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