Part II of the "Boundary Rigidity in Fluid Dynamics" Series - Boundary-Induced Regularity in Band-Limited Navier–Stokes Equations.

Boundary-Induced Regularity in Band-Limited Navier–Stokes Equations
(Part II of the “Boundary Rigidity in Fluid Dynamics” series)

This short note considers the three-dimensional incompressible Navier–Stokes equations under a fixed Fourier spectral truncation. When the velocity field is constrained to a finite frequency band, the system reduces to a finite-dimensional Galerkin model. In this setting, standard energy estimates together with the Bernstein inequality imply uniform enstrophy bounds and global-in-time regularity.

The result provides a clear illustration of a structural mechanism: exclusion of high-frequency degrees of freedom eliminates the possibility of uncontrolled energy cascade to arbitrarily small scales. Regularity in the band-limited system follows directly from bounded spectral support rather than from delicate nonlinear cancellations.

This work does not address the global regularity problem for the full three-dimensional Navier–Stokes equations. It serves as a pedagogical and structural example within the broader “Boundary Rigidity” framework developed in this series.