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

TL;DR

This paper proves finite-time blowup of classical solutions to radially symmetric Euler and Euler-Poisson equations with repulsive forces, including pressure effects, using a novel integration method, extending previous results to systems with pressure.

Contribution

It introduces a new integration approach to demonstrate blowup in Euler and Euler-Poisson systems with pressure, covering cases previously unhandled by existing methods.

Findings

Solutions blow up before or at time T=R^3/(2H_0).

Results apply to pressureless and pressure-including cases with repulsive forces.

Extends blowup analysis to systems with pressure, not covered by prior work.

Abstract

In this paper, we study the blowup of the $N$-dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant and in the sense which $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition% $H_{0}=\int_{0}^{R}rV_{0}dr>0$ blow up on or before the finite time $T=R^{3}/(2H_{0})$ for pressureless fluids or $\gamma>1.$ The main contribution of this article provides the blowup results of the Euler $(\delta=0)$ or Euler-Poisson $(\delta=1)$ equations with repulsive forces, and with pressure $(\gamma>1)$, as the previous blowup papers (\cite{MUK} \cite{MP}, \cite{P} and \cite{CT}) cannot handle the systems with the pressure term, for $C^{1}$ solutions.