Steady asymmetric vortex pairs for Euler equations

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Steady asymmetric vortex pairs for Euler equations

Zineb Hassainia, Taoufik Hmidi

To cite this version:

Zineb Hassainia, Taoufik Hmidi. Steady asymmetric vortex pairs for Euler equations. Discrete & Con-

tinuous Dynamical Systems - A, 2021, 41 (4), pp.1939-1969. �10.3934/dcds.2020348�. �hal-03163100�

DYNAMICAL SYSTEMS

Volume41, Number4, April2021 pp.1939–1969

STEADY ASYMMETRIC VORTEX PAIRS FOR EULER EQUATIONS

Zineb Hassainia

NYU, Abu Dhabi Saadiyat Marina District P.O. Box 129188, Abu Dhabi, UAE

Taoufik Hmidi

IRMAR, Universit´e de Rennes 1 Campus de Beaulieu 35 042 Rennes cedex, France

(Communicated by Thomas Bartsch)

Abstract. In this paper, we study the existence of co-rotating and counter- rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counter- rotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations. We also provide a careful study of the asymptotic behav- ior for the angular velocity and the translating speed close to the point vortex pairs.

1. Introduction. In this paper we shall be concerned with the dynamics of asym- metric vortex pairs for the two-dimensional incompressible Euler equations. These equations describe the motion of an ideal fluid and take the form







∂tω+v· ∇ω= 0 (t, x)∈R+×R², v=−∇^⊥(−∆)⁻¹ω,

ω_|t=0=ω0,

(1) where v = (v1, v2) refers to the velocity field and its vorticity ω is given by the scalar ω=∂1v2−∂2v1. The second equation in (1) is nothing but the Biot-Savart law which can be written in the complex form:

v(t, z) = i 2π

ˆ

R²

ω(t, ζ)

ζ−z dA(ζ), ∀z∈C,

where we identify v = (v1, v2) with the complex valued-function v1+iv2 and dA denotes the planar Lebesgue measure. The global existence and uniqueness of solu- tions with initial integrable and bounded vorticity was established a long time ago by Yudovich [37]. He proved that the system (1) admits a unique global solution in the weak sense, provided that the initial vorticity ω0 lies in L¹∩L^∞. In this setting, we can rigorously deal with the so-called vortex patches, which are the char- acteristic function of bounded domains. This specific structure is preserved along

2020Mathematics Subject Classification. Primary: 35Q35, 35Q31; Secondary: 53C35, 76C05.

Key words and phrases. Euler equations, vortex dynamics, steady vortex pairs, desingulariza- tion, implicit function theorem.

1939

the time and the solutionω(t) is uniformly distributed in the bounded domainDt, which is the image by the flow mapping of the initial domain D0. When D0 is a disc then the vorticity is stationary. However, elliptical vortex patches undergo a perpetual rotation about their centers without changing the shape. This discovery goes back to Kirchhoff [27] and till now, the ellipses are the only explicit examples with such properties in the setting of vortex patches. The existence of general class of implicit rotating patches, called also V-states, was discovered numerically by Deem and Zabusky [6]. Few years later, Burbea [2] gave an analytical proof using complex analytical tools and bifurcation theory to show the existence of a countable family of V-states withm-fold symmetry for each integer m≥2. More precisely, the rotating patches appear as a collection of one dimensional branches bifurcating from Rankine vortices at the discrete angular velocities set_m−1

2m , m≥2 . These local branches were extended very recently to global ones in [17], where the mini- mum value on the patch boundary of the angular fluid velocity becomes arbitrarily small near the end of each branch. The regularity of the V-states boundary has been conducted in a series of papers [3, 4, 17,24]. The existence of small loops in the bifurcation diagram has been proved recently in [19]. From numerical point of view, Wu, Overman, and Zabusky [36] went further along the same branches and found singular limiting solutions with 90^◦ corner. We also refer to the paper [30]

where it is proved that corners with right angles is the only plausible scenario for the limiting V-states. In this context also see [33,28,29].

It is worth pointing out that Burbua’s approach has been intensively exploited in the few last years in different directions. For instance, this was implemented to prove the existence of rotating patches close to Kirchhoff’s ellipses [4, 21], multi- connected patches [20,22,25], patches in bounded domains [8], non trivial rotating smooth solutions [5] and rotating vortices with non uniform densities [13]. We mention that many of these results apply not only to the 2D Euler equations but also to more singular nonlinear transport equations, but with much more involved computations. In particular, the construction of simply connected V-states for the inviscid generalized surface quasi-geostrophic equations was accomplished in [16,3]

and their boundary analyticity was discussed in [4]. The existence of uniformly rotating doubly connected patches was conducted in [7, 32] and the stationary doubly connected patches in [14]. The radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler and gSQG equations was analyzed in [11, 18, 15]. Besides the gSQG, the bifurcation theory was also successfully implemented to the quasi-geostrophic shallow-water equations in [10] as well to a 3D quasi-geostrophic model [12] to show the existence of non trivial rotating patches.

It is important to emphasize that all of the aforementioned analytical results treat connected patches. However, for the disconnected ones the bifurcation arguments discussed above are out of use. The main objective of this paper is to deal with vortex pairs moving without changing the shape. One of the very simplest nontrivial vortex equilibria is given by a pair of point vortices with magnitudeγ1 andγ2 and far away at a distanced. It is well known that if the circulations are with opposite signsγ1=−γ2then the system travels in a rectilinear motion with a constant speed U0 = γ1

2πd, otherwise the pair of point vortices rotates steadily with the angular velocity Ω₀=γ₁+γ₂

2πd² .

