Axisymmetric Solutions of the Euler Equations for Square Polytropic
Yuxi Zheng and Tong Zhang
We construct rigorously 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 polytropic gases
with square pressure-density law. 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.
These solutions include the one-parameter family of explicit solutions
reported in a recent article of the authors.