y), t))}_{i, j}$ are differentiable and their differentials are uniformly bounded with respect to ε, which implies the estimate $\begin{matrix} (5.7) & {‖ ψ (z + ε y, t) - ψ (z, t) ‖}_{L^{2} (0, T; H^{3 / 2} (\partial O))} ⩽ c ε . \end{matrix}$ Due to Lemma 5.3, one can deduce $\begin{array}{l} (5.8) & {‖ \frac{\partial U_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} ⩽ c ε . \end{array}$ Exploiting the weak formulation of (5.3) using an adequate choice of the test function, one can deduce that $\begin{array}{l} ‖ U_{ε} ‖_{L^{\infty} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} + ‖ \nabla U_{ε} ‖_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} \\ ⩽ {‖ ψ (z + ε y, t) - ψ (z, t) ‖}_{L^{2} (0, T; H^{1 / 2} (\partial O))} ⩽ c ε . \end{array}$ □

In the last part of this section, we consider the trilinear form $T : H^{1} (D) \times H^{1} (D) \times H^{1} (D)$ , which is defined as $\begin{matrix} T (u, v, w) = \int_{D} (u \cdot \nabla) v \cdot w d x, \forall u, v, w \in H^{1} (D) . \end{matrix}$ Let $p, q \in N$ such that $1 / p + 1 / q = 1 / 2$ . By Hölder inequality, one can check that if $u \in L^{p} (D)$ and $w \in L^{q} (D)$ , we have $\begin{matrix} | T (u, v, w) | ⩽ ‖ u ‖_{L^{p} (D)} ‖ \nabla v ‖_{L^{2} (D)} ‖ u ‖_{L^{q} (D)} . \end{matrix}$ Due to the Sobolev embedding Theorem (i.e. $H^{1} (D) ↪ L^{4} (D)$ , see Adam’s [3]), when $p = q = 4$ we have $\begin{array}{rcl} (5.9) & | T (u, v, w) | ⩽ ‖ u ‖_{L^{4} (D)} ‖ \nabla v ‖_{L^{2} (D)} ‖ u ‖_{L^{4} (D)}, \forall u, v, w \in H^{1} (D) . \end{array}$ Let $u \in H^{1} (D)$ such that $div u = 0$ in $D$ . Then, using the Green formula one can derive the identity $\begin{array}{rcl} (5.10) & \int_{D} (u \cdot \nabla) v \cdot w d x = \int_{\partial D} (u \cdot n) (v \cdot w) d s - \int_{D} (u \cdot \nabla) w \cdot v d x \forall v, w \in H^{1} (D) \end{array}$ where n is the outward unit normal vector to the boundary $\partial D$ .
Lemma 5.5 (see [13], Lemma IX.1.1, p. 588).

Let $D \subset R^{3}$ be a bounded and locally Lipschitz domain. Then, the trilinear form $T$ is continuous in the space $H^{1} (D_{z, ε}) \times H^{1} (D_{z, ε}) \times H^{1} (D_{z, ε})$ and satisfies $\begin{array}{rcl} (5.11) & | T (u, v, w) | = | \int_{D_{z, ε}} (u \cdot \nabla) v \cdot w d x | ⩽ c (D) ‖ u ‖_{H^{1} (D_{z, ε})} ‖ \nabla v ‖_{L^{2} (D_{z, ε})} ‖ w ‖_{H^{1} (D_{z, ε})} . \end{array}$ Moreover, if $u = w = 0$ on $\partial D$ we have $\begin{array}{rcl} (5.12) & | \int_{D_{z, ε}} (u \cdot \nabla) v \cdot w d x | ⩽ κ (D) ‖ \nabla u ‖_{L^{2} (D_{z, ε})} ‖ \nabla v ‖_{L^{2} (D_{z, ε})} ‖ \nabla w ‖_{L^{2} (D_{z, ε})}, \end{array}$ with $κ (D) = \frac{2 \sqrt{2}}{3} meas {(D)}^{1 / 6}$ is the constant defined in Section 2 .

Based on the relation (5.10), one can deduce the following lemma, see [[13], Lemma IX.2.1, p. 591] for similar result.

Lemma 5.6.
Let $v, w \in H^{1} (D_{z, ε})$ and $u \in H^{1} (D_{z, ε})$ such that $div u = 0$ in $D_{z, ε}$ . Then, we have the following relations:
If $v \in H_{0}^{1} (D_{z, ε})$ or $u \cdot n = 0$ on the boundary $\partial D \cup \partial O_{z, ε}$ , it follows that $\begin{matrix} \int_{D_{z, ε}} (u \cdot \nabla) v \cdot v d x = 0 . \end{matrix}$

If $w \in H_{0}^{1} (D_{z, ε})$ , we get $\begin{matrix} \int_{D_{z, ε}} (u \cdot \nabla) v \cdot w d x = - \int_{D_{z, ε}} (u \cdot \nabla) w \cdot v d x . \end{matrix}$

5.2. Proof of Theorem 4.1

In order to overcome the difficulties coming from the nonlinear term and the non-homogeneous condition imposed on the variable boundary $\partial O_{z, ε}$ , we split the solution $(R_{ε}, s_{ε})$ of the auxiliary problem (4.7) into three parts as follows $\begin{matrix} R_{ε} = R_{ε}^{1} + R_{ε}^{2} + R_{ε}^{3} and s_{ε} = s_{ε}^{1} + s_{ε}^{2} + s_{ε}^{3}, \end{matrix}$ and we will estimate each term separately.

