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 C1 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 \rho (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 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.