Weighted Sobolev Estimates and Conditional Regularity for the Three-Dimensional Navier–Stokes Equations: Energy, Pressure, and Vorticity–Strain Identities

Abstract

We analyze spatially weighted energy and regularity identities for the incompressible Navier–Stokes equations. For strictly positive weights bounded above and below, we prove equivalence of weighted and ordinary Sobolev norms and establish a corresponding Poincaré estimate. We derive the exact weighted kinetic-energy balance, showing explicitly why pressure and transport corrections persist when the weight is nonconstant. The curl of the weighted pressure gradient and the evolution of a vorticity–strain functional are also calculated; neither yields an unconditional conservation law or a proof of global regularity in three dimensions. To validate the derivations, we use a reproducible two-dimensional Taylor–Green exact-solution benchmark on periodic meshes and report velocity-gradient errors, discrete momentum residuals, observed convergence orders, and energy-balance defects. A second family of positive weights has a nonvanishing viscous weight-gradient contribution, for which we derive closed-form energy, dissipation, and correction integrals and test their numerical values. A self-contained MATLAB script regenerates the data tables and visualizations without specialist toolboxes. The results establish second-order spatial consistency for this smooth benchmark and verify an exact weighted balance; they do not demonstrate superior weighted-solver performance or resolve the three-dimensional strong-solution problem.