Coming back to the emergence of steady disconnected vortex patches, translat- ing vortex pairs of symmetric patches were discovered numerically by Deem and

Zabusky [6] and Pierrehumbert [31] where they conjectured the existence of a curve of translating symmetric pair of simply connected patches emerging from two point vortices. A similar study was established by Saffman and Szeto in [34] for the co-rotating vortex pairs, where two symmetric patches with the same circulations rotate about the centroid of the system with constant angular velocity. In the same direction Dritschel [9] calculated numerically the V-states of asymmetric vortex pairs and discussed their linear stability.

The analytical study of the corotating vortex pairs of patches was conducted by Turkington [35] using variational principle. However, this approach does not give sufficient information on the topological structure of each vortex patch and the uniqueness problem is left open. In the same direction Keady [27] implemented the same approach to prove the existence part of translating vortex pairs of symmetric patches. Very recently, Hmidi and Mateu [23] gave direct proof confirming the nu- merical experiments and showing the existence of co-rotating and counter-rotating vortex pairs using the contour dynamics equations combined with a desingulariza- tion of the point vortex pairs and the application of the implicit function theorem.

The main concern of this work is to explore the existence of of co-rotating and counter-rotating asymmetric vortex pairs using the contour dynamics equations.

Before stating our result we need to make some notation. Let ε∈ (0,1), γ1, γ2 ∈ R, b1, b2 ∈R+ and d >2(b1+b2). Consider two small simply connected domains D₁^εandD^ε₂containing the origin and contained in the open ballB(0,2) centered at the origin and with radius 2. Define

ω0,ε= γ1

ε²b²₁χD˜^ε₁+ γ2

ε²b²₂χD˜^ε₂, with

D˜₁^ε=εb1D₁^ε and D˜₂^ε=−εb2D₂^ε+d.

Informally stated, our main existence result is the following

Theorem 1.1. There existsε0>0such that the following results hold true.

1. For any γ1, γ2 ∈ R such that γ1+γ2 6= 0 and any ε ∈ (0, ε0] there exists two strictly convex domains D^ε₁ and D₂^ε at least of class C¹ such that ω0,ε

generates a co-rotating vortex pair for (1).

2. For γ1∈Rand anyε∈(0, ε0]there exists γ2=γ2(ε)and two strictly convex domainsD^ε₁,D^ε₂at least of classC¹such thatω0,εgenerates a counter-rotating vortex pair for (1).

Remark 1. The domains D_j^ε, j = 1,2 are small perturbations of the unit disc.

Moreover, as a by-product of the proofs the corresponding conformal parametriza- tionφ^ε_j:T→∂D^ε_j belongs toC^1+β for anyβ ∈(0,1), and has the Fourier asymp- totic expansion

φ_j(ε, w) =w+δ_jεb_j d

² w+δ_j

2 εb_j

d ³

w²+δ_j 3

εb_j d

⁴

w³+ 6 1 +δ_j w +δj

4 εbj

d 5

w⁴+ 3 1 +δj w²

+o(ε⁵), j= 1,2, where δj = ^γ3−j_γ

j in the co-rotating case and δj = −1 in the translating one. In addition, the angular velocity has the expansion

Ω(ε) =γ1+γ2

2d² + ε⁴ 2d⁶

γ₁b⁴₂+γ₂b⁴₁

+o(ε⁴),

and the center of rotation has the expansion Z(ε) = γ2d

γ₁+γ₂ + ε⁴ d³(γ₁+γ₂)²

γ₂³ γ₁b⁴₁−γ₁³

γ₂b⁴₂

+o(ε⁴).

In the case of translation, the speed has the expansion U(ε) = γ₁

2d

1 + ε⁴

d⁴(2b⁴₁+b⁴₂)

+o(ε⁴) and the vorticityγ2 has the expansion

γ2(ε) =γ1

1 + ε⁴

d⁴(b⁴₁−b⁴₂)

+o(ε⁴).

Remark 2. As we shall see later, the caseε= 0 corresponds to the vortex point system. This allows to recover the classical result stating that two point vortices at distance 2d and magnitudes 2πγ1 and 2πγ2 rotate uniformly about their centroid with the angular velocity Ω0 = ^γ1_2d^+γ2² provided that γ1+γ2 6= 0 . However, they translate uniformly with the speedU = ^γ_2d¹ whenγ1andγ2 are opposite.

Remark 3. The interval (0, ε0] is uniform inb1andb2and therefore we may recover the point vortex-vortex patch configuration by lettingb1 andb2 go to 0.

The proof of this theorem is done in the spirit of the work [23] using contour dynamics reformulation. It is a known fact that the initial vorticity ω0,ε with the velocityv0,ε generates a rotating solution, with constant angular velocity Ω around the pointZ on the real axis, if and only if

Re

−iΩ z−Z

−v(z)

~n

= 0 ∀z∈∂D˜^ε₁∪∂D˜₂^ε, (2) where~nis the exterior unit normal vector to the boundary at the pointz. According to the Biot-Savart law combined with Green-Stokes formula , we can write

v(z) = iγ1

2ε²b²_{1 ∂}D˜^ε₁

ξ−z

ξ−zdξ+ iγ2

2ε²b²_{2 ∂}D˜^ε₂

ξ−z

ξ−zdξ, ∀z∈C.

