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A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations
Author(s) -
Constantin Peter,
Iyer Gautam
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20192
Subject(s) - inviscid flow , mathematics , euler equations , navier–stokes equations , nonlinear system , mathematical analysis , compressibility , representation (politics) , burgers' equation , non dimensionalization and scaling of the navier–stokes equations , classical mechanics , partial differential equation , physics , mechanics , politics , political science , law , quantum mechanics
In this paper we derive a probabilistic representation of the deterministic three‐dimensional Navier‐Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self‐contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic‐type equations, including viscous Burgers equations and Lagrangian‐averaged Navier‐Stokes alpha models. © 2007 Wiley Periodicals, Inc.