Martingale Problem Approach to the Representations of the Navier-Stokes Equations on Smooth Manifolds with Smooth Boundary (original) (raw)
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Random diffeomorphisms and integration of the classical Navier—Stokes equations
Reports on Mathematical Physics, 2002
We derive random implicit representations for the solutions of the classical Navier—Stokes equations for an incompressible viscous fluid. This program is carried out for Riemannian manifolds (without boundary) which are isometrically embedded in a Euclidean space (spheres, tori, n, etc.). Our results appear as an extension to smooth manifolds of the random vortex method of computational fluid dynamics. We derive these representations from gauge-theoretical considerations and the Ito formula for differential forms of stochastic analysis.
Summary: In this article we integrate in closed form, the Navier- Stokes equation for an incompressible ßuid on a compact manifold which is isometrically immersed in Euclidean space. We carry out this integration through the application of the methods of Stochastic Dif- ferential Geometry, i.e. the theory of diusion processes on smooth manifolds. Thus we start by deÞning the invariant inÞnitesimal generators of diusion processes of dierential forms on smooth compact manifolds, in terms of the laplacians (on dierential forms) associated with the Riemann-Cartan-Weyl (RCW) connections. These geometries have a torsion tensor which reduces to a trace 1-form, whose conjugate vector Þeld is the drift of the diusion of scalar Þelds. We construct the diusion processes of dierential forms associated with these laplacians by using the property that the solution ßow of the stochastic dierential equation correspond- ing to the scalar diusion is -under Holder regularity conditions- a (random) die...
Reports on Mathematical Physics, 2002
Extending our previous work 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). 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 existence 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.