  The aim of this note is to present the recent results in [Buckmaster,
Cao-Labora, G\'omez-Serrano, arXiv:2208.09445, 2022], concerning the existence
of "imploding singularities" for the 3D isentropic compressible Euler and
Navier-Stokes equations. Our work builds upon the pioneering work of Merle,
Rapha\"el, Rodnianski, and Szeftel [Merle, Rapha\"el, Rodnianski, and Szeftel,
Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022,
Invent. Math., 227(1):247-413, 2022] and proves the existence of self-similar
profiles for all adiabatic exponents $\gamma>1$ in the case of Euler; as well
as proving asymptotic self-similar blow-up for $\gamma=\frac75$ in the case of
Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is
constructed to have density bounded away from zero and constant at infinity,
the first example of blow-up in such a setting. For simplicity, we will focus
our exposition on the compressible Euler equations.
