# On the implosion of a compressible fluid II: Singularity formation

### Abstract

In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of $\mathcal C^\infty$ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper (part I). All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

## Authors

Frank Merle

CY Cergy Paris Université and IHES, France

Pierre Raphaël

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK

Igor Rodnianski

Department of Mathematics, Princeton University, Princeton, NJ, USA

Jeremie Szeftel

CNRS & Laboratoire Jacques Louis Lions, Sorbonne Université, Paris, France