# On the blow-up of some complex solutions of the 3-d Navier-Stokes Equations: Theoretical Predictions and Computer simulations

## TL;DR

This paper investigates complex solutions to the 3D Navier-Stokes equations that blow up in finite time, combining theoretical analysis and computer simulations to reveal detailed blow-up behaviors and energy concentration phenomena.

## Contribution

It identifies two distinct types of blow-up solutions and provides detailed simulation results showing energy concentration and new features not previously predicted.

## Key findings

- Two types of blow-up solutions with different divergence rates
- Energy and enstrophy concentrate around singular points
- Simulations reveal features beyond current theoretical predictions

## Abstract

We consider some complex-valued solutions of the Navier-Stokes equations in $R^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the "fluid" remains quiet.