Explicit and exact swirling solutions of the 2-D Euler equations

We consider a Riemann-type initial value problem for the compressible and polytropic two-dimensional Euler equations. The initial states are such that the flows have constant densities, and all fluid particles rotate clockwise (or counter clockwise) around the origin in circles with a constant speed. The equations are reduced to a system of ordinary differential equations for self-similar and axisymmetric solutions. We have established the existence of a two-parameter family of such solutions. All these solutions are globally bounded and continuous, have finite local energy and vorticity with finite boundary values at infinity. A one-parameter family of such solutions is in explicit form. The incompressible Euler equations with similar initial data are also considered.