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Multipole vortex patch equilibria for active scalar equations

We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu \cite{Hmidi-Mateu} for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point-vortex limit. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem.

1. Introduction 1.1. Historical discussion. In this note we consider the generalized surface quasi-geostrophic (gSQG) equations, which describe the evolution of the potential temperature ω through the transport equation (1.1) Here ∇ ⊥ = (−∂ 2 , ∂ 1 ) while α ∈ [0, 1) is a real parameter. The vector field v is the flow velocity and the fractional Laplacian operator (−∆) −1+ α 2 is of convolution type, defined by , (1.2) where Γ is the gamma function. This model was proposed by Córdoba et al. [12] as an interpolation between Euler equations and the surface quasi-geostrophic model, which correspond to α = 0 and α = 1, respectively. The main purpose of this note is to show the existence of new families of periodic global solutions of (1.1) in the vortex patch setting, namely when ω( · , t) is the characteristic function of finite collection of bounded domains. Such patterns are a special class of Yudovich solutions where ω( · , t) is merely bounded and integrable. Yudovich solutions are known to be unique and to exist globally in time in the case of Euler equations [54], but for α > 0 the situation is more delicate because the velocity field v is in general not Lipschitz. Nonetheless, when the initial datum has a patch structure, one can locally construct a unique solution which remains a patch. The motion of the boundary of the patch is governed by so-called contour dynamics equations; see [20,47]. It is worth mentioning that, while the boundary's regularity is globally preserved for α = 0 [11,3], for α > 0 numerical evidence [12] suggests singularity formation in finite time.
There are very few explicit solutions to the gSQG and Euler equations. The only known explicit simply-connected vortex patch solutions are the Rankine vortex, which is stationary, and the Kirchhoff ellipses [29] for the Euler equation, which are rotating. Nevertheless, a family of uniformly rotating patches with m-fold symmetry, called V-states, was numerically computed by Deem and Zabusky [17]. Later, Burbea [4] gave an analytical proof of their existence, based on a conformal mapping parametrization and local bifurcation theory. Recently, Burbea's branches of solutions were extended to global ones [32]. The regularity and the convexity of the V-states have been investigated in [37,32,8]. Similar research has been carried out for the gSQG equations: The construction of simply connected V-states was established in [30,7], and their boundary regularity was discussed in [8].
We point out that there is a large literature dealing with rotating vortex patches and related problems. For instance, we mention the existence results of rotating patches close to Kirchhoff's ellipses [34,8], multiply-connected patches [38,16,35,14,46,26], patches in bounded domains [15], non-trivial rotating smooth solutions [9] and rotating vortices with non-uniform densities [24]. The radial symmetry properties of stationary and uniformly-rotating solutions was studied in a series of works [19,33,28].
All of the above analytical results treat connected patches. The first numerical works revealing the existence of translating symmetric pairs of simply connected patches for Euler equation are due to Deem and Zabusky [17] and Pierrehumbert [45]. Similar studies were performed by Saffman and Szeto in [50] for the symmetric co-rotating vortex pairs and by Dritschel [18] for asymmetric pairs. Later, Turkington gave in [51] an analytical proof using variational arguments, where he considered an unbounded fluid domain with N symmetrically arranged vortex patches rotating about the origin. Implementing the same approach, Keady [39] proved the existence of translating pairs of symmetric patches and Wan [52] studied the existence and stability of desingularizations of a general system of rotating point vortices. Very recently, Godard-Cadillac, Gravejat and Smets [25] extended Turkington's result to the gSQG equations, while Ao, Dávila, Del Pino, Musso and Wei [1] have obtained related families of smooth solutions via gluing techniques. See [41,42,44,49,53] for additional references on multiply connected patches.
The variational arguments [51,39,25] do not give much information about the shape of the vortex patches, or about the uniqueness of solutions. In [36], however, Hmidi and Mateu gave a direct proof showing the existence of co-rotating and counter-rotating vortex pairs, using an elegant desingularization of the contour dynamics equations and an application of the implicit function theorem. The same technique was implemented for the desingularization of the asymmetric pairs [31], Kármán street [21] and the vortex polygon [22]. See [6] for related results where point vortices are instead desingularized into doubly-connected patches. We mention that, using more sophisticated Nash-Moser techniques, Gómez-Serrano, Park and Shi [27] have very recently constructed stationary configurations of multi-layered patches with finite kinetic energy. Also, García and Haziot [23] have combined ideas from [36] and [32] to prove a global bifurcation result for co-rotating and counter-rotating pairs.
