Asymptotic behavior of 2D incompressible ideal flow around small disks

Abstract

In this article, we study the limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of $n_{k}$ disjoint disks with centers ${z_{i}^{k}}$ and radii $ε_{k}$ . We assume that the initial velocities $u_{0}^{k}$ are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, $n_{k} \to \infty$ , and we assume $ε_{k} \to 0$ as $k \to \infty$ .

Let $γ_{i}^{k}$ be the circulation of $u_{0}^{k}$ around the circle ${| x - z_{i}^{k} | = ε_{k}}$ . We prove that the limit as $k \to \infty$ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) $ω_{0}^{k} = curl u_{0}^{k}$ has a uniform compact support and converges weakly in $L^{p_{0}}$ , for some $p_{0} > 2$ , to $ω_{0} \in L_{c}^{p_{0}} (R^{2})$ , (2) $\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ$ weak-∗ in $BM (R^{2})$ for some bounded Radon measure μ, and (3) the radii $ε_{k}$ are sufficiently small. Then the corresponding solutions $u^{k}$ converge strongly to a weak solution u of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity $ω = curl u$ , with initial data $ω_{0}$ , where the transporting velocity field is generated from ω, so that its curl is $ω + μ$ . As a byproduct, we obtain a new existence result for this modified Euler system.

Keywords

Euler equations homogenization in perforated domain shrinking obstacles and porous medium

1. Introduction

In this article we are concerned with the asymptotic behavior of solutions of the two-dimensional incompressible Euler equations in the exterior of a finite number of vanishingly small disks while, possibly, the number of disks simultaneously tends to infinity. For example, let us say we were interested in comparing fluid flow in the exterior of a rigid wall with flow in the exterior of a row of small disks following the shape of the wall. Fluid flow in the exterior of a rigid wall was studied by one of the authors in [16]; it can be described as the flow due to a vortex sheet whose position is stuck at the wall, but with a vortex sheet strength which is time-dependent. On the other hand, as we will see, flow in the exterior of a finite number of obstacles preserves circulation around each obstacle, according to Kelvin’s circulation theorem. Under appropriate hypothesis, as the size of each obstacle vanishes and the number of obstacles increases to approximate the wall, the flows converge to a vortex sheet flow whose position is stuck at the wall but with a time-independent sheet strength, which is, therefore given by the initial data. In other words, conservation of circulation around the small obstacles implies that the “homogenization” limit does not yield solutions of the Euler equations in the exterior of the wall, but, instead, produces solutions of a suitably modified Euler system, in the full plane. In its simplest form this modified Euler system, in vorticity formulation, is given by a transport equation for the vorticity ω by a velocity whose curl is the sum of ω and a Dirac delta supported on the wall. In fact, we will prove that, given any bounded, compactly supported, time-independent, Radon measure μ, the vorticity flow whose velocity is modified by μ can be approximated by an exterior domain flow, exterior to a finite number of small disks.

More precisely, for each $k = 1, 2, \dots$ we consider a family of $n_{k}$ disjoint disks with centers $z_{i}^{k}$ and radii $ε_{k}$ , $i = 1, \dots, n_{k}$ . We denote the disks by $B (z_{i}^{k}, ε_{k}) : = {| x - z_{i}^{k} | < ε_{k}}$ and the fluid domain by: $\begin{matrix} Ω^{k} : = R^{2} ∖ (⋃_{i = 1}^{n_{k}} \overline{B (z_{i}^{k}, ε_{k})}) . \end{matrix}$ Let v be smooth, divergence-free vector field in $Ω^{k}$ , tangent to the boundary, with vorticity, $w : = curl v$ , having bounded support. In non-simply connected domains there are infinitely many vector fields whose curl is w. To recover the velocity v from its vorticity w it is necessary to introduce the circulation around each disk: $\begin{matrix} (1.1) & γ_{i}^{k} [v] : = \oint_{\partial B (z_{i}^{k}, ε_{k})} v \cdot \hat{τ} d s . \end{matrix}$ The above integral is considered in the counter-clockwise sense, hence $\hat{τ} = - {\hat{n}}^{⊥}$ , where $\hat{n}$ denotes the outward unit normal vector at $\partial Ω^{k}$ . Vorticity together with the circulations uniquely determine the velocity, see, for example, [14] for a full discussion.

Given $ω_{0}^{k} \in C_{c}^{\infty} (\overline{Ω^{k}})$ and ${γ_{i}^{k}}_{i = 1}^{n_{k}} \subset R$ , the corresponding initial velocity $u_{0}^{k}$ is the unique smooth solution of $\begin{matrix} (1.2) & \{\begin{matrix} div u_{0}^{k} = 0 & in Ω^{k} \\ curl u_{0}^{k} = ω_{0}^{k} & in Ω^{k} \\ u_{0}^{k} \cdot \hat{n} = 0 & on \partial Ω^{k} \\ γ_{i}^{k} [u_{0}^{k}] = γ_{i}^{k} & for i = 1, \dots, n_{k} \\ | u_{0}^{k} (x) | \to 0 & as x \to \infty . \end{matrix} \end{matrix}$

The IBVP for the Euler equations in $Ω^{k}$ , in velocity formulation, takes the form: $\begin{matrix} (1.3) & \{\begin{matrix} \partial_{t} u^{k} + u^{k} \cdot \nabla u^{k} = - \nabla p^{k} & in (0, \infty) \times Ω^{k} \\ div u^{k} = 0 & in [0, \infty) \times Ω^{k} \\ u^{k} \cdot \hat{n} = 0 & in [0, \infty) \times \partial Ω^{k} \\ {lim}_{x \to \infty} | u^{k} | = 0 & for t \in [0, \infty) \\ u^{k} (0, x) = u_{0}^{k} (x) & in Ω^{k}, \end{matrix} \end{matrix}$ where $p^{k} = p^{k} (t, x)$ is the pressure. Existence and uniqueness of a smooth solution $u^{k} = u^{k} (t, x), p^{k} = p^{k} (t, x)$ of system (1.3) is due to K. Kikuchi in [15]. We note that, by Kelvin’s circulation theorem, the circulations $γ_{i}^{k} [u^{k} (t, \cdot)]$ , $i = 1, \dots, n_{k}$ , as defined through (1.1), are conserved quantities.

Taking the curl of the momentum equation in (1.3) yields a transport equation for the corresponding vorticity $ω^{k} : = curl u^{k} = \partial_{1} u_{2}^{k} - \partial_{2} u_{1}^{k}$ . This transport equation, together with an elliptic system relating vorticity to velocity and (initial) circulations, comprises the vorticity formulation of the 2D Euler equations: $\begin{matrix} (1.4) & \{\begin{matrix} \partial_{t} ω^{k} + u^{k} \cdot \nabla ω^{k} = 0 & in (0, \infty) \times Ω^{k} \\ div u^{k} = 0 & in [0, \infty) \times Ω^{k} \\ curl u^{k} = ω^{k} & in [0, \infty) \times Ω^{k} \\ u^{k} \cdot \hat{n} = 0 & on [0, \infty) \times \partial Ω^{k} \\ γ_{i}^{k} [u^{k} (t, \cdot)] = γ_{i}^{k} & for i = 1, \dots, n_{k}, for t \in [0, \infty) \\ | u^{k} (t, x) | \to 0 & as x \to \infty, for t \in [0, \infty) \\ ω^{k} (0, \cdot) = ω_{0}^{k} & in Ω^{k} . \end{matrix} \end{matrix}$

We emphasize that $γ_{i}^{k}$ and $ω_{0}^{k}$ are initial data. We also note, in passing, that, as the transporting velocity is divergence-free, the $L^{p}$ norm of vorticity is a conserved quantity, for any $p ⩾ 1$ .

Throughout we will adopt the convention that, whenever it is needed to extend $u^{k} (t, \cdot)$ or $ω^{k} (t, \cdot)$ to all of $R^{2}$ , they will be set to vanish in the interior of the disks $B (z_{i}^{k}, ε_{k})$ .

The main purpose of this article is to study the asymptotic behavior of the sequence ${u^{k}}$ when the radii of the disks tend to zero, both when the number of disks $n_{k}$ is finite and independent of k and when $n_{k} \to \infty$ as $k \to \infty$ .

It is convenient to fix a scale with respect to which we consider this asymptotic limit. To this end we fix an initial compactly supported vorticity, or eddy, $ω_{0}$ .

We denote the space of all bounded Radon real measures on $R^{2}$ by $BM (R^{2})$ . If $μ \in BM (R^{2})$ , the total variation of μ is defined by $\begin{matrix} ‖ μ ‖_{BM} = sup {\int_{R^{2}} φ d μ | φ \in C_{0} (R^{2}), ‖ φ ‖_{L^{\infty}} ⩽ 1}, \end{matrix}$ where $C_{0} (R^{2})$ is the set of all real-valued continuous functions on $R^{2}$ vanishing at infinity. We recall that $BM (R^{2})$ equipped with the total variation norm is a Banach space. We say that a sequence $(μ_{n})$ in $BM (R^{2})$ converges weak-∗ to μ if $\int_{R^{2}} φ d μ_{n} \to \int_{R^{2}} φ d μ$ as $n \to \infty$ for all $φ \in C_{0} (R^{2})$ .

We assume that the sequence of measures $\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}}$ converges, weak-∗ in the sense of measures, to a compactly supported bounded Radon measure μ. This measure μ represents the homogenized limit of the circulations $γ_{i}^{k}$ . We assume, without loss of generality, that, for every $k \in N$ , ${z_{i}^{k}}_{i = 1}^{n_{k}}$ is a set of distinct points.

Our main theorem reads:

Theorem 1.1.
Fix $ω_{0} \in L_{c}^{p_{0}} (R^{2})$ for some $p_{0} \in (2, \infty]$ . Set $R_{0} > 0$ so that $supp ω_{0} \subset B (0, R_{0})$ . Let ${ω_{0}^{k}} \subset C_{c}^{\infty} (B (0, R_{0}))$ be such that $\begin{matrix} ω_{0}^{k} ⇀ ω_{0} weakly in L^{p_{0}} (R^{2}) . \end{matrix}$ Consider ${γ_{i}^{k}}_{i = 1, \dots, n_{k}} \subset R$ , ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset B (0, R_{0})$ and suppose that $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ weak- * in BM (R^{2}), as k \to \infty, \end{matrix}$ for some $μ \in {BM}_{c} (R^{2})$ .

Then there exists a subsequence, still denoted by k, a sequence ${ε_{k}} \subset R_{+}^{}$ , with* $ε_{k} \to 0$ , together with a vector field $u \in L_{loc}^{p} (R_{+} \times R^{2})$ , for any $p \in [1, 2)$ , and a function $ω \in L^{\infty} (R_{+}; L^{1} \cap L^{p_{0}} (R^{2}))$ , such that the solutions $(u^{k}, ω^{k})$ of the Euler equations in $\begin{matrix} Ω^{k} = R^{2} ∖ (⋃_{i = 1}^{n_{k}} \overline{B (z_{i}^{k}, ε_{k})}), \end{matrix}$ with initial vorticity $ω_{0}^{k} |_{Ω^{k}}$ and initial circulations $(γ_{i}^{k})$ around the disks, verify
$u^{k} \to u$ strongly in $L_{loc}^{p} (R_{+} \times R^{2})$ for any $p \in [1, 2)$ ;

$ω^{k} ⇀ ω$ weak ∗ in $L^{\infty} (R_{+}; L^{q} (R^{2}))$ for any $q \in [1, p_{0}]$ ;

the limit pair $(u, ω)$ verifies, in the sense of distributions: $\begin{array}{l} \{\begin{matrix} \partial_{t} ω + u \cdot \nabla ω = 0 & in (0, \infty) \times R^{2} \\ div u = 0 & in [0, \infty) \times R^{2} \\ curl u = ω + μ & in [0, \infty) \times R^{2} \\ {lim}_{x \to \infty} | u | = 0 & for t \in [0, \infty) \\ ω (0, x) = ω_{0} (x) & in R^{2} . \end{matrix} \end{array}$

In view of this theorem, the limit behavior of $(u^{k}, ω^{k})$ is described by solutions of a modification of the incompressible Euler equations: namely, the vorticity ω is transported by a divergence-free vector field u which satisfies $curl u = ω + μ$ . The additional term μ is a reminder of the circulation around the evanescent obstacles. We emphasize that the measure μ above does not depend on time and, also, it is not necessary single-signed, differently from the work of Delort [8].

As a byproduct, Theorem 1.1 gives the existence of a global weak solution to this modified Euler problem, where the velocity is obtained by the standard Biot–Savart law of the plane applied to $ω + μ$ , for μ a fixed, compactly supported, bounded Radon measure. Therefore, our result may be understood as an extension of the work of Marchioro [23], in which the author obtained existence of global weak solution when the velocity is the standard Biot–Savart law of the plane applied to $ω + α δ_{0}$ . Indeed, given any $μ \in {BM}_{c} (R^{2})$ , it is easy to find an approximation $\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}}$ , for example by discretization on a grid.

Uniqueness for weak solutions of this modified Euler system is an interesting open problem. In the case $μ, ω_{0} \in L_{c}^{\infty} (R^{2})$ , one can adapt the argument in Yudovich’s theorem, see [28], to prove uniqueness and, in addition, it holds that $u \in ⋂_{p > 2} W^{1, p} (R^{2})$ is a strong solution of the velocity equation in $R^{2}$ (see (1.3)). In the special case where μ is a finite sum of Dirac masses, $\sum_{i = 1}^{n} γ_{i} δ_{z_{i}}$ , and when $ω_{0} \in L_{c}^{\infty} (R^{2})$ , uniqueness was proved by Lacave and Miot in [19] under the assumption that $ω_{0}$ is initially constant around each of the vortex points $z_{i}$ . A key point of their argument was to prove that the non-constant part of ω never meets the trajectories of the vortex points. Uniqueness for general bounded vorticity $ω_{0}$ , not necessarily constant around $z_{i}$ , is already a challenging open question. For more general μ, it is not clear that a vorticity which initially vanishes around the support of μ does not intersect this support in finite time. However, we have local uniqueness, up to the time of collision (see Section 5). In all cases for which uniqueness holds for the limit equation we note that the convergences stated in Theorem 1.1 hold for the full sequence, without the need to pass to a subsequence.

The asymptotic behavior of an inviscid fluid around obstacles shrinking to points was first studied by Iftimie, Lopes Filho and Nussenzveig Lopes in [13], where the authors considered only one obstacle which shrinks to one point. Their result can be viewed as a special case of Theorem 1.1: $n_{k} = 1$ , $z_{1}^{k} = 0$ , $μ = γ δ_{0}$ , $ω_{0}^{k} = ω_{0} \in C_{c}^{\infty} (R^{2} ∖ {0})$ . It was crucial, in [13], that there be only one obstacle, because the authors used the explicit form of the Biot–Savart law (giving the velocity in terms of the vorticity) in the exterior of a simply connected compact set. Later, Lopes Filho [22] treated the case of several obstacles where one of them shrinks to a point. However, the fluid domain was assumed to be bounded: the use of the explicit Biot–Savart law was replaced by standard elliptic ideas in bounded domains. The initial motivation for the present work was to understand how, and whether it was possible, to approximate ideal flow around a curve by flow outside a finite number of islands approaching the curve. We refer to Sections 5.3 and 5.4 for further discussion.

From a technical point of view, our main difficulties are how to treat several obstacles without an explicit formula for the Biot–Savart law, together with issues related to unbounded domains (for example: integrability at infinity and analysis of boundary terms which arise when integrating by parts). In particular, we develop in Section 3.2 (Step 2) an argument based on the inversion map $i (x) : = x / | x |^{2}$ , which sends exterior domains to bounded sets, allowing us to use elliptic tools, such as the maximum principle. For simplicity, we have decided to present all the details in the case where the obstacles are disks, but it is possible to substitute the disks for a more general shape. Let $K$ be a smooth, simply connected compact set containing zero and consider obstacles of the form $z_{i}^{k} + ε_{k} K$ . Then, in [3], these results have been extended to these kinds of obstacles and it was shown that the limit does not depend on the shape of $K$ if the distance between the obstacles is large enough.

The remainder of this article is organized in four sections. In the next section, we recall precisely how to recover velocity from vorticity and circulation around boundary components and we introduce basic notation, to be used throughout the paper. Section 3 is dedicated to deriving uniform estimates on velocity in $L^{p}$ , for all $p \in [1, 2)$ . The method we use is to construct an explicit correction of the Biot–Savart law from the formulas in the exterior of a single disk, and then to compare this with $u^{k}$ and with the standard Biot-Savard law in the full plane, applied to $ω + μ$ . Comparing this correction with the Biot–Savart formula in the full plane (Step 1) will not be too hard, because we have explicit expressions. However, the comparison with $u^{k}$ (Step 2) will be more complicated, and will justify the use of the inversion mapping. The additional difficulty comes from the fact that we are considering non-zero circulations. Indeed, when an obstacle shrinks to a point, a Dirac mass in the curl of u appears as an asymptotic limit, and the velocity associated to $γ δ_{z}$ is of the form $γ \frac{{(x - z)}^{⊥}}{2 π | x - z |^{2}}$ , which only belongs to $L_{loc}^{p} (R^{2})$ for $p < 2$ , but not to $L_{loc}^{2} (R^{2})$ . The stability of the Euler equations under Hausdorff approximation of the fluid domain is a recent result of Gérard-Varet and Lacave [10,11], but the authors considered obstacles with $H^{1}$ positive capacity, in order to use $L^{2}$ arguments (the Sobolev capacity of a material point is zero, see [10,11] for details). This difficulty also explains why the authors in [3,18] treat the case of zero circulations (see Section 5.4).

