We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes equations for fully three-dimensional initial data characterized by uniformly large vorticity; smoothness assumptions for initial data are the same as in local existence theorems. The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. Infinite time regularity is obtained by bootstrapping from global regularity of the limit equations. Algebraic geometry of resonant Poincaré curves is also used to obtain regularity results in generic cases, for solutions of 3D Euler equations with initial data characterized by uniformly large vorticity. The existence of a countable set of finite dimensional manifolds invariant under the nonlinear dynamics is demonstrated for the limit "2 1/2-dimensional" Euler equations in generic cases.
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