Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve

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Two dimensional incompressible ideal flow around a thin obstacle tending to a curve

Christophe Lacave

Université de Lyon, Université Lyon 1, INSA de Lyon, F-69621, Ecole Centrale de Lyon, CNRS, UMR 5208, Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Received 18 November 2007; received in revised form 7 May 2008; accepted 30 June 2008 Available online 30 July 2008

Abstract

In this work we study the asymptotic behavior of solutions of the incompressible two dimensional Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to a curve. We extend results by Iftimie, Lopes Filho and Nussenzveig Lopes, obtained in the context of an obstacle tending to a point, see [D. Iftimie, M.C. Lopes Filho, H.J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Partial Differential Equations 28 (1–2) (2003) 349–379].

©2008 Elsevier Masson SAS. All rights reserved.

Keywords:Incompressible flow; Ideal flow; Exterior flow

1. Introduction

The purpose of this work is to study the influence of a thin material obstacle on the behavior of two dimensional incompressible ideal flows. More precisely, we consider a family of obstaclesΩ_ε which are smooth, bounded, open, connected, simply connected subsets of the plane, contracting to a smooth curveΓ asε→0. Given the geometry of the exterior domainR²\Ω_ε, a velocity field (divergence free and tangent to the boundary) on this domain is uniquely determined by the two following (independent) quantities: vorticity and circulation of velocity on the boundary of the obstacle. Throughout this paper we assume that initial vorticityω₀is independent ofε, smooth, compactly supported outside the obstaclesΩ_εand thatγ, the circulation of the initial velocity on the boundary, is independent ofε. From the work of K. Kikuchi [4], we know that there existsu^ε=u^ε(x, t )a unique global solution to the Euler equation in the exterior domainR²\Ω_ε associated to the initial data described above. Our aim is to determine the limit ofu^ε asε→0. As a consequence, we also obtain the existence of a solution of the Euler equations in the exterior of the curveΓ.

The study of incompressible fluid flows in presence of small obstacles was initiated by Iftimie, Lopes Filho and Nussenzveig Lopes [2,3]. The paper [2] treats the same problem as above but with obstacles that shrink homothetically to a pointP, instead of a curve. The case of Navier–Stokes is considered in [3]. In the inviscid case, these authors prove that if the circulationγvanishes, then the limit velocity verifies the Euler equation inR²(with the same initial

E-mail address:lacave@math.univ-lyon1.fr.

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.06.004

vorticity). If the circulation is non-zero, then the limit equation involves a new term that looks that a fixed Dirac mass in the pointPof strengthγ; the initial vorticity also acquires a Dirac mass inP. In the case of Navier–Stokes, the limit equation is always Navier–Stokes but the initial vorticity of the limit equation still has an additional Dirac mass inP. Here we will show that, in the inviscid case, the limit equation is the Euler equation inR²\Γ. The initial velocity for the limit equation is a velocity field which is divergence free inR², tangent toΓ such that the curl computed in R²\Γ isω0and the curl computed inR²isω0+gω₀δΓ wheregω₀ is a density given explicitly in terms ofω0andγ. Alternatively,gω₀is the jump of the tangential velocity acrossΓ.

More precisely, letΦε be a cut-off function in a smallε-neighborhood of the boundary (the precise definition of Φ_εis given in Section 4.2) and setω_ε=curlu^ε. Our main theorem may be stated as follows:

Theorem 1.1.There exists a subsequenceε=ε_k→0such that (a) Φ^εu^ε→ustrongly inL²_loc(R₊×R²);

(b) Φ^εω^ε→ωweak-∗inL^∞(R+;L⁴_loc(R²));

(c) uis related toωby means of relation(5.2);

(d) uandωare weak solutions ofω_t+u· ∇ω=0inR²×(0,∞).

The limit velocityuis explicitly given in terms ofωandγ(see Theorem 5.6) and can be viewed as the divergence free vector field which is tangent toΓ, vanishing at infinity, with curl inR²\Γ equal toωand with circulation around the curveΓ equal toγ. This velocity is blowing up at the endpoints of the curveΓ as the inverse of the square root of the distance and has a jump acrossΓ. Moreover, we have curlu=ω+g_ω(s)δ_Γ inR²× [0,∞), whereδ_Γ is the Dirac function of the curveΓ, and theg_ω which is defined in Lemma 5.8 depends onωand the circulationγ. The functiongω is continuous onΓ and blows up at the endpoints of the curveΓ as the inverse of the square root of the distance. One can also characterizegω as the jump of the tangential velocity acrossΓ. The presence of the additional termgωin the expression of curlu, compared of the Euler equation in the full plane, is compulsory to obtain a vector field tangent to the curve, with a circulationγ around the curve.