In this note we show how the technique in [36] can be extended to arbitrary configurations of finitely-many point vortices. For a general configuration, the problem reduces via Lyapunov-Schmidt to a finite-dimensional nonlinear equation. Under an natural non-degeneracy assumption on the point vortex configuration alone, one can instead simply apply a modified version of the implicit function theorem. For highly symmetric configurations, we may also simplify the problem by reformulating it in spaces which take these symmetries into account.  The case α = 0 corresponds to the classical point vortex Eulerian interaction. A general review about the N -vortex problem and vortex statics can be found in [2] for the Newtonian interaction and [48] for gSQG interactions. We are concerned with periodic solutions for which the configuration of vortices is instantaneously moving as a rigid body, so that d dt z j (t) = iU + iΩz j (t), (1.5) where here U ∈ R is the constant linear velocity and Ω ∈ R is the constant angular velocity. Such solutions are known as relative equilibria or vortex crystals. Explicitly solving (1.5), we see that whenever Ω = 0 we can shift coordinates so that U = 0. Thus there is no loss of generality in restricting our attention to equilibria which are either rotating with Ω = 0 and V = 0, translating with U = 0 and Ω = 0, or stationary with U = Ω = 0. Figure 1. (a) A solution of the point vortex system (1.6). The vortices are located at the points z j (0) := z j and have circulations πγ j . (b) A desingularization into vortex patches. The vortex at z j has become a small nearly-circular patch D ε j . To leading order in the small parameter ε, the radius of the patch is εb j , and the net circulation is πγ j .
Theorem 1.2. Let α ∈ [0, 1). Then any non-degenerate solution λ of (1.6) can be desingularized into a family of vortex patch equilibria depending smoothly on a small parameter ε > 0 measuring the size of the patches.
See Figure 1 for an illustration. Remark 1.3. As we shall see later in Theorems 2.7 and 2.8, the vortex patches in Theorem 1.2 are small C 1+β perturbations of the unit disk, whose boundaries are given by conformal parametrizations which can be explicitly expanded to any order in the small parameter ε. In particular, the conformal parametrizations of the boundaries φ j : T → ∂O ε j have the explicit Fourier asymptotic expansions Furthermore, with slight modifications in the proof we can show that the boundary of each vortex patches belongs to C n+β for any fixed n ∈ N. The range of ε would be not uniform with respect to n but would shrink to zero as n goes to infinity. However, using the approach developed in [8], we expect that the conformal mappings possess holomorphic extensions outside of a small disc and thus that the boundaries are analytic. In the Euler case, the analyticity of sufficiently smooth patch boundaries could alternatively be proved using the same technique as in [32].
Remark 1.4. The leading-order ratios between the sizes of the patches can be specified a priori. Moreover, the range of ε is uniform as some of these ratios are sent to zero, allowing us to recover solutions involving a combination of point vortices and vortex patches. See Remark 2.9 for more details.
Remark 1.5. While the proof is valid for α ∈ [0, 1), we expect that a similar result can be proved for α ∈ [1, 2) using the spaces introduced in [7]; see [5].  (1.6) are co-rotating and counter-rotating vortex pairs. A family of asymmetric co-rotating pairs is given by where γ ∈ R \ {0}, d > 0 and 0 < |c| < 1; see Figure 2(a). In the time-dependent problem, the two vortices steadily rotate about the origin with angular velocity Ω * . Counter-rotating pairs given by instead steadily translate along the y-axis; see Figure 2(b). We also consider asymmetric stationary tripoles of the form where a ∈ (0, 1); see Figure 2(c). Note that all of these above configurations are invariant under reflections about the x-axis. Furthermore, horizontal translations of (1.10) and rotations of (1.9) are also solutions. As we shall see in Section 3, these configurations are non-degenerate in the sense of Definition 1.1. Using Theorem 1.2, they can therefore be desingularized into steady vortex patch equilibria. As no two vortices in (1.9) or (1.11) can be identified with one another, the same is true of the corresponding vortex patches. While the two vortices in (1.10) can be identified, this symmetry is broken if we require the leading-order ratio between the sizes of the patches to be different from 1. For the pairs, this extends the desingularization result of [31], obtained in the Eulerian case α = 0, to gSQG equations (1.1) with α ∈ (0, 1). To the best of our knowledge, the asymmetric tripole patch solutions are new both for the Euler and gSQG equations. Furthermore, this appears to be the first existence proof for stationary solutions to the gSQG equations involving multiple patches; see [27] for stationary solutions to the Euler equations with multiple multi-layered patches and [26] for stationary doubly connected solutions to the gSQG equations. Theorem 1.6. Let α ∈ [0, 1) and b 1 , b 2 , b 3 ∈ (0, ∞), and let γ, d, c, a be as above. Then, the following results hold true.
