Axisymmetric Solutions of the Euler Equations for Sub-Square Polytropic
Gases
Yuxi Zheng and Tong Zhang
Abstract
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 1 < gamma < 2 and gamma =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.