{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T23:05:57Z","timestamp":1774479957769,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:00:00Z","timestamp":1584576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Water"],"abstract":"<jats:p>We present a stochastic Lagrangian view of fluid dynamics. The velocity solving the deterministic Navier\u2013Stokes 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\u2013Stokes equation is obtained via a variational principle. The pressure can be regarded as a Lagrange multiplier. The approach is based on It\u00f4\u2019s stochastic calculus. Different related probabilistic methods to study the Navier\u2013Stokes equation are discussed. We also consider Navier\u2013Stokes equations perturbed by random terms, which we derive by means of a variational principle.<\/jats:p>","DOI":"10.3390\/w12030864","type":"journal-article","created":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T07:29:07Z","timestamp":1584689347000},"page":"864","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5256-1174","authenticated-orcid":false,"given":"Ana Bela","family":"Cruzeiro","sequence":"first","affiliation":[{"name":"Departamento Matem\u00e1tica I.S.T. and Grupo de F\u00edsica-Matem\u00e1tica University, Av. Rovisco Pais, 1049-001 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,19]]},"reference":[{"key":"ref_1","unstructured":"Marchioro, C., and Pulvirenti, M. (1984). 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