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In this paper we justify the hydrostatic approximation of the primitiveequations in the maximal $L^p$-$L^q$-setting in the three-dimensional layerdomain $\Omega = \Torus^2 \times (-1, 1)$ under the no-slip (Dirichlet)boundary condition in any time interval $(0, T)$ for $T>0$. We show that thesolution to the scaled Navier-Stokes equations with Besov initial data $u_0 \inB^{s}_{q,p}(\Omega)$ for $s > 2 - 2/p + 1/ q$ converges to the solution to theprimitive equations with the same initial data in $\mathbb{E}_1 (T) = W^{1,p}(0, T ; L^q (\Omega)) \cap L^p(0, T ; W^{2, q} (\Omega)) $ with order$O(\epsilon)$ where $(p,q) \in (1,\infty)^2$ satisfies $ \frac{1}{p} \leq \min\bracket{ 1 - 1/q, 3/2 - 2/q }$. The global well-posedness of the scaledNavier-Stokes equations in $\mathbb{E}_1 (T)$ is also proved for sufficientlysmall $\epsilon>0$. Note that $T = \infty$ is included.

The Hydrostatic Approximation for the Primitive Equations by the Scaled Navier-Stokes Equations under the No-Slip Boundary Condition