Following the same line of [23] we can remove the singularity inε by slightly per- turbing the unit disc with a small amplitude of order ε and taking advantage of its symmetry. Indeed, we seek for conformal parametrization of the boundaries φj:D^c →[D_j^ε]^c, in the form

φ_j(w) =w+εb_jf_j(w), with f_j(w) =X

n≥1

a^j_n

wⁿ, a^j_n∈R, j= 1,2.

Straightforward computations and substitutions we find that the conformal map- pings are subject to two coupled nonlinear equations defined as follows: for all w∈Tandj ∈ {1,2},

Fj ε,Ω, Z, f1, f2

:= Imn

2Ω εbjφj(w) + (−1)^jZ−(j−1)d

+J_j^ε(w)

wφ⁰_j(w)o

= 0, (3)

whereJεranges in the H¨older spaceC^αand can be extended forε∈(−ε0, ε0) with ε0 >0. In addition, the functional F = (F1, F2) : (−¹₂,¹₂)×R×R×X → Y is well-defined and it is of classC¹where

X:=

f ∈ C^1+α(T)²

, f(w) =X

n≥1

Anwⁿ, An ∈R², w∈T

and Y =

g∈ C^α(T)2

, g=X

n≥1

C_ne_n, C_n∈R², w∈T

, e_n(w) = Im wⁿ . In order to apply the Implicit Function Theorem we compute the linearized operator around the point vortex pairs leading to

D((f₁,f₂)F(0,Ω, Z,0,0)(h1, h2)(w) =−



 γ1Imn

h⁰₁(w)o γ2Imn

h⁰₂(w)o



, This operator is not invertible fromX toY but it does from X to ˜Y, where

Y˜ :=

g∈Y : C1= 0

0 .

The idea to remedy to this defect is to consider the following operator: for any h= (α₁, α₂, h₁, h₂)∈R×R×X

D_{(Ω,Z,f1,f2)}F(0,Ω0, Z0,0,0)h(w) =− 2α1d γ1+γ2

γ2

γ1

Im

w

−α₂(γ₁+γ₂) d²

1

−1

Im w −



 γ1Imn

h⁰₁(w)o γ2Imn

h⁰₂(w)o



. which is invertible fromR×R×X toY.

This allows to to apply the Implicit Function Theorem and complete the proof of the main theorem.

Notation. We need to fix some notation that will be frequently used along this paper. We shall use the symbol :=, to define an object. We denote by D the unit disc and its boundary, the unit circle, is denoted by T. Let f : T→ Cbe a continuous function. We define its mean value by,

T

f(τ)dτ := 1 2iπ

ˆ

T

f(τ)dτ, wheredτ stands for the complex integration.

2. Boundary equation. In what follows, we shall describe the motion of a co- rotating and a counter-rotating asymmetric pair of patches in the plane. The equa- tions governing the dynamics of the boundaries can be thought of as a system of two steady nonlocal equations of nonlinear type coupling the conformal parametrization of the two domains.

2.1. Co-rotating vortex pairs. Letε∈(0,1) and consider two bounded simply connected domains D₁^ε and D^ε₂, containing the origin and contained in the ball B(0,2). Forb1, b2∈R+ andd >2(b1+b2) we define the domains

D˜^ε₁:=εb₁D₁^ε and D˜₂^ε:=−εb₂D₂^ε+d. (4) Givenγ1, γ2∈R, we consider the initial vorticity

ω0,ε= γ1

ε²b²₁χD˜^ε₁+ γ2

ε²b²₂χD˜^ε₂. (5) As we can readily observe, this initial data is composed of unequal-sized pair of simply connected patches with vorticity magnitudes γ1 and γ2. Assume that the initial vorticity ω_0,ε gives rise to a rotating pair of patches about some point Z in

the complex plan. Since the boundary of ˜D₁^ε∪D˜^ε₂is advected by the flow, then it is folklore (see,for instance, [20]) that this condition can be expressed by the equation

Re

−iΩ z−Z

~ n

= Re v(z)~n

∀z∈∂D˜^ε₁∪∂D˜^ε₂, (6) where ~n is the exterior unit normal vector to the boundary at the pointz. From the Biot-Savart law, the velocity can be recovered from the vorticity by

v(z) = iγ₁ 2πε²b²₁

ˆ

D˜^ε₁

dA(ζ)

ζ−z + iγ₂ 2πε²b²₂

ˆ

D˜^ε₂

dA(ζ)

ζ−z, ∀z∈C. Therefore, by using Green-Stokes formula we get

v(z) = iγ1

2ε²b²_{1 ∂}D˜^ε₁

ξ−z

ξ−zdξ+ iγ2

2ε²b²_{2 ∂}D˜^ε₂

ξ−z

ξ−zdξ, ∀z∈C. Changingξby−ξ+din the last integral gives

v(z) = iγ1

2ε²b1 ∂D˜₁^ε

ξ−z

ξ−zdξ− iγ2

2ε²b²_{2 ∂}D˜^ε₃

ξ+z−d

ξ+z−ddξ (7) where we have used the notation

D˜^ε₃:=−D˜^ε₂+d=εb₂D^ε₂.

