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Consider the $3$-d primitive equations in a layer domain $\Omega=G \times(-h,0)$, $G=(0,1)^2$, subject to mixed Dirichlet and Neumann boundaryconditions at $z=-h$ and $z=0$, respectively, and the periodic lateral boundarycondition. It is shown that this equation is globally, strongly well-posed forarbitrary large data of the form $a=a_1 + a_2$, where $a_1\inC(\overline{G};L^p(-h,0))$, $a_2\in L^{\infty}(G;L^p(-h,0))$ for $p>3$, andwhere $a_1$ is periodic in the horizontal variables and $a_2$ is sufficientlysmall. In particular, no differentiability condition on the data is assumed.The approach relies on $L^\infty_HL^p_z(\Omega)$-estimates for terms of theform $t^{1/2} \lVert \partial_z e^{tA_{\overline{\sigma}}}\mathbb{P}f\rVert_{L^\infty_H L^p_z(\Omega)}\le C e^{t\beta} \lVert f \rVert_{L^\infty_HL^p_z (\Omega)}$ for $t>0$, where $e^{t A_{\overline{\sigma}}}$ denotes thehydrostatic Stokes semigroup. The difficulty in proving estimates of this formis that the hydrostatic Helmholtz projection $\mathbb{P}$ fails to be boundedwith respect to the $L^\infty$-norm. The global strong well-posedness result isthen obtained by an iteration scheme, splitting the data into a smooth and arough part and by combining a reference solution for smooth data with anevolution equation for the rough part.

The Hydrostatic Stokes Semigroup and Well-Posedness of the Primitive Equations on Spaces of Bounded Functions