In this article, we study the limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family ofnk disjoint disks with centers{zki}and radii εk

SMALL DISKS

C. LACAVE, M. C. LOPES FILHO AND H. J. NUSSENZVEIG LOPES

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 ofnk disjoint disks with centers{z^ki}and radii εk. We assume that the initial velocitiesu^k₀ are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require,nk→ ∞, and we assumeεk→0 ask→ ∞.

Letγ_i^kbe the circulation ofu^k0 around the circle{|x−z_i^k|=εk}. We prove that the limit ask→ ∞ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1)ω^k0 = curlu^k0 has a uniform compact support and converges weakly inL^p0, for somep0>2, toω0∈L^pc⁰(R²), (2)Pn_k

i=1γi^kδ_zk

i * µweak-∗inBM(R²) for some bounded Radon measureµ, and (3) the radiiεk are sufficiently small. Then the corresponding solutionsu^k converge strongly to a weak solutionuof 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 quantityω= curlu, with initial dataω0, where the transporting velocity field is generated fromω, so that its curl isω+µ. As a byproduct, we obtain a new existence result for this modified Euler system.

Contents

1. Introduction 1

2. Preliminaries about the Biot-Savart law 5

2.1. Harmonic part 6

2.2. The Laplacian-inverse 8

2.3. Biot-Savart law 9

3. Convergence for fixed time 9

3.1. The number of obstacles (proof of Proposition 3.1) 10

3.2. The harmonic part (proof of Proposition 3.2) 11

3.3. The Laplacian-inverse (proof of Proposition 3.3) 18

3.4. Construction of εk 20

4. Time evolution 21

4.1. Euler equations and vorticity estimates 22

4.2. Velocity estimates 22

4.3. Strong convergence for the velocities 23

4.4. Passing to the limit in the Euler equations 25

5. Final remarks and comments 25

5.1. Fixed Dirac masses 25

5.2. Uniqueness results 26

5.3. Flow around a curve 26

5.4. Rate size of the disks/space between the disks 28

References 29

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,

Date: October 23, 2017.

1

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 istime-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 atime-independentsheet 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 vorticityω by a velocity whose curl is the sum ofω 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 byµcan be approximated by an exterior domain flow, exterior to a finite number of small disks.

More precisely, for each k= 1,2, . . . we consider a family of n_k disjoint disks with centers z^k_i and radiiε_k,i= 1, . . . , n_k. We denote the disks byB(z_i^k, ε_k) :={|x−z^k_i|< ε_k}and the fluid domain by:

Ω^k:=R²\[^nk

i=1

B(z_i^k, ε_k) .

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 isw. To recover the velocity v from its vorticity w it is necessary to introduce the circulation around each disk:

γ_i^k[v] :=

I

∂B(z_i^k,εk)

v·τ ds.ˆ (1.1)

The above integral is considered in the counter-clockwise sense, hence ˆτ =−ˆn^⊥, where ˆndenotes the outward unit normal vector at∂Ω^k. Vorticity together with the circulations uniquely determine the velocity, see, for example, [14] for a full discussion.

Given ω^k₀ ∈C_c^∞(Ω^k) and {γ_i^k}ⁿ_i=1^k ⊂ R, the corresponding initial velocity u^k₀ is the unique smooth solution of

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divu^k₀ = 0 in Ω^k curlu^k₀ =ω₀^k in Ω^k u^k₀ ·nˆ= 0 on∂Ω^k γ^k_i[u^k₀] =γ_i^k fori= 1, . . . , n_k

|u^k₀(x)| →0 asx→ ∞.

(1.2)

The IBVP for the Euler equations in Ω^k, in velocity formulation, takes the form:

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∂tu^k+u^k· ∇u^k=−∇p^k in (0,∞)×Ω^k

divu^k= 0 in [0,∞)×Ω^k

u^k·nˆ= 0 in [0,∞)×∂Ω^k

x→∞lim |u^k|= 0 fort∈[0,∞)

u^k(0, x) =u^k₀(x) in Ω^k,

(1.3)

wherep^k=p^k(t, x) is the pressure. Existence and uniqueness of a smooth solutionu^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 circulationsγ_i^k[u^k(t,·)], i= 1, . . . , nk, 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 := curlu^k = ∂₁u^k₂ −∂₂u^k₁. This transport equation, together with an elliptic system relating vorticity to velocity and (initial) circulations, comprises the vorticity formulation of the 2D Euler equations:

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∂tω^k+u^k· ∇ω^k= 0 in (0,∞)×Ω^k

divu^k = 0 in [0,∞)×Ω^k

curlu^k =ω^k in [0,∞)×Ω^k

u^k·nˆ= 0 on [0,∞)×∂Ω^k

γ_i^k[u^k(t,·)] =γ_i^k fori= 1, . . . , nk, fort∈[0,∞)

|u^k(t, x)| →0 asx→ ∞, fort∈[0,∞)

ω^k(0,·) =ω₀^k in Ω^k.

(1.4)

We emphasize that γ_i^k and ω^k₀ are initial data. We also note, in passing, that, as the transporting velocity is divergence-free, theL^p norm of vorticity is a conserved quantity, for anyp≥1.

Throughout we will adopt the convention that, whenever it is needed to extend u^k(t,·) or ω^k(t,·) to all ofR², they will be set to vanish in the interior of the disks B(z_i^k, εk).

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 whenn_k → ∞ask→ ∞.

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,ω0.

We denote the space of all bounded Radon real measures on R² by BM(R²). If µ∈ BM(R²), the total variation ofµis defined by

kµk_BM= supnZ

R²

ϕ dµ |ϕ∈C₀(R²), kϕk_L^∞ ≤1o ,

whereC0(R²) is the set of all real-valued continuous functions on R² vanishing at infinity. We recall thatBM(R²) equipped with the total variation norm is a Banach space. We say that a sequence (µ_n) inBM(R²) converges weak-∗ toµifR

R²ϕ dµ_n→R

R²ϕ dµasn→ ∞for all ϕ∈C₀(R²).

We assume that the sequence of measuresPnk

i=1γ_i^kδ_zk

i converges, weak-∗ in the sense of measures, to a compactly supported bounded Radon measure µ. This measure µ represents the homogenized limit of the circulationsγ_i^k. We assume, without loss of generality, that, for everyk∈N,{z^k_i}ⁿ_i=1^k is a set of distinct points.

Our main theorem reads:

Theorem 1.1. Fix ω₀ ∈L^p_c⁰(R²) for some p₀ ∈(2,∞]. Set R₀ >0 so that suppω₀ ⊂B(0, R₀). Let {ω₀^k} ⊂C_c^∞(B(0, R₀))be such that

ω^k₀ * ω0 weakly inL^p0(R²).

Consider{γ_i^k}_i=1...nk ⊂R, {z^k_i}_i=1...nk ⊂B(0, R₀) and suppose that

nk

X

i=1

γ_i^kδ_z^k

i * µweak-∗ in BM(R²), as k→ ∞, for some µ∈ BM_c(R²).

Then there exists a subsequence, still denoted by k, a sequence {ε_k} ⊂ R^∗+, with ε_k → 0, together with a vector field u∈ L^p_loc(R+×R²), for any p∈[1,2), and a function ω ∈L^∞(R+;L¹∩L^p0(R²)), such that the solutions(u^k, ω^k) of the Euler equations in

Ω^k=R²\[^nk

i=1

B(z_i^k, ε_k) ,

with initial vorticityω^k₀|_Ωk and initial circulations (γ_i^k) around the disks, verify (a) u^k→u strongly in L^p_loc(R+×R²) for any p∈[1,2);

(b) ω^k* ω weak ∗ in L^∞(R+;L^q(R²)) for anyq ∈[1, p₀];

(c) the limit pair (u, ω) verifies, in the sense of distributions:

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∂_tω+u· ∇ω= 0 in (0,∞)×R² divu= 0 in [0,∞)×R² curlu=ω+µ in [0,∞)×R²

x→∞lim |u|= 0 for t∈[0,∞) ω(0, x) =ω0(x) in R².

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 vorticityω is transported by a divergence-free vector field u which satisfies curlu =ω+µ. The additional termµ is a reminder of the circulation around the evanescent obstacles. We emphasize that the measureµabove 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 toω+µ, forµa 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+αδ₀. Indeed, given anyµ∈ BM_c(R²), it is easy to find an approximationPnk

i=1γ_i^kδ_z^k

i, for example by discretization on a grid.