There is a sharp contrast between the behavior of ideal flows around a small and thin obstacle. In [2], the authors studied the vanishing obstacle problem when the obstacle tends homothetically to a pointP. Their main result is that the limit vorticity satisfies a modified vorticity equation of the form ω_t+u· ∇ω=0, with divu=0 and curlu= ω+γ δ(x−P ). In other words, for small obstacles the correction due to the vanishing obstacle appears as time- independent additional convection centered atP, whereas in the thin obstacle case, the correction term depends on the time. Although treating a related problem, the present work requires a different approach. Indeed, in [2], the proofs are simplified by the fact that the obstacles are homothetic to a fixed domain. Indeed, an easy change of variablesy=x/ε allows in that case to return to a fixed obstacle and to deduce the required estimates. This argument clearly does not work here and a considerable amount of work is needed to characterize the conformal mapping that sends the exterior of a small obstacle into the exterior of the unit disk. Moreover, in [2] the authors use the div–curl Lemma to obtain strong convergence for velocity. This is made possible by the validity of some bounds on the divergence and the curl of the velocity. A consequence of our work is that these estimates are no longer valid in our case, so this approach cannot work. We will be able to prove directly strong convergence for the velocity through several applications of the Lebesgue dominated convergence theorem. We finally observe that, in contrast to the case of [2], the vanishing of the circulationγ plays no role in our result. The limit velocity will always verify the same type of equation.

We also mention that Lopes Filho treated in [5] the case of several obstacles with one of the obstacles tending to a point, but the author had to work on a bounded domain. In this case, we do not have explicit formulas anymore, and the conformal mapping technique is replaced by qualitative analysis using elliptic techniques, including variational methods and the maximum principle.

The remainder of this work is organized in five sections. We introduce in Section 2 a family of conformal mappings between the exterior ofΩ_εand the exterior of the unit disk, allowing the use of explicit formulas for basic harmonic fields and the Biot–Savart law, which will be really helpful to obtain sharp estimations. In the third part, we precisely formulate the flow problem in the exterior of a vanishing obstacle. In Section 4, we collect a priori estimates in order to find the equation limit in Section 5 of this article. The last subsection concerns an existence result of the Euler equations on the exterior of a curve.

For the sake of clarity, the main notations are listed in an appendix at the end of the paper.

2. The Laplacian in an exterior domain 2.1. Conformal maps

LetD=B(0,1)andS=∂D. In what follows we identifyR²with the complex planeC.

We begin this section by recalling some basic definitions on the curve.

Definition 2.1.We call aJordan arca curveC given by a parametric representationC: ϕ(t ), 0t1, withϕ an injective (=one-to-one) function, continuous on[0,1]. Anopen Jordan archas a parametrizationC: ϕ(t ), 0< t <1, withϕ continuous and injective on(0,1).

We call aJordan curvea curve C given by a parametric representationC: ψ (t ),t∈R, 1-periodic, with ψ an injective function on[0,1), continuous onR.

Thus a Jordan curve is closed (ϕ(0)=ϕ(1)) whereas a Jordan arc has distinct endpoints. IfJ is a Jordan curve inC, then the Jordan Curve Theorem states thatC\J has exactly two componentsG₀ andG₁, and these satisfy

∂G₀=∂G₁=J.

The Jordan arc (or curve) is of classC^n,α (n∈N^∗,0< α1) if its parametrization ϕ isn times continuously differentiable, satisfyingϕ(t )=0 for allt, and if|ϕ⁽ⁿ⁾(t1)−ϕ⁽ⁿ⁾(t2)|C|t1−t2|^αfor allt1andt2.

LetΓ: Γ (t ), 0t1, be a Jordan arc. Then the subsetR²\Γ is connected and we will denote it byΠ. The purpose of this part is to obtain some properties of a biholomorphismT:Π→int D^c. After applying a homothetic transformation, a rotation and a translation, we can suppose that the endpoints of the curve are−1=Γ (0) and 1=Γ (1).

Proposition 2.2. If Γ is a C² Jordan arc, such that the intersection with the segment [−1,1] is a finite union of segments and points, then there exists a biholomorphismT:Π→intD^cwhich verifies the following properties:

• T⁻¹andDT⁻¹extend continuously up to the boundary, andT⁻¹mapsStoΓ,

• DT⁻¹is bounded,

• T and DT extend continuously up toΓ with different values on each side of Γ, except at the endpoints of the curve whereT behaves like the square root of the distance andDT behaves like the inverse of the square root of the distance,

• DT is bounded in the exterior of the diskB(0, R), withΓ ⊂B(0, R),

• DT is bounded inL^p(Π∩B(0, R))for allp <4andR >0.

