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Consider the primitive equations on    $$\mathbb {R}^2\times (z_0,z_1)$$  R 2 × ( z 0 , z 1 ) with initial data  a  of the form    $$a=a_1+a_2$$  a = a 1 + a 2 , where    $$a_1 \in \mathrm{BUC}_\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$  a 1 ∈ BUC σ ( R 2 ; L 1 ( z 0 , z 1 ) ) and    $$a_2 \in L^\infty _\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$  a 2 ∈ L σ ∞ ( R 2 ; L 1 ( z 0 , z 1 ) ) . These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for    $$a_1$$  a 1 arbitrary large and    $$a_2$$  a 2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided  a  is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the    $$L^\infty (L^1)$$  L ∞ ( L 1 ) -setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the    $$L^\infty (L^1)$$  L ∞ ( L 1 ) -setting.

The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$