Uniqueness for weak solutions of this modified Euler system is an interesting open problem. In the caseµ, ω₀ ∈L^∞_c (R²), one can adapt the argument in Yudovich’s theorem, see [28], to prove uniqueness and, in addition, it holds thatu∈ ∩_p>2W^1,p(R²) is a strong solution of the velocity equation inR²(see (1.3)). In the special case whereµis a finite sum of Dirac masses,Pn

i=1γ_iδ_zi, and whenω₀∈L^∞_c (R²), uniqueness was proved by Lacave and Miot in [19] under the assumption thatω0 is initially constant around each of the vortex pointsz_i. A key point of their argument was to prove that the non-constant part ofω never meets the trajectories of the vortex points. Uniqueness for general bounded vorticity ω₀, not necessarily constant aroundz_i, is already a challenging open question. For more generalµ, 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, z₁^k = 0, µ= γδ0, ω₀^k = ω0 ∈ C_c^∞(R² \ {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 Subsection 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. LetK be a smooth, simply connected compact set containing zero and consider obstacles of the formz_i^k+ε_kK.

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 ofK 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 ∈[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 toω+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 withu^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 γδ_z is of the form γ_2π|x−z|^(x−z)⊥2, which only belongs to L^p_loc(R²) for p < 2, but not to L²_loc(R²). The stability of the Euler equations under Hausdorff approximation of the fluid domain is a recent result of G´erard-Varet and Lacave [10, 11], but the authors considered obstacles with H¹ positive capacity, in order to use L² 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 theseL^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 whereµis a bounded function, the case of a finite number of Dirac masses, and local uniqueness if the initial vorticity is supported far from suppµ). 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→ ∞, 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 radiiε_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), Mikeli´c-Paoli [24] and Lions-Masmoudi [21] also obtained a homogenized limit problem.

Notation: we adopt the convention (a, b)^⊥ = (−b, a) and ∇^⊥f = (∇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 [15, 22, 14].

Let Ω be the exterior ofndisjoint disks:

Ω =R²\[ⁿ

i=1

B(z_i, r_i) .

Consider also a disk withn(circular) holes:

Ω =e B(z₀, r₀)\[ⁿ

i=1

B(z_i, r_i) whereSn

i=1B(zi, ri)⊂B(z0, r0).

LetU ⊆R²be a smooth domain, which can be either Ω orΩ. Given a smooth functione f, compactly supported inU, and (γ_i)∈Rⁿ, we consider the following elliptic system:

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divu= 0 inU

curlu=f inU

u·nˆ= 0 on ∂U γi[u] =γi fori= 1, . . . , n,

(2.1)

whereγi[u] was defined in (1.1).

WhenU = Ω we will assume an additional condition at infinity, namely, limx→∞|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...n, the unique solutions of

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∆Φi= 0 in Ω

Φi =δij on∂B(zj, rj) Φ_i has a finite limit at∞,

(where δij is the Kronecker delta) ;

• in bounded domains, we introduce the functions {eΦi}_i=0...n as the unique solutions of (∆Φei = 0 inΩe

Φe_i=δ_ij on ∂B(z_j, r_j). (2.2)

By uniqueness of the Dirichlet problem for the Laplacian, we have:

n

X

i=1

Φ_i ≡1 in Ω,

n

X

i=0

Φe_i ≡1 in Ω,e and for any i

Φ_i(x)∈[0,1] ∀x∈Ω, Φe_i(x)∈[0,1]∀x∈Ω.e 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 z₁ = 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 inB(0,1/r1)) withn−1 obstacles. Next, we introduce Φe_i(x) := Φ_i◦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 {∇^⊥Φei}_i=1...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, letH be a harmonic vector field, then:

• the divergence-free condition reads as H(x) =∇^⊥ψ(x) for some stream functionψ;

• the tangency condition meansψis constant on each ∂B(z_i, r_i) fori= 0. . . n. We denote byc_i these constants. As ψ is defined up to a constant, we can determine uniquely ψ by assuming c₀ = 0;

• the curl-free condition implies ∆ψ= 0 inΩ.e

By uniqueness, it follows that

ψ=

n

X

i=1

c_iΦe_i, H=

n

X

i=1

c_i∇^⊥Φe_i.

