On Bridging Analyticity and Sparseness in Hyperdissipative Navier–Stokes Systems

Abstract

We study the three-dimensional hyperdissipative Navier–Stokes system in the super-critical regime (below the Lions threshold). Leveraging the analyticity–sparseness gap in the case where the ratios of the lower-order derivatives with respect to the higher-order derivatives are bounded by constants, we introduce time-weighted bridge inequalities quantifying these ratios across all derivative levels. On one hand, the time weights are chosen in a way that the resulting lower bound on the radius of spatial analyticity still dominates the scale of sparseness of the super-level sets indicating that the flow entered the dissipation regime. On the other hand, they provide a less restrictive environment for the ratios giving the construction more flexibility. As an illustration of its utility, it is shown that – together with a harmonic-measure contraction on one-dimensional sparse sets – this mechanism rules out a more general class of the blow-up scenarios compared to the previous work by Farhat and Grujić (essentially a perturbation of the multidimensional geometric series).