We present an explicit upper-bound for the life-span of C3-smooth solutions to the multi-dimensional compressible Euler equations with a certain class of initial data containing compression vacuum states. We also show that the divergence of a fluid velocity will blow up along the particle trajectories issued from the compression vacuum states, which represent a shock formation at the vacuum states.
"On the formation of shocks to the compressible Euler equations." Commun. Math. Sci. 7 (3) 627 - 634, September 2009.