Blowup for the C^1 Solutions of the Euler-Poisson Equations of Gaseous Stars in R^N
arXiv:1012.5364 · doi:10.1016/j.jmaa.2011.05.048
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 -dimensional system with adiabatic exponent , in radial symmetry. We could show that the non-trivial classical solutions , with compact support in , where is a positive constant with and for , 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 \newline blow up before a finite time for pressureless fluids or Our results could fill some gaps about the blowup phenomena to the classical 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 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".
12 pages, We merged the result in "M.W. Yuen, Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces II, Pre-print, arXiv:1012.5143" in the updated version. Key Words: Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, Solutions, No-Slip Boundary Condition, Compact Support, Initial Value Problem, First Boundary Condition