∙ The term $(R_{ε}^{1}, s_{ε}^{1})$ is concerned with the non-homogeneous boundary condition on $\partial O_{z, ε}$ . It satisfies the evolutionary linear Stokes system $\begin{matrix} (5.13) & \{\begin{matrix} \frac{\partial R_{ε}^{1}}{\partial t} - ν Δ R_{ε}^{1} + \nabla s_{ε}^{1} = 0 & in D_{z, ε} \times (0, T), \\ div R_{ε}^{1} = 0 & in D_{z, ε} \times (0, T), \\ R_{ε}^{1} = 0 & on Γ \times (0, T), \\ R_{ε}^{1} = - u_{0} (x, t) + u_{0} (z, t) & on \partial O_{z, ε} \times (0, T), \\ R_{ε}^{1} (\cdot, 0) = 0 & in D_{z, ε} . \end{matrix} \end{matrix}$ For this first component, the aim is to prove that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} {‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε . \end{matrix}$ To this end, we start by setting $V_{ε}^{u_{0}} (x, t) = U_{ε} (\frac{x - z}{ε}, t) \forall (x, t) \in D_{z, ε} \times (0, T)$ , with $U_{ε}$ is the velocity field defined in Lemma 5.4 when $ψ = u_{0}$ . Since $supp (U_{ε}) \subset \overline{B (0, R) ∖ O} \times [0, T]$ , one can deduce that the support of the function $V_{ε}^{u_{0}}$ is included in the set $(\overline{D_{z, ε}} \cap \overline{B (z, R ε)}) \times [0, T]$ . Extending $V_{ε}^{u_{0}}$ by zero in $D ∖ B (z, R ε) \times [0, T]$ and setting $\begin{matrix} \tilde{u_{ε}} = R_{ε}^{1} - V_{ε}^{u_{0}} . \end{matrix}$ Then, the weak formulation of the system (5.13), with choosing $\tilde{u_{ε}}$ as a test function, gives $\begin{matrix} (5.14) & \begin{matrix} \frac{1}{2} \int_{D_{z, ε}} {| R_{ε}^{1} (\cdot, t_{0}) |}^{2} d x + ν \int_{0}^{t_{0}} \int_{D_{z, ε}} {| \nabla R_{ε}^{1} |}^{2} d x d t \\ = ν \int_{0}^{t_{0}} \int_{D_{z, ε}} \nabla R_{ε}^{1} . \nabla V_{ε}^{u_{0}} d x d t + \int_{0}^{t_{0}} \int_{D_{z, ε}} \frac{\partial R_{ε}^{1}}{\partial t} V_{ε}^{u_{0}} d x d t, \end{matrix} \end{matrix}$ for almost $t_{0} \in [0, T]$ .

By the Cauchy–Schwartz inequality, the first term on the right-hand-side of (5.14) can be estimated as $\begin{matrix} (5.15) & | ν \int_{0}^{t_{0}} \int_{D_{z, ε}} \nabla R_{ε}^{1} . \nabla V_{ε}^{u_{0}} d x d t | ⩽ ν {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} {‖ \nabla V_{ε}^{u_{0}} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{matrix}$ Using integration by parts in time, the second term on the right-hand-side of (5.14) may rewritten as $\begin{array}{l} \int_{0}^{t_{0}} \int_{D_{z, ε}} \frac{\partial R_{ε}^{1}}{\partial t} V_{ε}^{u_{0}} d x d t = \int_{D_{z, ε}} R_{ε}^{1} (\cdot, t_{0}) V_{ε}^{u_{0}} (\cdot, t_{0}) d x - \int_{0}^{t_{0}} \int_{D_{z, ε}} R_{ε}^{1} \frac{\partial V_{ε}^{u_{0}}}{\partial t} d x d t . \end{array}$ Taking the supremum over $[0, T]$ after using the Cauchy–Schwartz inequality, one can deduce $\begin{array}{l} (5.16) & \begin{matrix} | \int_{0}^{t_{0}} \int_{D_{z, ε}} \frac{\partial R_{ε}^{1}}{\partial t} V_{ε}^{u_{0}} d x d t | ⩽ & {‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} {‖ V_{ε}^{u_{0}} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} \\ + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} {‖ \frac{\partial V_{ε}^{u_{0}}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{matrix} \end{array}$ On other hand, by the change of variables $x = z + ε y$ and Lemma 5.4, one can check that there exists $c > 0$ , independent of ε such that $\begin{matrix} {‖ V_{ε}^{u_{0}} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ \frac{\partial V_{ε}^{u_{0}}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} + {‖ \nabla V_{ε}^{u_{0}} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} ⩽ c ε, \end{matrix}$ which implies that for almost $t_{0} \in [0, T]$ , we have $\begin{array}{l} | \int_{0}^{t_{0}} \int_{D_{z, ε}} \frac{\partial R_{ε}^{1}}{\partial t} V_{ε}^{u_{0}} d x d t + ν \int_{0}^{t_{0}} \int_{D_{z, ε}} \nabla R_{ε}^{1} . \nabla V_{ε}^{u_{0}} d x d t | \\ ⩽ c ε ({‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))}) . \end{array}$ Moreover, from (5.14), (5.15) and (5.16), one can obtain $\begin{array}{l} (5.17) & \frac{1}{2} \int_{D_{z, ε}} {| R_{ε}^{1} (\cdot, t_{0}) |}^{2} d x ⩽ c ε ({‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))}), \\ (5.18) & {‖ \nabla R_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ c ε ({‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))}) . \end{array}$ It is easy to show that from (5.17), it follows $\begin{array}{l} (5.19) & {‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ 2 c ε ({‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))}) . \end{array}$ Again from the inequality (5.17), one can derive $\begin{array}{l} (5.20) & {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ 2 c ε T ({‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))}) . \end{array}$ Finally, combining (5.18), (5.19) and (5.20), we deduce the desired estimate for the term $R_{ε}^{1}$ $\begin{matrix} (5.21) & {‖ R_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε . \end{matrix}$

∙ The term $(R_{ε}^{2}, s_{ε}^{2})$ is constructed in such a way that $R_{ε}^{2} = - U_{z, ε}$ on $\partial D$ . It is defined as $\begin{matrix} (5.22) & R_{ε}^{2} (x, t) = \sum_{i = 1}^{3} u_{0}^{j} (z, t) χ_{ε}^{j} (x), s_{ε}^{2} (x, t) = \sum_{i = 1}^{3} u_{0}^{j} (z, t) δ_{ε}^{j} (x), \end{matrix}$ where $u_{0}^{j} (z, t)$ is the $j^{th}$ component of the velocity field $u_{0} (z, t)$ and $(χ_{ε}^{j}, δ_{ε}^{j})$ is the solution to the following stationary Stokes problem $\begin{matrix} (5.23) & \{\begin{matrix} - ν Δ χ_{ε}^{j} + \nabla δ_{ε}^{j} = 0 & in D_{z, ε}, \\ div χ_{ε}^{j} = 0 & in D_{z, ε}, \\ χ_{ε}^{j} = - Ψ^{j} ((x - z) / ε) & on \partial D, \\ χ_{ε}^{j} = 0 & on \partial O_{z, ε} . \end{matrix} \end{matrix}$ In the previous system $y \mapsto Ψ^{j} (y)$ is the velocity field solution to the exterior Stokes problem (4.3).

