Blowup for the Euler and Euler-Poisson equations with repulsive forces

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 >= R, under the initial condition H-0 = integral(R)(0) rV(0)dr > 0, (1) blow up on or before the finite time T = R-3/H-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 (Makino et al., 1987 [18], Makino and Perthame, 1990 [19], Perthame, 1990 [20] and Chae and Tadmor, 2008 [24]) cannot handle the systems with the pressure term, for C-1 solutions. (C) 2010 Elsevier Ltd. All rights reserved.