{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:11:02Z","timestamp":1787321462357,"version":"3.56.0"},"reference-count":52,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>This paper concerns the fluid dynamics modelled by the stochastic flow \\left\\{ \\begin{array}{l} \\boldsymbol{\\dot{\\eta}}\\left( t,x\\right) =\\boldsymbol{u}\\left( t,\\boldsymbol{\\eta} \\left( t,x\\right) \\right) +\\boldsymbol{\\sigma}\\left( t,\\boldsymbol{\\eta}\\left( t,x\\right) \\right) \\circ\\dot{W}, \\\\ \\\\ \\boldsymbol{\\eta}(0,x)=x, \\end{array} \\right. where the turbulent term is driven by the white noise $\\dot{W}$. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler equations for the undetermined components $\\boldsymbol{u}(t,x)$ and $\\boldsymbol{\\sigma}(t,x)$ of the spatial velocity field are derived from the first principles. The resulting equations include as particular cases the deterministic and randomly forced counterparts of these equations.<\/jats:p>\n                  <jats:p>In the second part of the paper, we prove the existence and uniqueness of a strong local solution to the stochastic Navier--Stokes equation in $W_{p}^{1}(\\boldsymbol{R}^{d}),d &gt;1,p &gt; d. In the two-dimensional case, the existence and uniqueness of a global strong solution is shown.<\/jats:p>\n                  <jats:p>In the third part, we deal with the propagation of Wiener chaos by the stochastic Navier--Stokes equation and its relation to statistical moments of the solution.<\/jats:p>","DOI":"10.1137\/s0036141002409167","type":"journal-article","created":{"date-parts":[[2004,2,9]],"date-time":"2004-02-09T21:00:50Z","timestamp":1076360450000},"page":"1250-1310","source":"Crossref","is-referenced-by-count":216,"title":["Stochastic Navier--Stokes Equations for Turbulent Flows"],"prefix":"10.1137","volume":"35","author":[{"given":"R.","family":"Mikulevicius","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B. 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