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Bifurcation of rotating patches from Kirchhoff vortices

In this paper we prove the existence of countable branches of rotating patches bifurcating from the ellipses at some implicit angular velocities.


Introduction
In this paper we deal with the vortex motion for incompressible Euler equations in two-dimensional space. The formulations velocity-vorticity is given by the nonlinear transport equation where ω denotes the vorticity of the velocity field v = (v 1 , v 2 ) and it is given by ω = ∂ 1 v 2 − ∂ 2 v 1 . The second equation in (1) is nothing but the Biot-Savart law which can be written with a singular operator as follows: by identifying v = (v 1 , v 2 ) with v 1 + iv 2 , we write (2) v(t, z) = i 2πˆC with dA being the planar Lebesgue measure. Global existence of classical solutions is a consequence of the transport structure of the vorticity equation, for more details about this subject we refer to [1,7]. For less regular initial data Yudovich proved in [31] that the system (1) admits a unique global solution in the weak sense when the initial vorticity ω 0 lies in L 1 ∩ L ∞ . This allows to deal rigorously with the vortex patches which are the characteristic function of bounded domains. Therefore, it follows that when ω 0 = χ D 0 with D 0 a bounded domain then the solution of (1) preserves this structure and ω(t) = χ Dt , with D t = ψ(t, D 0 ) being the image of D 0 by the flow. In the special case where D 0 is the open unit disc the vorticity is radial and thus we get a steady flow. Another remarkable exact solution was discovered by Kirchhoff [20] who proved that an ellipse D 0 performs a steady rotation about its center. More precisely, if the center is assumed to be the origin then D t = e itΩ D 0 , where the angular velocity Ω is determined by the semi-axes a and b of the ellipse through the formula Ω = ab/(a + b) 2 . These ellipses are often referred in the literature as Kirchhoff vortices. For a proof, see for instance [1, p.304] and [21, p.232]. The existence of general class of rotating patches, called also V-states, was discovered numerically by Deem and Zabusky [9]. Later on, Burbea gave an analytical proof and showed the existence of the m-fold symmetric V-states for each integer m ≥ 2 and in this countable family the case m = 2 corresponds to the known Kirchhoff's ellipses. Burbea's approach consists in using some complex analysis tools combined with the bifurcation theory. Notice that in this framework, the rotating patches appear as a countable collection of curves bifurcating from Rankine vortices (trivial solution) at the discrete angular velocities set m−1 2m , m ≥ 2 . It is extremely interesting to look at the pictures of the limiting V-states done in [30], which are the end points of each branch. The boundary develops corners at right angles. Recently, the authors studied in [15] the boundary regularity of the V-states close to the disc and proved that the boundaries are in fact of class C ∞ and convex. More recently, Castro, Córdoba and Gómez-Serrano proved in [5] the analyticity of the V-states close to the disc. Notice that the existence and the regularity of the V-states for more singular nonlinear transport equations arising in geophysical flows as the surface quasi-geostrophic equations has been studied recently in [4,5,13]. Another connected subject which has been investigated very recently in a series of paper [12,14,17,16] is the existence of doubly connected V-states (patches with one hole). The main goal of this paper is to study the second bifurcation of rotating patches from Kirchhoff ellipses corresponding to m = 2. This subject was first examined by Kam in [19], who gave numerical evidence of the existence of some branches bifurcating from the ellipses, see also [27]. We mention that the first bifurcation occurs at the aspect ratio 3 corresponding to the transition regime from stability to instability. In the paper [23] of Luzzatto-Fegiz and Willimason one can find more details about the diagram for the first bifurcations and some illustrations of the limiting V-states. Another central problem which has been studied since the work of Love [22] is the linear and nonlinear stability of the ellipses. For instance, we mention the following papers [18,28]. As to the linear stability of the m-folds symmetric V-states, it was conducted by Burbea and Landau in [3]. However the nonlinear stability of these structures in a small neighborhood of Rankine vortices was done by Wan in [29]. For further numerical discussions, see also [6,11,25]. In the current paper we intend to give an analytical proof of the bifurcation from the ellipses. Our result reads as follows.
Theorem 1. Consider the family of the ellipses E : Q ∈]0, 1[ → E Q given by the parametrization Then for each Q ∈ S there exists a nontrivial curve of rotating vortex patches bifurcating from the curve E at the ellipse E Q . Moreover the boundary of these V-states are C 1+α , ∀α ∈]0, 1[.
Before giving some details about the proof, we shall give first some remarks.
Remark. In a very recent paper [5], Castro, Córdoba and Gómez-Serrano proved the analyticity of the V-states close to the ellipses. Our approach sounds to be more easy but not so deep and cannot lead to the analyticity of the V-states. Notice that what one could expect from iterating our method is to get the regularity C n+α for each n ∈ N but the proof does not guarantee a uniform existence interval with respect to the parameter n.
Remark. In contrast to the bifurcation from the disc where we get a collection of m folds, the V-states of Theorem 1 are in general one or two-folds. For m even we can show from the proof that the V-states are symmetric with respect to the origin. Now we shall sketch the proof of Theorem 1 which is mainly based upon the bifurcation theory via Crandall-Rabinowitz theorem. We shall look for a parametrization of the boundary ∂D of the rotating patches as a small perturbation of a given ellipse. This parametrization takes the form Φ : T → ∂D, with T is the unit circle and Φ(w) = w + Qw + n≥2 a n w n , Q ∈]0, 1[, a n ∈ R.
Observe that when all the coefficients a n vanish then this parametrization corresponds to an ellipse, where Q = a−b a+b , with a and b being the major axis and the minor axis, respectively. As we shall see in the next section, the function Φ satisfies the nonlinear equation Setting α Q (w) = w + Qw then we retrieve the Kirchhoff solutions, meaning that, , α Q (w)) = 0, ∀w ∈ T.

