The Primitive Equations in the scaling invariant space $L^{\infty }(L^1)$

Abstract

Consider the primitive equations on ${\mathbb{R}}^2\times (z_0,z_1)$ with initial data $a$ of the form $a=a_1+a_2$, where $a_1\in BU{C}_{\sigma }({\mathbb{R}}^2;L^1(z_0,z_1))$ and $a_2\in L_{\sigma }^{\infty }({\mathbb{R}}^2;L^1(z_0,z_1))$ and where $BU{C}_{\sigma }(L^1)$ and $L_{\sigma }^{\infty }(L^1)$ denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on ${\mathbb{R}}^2$, 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 $\Vert a_2{\Vert }_{L_{\sigma }^{\infty }(L^1)}$ 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 $L^{\infty }(L^1)$-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.

17 pages