Asymptotic Criticality of the Navier-Stokes Regularity Problem
Zoran Grujic, Liaosha Xu
TL;DR
The paper tackles the Navier–Stokes regularity problem by introducing a sparseness-based framework that connects high-order derivative structures to spatial analyticity, thereby shrinking the scaling gap between a priori bounds and regularity criteria. It develops spatial analyticity results for $D^{(k)}u$ and $D^{(k)}\omega$ and defines the $Z^{(k)}_{\alpha_k}$ classes with $\alpha_k=1/(k+1)$, linking sparseness of super-level sets to regularity through harmonic measure. The main contribution is the demonstration of asymptotic criticality: as $k\to\infty$, the gap collapses so that $D^{(k)}u$ (and $D^{(k)}\omega$) in the corresponding $Z^{(k)}_{\alpha_k}$ framework implies regularity up to the potential blow-up time, via dynamic interpolation of ascending/descending derivative chains. This approach provides a novel path toward understanding the critical balance between nonlinearity and diffusion in 3D NSE and points to extensions to hyper-dissipative variants.
Abstract
The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a scaling gap between any regularity criterion and the corresponding \emph{a priori} bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework--based on a suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field--in which the scaling gap between the regularity class and the corresponding \emph{a priori} bound vanishes as the order of the derivative goes to infinity.