Converging/diverging self-similar shock waves: from collapse to reflection

Abstract

We solve the continuation problem for the non-isentropic Euler equations following the collapse of an imploding shock wave. More precisely, we prove that the self-similar Güderley imploding shock solutions for a perfect gas with adiabatic exponent ${\hat{I}}^3\in (1,3]$ admit a self-similar extension consisting of two regions of smooth flow separated by an outgoing spherically symmetric shock wave of finite strength. In addition, for ${\hat{I}}^3\in (1,\frac{5}{3}]$, we show that there is a unique choice of shock wave that gives rise to a globally defined self-similar flow with physical state at the spatial origin.

37 pages, 3 figures