120 research outputs found
Constantin and Iyer's representation formula for the Navier--Stokes equations on manifolds
The purpose of this paper is to establish a probabilistic representation
formula for the Navier--Stokes equations on compact Riemannian manifolds. Such
a formula has been provided by Constantin and Iyer in the flat case of or of . On a Riemannian manifold, however, there are several
different choices of Laplacian operators acting on vector fields. In this
paper, we shall use the de Rham--Hodge Laplacian operator which seems more
relevant to the probabilistic setting, and adopt Elworthy--Le Jan--Li's idea to
decompose it as a sum of the square of Lie derivatives.Comment: 26 pages. We add Section 4 discussing the Killing vector fields on
Riemannian symmetric spaces which satisfy the conditions in Section
Euler-Lagrangian approach to 3D stochastic Euler equations
3D stochastic Euler equations with a special form of multiplicative noise are
considered. A Constantin-Iyer type representation in Euler-Lagrangian form is
given, based on stochastic characteristics. Local existence and uniqueness of
solutions in suitable Hoelder spaces is proved from the Euler-Lagrangian
formulation
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