Hence by combining the last identity with (7) and (6) we obtain Re

2Ω z−Z

+ γ1

ε²b²_{1 ∂}D˜^ε₁

ξ−z ξ−zdξ

− γ₂ ε²b²_{2 ∂}D˜^ε₃

ξ+z−d ξ+z−ddξ

~τ

= 0 ∀z∈∂D˜^ε₁, Re

2Ω z+Z−d

− γ₁ ε²b²_{1 ∂}D˜^ε₁

ξ+z−d ξ+z−ddξ + γ2

ε²b²_{2 ∂}D˜^ε₃

ξ−z ξ−zdξ

~τ

= 0 ∀z∈∂D˜^ε₃

where ~τ denotes a tangent vector to the boundary of ˜D^ε₁∪D˜₃^ε at the point z.

Observe that whenz∈∂D˜^ε₁then−z+d6∈D˜₃^εand conversely; whenz∈∂D˜₃^ε, then

−z+d6∈D˜^ε₁. Hence according to residue theorem, we find that forj∈ {1,3},

∂D˜^ε_j

ξ+z−d ξ+z−ddξ=

∂D˜^ε_j

ξ

ξ+z−ddξ ∀z∈∂D˜_3−j^ε ·

Combining this identity with the change of variablesz→εb₁zandz→εb₂z, which send ˜D^ε₁to D^ε₁ and ˜D₃^εtoD^ε₂respectively, we get

Re

2Ω εz−Z + γ₁

εb1 ∂D^ε₁

ξ−z ξ−zdξ

−γ2

∂D₂^ε

ξ

εb2ξ+εb1z−ddξ

~τ

= 0 ∀z∈∂D^ε₁,

Re

2Ω εbz+Z−d

−γ₁

∂D^ε₁

ξ

εb1ξ+εb2z−ddξ + γ2

εb2 ∂D^ε₂

ξ−z ξ−zdξ

~ τ

= 0 ∀z∈∂D^ε₂. The structure of the last system may be unified as follows

Re

2Ω

εb_jz+ (−1)^jz₀−(j−1)d + γ_j

εbj ∂D^ε_j

ξ−z ξ−zdξ

−γ_3−j

∂D^ε_3−j

ξ

εb3−jξ+εbjz−ddξ

~ τ

= 0, ∀z∈∂D_j, j= 1,2.

(8)

We shall look for domains{Dj, j = 1,2} which are slight perturbation of the unit disc of orderε. In other words, we shall impose the conformal mappingφj :D^c→ [D^ε_j]^c to satisfy the expansions

φ_j(w) =w+εb_jf_j(w), with f_j(w) =X

n≥1

a^j_n

wⁿ, a^j_n∈R, j= 1,2.

Because a tangent vector to the boundary atz=φj(w) is determined by

~

τ φj(w)

=iwφ⁰_j(w),

then by a change of variables the steady vortex pairs system (8) becomes: for all w∈T,j = 1,2,

Im

2Ω

εbjφj(w)+(−1)^jZ−(j−1)d

+γjI_j^ε(w)−γiK_3−jj^ε (w)

wφ⁰_j(w)

= 0, (9) where

I_j^ε(w) := 1 εbj T

φj(τ)−φj(w)

φj(τ)−φj(w)φ⁰_j(τ)dτ and

K_ij^ε(w)

T

φi(τ)φ⁰_i(τ)

εb_iφ_i(τ) +εb_jφ_j(w)−ddτ.

Now, we shall see how to remove the singularity in ε from the full non linearity I_j^ε(w). To alleviate the discussion, we use the notation

A=τ−w and B_j =f_j(τ)−f_j(w).

Thus

I_j^ε(w) = 1 εb_{j T}

A+εbjBj

A+εb_jB_j h

1 +εb_jf_j⁰(τ)i dτ

=

T

A

Af_j⁰(τ)dτ +εbj T

ABj−ABj

A(A+εb_jB_j)f_j⁰(τ)dτ+

T

ABj−ABj

A² dτ

−εbj T

ABj−ABj

Bj

A²(A+εbjBj) dτ + 1 εbj T

A Adτ· From the identity

T

A Adτ =

T

w−τ

w−τdτ =−w

and by the residue theorem

T

A

Af_j⁰(τ)dτ = 0,

T

ABj−ABj

A² dτ = 0.

Therefore Imn

I_j^ε(w)wφ⁰_j(w)o

=−Imn f_j⁰(w)o +εbjIm

w

1 +εbjf_j⁰(w)

T

ABj−ABj

A(A+εb_jB_j) h

f_j⁰(τ)−Bj

A i

dτ

(10)

Inserting the last identity in the system (9) we get Fj(ε, g)(w) := Imn

F1j(ε, g)(w) +F2j(ε, g)(w) +F3j(ε, g)(w)o

= 0, for allw∈Twhere

g:= (Ω, Z, f1, f2) and

F1j(ε, g)(w) := 2Ω

εbj w+εbjfj(w)

+ (−1)^jZ−(j−1)d

w

1 +εbjf_j⁰(w)

−γjf_j⁰(w) F_2j(ε, g)(w) :=εb_jγ_jw

1 +εb_jf_j⁰(w)

T

ABj−ABj

A(A+εb_jB_j) h

f_j⁰(τ)−Bj

A i

dτ F3j(ε, g)(w) :=−γ_3−jw

1 +εbjf_j⁰(w)

×

T

τ+εb_3−jf_3−j(τ)

1 +εb_3−jf_3−j⁰ (τ) ε b3−jτ+bjw

+ε² b²_3−jf3−j(τ) +b²_jfj(w)

−ddτ.