Next, we seek a family of harmonic vector fields such that the circulation around B(z_j, r_j) isδ_ij. In bounded domains, we introduce the vector field{Xei}_i=1...n as the unique solution of

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divXei= curlXei = 0 inΩe Xe_i·nˆ = 0 on∂Ωe

I

∂B(zj,rj)

Xei·τ dsˆ =δij forj = 1. . . n.

It is easy to see that the family {Xei}_i=1...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 Xei’s in terms of stream functions Xe_i =∇^⊥Ψe_i, where Ψe_i is the unique solution of

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∆Ψei = 0 inΩe

∂_τΨe_i = 0 on ∂Ωe

Ψe_i= 0 on ∂B(z₀, r₀) I

∂B(zj,rj)

∂nΨeids=−δ_ij forj = 1. . . n.

The condition ∂τΨei = 0 means that Ψei is constant on each ∂B(zi, ri) for i = 0. . . n. Then, we can uniquely determine a change of basis matrix, i.e. constantsc_i,j such that

Ψe_i =

n

X

j=1

c_i,jΦe_j, Xe_i=

n

X

j=1

c_i,j∇^⊥Φe_j.

In unbounded domains, we introduce the vector fields{X_i}_i=1...n as the unique solutions of

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divX_i= curlX_i = 0 in Ω

X_i·nˆ = 0 on∂Ω

|X_i(x)| →0 when x→ ∞

I

∂B(zj,rj)

Xi·τ dsˆ =δij forj = 1. . . n.

Clearly, the family {X_i}_i=1...n is a basis for the harmonic vector fields. In addition, we have the following asymptotic expansion by Laurent series:

X_i(x) = 1 2π

x^⊥

|x|² +O 1

|x|²

, asx→ ∞. (2.3)

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

Ψ_i= 1

2πln|x|+c_0,i+O 1

|x|

.

Consequently, we can reformulate our description of the space of harmonic vector fields in terms of the stream functionsXi=∇^⊥Ψi, where, for eachi= 1, . . . , n, Ψi is the unique solution of

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∆Ψ_i = 0 in Ω

∂τΨi = 0 on ∂Ω

|Ψ_i(x)− 1

2π ln|x|| →0 when x→ ∞ I

∂B(zj,rj)

∂nΨids=−δ_ij forj = 1. . . n.

The stream functions Ψi above satisfy:

Ψi(x) = 1

2πln|x|+O 1

|x|

, asx→ ∞. (2.4)

We remark that the precise form of the termO(|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 {∇^⊥Φ_i}_i=1...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 ∆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

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∆ψ=f in Ω

ψ= 0 on ∂Ω

∇ψ(x)→0 when x→ ∞,

and we denote by ΨeD =ΨeD[f] the corresponding solution operator in Ω, so thate ψe= ΨeD[f] is the unique solution of

(∆ψe=f inΩe ψe= 0 on ∂Ω.e We also define

K[f] :=∇^⊥ΨD[f], K[fe ] :=∇^⊥ΨeD[f], which are divergence free, tangent to the boundary and verify

curlK[f] =f on Ω, curlK[f] =e f on Ω.e Moreover, we have for anyi= 1. . . n:

I

∂B(zi,ri)

K[f]·τ dsˆ =− Z

Ω

Φ_i(y)f(y)dy, I

∂B(zi,ri)

K[f]e ·τ dsˆ =− Z

Ωe

Φe_i(y)f(y)dy. (2.5) One relevant issue in the remainder of this article is the behavior at infinity of Ψ_D[f] and ofK[f].

We observe that if suppf ⊂B(0, R), then there exists CR such that, for all |x| ≥2R we have

|Ψ_D[f](x)| ≤C_Rkfk_L1, |K[f](x)| ≤ C_Rkfk_L1

|x|² . (2.6)

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

2.3. Biot-Savart law.

Next we introduce notation for the Biot-Savart law. Iff is a smooth function, compactly supported in Ω (or supported inΩ) and (γe _i)∈Rⁿ, the unique solutionu of the elliptic system (2.1) is given by:

u(x) =K[f](x) +Pn

i=1αi[f]Xi(x) in Ω (2.7)

u(x) =K[fe ](x) +P_n

i=1αe_i[f]Xe_i(x) inΩ,e where

αi[f] = Z

Ω

Φi(y)f(y)dy+γi and αei[f] = Z

Ωe

Φei(y)f(y)dy+γi, see [14] and [22].

