Global existence results for the Navier-Stokes equations in the rotational framework

Consider the equations of Navier-Stokes in ${\mathbb{R}}^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided the initial data is small with respect to the norm the Fourier-Besov space ${\dot{FB}}_{p,r}^{2-3\mathrm{/}p}({\mathbb{R}}^3)$, where $p \in (1,\infty]$and$r \in [1,\infty]$. In the two-dimensional setting, a unique, global mild solution to this set of equations exists for {\em non-small} initial data $u_0\in L_{\sigma }^p({\mathbb{R}}^2)$ for $p\in [2,\mathrm{\infty })$.