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 (${\hat{I}}^3>1$) arises from classical, radially symmetric, shock--free data. For such data prescribed at initial time ${\mathrm{T}}_{\mathrm{in}}<0$, we prove that the flow remains smooth up to a first singular time $t={\mathrm{T}}_{\ast }\in ({\mathrm{T}}_{\mathrm{in}},0)$, 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 $t={\mathrm{T}}_{\mathrm{fin}}\in ({\mathrm{T}}_{\ast },0)$ such that on $[{\mathrm{T}}_{\mathrm{fin}},0)$ the solution coincides exactly with the classical Guderley self--similar profile, and at $t={\mathrm{T}}_{\mathrm{fin}}$ 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.

76 pages, 7 figures, minor typos corrected