Yoshikazu Giga, Mathis Gries, Matthias Hieber, Amru Hussein, Takahito Kashiwabara

• Consider the primitive equations on ◂+▸R2×(◂,▸z0,z1) with initial data a of the form a=◂+▸a1+a2, where ◂+▸a1∈◂◽.▸BUCσ(◂,▸R2;L1(◂,▸z0,z1)) and ◂+▸a2∈L ∞ σ (◂,▸R2;L1(◂,▸z0,z1)). These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for a1 arbitrary large and a2 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∞(L1)-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∞(L1)-setting.