Steady vortex patch with polygonal symmetry for the planar Euler equations in a disc

In this paper, we construct two types of vortex patch equilibria for the two-dimensional Euler equations in a disc. The first type is called the “N+1 type” equilibrium, in which a central vortex patch is surrounded by N identical patches with opposite signs, and the other type is called the “2N type” equilibrium, in which the centers of N identical positive patches and N negative patches lie evenly on a circle. The construction is performed by solving a variational problem for the vorticity in which the kinetic energy is maximized subject to some symmetry constraints, and then analyzing the asymptotic behavior as the vorticity strength goes to infinity.