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Abstract In the supercritical range of the polytropic indices$$\gamma \in (1,\frac{4}{3})$$ $\gamma \in (1,\frac{4}{3})$we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case$$\gamma =1$$ $\gamma =1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof. 

Abstract Upon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960’s Zel’dovich (Voprosy Kosmogonii 9:157–170, 1963) and Harrison et al. (Gravitation Theory and Gravitational Collapse. The University of Chicago press, Chicago, 1965) formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity. 

Abstract Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state$$p=k\varrho $$ $p=k\varrho$,$$k>0$$ $k>0$, and subject to Newtonian gravity. We rigorously prove the existence of such a Larson–Penston solution. 

Abstract We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein–Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein–Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift$$\kappa >0$$ $\kappa >0$. We prove their linear instability when$$\kappa $$ $\kappa$is sufficiently large, i.e., when they are strongly relativistic, and prove that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel’dovich & Podurets (for the Einstein–Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein–Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein–Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.