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).