Abstract:We study the global-in-time continuation, past the singularity, of the smooth, non-isentropic, radially symmetric imploding solutions of the compressible Euler equations recently constructed by Chen, Shkoller, and Vicol. In three space dimensions, for all physically relevant adiabatic exponents $\gamma>1$, we consider the Euler solution that evolves smoothly until an implosion singularity forms at the origin at time $t=0$. We then prove that this solution can be uniquely continued for $t>0$ as a reflected outward-propagating shock, sometimes called a reflected blast wave. For $t>0$, the continuation is a globally forward self-similar weak solution of the Euler equations, selected by the Rankine--Hugoniot conditions and the Lax entropy inequality; it is smooth away from the expanding shock sphere and the spatial origin. The structure at the center of symmetry distinguishes these explosions from the classical Guderley reflected shock. In Guderley's continuation, the reflected blast wave leaves a point vacuum at the origin, where the density vanishes. The solutions constructed here exhibit the opposite behavior: for every fixed $t>0$ the density is unbounded at $r=0$ (though it remains locally integrable), while the pressure stays bounded and the temperature vanishes there.
From: Giorgio Cialdea [view email]
[v1]
Tue, 16 Jun 2026 16:52:06 UTC (663 KB)
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