Self-similar solutions to the compressible Euler equations and their instabilities.

• Numerical construction of smooth self-similar solutions to the Euler equations. • First construction of smooth linear perturbations. • Numerical evidence that the singularity formation is unstable. • Development of a numerical strategy to solve technical mathematical problems. • Our techniques can be adapted to other PDEs. This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable. [ABSTRACT FROM AUTHOR]