(i) For any ε > 0 sufficiently small, there are two strictly convex domains O ε 1 , O ε 2 , 1-fold symmetric, C 1+β perturbations of the unit disc, and real numbers generates a co-rotating vortex pair for (1.1) with angular velocity For any ε > 0 sufficiently small, there are two strictly convex domains O ε 1 , O ε 2 , 1-fold symmetric, C 1+β perturbations of the unit disc, and real numbers generates a counter-rotating vortex pair for perturbations of the unit disc, and two real numbers  While the above examples are asymmetric and have only two or three vortices, there are other well-known point vortex equilibria which are highly symmetric and have many vortices. When seeking similarly symmetric desingularizations of such equilibria, it convenient to integrate these additional symmetries into the statement of the problem. In particular, the relevant non-degeneracy conditions on the point vortex equilibria can be much simpler to verify than Definition 1.1.
As a concrete example, we consider two concentric regular m-gons with a vortex at each vertex, and assume that the vortices of a same polygon have the same vorticity γ 1 ∈ R \ {0} or γ 2 ∈ R. We place, in addition, a point vortex at the center of the regular m-gons with intensity γ 0 ∈ R. More specifically, we are concerned with the system of point vortices The value ϑ = 0 corresponds to the configuration where the vertices of the polygons are radially aligned with each other and ϑ = 1 refers to the case where the vertices are out-of-phase by an angle π/m. If we assume that (1.12) performs a perpetual uniform rotation, then we may easily check that the system of 2m + 1 equations in (1.6) can be reduced to a system of two real equations: Observe that the last system is linear in Ω and γ 2 , and, thus, under a simple non-degeneracy condition we can explicitly solve the system (1.13) for Ω and γ 2 = 0. For the sake of clarity we shall give an elementary statement about the desingularization of these configurations; for a complete statement see Theorem 4.3. Theorem 1.9. Let α ∈ [0, 1), ϑ ∈ {0, 1}, b 1 , b 2 ∈ (0, 1), γ 0 , γ 1 ∈ R \ {0}, d 1 , d 2 ∈ (0, ∞) and let (Ω * , γ * 2 ) ∈ (R \ {0}) 2 be a solution of (1.13) satisfying the non-degeneracy conditions (4.7) and (4.9). Then, for any ε > 0 sufficiently small, there are three strictly convex domains O ε 0 , O ε 1 , O ε 2 , C 1+β perturbations of the unit disc, and a real number γ 2 = γ 2 (ε) such that (1.14) generates a rotating solution for (1.1) with some constant angular velocity Ω(ε). Moreover O ε 0 is m-fold symmetric and O ε 1 , O ε 2 are 1-fold symmetric.
See Figure 3 for an illustration. Remark 1.11. The parameter ε > 0 can be chosen uniformly as any of the parameters b 0 , b 1 and b 2 tend to 0, and therefore we may recover the point vortex-vortex patch configurations discussed in [13,53]; see Remark 2.9.
