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

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 epsilon > 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 (epsilon(alpha)), respectively, with alpha > 2, for which the limiting system is the primitive equations with only horizontal viscosity as epsilon 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 epsilon tends to zero, and that the convergence rate is of order O (epsilon(beta/2)) , where beta= min{alpha - 2, 2}. Note that this result is different from the case alpha = 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 (epsilon). (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper rigorously justifies the hydrostatic approximation and derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in anisotropic horizontal viscosity regime. The study shows that for well-prepared initial data, the solutions converge strongly to corresponding solutions of anisotropic primitive equations with only horizontal viscosities as epsilon tends to zero, with a convergence rate of O (epsilon(beta/2)). The convergence rate is determined by the relationship between the parameters alpha and beta.