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Fluids, geometry, and the onset of Navier-Stokes turbulence in three space dimensions. (English) Zbl 1398.76060

Summary: 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.

MSC:

76F02 Fundamentals of turbulence
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76F10 Shear flows and turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
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References:

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