In Section 4, we use these $L^{p}$ estimates to prove our main theorem.

Finally, we collect in the last section some final comments and remarks. In particular, we prove a series of uniqueness results for the limit problem, as mentioned above (the case where μ is a bounded function, the case of a finite number of Dirac masses, and local uniqueness if the initial vorticity is supported far from $supp μ$ ). We will also discuss the case where the balls are uniformly distributed on a curve. In particular, we make rigorous the observation that the asymptotic behavior of solutions, as $n_{k} \to \infty$ , is not a solution of the Euler equations around a curve, as described in [16]. This disparity can be attributed to the fact that the disk radii $ε_{k}$ in Theorem 1.1 can be very small compared to the distance between the centers of the obstacles. Recently, the sharp control of the ratio between the size of the obstacles and the distance between obstacles was investigated in [3,18], in the particular case of zero circulations.

We conclude this introduction by recalling that the study of the flow through a porous medium has a long history in the homogenization framework. We refer to Cioranescu and Murat [5] for the Laplace problem, and to Tartar [27] and Allaire [1,2] for the Stokes and Navier–Stokes equations. There are also many works concerning viscous flow through a sieve, see, e.g., [6,7,9,25] and references therein. For a modified Euler equations (weakly non-linear), Mikelić-Paoli [24] and Lions-Masmoudi [21] also obtained a homogenized limit problem. Notation.
We adopt the convention ${(a, b)}^{⊥} = (- b, a)$ and $\nabla^{⊥} f = {(\nabla f)}^{⊥}$ .

2. Preliminaries about the Biot–Savart law

The purpose of this section is to introduce basic notation and recall facts about the Biot–Savart law, which expresses the velocity in terms of the vorticity and circulations. The details can be found, for example, in [14,15,22].

Let Ω be the exterior of n disjoint disks: $\begin{matrix} Ω = R^{2} ∖ (⋃_{i = 1}^{n} \overline{B (z_{i}, r_{i})}) . \end{matrix}$

Consider also a disk with n (circular) holes: $\begin{matrix} \tilde{Ω} = B (z_{0}, r_{0}) ∖ (⋃_{i = 1}^{n} \overline{B (z_{i}, r_{i})}) \end{matrix}$ where $⋃_{i = 1}^{n} \overline{B (z_{i}, r_{i})} \subset B (z_{0}, r_{0})$ .

Let $U \subseteq R^{2}$ be a smooth domain, which can be either Ω or $\tilde{Ω}$ . Given a smooth function f, compactly supported in U, and $(γ_{i}) \in R^{n}$ , we consider the following elliptic system: $\begin{matrix} (2.1) & \{\begin{matrix} div u = 0 & in U \\ curl u = f & in U \\ u \cdot \hat{n} = 0 & on \partial U \\ γ_{i} [u] = γ_{i} & for i = 1, \dots, n, \end{matrix} \end{matrix}$ where $γ_{i} [u]$ was defined in (1.1).

When $U = Ω$ we will assume an additional condition at infinity, namely, ${lim}_{x \to \infty} | u | = 0$ .

2.1. Harmonic part

We introduce families of harmonic functions called harmonic measures.

in the unbounded domains, the harmonic measures are the functions ${Φ_{i}}_{i = 1, \dots, n}$ , the unique solutions of $\begin{matrix} \{\begin{matrix} Δ Φ_{i} = 0 & in Ω \\ Φ_{i} = δ_{i j} & on \partial B (z_{j}, r_{j}) \\ Φ_{i} & has a finite limit at \infty, \end{matrix} \end{matrix}$ (where $δ_{i j}$ is the Kronecker delta);

in bounded domains, we introduce the functions ${{\tilde{Φ}}_{i}}_{i = 0, \dots, n}$ as the unique solutions of $\begin{matrix} (2.2) & \{\begin{matrix} Δ {\tilde{Φ}}_{i} = 0 & in \tilde{Ω} \\ {\tilde{Φ}}_{i} = δ_{i j} & on \partial B (z_{j}, r_{j}) . \end{matrix} \end{matrix}$

By uniqueness of the Dirichlet problem for the Laplacian, we have: $\begin{matrix} \sum_{i = 1}^{n} Φ_{i} \equiv 1 in Ω, \sum_{i = 0}^{n} {\tilde{Φ}}_{i} \equiv 1 in \tilde{Ω}, \end{matrix}$ and for any i $\begin{matrix} Φ_{i} (x) \in [0, 1] \forall x \in Ω, {\tilde{Φ}}_{i} (x) \in [0, 1] \forall x \in \tilde{Ω} . \end{matrix}$ Indeed,

the bounded case follows directly from the maximum principle;

in the unbounded case, let us assume that one of the disks is centered at the origin, let say $z_{1} = 0$ . Next, we use the inversion $i (x) : = x / | x |^{2}$ (see Lemma 3.7 for its properties), which maps Ω to a bounded domain (included in $B (0, 1 / r_{1})$ ) with $n - 1$ obstacles. Next, we introduce ${\tilde{Φ}}_{i} (x) : = Φ_{i} \circ i^{- 1} (x)$ which verifies (2.2) in $i (Ω)$ (see Lemma 3.7). Hence, the result follows from the bounded case.

In the bounded case, we will use later that ${\nabla^{⊥} {\tilde{Φ}}_{i}}_{i = 1, \dots, n}$ is a basis for the harmonic vector fields (i.e, the finite-dimensional vector space of vector fields which are both solenoidal and irrotational and which are tangent to the boundary). Indeed, let H be a harmonic vector field, then:

the divergence-free condition reads as $H (x) = \nabla^{⊥} ψ (x)$ for some stream function ψ;

the tangency condition means ψ is constant on each $\partial B (z_{i}, r_{i})$ for $i = 0, \dots, n$ . We denote by $c_{i}$ these constants. As ψ is defined up to a constant, we can determine uniquely ψ by assuming $c_{0} = 0$ ;

the curl-free condition implies $Δ ψ = 0$ in $\tilde{Ω}$ .

By uniqueness, it follows that

\begin{matrix} ψ = \sum_{i = 1}^{n} c_{i} {\tilde{Φ}}_{i}, H = \sum_{i = 1}^{n} c_{i} \nabla^{⊥} {\tilde{Φ}}_{i} . \end{matrix}

Next, we seek a family of harmonic vector fields such that the circulation around $B (z_{j}, r_{j})$ is $δ_{i j}$ .

In bounded domains, we introduce the vector field ${{\tilde{X}}_{i}}_{i = 1, \dots, n}$ as the unique solution of $\begin{matrix} \{\begin{matrix} div {\tilde{X}}_{i} = curl {\tilde{X}}_{i} = 0 & in \tilde{Ω} \\ {\tilde{X}}_{i} \cdot \hat{n} = 0 & on \partial \tilde{Ω} \\ \oint_{\partial B (z_{j}, r_{j})} {\tilde{X}}_{i} \cdot \hat{τ} d s = δ_{i j} & for j = 1, \dots, n . \end{matrix} \end{matrix}$ It is easy to see that the family ${{\tilde{X}}_{i}}_{i = 1, \dots, n}$ is also a basis for the harmonic vector fields, since this family is clearly linearly independent and we already know that the dimension of the space of harmonic vector fields is precisely n. Alternatively, we can also express the ${\tilde{X}}_{i}$ ’s in terms of stream functions ${\tilde{X}}_{i} = \nabla^{⊥} {\tilde{Ψ}}_{i}$ , where ${\tilde{Ψ}}_{i}$ is the unique solution of $\begin{matrix} \{\begin{matrix} Δ {\tilde{Ψ}}_{i} = 0 & in \tilde{Ω} \\ \partial_{τ} {\tilde{Ψ}}_{i} = 0 & on \partial \tilde{Ω} \\ {\tilde{Ψ}}_{i} = 0 & on \partial B (z_{0}, r_{0}) \\ \oint_{\partial B (z_{j}, r_{j})} \partial_{n} {\tilde{Ψ}}_{i} d s = - δ_{i j} & for j = 1, \dots, n . \end{matrix} \end{matrix}$ The condition $\partial_{τ} {\tilde{Ψ}}_{i} = 0$ means that ${\tilde{Ψ}}_{i}$ is constant on each $\partial B (z_{i}, r_{i})$ for $i = 0, \dots, n$ . Then, we can uniquely determine a change of basis matrix, i.e. constants $c_{i, j}$ such that $\begin{matrix} {\tilde{Ψ}}_{i} = \sum_{j = 1}^{n} c_{i, j} {\tilde{Φ}}_{j}, {\tilde{X}}_{i} = \sum_{j = 1}^{n} c_{i, j} \nabla^{⊥} {\tilde{Φ}}_{j} . \end{matrix}$

In unbounded domains, we introduce the vector fields ${X_{i}}_{i = 1, \dots, n}$ as the unique solutions of $\begin{matrix} \{\begin{matrix} div X_{i} = curl X_{i} = 0 & in Ω \\ X_{i} \cdot \hat{n} = 0 & on \partial Ω \\ | X_{i} (x) | \to 0 & when x \to \infty \\ \oint_{\partial B (z_{j}, r_{j})} X_{i} \cdot \hat{τ} d s = δ_{i j} & for j = 1, \dots, n . \end{matrix} \end{matrix}$ Clearly, the family ${X_{i}}_{i = 1, \dots, n}$ is a basis for the harmonic vector fields. In addition, we have the following asymptotic expansion by Laurent series: $\begin{matrix} (2.3) & X_{i} (x) = \frac{1}{2 π} \frac{x^{⊥}}{| x |^{2}} + O (\frac{1}{| x |^{2}}), as x \to \infty . \end{matrix}$

To reformulate this discussion in terms of stream functions, we first observe that we cannot assume that $Ψ_{i}$ goes to zero at infinity. One way to choose $Ψ_{i}$ uniquely is to assume that $c_{0, i} = 0$ in the Laurent expansion $\begin{matrix} Ψ_{i} = \frac{1}{2 π} ln | x | + c_{0, i} + O (\frac{1}{| x |}) . \end{matrix}$ Consequently, we can reformulate our description of the space of harmonic vector fields in terms of the stream functions $X_{i} = \nabla^{⊥} Ψ_{i}$ , where, for each $i = 1, \dots, n$ , $Ψ_{i}$ is the unique solution of $\begin{matrix} \{\begin{matrix} Δ Ψ_{i} = 0 & in Ω \\ \partial_{τ} Ψ_{i} = 0 & on \partial Ω \\ | Ψ_{i} (x) - \frac{1}{2 π} ln | x | | \to 0 & when x \to \infty \\ \oint_{\partial B (z_{j}, r_{j})} \partial_{n} Ψ_{i} d s = - δ_{i j} & for j = 1, \dots, n . \end{matrix} \end{matrix}$ The stream functions $Ψ_{i}$ above satisfy: $\begin{matrix} (2.4) & Ψ_{i} (x) = \frac{1}{2 π} ln | x | + O (\frac{1}{| x |}), as x \to \infty . \end{matrix}$

We remark that the precise form of the term $O (| x |^{- 1})$ above depends on the shape of the domain, so the expansions (2.3) and (2.4), will not be uniform in k, when we go back to our original problem. However, these estimates will be only used to justify certain integrations by parts. Moreover, we note that $Ψ_{i}$ and $Φ_{i}$ do not have the same behavior at infinity, and that ${\nabla^{⊥} Φ_{i}}_{i = 1, \dots, n}$ is not a basis for the harmonic vector fields in unbounded domains.

2.2. The Laplacian-inverse

In this subsection, we focus on the solution of $Δ Ψ = f$ with Dirichlet boundary condition. We denote by $Ψ_{D} = Ψ_{D} [f]$ the solution operator of the Dirichlet Laplacian in Ω. More precisely, $ψ = Ψ_{D} [f]$ is the unique solution of $\begin{matrix} \{\begin{matrix} Δ ψ = f & in Ω \\ ψ = 0 & on \partial Ω \\ \nabla ψ (x) \to 0 & when x \to \infty, \end{matrix} \end{matrix}$ and we denote by ${\tilde{Ψ}}_{D} = {\tilde{Ψ}}_{D} [f]$ the corresponding solution operator in $\tilde{Ω}$ , so that $\tilde{ψ} = {\tilde{Ψ}}_{D} [f]$ is the unique solution of $\begin{matrix} \{\begin{matrix} Δ \tilde{ψ} = f & in \tilde{Ω} \\ \tilde{ψ} = 0 & on \partial \tilde{Ω} . \end{matrix} \end{matrix}$

We also define $\begin{matrix} K [f] : = \nabla^{⊥} Ψ_{D} [f], \tilde{K} [f] : = \nabla^{⊥} {\tilde{Ψ}}_{D} [f], \end{matrix}$ which are divergence free, tangent to the boundary and verify $\begin{matrix} curl K [f] = f on Ω, curl \tilde{K} [f] = f on \tilde{Ω} . \end{matrix}$ Moreover, we have for any $i = 1, \dots, n$ : $\begin{matrix} (2.5) & \begin{matrix} \oint_{\partial B (z_{i}, r_{i})} K [f] \cdot \hat{τ} d s = - \int_{Ω} Φ_{i} (y) f (y) d y, \\ \oint_{\partial B (z_{i}, r_{i})} \tilde{K} [f] \cdot \hat{τ} d s = - \int_{\tilde{Ω}} {\tilde{Φ}}_{i} (y) f (y) d y . \end{matrix} \end{matrix}$

One relevant issue in the remainder of this article is the behavior at infinity of $Ψ_{D} [f]$ and of $K [f]$ . We observe that if $supp f \subset B (0, R)$ , then there exists $C_{R}$ such that, for all $| x | ⩾ 2 R$ we have $\begin{matrix} (2.6) & | Ψ_{D} [f] (x) | ⩽ C_{R} ‖ f ‖_{L^{1}}, | K [f] (x) | ⩽ \frac{C_{R} ‖ f ‖_{L^{1}}}{| x |^{2}} . \end{matrix}$

In these inequalities, the constant $C_{R}$ depends on the size of the support of f and of the domain Ω. As the support of the vorticity moves, the above estimates, with $f = ω^{k} (t, \cdot)$ , will depend on the time. But, as remarked in the previous subsection, even if the constant in (2.6) depends on t and k, it will be useful to justify certain integrations by parts, at t and k fixed. For a proof of (2.6), see, for example, [14].

2.3. Biot–Savart law

Next we introduce notation for the Biot–Savart law. If f is a smooth function, compactly supported in $\overline{Ω}$ (or supported in $\tilde{Ω}$ ) and $(γ_{i}) \in R^{n}$ , the unique solution u of the elliptic system (2.1) is given by: $\begin{array}{rcl} (2.7) & \begin{matrix} u (x) = K [f] (x) + \sum_{i = 1}^{n} α_{i} [f] X_{i} (x) in Ω \\ u (x) = \tilde{K} [f] (x) + \sum_{i = 1}^{n} {\tilde{α}}_{i} [f] {\tilde{X}}_{i} (x) in \tilde{Ω}, \end{matrix} \end{array}$ where $\begin{matrix} α_{i} [f] = \int_{Ω} Φ_{i} (y) f (y) d y + γ_{i} and {\tilde{α}}_{i} [f] = \int_{\tilde{Ω}} {\tilde{Φ}}_{i} (y) f (y) d y + γ_{i}, \end{matrix}$ see [14] and [22].

In what follows, we use the notation $m_{i} [f] : = \int_{Ω} Φ_{i} (y) f (y) d y$ . We add a superscript k to all the functions defined above, in order to keep in mind the dependence on the domain. For example, the unique vector field $u_{0}^{k}$ verifying (1.2) is given by: $\begin{matrix} u_{0}^{k} (x) = K^{k} [ω_{0}^{k} |_{Ω^{k}}] (x) + \sum_{i = 1}^{n_{k}} α_{i}^{k} [ω_{0}^{k} |_{Ω^{k}}] X_{i}^{k} (x) with α_{i}^{k} [ω_{0}^{k} |_{Ω^{k}}] = \int_{Ω^{k}} Φ_{i}^{k} (y) ω_{0}^{k} (y) d y + γ_{i}^{k} . \end{matrix}$

Our basic strategy is to deduce strong convergence for the velocity from (2.7) and from $L^{p}$ estimates of the vorticity.

