dc.contributor.author Li, J dc.contributor.author Titi, ES dc.contributor.author Yuan, G dc.date.accessioned 2021-11-25T16:32:39Z dc.date.available 2021-11-25T16:32:39Z dc.date.issued 2022 dc.identifier.issn 0022-0396 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/331166 dc.description.abstract In 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.sponsorship The 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.iso en dc.publisher Elsevier BV dc.title The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations dc.type Article prism.endingPage 524 prism.publicationDate 2022 prism.publicationName Journal of Differential Equations prism.startingPage 492 prism.volume 306 dc.identifier.doi 10.17863/CAM.78613 dc.identifier.doi 10.17863/CAM.78613 dcterms.dateAccepted 2021-10-26 rioxxterms.versionofrecord 10.1016/j.jde.2021.10.048 rioxxterms.version NA rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2022-01-05 dc.contributor.orcid Titi, ES [0000-0002-5004-1746] dc.contributor.orcid Yuan, G [0000-0003-4811-136X] dc.identifier.eissn 1090-2732 rioxxterms.type Journal Article/Review cam.issuedOnline 2021-11-04 cam.orpheus.success 2022-05-11: embargo success field applied cam.orpheus.counter 5 rioxxterms.freetoread.startdate 2023-11-05
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