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

Abstract

We consider concentrated vorticities for the Euler equation on a smooth domain $Ω\subset \mathbf{R}^2$ in the form of \[ ω= \sum_{j=1}^N ω_j χ_{Ω_j}, \quad |Ω_j| = πr_j^2, \quad \int_{Ω_j} ω_j dμ=μ_j \ne 0, \] supported on well-separated vortical domains $Ω_j$, $j=1, \ldots, N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with $Ω_j$ being part of unknowns. For any given vorticities $μ_1, \ldots, μ_N$ 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 $\partial Ω_j$, 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.