Smoothness of Leray solutions for the Navier-Stokes equations

We introduce a frequency-decomposition framework for the three-dimensional incompressible Navier-Stokes equations with smooth compactly supported initial data. The velocity field $\mathbf{w}$ is represented as the sum $\mathbf{w} = \mathbf{u} + \mathbf{v}$ of high- and low-frequency components, coupled through complementary Fourier multipliers. The low-frequency part $\mathbf{v}$ is shown to be globally smooth, while the high-frequency part $\mathbf{u}$ is proved to remain uniformly bounded via a localized Picard iteration on short time windows, combined with high-frequency damping. Since the cutoff parameter is fixed and disappears upon summation, the total velocity $\mathbf{w}$ satisfies the classical Navier-Stokes equations exactly. Consequently, $\mathbf{w}$ is globally bounded and therefore smooth for all times. This provides a constructive analytic mechanism establishing global regularity for smooth initial data.