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Periodic solutions of \(N\)-vortex type Hamiltonian systems near the domain boundary. (English) Zbl 1386.37060

Summary: The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of \(N\)-vortex type in a domain \(\Omega\subset\mathbb R^2\). These are solutions \(z(t)=(z_1(t),\dots,z_N(t)) \in \Omega^N\) of \(\dot{z}_j(t)=J\nabla_{z_j} H(z(t)), j=1,\dots,N\), where the Hamiltonian \(H\) has the form \(H(z_1,\dots,z_N) = -\sum_{j,k=1, j\neq k}^N \frac{1}{2\pi}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k).\) Here \(J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) is the standard symplectic matrix in \(\mathbb{R}^2\). The function \(g:\Omega\times\Omega \rightarrow\mathbb{R}\) is required to be of class \(\mathcal C^3\) and symmetric, the regular part of the Dirichlet Green’s function being our model. The Hamiltonian is unbounded from above and below, and the associated action integral is not defined on an open subset of the space of periodic \(H^{1/2}\) functions. Given a compact connected component \(\Gamma\subset\partial\Omega\) of class \(\mathcal C^3\) we are interested in periodic solutions of the Hamiltonian system with trajectories near \(\Gamma\). We present conditions on the behavior of \(g\) near \(\Gamma\) which imply that there exists a family of periodic solutions \(z^{(r)}(t)\), \(0<r<\overline{r}\), with arbitrarily small minimal period \(T_r\rightarrow0\) as \(r\rightarrow0\), and such that the “point vortices” \(z_j^{(r)}(t)\) approach \(\Gamma\) as \(r\rightarrow0\). The solutions are choreographies, i.e., \(z_j^{(r)}(t)\) moves on the same trajectory as \(z_1^{(r)}(t)\) with a phase shift. We can also relate the speed of each vortex with the curvature of \(\Gamma\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76B47 Vortex flows for incompressible inviscid fluids
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
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