Due to the smoothness of $u_{0}$ (of class $C^{1}$ near the point z), one can prove that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} (5.24) & {‖ R_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c \sum_{i = 1}^{3} {‖ χ_{ε}^{j} ‖}_{H^{1} (D_{z, ε})} . \end{matrix}$ By Lemma 5.1 and trace Theorem, one can check $\begin{array}{rcl} {‖ χ_{ε}^{j} ‖}_{H^{1} (D_{z, ε})} & ⩽ & c {‖ Ψ^{j} ((x - z) / ε) ‖}_{H^{1 / 2} (\partial D)} \\ (5.25) & ⩽ & c [{‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, R})} + {‖ \nabla_{x} Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, R})}] . \end{array}$ Using the change of variable $y = (x - z) / ε$ and Lemma 5.2, one can derive $\begin{array}{l} (5.26) & {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, R})} = ε^{3 / 2} {‖ Ψ^{j} (y) ‖}_{L^{2} ((D_{z, R} - z) / ε)} ⩽ c ε, \\ (5.27) & {‖ \nabla_{x} Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, R})} = ε^{1 / 2} {‖ \nabla_{y} Ψ^{j} (y) ‖}_{L^{2} ((D_{z, R} -) / ε)} ⩽ c ε . \end{array}$ Then, inserting (5.27) and (5.26) into (5.25) and using (5.24), it follows that there exists $c > 0$ , independent of ε, such that $\begin{matrix} (5.28) & {‖ R_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε . \end{matrix}$

∙ The third part $R_{ε}^{3}$ taken into account the convective and nonlinear terms. It is chosen such that $\begin{matrix} R_{ε}^{3} = R_{ε} - (R_{ε}^{1} + R_{ε}^{2}) . \end{matrix}$ Using (4.7), (5.13) and (5.22) it follows that $(R_{ε}^{3}, s_{ε}^{3})$ satisfies $\begin{matrix} (5.29) & \{\begin{matrix} \frac{\partial R_{ε}^{3}}{\partial t} - ν Δ R_{ε}^{3} + N (R_{ε}^{3}) + ⟨ D N (R_{ε}^{3}), W_{z, ε} ⟩ + \nabla s_{ε}^{3} = \sum_{k = 1}^{4} F_{ε}^{k} & in D_{z, ε} \times (0, T), \\ div R_{ε}^{3} = 0 & in D_{z, ε} \times (0, T), \\ R_{ε}^{3} = 0 & on \partial D \times (0, T), \\ R_{ε}^{3} = 0 & on \partial O_{z, ε} \times (0, T), \\ R_{ε}^{3} (\cdot, 0) = 0 & in D_{z, ε}, \end{matrix} \end{matrix}$ where $N (R_{ε}^{3}) = (R_{ε}^{3} \cdot \nabla) R_{ε}^{3}$ and $⟨ D N (R_{ε}), W_{z, ε} ⟩ = (W_{z, ε} \cdot \nabla) R_{ε}^{3} + (R_{ε}^{3} \cdot \nabla) W_{z, ε}$ with $\begin{matrix} W_{z, ε} = u_{0} + U_{z, ε} + R_{ε}^{1} + R_{ε}^{2} . \end{matrix}$ The source terms $F_{ε}^{k}$ , $k = 1, 4$ , are defined for almost every $(x, t) \in D_{z, ε} \times (0, T)$ as $\begin{array}{l} F_{ε}^{1} (x, t) = - \sum_{i = 1}^{3} \frac{\partial u_{0}^{j}}{\partial t} (z, t) [Ψ_{ε}^{j} ((x - z) / ε) + χ_{ε}^{j} (x)], \\ F_{ε}^{2} (x, t) = - [(u_{0} \cdot \nabla) U_{z, ε} + (U_{z, ε} \cdot \nabla) u_{0} + (U_{z, ε} \cdot \nabla) U_{z, ε}], \\ F_{ε}^{3} (x, t) = - [([R_{ε}^{1} + R_{ε}^{2}] \cdot \nabla) [u_{0} + U_{z, ε}] + ([u_{0} + U_{z, ε}] \cdot \nabla) [R_{ε}^{1} + R_{ε}^{2}]], \\ F_{ε}^{4} (x, t) = - [([R_{ε}^{1} + R_{ε}^{2}] \cdot \nabla) [R_{ε}^{1} + R_{ε}^{2}]] . \end{array}$ Aiming to derive the behavior of $R_{ε}^{3}$ with respect to ε, we begin by exploiting the variational formulation of (5.29). Then, using $R_{ε}^{3}$ as a test function, we obtain $\begin{array}{l} \int_{0}^{τ} \int_{D_{z, ε}} \frac{\partial R_{ε}^{3}}{\partial t} R_{ε}^{3} d x d t + ν \int_{0}^{τ} \int_{D_{z, ε}} {| \nabla R_{ε}^{3} |}^{2} d x d t + \int_{0}^{τ} \int_{D_{z, ε}} N (R_{ε}^{3}) \cdot R_{ε}^{3} d x d t \\ + \int_{0}^{τ} \int_{D_{z, ε}} [(W_{z, ε} \cdot \nabla) R_{ε}^{3} + (R_{ε}^{3} \cdot \nabla) W_{z, ε}] \cdot R_{ε}^{3} d x d t \\ (5.30) & = \sum_{k = 1}^{4} \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{k} \cdot R_{ε}^{3} d x d t . \end{array}$ Since $R_{ε}^{3} = 0$ on $\partial D$ , by Lemma 5.6 we have $\begin{array}{l} \int_{0}^{τ} \int_{D_{z, ε}} N (R_{ε}^{3}) \cdot R_{ε}^{3} d x d t = \int_{0}^{τ} \int_{D_{z, ε}} (R_{ε}^{3} \cdot \nabla) R_{ε}^{3} \cdot R_{ε}^{3} d x d t = 0 and \\ \int_{0}^{τ} \int_{D_{z, ε}} (W_{z, ε} \cdot \nabla) R_{ε}^{3} \cdot R_{ε}^{3} d x d t = 0 . \end{array}$ Integrating in time and using the initial condition $R_{ε}^{3} (\cdot, 0) = 0$ in $D_{z, ε}$ , it follows $\begin{array}{rcl} \int_{0}^{τ} \int_{D_{z, ε}} \frac{\partial R_{ε}^{3}}{\partial t} R_{ε}^{3} d x d t = \frac{1}{2} \int_{D_{z, ε}} {| R_{ε}^{3} (x, τ) |}^{2} d x . \end{array}$ Then, the relation (5.30) becomes $\begin{array}{l} \frac{1}{2} \int_{D_{z, ε}} {| R_{ε}^{3} (x, τ) |}^{2} d x + ν \int_{0}^{τ} \int_{D_{z, ε}} {| \nabla R_{ε}^{3} |}^{2} d x d t + \int_{0}^{τ} \int_{D_{z, ε}} (R_{ε}^{3} \cdot \nabla) W_{z, ε} \cdot R_{ε}^{3} d x d t \\ (5.31) & = \sum_{k = 1}^{4} \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{k} \cdot R_{ε}^{3} d x d t . \end{array}$ In order to derive the desired estimate of $R_{ε}^{3}$ , we need to examine the trilinear term as well as the integral terms involving $F_{ε}^{k}$ , $1 ⩽ k ⩽ 4$ .

