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

Gui-Qiang G. Chen; Marshall Slemrod; Dehua Wang

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arXiv · 1612.08757

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

Gui-Qiang G. Chen·Marshall Slemrod·Dehua Wang

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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.

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BibTeX RIS

Gui-Qiang G. Chen, Marshall Slemrod, Dehua Wang. 2016-12-27. Fluids, Geometry, and the Onset of Navier-Stokes Turbulence in Three Space Dimensions. https://arxiv.org/abs/1612.08757

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