The hydrostatic approximation for the primitive equations by the scaled Navier–Stokes equations under the no-slip boundary condition

In this paper, we justify the hydrostatic approximation of primitive equations in maximal $$L^p$$ - $$L^q$$ -settings three-dimensional layer domain $$\varOmega = \mathbb {T} ^2 \times (-1, 1)$$ under no-slip (Dirichlet) boundary condition any time interval (0, T) for $$T>0$$ . We show that solution to $$\epsilon $$ -scaled Navier–Stokes with Besov initial data $$u_0 \in B^{s}_{q,p}(\varOmega )$$ $$s > 2 2/p + 1/ q$$ converges same $$\mathbb {E}_1 (T) W^{1, p}(0, T ; L^q (\varOmega )) \cap L^p(0, W^{2, q} order $$O(\epsilon , where $$(p,q) (1,\infty )^2$$ satisfies \frac{1}{p} \le \min ( 1 1/q, 3/2 2/q ) and has length scale. The global well-posedness scaled by (T)$$ is also proved sufficiently small >0$$ Note $$T \infty included.