Fluids, geometry, and the onset of Navier–Stokes turbulence in three space dimensions.

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. [ABSTRACT FROM AUTHOR]