1.4. Idea of the proof. We shall briefly explain the basic ideas behind Theorem 1.2 for the Euler equations in the rotating case; a similar strategy is followed for the gSQG equations and for traveling or stationary patches. We seek simply connected bounded domains O ε j such that the initial datum performs a uniform rotation around the center of mass of the system, taken to be the origin, with an angular velocity Ω. Here the parameters b j will allow us to specify the relative sizes of the patches; see Remark 1.4. After moving to the rotating frame, the boundaries of the system are subject to the following stationary system, see for instance [16, Page 1896], where n is the exterior unit normal vector to the boundary at the point z. In virtue of the Biot-Savart law and Green's theorem, we may write for all z ∈ C. Following Hmidi and Mateu [36], we reformulate (1.15) in terms of the conformal parametrizations of the boundaries φ j : T → ∂O ε j , which we assume have the form In other words, we shall look for domains O ε j which are small perturbations of the unit disc with an amplitude of order εb j . While the resulting problem initially appears to have terms which are singular in ε, as in [36] there is a cancellation -essentially due to the symmetry of the diskwhich eliminates these terms. This leads to a nonlinear system for the perturbations f := (f 1 , . . . , f N ) of the patch boundaries, where the nonlinear operator G = (G 1 , . . . , G N ) : (−ε 0 , ε 0 ) × B 0 × Λ → W 0 is well-defined and of class C 1 . Here Λ in a small neighborhood of λ * , which is the solution to the point vortex system (1.6) with α = 0, B 0 is the unit ball in V 0 and a n w n , a n ∈ C , for some fixed Hölder exponent β ∈ (0, 1). Moreover, for (ε, f ; λ) = (0, 0; λ) we find Intending to apply the implicit function theorem, we next linearize G about the point vortex solution (0, 0; λ * ). The linearized operator with respect to the patch boundaries f is which has a trivial kernel and a range with finite codimension 2N . To deal with this deficiency, we also linearize with respect to the point vortex parameters λ using (1. 16). Assuming that λ * is nondegenerate in the sense of Definition 1.1, we deduce that the linearized operator D (f ;λ 1 ) G(0, 0; λ * ) has trivial kernel and a range with codimension 1. As this final deficiency is caused by a nonlinear identity satisfied by the functional G, we can conclude by applying a modified version of the implicit function theorem (Lemma 2.6).
1.5. Notation. Let us end this part by summarizing some notation to be used in the paper. We will denote the unit disc by D and its boundary by T. For continuous functions f : where dτ stands for complex integration. For any x ∈ R and n ∈ N, we use the notation (x) n to denote the Pochhammer symbol defined by Finally, we use the notation δ ij to denote the Kronecker delta defined by 1.6. Outline of the paper. In Section 2 we consider completely general vortex equilibria, without any symmetry assumptions, show how the problem is desingularized and prove Theorem 1.2. Section 3 is devoted to the proof of Theorem 1.6. Finally, in Section 4, we prove Theorem 1.9 by imposing suitable rotation and reflection symmetries.

Desingularization of general vortex equilibria
In this section we consider a general configuration of finitely many point vortices in uniform rotation or translation, and show how these vortices can be desingularized into small vortex patches. Using the approach developed in [36], we first write down the contour dynamics equations governing the N steady vortex patches, and then find the suitable function spaces where the problem is wellposed. Finally, we prove Theorem 1.2 using a simple extension of the implicit function theorem.

2.1.
Integral identities for the stream function. We shall give in this subsection some identities related to the (non-relative) stream function, As we shall see in the following subsections, these identities will be useful to explain the degeneracy of the functional defining the V-states. While they can be derived using the symmetries and variational structure of the problem, we shall give here a simple proof using the structure of the kernel K α .
Lemma 2.1. For all ε ∈ (0, ε 0 ), the stream function (2.4) satisfies the following identities: Proof. Applying the complex version of Green's theorem, The integrand K α (z − ξ) changes sign when the roles of z and ξ are reversed, and so (i) follows. Next we apply formula (2.5) tozψ ε , which yields Since ψ ε (z) is real, the last identity implies Differentiating (2.4) with respect to z we get Inserting this into (2.6) and interchanging the roles of ξ and z we obtain The desired identity (ii) now follows from the fact that the integrand is purely real, and the proof is complete.

Boundary equations.
First suppose that ω ε 0 gives rise to N rotating patches for the model (1.1) about the centroid of the system, assumed to be the origin, with an angular velocity Ω. More precisely, we are looking for a solution ω ε (t) of (1.1) of the form Using the patch structure, we conclude that [16, Page 1896] Re γ j v ε (z) + iΩ z n = 0 for all z ∈ ∂D ε j , j = 1, . . . , N, (2.7) where n is the exterior unit normal vector to the boundary at the point z.