Estimate of the trilinear term: Taking into account the fact that $R_{ε}^{3} = 0$ on $\partial D$ , by Lemma 5.5 one can obtain $\begin{matrix} | \int_{0}^{τ} \int_{D_{z, ε}} (R_{ε}^{3} \cdot \nabla) W_{z, ε} \cdot R_{ε}^{3} d x d t | & ⩽ κ (D) \int_{0}^{T} {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (D_{z, ε})}^{2} ‖ \nabla W_{z, ε} ‖_{L^{2} (D_{z, ε})} d t \\ ⩽ κ (D) {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} ‖ \nabla W_{z, ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} . \end{matrix}$ Recall that $W_{z, ε} = u_{0} + U_{z, ε} + R_{ε}^{1} + R_{ε}^{2}$ , with $U_{z, ε} (x, t) = \sum_{j = 1}^{3} u_{0}^{j} (z, t) Ψ^{j} ((x - z) / ε)$ . Then, exploiting the estimates of $R_{ε}^{1}$ and $R_{ε}^{2}$ , it follows $\begin{array}{l} ‖ \nabla W_{z, ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} ⩽ ‖ \nabla u_{0} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + ‖ \nabla U_{z, ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + c ε . \end{array}$ By the change of variable $y = (x - z) / ε$ and Lemma 5.2, one can obtain $\begin{array}{l} (5.32) & ‖ \nabla U_{z, ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} ⩽ {‖ u_{0} (z, t) ‖}_{L^{\infty} (0, T)} \sum_{j = 1}^{3} {‖ \nabla Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, ε}))} ⩽ c ε^{1 / 2} . \end{array}$ With the help of Assumptions 3.4 (i.e. $‖ \nabla u_{0} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} ⩽ β < ν / κ (D)$ ) and the fact that $c ε + c ε^{1 / 2}$ goes to zero with ε, one can check that there exist $ε_{0} > 0$ and a constant $\tilde{β} > 0$ , such that $\begin{array}{l} ‖ \nabla W_{z, ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} ⩽ \tilde{β} < ν / κ (D), \forall ε \in] 0, ε_{0} [. \end{array}$ Here, one can choose $\tilde{β} = \frac{ν / κ (D) + β}{2}$ . Consequently, the trilinear term satisfies the estimate $\begin{array}{l} (5.33) & | \int_{0}^{τ} \int_{D_{z, ε}} (R_{ε}^{3} \cdot \nabla) W_{z, ε} \cdot R_{ε}^{3} d x d t | ⩽ \tilde{β} κ (D) {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} . \end{array}$ Next, we will examine the integral terms involving the functions $F_{ε}^{k}$ , $1 ⩽ k ⩽ 4$ .

Estimate of the first integral term: Due to the smoothness of $u_{0}$ and Cauchy–Schwartz inequality, it follows $\begin{array}{l} | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{1} \cdot R_{ε}^{3} d x d t | & ⩽ \sum_{i = 1}^{3} | \int_{0}^{T} \frac{\partial u_{0}^{j}}{\partial t} (z, t) \int_{D_{z, ε}} [Ψ_{ε}^{j} ((x - z) / ε) + χ_{ε}^{j} (x)] \cdot R_{ε}^{3} d x d t | \\ ⩽ c [{‖ Ψ_{ε}^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, ε})} + {‖ χ_{ε}^{j} ‖}_{L^{2} (D_{z, ε})}] {‖ R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$ By the change of variable $x = z + ε y$ , the relation (5.25) and Lemma 5.2, one can conclude that there exists $c > 0$ , independent of ε, such that $\begin{array}{l} (5.34) & | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{1} \cdot R_{ε}^{3} d x d t | ⩽ c ε {‖ R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$

Estimate of the second integral term: Since $R_{ε}^{3} = 0$ on $\partial D \cup \partial O_{z, ε}$ , by Lemma 5.6 the integral term involving $F_{ε}^{2}$ can be rewritten as $\begin{array}{l} \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{2} \cdot R_{ε}^{3} d x d t \\ = - \int_{0}^{τ} \int_{D_{z, ε}} [(u_{0} \cdot \nabla) U_{z, ε} + (U_{z, ε} \cdot \nabla) u_{0} + (U_{z, ε} \cdot \nabla) U_{z, ε}] \cdot R_{ε}^{3} d x d t \\ = \int_{0}^{τ} \int_{D_{z, ε}} [(u_{0} \cdot \nabla) R_{ε}^{3} \cdot U_{z, ε} + (U_{z, ε} \cdot \nabla) R_{ε}^{3} \cdot u_{0} + (U_{z, ε} \cdot \nabla) R_{ε}^{3} \cdot U_{z, ε}] d x d t . \end{array}$ Using (5.9), one can obtain $\begin{array}{l} | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{2} \cdot R_{ε}^{3} d x d t | \\ ⩽ \int_{0}^{T} [2 ‖ u_{0} ‖_{L^{\infty} (D)} ‖ U_{z, ε} ‖_{L^{2} (D_{z, ε})} + ‖ U_{z, ε} ‖_{L^{4} (D_{z, ε})}^{2}] {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (D_{z, ε})} d t \\ ⩽ 2 ‖ u_{0} ‖_{L^{\infty} (0, T; L^{\infty} (D))} ‖ U_{z, ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} \\ (5.35) & + ‖ U_{z, ε} ‖_{L^{\infty} (0, T; L^{4} (D_{z, ε}))} ‖ U_{z, ε} ‖_{L^{2} (0, T; L^{4} (D_{z, ε}))} {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} \end{array}$ Exploiting the definition of $U_{z, ε}$ and the smoothness of $u_{0}$ , one get $\begin{array}{l} ‖ U_{z, ε} ‖_{L^{2} (D_{z, ε})} ⩽ \sum_{j = 1}^{3} | u_{0}^{j} (z, t) | {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, ε})} ⩽ c \sum_{j = 1}^{3} {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, ε})} \\ ‖ U_{z, ε} ‖_{L^{4} (D_{z, ε})} ⩽ \sum_{j = 1}^{3} | u_{0}^{j} (z, t) | {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{4} (D_{z, ε})} ⩽ c \sum_{j = 1}^{3} {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{4} (D_{z, ε})} . \end{array}$ Using the change of variable $y = (x - z) / ε$ and Lemma 5.2, one can derive $\begin{array}{l} {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{2} (D_{z, ε})} = ε^{3 / 2} {‖ Ψ^{j} ‖}_{L^{2} ((D_{z, ε} - z) / ε)} ⩽ c ε \\ {‖ Ψ^{j} ((x - z) / ε) ‖}_{L^{4} (D_{z, ε})} = ε^{3 / 4} {‖ \nabla Ψ^{j} ‖}_{L^{4} ((D_{z, ε} - z) / ε)} ⩽ c ε^{3 / 4}, \end{array}$ which implies that there exists $c > 0$ , independent of ε, such that $\begin{array}{l} (5.36) & \begin{aligned} ‖ U_{z, ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} ⩽ c ε, ‖ U_{z, ε} ‖_{L^{2} (0, T; L^{4} (D_{z, ε}))} ⩽ c ε^{3 / 4} and \\ ‖ U_{z, ε} ‖_{L^{\infty} (0, T; L^{4} (D_{z, ε}))} ⩽ c ε^{3 / 4} . \end{aligned} \end{array}$ Since $‖ u_{0} ‖_{L^{\infty} (0, T; L^{\infty} (D))}$ is uniformly bounded, then from (5.35) and (5.36) one can deduce that the integral term involving $F_{ε}^{2}$ satisfies the estimate $\begin{array}{l} (5.37) & | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{2} \cdot R_{ε}^{3} d x d t | ⩽ c ε {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$