Proof. We first study the case where the arc is the segment[−1,1]. We can have here an explicit formula for T. Indeed, the Joukowski function

G(z)=1 2

z+1 z

is a biholomorphism between the exterior of the disk and the exterior of the segment. It maps the circleC(0, R)on the ellipse parametrized by ¹₂(R+1/R)cosθ+¹₂(R−1/R)isinθwithθ∈ [0,2π ), and it maps the unit circle on the segment.

Remarking thatG(z)=G(1/z)we can conclude thatGis also a biholomorphism between the interior of the disk minus 0 and the exterior of the segment.

Therefore, anyz /∈ [−1,1]has one antecedent ofGinDand another one in intD^c. Forz∈ [−1,1]the antecedents are exp(±iarccosz)=z±i√

1−z². Therefore, there are exactly two antecedents except for−1 and 1. In fact, we have considered the double coveringGfromC^∗toC, which is precisely ramified over−1 and 1.

LetT˜be the biholomorphism between the exterior of the segment and intD^c, such thatT˜⁻¹=G, and letT˜int=1/T˜ be the biholomorphism between the exterior of the segment andD\ {0}, such thatT˜_int⁻¹=G.

To find an explicit formula ofT˜, we have to solve an equation of degree two. We find two solutions:

T˜₊=z+

z²−1 and T˜₋=z−

z²−1. (2.1)

We consider that the function square-root is defined by √ z=√

|z|e^iθ/2 with θ, the argument of z, verifying

−π < θ π. It is easy to observe that T˜ = ˜T₊ on {z| (z) >0} ∪iR+ andT˜ = ˜T₋ on {z| (z) <0} ∪iR−. Despite this,T˜ isC^∞(C\ [−1,1],intD^c)becauseT˜=G⁻¹.

Therefore in the segment case,T = ˜T and the first two points are straightforward. An obvious calculation allows us to find an explicit formula forT˜:

T˜(z)=1± z

√z²−1, (2.2)

with the choice of sign as above. This form shows us thatDT˜ blows up at the endpoints of the segment as the inverse of the square root of the distance, which is bounded inL^p_locforp <4. Moreover, a mere verification can be done to find that for everyx∈(−1,1), we have

z→x,lim(z)>0T (z)=x+i

1−x²= ˜T₊(x) even if(z) <0, and

z→x,lim(z)<0T (z)=x−i

1−x²= ˜T₋(x).

In the same way,DT extends continuously up to each side ofΓ, which concludes the proposition in the segment case.

Now, we come back to our problem, with any curve Γ. We consider the curveΓ˜ ≡ ˜T (Γ )∪ ˜T_int(Γ )= ˜T₊(Γ )∪ T˜₋(Γ ). We now show thatΓ˜ is aC^1,1Jordan curve.

We consider first the case where Γ does not intersect the segment (−1,1). Then γ₁≡ ˜T (Γ )⊂D^c andγ₂≡ T˜int(Γ )⊂DareC²open Jordan arcs, with the endpoints on−1 and 1 (see Fig. 1). SoT (˜ −1)= −1= ˜Tint(−1)and we can observe thatΓ˜ is a Jordan curve.

We wroteopenJordan curve because the problem with−1 and 1 is thatT˜(±1)is not defined. However, if we use the arclength coordinates

s(t )= t 0

γ₁(τ )dτ= t 0

T˜Γ (τ )Γ(τ )dτ, (2.3)

which are well-defined and bounded becauseT˜is bounded inL¹_loc, then we have^dγ_ds¹ =_|^γ_γ¹

1|. So to prove the derivative continuity, we should show that limt→0

γ₁

|γ₁| and limt→0 γ₂

|γ₂| exist and are opposite. For that, we will use the following lemma:

Lemma 2.3.If there exists a neighborhood of0whereΓ (t )does not intersect the segment(−1,1), then ^{T˜(Γ )}

| ˜T(Γ )|(t )has a limit ast→0.

Proof. First, sinceT˜(z)=1−z/√

z²−1 in a neighborhood of−1, we compute T˜(Γ )

| ˜T(Γ )|(t )=|√ Γ²−1|

√Γ²−1 (t )

√Γ²−1−Γ

|√

Γ²−1−Γ|(t ).

The second fraction tends to 1 ast→0. We haveΓ(0)=0, so we can writeΓ²(t )=1+at+o(t )for a∈C^∗. If a /∈R⁻then fort small enough{Γ²(t )−1} ⊂C\R⁻and

tlim→0

T˜(Γ )

| ˜T(Γ )|(t )=

√√|a|

a =e^−iθ/2 withθ≡arga∈(−π, π ).

