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On Multidimensional Axisymmetric Oscillations of a Collisional Cold Plasma

Olga S. Rozanova, Maria I. Delova

TL;DR

This work analyzes axisymmetric multidimensional oscillations of a collisional cold plasma by studying the frictional Euler–Poisson system. By reducing to radial variables and applying the Radon lemma to the derivative dynamics, the authors derive global regularity criteria and asymptotic decay under friction: (i) small initial data yield globally smooth, exponentially decaying solutions for any $\nu>0$ (Theorem 1); (ii) explicit sufficient conditions and blow-up criteria expressible via initial data (Theorem 2); (iii) for sufficiently large friction, $\nu$ can stabilize the system for arbitrary data, yielding global smoothness and decay (Theorem 3). Numerical experiments quantify practical friction values needed to suppress singularity formation, showing that higher spatial dimension lowers the required friction and that physically realistic $\nu$ values still allow substantial safe amplitudes for initial pulses. These results illuminate how collisional damping regularizes nonlinear axisymmetric plasma oscillations and provide a framework for assessing stability in multidimensional settings.

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.

On Multidimensional Axisymmetric Oscillations of a Collisional Cold Plasma

TL;DR

This work analyzes axisymmetric multidimensional oscillations of a collisional cold plasma by studying the frictional Euler–Poisson system. By reducing to radial variables and applying the Radon lemma to the derivative dynamics, the authors derive global regularity criteria and asymptotic decay under friction: (i) small initial data yield globally smooth, exponentially decaying solutions for any (Theorem 1); (ii) explicit sufficient conditions and blow-up criteria expressible via initial data (Theorem 2); (iii) for sufficiently large friction, can stabilize the system for arbitrary data, yielding global smoothness and decay (Theorem 3). Numerical experiments quantify practical friction values needed to suppress singularity formation, showing that higher spatial dimension lowers the required friction and that physically realistic values still allow substantial safe amplitudes for initial pulses. These results illuminate how collisional damping regularizes nonlinear axisymmetric plasma oscillations and provide a framework for assessing stability in multidimensional settings.

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 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 . 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 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.
Paper Structure (10 sections, 8 theorems, 82 equations, 1 figure)

This paper contains 10 sections, 8 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Behavior of solutions along characteristics
  3. Physical constraints on solution components
  4. Boundedness of the solution
  5. Study of the behavior of derivatives
  6. Proof of Theorem \ref{['T1']}
  7. Proof of Theorem \ref{['T21']}
  8. Proof of Theorem \ref{['T22']}
  9. Examples and numerical experiment
  10. Discussion

Key Result

Theorem 1

For arbitrary small $\nu>0$ there exists $\varepsilon(\nu)>0$, such that the solution of the problem 4 with the initial data CD1, satisfying keeps $C^1$ - smoothness for all $t>0.$ Moreover,

Figures (1)

  • Figure 1: The behavior of $Q(t)$ for $d=2$. Left: $a=0.499$, $\nu=0.9315$, $r=0.03$. Right: $a=0.41$, $\nu=0.018$, $r=0.35$

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 2
  • Theorem 4