Formation of Singularities in Plasma Ion Dynamics

TL;DR

This paper investigates the conditions under which smooth solutions to the Euler-Poisson system for plasma ion dynamics develop singularities, providing new criteria that do not always require large initial velocity gradients.

Contribution

It establishes novel blow-up criteria for the Euler-Poisson system, including cases where initial velocity gradients are not large, advancing understanding of plasma singularity formation.

Findings

Blow-up criteria for pressureless Euler-Poisson system without large velocity gradients.

Finite-time breakdown of smooth solutions in the isothermal case with large initial Riemann gradients.

Singularity formation can occur even with initially trivial velocity gradients.

Abstract

We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for $C^1$ blow-up of the Euler-Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the presureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave $C^1$ class in a finite time when the gradients of the Riemann functions are initially large.