Now we introduce the function
then by this transformation the ellipses lead to a family of trivial solutions: F (Q, 0) = 0, ∀Q ∈]0, 1[. Therefore it is legitimate at this stage to look for non trivial solutions by using the bifurcation techniques in the spirit of Burbea's work [2]. As we shall see, the computations of the linearized operator L Q ∂ f F (Q, 0) are a little bit more involved that the radial case but they can still be done in an explicit way. We shall see from this part that the dispersion set S introduced in Theorem 1 corresponds in fact to the values of Q such that the kernel of the operator L Q is one -dimensional. We shall also check that all the assumptions of Crandall-Rabinowitz theorem are satisfied and therefore the proof of the main result will follow immediately. Notation. We need to fix some notation that will be frequently used along this paper. We denote by C any positive constant that may change from line to line. We denote by D the unit disc and its boundary, the unit circle, is denoted by T. Let f : T → C be a continuous function, we define its mean value by, where dτ stands for the complex integration. Let X and Y be two normed spaces. We denote by L(X, Y ) the space of all continuous linear maps T : X → Y endowed with its usual strong topology. We shall denote by N (T ) and R(T ) the kernel and the range of T , respectively. Finally, if F is a subspace of Y , then Y /F denotes the quotient space.

Formulation of the problem
Following [16,17] one can see that the boundary of any smooth V-states χ Dt , with D t = e itΩ D is subject to the equation Recall that a curve γ of the complex plane C is said a regular Jordan curve if it admits a parametrization Φ : T → γ which is simple and of class C 1 such that Φ ′ (w) = 0, ∀w ∈ T. Note that in this case the curve γ encloses a simply connected domain. Now to solve the equation (3) we shall restrict ourselves to domains whose boundaries are parametrized by a regular Jordan curve Φ : T → C. A tangent vector to the boundary at the point Φ(w) is determined by z ′ = iwΦ ′ (w) and therefore (3) becomes We shall define the object G by It is easily seen that the equation (4) is invariant by rotation and dilation. Moreover, one can deduce from this formulation Kirchhoff's result which states that an ellipse of the semi-axes a and b rotates with the angular velocity Ω = ab (a+b) 2 . Indeed, note that in this case the ellipse may be parametrized by the conformal parametrization, In the sequel we shall use the notation By straightforward computations we get . Using residue theorem and taking r > 1 we get and consequently (4) is satisfied provided that This can be written in the form Now we shall introduce the function From the preceding discussion we readily get To prove Theorem 1 we need to show the existence of nontrivial solutions of the equation defining the V-states : F (Q, f (w)) = 0, ∀ w ∈ T. It will be done using the bifurcation theory through Crandall-Rabinowitz theorem [8]. For the completeness of the paper we recall this basic theorem and it will referred to as sometimes by C-R theorem.
Theorem 2. Let X, Y be two Banach spaces, V a neighborhood of 0 in X and let F : R × V → Y with the following properties: (1) F (λ, 0) = 0 for any λ ∈ R.
(2) F is C 1 and F λx exists and are continuous.
If Z is any complement of N (L 0 ) in X, then there is a neighborhood U of (0, 0) in R × X, an interval (−a, a) and continuous functions ϕ : (−a, a) → R, ψ : (−a, a) → Z such that ϕ(0) = 0, ψ(0) = 0 and Now we shall give a precise statement of Theorem 1. For this purpose we should fix the spaces X and Y used in C-R theorem. They are given by, a n w n , a n ∈ R and (10) g n e n , g n ∈ R, w ∈ T , e n (w) Im(w n ).
