Axisymmetric Solutions of the Euler Equations for Sub-Square Polytropic Gases

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 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.