Delta-shock for the pressureless Euler-Poisson system

TL;DR

This paper investigates the formation of singularities, specifically delta-shocks, in the pressureless Euler-Poisson system, providing detailed asymptotic profiles and demonstrating how solutions blow up near critical points.

Contribution

It offers a constructive proof of singularity formation, revealing the precise blow-up profile and asymptotic behavior of solutions in the pressureless Euler-Poisson system.

Findings

Density becomes unbounded with a (x-x_*)^{-2/3} profile at blow-up

Velocity exhibits C^{1/3} regularity at the blow-up point

Provides exact blow-up profiles for pressureless Euler equations

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 $\rho(x,T_\ast) \sim (x-x_*)^{-2/3}$ 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.