Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C^{1-\mu }$ regularity for $\mu \in [1\mathrm{/}2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.