3. Convergence for fixed time

For a function v defined in the domain $\begin{matrix} Ω^{ε, k} : = R^{2} ∖ (⋃_{i = 1}^{n_{k}} \overline{B (z_{i}^{k}, ε)}), \end{matrix}$ we denote by $E v$ its extension to the full plane by setting $E v$ to vanish outside $Ω^{ε, k}$ . Let $K_{R^{2}}$ be the Biot–Savart operator in the full plane. For h regular enough, we write: $\begin{matrix} K_{R^{2}} [h] = \frac{1}{2 π} \frac{x^{⊥}}{| x |^{2}} * h . \end{matrix}$

We seek to prove that the Biot–Savart formula (2.7) converges to $K_{R^{2}} [μ + f]$ when $k \to \infty$ . It is natural to study the following decomposition: $\begin{array}{l} K_{R^{2}} [μ + E f |_{Ω^{ε, k}}] (x) - E (K^{ε, k} [f |_{Ω^{ε, k}}] (x) + \sum_{i = 1}^{n_{k}} α_{i}^{ε, k} [f] X_{i}^{ε, k} (x)) \\ = (K_{R^{2}} [μ] - K_{R^{2}} [\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}}]) (x) \\ + \sum_{i = 1}^{n_{k}} γ_{i}^{k} (\frac{{(x - z_{i}^{k})}^{⊥}}{2 π | x - z_{i}^{k} |^{2}} - E X_{i}^{ε, k} (x)) \\ + (K_{R^{2}} [E f |_{Ω^{ε, k}}] - E K^{ε, k} [f |_{Ω^{ε, k}}] - \sum_{i = 1}^{n_{k}} m_{i}^{ε, k} [f] E X_{i}^{ε, k}) (x) \\ = : A_{k} (x) + B_{ε, k} (x) + C_{ε, k} [f] (x) . \end{array}$

The goal of this section is to prove a series of three propositions associated to this decomposition.

The first proposition is concerned with the first term, where we convert the weak-∗ convergence in ${BM}_{c} (R^{2})$ to strong convergence of the associated velocities.

Proposition 3.1.
Let $R_{0} > 0$ fixed. If a sequence ${γ_{i}^{k}}_{i = 1, \dots, n_{k}} \subset R$ and ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset B (0, R_{0})$ verifies $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ weak- * in BM (R^{2}) as k \to \infty \end{matrix}$ for some $μ \in {BM}_{c} (R^{2})$ , then for any $p \in (1, 2)$ we can decompose $A_{k}$ in two parts $A_{k} = A_{k, 1} + A_{k, 2}$ such that $\begin{matrix} ‖ A_{k, 1} ‖_{L^{p} (R^{2})} + ‖ A_{k, 2} ‖_{L^{p^{}} (R^{2})} \to 0 as k \to \infty, \end{matrix}$ where* $p^{} = {(\frac{1}{p} - \frac{1}{2})}^{- 1} \in (2, \infty)$ .

The result in Proposition 3.1 fixes k such that $K_{R^{2}} [μ]$ is well approximated. In the following proposition, the k is fixed, and we study the limit $ε \to 0$ for the harmonic vector fields.
Proposition 3.2.
Let* $k \in N$ fixed and ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset B (0, R_{0})$ given. Then, for any $i = 1, \dots, n_{k}$ , there exists $v_{i}^{ε} \in C^{\infty} (\overline{Ω^{ε, k}})$ such that $\begin{matrix} {‖ \frac{{(x - z_{i})}^{⊥}}{2 π | x - z_{i} |^{2}} - E v_{i}^{ε} (x) ‖}_{L^{p} (R^{2})} + {‖ E v_{i}^{ε} - E X_{i}^{ε, k} ‖}_{L^{2} (R^{2})} \to 0 as ε \to 0, \end{matrix}$ for any $p \in [1, 2)$ .

To be more precise for $p = 1$ , if we define $d_{k} : = {min}_{i \neq j} | z_{i}^{k} - z_{j}^{k} |$ , then there exist $C, C_{1} > 0$ independent of k and ε (depending only on $R_{0}$ ) such that for all $ε ⩽ \frac{d_{k}}{n_{k}^{2} \sqrt{2 C_{1}}}$ , we have $\begin{array}{l} \sum_{i = 1}^{n_{k}} {‖ \frac{{(x - z_{i})}^{⊥}}{2 π | x - z_{i} |^{2}} - E v_{i}^{ε} (x) ‖}_{L^{1} (R^{2})} + \sum_{i = 1}^{n_{k}} {‖ E v_{i}^{ε} - E X_{i}^{ε, k} ‖}_{L^{2} (R^{2})} \\ ⩽ C ({(n_{k}^{5} ε^{2})}^{1 / 4} + {(\frac{ε n_{k}^{2}}{d_{k}})}^{2} | ln ε |) . \end{array}$

Next, we are interested in the convergence of $C_{ε, k}$ . We will see later that it is important to establish this convergence uniformly in the $(L^{1} \cap L^{p_{0}})$ -norm of f.
Proposition 3.3.
Let $M_{0} > 0$ , $p_{0} \in (2, \infty]$ , $k \in N$ , ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset R^{2}$ fixed. We have that $\begin{matrix} {‖ K_{R^{2}} [E f |_{Ω^{ε, k}}] - E K^{ε, k} [f |_{Ω^{ε, k}}] - \sum_{i = 1}^{n_{k}} m_{i}^{ε, k} [f] E X_{i}^{ε, k} ‖}_{L^{2} (R^{2})} \to 0 as ε \to 0, \end{matrix}$ uniformly in f verifying: $\begin{matrix} ‖ f ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0} and f is compactly supported. \end{matrix}$

To be more precise, there exist $C > 0$ independent of k and ε (depending only on $R_{0}$ ) such that for all $ε ⩽ \frac{d_{k}}{8}$ (with $d_{k}$ is defined in Proposition 3.2 ), we have $\begin{matrix} {‖ K_{R^{2}} [E f |_{Ω^{ε, k}}] - E K^{ε, k} [f |_{Ω^{ε, k}}] - \sum_{i = 1}^{n_{k}} m_{i}^{ε, k} [f] E X_{i}^{ε, k} ‖}_{L^{2} (R^{2})} ⩽ C ‖ f ‖_{L^{1} \cap L_{0}^{p}} ε | ln ε | n_{k}^{1 / 2} \end{matrix}$ for any $f \in L_{c}^{p_{0}} (R^{2})$ .

In the last part of this section, we will construct $ε_{k}$ such that Theorem 1.1 can be proved.
3.1. The number of obstacles (proof of Proposition 3.1)

Since ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset B (0, R_{0})$ , we have that $supp μ \subset \overline{B (0, R_{0})}$ . From the weak-∗ convergence $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ weak- * in BM (R^{2}) when k \to \infty, \end{matrix}$ we infer that $‖ \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ‖_{BM (R^{2})}$ is uniformly bounded in k (see [4, Prop. 3.13], which is a consequence of the Banach–Steinhaus theorem). We have, by duality, that $BM (B (0, R_{0} + 1))$ is compactly imbedded in $W^{- 1, p} (B (0, R_{0} + 1))$ for any $p \in (1, 2)$ , since $W_{0}^{1, p^{'}} (B (0, R_{0} + 1))$ is compactly imbedded in $C_{0} (B (0, R_{0} + 1))$ , where $p^{'} = p / (p - 1)$ . Hence we get that $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} - μ \to 0 strongly in W^{- 1, p} (B (0, R_{0} + 1)) when k \to \infty, \end{matrix}$ for any $p \in (1, 2)$ . Let us fix $p \in (1, 2)$ . We recall that a distribution h in $W^{- 1, p} (Ω)$ can be decomposed as $h = f_{0} + \partial_{1} f_{1} + \partial_{2} f_{2}$ , where $f_{0}$ , $f_{1}$ and $f_{2}$ are in $L^{p} (Ω)$ and ${max}_{l = 0, \dots, 2} ‖ f_{l} ‖_{L^{p} (Ω)} ⩽ ‖ h ‖_{W^{- 1, p} (Ω)}$ (see, for example, [4, Prop. 9.20] for this decomposition). Consequently, there exists $f_{0}^{k}$ , $f_{1}^{k}$ and $f_{2}^{k}$ belonging in $L^{p} (R^{2})$ such that $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} - μ = f_{0}^{k} + \partial_{1} f_{1}^{k} + \partial_{2} f_{2}^{k} in R^{2}, \end{matrix}$ with ${max}_{l = 0, 1, 2} ‖ f_{l}^{k} ‖_{L^{p} (R^{2})} \to 0$ .

Now, we apply the Biot–Savart law in $R^{2}$ : $\begin{matrix} (3.1) & K_{R^{2}} [\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}}] - K_{R^{2}} [μ] = K_{R^{2}} [f_{0}^{k}] + \partial_{1} K_{R^{2}} [f_{1}^{k}] + \partial_{2} K_{R^{2}} [f_{2}^{k}] . \end{matrix}$ For the first right hand side term, we apply the Hardy–Littlewood–Sobolev Theorem (see e.g. [26, Theo. V.1] with $α = 1$ ) where $\frac{1}{p} = \frac{1}{p^{*}} + \frac{1}{2}$ to get: $\begin{matrix} {‖ K_{R^{2}} [f_{0}^{k}] ‖}_{L^{p^{*}} (R^{2})} ⩽ C_{p} {‖ f_{0}^{k} ‖}_{L^{p} (R^{2})} . \end{matrix}$ This inequality holds for $p \in (1, 2)$ , and $p^{*}$ belongs to $(2, \infty)$ .

Concerning the other terms in (3.1), we infer from the Calderón–Zygmund inequality that $\begin{array}{rcl} {‖ \partial_{1} K_{R^{2}} [f_{1}^{k}] + \partial_{2} K_{R^{2}} [f_{2}^{k}] ‖}_{L^{p} (R^{2})} & ⩽ & {‖ \nabla K_{R^{2}} [f_{1}^{k}] ‖}_{L^{p} (R^{2})} + {‖ \nabla K_{R^{2}} [f_{2}^{k}] ‖}_{L^{p} (R^{2})} \\ ⩽ & C_{p} ({‖ f_{1}^{k} ‖}_{L^{p} (R^{2})} + {‖ f_{2}^{k} ‖}_{L^{p} (R^{2})}) . \end{array}$

Setting $A_{k, 1} : = \partial_{1} K_{R^{2}} [f_{1}^{k}] + \partial_{2} K_{R^{2}} [f_{2}^{k}]$ and $A_{k, 2} : = K_{R^{2}} [f_{0}^{k}]$ ends the proof of Proposition 3.1.

3.2. The harmonic part (proof of Proposition 3.2)

In this subsection k is fixed, so to simplify the notations we will omit the parameter k in all the functions and domains. As $n_{k}$ is fixed, then we study the behavior of the flow around $n (= n_{k})$ obstacles ${B (z_{i}, ε)}_{i = 1, \dots, n}$ which shrink to n points as $ε \to 0$ . We denote by $d = {min}_{i \neq j} | z_{i} - z_{j} |$ and $B_{i}^{ε} : = B (z_{i}, ε)$ with $ε < d / 2$ . Let i be fixed, the goal of this subsection is to compare $\begin{matrix} \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} and E X_{i}^{ε} . \end{matrix}$

The first vector field is not tangent to $B_{j}^{ε}$ for $j \neq i$ , whereas $X_{i}^{ε}$ is. We introduce a vector field $v_{i}^{ε}$ which has an explicit form and verifies some of the properties of $X_{i}^{ε}$ . If we denote by $φ : [0, \infty) \to [0, \infty)$ a non-increasing function such that $φ (s) = 1$ if $s < 1 / 4$ and $φ (s) = 0$ if $s > 1 / 2$ , we introduce cut-off functions: $φ_{j} (x) : = φ (\frac{| x - z_{j} |}{d})$ which verifies $\begin{matrix} (3.2) & \{\begin{matrix} φ_{j} (x) \equiv 1 & in B (z_{j}, d / 4) \\ φ_{j} (x) \equiv 0 & in B {(z_{j}, d / 2)}^{c} . \end{matrix} \end{matrix}$ We use the inversion with respect to $B_{j}^{ε}$ to define, for each $i \neq j$ , $\begin{matrix} (3.3) & z_{i}^{* j, ε} : = z_{j} + ε^{2} \frac{z_{i} - z_{j}}{| z_{i} - z_{j} |^{2}} \in B (z_{j}, ε), \end{matrix}$ which allows us to introduce the vector fields $v_{i}^{ε}$ by: $\begin{matrix} (3.4) & v_{i}^{ε} (x) : = \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} - \nabla^{⊥} (\sum_{j \neq i} \frac{φ_{j} (x)}{2 π} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |}) . \end{matrix}$ The properties of such a vector field are listed here:

Lemma 3.4.
We have that:
$v_{i}^{ε}$ is divergence free;

$v_{i}^{ε}$ is tangent to the boundary $\partial Ω^{ε}$ ;

the circulations of $v_{i}^{ε}$ around $B_{j}^{ε}$ are equal to $δ_{i j}$ , for all $j = 1, \dots, n$ ;

$curl v_{i}^{ε} (x) = - \frac{1}{2 π} \sum_{j \neq i} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |} Δ φ_{j} (x) - \frac{1}{π} \sum_{j \neq i} (\frac{x - z_{i}^{* j, ε}}{| x - z_{i}^{* j, ε} |^{2}} - \frac{x - z_{j}}{| x - z_{j} |^{2}}) \cdot \nabla φ_{j} (x)$ in $Ω^{ε}$ .

Proof.
The first point is obvious, because this vector field is a perpendicular gradient. The last point is just a basic computation, noting that $Δ \frac{1}{2 π} ln | x - P | = δ_{P}$ (where $δ_{P}$ denotes the Dirac mass at the point P) and that $z_{i}^{* j, ε} \in B (z_{j}, ε)$ , so $Δ ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |}$ is equal to zero outside $B (z_{j}, ε)$ .

Concerning (2) and (3), we note that $v_{i}^{ε}$ is equal to $\begin{matrix} \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} \end{matrix}$ in a neighborhood of $B_{i}^{ε}$ (namely for any $x \in B (z_{i}, d / 4) ∖ B (z_{i}, ε)$ ), whereas it is equal to $\begin{matrix} \frac{1}{2 π} {(\frac{x - z_{i}}{| x - z_{i} |^{2}} - \frac{x - z_{i}^{* j, ε}}{| x - z_{i}^{* j, ε} |^{2}} + \frac{x - z_{j}}{| x - z_{j} |^{2}})}^{⊥} = - \frac{1}{2 π} \nabla^{⊥} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{i} | | x - z_{j} |} \end{matrix}$ in a neighborhood of $B_{j}^{ε}$ for $j \neq i$ (namely for any $x \in B (x_{j}, d / 4) ∖ B (x_{j}, ε)$ ). With the first form, it is clear that $v_{i}^{ε}$ is tangent to $B_{i}^{ε}$ and that $\begin{matrix} \oint_{\partial B_{i}^{ε}} v_{i}^{ε} \cdot \hat{τ} d s = 1 . \end{matrix}$ Thanks to the definition of $z_{i}^{* j, ε}$ (3.3), we can easily prove that for all $x \in \partial B_{j}^{ε}$ we have $\begin{array}{l} \frac{| x - z_{i}^{* j, ε} |}{| x - z_{i} | | x - z_{j} |} & = \frac{| (x - z_{j}) - ε^{2} \frac{z_{i} - z_{j}}{| z_{i} - z_{j} |^{2}} |}{| (x - z_{j}) - (z_{i} - z_{j}) | ε} \\ = \frac{{(| x - z_{j} |^{2} - 2 ε^{2} \frac{(x - z_{j} | z_{i} - z_{j})}{| z_{i} - z_{j} |^{2}} + ε^{4} \frac{1}{| z_{i} - z_{j} |^{2}})}^{1 / 2}}{{(| x - z_{j} |^{2} - 2 (x - z_{j} | z_{i} - z_{j}) + | z_{i} - z_{j} |^{2})}^{1 / 2} ε} \\ = \frac{1}{| z_{i} - z_{j} |}, \end{array}$ hence we deduce that the normal component of the perpendicular gradient is equal to zero, i.e. $v_{i}^{ε}$ is tangent to $B_{j}^{ε}$ . Moreover, there exists $r > 0$ such that $B (z_{i}^{* j, ε}, r) ⋐ B (z_{j}, ε)$ , so we get by Stokes formula that: $\begin{array}{rcl} \oint_{\partial B_{j}^{ε}} v_{i}^{ε} \cdot \hat{τ} d s & = & \frac{1}{2 π} \int_{B_{j}^{ε}} curl \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} d x + \frac{1}{2 π} \oint_{\partial B_{j}^{ε}} \frac{{(x - z_{j})}^{⊥}}{| x - z_{j} |^{2}} \cdot \hat{τ} d s \\ - \frac{1}{2 π} \oint_{\partial B (z_{i}^{* j, ε}, r)} \frac{{(x - z_{i}^{* j, ε})}^{⊥}}{| x - z_{i}^{* j, ε} |^{2}} \cdot \hat{τ} d s \\ = & 0 + 1 - 1 = 0, \end{array}$ which ends the proof of the lemma. □
Remark 3.5.
The fact that the obstacles are disks simplifies the expression of $v_{i}^{ε}$ . Otherwise, we would have to consider $T_{j}^{ε}$ the biholomorphism between ${(K_{j}^{ε} - z_{j})}^{c}$ and the exterior of the unit disk, and we would define: $\begin{matrix} v_{i}^{ε} (x) : = \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} + \nabla^{⊥} (\sum_{j \neq i} φ_{j} \frac{1}{2 π} ln \frac{| T_{j}^{ε} (x - z_{j}) - T_{j}^{ε} (z_{i} - z_{j}) | | T_{j}^{ε} (x - z_{j}) |}{\frac{| x - z_{i} |}{ε} | T_{j}^{ε} (x - z_{j}) - \frac{T_{j}^{ε} (z_{i} - z_{j})}{| T_{j}^{ε} (z_{i} - z_{j}) |^{2}} |}) \end{matrix}$ and we would have to prove that the modified $v_{i}^{ε}$ behaves as in (3.4) because $T_{j}^{ε} (x) = β_{j}^{ε} \frac{x}{ε} + h (\frac{x}{ε})$ with $h^{'} (x) = O (\frac{1}{| x |^{2}})$ at infinity.

