Structure of singularities for the Euler-Poisson system of ion dynamics

TL;DR

This paper constructs a detailed description of how smooth solutions to the Euler-Poisson system in plasma physics develop cusp-like singularities in finite time, advancing understanding of blow-up behavior beyond previous limited results.

Contribution

It provides a constructive proof of singularity formation from smooth initial data using self-similar variables and stability estimates, revealing detailed blow-up profiles and regularity loss.

Findings

Solutions develop cusp-type singularities in finite time

Blow-up solutions exhibit $C^1$ blow-up and are $C^{1/3}$ at singularity

Analysis applies to both isothermal and isentropic cases

Abstract

We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for the blow-up and the temporal blow-up rate along the characteristic curve, we rather give a constructive proof of singularity formation from smooth initial data. More specifically, employing the stable blow-up profile of the Burgers equation in the self-similar variables, we establish the global stability estimate in the self-similar time, which yields the asymptotic behavior of blow-up solutions near the singularity point. Our analysis indicates that the smooth solution to the Euler-Poisson system can develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time, while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial data are sufficiently close to the blow-up profile in some weighted $C^4$-topology. We also present a similar result for the isentropic case, and discuss noteworthy differences in the analysis.