Student Analysis Seminar
Classical Euler flows generate the strong Guderley imploding shock wave
Speaker: Giorgio Cialdea, NYU Courant
Location: Warren Weaver Hall 517
Date: Tuesday, October 28, 2025, 11 a.m.
Synopsis:
We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal gas law arises from classical, radially symmetric, shock-free data. For such data prescribed at initial time t<0, we prove that the flow remains classical up to a first singular time, 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. For t>0 the solution continues as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.
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. For t>0 the solution continues as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.
This is a joint work with S. Shkoller and V. Vicol.