Estimate of the third integral term: Similarly to the previous case, by Lemma 5.6 this integral term can be rewritten as $\begin{array}{l} \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{3} \cdot R_{ε}^{3} d x d t = & \int_{0}^{τ} \int_{D_{z, ε}} ([R_{ε}^{1} + R_{ε}^{2}] \cdot \nabla) R_{ε}^{3} \cdot [u_{0} + U_{z, ε}] d x d t \\ + \int_{0}^{τ} \int_{D_{z, ε}} ([u_{0} + U_{z, ε}] \cdot \nabla) R_{ε}^{3} \cdot [R_{ε}^{1} + R_{ε}^{2}] d x d t . \end{array}$ Using Lemma 5.5, we have $\begin{array}{l} | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{3} \cdot R_{ε}^{3} d x d t | \\ ⩽ c (D) ‖ u_{0} + U_{z, ε} ‖_{L^{\infty} (0, T; H^{1} (D_{z, ε}))} {‖ R_{ε}^{1} + R_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$ According to (5.32) and (5.36), the quantity $‖ u_{0} + U_{z, ε} ‖_{L^{\infty} (0, T; H^{1} (D_{z, ε}))}$ is uniformly bounded. The established results in (5.21) and (5.28) give $\begin{array}{l} (5.38) & {‖ R_{ε}^{1} + R_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ {‖ R_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} + {‖ R_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε, \end{array}$ which implies that the third integral term admits the estimate $\begin{array}{l} (5.39) & | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{3} \cdot R_{ε}^{3} d x d t | ⩽ c ε {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$

Estimate of the fourth integral term: By Lemma 5.5 and the estimate (5.38), one can establish that the source term involving $F_{ε}^{4}$ satisfies $\begin{array}{l} | \int_{0}^{τ} \int_{D_{z, ε}} F_{ε}^{4} \cdot R_{ε}^{3} d x d t | & ⩽ c (D) {‖ R_{ε}^{1} + R_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))}^{2} {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} \\ (5.40) & ⩽ c ε^{2} {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$

Now, returning to the relation (5.31) and exploiting the obtained approximations (5.33), (5.34), (5.37), (5.39) and (5.40), we obtain

\begin{array}{l} \frac{1}{2} \int_{D_{z, ε}} {| R_{ε}^{3} (x, τ) |}^{2} d x + ν \int_{0}^{τ} \int_{D_{z, ε}} {| \nabla R_{ε}^{3} |}^{2} d x d t \\ ⩽ \tilde{β} κ (Ω) {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} + c ε {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}, \end{array}

for almost all

τ \in [0, T]

. Thus, taking the supremum in time, we deduce

\begin{array}{rcl} (5.41) & {‖ R_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))}^{2} + 2 [ν - \tilde{β} κ (D)] {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ 2 c ε {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}

From the fact that

ν - \tilde{β} κ (D) > 0

, it follows

\begin{array}{rcl} (5.42) & {‖ R_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ 2 c ε {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}, \end{array}

which gives

\begin{array}{rcl} (5.43) & {‖ R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ T {‖ R_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))}^{2} ⩽ 2 c T ε {‖ \nabla R_{ε}^{3} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}

Then, combining the inequalities (5.41), (5.42) and (5.43) one deduce that the term

R_{ε}^{3}

satisfies the approximation

\begin{array}{rcl} (5.44) & {‖ R_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ R_{ε}^{3} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε . \end{array}

Finally, using the preliminaries estimates (5.21), (5.28) and (5.44), one can check that there exists

c > 0

, independent of ε, such that

\begin{array}{rcl} (5.45) & ‖ R_{ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + ‖ R_{ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε \end{array}

which completes the proof of Theorem 4.1.

6. Shape functional variation

This section is concerned with the proof of Theorem 4.3. Firstly, we derive a preliminary approximation describing the shape functional variation with respect to the insertion of a small obstacle $O_{z, ε}$ .

