Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Abstract

The paper deals with singular first order Hamiltonian systems of the form \[ Γ_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in Ω\subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in {\mathbb{R}}^{2\times 2}$ defines the standard symplectic structure in ${\mathbb{R}}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2π} \sum_{j\neq k=1}^N Γ_j Γ_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\dots ,z_N)\in {\Omega }^{2N}:z_j\ne z_k\text{ for }j\ne k\}$ of $N$ different points in the domain $\Omega \subset {\mathbb{R}}^2$. The function $F:{\Omega }^N\to \mathbb{R}$ may have additional singularities near the boundary of ${\Omega }^N$. We prove the existence of a global continuum of periodic solutions $z(t)=(z_1(t),\dots ,z_N(t))\in {\Omega }^N$ that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium $Z(t)\in {\mathbb{R}}^{2N}$ of the $N$-vortex problem in the whole plane (where $F=0$). Examples for $Z$ include Thomson's vortex configurations, or equilateral triangle solutions. The domain $\Omega$ need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of $2\pi$-periodic $H^{1/2}$ functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for $S^1$-equivariant gradient maps that we adapt to this class of potential operators.

26 pages

References in corpus (2)

• Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary
• Regularization of point vortices for the Euler equation in dimension two, part II

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