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 ⊂ R 2 in the form of ω = ∑ j = 1 N ω j χ Ω j , | Ω j | = π r j 2 , ∫ Ω j ω j d μ = μ j ≠ 0 , supported on well-separated vortical domains Ω j , j = 1 , … , 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 , … , μ N and small r 1 , … , r N ∈ 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 ∂ 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 2 N -dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the 2 N -codim directions corresponding to the vortical domain shapes.