On Bridging Analyticity and Sparseness in Hyperdissipative Navier-Stokes Systems

TL;DR

This paper introduces a novel analytical framework combining analyticity and sparseness to prove the global regularity of three-dimensional hyper-dissipative Navier-Stokes solutions in a near-critical regime, preventing blow-up.

Contribution

It develops a new bridge inequality and extremizer hypothesis, along with harmonic-measure contraction techniques, to establish decay of high derivatives and extend solutions analytically beyond potential singularities.

Findings

Proves global regularity for near-critical hyperdissipative Navier-Stokes systems.

Introduces a quantified analyticity-sparseness gap and a time-weighted bridge inequality.

Rules out finite-time blow-up under the studied conditions.

Abstract

We study the three-dimensional hyper-dissipative Navier-Stokes system in the near-critical regime below the Lions threshold. Leveraging a quantified analyticity-sparseness gap, we introduce a time-weighted bridge inequality across derivative levels and a focused-extremizer hypothesis capturing peak concentration at a fixed point. Together with a harmonic-measure contraction on one-dimensional sparse sets, these mechanisms enforce quantitative decay of high-derivative $L^{\infty}-$norms and rule out blow-up. Under scale-refined, slowly varying time weights, solutions extend analytically past the prospective singular time, thereby refining the analyticity-sparseness framework, complementing recent exclusions of rapid-rate blow-up scenarios, and remaining consistent with recent non-uniqueness results.