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

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  • We examine the $N$ -vortex problem on general domains $Ω\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form

    $Γ_k\dot{z}_k = J\nabla_{z_k}H(z_1,...,z_N),\ \ \ \ k = 1,...,N,$

    where $Γ_k∈\mathbb{R}\setminus\{0\}$ is the strength of the $k$ th vortex at position $z_k(t)∈Ω$ , $J∈\mathbb{R}^{2× 2}$ is the standard symplectic matrix and

    $H(z_1,...,z_N) = -\frac{1}{2π}\sum\limits_{\underset{k≠ j}{k,j = 1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum\limits_{k,j = 1}^NΓ_jΓ_k g(z_k,z_j)$

    with some regular and symmetric, but in general not explicitely known function $g:Ω×Ω \to \mathbb{R}$ . 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 explicitly verified for an example in the unit disc $Ω = B_1(0)$ . In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.

    Mathematics Subject Classification: Primary: 37J45; Secondary: 37N10, 76B47.


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  • Figure 1.  This diagram illustrates the superposition idea. The $2$-vortex problem in the unit disc admits a stationary solution with $\Gamma^1 = -\Gamma^2$, cf. Example 1.2, say $\Gamma^1 = -2$ (blue star), $\Gamma^2 = 2$ (red star). As rigidly rotating configurations on $\mathbb{R}^2$ we take here for simplicity two identical vortices for $\Gamma^1$ and $\Gamma^2$, i.e. $\Gamma^1_1 = \Gamma^1_2 = -1$ rotate on the blue circle in clockwise direction and $\Gamma^2_1 = \Gamma^2_2 = 1$ 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 $\Gamma^1_1,\Gamma^1_2,\Gamma^2_1,\Gamma^2_2$, 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

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