Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

TL;DR

This paper studies concentrated steady vorticities in 2D Euler flows, establishing existence and stability of vortex patches near critical points of a Hamiltonian, linking their stability to point vortex dynamics.

Contribution

It introduces a conformal mapping framework for free boundary vortex patches and connects their linear stability to finite-dimensional point vortex models.

Findings

Existence of steady vortex patches near critical points of the Kirchhoff-Routh Hamiltonian.

Linear stability of patches is governed by the stability of corresponding point vortex systems.

Oscillatory boundary shape dynamics are decoupled from stability analysis.

Abstract

We consider concentrated vorticities for the Euler equation on a smooth domain $\Omega \subset \mathbf{R}^2$ in the form of \[ \omega = \sum_{j=1}^N \omega_j \chi_{\Omega_j}, \quad |\Omega_j| = \pi r_j^2, \quad \int_{\Omega_j} \omega_j d\mu =\mu_j \ne 0, \] supported on well-separated vortical domains $\Omega_j$, $j=1, \ldots, N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with $\Omega_j$ being part of unknowns. For any given vorticities $\mu_1, \ldots, \mu_N$ and small $r_1, \ldots, r_N\in \mathbf{R}^+$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff-Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $\partial \Omega_j$, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the $2N$-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the $2N$-codim directions corresponding to the vortical domain shapes.