In what follows, we use the notation m_i[f] :=

Z

Ω

Φ_i(y)f(y)dy. 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^k₀ verifying (1.2) is given by:

u^k₀(x) =K^k[ω₀^k|_Ωk](x) +

nk

X

i=1

α^k_i[ω^k₀|_Ωk]X_i^k(x) with α_i^k[ω₀^k|_Ωk] = Z

Ω^k

Φ^k_i(y)ω₀^k(y)dy+γ_i^k.

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

3. Convergence for fixed time For a functionv defined in the domain

Ω^ε,k :=R²\[^nk

i=1

B(z_i^k, ε)

,

we denote byEv its extension to the full plane by settingEv to vanish outside Ω^ε,k. Let K_R² be the Biot-Savart operator in the full plane. Forh regular enough, we write:

K_R²[h] = 1 2π

x^⊥

|x|² ∗h.

We seek to prove that the Biot-Savart formula (2.7) converges to K_R²[µ+f] when k→ ∞. It is natural to study the following decomposition:

K_R²[µ+Ef|_Ωε,k](x)−E

K^ε,k[f|_Ωε,k](x) +

nk

X

i=1

α^ε,k_i [f]X_i^ε,k(x)

=

K_R2[µ]−K_R2

"_nk X

i=1

γ_i^kδ_zk i

#

(x)

+

nk

X

i=1

γ_i^k

(x−z_i^k)^⊥

2π|x−z_i^k|² −EX_i^ε,k(x)

+

K_R2[Ef|_Ωε,k]−EK^ε,k[f|_Ωε,k]−

nk

X

i=1

m^ε,k_i [f]EX_i^ε,k (x)

=:A_k(x) +B_ε,k(x) +C_ε,k[f](x).

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(R²) to strong convergence of the associated velocities.

Proposition 3.1. LetR₀>0fixed. If a sequence{γ_i^k}_i=1...nk ⊂Rand{z^k_i}_i=1...nk ⊂B(0, R₀)verifies

nk

X

i=1

γ_i^kδ_zk

i * µweak-∗ in BM(R²) as k→ ∞

for some µ∈ BM_c(R²), then for any p ∈(1,2)we can decompose A_k in two parts A_k =A_k,1+A_k,2 such that

kA_k,1k_Lp(R²)+kA_k,2k_Lp∗

(R²)→0 as k→ ∞, where p^∗ = (¹_p −¹₂)⁻¹∈(2,∞).

The result in Proposition 3.1 fixes k such that K_R2[µ] is well approximated. In the following proposition, the kis fixed, and we study the limit ε→0 for the harmonic vector fields.

Proposition 3.2. Let k∈Nfixed and {z_i^k}_i=1...nk ⊂B(0, R₀) given. Then, for anyi= 1. . . n_k, there exists v_i^ε∈C^∞(Ω^ε,k) such that

(x−zi)^⊥

2π|x−z_i|² −Ev_i^ε(x)

L^p(R²)+kEv_i^ε−EX_i^ε,kk_L2(R²)→0 as ε→0, for anyp∈[1,2).

To be more precise for p = 1, if we define d_k := mini6=j|z_i^k −z_j^k|, then there exist C, C₁ > 0 independent of k andε (depending only on R0) such that for all ε≤ ^dk

n²_k√

2C1, we have

nk

X

i=1

(x−zi)^⊥

2π|x−z_i|² −Ev_i^ε(x) _L1(

R²)+

nk

X

i=1

kEv_i^ε−EX_i^ε,kk_L2(R²)≤C

(n⁵_kε²)^1/4+εn²_k d_k

2

|lnε|

.

Next, we are interested in the convergence ofC_ε,k. We will see later that it is important to establish this convergence uniformly in the (L¹∩L^p0)-norm of f.

Proposition 3.3. Let M₀ >0, p₀∈(2,∞], k∈N, {z^k_i}_i=1...nk ⊂R² fixed. We have that

K_R²[Ef|_Ωε,k]−EK^ε,k[f|_Ωε,k]−

nk

X

i=1

m^ε,k_i [f]EX_i^ε,k

L²(R²)→0 as ε→0, uniformly inf verifying:

kfk_L1∩L^p0(R²)≤M₀ and f is compactly supported.

