A New Approach to Three-Dimensional Navier-Stokes Equations via Antisymmetric Decomposition for a Restricted Class of Initial Data

We present a novel mathematical framework for analyzing the three-dimensional incompressible
Navier-Stokes equations through antisymmetric decomposition on domains invariant
under bilateral reflection σ(x, y, z) = (x,−y, z). Important: This work provides
substantial progress for a specific class of initial data (approximately 20% coverage) but
does not solve the complete Clay Institute Millennium Problem.

By establishing a fundamental orthogonal decomposition V = Vantisym ⊕ Vsym on σ-
admissible domains, we reveal hidden cancellation properties. Our main discovery is the
exact cancellation ⟨(ua · ∇)ua, ua⟩ = 0 for antisymmetric components, which dramatically
relaxes global existence conditions for antisymmetric-dominant initial data.

We prove global existence and uniqueness for initial data satisfying: (1) ∥u0∥H1 < 1.003,
and (2) antisymmetric dominance ∥Pau0∥L2 ≥ 0.6∥u0∥L2 . This represents a 5.25× improvement
over classical Leray-Hopf methods (∥u0∥H1 < 0.191) but applies only to antisymmetric-dominant
data on periodic cubic domains or R3.

Additionally, we discover that the antisymmetric pressure component satisfies Δpa = 0,
revealing an unexpected harmonic structure. Our results include explicit constructions
of initial data that violate classical thresholds but guarantee global regularity under our
framework, supported by rigorous computational validation with 100% success rate.