[Paper Review] Periodic solutions with prescribed minimal period of the 2-vortex problem in domains

Thomas Bartsch, Matteo Sacchet|arXiv (Cornell University)|Aug 24, 2016

Advanced Mathematical Modeling in Engineering3 citations

TL;DR

This paper establishes the existence of infinitely many periodic solutions with prescribed minimal period for the 2-vortex problem in planar domains, using a superposition of slow center-of-vorticity motion along level sets of the Robin function and fast rotation around the center of vorticity. The results apply to subsets 𝒜⊂Ω satisfying a geometric condition and hold for any minimal period T in an interval I(𝒜), leveraging a higher-dimensional Poincaré-Birkhoff theorem.

ABSTRACT

We consider the Hamiltonian system \[ \dot{z}_k = J abla_{z_k} H_\Omega(z_1,z_2), \quad k=1,2, \] for two point vortices $z_1,z_2\in\Omega$ in a domain $\Omega\subset\mathbb{R}^2$. The Hamiltonian $H_\Omega$ is of the form \[ H_\Omega(z_1,z_2) = -\frac{1}{2\pi} \log |z_1-z_2| - 2g(z_1,z_2) - h(z_1) - h(z_2), \] where $g:\Omega imes\Omega o\mathbb{R}$ is the regular part of a hydrodynamic Green's function in $\Omega$, and $h:\Omega o\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$. The system is singular and not integrable, except when $\Omega$ is a disk or an annulus. We prove the existence of infinitely many periodic solutions with minimal period $T$ which are a superposition of a slow motion of the center of vorticity along a level line of $h$ and of a fast rotation of the two vortices around their center of vorticity. These vortices move in a prescribed subset $\mathcal{A}\subset\Omega$ that has to satisfy a geometric condition. The minimal period can be any $T$ in an interval $I(\mathcal{A})\subset\mathbb{R}$. Subsets $\mathcal{A}$ to which our results apply can be found in any generic bounded domain. The proofs are based on a recent higher dimensional version of the Poincar\'e-Birkhoff theorem due to Fonda and Ure\~na.

Motivation & Objective

• To establish the existence of periodic solutions with prescribed minimal period for two point vortices in a bounded domain Ω⊂ℝ².
• To analyze the dynamics of two vortices under a singular, non-integrable Hamiltonian system in general domains.
• To identify geometric conditions on a subset 𝒜⊂Ω that allow for the existence of such periodic solutions.
• To demonstrate that the minimal period T can be any value in a specific interval I(𝒜) depending on 𝒜.
• To extend the applicability of periodic solution results beyond integrable cases like disks or annuli.

Proposed method

• Formulate the Hamiltonian system for two point vortices in Ω⊂ℝ² using the regular part of the hydrodynamic Green's function and the Robin function.
• Decompose the motion into a slow drift of the center of vorticity along a level line of the Robin function h(z) and a fast rotation around the center of vorticity.
• Apply a recent higher-dimensional version of the Poincaré-Birkhoff theorem by Fonda and Ureña to prove the existence of periodic solutions.
• Identify a geometric condition on the subset 𝒜⊂Ω that ensures the applicability of the Poincaré-Birkhoff argument.
• Show that the minimal period T of the solutions can be any value in an interval I(𝒜) determined by the geometry of 𝒜.
• Use the structure of the Hamiltonian HΩ involving logarithmic and regular parts to model vortex interactions and derive the dynamical system.

Experimental results

Research questions

• RQ1 Can periodic solutions with prescribed minimal period be constructed for the 2-vortex problem in arbitrary bounded domains Ω⊂ℝ²?
• RQ2 What geometric constraints on a subset 𝒜⊂Ω are necessary and sufficient for the existence of such periodic solutions?
• RQ3 How does the interplay between slow center-of-vorticity motion and fast rotational motion contribute to the periodicity of the solution?
• RQ4 Is it possible to achieve any minimal period T∈I(𝒜) for solutions confined to a given subset 𝒜⊂Ω?
• RQ5 To what extent do the results extend beyond integrable cases such as disks and annuli?

Key findings

• Infinitely many periodic solutions with prescribed minimal period T exist for the 2-vortex problem in a generic bounded domain Ω⊂ℝ².
• The solutions are a superposition of slow motion of the center of vorticity along a level line of the Robin function h and fast rotation around the center of vorticity.
• The minimal period T can be any value in an interval I(𝒜)⊂ℝ, which depends on the geometric properties of the subset 𝒜⊂Ω.
• The subset 𝒜⊂Ω must satisfy a specific geometric condition for the existence of such solutions, and such subsets can be found in any generic bounded domain.
• The results are established via a higher-dimensional version of the Poincaré-Birkhoff theorem, extending its applicability to non-integrable, singular Hamiltonian systems.
• The method applies beyond integrable cases such as disks and annuli, significantly broadening the scope of known periodic solution existence.