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 (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

h0 =∫ ^r₀rv₀dr > 0, (1)

blow up on or before the finite time t = r³/h₀ 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, makino and perthame, 1990 , perthame, 1990  and chae and tadmor, 2008) cannot handle the systems with the pressure term, for c1 solutions.