To be more precise, there existC >0 independent of k andε (depending only onR0) such that for allε≤ ^d₈^k (with d_k is defined in Proposition 3.2), we have

K_R2[Ef|_Ωε,k]−EK^ε,k[f|_Ωε,k]−

nk

X

i=1

m^ε,k_i [f]EX_i^ε,k L²(R²)

≤Ckfk_L1∩L^p₀ε|lnε|n^1/2_k for anyf ∈L^pc⁰(R²).

In the last part of this section, we will constructε_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...nk ⊂B(0, R0), we have that suppµ⊂B(0, R0). From the weak-∗ convergence

nk

X

i=1

γ_i^kδ_zk

i * µweak-∗in BM(R²) when k→ ∞, we infer that kPnk

i=1γ_i^kδ_zk ik_BM(

R²) is uniformly bounded in k (see [4, Prop. 3.13], which is a conse- quence of the Banach-Steinhaus theorem). We have, by duality, thatBM(B(0, R₀+ 1)) is compactly

imbedded inW^−1,p(B(0, R₀+ 1)) for anyp∈(1,2), since W₀^1,p0(B(0, R₀+ 1)) is compactly imbedded inC0(B(0, R0+ 1)), where p⁰ =p/(p−1). Hence we get that

n_k

X

i=1

γ_i^kδ_zk

i −µ→0 strongly in W^−1,p(B(0, R₀+ 1)) when k→ ∞,

for anyp∈(1,2). Let us fixp∈(1,2). We recall that a distributionhinW^−1,p(Ω) can be decomposed ash=f₀+∂₁f₁+∂₂f₂, wheref₀,f₁andf₂are inL^p(Ω) and max_l=0...2kf_lk_Lp(Ω)≤ khk_W−1,p(Ω)(see, for example, [4, Prop. 9.20] for this decomposition). Consequently, there existsf₀^k,f₁^k and f₂^k belonging inL^p(R²) such that

nk

X

i=1

γ_i^kδ_zk

i −µ=f₀^k+∂1f₁^k+∂2f₂^k inR², with max_l=0,1,2kf_l^kk_Lp(R²)→0.

Now, we apply the Biot-Savart law inR²: K_R²[

nk

X

i=1

γ_i^kδ_z^k

i]−K_R²[µ] =K_R²[f₀^k] +∂1K_R²[f₁^k] +∂2K_R²[f₂^k]. (3.1) For the first right hand side term, we apply the Hardy-Littlewood-Sobolev Theorem (see e.g. [26, Theo. V.1] withα= 1) where ¹_p = _p¹∗ +¹₂ to get:

kK_R2[f₀^k]k_Lp∗

(R²) ≤C_pkf₀^kk_Lp(R²). This inequality holds forp∈(1,2), andp^∗ belongs to (2,∞).

Concerning the other terms in (3.1), we infer from the Calder´on-Zygmund inequality that k∂₁K_R²[f₁^k] +∂2K_R²[f₂^k]k_Lp(R²) ≤ k∇K_R2[f₁^k]k_Lp(R²)+k∇K_R2[f₂^k]k_Lp(R²)

≤ Cp(kf₁^kk_Lp(R²)+kf₂^kk_Lp(R²)).

SettingA_k,1:=∂1K_R²[f₁^k] +∂2K_R²[f₂^k] and A_k,2 :=K_R²[f₀^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...n which shrink to n points as ε→ 0. We denote by d= mini6=j|z_i−zj|and B_i^ε:=B(z_i, ε) withε < d/2. Let ibe fixed, the goal of this subsection is to compare

1 2π

(x−zi)^⊥

|x−zi|² and EX_i^ε.

