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 ℝ n or of 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.
KeywordsNavier–Stokes equations Stochastic representation de Rham–Hodge Laplacian Stochastic flow Pull-back vector field
Mathematics Subject Classification (2010)35Q30 58J65
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The two authors would like to thank D. Elworthy for his interest on this work, and for drawing their attentions to Riemannian symmetric spaces. The second author is grateful to the financial supports of the National Natural Science Foundation of China (Nos. 11431014, 11571347), the Seven Main Directions (Y129161ZZ1) and the Special Talent Program of the Academy of Mathematics and Systems Science, Chinese Academy of Sciences.
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