The authors construct self-similar solutions of 1D compressible Euler in which a stationary vacuum boundary waits a finite time, then moves with the physical vacuum condition.
On self-similar converging shock waves
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abstract
In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for $\gamma\in (1,3]$. These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.
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Waiting Time Solutions in gas dynamics
The authors construct self-similar solutions of 1D compressible Euler in which a stationary vacuum boundary waits a finite time, then moves with the physical vacuum condition.