Theorem 3. Consider the family of ellipses E : Q ∈ (0, 1) → E Q given by the parametrization Then for each Q = Q m ∈ S there exists a nontrivial curve of rotating vortex patches bifurcating from the curve E at the ellipse E Q . Moreover the boundary of these V-states are C 1+α . More precisely, let Z m any complement of the vector v m = w m+1 1−Qw 2 in the space X. Then there exist a > 0 and continuous functions Q : (−a, a) → R, ψ : (−a, a) → Z m satisfying Q(0) = Q m , ψ(0) = 0, such that the bifurcating curve at this point is described by, In particular the boundary of the V-states rotating is described by The proof consists in checking all the assumptions of Theorem 2. This will be done in details in the next sections.

Regularity of the functional
This section is devoted to the study of the regularity assumptions stated in C-R theorem. We shall study the nonlinear functional F defining the V-states already seen in (7). It is given through the functional G as follows, For r ∈ (0, 1) we denote by B r the open ball of X (this space was introduced in (9)) with center 0 and radius r, We shall make use at several stages of the following lemma, for more details see [24, p. 419].
Lemma 1. Let T be a singular operator defined by Assume that the kernel of the operator T satisfies (1) K is measurable on T × T and (2) For each ξ ∈ T, w → K(ξ, w) is differentiable in T\{ξ} and Then for every 0 < α < 1 The main result of this section reads as follows.
Proof. (1) To get this result it suffices to prove that ∂ Q F, ∂ f F :]0, ε[×B rε → Y exist and are continuous. We shall first compute ∂ f F (Q, f ). This will be done by showing first the existence of the Gâteaux derivative and second its continuity in the strong topology. Before dealing with this problem we should first show that the functional F is well-defined. For this purpose it suffices to show that the functional G sends X into C α (T) and the Fourier coefficients of G((1−Q 2 )/4, α Q +f )) are real when f belongs to X. As to the second claim we follow the Arxiv version of the paper [15] and for the sake of simplicity we shall skip the details and sketch just the basic ideas of the proof. First, we write with the notation Φ f = α Q + f. It is clear that G 1 is polynomial in the variable Q and bilinear on f and f ′ . Therefore using the algebra structure of C α (T) one gets This implies in particular that G 1 : (0, 1) × X → Y is of class C ∞ . Now we shall focus on the second part G 2 . Fix Q ∈ (0, ε) and put r ε = 1−ε 2 , then for f ∈ B rε we get Indeed, 6 We combine this with the mean-value theorem applied to f which is holomorphic inside the unit disc Now using Lemma 1 we get that G 2 (Q, f ) ∈ C α (T). This concludes the fact that F is well-defined. Next, we shall prove that for f ∈ X with f 1+α < r ε the Gâteaux derivative ∂ f G 2 exists and is continuous. Straightforward computations show that this derivative is given by: for h ∈ X, Notice that we have used the fact that the Fourier coefficients of Φ f are real and therefore Φ f (w) = Φ f (w). It is easy to check according to (12) that |K 1 (ξ, w)| = 1, |∂ w K(ξ, w)| ≤ C 0 |w − ξ| −1 and thus we deduce from Lemma 1 that For the second term I 2 we have the following estimates for the kernel Once again from Lemma 1 one gets, The last term can be estimated similarly to the previous one and we get Putting together the preceding estimates we get This shows the existence of Gâteaux derivative and now we intend to prove the continuity of the map f → ∂ f G 2 (Q, f ) from X to L(X, Y ). This is a consequence of the following estimate that we shall prove now: for f, g ∈ B rε , one has First we write By the mean value theorem we may check that Thus we obtain by using Lemma 1 To estimate I 2 (Q, f )h − I 2 (Q, g)h we shall use the identity and thus The kernels can be estimated as follows