Fora∈R⁻, asa=(Γ²)(0)=2Γ (0)Γ(0), then we haveΓ(0)∈R⁺and the curve is tangent to the segment [−1,1]. We have here two cases:

• ifΓ is over the segment on the neighborhood, then(Γ²(t )−1) <0 and

tlim→0

T˜(Γ )

| ˜T(Γ )|(t )=i,

• ifΓ is below the segment on the neighborhood, then(Γ²(t )−1) >0 and

tlim→0

T˜(Γ )

| ˜T(Γ )|(t )= −i. 2

Fig. 1.Γ does not intersect[−1,1].

Let us continue the proof of Proposition 2.2. Lemma 2.3 allows us to observe thatΓ˜ isC¹in−1, because

tlim→0

γ₁

|γ₁|(t )=lim

t→0

T˜(Γ )

| ˜T(Γ )|(t )Γ

|Γ|(t )= −lim

t→0−| ˜T²(Γ )|

T˜²(Γ ) (t )T˜(Γ )

| ˜T(Γ )|(t )Γ

|Γ|(t )= −lim

t→0

γ₂

|γ₂|(t ), becauseT˜_int=1/T˜.

To prove thatΓ˜ is Lipschitz, we will show thatγ₁andγ₂ areC¹with the arclength parametrization denoted by sand defined in (2.3) (t denotes the variable of the original parametrization). Letf₁(s)=^dγ_ds¹(s)=_|^γ_γ^{1(t )}

1(t )|, where the primes denote derivatives with respect tot, then we need to prove that ^df_ds¹ has a limit whens→0. We have

df₁

ds (s)= γ₁

|γ₁|²− γ₁

|γ₁|⁴ γ₁, γ₁

= 1

|γ₁|²

γ₁− γ₁

|γ₁| γ₁

|γ₁|, γ₁

≡ 1

|γ₁|²A.

We compute γ₁= ˜T(Γ )Γ,

γ₁= ˜T(Γ )(Γ)²+ ˜T(Γ )Γ, with

T (z)˜ =z− z²−1, T˜(z)=1−z/

z²−1, T˜(z)= −1/

z²−1+z²/ z²−1³.

We do some Taylor expansions in a neighborhood of zero:

Γ (t )= −1+at+bt²+O t³

, Γ²(t )=1−2at+

a²−2b t²+O

t³ , 1/

Γ²−1(t )= 1

√−2a

√1 t

1−t

a²−2b

/(−4a) +O

t^3/2 .

The last expansion holds in any case, except whenΓ is tangent to the segment (a∈R⁺) and over the segment on a neighborhood of−1. In this last case, we should replace 1/√

−2abyiinstead of−i.

Then

T˜(Γ )= 1

√−2a

√1

t +1+O t^1/2

, T˜(Γ )= 1

√−2a³

√1 t³+C1

√t +O t^1/2

. and

γ₁= a

√−2a

√1

t +a+O t^1/2

, γ₁

|γ₁| = a

|a|

|√

−2a|

√−2a +C₂√

t+O(t ), γ₁= a²

√−2a³

√1 t³+C3

√1

t +O(1).

Now, we can evaluateA:

γ₁− γ₁

|γ₁| γ₁

|γ₁|, γ₁

= 1

√t³ a²

√−2a³− a

|a|

|√

−2a|

√−2a a

|a|

|√

−2a|

√−2a , a²

√−2a³

+C₄1 t +O

t^−1/2 . We can easily see that arg(a²/√

−2a³)= ±π+arg(a/√

−2a), so a²

√−2a³− a

|a|

|√

−2a|

√−2a a

|a|

|√

−2a|

√−2a , a²

√−2a³

=0

anddf₁/ds=C₅+O(t^1/2), which means thatdf₁/ds has a limit ass→0. This argument holds forγ₂, doing the calculations with T˜_int(z)=z+√

z²−1. So dγ₁/ds anddγ₂/ds are C¹ on[0,1] and we see thatΓ˜ is Lipschitz becauseΓ˜ =γ₁∪γ₂.

Finally, ifΓ intersects[−1,1]at one pointx=Γ (t0), thenT (Γ )˜ is the union of two Jordan curves with a jump:

˜

T (Γ (t₀⁻))=1/T (Γ (t˜ ₀⁺))(see Fig. 2). In this case,T˜_int(Γ )is also the union of two Jordan arcs which extendT (Γ ),˜ indeed

˜ T_int

Γ (t₀⁺)

=1/T˜ Γ (t₀⁺)

= ˜T Γ (t₀⁻)

.

