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

TL;DR

This paper extends previous results on finite-time blowup of solutions to the radial compressible Euler and Euler-Poisson equations with repulsive forces, applying an integration method to more general initial conditions.

Contribution

It generalizes blowup conditions for classical solutions with compact support, covering cases with arbitrary positive n, and extends prior work for n=1.

Findings

Solutions blow up before a finite time depending on initial conditions.

Results include pressureless fluids and gamma > 1 cases.

Generalizes previous blowup criteria to broader initial data.

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 $n=1$.