So Lemma 1 implies that
It remains to check the continuity of I 3 . We write The first term can be written in the form where the kernel K 8 satisfies

This yields in view of Lemma 1
The second term can written under the form and the kernel K 9 satisfies which yields in view of Lemma 1 For the third term we can check that the kernel K 7 satisfies which gives according to Lemma 1 Putting together the preceding estimates we get This achieves the proof of (14) and therefore the Gâteaux derivative is Lipschitz and thus it is continuous on the variable f . Therefore we conclude at this stage that the Fréchet derivative exists and coincides with Gâteaux derivative. See [10] for more information.
We shall now study the regularity of G 2 with respect to Q. This reduces to studying the regularity of Q → G 2 (Q, f ) given by Easy computations yields As before, using Lemma 1 we get for Reproducing the same analysis we get for any k ∈ N (2) We shall check that ∂ Q ∂ f G 2 : (0, ε) × B rε → C α (T) is continuous. According to (13) we obtain • Estimate of ∂ Q I 1 (Q, f )h. From its expression we write

Straightforward computations yield
Using Lemma 1 we get • Estimate of ∂ Q I 2 (Q, f )h. One may write Using again Lemma 1 we find We can check that and |∂ w K 9 (ξ, w)| + |∂ w K 10 (ξ, w)| ≤ C h ′ L ∞ 1 |w − ξ| and therefore we get by Lemma 1, Finally we obtain Reproducing the same analysis we get for any k ∈ N On the other hand the same analysis used for proving (14) shows that and thus Combining (15) for the case k = 1 with (16) we conclude that ∂ Q ∂ f G 2 : (0, ε) × B rε → C α (T) is continuous. This achieves the proof of the proposition.

