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

Abstract

The global regularity problem of the three-dimensional incompressibleNavier–Stokes equations is one of the most famous unsolved mathematicalproblems listed on the Clay Mathematics Institute Millennium Prize list.Over the past decades, a large number of scholars have carried out extensiveresearch on the existence of weak solutions, yet the key issue whether finitetime velocity gradient blow-up may occur remains unresolved. The mainanalytical obstacle of direct analysis on unbounded R3 lies in the uncontrollable energy accumulation at spatial infinity, which cannot be eliminatedby conventional local a priori estimates.In this paper, we construct a stereographic compactification mappingthat transforms the initial value problem defined on unbounded Euclideanspace into an equivalent partial differential equation system on closed compact three-sphere S3. This geometric transformation fundamentally removes the difficulty of far-field energy divergence and provides a unifiedglobal embedding constant for all Sobolev spaces. We define the spectralNS operator SNO acting on the Hilbert subspace of divergence-free vectorfields, and adopt Kato’s relative bounded perturbation theory to give acomplete proof of its essential self-adjointness.We establish Lemma E to characterize the asymptotic behavior of velocity fields at spatial infinity, derive rigorous pointwise decay bounds, andstrictly verify the optimality of the |x|−1 decay exponent. The decay theoryis further extended to physical flows with non-zero far-field constant velocity. A complete system of elliptic a priori estimates for pressure is derivedbased on the Laplace–Beltrami equation on S3. After systematic analysisof Leray–Hopf weak solutions, we introduce the concept of velocity singularset and prove that such a set must be empty via rigorous contradictionargument. Under the admissible initial data class proposed in this work,finite-time blow-up cannot take place, and the Sobolev bootstrap techniqueyields infinitely smooth velocity fields over the entire time axis.An independent chapter is specially arranged to clarify the auxiliary status and strict application boundary of finite element numerical experiments.All mesh files, discrete source codes and raw time series data are packagedas supplementary materials, so that other researchers can independentlyreproduce all numerical observations for cross-verification.