SMOOTH SELF-SIMILAR IMPLODING PROFILES TO 3D COMPRESSIBLE EULER

The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gomez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint 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, Raphael, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247-413; Ann. of Math. (2) 196 (2022), pp. 567-778; Ann. of Math. (2) 196 (2022), pp. 779-889] 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 = 7/5 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.