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

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 ( ρ , V ) , with compact support in [ 0 , R ] , where R > 0 is a positive constant and in the sense which ρ ( t , r ) = 0 and V ( t , r ) = 0 for r ≥ R , under the initial condition (1) H 0 = ∫ 0 R r V 0 d r > 0 , blow up on or before the finite time T = R 3 / H 0 for pressureless fluids or γ > 1 . The main contribution of this article provides the blowup results of the Euler ( δ = 0 ) or Euler–Poisson ( δ = 1 ) equations with repulsive forces, and with pressure ( γ > 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.