If instead ω ε 0 translates vertically with uniform velocity U , that is For the sake of abbreviation and simplicity we shall unify (2.7) and (2.8) as follows and assume that either Ω or U vanishes.

Euler equation. In view of the Biot-Savart law one has
for all z ∈ C. By the complex form of Green's theorem (2.5), we may replace the integral over D k in (2.10) with an integral along ∂D k : Inserting the last identity into (2.9) leads to where z denotes a tangent vector to the boundary at the point z and In view of (2.1), a suitable change of variables gives Observe, from (2.2), that for any j = k and z ∈ ∂O ε j one has Thus, by the residue theorem, for every z ∈ ∂O ε j , we may write Replacing z by εb j z + z j in (2.11) and using the last identity we get (2.12) We shall look for domains O ε j , which are perturbations of the unit disc with an amplitude of order εb j . More precisely, we shall consider φ j : C\D → C\O ε j the unique conformal map with the expansion By the Kellogg-Warschawski theorem [45,Theorem 3.6], since the boundary ∂O ε j is assumed to be a smooth Jordan Curve, φ j extends to a smooth mapping C\D → C\O ε j , and its trace, that we shall also denote by φ j , is a smooth parametrization of ∂O ε j . Thus, making the change of variable for any w ∈ T and j = 1, . . . , N .
To desingularize this system in ε, we follow the ideas of [36] and write, by virtue of (2.13), From the obvious identity the expression of φ j in (2.13), and the symmetry of the disc we can get rid of the singular term from the full nonlinearity, Inserting the last equation into (2.14), we conclude that Here we have used the notation I 0 and J 0 kj with a complex conjugate in order to unify the notation with the functions I α , J α kj and G α j that we shall introduce in next subsection for the gSQG equations. Furthermore, both sides of (2.15) are multiplied by γ j to ensure that the system is valid even if we set some of the γ j equal to zero.

gSQG equations. The velocity can be recovered from the boundary as follows
for all z ∈ C, see for instance [30]. In view of (2.1), suitable change of variables gives Inserting the last identity into (2.9) and taking a complex conjugate inside the real part leads to where z denotes a tangent vector to the boundary at the point z. Replacing z by εb j z + z j in the last system gives (2.20) We shall look for conformal parametrizations φ j : T → ∂O ε j having the expansions Here, the coefficient |ε| α in the definition of the conformal mapping φ j comes from the singularity of the gSQG kernel. For every w ∈ T, the tangent vector is given by z = iwφ j (w) and therefore (2.20) becomes In order to desingularize the system (2.22) we shall use the following Taylor formula, which is true for any complex numbers A, B such that |B| < |A|.

(2.24)
On the other hand, from (2.21), one has Using the identity [30, Page 337], and applying the formula (2.23) with A = τ − w and B = ε|ε| α b 1+α As in the Euler case, by (2.21) and the symmetry of the disc the singular term disappears from the nonlinearity, for all w ∈ T and j = 1, . . . , N , where f = (f 1 , . . . , f N ), λ denotes the point vortex parameters (2.16) and (2.29) 2.3. Regularity and linearization of the functional G α . For any β ∈ (0, 1), we denote by C β (T) the space of continuous functions f : T → C such that For any integer n, the space C n+β (T) stands for the set of functions f of class C n whose n-th order derivatives are Hölder continuous with exponent β. It is equipped with the usual norm, We henceforth fix some β ∈ (0, 1), and for α ∈ [0, 1) define Consider the Banach spaces a n w n , a n ∈ C , Here the subscript 0 refers to the fact that no a priori symmetry assumption is made on the patch boundaries. The restriction on the Fourier coefficients of functions in V α 0 guarantees that these functions can be extended to holomorphic functions on C \ D. The restriction on the Fourier coefficients of functions in W α 0 , on the other hand, simply says that these functions are real-valued. We denote by B α 0 the unit ball in V α 0 , B α 0 := f ∈ V α : f C ηα+1 (T) < 1 and by B α 0 the unit ball in W α 0 , B α 0 := f ∈ W α : f C ηα (T) < 1 . We shall unify the expression of the function G α j in (2.27) and the function G 0 j in (2.15) as follows: For every f ∈ V α 0 and w ∈ T, (i) There exists ε 0 > 0 and a small neighborhood Λ of λ * such that G α can be extended to a C 1 mapping (−ε 0 , ε 0 ) × B α 0 × Λ → W α 0 .
