Radon measure solutions of spherically symmetric isentropic compressible Euler equations

We consider the well-posedness of a class of initial-boundary value problems of the spherically symmetric isentropic compressible Euler equations for polytropic gas. We find that a singularity always occurs on the sonic circle near the origin, even in the case of steady flow, regardless of whether the speed of the incoming flow at far field is subsonic or supersonic. This inherent difficulty makes it impossible to extend the solution towards the origin pass across the sonic circle in the sense of classical solutions. To address the singularity, our idea is to fix a delta-wave on the sonic circle in the framework of the Radon measure solution. As a result, we obtain the existence of the Radon measure solution to the initial-boundary value problem and prove that a concentration will form on the sonic circle. More importantly, we deduce an exact formula for pressure distribution on the sonic circle. In particular, there is a concentration at the origin as the Mach number of incoming flow goes to infinity, which corresponds to the case of pressureless gas. Additionally, we investigate the case of Chaplygin gas.