In this paper, we justify the hydrostatic approximation of the primitive equations in maximal Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}-Lq\documentclass[12pt]{minimal}
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\begin{document}$$L^q$$\end{document}-settings in the three-dimensional layer domain Ω=T2×(-1,1)\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega = \mathbb {T} ^2 \times (-1, 1)$$\end{document} under the no-slip (Dirichlet) boundary condition in any time interval (0, T) for T>0\documentclass[12pt]{minimal}
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\begin{document}$$T>0$$\end{document}. We show that the solution to the ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon $$\end{document}-scaled Navier–Stokes equations with Besov initial data u0∈Bq,ps(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$u_0 \in B^{s}_{q,p}(\varOmega )$$\end{document} for s>2-2/p+1/q\documentclass[12pt]{minimal}
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\begin{document}$$s > 2 - 2/p + 1/ q$$\end{document} converges to the solution to the primitive equations with the same initial data in E1(T)=W1,p(0,T;Lq(Ω))∩Lp(0,T;W2,q(Ω))\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {E}_1 (T) = W^{1, p}(0, T ; L^q (\varOmega )) \cap L^p(0, T ; W^{2, q} (\varOmega )) $$\end{document} with order O(ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$O(\epsilon )$$\end{document}, where (p,q)∈(1,∞)2\documentclass[12pt]{minimal}
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\begin{document}$$(p,q) \in (1,\infty )^2$$\end{document} satisfies 1p≤min(1-1/q,3/2-2/q)\documentclass[12pt]{minimal}
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\begin{document}$$ \frac{1}{p} \le \min ( 1 - 1/q, 3/2 - 2/q ) $$\end{document} and ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon $$\end{document} has the length scale. The global well-posedness of the scaled Navier–Stokes equations by ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon $$\end{document} in E1(T)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {E}_1 (T)$$\end{document} is also proved for sufficiently small ϵ>0\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon >0$$\end{document}. Note that T=∞\documentclass[12pt]{minimal}
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\begin{document}$$T = \infty $$\end{document} is included.
