Communications in Mathematical Sciences

Volume 7 (2009)

Number 3

On the formation of shocks to the compressible Euler equations

Pages: 627 – 634



Dongho Chae

Seung-Yeal Ha


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.


Compressible Euler equations, blow-up, shock wave, particle trajectory, vacuum boundary

2010 Mathematics Subject Classification

35B30, 35Q35

Published 1 January 2009