# Spherically symmetric solutions of the multi-dimensional, compressible, isentropic Euler equations

## TL;DR

This paper proves that spherically symmetric solutions to the compressible Euler equations are valid weak solutions in multiple dimensions, using new uniform estimates that improve understanding of solution behavior near the origin.

## Contribution

It introduces new uniform estimates for artificial viscosity approximations, removing previous restrictions and ensuring solutions are valid weak solutions without boundary layer issues.

## Key findings

- Solutions are weak solutions of multi-dimensional Euler equations
- Uniform estimates prevent artificial boundary layer formation
- Results inform on potential blow-up rates at the origin

## Abstract

In this note, we prove that the solutions obtained to the spherically symmetric Euler equations in the recent works [2, 3] are weak solutions of the multi-dimensional compressible Euler equations. This follows from new uniform estimates made on the artificial viscosity approximations up to the origin, removing previous restrictions on the admissible test functions and ruling out formation of an artificial boundary layer at the origin. The uniform estimates may be of independent interest as concerns the possible rate of blow-up of the density and velocity at the origin for spherically symmetric flows.