  We establish rigorously the existence of a three-parameter family of
self-similar,globally bounded, and continuous weak solutions in two space
dimensions to the compressible Euler equations with axisymmetry for gamma-law
polytropic gases with gamma between 1 and 2, including 1. The initial data of
these solutions have constant densities and outward-swirling velocities. We use
the axisymmetry and self-similarity assumptions to reduce the equations to a
system of three ordinary differential equations, from which we obtain detailed
structures of solutions besides their existence. These solutions exhibit
familiar structures seen in hurricanes and tornadoes. They all have finite
local energy and vorticity with well-defined initial and boundary values.