For the sake of simplicity, we will work with the disks (see [3,13,18], where an extension to more general domains was considered).

To prove Proposition 3.2, we write: $\begin{matrix} (3.5) & \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} - E X_{i}^{ε} = (\frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} - E v_{i}^{ε}) + E (v_{i}^{ε} - X_{i}^{ε}) . \end{matrix}$

Step 1: Convergence of the first term on the right hand side of ( 3.5 ).

Let us fix $p \in [1, 2)$ . As this term is explicit, we can compute $\begin{array}{rcl} \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} - E v_{i}^{ε} (x) & = & (1 - E) \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} \\ + \sum_{j \neq i} (\nabla^{⊥} φ_{j}) (x) \frac{1}{2 π} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |} \\ + E \sum_{j \neq i} φ_{j} (x) \frac{1}{2 π} (\frac{{(x - z_{i}^{* j, ε})}^{⊥}}{| x - z_{i}^{* j, ε} |^{2}} - \frac{{(x - z_{j})}^{⊥}}{| x - z_{j} |^{2}}) . \end{array}$

This first term above tends to zero strongly in $L^{p}$ because the map $x \mapsto 1 / | x |$ belongs to $L_{loc}^{q}$ for $q \in (p, 2)$ and the Lebesgue measure of the support of $1 - E$ is equal to $π n ε^{2}$ , so we have $\begin{matrix} {‖ (1 - E) \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} ‖}_{L^{p}} ⩽ C {(n ε^{2})}^{α} \end{matrix}$ where $α = \frac{1}{p} - \frac{1}{q} \in (0, \frac{1}{p} - \frac{1}{2})$ and with C is independent of n and d (it depends only on p, q and $R_{0}$ such that ${z_{i}} \subset B (0, R_{0})$ ).

For the second term, we note from the definition of $z_{i}^{* j, ε}$ (3.3) that $\begin{matrix} | \frac{z_{i}^{* j, ε} - z_{j}}{x - z_{j}} | = \frac{ε^{2}}{| z_{i} - z_{j} | | x - z_{j} |} ⩽ \frac{ε^{2}}{d^{2} / 4} \forall x \in supp \nabla φ_{j} . \end{matrix}$ As $\begin{matrix} (3.6) & | ln \frac{| b + c |}{| b |} | ⩽ 2 \frac{| c |}{| b |}, if \frac{| c |}{| b |} ⩽ \frac{1}{2}, \end{matrix}$ we infer that if $ε ⩽ d / 2^{3 / 2}$ , then $\begin{matrix} {‖ \sum_{j \neq i} (\nabla^{⊥} φ_{j}) (x) \frac{1}{2 π} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |} ‖}_{L^{p}} ⩽ \frac{C ε^{2}}{d^{3}} {(n \int_{B (0, d / 2) ∖ B (0, d / 4)} 1 d x)}^{1 / p} ⩽ \frac{C ε^{2} n^{1 / p} d^{\frac{2}{p}}}{d^{3}} \end{matrix}$ where C depends only on $p \in [1, \infty]$ .

Concerning the last term, as $| \frac{z_{i}^{* j, ε} - z_{j}}{x - z_{j}} | ⩽ \frac{ε^{2}}{d ε}$ for all $x \in supp φ_{j}$ , we infer that $ε ⩽ d / 2$ implies $\begin{matrix} | x - z_{i}^{* j, ε} | ⩾ | x - z_{j} | - | z_{i}^{* j, ε} - z_{j} | ⩾ \frac{| x - z_{j} |}{2} \forall x \in supp φ_{j} . \end{matrix}$ By the relation $| \frac{a}{| a |^{2}} - \frac{b}{| b |^{2}} | = \frac{| a - b |}{| a | | b |}$ , we deduce that $\begin{matrix} {‖ E \sum_{j \neq i} φ_{j} (x) \frac{1}{2 π} (\frac{{(x - z_{i}^{* j, ε})}^{⊥}}{| x - z_{i}^{* j, ε} |^{2}} - \frac{{(x - z_{j})}^{⊥}}{| x - z_{j} |^{2}}) ‖}_{L^{p}} ⩽ \frac{ε^{2}}{2 π d} {(n \int_{B (0, d / 2) ∖ B (0, ε)} \frac{2^{p}}{| x |^{2 p}} d x)}^{1 / p} \end{matrix}$ which is bounded by $\frac{C ε^{2 / p} n^{1 / p}}{d}$ if $p > 1$ and by $\frac{C ε^{2} n}{d} ln \frac{d}{ε}$ for $p = 1$ . This allows us to get the convergence for the first term on the right hand side in (3.5): for all $ε ⩽ d / 2^{3 / 2}$ we have $\begin{array}{l} (3.7) & {‖ \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} - E v_{i}^{ε} ‖}_{L^{p} (R^{2})} & ⩽ C ({(n ε^{2})}^{(2 - p) / 4 p} + \frac{ε^{2} n^{1 / p} d^{\frac{2}{p}}}{d^{3}} + \frac{ε^{2 / p} n^{1 / p}}{d}) if p \in (1, 2) \\ (3.8) & ⩽ C ({(n ε^{2})}^{1 / 4} + \frac{ε^{2} n}{d} + \frac{ε^{2} n}{d} | ln ε |) if p = 1 \end{array}$ where C depends only on p and $R_{0}$ .

Step 2: convergence of the second term on the right hand side of ( 3.5 ).

If we denote ${\hat{v}}_{i}^{ε} : = v_{i}^{ε} - X_{i}^{ε}$ , we deduce from Lemma 3.4 that: $\begin{matrix} (3.9) & \{\begin{matrix} div {\hat{v}}_{i}^{ε} = 0 & in Ω^{ε} \\ curl {\hat{v}}_{i}^{ε} = {\hat{ω}}^{ε} & in Ω^{ε} \\ {\hat{v}}_{i}^{ε} \cdot \hat{n} = 0 & on \partial Ω^{ε} \\ \oint_{\partial B_{j}^{ε}} {\hat{v}}_{i}^{ε} \cdot \hat{τ} d s = 0 & for all j = 1, \dots, n \end{matrix} \end{matrix}$ where $\begin{matrix} {\hat{ω}}^{ε} (x) = - \frac{1}{2 π} \sum_{j \neq i} [Δ φ_{j} (x) ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |} + 2 \nabla φ_{j} (x) \cdot (\frac{x - z_{i}^{* j, ε}}{| x - z_{i}^{* j, ε} |^{2}} - \frac{x - z_{j}}{| x - z_{j} |^{2}})] . \end{matrix}$ Thanks to (2.3), we note that ${\hat{v}}_{i}^{ε} (x) = O (\frac{1}{| x |^{2}})$ at infinity, therefore, $\begin{matrix} \int_{Ω^{ε}} {\hat{ω}}^{ε} = - \sum_{j = 1}^{n} \oint_{\partial B_{j}^{ε}} {\hat{v}}_{i}^{ε} \cdot \hat{τ} d s + lim_{R \to \infty} \oint_{\partial B (0, R)} {\hat{v}}_{i}^{ε} \cdot \hat{τ} d s = 0 . \end{matrix}$

One of the main ideas in this proof is to perform an integration by parts in order to get the following.
Lemma 3.6.
Let ${\hat{v}}_{i}^{ε}$ be a vector field which verifies System ( 3.9 ) and such that $\int {\hat{ω}}^{ε} = 0$ . If we denote by $ψ_{i}^{ε}$ any stream function of ${\hat{v}}_{i}^{ε}$ (i.e. ${\hat{v}}_{i}^{ε} = \nabla^{⊥} ψ_{i}^{ε}$ ), then we have: $\begin{matrix} {‖ {\hat{v}}_{i}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} ⩽ (sup_{supp ({\hat{ω}}^{ε})} | ψ_{i}^{ε} |) {‖ {\hat{ω}}^{ε} ‖}_{L^{1} (Ω^{ε})} . \end{matrix}$
Proof.
We write the Biot–Savart law (2.7) for ${\hat{v}}_{i}^{ε}$ : $\begin{matrix} {\hat{v}}_{i}^{ε} = K^{ε} [{\hat{ω}}^{ε}] + \sum_{j = 1}^{n} α_{j} [{\hat{ω}}^{ε}] X_{j}^{ε} \end{matrix}$ where $α_{j} [{\hat{ω}}^{ε}] = \int_{Ω^{ε}} Φ_{j}^{ε} {\hat{ω}}^{ε} d x$ . We can rewrite this equation with the stream functions: $\begin{matrix} ψ_{i}^{ε} = Ψ_{D}^{ε} [{\hat{ω}}^{ε}] + \sum_{j = 1}^{n} α_{j} [{\hat{ω}}^{ε}] Ψ_{j}^{ε} + C, \end{matrix}$ with C a constant. From (2.6) we recall that $Ψ_{D}^{ε} [{\hat{ω}}^{ε}]$ is bounded at infinity. Using the behavior at infinity of $Ψ_{j}^{ε}$ (2.4) we compute: $\begin{matrix} \sum_{j = 1}^{n} α_{j} [{\hat{ω}}^{ε}] Ψ_{j}^{ε} = \sum_{j = 1}^{n} α_{j} [{\hat{ω}}^{ε}] (\frac{ln | x |}{2 π} + O (\frac{1}{| x |})) = \frac{ln | x |}{2 π} \sum_{j = 1}^{n} \int Φ_{j}^{ε} {\hat{ω}}^{ε} + O (\frac{1}{| x |}) = O (\frac{1}{| x |}), \end{matrix}$ where we have used that $\sum_{j = 1}^{n} Φ_{j}^{ε} \equiv 1$ and $\int {\hat{ω}}^{ε} = 0$ . Then, we have obtained that $ψ_{i}^{ε}$ is bounded at infinity. In the same way, we can prove that ${\hat{v}}_{i}^{ε} = O (\frac{1}{| x |^{2}})$ at infinity. Recalling that the stream functions are constant on each boundary component, we infer from the vanishing circulations that $\begin{array}{rcl} {‖ {\hat{v}}_{i}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} & = & \int_{Ω^{ε}} {\hat{v}}_{i}^{ε} \cdot {\hat{v}}_{i}^{ε} d x \\ = & - \int_{Ω^{ε}} \nabla ψ_{i}^{ε} \cdot {\hat{v}}_{i}^{ε ⊥} d x \\ = & - \int_{Ω^{ε}} ψ_{i}^{ε} {\hat{ω}}^{ε} d x - \sum_{j = 1}^{n} \oint_{\partial B_{j}^{ε}} ψ_{i}^{ε} {\hat{v}}_{i}^{ε} \cdot \hat{τ} d s + lim_{R \to \infty} \oint_{\partial B (0, R)} ψ_{i}^{ε} {\hat{v}}_{i}^{ε} \cdot \hat{τ} d s \\ = & - \int_{Ω^{ε}} ψ_{i}^{ε} {\hat{ω}}^{ε} d x \\ ⩽ & (sup_{supp ({\hat{ω}}^{ε})} | ψ_{i}^{ε} |) {‖ {\hat{ω}}^{ε} ‖}_{L^{1} (Ω^{ε})}, \end{array}$ which concludes the proof of the lemma. □

Using that the support of $\nabla φ_{j}$ is contained in $B (z_{j}, d / 2) ∖ B (z_{j}, d / 4)$ , estimating $‖ {\hat{ω}}^{ε} ‖_{L^{1}}$ is not too difficult. Indeed, following the reasoning in Step 1, we deduce that for all $ε ⩽ d / 2^{3 / 2}$ : $\begin{matrix} (3.10) & \begin{matrix} {‖ {\hat{ω}}^{ε} ‖}_{L^{1} (Ω^{ε})} & ⩽ \frac{2 ε^{2}}{2 π d^{2} / 4} \int_{supp \sum Δ φ_{j}} \frac{C}{d^{2}} d x + \frac{2 ε^{2}}{2 π d} \sum_{j} \int_{supp \nabla φ_{j}} \frac{C}{d | x - z_{j} |^{2}} d x \\ ⩽ C n {(\frac{ε}{d})}^{2} . \end{matrix} \end{matrix}$

Now we need an estimate of $ψ_{i}^{ε}$ on the support of ${\hat{ω}}^{ε}$ , i.e. on the support of $\nabla φ_{j}$ . For that, we use the explicit formula of $v_{i}^{ε}$ (3.4) to introduce the following stream function of ${\hat{v}}_{i}^{ε} = v_{i}^{ε} - X_{i}^{ε}$ : $\begin{matrix} ψ_{i}^{ε} (x) : = \frac{1}{2 π} ln | x - z_{i} | - \sum_{j \neq i} (φ_{j} \frac{1}{2 π} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |}) - (Ψ_{i}^{ε} - c_{i, i}^{ε}), \end{matrix}$ where $c_{i, i}^{ε}$ is the value of $Ψ_{i}^{ε}$ on $\partial B_{i}^{ε}$ . The first term on the right hand side is bounded by $\frac{1}{2 π} (| ln d / 4 | + ln R_{0})$ and the second by $4 ε^{2} / (π d^{2})$ (for $ε ⩽ d / 2^{3 / 2}$ ). The last term is the hardest to treat. The other main idea in this proof is to use the tools available for bounded domains, such as the maximum principle. To do this, we will use $\begin{matrix} i = i (x) : = x / | x |^{2} \end{matrix}$ the inversion with respect to the unit circle, which sends the exterior of the unit disk to the interior of the unit disk. Without loss of generality, let us assume that the obstacle $B_{i}^{ε}$ is centered at the origin: $B_{i}^{ε} = B (0, ε)$ . As the radius of $B_{i}^{ε}$ is ε, we should use $\begin{matrix} i_{ε} (x) : = i (x / ε) = ε i (x) . \end{matrix}$ The image of $Ω^{ε}$ by the last inversion, denoted by ${\tilde{Ω}}^{ε}$ , is the unit disk with $n - 1$ holes. These holes ${\tilde{B}}_{j}^{ε}$ correspond to the image of $B_{j}^{ε}$ by $i_{ε}$ , for $j \neq i$ . We give here some properties of the inversion $i_{ε}$ :
Lemma 3.7.
We have that
$i_{ε}^{- 1} (x) = i_{ε} (x) = ε i (x)$ ;

$i_{ε} (B ((ρ, 0), ε)) = B ((\frac{ε}{ρ (1 - ε^{2})}, 0), \frac{ε^{2}}{ρ (1 - ε^{2})})$ ;

$det D i (x) = - 1 / | x |^{4}$ .
Let $f : Ω^{ε} \to R$ , and $g (x) : = f (i_{ε}^{- 1} (x))$ . If $f (x) = O (1)$ at infinity and $Δ f \equiv 0$ in a neighborhood of infinity, then
$\nabla^{⊥} g (x) = - ε D i (x) (\nabla^{⊥} f) (ε i (x))$ ;

$Δ g (x) = ε^{2} | det (D i (x)) | (Δ f) (ε i (x))$ .
If $f (x) = \frac{α}{2 π} ln | x | + O (1)$ and $Δ f = 0$ , then $Δ g (x) = - α δ_{0}$ .

