  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 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 $a_1\in
C(\overline{G};L^p(-h,0))$, $a_2\in L^{\infty}(G;L^p(-h,0))$ 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 $L^\infty_HL^p_z(\Omega)$-estimates for terms of the
form $t^{1/2} \lVert \partial_z e^{tA_{\overline{\sigma}}}\mathbb{P}f
\rVert_{L^\infty_H L^p_z(\Omega)}\le C e^{t\beta} \lVert f \rVert_{L^\infty_H
L^p_z (\Omega)}$ for $t>0$, where $e^{t A_{\overline{\sigma}}}$ denotes the
hydrostatic Stokes semigroup. The difficulty in proving estimates of this form
is that the hydrostatic Helmholtz projection $\mathbb{P}$ fails to be bounded
with respect to the $L^\infty$-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.
