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. Had？i？, 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.
Publication:
SIAM Journal on Mathematical Analysis. Volume. 54, Issue. 3, pp: 3139-3160.
Author:
Feimin Huang
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email: fhuang@amt.ac.cn
Yue Yao
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email: yaoyue17@mails.ucas.ac.cn
地址 北京市海淀区中关村东路55号 思源楼6-7层 南楼5-6、8层 邮编：100190 电子邮箱：iam@amss.ac.cn