Fig. 2.Γ intersects[−1,1]at one point.

To show the continuity ofΓ˜ onT (Γ (t˜ ₀)), we consider for example thatx∈(0,1)and that(Γ (t₀⁻)) >0 and (Γ (t₀⁺)) <0. We can compute

T˜ Γ (t₀⁻)

=x+i 1−x², T˜

Γ (t₀⁺)

=1+xi/

1−x², becauseT˜=T₊in a neighborhood ofx T˜

Γ (t₀⁻)

=1−xi/

1−x²,

to check that− ˜T (Γ (t₀⁻))²T˜(Γ (t₀⁺))= ˜T(Γ (t₀⁻)), which allows us to conclude that T˜_int

Γ t₀⁺

= −1/T˜ Γ

t₀⁺2T˜ Γ

t₀⁺

= ˜T Γ

t₀⁻ .

We leave to the reader the other cases. Let us do just another case:x=0 then T˜

Γ (t₀⁻)

= ±i, T˜

Γ (t₀⁺)

=1, T˜

Γ (t₀⁻)

=1,

and as−(±i)²=1 we have the continuity ofΓ˜. AsΓ˜is bounded,Γ˜is Lipschitz, so the curveΓ˜ isC^1,1and closed.

We have just studied the case of one or zero intersection ofΓ with the segment(−1,1)but this argument works in the general case because we have a finite number of intersection. For example, ifΓ ⊂ [−1,1]in a neighborhood of−1, thenG˜ ⊂S, soG˜ is obviouslyC^1,1in this neighborhood.

We denote byΠ˜ the unbounded connected component ofR²\ ˜Γ. We claim that we can constructT₂, a biholo- morphism betweenΠ andΠ˜, such thatT₂⁻¹=G. Indeed, if we introduceA= ˜Π∩ ¯DandB=(intΠ˜^c)∩D^c (see Fig. 1), we observe thatB=1/A, becauseγ₂=1/γ1and 1/S=S. AsG(1/z)=G(z),Gis bijective on intD^cand 1/(∂D∩ ˜Π )⊂ ˜Π^c thenGis bijective onΠ˜ andG(Π )˜ =R²\Γ. Therefore, we have an functionT2 mapping the exterior of the Jordan arc to the exterior of the inner domain of aC^1,1Jordan curve, such thatT₂⁻¹(z)=1/2(z+1/z).

Next, we just have to use the Riemann mapping Theorem and we find a conformal mappingF betweenΠ˜ andD^c, such thatT ≡F◦T₂mapsΠtoD^candF (∞)= ∞. To finish the proof, we use the Kellogg–Warschawski Theorem (see Theorem 3.6 of [6], which can be applied for the exterior problems), to observe thatF andFhave a continuous extension up to the boundary. Therefore, adding the fact thatDFandDF⁻¹are bounded at infinity (see Remark 2.5), we find the same properties as in the segment case, in particular thatDT blows up at the endpoints of the curve like the inverse of the square root of the distance (see (2.2)). 2

Remark 2.4.IfΓ intersects the segment[−1,1]infinitely many times, the curveΓ˜ may not be even C¹. For ex- ample a curve which starts ast →(t−1;e^1/t2sin(1/t )), t∈ [0,1/4], has two sequencest_n→0 and t˜_n→0 such T˜(Γ )/| ˜T(Γ )|tends toiand−i.

Remark 2.5. If we have a biholomorphism H between the exterior of an open connected and simply connected domainAandD^c, such thatH (∞)= ∞, then there exists a non-zero real numberβ and a holomorphic function h:Π→Csuch that:

H (z)=βz+h(z) with

h(z)=O 1

|z|²

, as|z| → ∞.

This property can be applied for theF above to see thatDF andDF⁻¹are bounded.

Proof. Indeed, after a translation we can suppose that 0∈intA, and we considerW (z)=1/H (1/z). The function W is holomorphic in a neighborhood of 0 and can be written asW (z)=a₀+a₁z+a₂z²+ · · ·.We haveW (0)=0 soa₀=0. Now we want to prove thata₁=0 thanks to the univalence. Indeed, ifa₁=0, we consider the first non- zeroa_k, and we observe that there existsR >0 such that|W (z)−a_kz^k||a_k||z^k|inB(0, R). Next, we denote by g(z)=a_kz^k. On∂B(0, R),|W (z)−g(z)||a_k|R^k|g(z)|. Then we can apply the Rouché Theorem to conclude that W andghave the same number of zeros inB(0, R), which is a contradiction with the fact thatWis a biholomorphism andgnot. Thereforea1=0 andH (z)=z/a1+b0+b1/z+ · · ·,which ends the proof. MultiplyingH by|a1|/a¯1, we can assume thatβ=1/a1is real number. 2

2.2. The Biot–Savart law

LetΩ₀be a bounded, open, connected, simply connected subset of the plane, the boundary of which, denoted byΓ₀, is aC^∞Jordan curve. We will denote byΠ₀the unbounded connected component ofR²\Γ₀, so thatΩ₀^c=Π₀.

