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 $r\ge R$, 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".

arXiv comment12 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, $C^1$ Solutions, No-Slip Boundary Condition, Compact Support, Initial Value Problem, First Boundary Condition

References in corpus (4)

• Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars
• Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces
• A sharp local blow-up condition for Euler-Poisson equations with attractive forcing
• Analytically Periodic Solutions to the 3-dimensional Euler-Poisson Equations of Gaseous Stars with Negative Cosmological Constant

Cited by in corpus (3)

• Blow-up Phenomena for Compressible Euler Equations with Non-vacuum Initial Data
• Blowup of Regular Solutions for the Relativistic Euler-Poisson Equations
• Blowup for $C^2$ Solutions of the N-dimensional Euler-Poisson Equations in Newtonian Cosmology