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Shizan Fang - One of the best experts on this subject based on the ideXlab platform.
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geometry on the wasserstein space over a Compact Riemannian Manifold
arXiv: Mathematical Physics, 2021Co-Authors: Hao Ding, Shizan FangAbstract:We will revisit the intrinsic differential geometry of the Wasserstein space over a Riemannian Manifold, due to a series of papers by Otto, Villani, Lott, Ambrosio, Gigli, Savare and so on.
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Nash Embedding, Shape Operator and Navier-Stokes Equation on a Riemannian Manifold
Acta Mathematicae Applicatae Sinica English Series, 2020Co-Authors: Shizan FangAbstract:What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian Manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden’s Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general Compact Riemannian Manifold is obtained when de Rham-Hodge Laplacian is involved.
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Generalized stochastic Lagrangian paths for the Navier-Stokes equation
Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 2018Co-Authors: Marc Arnaudon, Ana Bela Cruzeiro, Shizan FangAbstract:In the note added in proof of the seminal paper [Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a Compact Riemannian Manifold. In the spirit of Brenier's generalized flows for the Euler equation, we introduce a class of semimartingales on a Compact Riemannian Manifold. We prove that these semimartingales are critical points to the corresponding kinetic energy if and only if its drift term solves weakly the Navier-Stokes equation defined with Ebin-Marsden's Laplacian. We also show that for the torus case, classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.
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a note on constantin and iyer s representation formula for the navier stokes equations
arXiv: Probability, 2015Co-Authors: Shizan FangAbstract:The purpose of this note is to establish a probabilistic representation formula for Navier–Stokes equations on a Compact Riemannian Manifold. To this end, we first give a geometric interpretation of Constantin and Iyer's representation formula for the Navier–Stokes equation, then extend it to a Compact Riemannian Manifold. We shall use Elworthy–Le Jan–Li's idea to decompose de Rham–Hodge Laplacian operator on a Manifold as a sum of the square of vector fields. MSC 2010: 35Q30, 58J65 Keywords: Navier–Stokes equations, stochastic representation, de Rham–Hodge Lapla-cian, stochastic flow, pull-back vector field
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stochastic analysis on the path space of a Riemannian Manifold i markovian stochastic calculus
Journal of Functional Analysis, 1993Co-Authors: Shizan Fang, P MalliavinAbstract:Abstract On the Brownian flow of a Compact Riemannian Manifold, an intrinsic stochastic calculus is defined. This calculus is Markovian. The stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool. The Bismut connection plays a paramount role in the final construction.
Fuzhou Gong - One of the best experts on this subject based on the ideXlab platform.
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the log sobolev inequality on loop space over a Compact Riemannian Manifold
Journal of Functional Analysis, 1998Co-Authors: Fuzhou GongAbstract:Abstract We obtain a log-Sobolev inequality with a neat and explicit potential for the gradient on a based loop space over a Compact Riemannian Manifold. The potential term relies only on the curvature of the Manifold and the Hessian of the heat kernel, and is L p -integrable for all p ⩾1. The log-Sobolev inequality is derived by a martingale representation theorem for the differentiable functions on loop space, which is a variation of the Clark–Ocone–Haussmann formula.
John Lott - One of the best experts on this subject based on the ideXlab platform.
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Eigenvalue estimates and differential form Laplacians on Alexandrov spaces
arXiv: Differential Geometry, 2017Co-Authors: John LottAbstract:We give upper bounds on the eigenvalues of the differential form Laplacian on a Compact Riemannian Manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a Compact resolvent. We identify its kernel with an intersection homology group.
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Some Geometric Calculations on Wasserstein Space
Communications in Mathematical Physics, 2007Co-Authors: John LottAbstract:We compute the Riemannian connection and curvature for the Wasserstein space of a smooth Compact Riemannian Manifold.
Xiangao Liu - One of the best experts on this subject based on the ideXlab platform.
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partial regularity for weak heat flows into a general Compact Riemannian Manifold
Archive for Rational Mechanics and Analysis, 2003Co-Authors: Xiangao LiuAbstract:In this paper the partial regularity of the weak heat flow of harmonic maps from a Riemannian Manifold M into a general Compact Riemannian Manifold N without boundary is considered. Partial results have been obtained for target Manifolds that are spheres [12, 4] or homogeneous spaces [6]. The proofs in these special cases relied heavily on the geometry of these Manifolds, and cannot be applied to the general case. We prove in this article that the singular set Sing(u) of the stationary weak heat flow satisfies H n ρ (Sing(u))=0, with n=dimension M, where H n ρ is the Hausdorff measure with respect to parabolic metric .
Emmanuel Humbert - One of the best experts on this subject based on the ideXlab platform.
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a limit equation associated to the solvability of the vacuum einstein constraint equations by using the conformal method
Duke Mathematical Journal, 2012Co-Authors: Mattias Dahl, Romain Gicquaud, Emmanuel HumbertAbstract:Let (M, g) be a Compact Riemannian Manifold on which a trace-free and divergence-free sigma is an element of W-1,W-p and a positive function tau is an element of W-1,W-p, p > n are fixed. In thi ...
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Positive mass theorem for the Paneitz-Branson operator
Calculus of Variations and Partial Differential Equations, 2009Co-Authors: Emmanuel Humbert, Simon RaulotAbstract:We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a Compact Riemannian Manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin Manifolds by Ammann-Humbert.
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positive mass theorem for the paneitz branson operator
Calculus of Variations and Partial Differential Equations, 2009Co-Authors: Emmanuel Humbert, Simon RaulotAbstract:We prove that under suitable assumptions, the constant term in the Green function of the Paneitz–Branson operator on a Compact Riemannian Manifold (M, g) is positive unless (M, g) is conformally diffeomorphic to the standard sphere. The proof is inspired by the positive mass theorem on spin Manifolds by Ammann and Humbert (Geom Func Anal 15(3):567–576, 2005 [1]).