6.1. Preliminary estimate

Denoting by $T^{j} (ε)$ , $1 ⩽ j ⩽ 3$ the following quantities: $\begin{array}{l} T^{1} (ε) = - ε \sum_{j = 1}^{3} \int_{0}^{T} u_{0}^{j} (z, t) \int_{\partial O} σ (Ψ^{j}, Π^{j}) (y) n [v_{0} (z + ε y, t) - v_{0} (z, t)] d s (y) d t, \\ T^{2} (ε) = \int_{0}^{T} \int_{D} [(u_{ε} - u_{0}) \cdot \nabla] (u_{ε} - u_{0}) \cdot v_{0} d x d t, \\ T^{3} (ε) = - \int_{0}^{T} \int_{\partial O_{z, ε}} σ (R_{ε}, s_{ε}) n v_{0} d s d t . \end{array}$

Proposition 6.1.
The variation of the shape functional $S$ , with respect to the presence of a small obstacle $O_{z, ε}$ inside the fluid flow domain $D$ , satisfies the following estimate $\begin{array}{l} S (D ∖ \overline{O_{z, ε}}) - S (D) = & ε [- \sum_{j = 1}^{3} \int_{0}^{T} u_{0}^{j} (z, t) \int_{\partial O} σ (Ψ^{j}, Π^{j}) (y) n v_{0} (z, t) d s (y) d t + δ F (z)] \\ + \sum_{j = 1}^{3} T^{j} (ε) + o (ε) . \end{array}$
Proof.
According to the Assumptions 4.2, we have $\begin{array}{rcl} S (D ∖ \overline{O_{z, ε}}) - S (D) & = & \int_{0}^{T} [F_{ε} (u_{ε} (\cdot, t)) - F_{0} (u_{0} (\cdot, t))] d t \\ = & \int_{0}^{T} D F_{0} (u_{0} (\cdot, t)) (u_{ε} (\cdot, t) - u_{0} (\cdot, t)) d t + ε δ F (z) + o (ε) . \end{array}$ From the weak formulation of the adjoint problem (4.9) and the fact that $u_{ε} = 0$ in $O_{z, ε}$ , one get $\begin{array}{l} \int_{0}^{T} D F_{0} (u_{0} (\cdot, t)) (u_{ε} (\cdot, t) - u_{0} (\cdot, t)) d t \\ = \int_{0}^{T} \int_{D_{z, ε}} \frac{\partial v_{0}}{\partial t} ξ_{ε} d x d t - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} u_{0} d x d t - ν \int_{0}^{T} \int_{D_{z, ε}} \nabla v_{0} \nabla ξ_{ε} d x d t \\ (6.1) & + ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla u_{0} d x d t - \int_{0}^{T} \int_{D} [G (u_{0}) v_{0} -^{t} G (v_{0}) u_{0}] \cdot ξ_{ε} d x d t, \end{array}$ where $ξ_{ε} = u_{ε} - u_{0}$ is the velocity field variation, solution to $\begin{matrix} (6.2) & \{\begin{matrix} \frac{\partial ξ_{ε}}{\partial t} - ν Δ ξ_{ε} + N (ξ_{ε}) + ⟨ D N (ξ_{ε}), u_{0} ⟩ + \nabla π_{ε} = 0 & in D_{z, ε} \times (0, T), \\ div ξ_{ε} = 0 & in D_{z, ε} \times (0, T), \\ ξ_{ε} = 0 & on Γ \times (0, T), \\ ξ_{ε} = - u_{0} & on \partial O_{z, ε} \times (0, T), \\ ξ_{ε} (\cdot, 0) = 0 & in D_{z, ε}, \end{matrix} \end{matrix}$ with $π_{ε} = p_{ε} - p_{0}$ represents the pressure field variation.

Since $ξ_{ε} (\cdot, 0) = 0$ and $v_{0} (\cdot, T) = 0$ , the integration by parts in time gives $\begin{array}{rcl} (6.3) & \int_{0}^{T} \int_{D_{z, ε}} \frac{\partial v_{0}}{\partial t} ξ_{ε} d x d t = - \int_{0}^{T} \int_{D_{z, ε}} \frac{\partial ξ_{ε}}{\partial t} v_{0} d x d t . \end{array}$ Using the fact that $v_{0} = ξ_{ε} = 0$ on $\partial D$ , by Lemma 5.6 one can check $\begin{array}{rcl} \int_{0}^{T} \int_{D} [G (u_{0}) v_{0} -^{t} G (v_{0}) u_{0}] \cdot ξ_{ε} d x d t & = & \int_{0}^{T} \int_{D} [(ξ_{ε} \cdot \nabla) u_{0} + (u_{0} \cdot \nabla) ξ_{ε}] \cdot v_{0} d x d t \\ (6.4) & = & \int_{0}^{T} \int_{D} ⟨ D N (ξ_{ε}), u_{0} ⟩ \cdot v_{0} d x d t . \end{array}$ The weak formulation of (6.2) implies $\begin{matrix} (6.5) & \begin{matrix} \int_{0}^{T} \int_{D_{z, ε}} \frac{\partial ξ_{ε}}{\partial t} v_{0} d x d t + ν \int_{0}^{T} \int_{D_{z, ε}} \nabla ξ_{ε} \nabla v_{0} d x d t + \int_{0}^{T} \int_{D_{z, ε}} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t \\ + \int_{0}^{T} \int_{D_{z, ε}} ⟨ D N (ξ_{ε}), u_{0} ⟩ \cdot v_{0} d x d t \\ = \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) n v_{0} d s d t . \end{matrix} \end{matrix}$ Combining the relations (6.1), (6.3), (6.4) and (6.5), one can deduce $\begin{array}{l} \int_{0}^{T} D F_{0} (u_{0} (\cdot, t)) (u_{ε} (\cdot, t) - u_{0} (\cdot, t)) d t \\ = - \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) n v_{0} d s d t + \int_{0}^{T} \int_{D} (ξ_{ε} \cdot \nabla) ξ_{ε} \cdot v_{0} d x d t \\ (6.6) & - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} u_{0} d x d t + ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla u_{0} d x d t - \int_{0}^{T} \int_{O_{z, ε}} (u_{0} \cdot \nabla) u_{0} \cdot v_{0} d x d t . \end{array}$ According to (4.6), we have $ξ_{ε} = U_{z, ε} + R_{ε}$ and $π_{ε} = P_{z, ε} + s_{ε}$ , which implies $\begin{array}{rcl} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (ξ_{ε}, π_{ε}) n v_{0} d s d t & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (R_{ε}, s_{ε}) n v_{0} d s d t \\ (6.7) & + \int_{0}^{T} \int_{\partial O_{z, ε}} σ (U_{z, ε}, P_{z, ε}) n v_{0} d s d t . \end{array}$ Then, using the definitions (4.1) and (4.2), the second integral term can be rewritten as $\begin{array}{l} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (U_{z, ε}, P_{z, ε}) n v_{0} d s d t \\ = \frac{1}{ε} \sum_{j = 1}^{3} \int_{0}^{T} u_{0}^{j} (z, t) \int_{\partial O_{z, ε}} σ_{x} (ε Ψ^{j} (\frac{x - z}{ε}), Π^{j} (\frac{x - z}{ε})) n v_{0} d s (x) d t . \end{array}$ By the change of variable $y = (x - z) / ε$ , one can obtain $\begin{array}{l} \int_{0}^{T} \int_{\partial O_{z, ε}} σ (U_{z, ε}, P_{z, ε}) n v_{0} d s (x) d t \\ (6.8) & = ε \sum_{j = 1}^{3} \int_{0}^{T} u_{0}^{j} (z, t) \int_{\partial O} σ (Ψ^{j}, Π^{j}) (y) n v_{0} (z + ε y, t) d s (y) d t . \end{array}$ From the smoothness of $u_{0}$ and $v_{0}$ (of class $C^{1}$ ) near the point z, one can check that there exists $c > 0$ (independent of ε) such that $\begin{array}{l} | - \int_{0}^{T} \int_{O_{z, ε}} \frac{\partial v_{0}}{\partial t} u_{0} d x d t + ν \int_{0}^{T} \int_{O_{z, ε}} \nabla v_{0} \nabla u_{0} d x d t - \int_{0}^{T} \int_{O_{z, ε}} (u_{0} \cdot \nabla) u_{0} \cdot v_{0} d x d t | \\ (6.9) & ⩽ c | O_{z, ε} |, \end{array}$ with $| O_{z, ε} | = ε^{3} | O |$ , is the Lebesgue measure of the subdomain $O_{z, ε}$ .

