  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 $B^{2/p}_{pq}$ for $p,q\in (1,\infty)$ with
$1/p+1/q \leq 1$. This solution regularize instantaneously and becomes even
real analytic for $t>0$.
