On the formation of shocks to the compressible Euler equations

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.

Keywords

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

2010 Mathematics Subject Classification

35B30, 35Q35

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Published 1 January 2009