## Viscous and Magneto Fluid-Dynamics, Torsion Fields, and Brownian Motions Representations on Compact Manifolds and the Random Symplectic Invariants

**Authors:** Diego L. Rapoport

We reintroduce the Riemann-Cartan-Weyl geometries with trace
torsion and their associated Brownian motions on spacetime to extend them to
Brownian motions on the tangent bundle and exterior powers of them. We
characterize the diffusion of differential forms, for the case of manifolds without
boundaries and the smooth boundary case. We present implicit representations
for the Navier-Stokes equations (NS) for an incompressible fluid in a smooth
compact manifold without boundary as well as for the kinematic dynamo equation
(KDE, for short) of magnetohydrodynamics. We derive these representations
from stochastic differential geometry, unifying gauge theoretical structures
and the stochastic analysis on manifolds (the Ito-Elworthy formula for differential
forms. From the diffeomorphism property of the random flow given by
the scalar lagrangian representations for the viscous and magnetized fluids, we
derive the representations for NS and KDE, using the generalized Hamilton and
Ricci random flows (for arbitrary compact manifolds without boundary), and
the gradient diffusion processes (for isometric immersions on Euclidean space of
these manifolds). We solve implicitly this equations in 2D and 3D. Continuing
with this method, we prove that NS and KDE in any dimension other than 1,
can be represented as purely (geometrical) noise processes, with diffusion tensor
depending on the fluid's velocity, and we represent the solutions of NS and KDE
in terms of these processes. We discuss the relations between these representations
and the problem of infinite-time existance of solutions of NS and KDE.
We finally discuss the relations between this approach with the low dimensional
chaotic dynamics describing the asymptotic regime of the solutions of NS. We
present the random symplectic theory for the Brownian motions generated by
these Riemann-Cartan-Weyl geometries, and the associated random Poincare-Cartan
invariants. We apply this to the Navier-Stokes and kinematic dynamo
equations. In the case of 2D and 3D, we solve the Hamiltonian equations.

**Comments:** recovered from sciprint.org

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### Submission history

[v1] 10 Mar 2007

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