Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces II

Abstract

In this paper, we continue to study the blowup problem of the $N$-dimensional compressible Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. In details, we extend the recent result of "M.W. Yuen, \textit{Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces}, Nonlinear Analysis Series A: Theory, Methods & Applications \textbf{74} (2011), 1465--1470.". We could further apply the integration method to obtain the more general results which the 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>0 \end{equation} where an arbitrary constant $n>0$, blow up on or before the finite time $T=2R^{n+2}/(n(n+1)H_{0})$ for pressureless fluids or $\gamma>1.$ The results obtained here fully cover the previous known case for $n=1$.