The Primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

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cam.issuedOnline2021-11-04
cam.orpheus.counter9
cam.orpheus.success2022-05-11: embargo success field applied
dc.contributor.authorLi, Jinkai
dc.contributor.authorTiti, Edriss
dc.contributor.authorYuan, Guozhi
dc.contributor.orcidTiti, Edriss [0000-0002-5004-1746]
dc.date.accessioned2021-11-02T00:31:10Z
dc.date.available2021-11-02T00:31:10Z
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 ε >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 ε^α, respectively, with α>2, for which the limiting system is the primitive equations with only horizontal viscosity as ε 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 ε tends to zero, and that the convergence rate is of order O(ε^(β/2)), where β=min{α-2,2}. Note that this result is different from the case α=2 studied in [Li, J.; Titi, E.S.,The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., Vol. 124, (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(ε).
dc.description.sponsorshipEinstein Stiftung/Foundation-Berlin, Einstein Visiting Fellowship No. EVF-2017-358.
dc.identifier.doi10.17863/CAM.77616
dc.identifier.issn0022-0396
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/330173
dc.publisherElsevier
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 International
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0/
dc.titleThe Primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations
dc.typeArticle
dcterms.dateAccepted2021-10-26
prism.publicationNameJournal of Differential Equations
rioxxterms.licenseref.startdate2021-10-28
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.typeJournal Article/Review
rioxxterms.versionAM
rioxxterms.versionofrecord10.1016/j.jde.2021.10.048
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