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This article presents the maximal regularity approach to the primitive equations. It is proved that the 3 D  primitive equations on cylindrical domains admit a unique global strong solution for initial data lying in the critical solonoidal Besov space    B   p  q    2  /  pfor    p  ,  q  ∈  (  1  ,  ∞  )    with    1  /  p  +  1  /  q  ≤  1   . This solution regularize instantaneously and becomes even real analytic for    t  >  0   .

Analyticity of solutions to the primitive equations