We denote byG_Π0=G_Π0(x, y)the Green’s function, whose the value is:

GΠ₀(x, y)= 1

2π log |T₀(x)−T₀(y)|

|T₀(x)−T₀(y)^∗||T₀(y)| (2.4)

writingx^∗=x/|x|². The Green’s function is the unique function which verifies:

⎧⎨

⎩

_yG_Π0(x, y)=δ(y−x) forx, y∈Π₀, G_Π0(x, y)=0 fory∈Γ₀,

G_Π0(x, y)=G_Π0(y, x).

(2.5)

Then the kernel of the Biot–Savart law isK_Π0=K_Π0(x, y)≡ ∇_x^⊥G_Π0(x, y). With(x₁, x₂)^⊥=_−x2

x1

, the explicit formula ofK_Π0 is given by

K_Π0(x, y)=((T0(x)−T0(y))DT0(x))^⊥

2π|T₀(x)−T₀(y)|² −((T0(x)−T0(y)^∗)DT0(x))^⊥

2π|T₀(x)−T₀(y)^∗|² . (2.6)

We require information on far-field behavior ofK_Π0. We will use several times the following general relation:

a

|a|²− b

|b|²

=|a−b|

|a||b| , (2.7)

which can be easily checked by squaring both sides.

We now find the following inequality:

KΠ₀(x, y)C |T₀(y)−T₀(y)^∗|

|T₀(x)−T₀(y)||T₀(x)−T₀(y)^∗|. Forf ∈C_c^∞(Π₀), we introduce the notation

K_Π0[f] =K_Π0[f](x)≡

Π0

K_Π0(x, y)f (y) dy.

It is easy to see, for large|x|, that|K_Π0[f]|(x)C₁/|x|²whereC₁depends on the size of the support off. We used here the explicit formulas for the biholomorphismT₀(Remark 2.5).

Lemma 2.6.The vector fieldu=K_Π0[f]is a solution of the elliptic system:

⎧⎪

⎨

⎪⎩

divu=0 inΠ0, curlu=f inΠ₀, u· ˆn=0 onΓ₀, lim_|x|→∞|u| =0.

The proof of this lemma is straightforward.

2.3. Harmonic vector fields

We will denote bynˆ the unit normal exterior toΠ0 atΓ0. In what follows all contour integrals are taken in the counter-clockwise sense, so that

Γ₀F ·ds= −

Γ₀F · ˆn^⊥ds.

Proposition 2.7.There exists a unique classical solutionH=H_Π0 of the problem:

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

divH=0 inΠ₀,

curlH=0 inΠ0, H· ˆn=0 onΓ₀, lim_|x|→∞|H| =0,

Γ0H·ds=1.

(2.8)

Moreover,H_Π0=O(1/|x|)as|x| → ∞.

To prove this, one can check thatH_Π0(x)=_2π¹∇^⊥log|T₀(x)|is the unique solution. The details can be found in [2].

3. Flow in an exterior domain

Let us formulate precisely here the small obstacle limit.

3.1. The initial-boundary value problem

Letu=u(x, t )=(u₁(x₁, x₂, t ), u₂(x₁, x₂, t ))be the velocity of an incompressible, ideal flow inΩ₀^c. We assume thatuis tangent toΓ₀andu→0 as|x| → ∞. The evolution of such a flow is governed by the Euler equations:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂tu+u· ∇u= −∇p inΠ0×(0,∞), divu=0 inΠ₀× [0,∞), u· ˆn=0 inΓ0× [0,∞), lim_|x|→∞|u| =0 fort∈ [0,∞), u(x,0)=u₀(x) inΩ^c,

(3.1)

wherep=p(x, t )is the pressure. An important quantity for the study of this problem is the vorticity:

ω=curl(u)=∂₁u₂−∂₂u₁.

The velocity and the vorticity are coupled by the elliptic system:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

divu=0 inΠ₀× [0,∞) curlu=ω inΠ₀× [0,∞), u· ˆn=0 inΓ0× [0,∞), lim_|x|→∞|u| =0 fort∈ [0,∞).

