Blowup of C^2 Solutions for the Euler Equations and Euler-Poisson Equations in R^N

Abstract

In this paper, we use integration method to show that there is no existence of global $C^{2}$ solution with compact support, to the pressureless Euler-Poisson equations with attractive forces in $R^{N}$. And the similar result can be shown, provided that the uniformly bounded functional:% \int_{Ω(t)}Kγ(γ-1)ρ^{γ-2}(\nablaρ)^{2}% dx+\int_{Ω(t)}Kγρ^{γ-1}Δρdx+ε\geq -δα(N)M, where $M$ is the mass of the solutions and $| Ω| $ is the fixed volume of $Ω(t)$. On the other hand, our differentiation method provides a simpler proof to show the blowup result in "D. H. Chae and E. Tadmor, \textit{On the Finite Time Blow-up of the Euler-Poisson Equations in}$R^{N}$, Commun. Math. Sci. \textbf{6} (2008), no. 3, 785--789.". Key Words: Euler Equations, Euler-Poisson Equations, Blowup, Repulsive Forces, Attractive Forces, $C^{2}$ Solutions