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dc.contributor.authorLi, J
dc.contributor.authorTiti, ES
dc.contributor.authorYuan, G
dc.date.accessioned2021-11-25T16:32:39Z
dc.date.available2021-11-25T16:32:39Z
dc.date.issued2022
dc.identifier.issn0022-0396
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/331166
dc.description.abstractIn this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $\varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(\varepsilon^\alpha)$, respectively, with $\alpha>2$, for which the limiting system is the primitive equations with only horizontal viscosity as $\varepsilon$ tends to zero. In particular we show that for "well prepared" initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as $\varepsilon$ tends to zero, and that the convergence rate is of order $O\left(\varepsilon^\frac\beta2\right)$, where $\beta=\min\{\alpha-2,2\}$. Note that this result is different from the case $\alpha=2$ studied in [Li, J.; Titi, E.S.: \emph{The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., \textbf{124} \rm(2019), 30--58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order $O\left(\varepsilon\right)$.
dc.description.sponsorshipThe work of J.L. was supported in part by the National Natural Science Foundation of China (11971009 and 11871005), the Guangdong Basic and Applied Basic Research Foundation (2019A1515011621, 2020B1515310005, 2020B1515310002, and 2021A1515010247), and the Key Project of National Natural Science Foundation of China (12131010). The work of E.S.T. was supported in part by the Einstein Stiftung/Foundation-Berlin, through the Einstein Visiting Fellow Program (EVF-2017-358).
dc.language.isoen
dc.publisherElsevier BV
dc.titleThe primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations
dc.typeArticle
prism.endingPage524
prism.publicationDate2022
prism.publicationNameJournal of Differential Equations
prism.startingPage492
prism.volume306
dc.identifier.doi10.17863/CAM.78613
dc.identifier.doi10.17863/CAM.78613
dcterms.dateAccepted2021-10-26
rioxxterms.versionofrecord10.1016/j.jde.2021.10.048
rioxxterms.versionNA
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.licenseref.startdate2022-01-05
dc.contributor.orcidTiti, ES [0000-0002-5004-1746]
dc.contributor.orcidYuan, G [0000-0003-4811-136X]
dc.identifier.eissn1090-2732
rioxxterms.typeJournal Article/Review
cam.issuedOnline2021-11-04
cam.orpheus.success2022-05-11: embargo success field applied
cam.orpheus.counter5
rioxxterms.freetoread.startdate2023-11-05


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