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

Abstract

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 $L_H^{\infty }{L}_z^p(\Omega )$-estimates for terms of the form $t^{1/2}\Vert {\partial }_z{e}^{t{A}_{\overline{\sigma }}}\mathbb{P}f{\Vert }_{L_H^{\infty }{L}_z^p(\Omega )}\le C{e}^{t\beta }\Vert f{\Vert }_{L_H^{\infty }{L}_z^p(\Omega )}$ 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.

30 pages