Next we shall use the following notation F(ε, g)(w) :=

F₁(ε, g)(w), F₂(ε, g)(w)

∀w∈T· (11) 2.2. Counter-rotating vortex pair. We shall consider two bounded simply con- nected domainsD^ε₁andD^ε₂, containing the origin and contained in the ballB(0,2).

Forb1, b2∈(0,∞) andd > b1+b2 we set

D˜^ε₁:=εb₁D₁^ε and D˜₂^ε:=−εb2D₂^ε+d. (12) Givenγ1, γ2∈R, we consider the initial vorticity

ω_0,ε= γ1

ε²b²₁χD˜^ε₁− γ2

ε²b²₂χD˜^ε₂. (13) We assume that this unequal-sized pair of simply connected patches with vorticity magnitudesγ1and−γ2travels steadily in (Oy) direction with uniform velocityU. Then in the moving frame the pair of the patches is stationary and consequently,

Ren

v(z) +iU

~no

= 0 ∀z∈∂D˜₁^ε∪∂D˜^ε₂, (14)

where~nis the exterior unit normal vector to the boundary of ˜D^ε₁∪D˜₂^εat the point z. From (7) one has

Re

2U+ γ1

ε²b²_{1 ∂}D˜^ε₁

ξ−z

ξ−zdξ+ γ2

ε²b²_{2 ∂}D˜₃^ε

ξ+z−d ξ+z−ddξ

~ τ

= 0 ∀z∈∂D˜₁^ε, Re

2U+ γ₁ ε²b²_{1 ∂}D˜^ε₁

ξ+z−d

ξ+z−ddξ+ γ₂ ε²b²_{2 ∂}D˜₃^ε

ξ−z ξ−zdξ

~ τ

= 0 ∀z∈∂D˜₃^ε where we have used the notation

D˜^ε₃:=εb2D₂^ε.

and ~τ denote for tangent vector to the boundary of ˜D^ε₁∪D˜^ε₃ at the point z. We can easily verify that when z ∈ ∂D˜^ε₁, −z+d 6∈ D˜₃^ε. Therefore, if z ∈ ∂D˜^ε₃ then

−z+d6∈D˜₁^ε. Thus, by the residue theorem we conclude that for all i, j ∈ {1,3}

andi6=j, we have

∂D˜^ε_j

ξ+z−d ξ+z−ddξ=

∂D˜^ε_j

ξ

ξ+z−ddξ ∀z∈∂D˜_3−j^ε ·

Inserting the last identity into the system above and changingz →εb₁z andz → εb₂z we find

Re

2U+ γ_j εbj ∂D^ε_j

ξ−z

ξ−zdξ+γ_3−j

∂D^ε_3−j

ξ

εb3−jξ+εbjz−ddξ

~ τ

= 0, (15) for all z∈∂Dj with j = 1,2. We shall now use the conformal parametrization of the boundariesφ_j:D^c →[D_j^ε]^c,

φj(w) =w+εbjfj(w), with fj(w) =X

n≥1

a^j_n

wⁿ, a^j_n∈R, j= 1,2.

Since the tangent vector to the boundary atz=φj(w) is given by

~

τ φ_j(w)

=iwφ⁰_j(w), then by a change of variables the system (15) becomes

Im

2U+γjI_j^ε(w) +γ_3−jK_j−3j^ε (w)

wφ⁰_j(w)

= 0, j= 1,2 (16) for allw∈T, where

I_j^ε(w) := 1 εbj T

φ_j(τ)−φ_j(w)

φj(τ)−φj(w)φ⁰_j(τ)dτ, K_ij^ε(w) :=

T

φ_i(τ)φ⁰_i(τ)

εbiφi(τ) +εbjφj(w)−ddτ.

As in the rotating case, from (10), one has Imn

I_j^ε(w)wφ⁰_j(w)o

=−Imn f_j⁰(w)o +εbjIm

w

1 +εbjf_j⁰(w)

T

AB_j−AB_j A(A+εbjBj)

h

f_j⁰(τ)−B_j A i

dτ

, where we have used the notation

A=τ−w and B_j =f_j(τ)−f_j(w).

Inserting the last identity into (16) we get F_j(ε, g)(w) := Imn

F_1j(ε, g)(w) +F_2j(ε, g)(w) +F_3j(ε, g)(w)o

= 0, ∀w∈T (17) with

g:= (U, γ₂, f₁, f₂) and

F1j(ε, g)(w) := 2U w

1 +εbjf_j⁰(w)

−γjf_j⁰(w) F2j(ε, g)(w) :=εbjγjw

1 +εbjf_j⁰(w)

T

ABj−ABj

A(A+εbjBj) h

f_j⁰(τ)−Bj

A i

dτ F_3j(ε, g)(w) :=γ_3−jw

1 +εb_jf_j⁰(w)

×

T

τ+εb_3−jf_3−j(τ)

1 +εf_3−j⁰ (τ) ε b_3−jτ+b_jw

+ε² b²_3−jf_3−j(τ) +b²_jf_j(w)

−ddτ )

In what follows we shall use the notation

F(ε, g)(w) := F1(ε, g)(w), F2(ε, g)(w)

