Delta-shock for the pressureless Euler-Poisson system

Abstract

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 $C^1$ blow-up, the $L^{\infty }$ 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 $t=T_{\ast }$, the density function is unbounded but is locally integrable with the profile of $\ddot{I}(x,T_{\ast })\sim (x-x_{\ast }{)}^{-2/3}$ near the blow-up point $x=x_{\ast }$. This profile is not yet a Dirac measure. On the other hand, the velocity function has $C^{1/3}$ regularity at the blow-up point. Loosely following our analysis, we also obtain an exact blow-up profile for the pressureless Euler equations.

31 pages, 2 figures. arXiv admin note: text overlap with arXiv:2405.02557