Motions of vortex patches

Abstract

An evolution equation describing the motion of vortrex patches is established. The existence of steady solutions of this equation is proved. These solutions arem-fold symmetric regions of constant vorticity ω₀ and are uniformly rotating with angular velocity Ω in the range

$$\tilde \Omega _{m - 1}< \tilde \Omega \leqslant \tilde \Omega _m (\tilde \Omega = \Omega /\omega _0 ,m \geqslant 2)$$

where$\tilde \Omega _m = (m - 1)/2m$. We call this class, ofm-fold symmetric rotating regionsD, the class

of them-waves of Kelvin. Any

may be regarded as a simply connected region which is a stationary configuration of the Euler equations in two dimensions. If

then any magnification, rotation or reflection is also in

with the same angular velocity Ω ofD. The angular velocity$\Omega _m = \tilde \Omega _m \omega _0$ corresponds only to the circle solution, which is a trivial member of every class

,m⩾2. The class

corresponds to the rotating ellipses of Kirchoff. Other properties of the class

are established.