Blowup for the C^1 Solutions of the Euler-Poisson Equations of Gaseous Stars in R^N

Abstract

The Newtonian Euler-Poisson equations with attractive forces are the classical models for the evolution of gaseous stars and galaxies in astrophysics. In this paper, we use the integration method to study the blowup problem of the $N$-dimensional system with adiabatic exponent $\gamma>1$, in radial symmetry. We could show that the $C^{1}$ non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant with $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition \begin{equation} H_{0}=\int_{0}^{R}r^{n}V_{0}dr>\sqrt{\frac{2R^{2n-N+4}M}{n(n+1)(n-N+2)}}% \end{equation} with an arbitrary constant $n>\max(N-2,0),$\newline blow up before a finite time $T$ for pressureless fluids or $\gamma>1.$ Our results could fill some gaps about the blowup phenomena to the classical $C^{1}$ solutions of that attractive system with pressure under the first boundary condition.\newline In addition, the corresponding result for the repulsive systems is also provided. Here our result fully covers the previous case for $n=1$ in "M.W. Yuen, \textit{Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces}, Nonlinear Analysis Series A: Theory, Methods & Applications 74 (2011), 1465--1470".