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

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 $\mathbf{\text{74}}$ (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 $r\ge R$, 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 $T=2R^{n+2}\mathrm{/}(n(n+1)H_0)$ for pressureless fluids or $\gamma >1.$ The results obtained here fully cover the previous known case for $n=1$.