Study of the linearized equation
The main goal of this section is to study some spectral properties of the linearized operator of the functional f ∈ X → F (Q, f ) in a neighborhood of zero. This operator is defined by where F was defined in (7) and the space X in (9). Now we introduce the following set For given m ≥ 3 the function f m : Q ∈ (0, 1) → 1 + Q m − 1−Q 2 2 m is strictly nondecreasing and satisfies Consequently, there is only one Q m ∈ (0, 1) with f m (Q m ) = 0. This allows to construct a function m → Q m . As the map n → f n (Q) is strictly decreasing then one can readily prove that the sequence m → Q m is strictly increasing. Moreover, it is not difficult to prove the asymptotic behavior where α is the unique solution of 1 + e −α − α = 0. Now we shall establish the following properties for L Q which yields immediately to Theorem 1 and Theorem 3 according to Crandall-Rabinowitz theorem.
Proposition 2. The following assertions hold true.
(1) Let h(w) = n≥2 a n w n ∈ X, then g n+1 e n ; e n (w) = Im(w n ), (2) The kernel of L Q is nontrivial if and only if Q = Q m ∈ S and it is a one-dimensional vector space generated by v m (w) = w m+1 1 − Qw 2 · (3) The range of L Q is of co-dimension one in Y and it is given by R(L Q ) = g ∈ C α (T), g = n≥1 n =m g n+1 e n , g n ∈ R .
(4) Transversality assumption: for any Q = Q m ∈ S, and Then we get successively, Applying once again residue theorem we obtain . It follows that Putting together the identities (17), (18), (19) and (20) one gets and consequently, This can also be written in the form It is easy to check that with c n = 1 − Q 2 2 (n + 1)a n+1 − Q(n − 1)a n−1 , ∀n ≥ 3.
As to the third term, we easily find Performing the same analysis yields According to (21) and (23) we get with the notation e n (w) = Im(w n ). It follows that Combining (22) and (24) we obtain (26) d n − d n = 1 − Q 2 2 n − 1 − Q n a n+1 − Qa n−1 .
Therefore the equation Im Lh(w) = 0 is equivalent to the linear system Since Q ∈ (0, 1) then necessarily a 2 = a 3 = 0.
The last equation is equivalent to Let m ≥ 3, then we know from the beginning of this section the existence of only one solution Q = Q m ∈ (0, 1) of the equation Moreover, the left part of this equality defines a strictly decreasing function in m implying that Thus (27) is equivalent to a n+1 = Qa n−1 , ∀n ≥ 3, n = m.
Thus the dynamical system (27) combined with the vanishing two first values admits the following solutions ∀n ≥ 0, a m+1+2n = Q n a m+1 and 0 otherwise This means that the associated kernel is one dimensional generated by the eigenfunction v m (w) = n≥0 Q n w m+1+2n = w m+1 1 − Qw 2 · (3) Let Q ∈ S and m being the frequency such that Q = Q m . We will show that the range R(L Qm ) coincides with the closed subspace Y g ∈ C α (T); g = n≥1 n =m g n+1 e n , g n ∈ R .
From (25) and (26) one sees that the range of L Q is contained in Y. Conversely, let g ∈ Y we shall look for h(w) = n≥2 a n w n ∈ C 1+α (T) such that L Q h = g. Once again from (25) this is equivalent to This determines uniquely the sequence (a n ) 2≤n≤m and for n ≥ m + 1 one has the recursive formula a n+1 − Qa n−1 = g n+1 The only free coefficient is a m+1 and therefore the solutions of the above system form onedimensional affine space. To prove that any pre-image h belongs to C 1+α (T) it suffices to show it for the function H(w) = n≥m+2 a n w n . Set Then = w 2 Q n≥m a n w n + wG(w) = w 2 QH(w) + w 2 Q(a m w m + a m+1 w m+1 ) + wG(w).
The problem reduces then to check that G ∈ C 1+α (T). We split G into two terms as follows Since the sequence (g n ) is bounded then for large n one gets This shows that G 2 ∈ C k (T) for all k ∈ N. Let us now prove that G 1 ∈ C 1+α (T). First from the embedding C α (T) ֒→ L ∞ (T) ֒→ L 2 (T) one obtains n |g n+1 | 2 g 2 C α .
Therefore by Cauchy-Scwharz It remains to prove that G ′ 1 ∈ C α (T). Differentiating term by term the series we get The function G 3 is clearly in C α (T) according to the assumption g ∈ C α . The function G 4 belongs to L ∞ (T) and G ′ 4 ∈ C α (T). Indeed, This gives (wG 4 ) ′ ∈ C α (T) and thus G 4 ∈ C 1+α (T). On the other hand Arguing as before we see that wG ′ 5 ∈ L ∞ (T) and belongs also to C α (T). This shows that G ′ 1 ∈ C α (T) which gives that G 1 ∈ C 1+α (T). This shows finally that any pre-image of g belongs to the space X.
(4) Let m ≥ 3 be an integer and Q = Q m the associated element in the set S. We have seen that the kernel is one-dimensional generated by v m (w) = w m+1 1 − Qw 2 = n≥0 Q n w m+1+2n .
We shall compute ∂ Q ∂ f F (Q m , 0)v m which coincides with ∂ Q L Q v m Q=Qm . The transversality condition that we shall check is From the structure of the range of L this is equivalent to prove that the coefficient of e m in the decomposition ∂ Q L Q v m Q=Qm is not zero. From (25) and (26) 19 We have used the fact that a m+1 = 1 and a m−1 = 0. This achieves the transversality assumption and therefore the proof of Proposition 2 is complete.