Formation of singularities in cold ion dynamics

We propose a criterion for the singularity formation of the pressureless Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of cold ions in an electrostatic plasma. Under the proposed criterion, we prove that the smooth solutions develop a $C^1$ blow-up in a finite time and obtain their temporal blow-up rates. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time when the gradient of initial velocity is negatively large. In contrast, our blow-up condition does not require the largeness of gradient of velocity, and our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial.