The primitive equations approximation of the anisotropic horizontally viscous 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 $varepsilon >0$ 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 three-dimensional Navier-Stokes 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 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 $varepsilon$ tends to zero, and that the convergence rate is of order $Oleft(varepsilon^fracbeta2right)$, where $beta=min{alpha-2,2}$. 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 Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., textbf{124} rm(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 $Oleft(varepsilonright)$.