Classical Euler flows generate the strong Guderley imploding shock wave

TL;DR

This paper proves that classical, smooth initial data for the Euler equations can generate the Guderley imploding shock wave, demonstrating the formation and evolution of shocks from smooth flows and their self-similar collapse.

Contribution

It establishes the rigorous derivation of the Guderley shock wave from classical initial conditions for the Euler equations, including shock formation and self-similar implosion.

Findings

Guderley's shock arises from smooth, radially symmetric initial data.

A preshock with a $C^{1/3}$ cusp forms before shock development.

The solution matches the Guderley self-similar profile at a finite time.

Abstract

We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal--gas law ($\gamma>1$) 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.