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

Yuxi Zheng and Tong Zhang

Abstract

We prove constructively 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 > 2. 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. The case gamma = 2, which is a bifurcation point of the system, and the cases gamma < 2 have been studied in our companion papers. A noticeable difference is that all solutions in the case gamma > 2 have nontrivial cavities at the center. These solutions exhibit familiar structures seen in hurricanes. All of the solutions have finite local energy and vorticity with well-defined initial and boundary values.