Classical Euler flows generate the strong Guderley imploding shock wave

Abstract

We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal--gas law ($γ>1$) arises from classical, radially symmetric, shock--free data. For such data prescribed at initial time $\mathrm{T_{in}} < 0$, we prove that the flow remains smooth up to a first singular time $t=\mathrm{T}_* \in (\mathrm{T_{in}}, 0)$, where a preshock forms with a $C^{1/3}$ cusp in the fast acoustic variable. From this preshock a unique, initially weak, regular shock is born, whose strength can be made arbitrarily large on a controlled time interval; the front then deforms onto the Guderley shock and implodes at the origin at the collapse time $t=0$. There exists a matching time $t=\mathrm{T_{fin}} \in (\mathrm{T}_*,0)$ such that on $[\mathrm{T_{fin}},0)$ the solution coincides exactly with the classical Guderley self--similar profile, and at $t=\mathrm{T_{fin}}$ the shock trajectory matches the self--similar front to all orders. As $t \to 0^-$, the Euler solution implodes at the center, and continues for $t>0$ as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.