The Primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations
View / Open Files
Journal Title
Journal of Differential Equations
ISSN
0022-0396
Publisher
Elsevier
Type
Article
This Version
AM
Metadata
Show full item recordCitation
Li, J., Titi, E., & Yuan, G. (2021). The Primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations. Journal of Differential Equations https://doi.org/10.1016/j.jde.2021.10.048
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 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).
Sponsorship
Einstein Stiftung/Foundation-Berlin, Einstein Visiting Fellowship No.
EVF-2017-358.
Identifiers
External DOI: https://doi.org/10.1016/j.jde.2021.10.048
This record's URL: https://www.repository.cam.ac.uk/handle/1810/330173
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International
Licence URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
Statistics
Total file downloads (since January 2020). For more information on metrics see the
IRUS guide.