The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes 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 three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting e > 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 ( e α ) , respectively, with α > 2 , for which the limiting system is the primitive equations with only horizontal viscosity as e 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 e tends to zero, and that the convergence rate is of order O ( e 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 ( e ) .