On Multidimensional Axisymmetric Oscillations of a Collisional Cold Plasma

TL;DR

This paper investigates how a small friction term affects the stability and smoothness of solutions in multidimensional cold plasma oscillations, showing that friction can prevent blow-up and ensure global stability.

Contribution

It demonstrates that even minimal friction stabilizes solutions of the Euler-Poisson equations, preventing singularities and ensuring global smoothness in multidimensional cold plasma models.

Findings

Small friction stabilizes solutions and prevents blow-up.

Solutions with friction tend to zero as time progresses.

Numerical experiments estimate friction values needed to suppress singularities.

Abstract

We study the influence of the friction term on the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations in many spatial dimensions. It is shown that for any arbitrarily small non-negative constant friction coefficient, there exists a neighborhood of the zero equilibrium in the $C^1$ norm such that the solution of the Cauchy problem with initial data belonging to this neighborhood remains globally smooth in time. Moreover, this solution stabilizes to zero as $t\to\infty$. This result contrasts with the situation of zero friction, where any small deviation from the zero equilibrium generally leads to a blow-up. Our method allows us to estimate the lifetime of smooth solutions. Further, we prove that for any initial data, one can find such coefficient of friction that the respective solution to the Cauchy problem keeps smoothness for all $t>0$ and stabilizes to zero. We also present the results of numerical experiments for physically reasonable situations, which allows us to estimate the value of the friction coefficient, which makes it possible to suppress the formation of singularities of solutions.