From anisotropic Navier-Stokes equations to primitive equations for the ocean and atmosphere

We study the well-posedness of the primitive equations for the ocean and atmosphere on two particular domains : a bounded domain ${\mathrm{\Omega }}_1:=(-1,1{)}^3$ with periodic boundary conditions and the strip ${\mathrm{\Omega }}_2:={\mathbb{R}}^2\times (-1,1)$ with a periodic boundary condition for the vertical coordinate. An existence theorem for global solutions on a suitable Besov space is derived. Then, in a second step, we rigorously justify the passage to the limit from the rescaled anisotropic Navier-Stokes equations to these primitive equations in the same functional framework as that found for the solutions of the primitive equations.