The primitive equations approximation of the anisotropic horizontally viscous 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 $\epsilon >0$ 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 $O\left({\epsilon }^{\frac{\beta }{2}}\right)$, 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 $O\left(\epsilon \right)$.

References in corpus (6)

• Local and Global Well-posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity
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• Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations with Heat and Salinity Diffusion
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• Primitive Equations with Horizontal Viscosity: The Initial Value and the Time-Periodic Problem for Physical Boundary Conditions

Cited by in corpus (1)

• Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity