On the properties of multidimensional electrostatic oscillations of an electron plasma

TL;DR

This paper analyzes the properties of multidimensional electrostatic oscillations in electron plasmas, introducing an iterative method to estimate blow-up times and providing refined conditions for smoothness preservation in symmetric cases.

Contribution

It presents a constructive iteration procedure to estimate blow-up times for 3D electrostatic oscillations, especially in symmetric and low-dimensional cases, advancing understanding of plasma oscillation dynamics.

Findings

Iterative procedure for blow-up time estimation

Refined conditions for smoothness in symmetric cases

Application to numerical data for blow-up time estimation

Abstract

We consider the classical Cauchy problem for a system of equations describing 3D arbitrary electrostatic oscillations of the cold plasma and introduce an iteration procedure that allows estimating the blow-up time from below. This procedure is constructive provided one succeed to obtain a two-sided estimate of an additional quantity depending on the solution. We show that this is possible in the case of one and two dimensions, as well as for solutions with zero vorticity. For the particular case of two-dimensional initial data with the radial symmetry, refined sufficient conditions for destruction and preservation of smoothness in the first period of oscillations are obtained. Moreover, we give the example of estimating the blow-up time for the data for which results of numerics exist and discuss a roughness of our estimate.