Stabilization and blow-up in the relativistic model of cold collisional plasma

TL;DR

This paper analyzes how electron-ion collisions affect the stability and blow-up behavior of solutions in a relativistic cold plasma model, providing analytical estimates and numerical validation of different scenarios.

Contribution

It offers the first analytical estimates for the lifespan of smooth solutions and the conditions under which collisions suppress blow-up in the relativistic plasma model.

Findings

Collisions can prevent blow-up of small oscillations.

The solution's lifespan can be estimated analytically.

Numerical experiments confirm theoretical predictions.

Abstract

We study the influence of the factor of electron-ion collisions on the solution of the Cauchy problem in the one-dimensional relativistic model of cold plasma and show that, depending on their intensity and initial data, two scenarios are possible: either the solution remains smooth and stabilizes to a stationary state, or during a finite time the oscillations blowup. In contrast to the nonrelativistic model, when exact conditions can be obtained separating the two behaviors, in a much more complicated relativistic situation, it turns out to be possible to analytically estimate from below the time during which the existence of a smooth solution and the guaranteed number of oscillations during this time. In addition, we show that in contrast to the relativistic case without taking into account collisions, when oscillations corresponding to arbitrarily small deviations from the zero equilibrium position blow up, the presence of electron collisions can suppress the blow-up of sufficiently small oscillations. Further, based on the analysis of characteristics, a numerical algorithm is constructed, the order of accuracy of which is determined only by the smoothness of the initial data. Numerical experiments are presented to illustrate the theoretical results. The initial conditions are chosen as reasonably as possible from the point of view of full-scale physical experiments.