Self-similar gravitational collapse

We discuss recent mathematical constructions of self-similar gravitational collapse for Newtonian stars governed by the Euler-Poisson system, known as Larson-Penston solutions for the isothermal stars and Yahil solutions for polytropic stars, and thereby show the existence of smooth initial data that lead to finite time collapse, characterized by the blow-up of the star density. If time permits, we will also discuss the relativistic Larson-Penston solutions and formation of naked singularities for the Einstein-Euler system.