Fluids, Geometry, and the Onset of Navier-Stokes Turbulence in Three Space Dimensions

TL;DR

This paper develops a geometric theory linking fluid dynamics to Riemannian geometry, providing insights into turbulence onset in Navier-Stokes equations through evolving metrics and wild manifolds.

Contribution

It introduces a novel geometric framework for analyzing fluid evolution and turbulence onset, applying Nash-Kuiper theorem to Navier-Stokes equations.

Findings

Computed critical initial data for turbulence onset.

Established a geometric approach to fluid dynamics.

Connected Riemannian metrics with turbulence phenomena.

Abstract

A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving $C^{1}$ manifold. The theory is applied to the incompressible Euler and Navier-Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier-Stokes equations.