On Bridging Analyticity and Sparseness in Hyperdissipative Navier-Stokes Systems
Moses Patson Phiri
TL;DR
The paper investigates the near-critical, three-dimensional hyper-dissipative Navier–Stokes system with dissipation exponent $\beta>1$, below the Lions critical threshold. It introduces a quantified analyticity-sparseness gap via time-weighted derivative bridges and a focused-extremizer/harmonic-measure framework to bound high-order derivatives and rule out finite-time blow-up. Under scale-refined, slowly varying time weights and a fixed-point focus condition, the authors prove that the putative blow-up time is in fact regular and the solution analytically extends past it, refining prior analyticity-sparseness approaches. The work thereby strengthens near-critical regularity results, providing a robust toolkit (bridge inequalities, sparseness controls, harmonic-measure contractions) compatible with ongoing non-uniqueness developments in this regime.
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.