Singularity formation for the supersonic inward wave of compressible Euler equations with radial symmetry

In this paper, we consider the singularity formation of smooth solutions for the compressible radially symmetric Euler equations. By applying the characteristic method and the invariant domain idea, we show that, for polytropic ideal gases with $\gamma \ge 3$, the smooth solution develops a singularity in finite time for a class of initial supersonic inward waves.