Exact spiral solutions of the two-dimensional Euler equations

We construct a two-parameter family of self-similar solutions to both the compressible and incompressible two-dimensional Euler equations with axisymmetry. The equations can be reduced under the situation to two systems of ordinary differential equations. In the compressible and polytropic case, the system in autonomous form consists of four ordinary differential equations with a two-dimensional set of stationary points, one of which is degenerate up to order four. Through asymptotic analysis and computations of numerical solutions, we are fortunate to be able to recognize a one-parameter family of exact solutions in explicit form. All the solutions (exact or numerical) are globally bounded and continuous, have finite local energy and vorticity, and have well-defined initial and boundary values at time zero and spatial infinity respectively. Particle trajectories of some of these solutions are spiral-like. In the incompressible case, we also find explicit self-similar axisymmetric spiral solutions, which are, however, somewhat less physical due to unbounded pressures or infinite local energy near their swirling centers.