Finally, exploiting the relations (6.6), (6.7), (6.8) and (6.9), one can conclude that the design function $S$ satisfies the following desired estimate $\begin{array}{rcl} S (D ∖ \overline{O_{z, ε}}) & = & S (D) + ε [- \sum_{j = 1}^{3} \int_{0}^{T} u_{0}^{j} (z, t) \int_{\partial O} σ (Ψ^{j}, Π^{j}) (y) n v_{0} (z, t) d s (y) d t + δ F (z)] \\ + \sum_{j = 1}^{3} T^{j} (ε) + o (ε) . \end{array}$ □

6.2. Proof of Theorem 4.3

According to the established estimate in Proposition 6.1, for proving Theorem 4.3 it suffices to demonstrate that the terms $T^{j} (ε)$ , $1 ⩽ j ⩽ 3$ can be neglected with respect to ε (i.e. $T^{j} (ε) = o (ε)$ , $j = 1, 3$ ).

∙ Estimate of the term $T^{1} (ε)$ : Let $R > 0$ such that $\overline{O} \subset B (0, R)$ . Posing $V_{ε}^{v_{0}} (y, t) = U_{ε} (y, t)$ , with $U_{ε}$ is the velocity field defined in Lemma 5.4 when $ψ = v_{0}$ . Since $- ν Δ Ψ^{j} + \nabla Π^{j} = 0$ in $B (0, R) ∖ \overline{O}$ , we have $\begin{matrix} \int_{\partial O} σ (Ψ^{j}, Π^{j}) (y) n [v_{0} (z + ε y, t) - v_{0} (z, t)] d s (y) = ν \int_{B (0, R) ∖ \overline{O}} \nabla_{y} Ψ^{j} (y) \nabla_{y} V_{ε}^{v_{0}} (y, t) d y . \end{matrix}$ Then, using the smoothness of $u_{0}$ and the fact that $‖ \nabla Ψ^{j} ‖_{L^{2} (B (0, R) ∖ \overline{O})}$ are uniformly bounded one can deduce that there exists $c > 0$ such that $\begin{array}{rcl} | T^{1} (ε) | & = & | ε \sum_{j = 1}^{3} \int_{0}^{T} u_{0}^{j} (z, t) \int_{\partial O} σ (Ψ^{j}, Π^{j}) (y) n [v_{0} (z + ε y, t) - v_{0} (z, t)] d s (y) d t | \\ ⩽ & c ε {‖ \nabla V_{ε}^{v_{0}} ‖}_{L^{2} (0, T; L^{2} (B (0, R) ∖ \overline{O}))} . \end{array}$ By Lemma 5.4, one can conclude that the term $T^{1} (ε)$ admits the estimate $\begin{array}{rcl} | T^{1} (ε) | ⩽ c ε^{2} = o (ε) . \end{array}$

