Smooth self-similar imploding profiles to 3D compressible Euler

TL;DR

This paper demonstrates the existence of smooth, self-similar imploding profiles leading to singularities in 3D compressible Euler equations, and constructs a novel blow-up solution for Navier-Stokes with bounded density.

Contribution

It proves the existence of self-similar imploding profiles for all adiabatic exponents in Euler and constructs a new blow-up solution for Navier-Stokes with bounded density.

Findings

Existence of self-similar profiles for Euler with all γ>1

Asymptotic self-similar blow-up for Navier-Stokes at γ=7/5

First example of Navier-Stokes blow-up with density bounded away from zero

Abstract

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.