(18) 3. Regularity of the nonlinear functional. This section is devoted to the reg- ularity study of the nonlinear functionalF introduced in (11) and (17) and which defines the V-states equations. We proceed first with the Banach spacesX andY of H¨older type to which the Implicit Function Theorem will be applied. Recall that for givenα∈(0,1), we denote byC^αthe space of continuous functionsf :T→C such that

kfk_Cα(T):=kfk_L^∞_(T)+ sup_τ6=w∈T|f(τ)−f(w)|

|τ−w|^α <∞·

For any integern∈N, the spaceC^n+α(T) stands for the set of functionsf of class Cⁿ whosen−th order derivative is H¨older continuous with exponentα. This space is equipped with the usual norm

kfk_Cα+n(T):=kfk_L^∞_(T)+

dⁿf dwⁿ _Cα(

T)· Now, consider the spaces

X:=

f ∈ C^1+α(T)2

, f(w) =X

n≥1

Anwⁿ, An∈R², w∈T

,

Y =

g∈ C^α(T)2

, g(w) =X

n≥1

Cnen, Cn ∈R², w∈T

, en= Im wⁿ ,

Y˜ :=

g∈Y : C1= 0

0 . (19)

Forr >0 we denote byBrthe open ball ofX centered at zero and of radiusr, Br:=

f ∈X, ||f||_Cα+n(T)≤r

·

It is straightforward that for any f ∈B_r the function w 7→φ(w) =w+εf(w) is conformal onC\Dprovided thatr, ε <1. Moreover according to Kellog-Warshawski result, the boundary ofφ(C\D) is a Jordan curve of classC^n+α.

3.1. Co-rotating vortex pairs. We propose to prove the following result con- cerning the regularity ofF.

Proposition 1. The following assertions hold true.

1. The functionFcan be extended toC¹function from −¹₂,¹₂

×R×R×B1→Y. 2. Two initial point vortex γ1πδ0 and γ2πδd with γ1 6= −γ2 rotate uniformly

about the point

Z0:= dγ2

γ1+γ2

with the angular velocity

Ω₀:= γ₁+γ₂ 2d² · 3. For allh= (α1, α2, h1, h2)∈R×R×X one has

DgF(0, g0)h(w) =− 2α1d γ1+γ2

γ2

γ1

Im

w −α2(γ1+γ2) d²

1

−1

Im w

−



 γ₁Imn

h⁰₁(w)o γ2Imn

h⁰₂(w)o





with

g:= (Ω, Z, f1, f2) and g0:= (Ω0, Z0,0,0).

4. The linear operator DgF(0, g0) :R×R×X→Y is an isomorphism.

Proof. (1)Notice that the partF1j+F2j defined in (11) appears identically in the boundary equation of co-rotating symmetric pairs and its regularity was discussed in the paper [23]. The only term that one should care about is F3j describing the interaction between the boundaries of the two patches which are supposed to be disjoint. Therefore the involved kernel is sufficiently smooth and it does not carry significant difficulties in the treatment. and can be dealt with in a very classical way.

(2) According to the formulation developed in Section 2 the two points vortex system is a solution to the equationF(0,Ω, Z,0,0) = 0. In this case we can easily check that

F_j(0,Ω, Z,0,0)(w) =

2Ω

(−1)^jZ+ (1−j)d +γ_3−j

d

Im w . ThereforeFj(0,Ω, Z,0,0) = 0 if and only if

−2ΩZ+γ2

d = 0 and 2Ω Z−d

+γ1

d = 0.

Thus,

Ω = Ω0:= γ1+γ2

2d² and Z=Z0:= dγ2

γ₁+γ₂· (20) (3)-(4)SinceF= (F1, F2) then for givenh= (h1, h2)∈X, we have

D_fF(0,Ω, Z,0,0)h(w) =

∂f₁F1(0,Ω, Z,0,0)h1(w) +∂2F1(0,Ω, Z,0,0)h2(w)

∂f₁F2(0,Ω, Z,0,0)h1(w) +∂f₂F2(0,Ω, Z,0,0)h2(w)

.

The Gˆateaux derivative of the function Fj at (0,Ω, b,0,0) in the direction hj is given by

∂f_jFj(0,Ω, Z,0,0)hj(w) = d dt h

Fj(0,Ω, Z, thj,0)i

t=0(w)

=−γ_jImn h⁰_j(w)o

. On the other hand, straightforward computations allow to get

∂f3−jFj(0,Ω, Z,0,0)h_3−j(w) = d dt h

Fj(0,Ω, Z,0, th_3−j)i

t=0(w)

= 0.

From (11) one can easily check that F_j(0,Ω, Z, f₁, f₂) = Im

2Ω

(−1)^jZ+ (1−j)d

w−γ_jf_j⁰(w) +γ_3−j d w

. (21) Differentiating the last identity with respect to Ω at (Ω₀, Z₀) gives

∂ΩFj(0,Ω0, Z0, f1, f2) = 2

(−1)^jZ0+ (1−j)d Im

w . (22)

Next, differentiating (21) with respect toZ at (Ω₀, Z₀) we get

∂ZFj(0,Ω0, Z0, f1, f2) = 2(−1)^jΩ0Im w . Therefore, for all (α₁, α₂)∈R²

D_(Ω,Z)F(0,Ω0, Z0, f1, f2)(α1, α2) =α1





 2γ2d γ₁+γ₂

2γ1d γ1+γ2





Im w

+α2





γ1+γ2

d²

−γ1+γ2

d²



Im w .

