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The research aims to address global regularity and finite-time blow-up in the 3D Navier–Stokes equations using spherical compactification methods.

August 9, 2026Open Access

Global Regularity via Spherical Compactification for the 3D Navier–Stokes System

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The research aims to address global regularity and finite-time blow-up in the 3D Navier–Stokes equations using spherical compactification methods.
Constructed a stereographic compactification mapping for the Navier–Stokes equations on the closed three-sphere S3.
Defined the spectral NS operator SNO in the context of divergence-free vector fields.
Applied Kato’s relative bounded perturbation theory and established a complete system of elliptic a priori estimates.
Proved that the velocity singular set is empty under the proposed initial data, ruling out finite-time blow-up.
Derived rigorous pointwise decay bounds for velocity fields at spatial infinity with optimality of the decay exponent |x|−1.
Showed that the Sobolev bootstrap technique yields infinitely smooth velocity fields over time.

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Demonstrates global regularity of 3D Navier–Stokes equations, indicating no finite-time blow-up occurs under new initial conditions.

Changmin Wei (2026) studied this question.

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