Periodic solutions for the N-vortex problem via a superposition principle

We examine the $N$-vortex problem on general domains $\Omega\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form $$ \Gamma_k\dot{z}_k=J\nabla_{z_k}H(z_1,\ldots,z_N),\quad k=1,\ldots,N, $$ where $\Gamma_k\in\mathbb{R}\setminus\{0\}$ is the strength of the $k$th vortex at position $z_k(t)\in\Omega$, $J\in\mathbb{R}^{2\times 2}$ is the standard symplectic matrix and $$ H(z_1,\ldots,z_N)=-\frac{1}{2\pi}\sum_{\underset{k\neq j}{k,j=1}}^N\Gamma_j\Gamma_k\log|z_k-z_j|-\sum_{k,j=1}^N\Gamma_j\Gamma_k g(z_k,z_j) $$ with some regular and symmetric, but in general not explicitely known function $g:\Omega\times\Omega\rightarrow \mathbb{R}$. The investigation relies on the idea to superpose a stationary solution of a system of less than $N$ vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple $T$-periodic solutions are shown to exist for every $T>0$ small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc $\Omega=B_1(0)$. In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.

with some regular and symmetric, but in general not explicitely known function g : Ω × Ω → R. The investigation relies on the idea to superpose a stationary solution of a system of less than N vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and

Introduction and statement of results
The N -vortex problem is a first order Hamiltonian system that describes the motion of N point vortices inside a planar domain Ω ⊂ R 2 . If z k (t) ∈ Ω denotes the position of the kth vortex at time t and Γ k ∈ R \ {0} its strength, the system is given by where J ∈ R 2×2 is the rotation by − π 2 and the Hamiltonian H Ω defined on F N (Ω) = (z 1 , . . . , z N ) ∈ Ω N : z j = z k for j = k reads H Ω (z 1 , . . . , z N ) = − 1 2π The function g Ω : Ω × Ω → R classically is defined by the requirement that is the Green's function of the Dirichlet Laplacian of Ω -or a more general hydrodynamic Green's function -and thus in almost all cases not explicitely known.
Similar Hamiltonian systems, in which g Ω in the definition of H Ω is replaced by a possibly different regular function, also appear in singular limits of other PDEs like the Ginzburg-Landau-Schrödinger (or Gross-Pitaevskii) equation and the Landau-Lifshitz-Gilbert equation, see [18,23] and references therein. In fact for our result it is enough that g : Ω×Ω → R is a sufficiently smooth and symmetric function and not necessarily the regular part of the Dirichlet or a hydrodynamic Green's function.
The present paper will address the question of existence of periodic solutions of (1.1) in an arbitrary domain. In special domains like Ω = R 2 , Ω = B 1 (0) quite a lot of periodic solutions of (1.1) can be found that rotate as a fixed configuration around a certain point, cf. section 1.3. This is possible because in those cases g Ω is explicitely known and invariant with respect to rotations. Besides the fact that the Hamiltonian is in almost all other cases not explicitely known, it is in general unbounded from both sides, not integrable, has singularities and non compact, not metrically complete energy surfaces. These difficulties cause the failure of standard theorems and methods for the existence of periodics.
