The structure of singularities arising in the course of the newtonian collapse of the dust-like matter

Abstract

We re-examine the spherically-symmetric collapse of the nonuniformly distributed dust-like cold matter. The necessary and sufficient condition for the shell-crossing spherical singularity arising is rigorously derived. The system of algebraic equations is obtained which determines the instant of appearance of the singular sphere and its radius. The explicit asymptotic solutions in the Eulerian variables describing the structure of the point-like central and spherical shell-crossing primordial singularities are found. Multiplication of flows arising after splitting the first shell-crossing singularity is investigated numerically.