(2.32) (iv) For any λ ∈ Λ, the linear operator D f G α (0, 0; λ) : V α 0 → W α 0 is an isomorphism. Proof. Note that the term appears identically in the study of the vortex pairs in [36], where it was shown that it is C 1 in ε and f . The spaces used in [36] are V α 0 with the reflection symmetry f j (w) = f j (w) and W α 0 with the symmetry g j (w) = −g j (w), but the study of the regularity can be easily generalized to our case without further difficulties. The terms N k=1,k =j describe the interaction between the boundaries of the patches, which are disjoint provided ε is sufficiently small. Therefore the kernel involved is sufficiently smooth and can be treated in a very classical way.
Next, we shall prove (ii) and (iii) in the case α = 0. Substituting ε = 0 in (2.17) and (2.18) gives where we have used the residue theorem in the last identity. Then, from (2.15), we get and hence = − Im P 0 j (λ)w , which shows (ii). Differentiating (2.33) with respect to f gives implying (iii) in the case α = 0.
To prove (ii) and (iii) in the case α ∈ (0, 1), we substitute ε = 0 in (2.28) and (2.29) and obtain Thus, by (2.27), one has Comparing the last expression with (1.6) concludes (ii). Next, differentiating (2.34) with respect to f gives The last expression was explicitly computed in [36, Pages 726-728] and takes the form Im a j n w n+1 , getting the announced result.
The proof of (iv) is elementary in the case α = 0, since For the case α ∈ (0, 1), we reproduce similar arguments to [36]. In particular, we observe from To check that is has a trivial kernel we can argue as in [36, page 728]. This concludes the proof of the proposition. Remark 2.3. Since W α 0 ⊂ W α 0 has codimension 2N , Proposition 2.2(iv) implies that D f G α (0, 0; λ * ) is Fredholm with finite index −2N . As we will see, even for non-generate equilibria the full linearized operator D (f ;λ) G α (0, 0; λ * ) will have a range with nonzero codimension. While this rules out an immediate application of the implicit function theorem, one can still, in a small neighborhood of (0, 0; λ * ), the nonlinear problem G α (ε, f ; λ) = 0 will always admit a Lyapunov-Schmidt reduction to an equation in finite dimensions; see for instance [40]. Of course, studying this reduced equation may still be quite challenging, and may in particular involve evaluating further Fréchet derivatives of G α .

2.4.
Integral identities for the functional G α . In this short section we prove several integral identities for the functional G α j which follow from Lemma 2.1. First, we show how that functional G α j defined in (2.27) can be written in terms of the relative stream function restricted to points on the boundary εb j φ j (w) + z j ∈ ∂D ε j . Writing w ∈ T as w = e iθ , we claim that To see this, we use ∂ θ φ j (w) = iwφ j (w) and v ε (z) = 2i∂ z ψ ε (z) to rewrite We can now prove the following lemma.
Then, the following identities hold: Proof. By continuity, it is enough to consider ε = 0, and thanks to the ε → −ε symmetry in (2.68), we can further restrict to ε > 0. We shall first prove (i). Averaging (2.36) over the boundaries, with Ω = 0, then summing and using Lemma 2.1 yields In view of (2.1) and (2.13), by making simple changes of variables we get Writing w ∈ T as w = e iθ then integrating by parts and using (2.37), the left hand side of (2.40) becomes Combining (2.40)-(2.41) and using the simple identity Then (i) follows from the identity As for (ii), integrating (2.36) with U = 0 over the boundaries then summing and taking the real part we obtain, by virtue of Lemma 2.1, According to (2.5), the quantity is purely imaginary. It follows that Then, in view of (2.1) and (2.13), by making simple changes of variables we get Writing w ∈ T as w = e iθ then integrating by parts and using (2.37) we find See, for instance, [43]. These identities can be seen as coming from the symmetries of the Hamiltonian system (1.3) under translations and rotations.
