Axisymmetric solutions of the two-dimensional Euler equations with a two-constant equation of state

A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional isentropic Euler equations with the equation of state p=A1ργ1+A2ργ2p=A1ργ1+A2ργ2 for initial pure radial velocity. The main difficulty is that the equations cannot be directly reduced to an autonomous system of ordinary differential equations. To deal with it, we first establish the global existence and the detailed structures of solutions to a new system under the axisymmetric and self-similar assumptions.