Delta-shock for the pressureless Euler-Poisson system

We study singularity formation for the pressureless Euler-Poisson system of cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when its smooth solutions experience C¹ blow-up, the L^∞ norm of the density becomes unbounded, which is often referred to as a delta-shock. We provide a constructive proof of singularity formation to obtain an exact blow-up profile and the detailed asymptotic behavior of the solutions near the blow-up point in both time and space. Our result indicates that at the blow-up time t=T_, the density function is unbounded but is locally integrable with the profile of ρ(x,T₎ ~ (x-x_*)-2/3 near the blow-up point x=x_. This profile is not yet a Dirac measure. On the other hand, the velocity function has C1/3 regularity at the blow-up point. Loosely following our analysis, we also obtain an exact blow-up profile for the pressureless Euler equations.