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.
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[v1] 10 Mar 2007
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