2.5. An abstract lemma. From Proposition 2.2, we know that the operator D f G α (0, 0; λ * ) is not onto. As discussed in Remark 2.3, the linearized operator D (f ;λ) G α (0, 0; λ * ) is also not onto. Thankfully, we will be able to "explain" this latter degeneracy using the nonlinear identities in Lemma 2.4. More precisely, for non-degenerate equilibria, we will be able to apply the following mild generalization of the usual implicit function theorem. Proof. We first claim that the linearizations of G and F at the origins in U × V and Z × U × V satisfy ran D y G(0) = ker D z F (0). (2.51) To see this, we differentiate (2.45) and (2.46) with respect to x and y to find This in particular implies that D z F (0)D y G(0) = 0, and hence ran D y G(0) ⊂ ker D z F (0). As (2.49) and (2.48) force codim ker D z F (0) = codim ran D y F (0) = n, the only possibility is that the two spaces are equal and (2.51) holds. Again using (2.49), we know that there is an n-dimensional subspace Z 1 ⊂ Z such that D z F (0) restricts to an invertible map Z 1 → R n . Moreover, by (2.51) we have (2.52)

Consider the augmented mapping
The linearized operator D (z 1 ,y) H(0) : Z 1 × Y → Z, given by is invertible. Indeed, it is onto by (2.52). If (z 1 , y) lies in its kernel, applying D z F (0) to (2.53) and using the definition of Z 1 (2.51) yields z 1 = 0, at which point y = 0 follows from (2.47). Thus, by the implicit function theorem there is a neighborhoodZ 1 ×Ũ ×Ṽ of the origin in Z 1 × U × V and C 1 mappings (h, g) :Ũ →Z 1 ×Ṽ satisfying (h(0), g(0)) = 0, Moreover, all solutions of H(z, x, y) = 0 inZ 1 ×Ũ ×Ṽ are of the form (h(x), x, g(x)), and the linearizations of (h, g) at the origin are uniquely determined by The proof will therefore be complete if we can show that h ≡ 0, for which will need the full force of the nonlinear conditions (2.45) and (2.46) and not just their linearizations at the origin. Consider the restriction By our choice of Z 1 , the linearized operator D z 1 F 1 (0) = D z F (0)| Z 1 is invertible. Applying the implicit function theorem and using (2.45) we conclude that, possibly after shrinkingZ 1 ×Ũ ×Ṽ , all solutions of F 1 (z 1 , x, y) = 0 inZ 1 ×Ũ ×Ṽ are of the form (0, x, y). By (2.46) and (2.54) we have , g(x)), x, g(x)) = F (h(x), x, g(x)) = F 1 (h(x), x, g(x)), for all x ∈Ũ , and so this forces h ≡ 0 as desired.
2.6. Existence of vortex patch equilibria. We shall give in this subsection a detailed statement of Theorem 1.2, as well as a proof based on Lemma 2.6. The stationary case is more degenerate than the rigid motion case due to the additional symmetries in the problem, and so we will treat it separately.
For rigidly rotating or translating vortex patch solutions, our main result is the following.
Theorem 2.7. Let α ∈ [0, 1) and let λ * be a non-degenerate solution, in the sense of Definition 1.1(i), to the N -vortex problem (1.6), with one of Ω, U nonzero. Then the following hold true.
By the uniqueness in (i), in order to prove (ii) it suffices to show that (2.69) Sincef j (w) = −f j (−w), by (2.28) we have Making the change of variable τ → −τ we get In a similar way we can check that Inserting the two last identities into (2.69) and using the fact thatf j (w) = f j (−w) yields (2.68) as desired.
As mentioned in Remark 1.3, with only minor modifications the above proof still holds when V α 0 and W α 0 in (2.30) are replaced by spaces with higher Hölder regularity C n+1+ηα and C n+ηα for any fixed n ∈ N, at the cost of possibly shrinking ε 1 . In particular, by uniqueness we may assume that f j and hence φ j are C 2 , which allows us to prove the convexity of the domains O ε j by following the same argument as in [36].
