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In this paper, we show mathematical justification of the data assimilation ofnudging type in $L^p$-$L^q$ maximal regularity settings.  We prove that the approximate solution of the primitive equations by dataassimilation converges to the true solution with exponential order on the Besovspace $B^{2/q}_{q,p}(\Omega)$ in the periodic layer domain $\Omega =\mathbb{T}^2 \times (-h, 0)$.

Data Assimilation to the Primitive Equations with $L^p$-$L^q$-based Maximal Regularity Approach