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

In this paper, we show a mathematical justification of the data assimilation of nudging type in \(L^p\) $L^p$ - \(L^q\) $L^q$ maximal regularity settings. We prove that the approximate solution of the primitive equations constructed by the data assimilation converges to the true solution with exponential order in the Besov space \(B^{2/q}_{q,p}(\Omega )\) $B_{q,p}^{2/q}(\mathrm{\Omega })$ for \(1/p + 1/q \le 1\) $1/p+1/q\le 1$ on the periodic layer domain \(\Omega = \mathbb {T}^2 \times (-h, 0)\) $\mathrm{\Omega }={\mathbb{T}}^2\times (-h,0)$ .