It is shown that the hydrostatic Stokes operator on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) , where Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 is a cylindrical domain subject to mixed periodic, Dirichlet and Neumann boundary conditions, admits a bounded H ∞ H^\infty -calculus on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) for p ∈ ( 1 , ∞ ) p\in (1,\infty ) of H ∞ H^\infty -angle 0 0 . In particular, maximal L q − L p L^q-L^p -regularity estimates for the linearized primitive equations are obtained.

This article presents the maximal regularity approach to the primitive equations. It is proved that the 3 primitive equations on cylindrical domains admit a unique global strong solution for initial data lying in the critical solonoidal Besov space 2∕ for , ∈ (1, ∞) with 1∕ + 1∕ ≤ 1. This solution regularize instantaneously and becomes even real analytic for > 0. K E Y W O R D S global strong well-posedness, maximal regularity, primitive equations, regularity of solutions M S C ( 2 0 1 0 ) Primary: 35Q35; Secondary: 76D03, 47D06, 86A05 284

Consider the 3-d primitive equations in a layer domain Ω = G × (−h, 0), G = (0, 1) 2 , subject to mixed Dirichlet and Neumann boundary conditions at z = −h and z = 0, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form a = a 1 + a 2 , where a 1 ∈ C(G; L p (−h, 0)), a 2 ∈ L ∞ (G; L p (−h, 0)) for p > 3, and where a 1 is periodic in the horizontal variables and a 2 is sufficiently small. In particular, no differentiability condition on the data is assumed. The approach relies on L ∞ H L p z (Ω)-estimates for terms of the form t 1/2 ∂z e tA σ Pf L ∞for t > 0, where e tA σ denotes the hydrostatic Stokes semigroup. The difficulty in proving estimates of this form is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L ∞ -norm. The global strong well-posedness result is then obtained by an iteration scheme, splitting the data into a smooth and a rough part and by combining a reference solution for smooth data with an evolution equation for the rough part.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 47D06, 76D03, 86A05.

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.

Consider the primitive equations on R 2 × (z 0 , z 1 ) with initial data a of the form a = a 1 + a 2 , where a 1 ∈ BU Cσ (R 2 ; L 1 (z 0 , z 1 )) and a 2 ∈ L ∞ σ (R 2 ; L 1 (z 0 , z 1 )) and where BU Cσ(L 1 ) and L ∞ σ (L 1 ) denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on R 2 , respectively, which take values in L 1 (z 0 , z 1 ). These spaces are scaling invariant and represent the anisotropic character of these equations. It is shown that, if a 2 L ∞ σ (L 1 ) is sufficiently small, then this set of equations has a unique, local, mild solution. If in addition a is periodic in the horizontal variables, then this solution is a strong one and extends to a unique, global, strong solution. The primitive equations are thus strongly and globally well-posed for these data. The approach depends crucially on mapping properties of the hydrostatic Stokes semigroup in the L ∞ (L 1 )-setting and can thus be seen as the counterpart of the classical iteration schemes for the Navier-Stokes equations for the situation of the primitive equations.