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A sufficient condition for the existence of smooth solutions of the relativistic cold plasma equations on any given time interval

Olga S. Rozanova, Evgeniy V. Chizhonkov

TL;DR

This work addresses ensuring smooth solutions for the one‑dimensional relativistic cold plasma equations on a prescribed time interval by deriving a data‑dependent sufficient condition that involves the initial momentum and electric field derivatives. The authors linearize an extended characteristic system via the Radon lemma and bound the onset of blow‑up by tracking a vanishing denominator, yielding a computable inequality in terms of $C(\rho)$, $K_-$, and $p_0,e_0$. The main contribution is a practical criterion that extends the nonrelativistic result to the relativistic setting and quantifies how initial data, beyond their size, influence solution longevity. They also provide asymptotic estimates for small disturbances and validate the theory through numerical experiments that reveal the blow‑up structure: the electric field forms a step while the momentum develops a cusp near the density singularity. This work thus offers a concrete tool for predicting and controlling the smooth behavior of relativistic plasma oscillations in both theory and application.

Abstract

In terms of initial data, a sufficient condition for the smoothness of the solution to the Cauchy problem for one-dimensional relativistic cold plasma equations over any given time interval is found. Unlike the non-relativistic case, such sufficient conditions take into account the smallness properties of not only the derivatives of the initial data but also the initial data themselves. The accuracy of the obtained initial condition is investigated using a numerical experiment. The structure of the emerging singularities is also studied.

A sufficient condition for the existence of smooth solutions of the relativistic cold plasma equations on any given time interval

TL;DR

This work addresses ensuring smooth solutions for the one‑dimensional relativistic cold plasma equations on a prescribed time interval by deriving a data‑dependent sufficient condition that involves the initial momentum and electric field derivatives. The authors linearize an extended characteristic system via the Radon lemma and bound the onset of blow‑up by tracking a vanishing denominator, yielding a computable inequality in terms of , , and . The main contribution is a practical criterion that extends the nonrelativistic result to the relativistic setting and quantifies how initial data, beyond their size, influence solution longevity. They also provide asymptotic estimates for small disturbances and validate the theory through numerical experiments that reveal the blow‑up structure: the electric field forms a step while the momentum develops a cusp near the density singularity. This work thus offers a concrete tool for predicting and controlling the smooth behavior of relativistic plasma oscillations in both theory and application.

Abstract

In terms of initial data, a sufficient condition for the smoothness of the solution to the Cauchy problem for one-dimensional relativistic cold plasma equations over any given time interval is found. Unlike the non-relativistic case, such sufficient conditions take into account the smallness properties of not only the derivatives of the initial data but also the initial data themselves. The accuracy of the obtained initial condition is investigated using a numerical experiment. The structure of the emerging singularities is also studied.
Paper Structure (9 sections, 2 theorems, 46 equations, 7 figures)

This paper contains 9 sections, 2 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem and the main result
  3. The proof of Theorem \ref{['T1']} and further theoretical analysis
  4. The proof of Theorem \ref{['T1']}
  5. Comments to Theorem \ref{['T1']}
  6. Asymptotic estimate of the blow-up time for small initial disturbances
  7. Numerical experiment
  8. Qualitative properties of the solution near the blow-up point
  9. Conclusion

Key Result

Theorem 1

Let the initial data cd1 belong to the class $C^1(\mathbb R)$ and for these data $C(\rho)$ is not identically constant. The condition where $p_0=P_0'$, $e_0=E_0'$, ensures that the solution to problem u1, cd1, remains smooth at least until time $t_*> n \pi$, $n\in \mathbb N$. In other words, if we need to ensure the smoothness of the solution of the problem u1, cd1 up to any time $T>0$, we need t

Figures (7)

  • Figure 1: Limitation of the integral curve on the $(q,p)$ plane. In the first step of the evaluation, the upper ($L_-$) and lower ($L_+$) limiters are shown, and the integral curve itself is highlighted by a dash line.
  • Figure 2: Dynamics of the sufficient condition \ref{['condn']} for $n=1$: the solid line represents the value on the left side of the condition as a function of the initial time, and the dotted line represents the background zero value.
  • Figure 3: Dynamics of the sufficient condition \ref{['condn']} for $n=2$: the solid line denotes the values of the expression on the left-hand side of \ref{['condn']} as a function of time, and the dotted line denotes the background zero value.
  • Figure 4: Electron density dynamics: maximum over the region (solid line) and at the origin (dashed line).
  • Figure 5: Spatial distribution of momentum and electric field at the moment of blow-up.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2: The Radon lemma