Existence of Steady Symmetric Vortex Patch in a Disk

Abstract

In this paper, we construct a family of symmetric vortex patches for the 2D steady incompressible Euler equations in a disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches “shrink” to a given minimum point of the corresponding Kirchhoff–Routh function as the vorticity strength parameter goes to infinity.