However in the past years three types of periodic solutions in almost arbitrary domains could be established. In the first one vortices with possibly different strengths and of arbitrary number are close to a critical point of the so called Robin function h Ω (z) = g Ω (z, z) and the configuration of vortices looks after rescaling like a rigidly rotating solution of the N -vortex system on R 2 , see [4,6]. In the second type of solutions, shown in [10], two identical vortices rotate around their center of vorticity while the center itself follows a level line of h Ω . The third result holds for an arbitrary number of identical vortices, which separated by time shifts follow the same curve close to the boundary of a simply connected bounded domain, [5]. The Here we will generalize the results of [4,6] in the following way: Instead of an equilibrium of the 1-vortex system on Ω, we consider a stationary solution of a system of m-vortices with strength Γ 1 , . . . , Γ m located at α 1 , . . . , α m ∈ Ω. For every vortex Γ k , k = 1, . . . , m take now a rigidly rotating configuration Z k (t) of the whole-plane system consisting of N k vortices with strengths Γ k 1 , . . . , Γ k N k , such that N k j=1 Γ k j = 0. In the case N k = 1 a stationary single vortex may also be considered as an admissible configuration. By a change of timescale we may assume that N k j=1 Γ k j = Γ k . We then ask for the existence of periodic solutions of the ( m k=1 N k )-vortex system on Ω, in which the vortices form m clusters (z k 1 , . . . , z k N k ), k = 1, . . . , m approximately satisfying with a small parameter r > 0. So we superpose a stationary solution of the m- (blue star), Γ 2 = 2 (red star). As rigidly rotating configurations on R 2 we take here for simplicity two identical vortices for Γ 1 and Γ 2 , i.e. Γ 1 1 = Γ 1 2 = −1 rotate on the blue circle in clockwise direction and Γ 2 1 = Γ 2 2 = 1 rotate on the red circle in counterclockwise direction. The result on the right-hand side is a periodic solution of the 4-vortex system in the disc with vorticities Γ 1 1 , Γ 1 2 , Γ 2 1 , Γ 2 2 , where each pair of vortices moves along a deformed circle in the same orientation as before. The shown trajectory is the actual numerically computed trajectory of the 4-vortex problem. Suitable initial conditions can in this case be found due to symmetry considerations.
vortex system on Ω and several rigidly rotating configurations of the whole-plane system. This is illustrated for a simple case in Figure 1.
The general idea of grouping vortices into different clusters plays a role in establishing the existence of quasi-periodic solutions via KAM theory, see [19,27].
In this paper we use it to provide general conditions that give rise to families of periodic solutions. The conditions will be verified for a concrete case in the unit disc Ω = B 1 (0) leading to examples of periodic solutions with an arbitrary number of N ≥ 3 vortices that are not rigidly rotating configurations, one of them is presented in Figure 1.
In the following subsections we will formulate two versions of our theorem Bevor we state our results we shortly like to mention the conclusions one can draw from solutions of the N -vortex system for the PDEs that give rise to this system as some sort of singular limit. By constructing appropriate stream functions it is possible to desingularize stationary solutions of the N -vortex problem to stationary solutions of the 2D Euler equations, see [11] and references therein. A similar result for the Euler equations and periodic solutions is so far not available.
Concerning other PDEs Venkatraman has shown in [33] that rigidly rotating solutions of (1.1) in the unit disc give rise to corresponding periodic solutions of the Gross-Pitaevskii equation. The same is true for rigidly rotating configurations on the sphere S 2 , see [17]. Apart from that the desingularization of general periodic solutions like the ones obtained here is also for the Gross-Pitaevskii equation an open problem.

Statement of results part 1
Let Ω ⊂ R 2 be a domain and fix a symmetric C 2 function g : Ω × Ω → R, for example the regular part of a hydrodynamic Green's function of Ω. We will investigate a point vortex like system similar to (1.1), which is induced by the generalized Green's and Robin functions At first we consider on the domain Ω a system of m ∈ N vortices with vorticities Γ 1 , . . . , Γ m ∈ R \ {0} and Hamiltonian defined on F m (Ω) = a = (a 1 , . . . , a m ) ∈ Ω m : a k = a k for all k = k . We require that the corresponding m-vortex system admits a stationary solution, cf.
To be more precise we assume (A1) H has a nondegenerate critical point α ∈ F m (Ω).
Next we fix a number l ∈ { 1, . . . , m }, which will be the number of vortices that are splitted into configurations consisting of more than a single vortex. Without restriction we take the first l vortices. I.e. for k = 1, . . . , l choose N k ≥ 2 vorticities We then define the Hamiltonian H k As mentioned in the introduction aÑ -vortex system on R 2 allows rigidly rotating solutions, also called relative equilibria, of the form Z(t) = e ωJÑ t z, ω = 0, cf.
