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

In this paper, we study the blowup of the NN-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)(ρ,V), with compact support in [0,R][0,R], where R>0R>0 is a positive constant and in the sense which ρ(t,r)=0ρ(t,r)=0 and V(t,r)=0V(t,r)=0 for r≥Rr≥R, under the initial condition equation(1)H0=∫0RrV0dr>0, blow up on or before the finite time T=R3/H0T=R3/H0 for pressureless fluids or γ>1γ>1.The main contribution of this article provides the blowup results of the Euler (δ=0)(δ=0) or Euler–Poisson (δ=1)(δ=1) equations with repulsive forces, and with pressure (γ>1)(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 C1C1 solutions.