The last item can be proved using that a harmonic function in $B (0, d) ∖ {0}$ which is bounded can be extended to a harmonic function in $B (0, d)$ . All the other items can be shown by basic computations. Now, we use this lemma to prove the following
Lemma 3.8.
There exists a constant C, independent of ε, d, n, such that $\begin{matrix} sup_{supp ({\hat{ω}}^{ε})} | Ψ_{i}^{ε} - c_{i, i}^{ε} | ⩽ C n^{2} (| ln d | + | ln ε | + n \frac{ε^{2}}{d^{2}} + \sum_{j = 1}^{n} {‖ {\hat{v}}_{j}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2}) . \end{matrix}$
Proof.
By the definition of $Ψ_{i}^{ε}$ , we know that there exists some constants $c_{i, j}^{ε}$ such that $\begin{matrix} Ψ_{i}^{ε} \equiv c_{i, j}^{ε} on \partial B_{j}^{ε}, for all j = 1, \dots, n . \end{matrix}$ Let us denote the inversion of $Ψ_{i}^{ε}$ by ${\tilde{Ψ}}^{ε} (x) : = Ψ_{i}^{ε} (i_{ε}^{- 1} (x)) - c_{i, i}^{ε} = Ψ_{i}^{ε} (ε i (x)) - c_{i, i}^{ε}$ . Then it verifies: $\begin{matrix} \{\begin{matrix} Δ {\tilde{Ψ}}^{ε} = - δ_{0} & in {\tilde{Ω}}^{ε} \\ {\tilde{Ψ}}^{ε} = 0 & on \partial B (0, 1) \\ {\tilde{Ψ}}^{ε} = c_{j}^{ε} : = c_{i, j}^{ε} - c_{i, i}^{ε} & on \partial {\tilde{B}}_{j}^{ε}, for all j \neq i \\ \oint_{\partial B (0, 1)} \nabla^{⊥} {\tilde{Ψ}}^{ε} \cdot \hat{τ} d s = - 1 \\ \oint_{\partial {\tilde{B}}_{j}^{ε}} \nabla^{⊥} {\tilde{Ψ}}^{ε} \cdot \hat{τ} d s = 0 & for all j = 1, \dots, n, j \neq i, \end{matrix} \end{matrix}$ after remarking, by a simple calculation, that the circulation changes sign upon inversion.

In a bounded domain, we can use the results from Section 2 to derive the following decomposition: $\begin{matrix} (3.11) & {\tilde{Ψ}}^{ε} = {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] + \sum_{j \neq i} c_{j}^{ε} {\tilde{Φ}}_{j}^{ε} . \end{matrix}$ We have already proved in Section 2 that $0 ⩽ {\tilde{Φ}}_{j}^{ε} ⩽ 1$ .

If we denote $f : = {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] + \frac{1}{2 π} ln | x |$ , we have: $\begin{matrix} Δ f = 0, f = 0 on \partial B (0, 1), and f = \frac{1}{2 π} ln | x | on \partial {\tilde{B}}_{j}^{ε} . \end{matrix}$ Using that the distance between the origin and any hole is $O (ε / d)$ (see Lemma 3.7), we can apply the maximum principle to f to say that $| f (x) | ⩽ C | ln (ε / d) |$ . Then we have $\begin{matrix} | {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] (x) | ⩽ C (| ln \frac{ε}{d} | + | ln | x | |) \end{matrix}$ which verifies: $\begin{matrix} | {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] | ⩽ C | ln \frac{ε}{d} | on i_{ε} (supp (\hat{ω})) . \end{matrix}$

To finish the estimate of ${\tilde{Ψ}}^{ε}$ in (3.11), we need to estimate the constants $c_{j}^{ε}$ . We compute the circulation of $\nabla^{⊥} {\tilde{Ψ}}^{ε}$ $\begin{matrix} 0 = \oint_{\partial {\tilde{B}}_{k}^{ε}} \nabla^{⊥} {\tilde{Ψ}}^{ε} \cdot \hat{τ} d s = \oint_{\partial {\tilde{B}}_{k}^{ε}} \nabla^{⊥} {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] \cdot \hat{τ} d s + \sum_{j \neq i} c_{j}^{ε} \oint_{\partial {\tilde{B}}_{k}^{ε}} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} \cdot \hat{τ} d s, \end{matrix}$ for any $k \neq i$ . By (2.5), we compute the circulation of $\nabla^{⊥} {\tilde{Ψ}}_{D}^{ε}$ $\begin{matrix} \oint_{\partial {\tilde{B}}_{k}^{ε}} \nabla^{⊥} {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] \cdot \hat{τ} d s = - \int_{{\tilde{Ω}}^{ε}} {\tilde{Φ}}_{k}^{ε} Δ {\tilde{Ψ}}_{D}^{ε} [- δ_{0}] = {\tilde{Φ}}_{k}^{ε} (0) . \end{matrix}$ If we denote by P the matrix with coefficients $p_{k, j} = \oint_{\partial {\tilde{B}}_{k}^{ε}} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} \cdot \hat{τ} d s = - \int_{{\tilde{Ω}}^{ε}} \nabla {\tilde{Φ}}_{j}^{ε} \cdot \nabla {\tilde{Φ}}_{k}^{ε}$ for $k, j \neq i$ , then we have $\begin{matrix} P (\begin{matrix} c_{1}^{ε} \\ ⋮ \\ c_{n}^{ε} \end{matrix}) = - (\begin{matrix} {\tilde{Φ}}_{1}^{ε} (0) \\ ⋮ \\ {\tilde{Φ}}_{n}^{ε} (0) \end{matrix}), \end{matrix}$ where we know that $| {\tilde{Φ}}_{j}^{ε} (0) | ⩽ 1$ . Moreover, if we expand the harmonic vector field $\nabla^{⊥} {\tilde{Φ}}_{j}^{ε}$ in the basis ${{\tilde{Ψ}}_{l}^{ε}}$ , we get $\begin{array}{l} (3.12) & \begin{aligned} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} = \sum_{l \neq i} (\oint_{\partial {\tilde{B}}_{l}^{ε}} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} \cdot \hat{τ} d s) {\tilde{X}}_{l}^{ε} \\ \int_{{\tilde{Ω}}^{ε}} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} \cdot {\tilde{X}}_{k}^{ε} = \sum_{l \neq i} (\int_{{\tilde{Ω}}^{ε}} {\tilde{X}}_{k}^{ε} \cdot {\tilde{X}}_{l}^{ε}) (\oint_{\partial {\tilde{B}}_{l}^{ε}} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} \cdot \hat{τ} d s) . \end{aligned} \end{array}$ Concerning the term on the left hand side, we integrate by parts: $\begin{matrix} \int_{{\tilde{Ω}}^{ε}} \nabla^{⊥} {\tilde{Φ}}_{j}^{ε} \cdot {\tilde{X}}_{k}^{ε} = - \int_{{\tilde{Ω}}^{ε}} {\tilde{Φ}}_{j} curl {\tilde{X}}_{k}^{ε} - \sum_{l \neq i} {\tilde{Φ}}_{j}^{ε} |_{\partial {\tilde{B}}_{l}^{ε}} \oint_{\partial {\tilde{B}}_{l}^{ε}} {\tilde{X}}_{k}^{ε} \cdot \hat{τ} d s = - δ_{j, k} . \end{matrix}$ Then, if we denote M by the matrix $m_{l, k} = \int_{{\tilde{Ω}}^{ε}} {\tilde{X}}_{k}^{ε} \cdot {\tilde{X}}_{l}^{ε}$ , identity (3.12) can be written: $\begin{matrix} - I_{n - 1} = P M \end{matrix}$ which means that $P^{- 1} = - M$ and that $| c_{j}^{ε} | ⩽ \sum_{k \neq i} | m_{k, j} | ⩽ C \sum_{k \neq i} ‖ {\tilde{X}}_{k}^{ε} ‖_{L^{2} ({\tilde{Ω}}^{ε})}^{2}$ .

The last step of this proof is to evaluate $‖ {\tilde{X}}_{k}^{ε} ‖_{L^{2} ({\tilde{Ω}}^{ε})}^{2}$ . Actually, we remark that $\begin{matrix} {\tilde{Ψ}}_{k}^{ε} (x) = - (Ψ_{k}^{ε} - c_{k, i}^{ε} - Ψ_{i}^{ε} + c_{i, i}^{ε}) (i_{ε}^{- 1} (x)) . \end{matrix}$ Using that $\begin{matrix} {\tilde{X}}_{k}^{ε} = \nabla^{⊥} {\tilde{Ψ}}_{k}^{ε} (x) = - D {(i_{ε}^{- 1})}^{t} (x) (X_{k}^{ε} - X_{i}^{ε}) (i_{ε}^{- 1} (x)), \end{matrix}$ by Lemma 3.7, we compute $\begin{array}{rcl} {‖ {\tilde{X}}_{k}^{ε} ‖}_{L^{2} ({\tilde{Ω}}^{ε})}^{2} & ⩽ & \int_{{\tilde{Ω}}^{ε}} {| D (i_{ε}^{- 1} (x)) |}^{2} {| X_{k}^{ε} - X_{i}^{ε} |}^{2} (i_{ε}^{- 1} (x)) d x \\ ⩽ & \int_{Ω^{ε}} {| X_{k}^{ε} - X_{i}^{ε} |}^{2} (y) d y \\ ⩽ & {‖ X_{k}^{ε} - X_{i}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} . \end{array}$ Recalling the definition of ${\hat{v}}_{i}^{ε}$ , we write $\begin{array}{rcl} {‖ X_{k}^{ε} - X_{i}^{ε} ‖}_{L^{2} (Ω^{ε})} & ⩽ & {‖ {\hat{v}}_{k}^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ {\hat{v}}_{i}^{ε} ‖}_{L^{2} (Ω^{ε})} \\ + \frac{1}{2 π} {‖ \frac{{(x - z_{k})}^{⊥}}{| x - z_{k} |^{2}} - \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} ‖}_{L^{2} (Ω^{ε})} \\ + {‖ \nabla^{⊥} (\sum_{j \neq k} φ_{j} \frac{1}{2 π} ln \frac{| x - z_{k}^{* j, ε} |}{| x - z_{j} |} - \sum_{j \neq i} φ_{j} \frac{1}{2 π} ln \frac{| x - z_{i}^{* j, ε} |}{| x - z_{j} |}) ‖}_{L^{2} (Ω^{ε})} . \end{array}$ The last term was already estimated in Step 1, where it was proved that for $ε ⩽ d / 2^{3 / 2}$ , this is bounded by $C n^{1 / 2} (\frac{ε^{2}}{d^{2}} + \frac{ε}{d})$ . Adding that $\frac{{(x - z_{k})}^{⊥}}{| x - z_{k} |^{2}} - \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}}$ is square integrable at infinity, the third term in the right hand side is also bounded by $C {(1 + | ln ε |)}^{1 / 2}$ , with C depending only on $R_{0}$ .

This allows us to conclude that:
for all j, and $ε ⩽ d / 2^{3 / 2}$ $\begin{array}{rcl} | c_{j}^{ε} | & ⩽ & C \sum_{k \neq i} ({‖ {\hat{v}}_{k}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + {‖ {\hat{v}}_{i}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + 1 + | ln ε | + n \frac{ε^{2}}{d^{2}}) \\ ⩽ & C n (\sum_{j = 1}^{n} {‖ {\hat{v}}_{j}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + 1 + | ln ε | + n \frac{ε^{2}}{d^{2}}), \end{array}$

for all $ε ⩽ d / 2^{3 / 2}$ $\begin{matrix} {‖ {\tilde{Ψ}}^{ε} ‖}_{L^{\infty} (i_{ε} (supp (\hat{ω})))} ⩽ C [| ln \frac{ε}{d} | + n^{2} (\sum_{j = 1}^{n} {‖ {\hat{v}}_{j}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + 1 + | ln ε | + n \frac{ε^{2}}{d^{2}})], \end{matrix}$
which ends the proof of the lemma. □

Putting together all the results of this subsection, we conclude that $\begin{matrix} {‖ {\hat{v}}_{i}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} ⩽ C_{1} n {(\frac{ε}{d})}^{2} n^{2} (| ln d | + | ln ε | + n \frac{ε^{2}}{d^{2}} + \sum_{j = 1}^{n} {‖ {\hat{v}}_{j}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2}), \end{matrix}$ where $C_{1}$ is independent of ε, d, i, n.

By summing in i, we infer that, for any ε such that $\begin{array}{l} ε ⩽ \frac{d}{n^{2} \sqrt{2 C_{1}}}, \\ \sum_{i = 1}^{n} {‖ {\hat{v}}_{i}^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} ⩽ 2 C_{1} {(\frac{ε n^{2}}{d})}^{2} (| ln d | + | ln ε | + n \frac{ε^{2}}{d^{2}}) ⩽ C {(\frac{ε n^{2}}{d})}^{2} | ln ε | . \end{array}$ In particular, we have that $‖ v_{i}^{ε} - X_{i}^{ε} ‖_{L^{2} (Ω^{ε})} \to 0$ , as $ε \to 0$ . Putting (3.7) together with (3.8), we conclude the proof of Proposition 3.2.
3.3. The Laplacian-inverse (proof of Proposition 3.3)

As in the previous subsection, k is fixed, so that, to simplify notation, we will omit the parameter k in all the functions and domains. Let $M_{0}$ , and ${z_{i}}_{i = 1, \dots, n} \subset B (0, R_{0})$ be fixed. The goal of this section is to prove that the difference between the portion of the Biot–Savart law with zero circulation around each $B_{i}^{ε}$ , namely: $\begin{matrix} E K^{ε} [f |_{Ω^{ε}}] + \sum_{i = 1}^{n} m_{i}^{ε} [f] E X_{i}^{ε} \end{matrix}$ and the Biot–Savart formula in $R^{2}$ : $\begin{matrix} K_{R^{2}} [E f |_{Ω^{ε}}] \end{matrix}$ converges strongly to zero in $L^{2} (R^{2})$ as $ε \to 0$ . Moreover, this convergence is uniform in f, for those f verifying: $\begin{matrix} ‖ f ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0} and f is compactly supported. \end{matrix}$

As before, we use the cut-off function $φ_{j}$ defined in (3.2), and the notation: $\begin{matrix} (3.13) & y^{* j, ε} : = z_{j} + ε^{2} \frac{y - z_{j}}{| y - z_{j} |^{2}}, \end{matrix}$ but now we introduce $\begin{array}{rcl} v^{ε} [f] (x) & : = & \frac{1}{2 π} \int_{Ω^{ε}} \frac{{(x - y)}^{⊥}}{| x - y |^{2}} f (y) d y \\ - \nabla^{⊥} (\sum_{i = 1}^{n} \frac{φ_{i} (x)}{2 π} \int_{Ω^{ε}} ln \frac{| x - y^{* i, ε} |}{| x - z_{i} |} f (y) d y) . \end{array}$ The properties of such a vector field are listed here:

Lemma 3.9.
For any $f \in L_{c}^{p_{0}} (R^{2})$ and $ε \in (0, d / 2)$ , we have that:
$v^{ε} [f]$ is divergence free;

$v^{ε} [f]$ is tangent to the boundary;

the circulations of $v^{ε} [f]$ around $B_{i}^{ε}$ are equal to zero, for all $i = 1, \dots, n$ ;

the curl of $v^{ε} [f]$ is equal to: $\begin{array}{l} f |_{Ω^{ε}} (x) - \sum_{i = 1}^{n} \frac{Δ φ_{i} (x)}{2 π} \int_{Ω^{ε}} ln \frac{| x - y^{* i, ε} |}{| x - z_{i} |} f (y) d y \\ - \sum_{i = 1}^{n} \frac{\nabla φ_{i} (x)}{π} \cdot \int_{Ω^{ε}} (\frac{x - y^{* i, ε}}{| x - y^{* i, ε} |^{2}} - \frac{x - z_{i}}{| x - z_{i} |^{2}}) f (y) d y . \end{array}$

Proof.
Concerning items (1), (2) and (4), the proof is exactly the same as that of Lemma 3.4. It remains to establish (3). For any $y \in Ω^{ε}$ fixed, we can consider $r_{y} > 0$ such that $B (y^{* i, ε}, r_{y}) ⋐ B (z_{i}, ε)$ , and we deduce by Stokes formula that: $\begin{array}{l} \frac{1}{2 π} \oint_{\partial B_{i}^{ε}} \nabla^{⊥} (ln \frac{| x - y^{* i, ε} |}{| x - z_{i} |}) \cdot \hat{τ} d s \\ = \frac{1}{2 π} \oint_{\partial B_{i}^{ε}} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} \cdot \hat{τ} d s \\ - \frac{1}{2 π} \oint_{\partial B (y^{* j, ε}, r_{y})} \frac{{(x - y^{* j, ε})}^{⊥}}{| x - y^{* j, ε} |^{2}} \cdot \hat{τ} d s \\ = 1 - 1 = 0 . \end{array}$ Moreover, as $f |_{Ω^{ε}}$ belongs to $L^{1} \cap L^{p_{0}} (R^{2})$ with $p_{0} > 2$ , we claim that $x \mapsto K_{R^{2}} [E f |_{Ω^{ε}}] (x)$ is continuous on $R^{2}$ . Indeed, by standard estimates, we know that it is bounded (see [12], for example), and, by the Calderón–Zygmund inequality we may infer that its gradient belongs to $L^{p_{0}}$ . So $K_{R^{2}} [E f |_{Ω^{ε}}]$ belongs to $W_{loc}^{1, p_{0}} (R^{2})$ which is embedded in $C (R^{2})$ . Moreover, $K_{R^{2}} [E f |_{Ω^{ε}}]$ is curl free on $B_{i}^{ε}$ , and therefore, $\begin{matrix} \frac{1}{2 π} \oint_{\partial B_{i}^{ε}} K_{R^{2}} [E f |_{Ω^{ε}}] (x) \cdot \hat{τ} d s = 0, \end{matrix}$ which concludes the proof of the lemma. □

To prove Proposition 3.3, we decompose as follows: $\begin{array}{l} K_{R^{2}} [E f |_{Ω^{ε}}] - E (K^{ε} [f |_{Ω^{ε}}] + \sum_{i = 1}^{n} m_{i}^{ε} [f] X_{i}^{ε}) \\ = (K_{R^{2}} [E f |_{Ω^{ε}}] - E v^{ε} [f]) + E (v^{ε} [f] - K^{ε} [f |_{Ω^{ε}}] - \sum_{i = 1}^{n} m_{i}^{ε} [f] X_{i}^{ε}) . \end{array}$

Step 1: Convergence of the first term on the right hand side.

