The primitive equations approximation of the anisotropic horizontally viscous 3D NavierStokes equations
dc.contributor.author  Li, J  
dc.contributor.author  Titi, ES  
dc.contributor.author  Yuan, G  
dc.date.accessioned  20211125T16:32:39Z  
dc.date.available  20211125T16:32:39Z  
dc.date.issued  2022  
dc.identifier.issn  00220396  
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 threedimensional NavierStokes 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 threedimensional NavierStokes 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 threedimensional NavierStokes 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 $\beta=\min\{\alpha2,2\}$. 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 NavierStokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., \textbf{124} \rm(2019), 3058], 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/FoundationBerlin, through the Einstein Visiting Fellow Program (EVF2017358).  
dc.language.iso  en  
dc.publisher  Elsevier BV  
dc.title  The primitive equations approximation of the anisotropic horizontally viscous 3D NavierStokes 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  20211026  
rioxxterms.versionofrecord  10.1016/j.jde.2021.10.048  
rioxxterms.version  NA  
rioxxterms.licenseref.uri  http://www.rioxx.net/licenses/allrightsreserved  
rioxxterms.licenseref.startdate  20220105  
dc.contributor.orcid  Titi, ES [0000000250041746]  
dc.contributor.orcid  Yuan, G [000000034811136X]  
dc.identifier.eissn  10902732  
rioxxterms.type  Journal Article/Review  
cam.issuedOnline  20211104  
cam.orpheus.success  20220511: embargo success field applied  
cam.orpheus.counter  5  
rioxxterms.freetoread.startdate  20231105 
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