# Charles Bouton and the Navier-Stokes Global Regularity Conjecture

## TL;DR

This paper explores the invariants and self-similar solutions of the Navier-Stokes equations using Bouton's theory, providing insights into criticality, blow-up prevention, and conserved quantities relevant to fluid turbulence.

## Contribution

It applies Bouton's Lie group invariants to derive all self-similar solutions and identifies new conserved quantities in higher-dimensional analyses of the Navier-Stokes equations.

## Key findings

- Derived all self-similar solutions of NSE.
- Used Beale-Kato-Majda theorem to rule out blow-up for certain solutions.
- Found scale-invariant conserved quantities like cavitation number.

## Abstract

This article examines the Bouton-Lie group invariants of the Navier-Stokes equation (NSE) for incompressible fluids. Bouton's theory is applied to the general scaling transformation admitted by the NSE and is used to derive all self-similar solutions. In light of these, the criticality of the standard NSE system is examined and criticality criteria are derived. The theorem of Beale-Kato-Majda is used to rule out blow-up for a subset of Bouton's self-similar solutions. For a subset of Leray's self-similar solutions, the cavitation number of the fluid is found to be a scale-invariant, conserved quantity. By extending the analysis of Bouton to higher-dimensioned manifolds, additional conserved quantities are found, which could further elucidate the physics of fluid turbulence.