Consequently the restricted linear operatorD_{(Ω,Z,f1,f2)}F(0,Ω0, Z0, f1, f2) is invert- ible if if and only if the determinant of the two vectors, given by −2

d(γ₁+γ₂), is non vanishing.

3.2. Counter-rotating vortex pairs. This section is devoted to the counter- rotating asymmetric pairs and our main result reads as follows.

Proposition 2. The following assertions hold true.

1. The functionFcan be extended toC¹function from −¹₂,¹₂

×R×R×B₁→Y. 2. Two initial point vortex γ₁πδ₀ and −γ₂πδ_(d,0) translate uniformly with the

speed

U0:= γ1

2d·

3. For allh= (α1, α2, h1, h2)∈R×R×X one has DgF(0, g0)h(w) =α1

2 2

Im

w −α2

1 0

Im

w −2U d



 Imn

h⁰₁(w)o Imn

h⁰₂(w)o





with

g:= (U, γ₂, f₁, f₂) and g₀:= (U₀, γ₁,0,0).

4. The linear operator DgF(U0, γ1,0,0) :R×R×X →Y is an isomorphism.

Proof. (1)Notice that the functionFj2+Fj3coincide with the functionFj2+Fj3

appearing in the rotating case andFj1enjoys abviously the required regularity.

(2)The point vortex configuration corresponds toF(0, U, γ2,0,0). Thus, forε= 0 one has

Fj(0, γ1, γ2,0,0)(w) =

2U−γ_3−j d

Im

w . ThereforeF(0, U, γ2,0,0) = 0 if and only if

γ1=γ2= 2U d.

(3)-(4)Takingε= 0 in (17) we get Fj(0, U, γ2, f1, f2) = Im

2U w−γjf_j⁰(w)−γ3−j

d w

. Thus, for given (h1, h2)∈X, we have

∂f_jFj(0, U, γ2,0,0)hj =−γjImn h⁰_j(w)o and

∂f3−jFj(0, U, γ2,0,0)h3−j(w) = 0.

On the other hand, differential ofF with respect (U, γ2) is given by D_(U,γ2)F(0, U, γ₂, f₁, f₂)(α₁, α₂) =α₁

2 2

Im

w +α₂ −¹_d

0

Im w . Thus, we conclude that the linear operator D_gF(0, g₀) is invertible for all γ₁ ∈ R.

4. Existence of asymmetric co-rotating patches. The goal of this section is to provide a full statement of the point (1) of Theorem 1.1. In other words, we shall describe the set of solutions of the equation F ε, g

= 0 around the point (ε, g

= (0, g₀) by a one-parameter smooth curve ε7→g(ε) using the implicit func- tion theorem.

4.1. Co-rotating vortex pairs. The main result of this section reads as follows.

Proposition 3. The following assertions hold true.

1. There existsε₀>0and a uniqueC¹functiong:= (Ω, Z, f₁, f₂) : [−ε₀, ε₀]−→

R×R×B₁ such that F

ε,Ω(ε), Z(ε), f1(ε), f2(ε)

= 0 with

Ω(0), Z(0), f₁(0), f₂(0)

= Ω₀, Z₀,0,0 . 2. For allε∈[−ε0, ε0]\{0} one has

f1(ε), f2(ε)

6= (0,0).

3. If (ε, f₁, f₂)is a solution then (−ε,f˜₁,f˜₂) is also a solution, where

∀w∈T, f˜j(w) =fj(−w), j= 1,2.

4. For allε∈[−ε0, ε]\{0}, the domainsD₁^ε andD^ε₂ are strictly convex.

Proof. (1)From Proposition1one hasF : (−¹₂,¹₂)R×R×B1→Y is aC¹function and D_gF 0, g₀

:X → Y is an isomorphism. Thus the implicit function theorem ensures the existence ofε0>0 and a uniqueC¹parametrizationg= (Ω, Z, f1, f2) : [−ε0, ε0]→B1 verifying the equationF ε,Ω(ε), Z(ε), f1(ε), f2(ε)

= 0. Moreover, this solution passes through the origin, that is,

Ω, Z, f₁, f₂

(0) = (0,0).

This completes the proof of the desired result.

(3)We shall prove that for anyεsmall enough and any Ω we can not get a rotating vortex pair withf1= 0 andf2= 0. In other words , we need to prove that

F(ε,Ω, Z,0,0)6= 0.

To this end we write

Fj(ε,Ω, Z,0,0)(w) = Im (

2Ω

(−1)^jZ−(j−1)d

−γ3−j T

τ ε b_3−jτ+bjw

−ddτ

w )

. By Taylor expansion, we get

T

τ ε b_3−jτ+bjw

−ddτ =−X

n∈N

εⁿ

dⁿ⁺¹ _Tτ b_3−jτ+bjwⁿ dτ

=−X

n∈N

εⁿ dⁿ⁺¹bⁿ_jwⁿ

= 1

εb_jw−d·

(23)

Consequently,

F_j(ε,Ω, Z,0,0)(w) = Im (

2Ω

(−1)^jZ−(j−1)d

− γ_3−j εbjw−d

w

)

and this quantity is not zero ifε6= 0 is small enough.

