120 research outputs found

    Constantin and Iyer's representation formula for the Navier--Stokes equations on manifolds

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    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 Rn\mathbb R^n or of Tn\mathbb T^n. 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

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    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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