Lemma 2.6 and Proposition 2.7 assure us that the general solution of this system is given by u=u(x, t )= K_Π0[ω(·, t )](x)+αH_Π0(x)for a functionα=α(t ). However, using the fact that the circulationγ of uaroundΓ is conserved, we prove thatαdoes not depend on the time, andα(t )=γ+

Π0curlu₀(x)(see [2]). Therefore, if we give the circulation, then we have the uniqueness of the solution of the previous system.

Finally, we can now write the vorticity formulation of this problem as:

⎧⎨

⎩

∂tω+u· ∇ω=0 inΠ0×(0,∞), u=K_Π0[ω] +αH_Π0 inΠ₀× [0,∞), ω(x,0)=curlu₀(x) inΠ₀.

(3.2)

3.2. The evanescent obstacle

We will formulate in this subsection a family of problems, parametrized by the size of the obstacle. Therefore, we fixω₀such that its support is compact and does not intersectΓ.

We will consider a family of domain Ω_ε, containingΓ, withε small enough, such that the support ofω₀does not intersectΩ_ε. If we denote byT_ε the biholomorphism betweenΠ_ε≡Ω_ε^candD^c, then we suppose the following properties:

Assumption 3.1.The biholomorphism family{T_ε}verifies (i) T_ε−TL^∞(B(0,R)∩Πε)→0 for anyR >0,

(ii) det(DT_ε⁻¹)is bounded inΠ_εindependently ofε, (iii) DT_ε−DTL³(B(0,R)∩Πε)→0 for anyR >0,

(iv) there existR >0 andC >0 such that|DTε(x)|C|x|onB(0, R)^c.

Remark 3.2.We can observe that point (iii) implies that for anyR,DT_ε is bounded inL^p(B(0, R)∩Π_ε)indepen- dently ofε, forp3.

Just before going on, we give here one example of an obstacle family.

Example 3.3.We considerΩε≡T⁻¹(B(0,1+ε)\D). In this case,Tε=₁₊¹_εT verifies the previous assumption. If Γ is a segment, thenΩ_εis the interior of an ellipse around the segment.

The problem of this example is that the shape of the obstacle is the same ofΓ.

We naturally denote byΓ_ε=∂Ω_ε andΠ_ε=intΩ_ε^c. We denote also byG_ε,Kε andH_ε the previous functions corresponding atΠ_ε.

Consider also the following problem:

∂_tω^ε+u^ε· ∇ω^ε=0 inΠ_ε×(0,∞), u^ε=K^ε[ω^ε] +αH^ε inΠ_ε× [0,∞), ω^ε(x,0)=ω₀(x) inΠ_ε.

It follows from the work of Kikuchi [4] that, for any ε >0, ifω0is sufficiently smooth then this system has a unique solution.

We now write the explicit formulas forK^εandH^ε: K^ε= 1

2πDT_ε^t(x)

(T_ε(x)−T_ε(y))^⊥

|T_ε(x)−T_ε(y)|² −(T_ε(x)−T_ε(y)^∗)^⊥

|T_ε(x)−T_ε(y)^∗|²

(3.3) and

H^ε= 1

2πDT_ε^t(x)

(T_ε(x))^⊥

|T_ε(x)|²

. (3.4)

We introduce in the same way,KandH, replacingT_εbyT.

The regularity ofT implies thatH^εis bounded inL²_loc, which is really better than the punctual limit for the obstacle (see [2]) whereH^εis justL¹_loc. In our case, the limit is easier to see whenTε→T, and this extra regularity will allow us the passing to the limit.

4. A priori estimates

These estimates are important to conclude on the asymptotic behavior of the sequences(u^ε)and(ω^ε). The transport nature of (3.3) gives us the classical estimates for the vorticity:ω^ε(·, t )L^p(Π_ε)= ω0L^p(R²)and forp∈ [1,+∞].

In this article, we suppose thatω0isL^∞and compactly supported. Moreover we chooseεsmall enough, so that the support ofω₀does not intersectΠ_ε.

4.1. Velocity estimate

We begin by recalling a result found in [1].

Lemma 4.1.Leta∈(0,2),S⊂R²andh:S→R₊be a function inL¹(S)∩L^∞(S). Then

S

h(y)

|x−y|^adyCh¹_L⁻1(S)^a/2h^a/2_L∞(S).

The goal of this subsection is to find a velocity estimate thanks to the explicit formula ofu^εin function ofω^εandγ (Section 3.2):

u^ε(x, t )= 1

2πDT_ε^t(x)(I1+I2)+αH^ε(x) (4.1)

with I₁=

Πε

(T_ε(x)−T_ε(y))^⊥

|T_ε(x)−T_ε(y)|²ω^ε(y, t ) dy, (4.2)

and

I₂= −

Πε

(Tε(x)−Tε(y)^∗)^⊥

|T_ε(x)−T_ε(y)^∗|²ω^ε(y, t ) dy. (4.3)

We begin by estimatingI₁andI₂.

