5.
We consider concentrated vorticities for the Euler equation on a smooth domain
in the form of
supported on well-separated vortical domains
,
, of small diameters
. A conformal mapping framework is set up to study this free boundary problem with
being part of unknowns. For any given vorticities
and small
, 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
, 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.
相似文献