Continued Gravitational Collapse for Gaseous Star and Pressureless Euler--Poisson System

Abstract

The gravitational collapse of an isolated self-gravitating gaseous star for $\gamma$-law pressure $p(\rho)=\rho^\gamma$ ($1<\gamma<\frac43$) in the mass-supcritical case is investigated. It was first shown in [Y. Guo, M. Hadžić, and J. Jang, Arch. Rational Mech. Anal., 239 (2021), pp. 431--552]. that there exists a kind of continued gravitational collapse, and the collapse is based on a special solution of the pressureless Euler--Poisson system. In this paper, all spherically symmetric solutions of the pressureless Euler--Poisson system are classified. Precisely speaking, for fixed radius $r$, there exists a unique critical velocity $v^*(r)>0$ depending on the mean density in the ball $B(0,r)$ for the pressureless Euler--Poisson system such that if the initial velocity $\chi_1(r)\geq v^*(r)$ (escape case), then the dust runs away from the gravitational force forever along an escape trajectory, and if the initial velocity $\chi_1(r)< v^*(r)$ (collapse case), then the dust collapses at the origin in a finite time $t^*(r)$ even if it expands initially, i.e., $\chi_1(r)>0$. Moreover, it is proved that there exists a class of spherically symmetric solutions of a gaseous star, which formulates a continued gravitational collapse in finite time, based on the background of the pressureless solutions if $\chi_1(r)< v^*(r)$ for all $r\in[0,1]$. It is noted that $\chi_1(r)$ could be positive; that is, the star might expand initially but finally collapse.