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

TL;DR

This paper proves that smooth solutions with compact support to pressureless Euler-Poisson equations cannot exist globally in time, using integration and differentiation methods to establish finite-time blowup under certain conditions.

Contribution

It introduces a simplified proof technique for finite-time blowup of solutions to Euler-Poisson equations, extending previous results to attractive force cases.

Findings

No global $C^{2}$ solutions with compact support exist for pressureless Euler-Poisson equations in $R^{N}$.

A new, simpler proof method for blowup results is provided.

Conditions under which solutions blow up in finite time are identified.

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_{\Omega(t)}K\gamma(\gamma-1)\rho^{\gamma-2}(\nabla\rho)^{2}% dx+\int_{\Omega(t)}K\gamma\rho^{\gamma-1}\Delta\rho dx+\epsilon\geq -\delta\alpha(N)M, where $M$ is the mass of the solutions and $| \Omega| $ is the fixed volume of $\Omega(t)$. 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