We compute $\begin{array}{l} K_{R^{2}} [E f |_{Ω^{ε}}] - E v^{ε} [f] \\ = (1 - E) K_{R^{2}} [E f |_{Ω^{ε}}] \\ + \sum_{i = 1}^{n} \frac{\nabla^{⊥} φ_{i} (x)}{2 π} \int_{Ω^{ε}} ln \frac{| x - y^{* i, ε} |}{| x - z_{i} |} f (y) d y \\ + E \sum_{i = 1}^{n} \frac{φ_{i} (x)}{2 π} \int_{Ω^{ε}} {(\frac{x - y^{* i, ε}}{| x - y^{* i, ε} |^{2}} - \frac{x - z_{i}}{| x - z_{i} |^{2}})}^{⊥} f (y) d y . \end{array}$

As $K_{R^{2}} [E f |_{Ω^{ε}}]$ is uniformly bounded by $C ‖ f ‖_{L^{1} (R^{2})}^{α} ‖ f ‖_{L^{p_{0}} (R^{2})}^{1 - α} ⩽ C M_{0}$ (with $α \in (0, 1)$ which depends only on $p_{0}$ , see [12], for example $α_{\infty} = 1 / 2$ ), it is obvious that the $L^{2}$ norm of first term on the right hand side is less than $C M_{0} {(n ε^{2})}^{1 / 2}$ uniformly in f verifying $‖ f ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0}$ .

For the second term, as $\begin{matrix} | \frac{y^{* i, ε} - z_{i}}{x - z_{i}} | = \frac{ε^{2}}{| y - z_{i} | | x - z_{i} |} ⩽ \frac{ε}{d / 4} \forall x \in supp \nabla φ_{i}, y \in B {(z_{i}, ε)}^{c}, \end{matrix}$ we remark from (3.6) that, for $ε ⩽ d / 8$ , we have $\begin{array}{l} | \sum_{i = 1}^{n} \frac{\nabla^{⊥} φ_{i} (x)}{2 π} \int_{Ω^{ε}} ln \frac{| x - y^{* i, ε} |}{| x - z_{i} |} f (y) d y | \\ ⩽ C \frac{ε^{2}}{d} \sum_{i = 1}^{n} | \nabla^{⊥} φ_{i} (x) | \int_{Ω^{ε}} \frac{| f (y) |}{| y - z_{i} |} d y \\ ⩽ C \frac{ε^{2}}{d} \sum_{i = 1}^{n} | \nabla^{⊥} φ_{i} (x) | M_{0} \end{array}$ which has an $L^{2} (R^{2})$ norm less than $C M_{0} \frac{ε^{2} n^{1 / 2}}{d}$ , uniformly in f verifying $‖ f ‖_{L^{1} (R^{2})} ⩽ M_{0}$ .

For the last term, we decompose the integrals in two parts: $\begin{array}{l} | \frac{φ_{i} (x)}{2 π} \int_{Ω^{ε}} {(\frac{x - y^{* i, ε}}{| x - y^{* i, ε} |^{2}} - \frac{x - z_{i}}{| x - z_{i} |^{2}})}^{⊥} f (y) d y | \\ ⩽ \frac{| φ_{i} (x) |}{2 π} (\int_{B (z_{i}, 2 ε) ∖ B_{i}^{ε}} (\frac{1}{| x - y^{* i, ε} |} + \frac{1}{| x - z_{i} |}) | f (y) | d y \\ + \int_{Ω^{ε} ∖ B (z_{i}, 2 ε)} | \frac{x - y^{* i, ε}}{| x - y^{* i, ε} |^{2}} - \frac{x - z_{i}}{| x - z_{i} |^{2}} | | f (y) | d y) . \end{array}$ For the first integral, we can verify that $| x - y^{* i, ε} |^{2} | y - z_{i} |^{2} = | y - x^{* i, ε} |^{2} | x - z_{i} |^{2}$ , hence we have $\begin{array}{l} \int_{B (z_{i}, 2 ε) ∖ B_{i}^{ε}} (\frac{1}{| x - y^{* i, ε} |} + \frac{1}{| x - z_{i} |}) | f (y) | d y \\ ⩽ \frac{1}{| x - z_{i} |} (\int_{B (z_{i}, 2 ε) ∖ B_{i}^{ε}} \frac{2 ε}{| y - x^{* i, ε} |} | f (y) | d y + ‖ f 1_{B (z_{i}, 2 ε)} ‖_{L^{1}}) \\ ⩽ \frac{1}{| x - z_{i} |} (C ε ‖ f ‖_{L^{1} (R^{2})}^{α} ‖ f ‖_{L^{p_{0}} (R^{2})}^{1 - α} + ‖ f ‖_{L^{p_{0}} (R^{2})} {(π 4 ε^{2})}^{1 / p_{0}^{'}}) \\ ⩽ \frac{C M_{0} ε}{| x - z_{i} |}, \end{array}$ where we have used again the estimate on $K_{R^{2}} [f]$ [12]. Concerning the second integral, we use the relation $| \frac{a}{| a |^{2}} - \frac{b}{| b |^{2}} | = \frac{| a - b |}{| a | | b |}$ and that for $y \in B {(z_{i}, 2 ε)}^{c}$ and $x \in Ω^{ε}$ , we have $| x - y^{* i, ε} | ⩾ | x - z_{i} | - \frac{ε^{2}}{| y - z_{i} |} ⩾ \frac{ε}{2}$ . So we compute: $\begin{array}{l} \int_{Ω^{ε} ∖ B (z_{i}, 2 ε)} | \frac{x - y^{* i, ε}}{| x - y^{* i, ε} |^{2}} - \frac{x - z_{i}}{| x - z_{i} |^{2}} | | f (y) | d y \\ = \int_{Ω^{ε} ∖ B (z_{i}, 2 ε)} \frac{ε^{2} | y - z_{i} |^{- 1}}{| x - y^{* i, ε} | | x - z_{i} |} | f (y) | d y \\ ⩽ \frac{2 ε}{| x - z_{i} |} \int_{Ω^{ε}} \frac{| f (y) |}{| y - z_{i} |} d y \\ ⩽ \frac{C M_{0} ε}{| x - z_{i} |} . \end{array}$ Therefore, the $L^{2} (R^{2})$ norm of the last term can be estimated as $\begin{array}{l} {‖ E \sum_{i = 1}^{n} \frac{φ_{i} (x)}{2 π} \int_{Ω^{ε}} {(\frac{x - y^{* i, ε}}{| x - y^{* i, ε} |^{2}} - \frac{x - z_{i}}{| x - z_{i} |^{2}})}^{⊥} f (y) d y ‖}_{L^{2}} \\ ⩽ {‖ E \sum_{i = 1}^{n} | φ_{i} (x) | \frac{C M_{0} ε}{| x - z_{i} |} ‖}_{L^{2}} \\ ⩽ C M_{0} ε n^{1 / 2} {‖ \frac{1}{| z |} ‖}_{L^{2} (B (0, d / 2) ∖ B (0, ε))} \\ ⩽ C M_{0} ε | ln ε | n^{1 / 2}, \end{array}$ which also tends to zero as $ε \to 0$ , uniformly in f verifying $‖ f ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0}$ .

Therefore, we have established that $‖ K_{R^{2}} [E f |_{Ω^{ε}}] - E v^{ε} [f] ‖_{L^{2} (R^{2})} \to 0$ as $ε \to 0$ , uniformly in f verifying $‖ f ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0}$ .

Step 2: Convergence of the second term on the right hand side.

We define ${\hat{v}}^{ε} [f] = v^{ε} [f] - K^{ε} [f |_{Ω^{ε}}] - \sum_{i = 1}^{n} m_{i}^{ε} [f] X_{i}^{ε}$ . By Lemma 3.9, this vector field verifies $\begin{matrix} \{\begin{matrix} div {\hat{v}}^{ε} [f] = 0 = div (v^{ε} [f] - K_{R^{2}} [E f |_{Ω^{ε}}]) & in Ω^{ε} \\ curl {\hat{v}}^{ε} [f] = curl v^{ε} [f] - f = curl (v^{ε} [f] - K_{R^{2}} [E f |_{Ω^{ε}}]) & in Ω^{ε} \\ \oint_{B_{i}^{ε}} {\hat{v}}^{ε} [f] \cdot \hat{τ} d s = 0 = \oint_{B_{i}^{ε}} (v^{ε} [f] - K_{R^{2}} [E f |_{Ω^{ε}}]) \cdot \hat{τ} d s & for all i = 1, \dots, n \\ lim {\hat{v}}^{ε} [f] = 0 = lim (v^{ε} [f] - K_{R^{2}} [E f |_{Ω^{ε}}]) & as x \to \infty \\ {\hat{v}}^{ε} [f] \cdot n = 0 & on \partial Ω^{ε} . \end{matrix} \end{matrix}$ This implies that ${\hat{v}}^{ε} [f]$ is the Leray projection1
¹
Projection on divergence free vector fields which are tangent to the boundary.

of $v^{ε} [f] - K_{R^{2}} [E f |_{Ω^{ε}}]$ . Therefore, by orthogonality of this projection in $L^{2}$ , we have $\begin{matrix} {‖ {\hat{v}}^{ε} [f] ‖}_{L^{2} (Ω^{ε})} ⩽ {‖ v^{ε} [f] - K_{R^{2}} [E f |_{Ω^{ε}}] ‖}_{L^{2} (Ω^{ε})} ⩽ C M_{0} ε | ln ε | . \end{matrix}$

This concludes the proof of Proposition 3.3. Remark 3.10.
This argument is already present in [3], and it explains why $L^{2}$ plays a special role: for every ε, the Leray projector is an operator in $L^{2}$ bounded by 1. This argument cannot be used for the harmonic part, because we only proved estimates in $L^{p}$ for $p < 2$ , and it is not clear that the Leray projection is uniformly continuous in $L^{p}$ (see [18, Section 5.3]). The Step 2 in Section 3.2 avoids such a consideration.

3.4. Construction of $ε_{k}$

The goal of this section is to construct suitable $ε_{k}$ , so that Theorem 1.1 will hold true.

So let us fix $ω_{0} \in L_{c}^{p_{0}} (R^{2})$ for some $p_{0} \in (2, \infty]$ . Let us also fix $R_{0} > 0$ so that $supp ω_{0} \subset B (0, R_{0})$ . We consider a sequence $ω_{0}^{k} \in C_{c}^{\infty} (B (0, R_{0}))$ , ${γ_{i}^{k}}_{i = 1, \dots, n_{k}} \subset R$ , ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset B (0, R_{0})$ such that $\begin{matrix} ω_{0}^{k} ⇀ ω_{0} weakly in L^{p_{0}} (R^{2}) \end{matrix}$ and $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ weak- * in BM (R^{2}), \end{matrix}$ for some $μ \in {BM}_{c} (R^{2})$ . The first limit allows us to define $M_{0} \in R_{+}$ as following: $\begin{matrix} (3.14) & M_{0} : = sup_{k \in N} {{‖ ω_{0}^{k} ‖}_{L^{1} \cap L^{p_{0}} (R^{2})}} . \end{matrix}$ The second limit implies that $‖ \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ‖_{BM (R^{2})}$ is uniformly bounded in k, so there exists $M_{1} \in R_{+}$ such that: $\begin{matrix} (3.15) & \sum_{i = 1}^{n_{k}} | γ_{i}^{k} | ⩽ M_{1}, \forall k \in N, \forall i = 1, \dots, n_{k} . \end{matrix}$

Now, we fix $k \in N^{*}$ , and we are looking for a precise choice of $ε_{k}$ .

By Proposition 3.2, there exists $ε_{1}^{k}$ such that for any $ε \in (0, ε_{1}^{k})$ , there is $v_{i}^{ε}$ such that $\begin{matrix} {‖ \frac{{(x - z_{i})}^{⊥}}{2 π | x - z_{i} |^{2}} - E v_{i}^{ε} (x) ‖}_{L^{1} \cap L^{2 - \frac{1}{k}} (R^{2})} + {‖ E v_{i}^{ε} - E X_{i}^{ε, k} ‖}_{L^{2} (R^{2})} ⩽ \frac{1}{M_{1} k}, \end{matrix}$ for any $i = 1, \dots, n_{k}$ .

By Proposition 3.3, with $M_{0}$ defined in (3.14), there exists $ε_{2}^{k}$ such that for any $ε \in (0, ε_{2}^{k})$ and f verifying: $\begin{matrix} ‖ f ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0} and f is compactly supported, \end{matrix}$ we have $\begin{matrix} {‖ K_{R^{2}} [E f |_{Ω^{ε, k}}] - E K^{ε, k} [f |_{Ω^{ε, k}}] - \sum_{i = 1}^{n_{k}} m_{i}^{ε, k} [f] E X_{i}^{ε, k} ‖}_{L^{2} (R^{2})} ⩽ \frac{1}{k} . \end{matrix}$

We recall that $ε_{1}^{k}$ and $ε_{2}^{k}$ are chosen small enough such that the disks are disjoints. We finally choose: $\begin{matrix} (3.16) & ε_{k} : = min {ε_{1}^{k}, ε_{2}^{k}, \frac{1}{k}} . \end{matrix}$

4. Time evolution

The goal of this section is to prove Theorem 1.1, using Propositions 3.1, 3.2 and 3.3.

So let us fix $ω_{0} \in L_{c}^{p_{0}} (R^{2})$ for some $p_{0} \in (2, \infty]$ . Let us also fix $R_{0} > 0$ so that $supp ω_{0} \subset B (0, R_{0})$ . We consider a sequence $ω_{0}^{k} \in C_{c}^{\infty} (B (0, R_{0}))$ , ${γ_{i}^{k}}_{i = 1, \dots, n_{k}} \subset R$ , ${z_{i}^{k}}_{i = 1, \dots, n_{k}} \subset B (0, R_{0})$ such that $\begin{matrix} ω_{0}^{k} ⇀ ω_{0} weakly in L^{p_{0}} (R^{2}) \end{matrix}$ and $\begin{matrix} \sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ weak- * in BM (R^{2}), \end{matrix}$ for some $μ \in {BM}_{c} (R^{2})$ . As in the previous subsection, we introduce: $\begin{matrix} M_{0} : = sup_{k \in N} {{‖ ω_{0}^{k} ‖}_{L^{1} \cap L^{p_{0}} (R^{2})}} . \end{matrix}$ and $M_{1} \in R_{+}$ by $\begin{matrix} M_{1} : = sup_{k \in N} {\sum_{i = 1}^{n_{k}} | γ_{i}^{k} |} . \end{matrix}$ For the sequence of radii $ε_{k}$ chosen in (3.16), we consider the domains $\begin{matrix} Ω^{k} : = R^{2} ∖ (\overline{⋃_{i = 1}^{n_{k}} B (z_{i}^{k}, ε_{k})}) . \end{matrix}$

4.1. Euler equations and vorticity estimates

As mentioned in Section 2.3, there exists a unique smooth vector field $u_{0}^{k}$ verifying (1.2). For such initial data, Kikuchi [15] established existence and uniqueness of a global strong solution $u^{k}$ of (1.3) in $Ω^{k}$ . Moreover, he proved that this solution verifies (1.4) in the sense of distributions. Thanks to the regularity of $u^{k}$ , the method of characteristics implies that

$ω^{k} (t, \cdot)$ is compactly supported for any $t \in R_{+}$ (not uniformly);

$\int_{Ω^{k}} ω^{k} (t, \cdot)$ is a conserved quantity;

the $L^{q}$ norms of the vorticity are conserved for any $q \in [1, \infty]$ .

