Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Abstract

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 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.