has only 3 linear independent 2π-periodic solutions. This is the minimal possible number due to the invariance under rotations and translations. Our third requirement is: Note that condition (A2) can always be achieved by a change of time scale provided one has a relative equilibrium solution of (1.2) with j Γ k j = 0. The remaining m − l vortices -which may be none -are not splitted into configurations. I.e. for k = l + 1, . . . , m we let The system under investigation is the generalized N := m k=1 N k -vortex system Here We will use the Sobolev spaces H 1 which is a l-dimensional submanifold, since Z l+1 = . . . = Z m = 0. And for a = (a 1 , . . . , a m ) ∈ R 2m we definê a = (a 1 , . . . , a 1 , a 2 , . . . , a 2 , . . . , a m , . . . , a m ) Now we are ready to formulate a first version of our theorem. • m ∈ N, Γ 1 = . . . = Γ m = 0 and Ω not simply connected [13] or dumbell shaped [14], • m ∈ N, conditions on Γ k (different from the ones in [8]) for Ω arbitrary and for Ω not simply connected [21], , Ω symmetric with respect to reflection at a line [9] or the action of a dihedral group [22].
None of the mentioned results addresses the question of nondegeneracy of the critical points, on which our proof relies. Indeed condition (A1) is for these solutions hard to check, since the Hamiltonian H Ω and the critical point α are not explicitely known. However a recent result of Bartsch, Micheletti and Pistoia shows that H Ω has only nondegenerate critical points for a generic bounded domain Ω, see [7].
So if the vorticities Γ 1 , . . . , Γ m allow the existence of a critical point of H Ω , as for example in one of the listed cases, then condition (A1) is satisfied at least after an arbitrarily small deformation of the domain.
In some cases also explicit stationary configurations are known, for example if Ω = R 2 or Ω = B 1 (0). But these are all degenerate due to the symmetries of is violated. But we will see that degeneracy induced by symmetries can still be handled, i.e. we may replace assumption (A1) by (A1 ) H has a critical point α ∈ F m (Ω) and one of the following properties holds: (i) α is nondegenerate, (ii) Ω and g are radial e λJ Ω = Ω, g e λJ x, e λJ y = g(x, y) for every λ ∈ R, x, y ∈ Ω and dim ker ∇ 2 H(α) = 1, (iii) Ω and g are in one direction translational invariant there exists ν ∈ R 2 \ {0} with λν + Ω = Ω, g(x + λν, y + λν) = g(x, y) for every λ ∈ R, x, y ∈ Ω and dim ker ∇ 2 H(α) = 1, Note that in the classical case g = g Ω always inherits the symmetries of the domain.
Let Ω be the unit disc B 1 (0) and g = g B 1 (0) be the regular part of the Dirichlet Green's function of B 1 (0), which is given by This will be shown in section 5.

Remark 1.3.
If Ω = R 2 , g = g R 2 ≡ 0 then critical points of H R 2 exist depending on the vorticities Γ 1 , . . . , Γ m . In the easiest case m = 3 vortices with strengths Γ k satisfying Γ 1 Γ 2 + Γ 1 Γ 3 + Γ 2 Γ 3 = 0 are stationary when placed at certain distances along a fixed line, see Theorem 2.2.1 in [29]. More on stationary configurations can also be found in [2]. However for every critical point α of H R 2 the inequality dim ker ∇ 2 H R 2 (α) ≥ 4 holds true. Here 3 dimensions of the kernel are induced by translations and rotations of the critical point α. A fourth dimension by scaling, since differentiation of λ → H R 2 (λα) at λ = 1 shows that k =k Γ k Γ k = 0 is a necessary condition for the existence of critical points. Therefore we have This means that (A1 ) never holds for critical points of the classical m-vortex  • N = 2, Γ 1 + Γ 2 = 0, Z(0) ∈ F 2 (R 2 ) arbitrary, cf. Example 2.3 in [6], Definition 1.6. A relative equilibrium solution Z(t) of the whole plane system is called σ-nondegenerate, provided σ * Z(· + 2π) = Z and (1.3) has only three linear independent solutions satisfying σ * w(· + 2π) = w.
In the case that only the first vortex is splitted up into a configuration with at least two vortices, i.e. when l = 1, we can slightly improve Theorem 1.8.
