Axisymmetric Solutions of the Euler Equations for Square Polytropic Gases

Yuxi Zheng and Tong Zhang

Abstract

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.