The primitive equations in the scaling-invariant space L∞(L1)

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.