New method for blowup of the Euler-Poisson system (Q2820922)

In this paper a new method for establishing the blowup of \(C^2\) solutions for the Euler-Poisson system NEWLINE\[NEWLINE \rho_t+\nabla\cdot(\rho u)=0,\quad\rho[u_t+(u\cdot\nabla)u]=\delta\rho\nabla\Phi,\quad\Delta\Phi(t,x)=\rho, NEWLINE\]NEWLINE where \(\rho=\rho(t,x)\) and \(u=u(t,x)\in \mathbb R^N\) are the density and the velocity, respectively, of the fluid is provided. Let \(\Omega_{0ij}(x_0)=\frac12[\partial_iu^j(0;x_0)-\partial_ju^i(0,x_0)]\) be the vorticity matrix at \(x_0\in \mathbb R^N\). Theorem 1 states: For the pressureless Euler-Poisson system with \(\rho(0,x_0)> 0\) and \(\Omega_{0ij}(x_0)=0\) at some point \(x_0\), (I) with attractive forces (\(\delta=-1\)), and one of the following conditions satisfied, i.e., (Ia) \(N=1\) or \(N=2\), or (Ib) \(N\geq 3\), satisfying \(\operatorname{div} u(0, x_0)<\sqrt{\frac{2N\rho(0,x_0)}{N-2}}\), or (II) with repulsive forces (\(\delta=1\)), and \(N=1\) and satisfying \(\operatorname{div} u(0, x_0)\leq-\sqrt{2\rho(0,x_0)}\), the \(C^2\) solutions blow up in finite time \(T\). In the proof the generalized Hubble transformation \(\operatorname{div} u(t, x_0(t))= \frac{N\dot{a}(t)}{a(t)}\) is used.