Analyticity of solutions to the primitive equations

This article presents the maximal regularity approach to the primitive equations. It is proved that the $3D$ primitive equations on cylindrical domains admit a unique, global strong solution for initial data lying in the critical solonoidal Besov space $B_{pq}^{2\mathrm{/}p}$ for $p,q\in (1,\mathrm{\infty })$ with $1\mathrm{/}p+1\mathrm{/}q\le 1$. This solution regularize instantaneously and becomes even real analytic for $t>0$.