Axisymmetric and Self-similar Solutions

Abstract

We solve the Riemann problem for the axisymmetric (including radially symmetric) 2-D polytropic Euler system. The solutions have explicit swirling structures, with finite local energy and vorticity. The key step is the reduction, due to axisymmetry, of the Euler system in the self-similar variables to a system of ordinary differential equations. All main calculations are on the system of ordinary differential equations. The structures of solutions differ substantially for γ = 1, 1 < γ < 2, γ =2, and γ > 2, and for initial data above or below a threshold manifold in the phase space.