# Computer simulations for the blow-up of complex solutions of the 3-d Navier-Stokes equations

## TL;DR

This paper uses computer simulations to analyze the blow-up behavior of complex solutions to the 3D Navier-Stokes equations, revealing energy concentration and singularity formation near finite-time blow-up.

## Contribution

It provides detailed computational insights into the blow-up mechanism of complex solutions of the 3D Navier-Stokes equations, supporting theoretical predictions.

## Key findings

- Energy and enstrophy concentrate near blow-up points
- Solutions exhibit singularity formation at finite time
- Outside the blow-up region, the fluid remains relatively calm

## Abstract

We present a study by computer simulations of a class of complex-valued solutions of the three-dimensional Navier-Stokes equations in the whole space, which, according to Li and Sinai, present a blow-up (singularity) at a finite time. The computer results allow a detailed study of the blow-up mechanism, and show interesting features of the behavior of the solutions near the blow-up time, such as the concentration of energy and enstrophy in a small region around a few points of physical space, while outside this region the "fluid" remains "quiet".