Fast singular oscillating limits of
the three-dimensional "primitive" equations of
geophysical fluid flows are analyzed.
We prove existence on infinite time intervals
of regular solutions to the
3D "primitive" Navier-Stokes equations for strong
stratification (large stratification parameter N).
This uniform existence is proven for
periodic or stress-free boundary conditions
for all domain aspect ratios,
including the case of three wave resonances
which yield nonlinear "$2\frac{1}{2}$ dimensional"
limit equations for N → +∞;
smoothness assumptions are the same as for local
existence theorems, that is initial data in Hα, α ≥ 3/4.
The global existence is proven using techniques of
the Littlewood-Paley dyadic decomposition.
Infinite time regularity for solutions of the
3D "primitive" Navier-Stokes equations is obtained by bootstrapping
from global regularity of the limit resonant
equations and convergence theorems.