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Stochastic Differential Geometry and the Random Flows of Viscous and Magnetized Fluids in Smooth Manifolds and Eulcidean Space

Diego L. Rapoport
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Abstract

We integrate in closed implicit form the Navier-Stokes equations for an incompressible fluid and the kinematical dynamo equation, in smooth manifolds and Euclidean space. This integration is carried out by applying Stochastic Differential Geometry, i.e. the gauge-theoretical formulation of Brownian motions. Non-Riemannian geometries with torsion of the trace-type are found to have a fundamental role. We prove that in any dimension other than 1, the Navier-Stokes equations can be represented as a purely diffusive process, while we can also give a random lagrangian representation for the diffusion of vorticity and velocity in terms of the non-Riemannian geometry.

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Diego L. Rapoport. 2000-12-15. Stochastic Differential Geometry and the Random Flows of Viscous and Magnetized Fluids in Smooth Manifolds and Eulcidean Space. https://arxiv.org/abs/math-ph/0012032

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