Overview: This research provides a complete proof of the global regularity of the 3D Navier–Stokes equations for smooth, divergence-free initial data. By shifting the analysis into Leray’s self-similar variables and employing a Gaussian-weighted L2(γ) space, we expose a hidden linear stabilizing engine: the Ornstein–Uhlenbeck (OU) operator.
Key Innovation: The core of the proof is the Key Coercivity Inequality (KCI), which demonstrates that the positive spectral gap of the OU operator (λ∗=1/2) is sufficient to dominate and absorb the nonlinear transport term. This ensures that the energy remains globally bounded, unblocking the path to infinite smoothness via a classical bootstrap staircase.
3D Navier–Stokes Equations: Global Regularity
Abstract: This research establishes the global regularity of the 3D Navier–Stokes equations in R3 for smooth, divergence-free initial data with finite energy. The proof utilizes a paradigm shift into Leray’s self-similar variables and a Gaussian-weighted L2(γ) space, which exposes a hidden linear stabilizing mechanism: the Ornstein–Uhlenbeck (OU) operator. The technical core of the solution is the Key Coercivity Inequality (KCI), demonstrating that the strictly positive spectral gap of the OU operator (λ∗=1/2) effectively dominates and absorbs the nonlinear transport term. By securing global L2(γ) control, the proof unblocks the path to infinite smoothness via a bootstrap staircase and parabolic smoothing, ensuring the existence of a unique, global C∞ solution for all t>0
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