The three limits of the hydrostatic approximation

The primitive equations are derived from the $3D$-Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an $\varepsilon$-thin domain and anisotropic viscosities with vertical viscosity $\nu_z=\mathcal{O}(\varepsilon^\gamma)$ where $\gamma=2$, one obtains the primitive equations with full viscosity as $\varepsilon\to 0$. Here, we take two more limit equations into consideration: For $\gamma<2$ the $2D$-Navier-Stokes equations are obtained. For $\gamma>2$ the primitive equations with only horizontal viscosity $-\Delta_H$ as $\varepsilon\to 0$. Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally $\nu_z=\varepsilon^2 \delta$ and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for $\varepsilon,\delta\to 0$. The flexibility of our methods is also illustrated by the convergence for $\delta\to \infty$ and $\varepsilon\to 0$ to the $2D$-Navier-Stokes equations.