Waiting Time Solutions in gas dynamics

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent $\ga\in(1,3)$ in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least $C^1$ regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only H\"{o}lder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.