The primitive equations approximation of the anisotropic horizontally viscous 3D NavierStokes equations
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 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 threedimensional NavierStokes equations are of orders O (1) and O (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 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 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 and Titi (2019) [38], 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) .
 Publication:

Journal of Differential Equations
 Pub Date:
 January 2022
 DOI:
 10.1016/j.jde.2021.10.048
 arXiv:
 arXiv:2106.00201
 Bibcode:
 2022JDE...306..492L
 Keywords:

 35Q30;
 35Q86;
 76D05;
 86A05;
 86A10;
 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35Q30;
 35Q86;
 76D05;
 86A05;
 86A10