Linearization method and sharp thresholds for spherically symmetric multidimensional pressureless Euler-Poisson equations

TL;DR

This paper develops a linearization approach to determine when singularities form in radially symmetric solutions of multidimensional pressureless Euler-Poisson equations, providing criteria and numerical methods for analyzing solution smoothness.

Contribution

It introduces a linearization method that simplifies the singularity formation criterion for these equations and offers a numerical procedure to assess solution smoothness.

Findings

Singularity formation criteria can be reduced to linear ODE solutions.

Initial data can determine the occurrence of singularities in some cases.

A numerical method is proposed for cases where explicit criteria are unavailable.

Abstract

We show that the question about the criterion of a singularity formation for radially symmetric solutions to the Cauchy problem for a fairly wide class of equations related to the pressureless Euler-Poisson equations can be reduced to the study of solutions to a linear homogeneous ordinary differential equation. In some cases, such a criterion can be obtained in terms of the initial data. In the remaining cases, it is possible to construct a simple numerical procedure, on the basis of which the question about preserving smoothness for any set of initial data can be solved.