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