Lemma 4.2.Leta∈(0,2)andh:Π_ε→R+be a function inL¹(Π_ε)∩L^∞(Π_ε). We introduce I1,a=

Πε

|h(y)|

|T_ε(x)−T_ε(y)|^ady, (4.4)

I˜₂=

Πε

|h(y)|

|T_ε(x)−T_ε(y)^∗|dy. (4.5)

There exists a constantC >0depending only on the shape ofΓ, such that

|I1,a|Ch¹_L⁻1^a/2h^a/2_L∞ and | ˜I2|C

h^1/2_L1h^1/2_L∞+ hL¹

.

Remark 4.3.It will be clear from the proof that similar estimates hold true withT_εreplaced byT.

Proof. We start with theI_1,a estimate. LetJ=J (ξ )≡ |det(DTε⁻¹)(ξ )|andz=T_ε(x). Making also the change of variablesη=T_ε(y), we find

|I1,a|

|η|1

1

|z−η|^ah

T_ε⁻¹(η)J (η) dη. (4.6)

Next, we introducef^ε(η)= |h(T_ε⁻¹(η))|J (η)χ_{|η|1}, withχ_E the characteristic function of the setE. Changing variables back, we get

f^ε

L¹(R²)= hL¹.

The second point of Assumption 3.1 allows us to write f^ε

L^∞(R²)ChL^∞.

So we apply the previous lemma forf and we finally find

|I1,a|

R²

1

|z−η|^af^ε(η) dηC1f^ε1−a/2

L¹ f^εa/2

L^∞. (4.7)

This concludes the estimate forI_1,a. Let us estimateI˜₂:

| ˜I₂|

Πε

1

|T_ε(x)−T_ε(y)^∗|h(y)dy.

We use, as before, the notationsJ,zand the change of variablesη

| ˜I2|

|η|1

1

|z−η^∗|h

T_ε⁻¹(η)J (η) dη. (4.8)

Next, we again change variables writingθ=η^∗, to obtain:

| ˜I₂|

|θ|1

1

|z−θ|h

T_ε⁻¹(θ^∗)J (θ^∗)dθ

|θ|⁴

=

|θ|1/2

+

1/2|θ|1

≡I21+I22.

First we estimateI21. Asz=Tε(x), one has that|z|1, and if|θ|1/2 then|z−θ|1/2. Hence

|I₂₁|

|θ|1/2

2h

T_ε⁻¹(θ^∗)J (θ^∗)dθ

|θ|⁴ (4.9)

=2

|η|2

h

T_ε⁻¹(η)J (η) dη2h_L¹. (4.10)

Finally, we estimateI₂₂. Letg^ε(θ )= |h(T_ε⁻¹(θ^∗))|J (θ^∗)/|θ|⁴. We have I₂₂=

1/2|θ|1

1

|z−θ|g^ε(θ ) dθ.

As above, we deduce by changing variables back that g^ε

L¹(1/2|θ|1)hL¹. It is also trivial to see that

g^ε

L^∞(1/2|θ|1)ChL^∞. By Lemma 4.1

|I₂₂| =

1/2|θ|1

1

|z−θ|g^ε(θ ) dθ (4.11)

Cg^ε1/2

L¹(1/2|θ|1)g^ε1/2

L^∞(1/2|θ|1)Ch^1/2_L1h^1/2_L∞. 2 (4.12) Since|Tε(x)|1, one can easily see from (3.4) that|H^ε(x)||DTε(x)|. Moreover, applying the previous lemma witha=1 andh=ω^ε∈L¹∩L^∞, we get that

|I₁|Cω^ε1/2

L¹ ω^ε1/2

L∞ and |I₂|Cω^ε1/2

L¹ ω^ε1/2

L∞+ω^ε

L¹

. Thanks to the explicit formula (4.1), we can deduce directly the following theorem:

Theorem 4.4.u^ε is bounded inL^∞(R⁺, L²_loc(Π_ε))independently ofε. More precisely, there exists a constantC >0 depending only on the shape ofΓ and the initial conditionsω₀L¹,ω₀L^∞, such that

u^ε(·, t )

L^p(S)CDTεL^p(S), for allp∈ [1,∞]and for any subsetSofΠε.

The difference with [2] is that we have an estimateL^p_locinstead ofL^∞, but in our case, this estimate concerns all the velocityu^ε. It is one of the reason of the use of a different method to the velocity convergence.