The first vector field is not tangent to B_j^ε for j6=i, whereasX_i^ε is. We introduce a vector field v_i^ε which has an explicit form and verifies some of the properties ofX_i^ε. If we denote byϕ: [0,∞)→[0,∞) a non-increasing function such that ϕ(s) = 1 if s <1/4 and ϕ(s) = 0 ifs >1/2, we introduce cut-off functions: ϕ_j(x) :=ϕ(^|x−z_d^j|) which verifies

(ϕ_j(x)≡1 inB(z_j, d/4)

ϕ_j(x)≡0 inB(z_j, d/2)^c. (3.2) We use the inversion with respect toB_j^ε to define, for eachi6=j,

z_i^∗j,ε:=zj +ε² z_i−z_j

|z_i−zj|² ∈B(zj, ε), (3.3) which allows us to introduce the vector fieldsv_i^ε by:

v^ε_i(x) := 1 2π

(x−zi)^⊥

|x−zi|² − ∇^⊥X

j6=i

ϕj(x)

2π ln|x−z^∗j,ε_i |

|x−zj|

. (3.4)

The properties of such a vector field are listed here:

Lemma 3.4. We have that:

(1) v_i^ε is divergence free ;

(2) v_i^ε is tangent to the boundary ∂Ω^ε ;

(3) the circulations of v^ε_i around B_j^ε are equal to δij, for all j= 1. . . n ; (4) curlv_i^ε(x) =− 1

2π X

j6=i

ln|x−z_i^∗j,ε|

|x−zj| ∆ϕ_j(x)− 1 π

X

j6=i

x−z^∗j,ε_i

|x−z_i^∗j,ε|² − x−z_j

|x−zj|²

·∇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 ∆_2π¹ ln|x−P|=δ_P (whereδ_P denotes the Dirac mass at the pointP) and thatz_i^∗j,ε∈B(z_j, ε), so ∆ ln^|x−z

∗j,ε i |

|x−zj| is equal to zero outside B(z_j, ε).

Concerning (2) and (3), we note thatv_i^ε is equal to 1

2π

(x−zi)^⊥

|x−zi|²

in a neighborhood ofB_i^ε (namely for anyx∈B(zi, d/4)\B(zi, ε)), whereas it is equal to 1

2π

x−z_i

|x−zi|² − x−z_i^∗j,ε

|x−z^∗j,ε_i |² + x−z_j

|x−zj|² ⊥

=− 1

2π∇^⊥ln |x−z_i^∗j,ε|

|x−zi||x−zj|

in a neighborhood of B_j^ε for j6=i(namely for any x ∈B(x_j, d/4)\B(x_j, ε)). With the first form, it is clear thatv^ε_i is tangent toB_i^ε and that

I

∂B_i^ε

v_i^ε·τ dsˆ = 1.

Thanks to the definition ofz^∗j,ε_i (3.3), we can easily prove that for all x∈∂B_j^ε we have

|x−z_i^∗j,ε|

|x−z_i||x−z_j| =

(x−z_j)−ε²_|z^zi−zj

i−zj|²

(x−z_j)−(z_i−z_j) ε

=

|x−z_j|²−2ε^{2 (x−z}_|z^j|zi−zj)

i−zj|² +ε⁴_|z ¹

i−zj|²

1/2

|x−z_j|²−2(x−z_j|z_i−z_j) +|z_i−z_j|²1/2

ε

= 1

|z_i−z_j|,

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^∗j,ε_i , r) b B(zj, ε), so we get by Stokes formula that:

I

∂B_j^ε

v_i^ε·τ dsˆ = 1 2π

Z

B^ε_j

curl(x−zi)^⊥

|x−z_i|² dx+ 1 2π

I

∂B_j^ε

(x−zj)^⊥

|x−z_j|² ·τ dsˆ − 1 2π

I

∂B(z_i^∗j,ε,r)

(x−z^∗j,ε_i )^⊥

|x−z_i^∗j,ε|² ·τ dsˆ

= 0 + 1−1 = 0,

which ends the proof of the lemma.

Remark 3.5. The fact that the obstacles are disks simplifies the expression ofv_i^ε. Otherwise, we would have to considerT_j^ε the biholomorphism between (K_j^ε−zj)^c and the exterior of the unit disk, and we would define:

v_i^ε(x) := 1 2π

(x−z_i)^⊥

|x−zi|² +∇^⊥X

j6=i

ϕ_j 1

2πln|T_j^ε(x−z_j)−T_j^ε(z_i−z_j)||T_j^ε(x−z_j)|

|x−zi|

ε |T_j^ε(x−z_j)− _|T^Tε^jε(zi−zj) j(zi−zj)|²|

.

and we would have to prove that the modified v_i^ε behaves as in (3.4) because T_j^ε(x) = β_j^εx ε +h(x

ε) withh⁰(x) =O(_|x|¹2) 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).