We are looking for a solution z : R → F N (Ω) where each subgroup of vortices (z k 1 (t), . . . , z k N k (t)) is located near α k and forms a configuration close to a scaled version of the relative equilibrium Z k (t).
In order to reformulate the problem we define together with the following Hamiltonians H 0 : and for r > 0, H r : O r := u ∈ R 2N : ru +α ∈ F N (Ω) → R, Observe that F is defined on an open subset of R 2N containing 0.
for any (k, j) and a ∈ R 2m .
Proof. Openess and smoothness are easy to check, since by (A2) indeed For the derivative of F with respect to z k j we have and therefore Next we turn to the functional setting. Let τ := 2π ord(σ). In order to find T -periodic solutions of (1.1) with T > 0 small, we use the variational structure of (2.1) to look for τ -periodic solutions of (2.1) with r > 0 small. We work on the Sobolev space H 1 τ as stated in section 1.1 and will also need the corresponding spaces L 2 τ and H 2 τ . The action functional associated to (2.1) is given by Then Λ is open in [0, ∞) × H 1 τ , since H 1 τ embeds into C 0 τ , Φ ∈ C 2 (Λ , R) due to Lemma 2.2 and we have to solve ∇Φ r (u) = 0 for (r, u) ∈ Λ with r > 0.
HereŻ k is meant to be the element (0, . . . , 0,Ż k , 0, . . . , 0) ∈ X. Whereas this degeneracy is natural for the limiting case r = 0, the functionals Φ r with r > 0 are in general neither invariant with respect to translations by elements of D nor under the action of T m -except for synchronous time shifts θ = (θ 1 , . . . , θ 1 ) ∈ T m . To deal with the degeneracy of the limiting problem we modify our equation ∇Φ r (u) = 0.
For a subspace Y ⊂ X we denote by P Y : X → Y the orthogonal projection onto Y and by Y ⊥ the orthogonal complement of Y in X. Let is continuous, C 1 on U ∩((0, r 0 )×X) with D u ψ continuous up to r = 0 and satisfies for (r, u) ∈ U, r > 0: Proof. As a first step observe that for positive r,ψ r : Λ r → X, ψ r (u) = (id −P D )∇Φ r (u) + 1 r 2 P D ∇Φ r (u) has the same zeroes as ∇Φ r . In the second equation we used that ∇Φ 0 maps into D ⊥ , since Φ 0 is invariant with respect to translations. Clearlyψ is C 1 as long as r > 0. Since F is C 2 and ∇F (0) = 0,ψ r extends as r → 0 continuously tō The partial derivative D uψ : Λ → L(X) is continuous as well and the regularity ofψ will carry over to ψ once we have defined it. and Hence we see that k λ kv k +â is an element of the kernel of Dψ 0 (v) if and only if a ∈ ker ∇ 2 H(α), which meansâ ∈ Y ⊥ . So if we restrictψ to ψ as stated in the Lemma, especially Dψ It remains to prove that ψ r (u) = 0 for r > 0 small, u ∈ Y close to M implies Assume first that (iii) of (A1 ) holds, i.e. λν +Ω = Ω, g(x+λν, y+λν) = g(x, y) for any x, y ∈ Ω, λ ∈ R and some ν ∈ R 2 \ {0}. Then H(α + λν) = H(α), whereν = (ν, . . . , ν) ∈ R 2m , and Φ r (u + λν) = Φ r (u) show thatν ∈ Y ⊥ and ∇Φ r (u),ν = 0 for any u ∈ Λ r . So if ν is the only direction, in which g is invariant, then X = Y ⊕ Rν by (A1 ) and P Y ∇Φ r (u) = 0 automatically gives ∇Φ r (u) = 0.
In the remaining case (A1 )(iv), where Ω = R 2 we have to choose the neigh- For Since M v → P v ∈ L(X) is C 1 , we haveψ ∈ C 1 where r > 0, as well as continuity ofψ, D vψ , D wψ on all ofŨ. For (r, v, w) ∈Ũ there holds  Note also thatψ viewed as a map into H 2 τ ∩ Y with · H 2 τ instead of Y has the same regularity as the originalψ. So the implicit function theorem yields local However the compactness of M and the uniqueness of the solution allow us to construct a global map W as requested by the Lemma. The equivariance with respect to synchronuous time shifts follows from the corresponding equivariance ofψ, i.e.ψ(r, θ * v, θ * w) = θ * ψ(r, v, w).