(2)We only need to check that

Fj(ε,Ω, Z, f1, f2)(−w) =−Fj(−ε,Ω, Z,f˜1,f˜2)(w) for j = 1,2.

whereF_j is defined by (11). By definition we have F_1j(−ε,Ω, Z,f˜₁,f˜₂) = 2Ω

−εb_j w−εb_jf˜_j(w)

+ (−1)^jZ

−(j−1)d

w

1−εb_jf˜_j⁰(w)

−γ_jf˜_j⁰(w).

Since ˜f_j⁰(w) =−f_j⁰(−w) we get F1j(−ε,Ω, Z,f˜1,f˜2) = 2Ω

εbj −w+εbjfj(−w)

+ (−1)^jZ

−(j−1)d

w

1 +εbjf_j⁰(−w)

+γjf_j⁰(−w)

=−F_ij(ε,Ω, Z, f₁, f₂)(−w).

Straightforward computations will lead to the same properties for the functionsF2j

andF3j. This completes the proof of(4).

(3)Now to prove the convexity of the domainsD₁^ε,D₂^εwe shall reproduce the same arguments of [23]. Recall that the outside conformal mapping associated to the domainD^ε_j is given by

φj=w+εbjfj(w).

and the curvature can be expressed by the formula κ(θ) = 1

|φ^ε_j(w)|Re

1 +wφ⁰⁰_j(w) φ⁰_j(w)

. We can easily verify that

1 +wφ⁰⁰_j(w)

φ⁰_j(w) = 1 +εbjw f_j⁰⁰(w) 1 +εbjf_j⁰(w) and so

Re

1 +wφ⁰⁰_j(w) φ⁰_j(w)

≥1− |ε|bj

|f_j⁰⁰(w)|

1− |ε|bj|f_j⁰(w)| ≥ |ε|bj

1− |ε|bj

·

The last quantity is non-negative if|ε|<1/2. Thus the curvature is strictly positive and therefore the domainD^ε_j is strictly convex.

4.2. Counter-rotating vortex pairs. In this section we shall give the complete statement for the existence of translating unequal-sized pairs of patches.

Proposition 4. The following holds true.

1. There existsε₀>0and a uniqueC¹functiong:= (U, γ₂, f₁, f₂) : [−ε0, ε₀]−→

R×R×B₁ such that F

ε, U(ε), γ2(ε), f1(ε), f2(ε)

= 0, with

U(0), γ2(0), f1(0), f2(0)

=γ1

2d, γ1,0,0 .

2. If (ε, f1, f2)is a solution then (−ε,f˜1,f˜2) is also a solution, where

∀w∈T, f˜j(w) =fj(−w), j= 1,2.

3. For allε∈[−ε0, ε]\{0}, the domainsD₁^ε andD^ε₂ are strictly convex Proof. Same arguments as in Proposition3.

5. Asymptotic behavior. In this section we study the asymptotic expansion for smallεof the conformal mappings, the angular velocity, the center of rotation and the circulations.

5.1. Co-rotating pairs. In the case of the co-rotating pairs we get the following expansion at higher order inε.

Proposition 5. The conformal mappings of the co-rotating domains, φj : T →

∂D_j^ε, have the expansions φ_j(ε, w) =w+δ_jεb_j

d ²

w+δ_j 2

εb_j d

³ w²+δ_j

3 εb_j

d ⁴

w³+ 6 1 +δ_j w +δj

4 εbj

d 5

w⁴+ 3 1 +δ_j w²

+o(ε⁵),

for allw∈T andj= 1,2whereδj =^γ3−j_γ

j . The angular velocity has the expansion Ω(ε) =γ1+γ2

2d² + ε⁴ 2d⁶

γ1b⁴₂+γ2b⁴₁ , and the center of rotation has the expansion

Z(ε) = γ₂d γ1+γ2

+ ε⁴

d³(γ1+γ2)² γ³₂

γ1

b⁴₁−γ₁³ γ2

b⁴₂ . Proof. Recall that

φ_j(ε, w) =w+εb_jf_j(ε, w), j= 1,2.

and from Proposition3one has F ε, g(ε)

:=F

ε,Ω(ε), Z(ε), f1(ε), f2(ε)

= 0. (24)

whereF is defined in (11). Moreover

g(0) = Ω₀, Z₀,0,0

. (25)

By the composition rule we get

∂_εg(0) =−D_gF(0, g₀)⁻¹∂_εF(0, g₀)· (26) In view of (11) one has

F_j ε, g₀

=−γ_3−jIm 1

d+

T

τ ε b_3−jτ+b_jw

−ddτ w

and from (23) we conclude that Fj ε, g0

=γ_3−j

∞

X

n=1

εⁿbⁿ_j dⁿ⁺¹Im

wⁿ⁺¹ . (27)

Thus

∂_εⁿF(0, g0)(w) = n!

dⁿ⁺¹ γ2bⁿ₁

γ1bⁿ₂

Im

wⁿ⁺¹ · (28)

Now, we shall write down the explicit expression of DgF(0, g0)⁻¹. Recall from Proposition1 that for allh= (α1, α2, h1, h2)∈R×R×X with

hj(w) =X

n≥1

a^j_nwⁿ, j = 1,2, one gets

DgF(0, g0)h(w) =− 2α1d γ₁+γ₂

γ2

γ1

Im

w −α2

γ1+γ2

d² 1

−1

Im

w (29)

−X

n≥1

n γ1a¹_n

γ2a²_n

Im wⁿ⁺¹ . Then, for anyk∈Y with the expansion

k(w) =X

n≥0

An

Bn

Im{wⁿ⁺¹},