Moreover, by Kelvin’s Circulation Theorem,

the circulation around each disk $B (z_{i}^{k}, ε_{k})$ is conserved: $\begin{matrix} \oint_{\partial B_{i}^{k}} u^{k} (t, \cdot) \cdot \hat{τ} d s = γ_{i}^{k}, \forall t ⩾ 0 . \end{matrix}$

Therefore, we infer that $\begin{matrix} (4.1) & {‖ E ω^{k} (t, \cdot) ‖}_{L^{1} \cap L^{p_{0}} (R^{2})} = {‖ E ω_{0}^{k} ‖}_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0}, \forall t \in R_{+}, \forall k \in N^{*} . \end{matrix}$

4.2. Velocity estimates

We use the Biot–Savart law: $\begin{array}{l} u^{k} (t, x) = K^{k} [ω^{k} (t, \cdot)] (x) + \sum_{i = 1}^{n_{k}} α_{i}^{k} [ω^{k} (t, \cdot)] X_{i}^{k} (x) with \\ α_{i}^{k} [ω^{k} (t, \cdot)] = \int_{Ω^{k}} Φ_{i}^{k} (y) ω^{k} (t, y) d y + γ_{i}^{k}, \end{array}$ together with Propositions 3.1, 3.2 and 3.3 in order to get $L^{p}$ estimates for the velocity on compact sets.

Proposition 4.1.
With the above definitions, $E u^{k}$ is uniformly bounded in $L^{\infty} (R_{+}, L_{loc}^{p} (R^{2}))$ , for any $p \in [1, 2)$ . More precisely, for any $p \in (1, 2)$ , we can decompose the velocity as $E u^{k} = u_{1}^{k} + u_{2}^{k} + u_{3}^{k}$ such that $\begin{matrix} {‖ u_{1}^{k} ‖}_{L^{\infty} (R_{+}, L^{p} (R^{2}))}, {‖ u_{2}^{k} ‖}_{L^{\infty} (R_{+}, L^{2} (R^{2}))}, {‖ u_{3}^{k} ‖}_{L^{\infty} (R_{+}, L^{p^{}} (R^{2}))} are uniformly bounded, \end{matrix}$ with* $p^{} = {(\frac{1}{p} - \frac{1}{2})}^{- 1} \in (2, \infty)$ .
Proof.
Fix $p \in [1, 2)$ . We choose $k_{1}$ , such that, for any $k ⩾ k_{1}$ , we have: $\begin{array}{rcl} (4.2) & 2 - \frac{1}{k} ⩾ p . \end{array}$

We compare $E u^{k}$ with $\begin{matrix} w^{k} (t, x) = K_{R^{2}} [E ω^{k} (t, \cdot) + μ] (x), \end{matrix}$ by decomposing $\begin{array}{rcl} (w^{k} - E u^{k}) (t, x) & = & (K_{R^{2}} [μ] - K_{R^{2}} [\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}}]) (x) \\ + \sum_{i = 1}^{n_{k}} γ_{i}^{k} (\frac{{(x - z_{i}^{k})}^{⊥}}{2 π | x - z_{i}^{k} |^{2}} - E X_{i}^{k} (x)) \\ + (K_{R^{2}} [E ω^{k} (t, \cdot)] - E K^{k} [ω^{k} (t, \cdot)] - \sum_{i = 1}^{n_{k}} m_{i}^{k} [ω^{k} (t, \cdot)] E X_{i}^{k}) (x) \\ = : & A_{k} (x) + B_{k} (x) + C_{k} [ω^{k} (t, \cdot)] (x) . \end{array}$

By Proposition 3.1, we have that $\begin{matrix} ‖ A_{k, 1} ‖_{L^{p} (R^{2})}, ‖ A_{k, 2} ‖_{L^{p^{}} (R^{2})} are uniformly bounded, \end{matrix}$ with $A_{k} = A_{k, 1} + A_{k, 2}$ .

Thanks to (4.2), we can interpolate $L^{p}$ between $L^{1}$ and $L^{2 - \frac{1}{k}}$ ; hence we deduce from the definition of $ε_{k}$ (3.16) that $\begin{matrix} ‖ B_{k, 1} ‖_{L^{p} (R^{2})} + ‖ B_{k, 2} ‖_{L^{2} (R^{2})} ⩽ \frac{1}{k}, \end{matrix}$ with $B_{k, 1} : = \sum_{i = 1}^{n_{k}} γ_{i}^{k} (\frac{{(x - z_{i}^{k})}^{⊥}}{2 π | x - z_{i}^{k} |^{2}} - E v_{i}^{k} (x))$ and $B_{k, 2} : = \sum_{i = 1}^{n_{k}} γ_{i}^{k} (E v_{i}^{k} (x) - E X_{i}^{k})$ .

We already know from (4.1) that $\begin{matrix} {‖ E ω^{k} (t, \cdot) ‖}_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0}, \forall t, \forall k . \end{matrix}$ As $E ω^{k}$ is compactly supported for all t, k, we deduce from the definition of $ε_{k}$ that $\begin{matrix} {‖ C_{k} [ω^{k} (t, \cdot)] ‖}_{L^{2} (R^{2})} ⩽ \frac{1}{k}, \forall t, \forall k . \end{matrix}$

By Hardy–Littlewood–Sobolev Theorem, we have $\begin{matrix} {‖ K_{R^{2}} [E ω^{k} (t, \cdot)] ‖}_{L^{p^{}} (R^{2})} ⩽ C_{p} {‖ E ω^{k} (t, \cdot) ‖}_{L^{p} (R^{2})} ⩽ C_{p} M_{0}, \forall t \in R_{+}, \forall k \in N^{}, \end{matrix}$ hence, $‖ K_{R^{2}} [E ω^{k} (t, \cdot)] ‖_{L^{\infty} (R_{+}; L^{p^{}} (R^{2}))}$ is uniformly bounded.

It is clear from Section 3.1 (which is the proof of Proposition 3.1) that $K_{R^{2}} [μ]$ is the sum of a function belonging to $L^{p} (R^{2})$ and a function belonging to $L^{p^{}} (R^{2})$ .

Finally, we can put together all the estimates to obtain that $\begin{matrix} E u^{k} (t, x) = K_{R^{2}} [μ] (x) + K_{R^{2}} [E ω^{k} (t, \cdot)] (x) - A_{k} (x) - B_{k} (x) - C_{k} [ω^{k} (t, \cdot)] (x) \end{matrix}$ verifies the statement of the proposition. □

4.3. Strong convergence for the velocities

First, we use standard compactness argument to extract a subsequence such that $K_{R^{2}} [E ω^{k} (t, \cdot)]$ converges strongly.

Proposition 4.2.
There exists a subsequence, still denoted by k, and a function $ω \in L^{\infty} (R_{+}; L^{1} \cap L^{p_{0}} (R^{2}))$ , such that
$ω^{k} ⇀ ω$ weak ∗ in $L^{\infty} (R_{+}; L^{q} (R^{2}))$ for any $q \in [1, p_{0}]$ ;

$K_{R^{2}} [E ω^{k}] \to K_{R^{2}} [ω]$ strongly in $L_{loc}^{2} (R_{+} \times R^{2})$ .

Proof.
From (4.1) and by the Banach–Alaoglu’s theorem, we can extract a subsequence which converges weak ∗ in $L^{\infty} (R_{+}; L^{1} \cap L^{p_{0}} (R^{2}))$ , which gives the first item.

We consider $p \in (1, 2)$ such that $p^{'} = {(1 + p^{- 1})}^{- 1} \in (2, p_{0}]$ (we can take $p = p_{0}^{'}$ if $p_{0} < \infty$ , and $p = 3 / 2$ otherwise). Let fix $T > 0$ , and we set $X = L^{3 / 2} (R^{2})$ and $Y = H^{- 3} (R^{2})$ . Then X is a separable reflexive Banach space and Y is a Banach space such that $X ↪ Y$ and $Y^{'}$ is separable and dense in $X^{'}$ . By (4.1), we already know that ${E ω^{k}}$ is a bounded sequence in $L^{\infty} (0, T; X)$ . Moreover, for any function $Φ \in Y^{'} = H^{3} (R^{2})$ , and any $t \in [0, T]$ , we use the equation verified by $ω^{k}$ to get $\begin{array}{rcl} (\partial_{t} E ω^{k}, Φ) & = & - \int_{Ω^{k}} div (u^{k} ω^{k}) Φ \\ = & \int_{Ω^{k}} ω^{k} u^{k} \cdot \nabla Φ \\ ⩽ & ({‖ ω^{k} ‖}_{L^{p^{'}}} {‖ u_{1}^{k} ‖}_{L^{p}} + {‖ ω^{k} ‖}_{L^{2}} {‖ u_{2}^{k} ‖}_{L^{2}} + {‖ ω^{k} ‖}_{L^{{(p^{})}^{'}}} {‖ u_{3}^{k} ‖}_{L^{p^{}}}) ‖ \nabla Φ ‖_{L^{\infty}} \\ ⩽ & C ‖ Φ ‖_{H^{3}}, \end{array}$ where we have used Proposition 4.1 together with (4.1). This means that $E ω^{k} \in C ([0, T]; Y)$ and that for all $Φ \in Y^{'}$ , ${(E ω^{k}, Φ)}_{Y \times Y^{'}}$ is uniformly continuous in $[0, T]$ , uniformly in k. Therefore, Lemma C.1 of [20] states that ${E ω^{k}}$ is relatively compact in $C ([0, T]; X - w)$ .

Fixing $p_{1} \in (2, p_{0})$ , this argument also holds with $X = L^{p_{1}} (R^{2})$ . Therefore, we can extract a subsequence such that $\begin{matrix} E ω^{k} \to ω in C ([0, T]; L^{3 / 2} \cap L^{p_{1}} (R^{2}) - w) . \end{matrix}$

Now, we prove that this implies that the function $\begin{matrix} (t, x) \mapsto K_{R^{2}} [(ω - E ω^{k}) (t, \cdot)] (x) \end{matrix}$ tends to zero in $L^{2} ([0, T]; L^{2} (K))$ when $k \to \infty$ , for any K compact subset of $R^{2}$ .

For $t \in [0, T]$ and $x \in K$ fixed, the map $y \mapsto \frac{{(x - y)}^{⊥}}{| x - y |^{2}} 1_{{y \in B (x, 1)}}$ belongs to $L^{p_{1}^{'}} (R^{2})$ whereas $y \mapsto \frac{{(x - y)}^{⊥}}{| x - y |^{2}} 1_{{y \in B {(x, 1)}^{c}}}$ belongs to $L^{3} (R^{2})$ , hence $\begin{matrix} | \int_{R^{2}} \frac{{(x - y)}^{⊥}}{| x - y |^{2}} (E ω^{k} - ω) (t, y) d y | \to 0 when k \to \infty . \end{matrix}$ Moreover, using the estimate of [12], we have the uniform bound: $\begin{matrix} | \int_{R^{2}} \frac{{(x - y)}^{⊥}}{| x - y |^{2}} (E ω^{k} - ω) (t, y) d y | ⩽ C 2 M_{0}, \forall t \in [0, T], \forall x \in R^{2}, \forall k \in N^{} . \end{matrix}$ Hence, applying twice the dominated convergence theorem, we obtain the convergence of $K_{R^{2}} [E ω^{k} - ω]$ in $L^{2} ([0, T]; L^{2} (K))$ .

By a diagonal extraction on $T = N \in N$ , we can choose a subsequence holding for all T. □

The first item of the above proposition gives the assertion (b) in Theorem 1.1.

The main result of this subsection is the following.
Theorem 4.3.
With the sequence* ${ω^{k}}$ constructed in Proposition 4.2 , we have that for any $p \in [1, 2)$ : $\begin{matrix} E u^{k} ⟶ u strongly in L_{loc}^{2} (R_{+}; L_{loc}^{p} (R^{2})) when k \to \infty, \end{matrix}$ with $u = K_{R^{2}} [ω + μ]$ .

In particular, this result implies part (a) of Theorem 1.1. Proof.
Let us fix $p \in [1, 2)$ , $T > 0$ and K a compact subset of $R^{2}$ . We also fix $d > 0$ and we are looking for $k_{d}$ such that for all $k ⩾ k_{d}$ we have: $\begin{matrix} {‖ u - E u^{k} ‖}_{L^{2} ([0, T]; L^{p} (K))} ⩽ d . \end{matrix}$ First, we note that there exists $k_{1}$ , such that for any $k ⩾ k_{1}$ , we have: $\begin{array}{rcl} (4.3) & 2 - \frac{1}{k} ⩾ p . \end{array}$

As usual, we write: $\begin{array}{rcl} (u - E u^{k}) (t, x) & = & (K_{R^{2}} [μ] - K_{R^{2}} [\sum_{i = 1}^{n_{k}} γ_{i}^{k} δ_{z_{i}^{k}}]) (x) \\ + \sum_{i = 1}^{n_{k}} γ_{i}^{k} (\frac{{(x - z_{i}^{k})}^{⊥}}{2 π | x - z_{i}^{k} |^{2}} - E X_{i}^{k} (x)) \\ + (K_{R^{2}} [E ω^{k} (t, \cdot)] - E K^{k} [ω^{k} (t, \cdot)] - \sum_{i = 1}^{n_{k}} m_{i}^{k} [ω^{k} (t, \cdot)] E X_{i}^{k}) (x) \\ + K_{R^{2}} [(ω - E ω^{k}) (t, \cdot)] (x) \\ = : & A_{k} (x) + B_{k} (x) + C_{k} [ω^{k} (t, \cdot)] (x) + K_{R^{2}} [(ω - E ω^{k}) (t, \cdot)] (x) . \end{array}$

By Proposition 3.1, there exists $k_{2}$ such that for any $k ⩾ k_{2}$ , we have $‖ A_{k} ‖_{L^{2} ([0, T]; L^{p} (K))} ⩽ d / 4$ . By the definition of $ε_{k}$ (see (3.16)) together with (4.3), we see that there exists $k_{3} ⩾ k_{1}$ such that for any $k ⩾ k_{3}$ $\begin{matrix} ‖ B_{k} ‖_{L^{2} ([0, T]; L^{p} (K))} ⩽ d / 4 . \end{matrix}$

Moreover, we already know from (4.1) that $‖ E ω^{k} (t, \cdot) ‖_{L^{1} \cap L^{p_{0}} (R^{2})} ⩽ M_{0}$ , for every $t \in [0, T]$ , and all k, so we deduce from the definition of $ε_{k}$ that there exists $k_{4}$ such that for any $k ⩾ k_{4}$ , we have: $\begin{matrix} {‖ C_{k} [ω^{k} (t, \cdot)] ‖}_{L^{2} ([0, T]; L^{p} (K))} ⩽ d / 4 . \end{matrix}$

By Proposition 4.2, it also clear that there exists $k_{5}$ such that for any $k ⩾ k_{5}$ , we have: $\begin{matrix} {‖ K_{R^{2}} [(ω - E ω^{k}) (t, \cdot)] (x) ‖}_{L^{2} ([0, T]; L^{p} (K))} ⩽ d / 4 . \end{matrix}$

Denoting $k_{d} = {max}_{i = 1, \dots, 5} {k_{i}}$ , we have proved that for any $k ⩾ k_{d}$ $\begin{matrix} {‖ u - E u^{k} ‖}_{L^{2} ([0, T]; L^{p} (K))} ⩽ d, \end{matrix}$ which concludes the proof. □

4.4. Passing to the limit in the Euler equations

We have obtained the convergence of the velocity and of the vorticity as required in items (a)–(b) of Theorem 1.1.

The purpose of the remainder of this section is to prove item (c), namely that u and ω verify, in an appropriate sense, the system: $\begin{matrix} \{\begin{matrix} \partial_{t} ω + u \cdot \nabla ω = 0 & in (0, \infty) \times R^{2} \\ div u = 0 & in (0, \infty) \times R^{2} \\ curl u = ω + μ & in (0, \infty) \times R^{2} \\ {lim}_{x \to \infty} | u | = 0 & for t \in [0, \infty) \\ ω (0, x) = ω_{0} (x) & in R^{2} . \end{matrix} \end{matrix}$

Definition 4.4.
The pair $(u, ω)$ is a weak solution of the previous system if
for any test function $φ \in C_{c}^{\infty} ([0, \infty) \times R^{2})$ we have $\begin{matrix} \int_{0}^{\infty} \int_{R^{2}} φ_{t} ω d x d t + \int_{0}^{\infty} \int_{R^{2}} \nabla φ \cdot u ω d x d t + \int_{R^{2}} φ (0, x) ω_{0} (x) d x = 0, \end{matrix}$

we have $div u = 0$ and $curl u = ω + μ$ in the sense of distributions of $R^{2}$ , with $| u | \to 0$ at infinity.

Theorem 4.5.
The pair $(u, ω)$ obtained is a weak solution of the previous system.
Proof.
Item (b) of the definition is directly verified thanks to the explicit form of u: $\begin{matrix} u (t, x) = K_{R^{2}} [ω (t, \cdot) + μ] (x), \end{matrix}$ because μ is compactly supported, and $K_{R^{2}} [ω (t, \cdot)]$ belongs to $L^{\infty} (R_{+}; W^{1, p} (R^{2}))$ for some $p \in (2, p_{0})$ (by Hardy–Littlewood–Sobolev Theorem and Calderón–Zygmund inequality).