Proof which the Lusternik-Schnirelmann category of T l−1 provides l as a minimal bound, see for example [12].
This way we have found for every r ∈ (0, r 1 ) l distinct critical points of Φ r . Let u = v + W (r, v) ∈ Y be one of them. Then z(t) = ru(t/r 2 ) +α is by construction a T (r) = τ r 2 = 2π ord(σ)r 2 -periodic solution of (1.1), for which the properties of Theorem 1.8 hold.
4 Additional information and the case l = 1 For now we just continue our investigation with l ∈ { 1, . . . , m } arbitrary. Higher order derivatives with respect to z are written as F , F (4) and so on.
Proof. Since M v → P v ∈ L(X) is C ∞ and since W is implicitly defined, the regularity of W is induced by ψ. With g ∈ C k we also have F ∈ C k and hence Φ ∈ C k . Then by the definition of ψ in 3.1 one sees that ψ is indeed of class C k−2 provided κ : U → L 2 (R/τ Z, R 2N ), In order to proove this observe that κ is C k as long as r > 0. The continuity up to r = 0 follows as in the proof of Lemma 3.1 from the fact that F is C 2 and that ∇F (0) = 0. Also the partial dervivatives that include at least one differentiation of κ with respect to u are easily seen to extend in a continuous way as r → 0. So we have to look at the partial derivative where (r, u) ∈ U with r > 0. Now a (pointwise) expansion of F (j+1) gives for some ξ = ξ(j, u, t) ∈ (0, r). But as r → 0 we obtain for the remainder with respect to · L 2 τ and uniformly in u ∈ B ρ (M). Thus So the partial derivatives ∂ j r κ, j = 1, . . . , k − 2 exist and are continuous on all of U.
For the second part assume that g ∈ C 3 . Now W is C 1 on all of [0, r 0 ) × M and we know by Lemma 3.3 that for r > 0 small, cf. equation (3.4). Differentiation of both equations with respect to r at r = 0 and the use of ∂ r ∇Φ 0 (v) = 0 as well as Proof of Theorem 1.9. Let now l = 1. In that case the reduced map ϕ r is in fact constant. Hence the demanded solutions of ∇Φ r (u) = 0 can be parameterized by where r 1 > 0 is taken from Lemma 3.4 and Z = (Z 1 , 0 . . . , 0) ∈ M. By 4.1 this parametrization is indeed C k−2 provided g ∈ C k , k ≥ 2 and ∂ r u (0) ∈ D when k ≥ 3.

An explicit example
With Examples 1.5 and 1.7 we have already seen some relative equilibrium solutions that are σ-nondegenerate or just nondegenerate and therefore can be choosen in (A3 ) for theorem 1.8. Independent of the relative equilibrium solutions we also need for (A1 ) a nondegenerate or not too degenerate critical point of the m-vortex Hamiltonian H. We will verify this for Example 1.2. I.e. we look at the 2-vortex system in the unit disc Ω = B 1 (0) with vorticities Γ 1 = 1, Γ 2 = −1. By combining for example a Thomson N 1 -Gon configuration with vorticities Γ 1 j = 1 N 1 , j = 1, . . . , N 1 and a collinear configuration of N 2 vortices of strengths Γ 2 j = − 1 N 2 , j = 1, . . . , N 2 or another Thomson configuration we obtain therefore periodic solutions of (1.1) in the unit disc for an arbitrary number of N = N 1 + N 2 ≥ 3 vortices that are not rigidly rotating around the center of the disc.
Let R(y) = y |y| 2 be the reflection at the unit circle, then This corresponds to the degeneracy induced by the rotational invariance, which means J 2 α = (0, −µ, 0, µ) ∈ ker ∇ 2 H(α). On the other hand one easily sees that the first three columns are linearly independent. This shows that α is a critical point of the 2 vortex Hamiltonian H satisfying condition (A2 )(ii) as it has been stated in Example 1.2.