Recall that the curvature can be expressed, in terms of the conformal mapping, by the formula As φ j (w) = w + ε|ε| α b 1+α j f j (w), we easily verify that uniformly in w. Thus the curvature is strictly positive and therefore the domain O ε j is strictly convex. Now, we treat the stationary case, where Ω = U = 0. We have the following result. Proof. We shall only give the proof of the existence and uniqueness of (i). The proof of the asymptotic expansion and (ii)-(iii) follow the same lines of Theorem 2.7.

Examples of asymmetric vortex equilibria
In this section we shall give some explicit examples of point vortex solutions to the N -vortex problem (1.6) satisfying the non-degeneracy condition in Definition 1.1. In particular, we prove Theorem 1.6 by simply applying Theorems 2.7 and 2.8.
Asymmetric co-rotating pairs. Set N = 2 and consider the rotating solution λ * given by (1.9) to the N -vortex problem (1.6). The differential of the mapping Eliminating the second row we get a matrix with Jacobian determinant (α + 2)(1 + c) 2 , which is nonzero if c = −1. Thus, this matrix has rank 3, which implies that the kernel is trivial and the image has codimension 1. Therefore, Theorem 2.7 applies yielding the existence of ε 1 > 0 and a unique C 1 function (f ; It remains only to check the reflection symmetry property. From (2.31), one has Since λ * 1 (0) =λ * 1 (0), by uniqueness of the solution of (3.1) we conclude that f j (w) = f j (w) and y 2 = 0, which implies that the Fourier coefficients of f j ∈ V α 0 are real and the domain associated to the conformal mapping φ j = Id +ε|ε| α b α+1 j f j is symmetric with respect to the real axis.
Asymmetric counter-rotating pairs. Set N = 2 and consider the translating solution λ * given by (1.10) to the N -vortex problem (1.6). The differential of the mapping By eliminating the last line we get a matrix with Jacobian determinant 2γd(α+1), which is nonzero. Therefore, this matrix has rank 3, which implies that the kernel is trivial and the image has codimension one. Hence, the existence of counter-rotating vortex patch pair follows from Theorem 2.7. The reflection symmetry property can be checked similarly to the co-rotating case.
Stationary tripole. Set N = 3 in (1.6) and consider the stationary tripole λ * given by (1.11). The differential of the mapping which has rank 3. Thus, Theorem 2.8 guarantees the existence of stationary vortex patch tripole. The reflection symmetry property with respect to the real axis can be checked similarly to the co-rotating pairs.

Nested polygonal vortex patch equilibria
In this section we shall construct 2m + 1 multipolar vortex equilibria in which a central patch is surrounded by 2m satellite patches centered at the vertices of two nested regular m-gons. The vertices of the polygons are either radially aligned with each other or out of phase by an angle π/m, and the patches on each polygon are identical with the same strength; see Figure 3.
More precisely, we shall desingularize the following system of point vortices where γ 0 , γ 1 , γ 2 ∈ R \ {0}, d 2 > d 1 > 0 and ϑ = 0 corresponds to the aligned configuration while ϑ = 1 refers to the staggered configuration. Assuming that z 0 (t) = z 0 (0) and z jk (t) = e iΩt z jk (0), one may easily check that the system of 2m + 1 equations in (1.6) can be reduced to where λ = (Ω, γ 2 ) and Thus, the identities in (4.4) become (4.5) Moreover, the differential of the mapping Q α := (Q α 1 , Q α 2 ) with respect to λ = (Ω, γ 2 ) is given by If the Jacobian determinant is non-trivial, then the system (4.5) has a unique solution λ * = (Ω * , γ * 2 ) given by , (4.8) In order to ensure that γ 2 is non-vanishing, one has to assume that γ 0 and γ 1 verify the condition Remark 4.1. For the Eulerian interaction α = 0, one may easily check that It follows that (4.7) and (4.9) can be written as This amounts to the study the of the roots of two polynomials of order m. A detailed analysis is given in [10, and [2,.
Remark 4.2. While for general non-degenerate equilibria the differential D λ P α (λ * ) was never onto, in this more symmetric setting the differential D λ F α (λ * ) is onto whenever (4.7) holds. This will enable us to directly apply the implicit function theorem to the vortex patch equations, avoiding Lemma 2.6 and the use of integral identities.