For any test function $φ \in C_{0}^{\infty} ([0, \infty) \times R^{2})$ , we have that $(u^{k}, ω^{k})$ verifies (1.4): $\begin{array}{l} \int_{0}^{\infty} \int_{Ω^{k}} (ω^{k} \partial_{t} φ + ω^{k} u^{k} \cdot \nabla φ) (t, x) d x d t + \int_{Ω^{k}} φ (0, x) ω_{0}^{k} (x) d x = 0 \\ \int_{0}^{\infty} \int_{R^{2}} (E ω^{k} \partial_{t} φ + E ω^{k} E u^{k} \cdot \nabla φ) (t, x) d x d t + \int_{R^{2}} φ (0, x) ω_{0}^{k} (x) 1_{Ω^{k}} d x = 0 . \end{array}$ Therefore, we can pass easily to the limit thanks to the weak-strong convergence of the pair vorticity-velocity: $\begin{matrix} \int_{0}^{\infty} \int_{R^{2}} (ω \partial_{t} φ + ω u \cdot \nabla φ) (t, x) d x d t + \int_{R^{2}} φ (0, x) ω_{0} (x) d x = 0, \end{matrix}$ which concludes the proof of the main theorem. □

5. Final remarks and comments

5.1. Fixed Dirac masses

As mentioned in the introduction, one application of our result is the case of the approximation of an arbitrary compactly supported bounded Radon measure $μ \in {BM}_{c} (R^{2})$ by a square grid discretization $\sum γ_{i}^{k} δ_{z_{i}^{k}} ⇀ μ$ , where $z_{i}^{k}$ is the center of a square $C_{i}^{k}$ (the side of the square has length $1 / k$ ) and $γ_{i}^{k} : = \int_{C_{i}^{k}} φ_{i}^{k} d μ$ for $(φ_{i}^{k})$ a partition of unity (with $supp φ_{i}^{k} \subset z_{i}^{k} + {[- 1 / k, 1 / k]}^{2}$ ).

However, let us present another interesting example: the case where μ is a finite sum of Dirac masses.

For $μ = γ δ_{0}$ , we can consider the following setting: $\begin{matrix} n_{k} = 1, Ω^{k} = R^{2} ∖ \overline{B (0, ε_{k})} and γ_{1}^{k} = γ . \end{matrix}$ In this case, Theorem 1.1 is precisely the main result in [13]. Actually, by the uniqueness result of [19], we claim that for initial vorticity constant in a small neighborhood of 0 (bounded and compactly supported), the limit holds for any sequence $ε_{k} \to 0$ , without extraction of a subsequence.

More generally, if μ is a finite sum of diracs: $μ = \sum_{i = 1}^{n} γ_{i} δ_{z_{i}}$ , we can choose: $\begin{matrix} n_{k} = n, Ω^{k} : = R^{2} ∖ (⋃_{i = 1}^{n} \overline{B (z_{i}, ε_{k})}) and γ_{i}^{k} = γ_{i}, \end{matrix}$ for $ε_{k} < {inf}_{i \neq j} | z_{i} - z_{j} |$ . Therefore, we have proved that for any sequence $ε_{k} \to 0$ , we can extract a subsequence such that the conclusion of Theorem 1.1 holds true, which is an extension of [13] to several disks. Similarly to what we argued in the case of a single dirac above, the uniqeness result in [19] implies that, in the case of initial vorticity constant around each $z_{i}$ and belonging to $L_{c}^{\infty} (R^{2})$ , the limit system has at most one solution, which implies that the convergence holds for all the sequence, without extracting a subsequence.

Concerning Dirac masses, another interesting consequence is the fusion of two Dirac masses. Considering $z_{1}^{k}$ and $z_{2}^{k}$ two sequences converging to the same point $z_{0}$ , we note that $\begin{matrix} γ_{1} δ_{z_{1}^{k}} + γ_{2} δ_{z_{2}^{k}} ⇀ (γ_{1} + γ_{2}) δ_{z_{0}} . \end{matrix}$ Therefore, we can apply our theorem in this setting.

5.2. Uniqueness results

Let us comment on some cases where the global weak solution of our limit system is unique. As in the celebrated result of Yudovich, [28] we assume $p_{0} = \infty$ .

If μ is a bounded function compactly supported, it is clear by the Calderón–Zygmund inequality that u verifies $\begin{matrix} ‖ \nabla u ‖_{L^{p}} ⩽ C p, \forall p \in (2, \infty) \end{matrix}$ with C independent of p. This allows us to follow exactly Yudovich’s original proof to establish uniqueness of a weak solution verifying $\begin{matrix} ω \in L_{loc}^{\infty} (R_{+}; L^{1} \cap L^{\infty} (R^{2})) and u \in L_{loc}^{\infty} (R_{+}; L_{loc}^{2} (R^{2})) . \end{matrix}$

As observed in the previous section, another case for which we have uniqueness is when μ is a finite sum of Dirac masses $\sum_{i = 1}^{N} γ_{i} δ_{z_{i}}$ and $ω_{0} \in L_{c}^{\infty} (R^{2})$ is constant around each of the vortex point $z_{i}$ , see [19]. Here, the key to prove global uniqueness is to show separation, i.e. that the vorticity never meets the support of μ.

For more general μ, obtaining such a separation estimate is a difficult problem. Still, we have a local-in-time uniqueness result if we assume $ω_{0} \in L_{c}^{\infty} (R^{2} ∖ supp μ)$ . Indeed, far from the support of μ, the velocity is bounded, and, as the vorticity is transported by the velocity, there exists a time $T_{0} > 0$ such that $ω (t, \cdot)$ is supported outside $supp μ$ for all $t \in [0, T_{0}]$ . Then, following [19, Section 3] we have easily that the solution of the limit system (see item (c)) in Theorem 1.1) is unique up to the time $T_{0}$ .

5.3. Flow around a curve

Let $Γ : Γ (s), 0 ⩽ s ⩽ 1$ be a $C^{2}$ -Jordan arc. The first author has proved in [16] the existence of a global solution in the exterior of a curve. This solution is obtained by a compactness argument, on the solution in the exterior of a smooth thin obstacle shrinking to the curve. Actually, a formulation in the full plane was found: for $ω_{0} \in L_{c}^{\infty} (R^{2} ∖ \partial Ω)$ and $γ \in R$ given, there exists a pair $(u, ω)$ $\begin{matrix} u \in L_{loc}^{\infty} (R_{+}; L_{loc}^{2} (R^{2})), ω \in L^{\infty} (R_{+} \times R^{2}) \end{matrix}$ verifying, in the sense of distributions, the system: $\begin{matrix} (5.1) & \{\begin{matrix} \partial_{t} ω + u \cdot \nabla ω = 0 & in R^{2} \times (0, \infty) \\ div u = 0 & in R^{2} \times (0, \infty) \\ curl u = ω + g_{ω} (s) δ_{Γ} & in R^{2} \times (0, \infty) \\ ω (x, 0) = ω_{0} (x) & in R^{2} \end{matrix} \end{matrix}$ where $δ_{Γ}$ is the Dirac delta on Γ, and $g_{ω}$ is explicitly given in terms of ω, γ (which can be viewed as the initial circulation) and the shape of Γ. In fact u is a vector field which is tangent to Γ (with different values on each side of the curve), vanishing at infinity, with circulation around the curve Γ equal to γ. This velocity is blowing up at the endpoints of the curve Γ as the inverse of the square root of the distance and has a jump across Γ. Moreover $g_{ω} = (u_{down} - u_{up}) \cdot \vec{τ}$ . This result was extended in [10] for any $ω_{0} \in L_{c}^{q} (R^{2})$ , $q \in (2, \infty]$ , and without assuming any regularity for the curve.

Moreover, for any solution of the above system in the case of $C^{2}$ curve and $ω_{0} \in L_{c}^{\infty} (R^{2})$ , it was established in [17] that the solution is a renormalized solution in the sense of DiPerna–Lions, hence we have the following extra properties

the $L^{p}$ norm of the vorticity is conserved, for any $p \in [1, \infty]$ ;

the circulation around Γ is conserved;

the vorticity is always compactly supported, but this support can grow;

around the curve, the velocity at times $t > 0$ belongs to $L_{loc}^{p} (R^{2}) \cap W_{loc}^{1, r} (R^{2})$ only for $p < 4$ and $r < 4 / 3$ . Actually we have $\begin{matrix} u \in L^{\infty} (R_{+}; L^{p} (B (0, R))) \cap L^{\infty} ([0, T]; L^{s} (R^{2})) \cap L^{\infty} (R_{+}; W^{1, r} (B (0, R))) \end{matrix}$ for any $R > 0$ , $T > 0$ , $p \in [1, 4)$ , $s \in (2, 4)$ and $r \in [1, 4 / 3)$ .

If we assume that

ω_{0}

belongs to

L_{c}^{\infty} (R^{2} ∖ Γ)

and

(ω_{0}, γ)

has a definite sign (namely, either

ω_{0}

non-negative and

γ_{0} < - \int ω_{0}

ω_{0}

non-positive and

γ_{0} > - \int ω_{0}

) then the uniqueness of the global weak solution is the main result of [17]. The crucial step therein is to prove that the sign condition implies that the vorticity never meets the boundary, which is the place where the velocity is not regular.

The natural question is to ask if an infinite number of material points has the same effect as a material curve on the behavior of an inviscid flow. Let us consider k disjoint disks uniformly distributed on the “imaginary” curve Γ, and we look for the limit as $k \to \infty$ . As initial data, we choose ${γ_{i}^{k}}$ such that $\sum_{i = 1}^{k} γ_{i}^{k} z_{i}^{k}$ tends to the measure $g_{ω_{0}} (s) δ_{Γ}$ when $k \to \infty$ (we set $γ_{i}^{k}$ the integral of $g_{ω_{0}}$ around $z_{i}^{k}$ ). Our main theorem then states that there exists $ε_{k} \to 0$ such that the Euler solution on $R^{2} ∖ (⋃_{i = 1, \dots, k} B (z_{i}^{k}, ε_{k}))$ to a pair $(u, ω)$ verifying, in the sense of distributions, the following system $\begin{matrix} (5.2) & \{\begin{matrix} \partial_{t} ω + u \cdot \nabla ω = 0 & in R^{2} \times (0, \infty) \\ div u = 0 & in R^{2} \times (0, \infty) \\ curl u = ω + g_{ω_{0}} (s) δ_{Γ} & in R^{2} \times (0, \infty) \\ ω (x, 0) = ω_{0} (x) & in R^{2} . \end{matrix} \end{matrix}$

The difference between system (5.2) above and system (5.1) found in [16] lies in the third equation: in (5.1) the density g depends on $ω (t, \cdot)$ , whereas it is time-independent in (5.2). Even if this difference seems small, it has significant consequences. Indeed, in the exterior of a curve, the flow is required to be tangent to the curve itself, and the presence of the term $g_{ω} (s) δ_{Γ}$ ensures this. In fact, the density $g_{ω}$ represents the jump in tangential velocity at the curve and it is a time-dependent variable of the model. A relevant conserved quantity is the circulation around the curve, i.e. the integral of $g_{ω}$ . In the case of an infinite number of points, we approximate the vortex sheet $g_{ω_{0}} δ_{Γ}$ by k point vortices, but now the circulation of the velocity around each obstacle is conserved, which means that the densities $γ_{i}^{k}$ of the points vortices are constant in time. Therefore, for any time, $\sum_{i} γ_{i}^{k} z_{i}^{k}$ approximate $g_{ω_{0}} δ_{Γ}$ , a time-independent quantity, instead of $g_{ω} (t, \cdot) δ_{Γ}$ .

To conclude, we have carefully chosen $γ_{i}^{k}$ such that the limit velocity is initially tangent to the curve, but there is no reason that it should remain tangent: $u = K_{R^{2}} [ω (t, \cdot) + g_{ω_{0}} (s) δ_{Γ}]$ in this case whereas $u = K_{R^{2}} [ω (t, \cdot) + g_{ω (t, \cdot)} (s) δ_{Γ}]$ in the case of the exterior of the curve. Without the tangency condition, it appears plausible that the vorticity ω would meet the curve in finite time, which would make it difficult to get the global uniqueness, even with a sign condition.

5.4. Rate size of the disks/space between the disks

Note that the radii $ε_{k}$ are constructed in the proof of the main theorem, and not given initially. In our analysis, we have obtained a family $ε_{k}$ such that the convergence holds true, but this could mean, for example, that the size of the obstacles tends much more rapidly to 0 than the average separation of the obstacles. This makes it natural that, in the limit of a continuous curve, the effective flow may, in fact, cross the curve.

Let us fix $(ε_{k})$ and $(d_{k})$ two sequences tending to zero and we study the limit $k \to \infty$ . The natural question is to look for the critical criterion where the convergence in Theorem 1.1 holds true. Actually, assuming the obstacles equally spaced along a line segment, with the estimates in terms of $d_{k}$ and $n_{k}$ (with $n_{k} ⩽ 1 / d_{k}$ ) in Propositions 3.2 and 3.3, it is possible to show that the convergence holds if $\begin{matrix} (5.3) & \frac{ε_{k} \sqrt{| ln ε_{k} |}}{d_{k}^{3}} \to 0 as k \to \infty . \end{matrix}$ Note that we do not need such a condition for Proposition 3.1, as $ε_{k}$ is not involved.

In contrast, the main result in [3] reads as follows: in this particular setting (uniformly distributed on a segment) and assuming that all the circulations are zeros, then the limit $u^{k}$ converges to the Euler solution in the full plane if $d_{k} = ε_{k}^{α}$ with $α < 2$ . This corresponds to distance much smaller than in the present paper, where we can only reach the condition $α < 1 / 3$ .

The main idea in [3] is to look for the critical α such that the Step 1 in Section 3.3 could give the convergence to zero in $L^{2}$ of $K_{R^{2}} [E f |_{Ω^{ε}}] - E v^{ε} [f]$ . Next, the authors remark that the term $v^{ε} [f] - K^{ε} [f |_{Ω^{ε}}] - \sum_{i = 1}^{n} m_{i}^{ε} [f] X_{i}^{ε}$ is the Leray projection of $- (K_{R^{2}} [E f |_{Ω^{ε}}] - E v^{ε} [f])$ , and they concluded with the orthogonality of this projection in $L^{2}$ . This argument allows them to skip the analysis on the harmonic part (Step 2 in Section 3.2), which is the hardest part in this article. However, we note that such an argument is simplified by the assumption of zero circulations. As recalled previously, if we assume that $γ_{1}^{k}$ is constant, then $γ_{1} δ_{z_{1}}$ appears in the limit, which implies that the velocity behaves like $γ_{1} \frac{{(x - z_{1})}^{⊥}}{2 π | x - z_{1} |^{2}}$ close to $z_{1}$ , which does not belong to $L^{2}$ . Therefore, $L^{2} (R^{2})$ estimates are not available in our case, and it is not clear that the Leray projection is uniformly continuous in $L^{p}$ for $p < 2$ (see a discussion in [18, Section 5.3]). This explains why the delicate part in our case is Step 2 of Section 3.2 ( $L^{p}$ framework).

We do not know whether the criterion (5.3) is optimal for our case, but this was not the main purpose of this article. Indeed, even replacing Step 2 of Section 3.2 by the orthogonality argument in $L^{2} (Ω^{k})$ , if we look carefully at the proof of (3.7), we note that we cannot obtain better than $\begin{matrix} {‖ \frac{1}{2 π} \frac{{(x - z_{i})}^{⊥}}{| x - z_{i} |^{2}} - E v_{i}^{k} ‖}_{L^{2} (Ω^{k})} ⩽ C \frac{ε_{k} n_{k}^{1 / 2}}{d_{k}} \end{matrix}$ which means than we can expect the convergence for this part only for $α < 2 / 3$ which is very far from the condition $α < 2$ of [3]. Such a difference is natural because to allow non-zero circulations implies dealing with a more singular limit. We have decided in Step 2 to develop an $L^{p}$ analysis because it exemplifies the interest of the inversion argument. Here, this allows us to apply to unbounded domains some tools developed for bounded domains (for instance, the maximum principle and the change of basis for harmonic vector fields). Such a technique was also used in some crucial lemmas in [11].

In [18], the authors have completed the result of [3]: they exhibit some regimes where the limit of $u^{k}$ is a solution to the Euler equations in the exterior of the unit segment. That result relies again on an $L^{2}$ argument, and holds only when all the initial circulations are zero. When the obstacles are distributed on the segment, the impermeable wall appears if the distance $d_{k}$ is much smaller than the size $ε_{k}$ , namely such that $\frac{d_{k}}{ε_{k}^{3}} \to 0$ if the obstacle is a disk (see [18] for more details).

Footnotes

Acknowledgements

C.L. is partially supported by the Agence Nationale de la Recherche, Project DYFICOLTI grant ANR-13-BS01-0003-01, and Project IFSMACS grant ANR-15. H.J.N.L.’s research was supported in part by CNPq grant # 307918/2014-9 and FAPERJ project # E-26/103.197/2012. M.C.L.F.’s work was partially funded by CNPq grant # 306886/2014-6. This work has been supported by the Réseau Franco-Brésilien en Mathématiques and by the project PICS-CNRS “FLAME”, both of which made several visits between the authors possible.

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