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Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review

Departamento Matemática I.S.T. and Grupo de Física-Matemática University, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
Water 2020, 12(3), 864; https://doi.org/10.3390/w12030864
Received: 20 January 2020 / Revised: 9 March 2020 / Accepted: 13 March 2020 / Published: 19 March 2020
(This article belongs to the Special Issue Physical and Mathematical Fluid Mechanics)
We present a stochastic Lagrangian view of fluid dynamics. The velocity solving the deterministic Navier–Stokes equation is regarded as a mean time derivative taken over stochastic Lagrangian paths and the equations of motion are critical points of an associated stochastic action functional involving the kinetic energy computed over random paths. Thus the deterministic Navier–Stokes equation is obtained via a variational principle. The pressure can be regarded as a Lagrange multiplier. The approach is based on Itô’s stochastic calculus. Different related probabilistic methods to study the Navier–Stokes equation are discussed. We also consider Navier–Stokes equations perturbed by random terms, which we derive by means of a variational principle. View Full-Text
Keywords: Navier–Stokes equation; stochastic Lagrangian flows; stochastic variational principles; stochastic geometric mechanics Navier–Stokes equation; stochastic Lagrangian flows; stochastic variational principles; stochastic geometric mechanics
MDPI and ACS Style

Cruzeiro, A.B. Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review. Water 2020, 12, 864.

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