Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics

Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. VVe 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 tor all domain aspect ratios, including the case of three wave resonances which yield nonlinear "21/2 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.