∙ Estimate of the term $T^{2} (ε)$ : From the fact that $div (u_{ε} - u_{0}) = 0$ in $D$ and $v_{0} = 0$ on $\partial D$ , one gets $\begin{array}{rcl} T^{2} (ε) & = & \int_{0}^{T} \int_{D} [(u_{ε} - u_{0}) \cdot \nabla] (u_{ε} - u_{0}) \cdot v_{0} d x d t \\ = & - \int_{0}^{T} \int_{D} [(u_{ε} - u_{0}) \cdot \nabla] v_{0} \cdot (u_{ε} - u_{0}) d x d t \\ = & - \int_{0}^{T} \int_{D_{z, ε}} [(u_{ε} - u_{0}) \cdot \nabla] v_{0} \cdot (u_{ε} - u_{0}) d x d t - \int_{0}^{T} \int_{O_{z, ε}} [u_{0} \cdot \nabla] v_{0} \cdot u_{0} d x d t . \end{array}$ Due to the smoothness of $u_{0}$ and $v_{0}$ near the point z, one can deduce $\begin{array}{rcl} | T^{2} (ε) | & = & | \int_{0}^{T} \int_{D_{z, ε}} [(u_{ε} - u_{0}) \cdot \nabla] v_{0} \cdot (u_{ε} - u_{0}) d x d t | + o (ε) . \end{array}$ By Cauchy–Schwartz inequality, one can justify that there exists $c > 0$ , independent of ε, such that $\begin{array}{rcl} | \int_{0}^{T} \int_{D_{z, ε}} [(u_{ε} - u_{0}) \cdot \nabla] v_{0} \cdot (u_{ε} - u_{0}) d x d t | ⩽ c ‖ \nabla v_{0} ‖_{L^{\infty} (0, T; L^{\infty} (D_{z, ε}))} ‖ u_{ε} - u_{0} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} . \end{array}$ Since $u_{ε} - u_{0} = R_{ε} + U_{z, ε}$ in $D_{z, ε}$ , one can derive $\begin{array}{rcl} ‖ u_{ε} - u_{0} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))}^{2} & ⩽ & {‖ R_{ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} + ‖ U_{z, ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))}}^{2} \\ (6.10) & ⩽ & c ε^{2}, \end{array}$ which implies that the term $T^{2} (ε)$ obeys the estimate $\begin{array}{rcl} T^{2} (ε) = o (ε) . \end{array}$ ∙ Estimate of the term $T^{3} (ε)$ : This term can be written as $\begin{array}{rcl} T^{3} (ε) & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (R_{ε}, s_{ε}) n [v_{0} (z, t) - v_{0} (x, t)] d s d t - \int_{0}^{T} \int_{\partial O_{z, ε}} σ (R_{ε}, s_{ε}) n v_{0} (z, t) d s d t, \\ = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (R_{ε}, s_{ε}) n [Y_{ε}^{1} + Y_{ε}^{2}] d s d t, \end{array}$ where $Y_{ε}^{1} (x, t) = \sum_{j = 1}^{3} v_{0}^{j} (z, t) Ψ^{j} ((x - z) / ε)$ and $Y_{ε}^{2}$ solves the system $\begin{matrix} (6.11) & \{\begin{matrix} \frac{\partial Y_{ε}^{2}}{\partial t} - ν Δ Y_{ε}^{2} + \nabla r_{ε}^{2} = 0 & in D_{z, ε} \times (0, T), \\ div Y_{ε}^{2} = 0 & in D_{z, ε} \times (0, T), \\ Y_{ε}^{2} = - Y_{ε}^{1} & on Γ \times (0, T), \\ Y_{ε}^{2} = - v_{0} (x, t) + v_{0} (z, t) & on \partial O_{z, ε} \times (0, T), \\ Y_{ε}^{2} (\cdot, 0) = 0 & in D_{z, ε} . \end{matrix} \end{matrix}$ By the smoothness of $v_{0}$ and the change of variable $y = (x - z) / ε$ , one can obtain $\begin{array}{l} {‖ \frac{\partial Y_{ε}^{1}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} + {‖ Y_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ Y_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} \\ ⩽ c \sum_{j = 1}^{3} {‖ Ψ^{j} ((x - z) / ε) ‖}_{H^{1} (D_{z, ε})} \\ ⩽ c \sum_{j = 1}^{3} [ε^{3 / 2} {‖ Ψ^{j} ‖}_{L^{2} ((D_{z, ε} - z) / ε)} + ε^{1 / 2} {‖ \nabla Ψ^{j} ‖}_{L^{2} ((D_{z, ε} - z) / ε)}] . \end{array}$ Using Lemma 5.2, it follows $\begin{array}{rcl} (6.12) & {‖ \frac{\partial Y_{ε}^{1}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} + {‖ Y_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ Y_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (D_{z, ε}))} ⩽ c ε^{1 / 2} . \end{array}$ The term $Y_{ε}^{2}$ can be estimated in a similar way as $R_{ε}^{1}$ . Then, by Lemma 5.4 one can check that there exists $c > 0$ such that $\begin{array}{rcl} (6.13) & {‖ \frac{\partial Y_{ε}^{2}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} + {‖ Y_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω_{z, ε}))} + {‖ Y_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω_{z, ε}))} ⩽ c ε . \end{array}$ Let $Y_{ε} = Y_{ε}^{1} + Y_{ε}^{2}$ . Since $Y_{ε} = 0$ on $\partial D$ , the weak formulation of (4.7) implies $\begin{array}{rcl} T^{3} (ε) & = & \int_{0}^{T} \int_{\partial O_{z, ε}} σ (R_{ε}, s_{ε}) (x, t) n Y_{ε} d s d t \\ = & \int_{D_{z, ε}} R_{ε} (. T) Y_{ε} (. T) d x - \int_{0}^{T} \int_{D_{z, ε}} \frac{\partial Y_{ε}}{\partial t} R_{ε} d x d t + ν \int_{0}^{T} \int_{D_{z, ε}} \nabla R_{ε} \nabla Y_{ε} d x d t \\ + \int_{0}^{T} \int_{D_{z, ε}} (R_{ε} \cdot \nabla) R_{ε} \cdot Y_{ε} d x d t + \int_{0}^{T} \int_{D_{z, ε}} ([u_{0} + U_{z, ε}] \cdot \nabla) R_{ε} \cdot Y_{ε} d x d t \\ + \int_{0}^{T} \int_{D_{z, ε}} (R_{ε} \cdot \nabla) [u_{0} + U_{z, ε}] \cdot Y_{ε} d x d t - \int_{0}^{T} \int_{R_{z, ε}} F_{ε} \cdot Y_{ε} d x d t . \end{array}$ Using Cauchy–Schwartz inequality and Lemma 5.5, one can obtain $\begin{array}{rcl} | T^{3} (ε) | & ⩽ & ‖ R_{ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} ‖ Y_{ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + {‖ \frac{\partial Y_{ε}}{\partial t} ‖}_{L^{2} (0, T; L^{2} (D_{z, ε}))} ‖ R_{ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} \\ + ‖ \nabla R_{ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} ‖ \nabla Y_{ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} \\ + c (D) ‖ R_{ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))}^{2} ‖ Y_{ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))} \\ + 2 c (D) ‖ u_{0} + U_{z, ε} ‖_{L^{\infty} (0, T; H^{1} (D_{z, ε}))} ‖ R_{ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))} ‖ Y_{ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))} \\ + ‖ F_{ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} ‖ Y_{ε} ‖_{L^{2} (0, T; L^{2} (D_{z, ε}))} . \end{array}$ Using the smoothness of $u_{0}$ and the fact that $‖ Ψ^{j} ((x - z) / ε) ‖_{H^{1} (D_{z, ε})} ⩽ ε^{1 / 2}$ , one can derive $\begin{matrix} ‖ F_{ε} ‖_{L^{2} (0, T; L^{2} (Ω_{z, ε}))} = O (ε^{1 / 2}) . \end{matrix}$

In other hand, we have (see Theorem 4.1) $\begin{matrix} ‖ R_{ε} ‖_{L^{\infty} (0, T; L^{2} (D_{z, ε}))} + ‖ R_{ε} ‖_{L^{2} (0, T; H^{1} (D_{z, ε}))} = O (ε) . \end{matrix}$ Therefore, one can deduce that $\begin{matrix} | T^{3} (ε) | = o (ε) . \end{matrix}$

Consequently, all shape functions $S$ satisfying the Assumptions 4.2 admit the asymptotic expansion $\begin{matrix} S (D ∖ \overline{O_{z, ε}}) = S (D) + ε [\int_{0}^{T} u_{0} (x, t) \cdot T_{O} v_{0} (x, t) d t + δ F (x)] + o (ε), \end{matrix}$ for all small $ε > 0$ .

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Topological sensitivity analysis for the 3D nonlinear Navier–Stokes equations

Abstract

Keywords

1. Introduction

2. Problem statement

3. Weak solutions of the full Navier–Stokes equations

Lemma 3.1 ([34], p. 237).

4.1. Asymptotic behavior of the perturbed velocity

5.1. Preliminary lemmas

6. Shape functional variation

6.1. Preliminary estimate

References