Global Well-Posedness of the 3D Navier-Stokes Equations under Multi-Level Logarithmically Improved Criteria
Rishabh Mishra
TL;DR
The paper develops a comprehensive multi-level logarithmic framework to address global regularity for the 3D Navier–Stokes equations. By constructing a hierarchy of commutator estimates with nested logarithmic corrections and coupling them to energy methods, it proves global well-posedness for initial data in $L^2(\mathbb{R}^3)\cap\dot H^s(\mathbb{R}^3)$ with $s\in(\tfrac12,1)$ under a nested logarithmic smallness condition on $\|(-\Delta)^{s/2}u_0\|_{L^q}$. It introduces a critical threshold function $\Phi(s,q,\{\delta_j\}_{j=1}^n)$, derives its asymptotics as $s\to\tfrac12$ and as $s\to1$, and provides a sharp bound on the Hausdorff dimension of potential singular sets, $\dim_H(\mathcal S_{T^*})\le 1-\sum_{j=1}^n \frac{\delta_j}{1+\delta_j}\cdot\frac{1}{j+1}$. The work also develops a multifractal/geometric analysis of velocity-gradients and a refined spectral picture showing energy-cascade corrections and modified spectra governed by multiple logarithmic factors. Together, these results push the boundary toward criticality by systematically lowering the gap via nested logarithmic improvements and outlining a path toward a possible universal regularity criterion in the infinite-nesting limit.
Abstract
This paper extends our previous results on logarithmically improved regularity criteria for the three-dimensional Navier-Stokes equations by establishing a comprehensive framework of multi-level logarithmic improvements. We prove that if the initial data $u_0 \in L^2(\mathbb{R}^3)$ satisfies a nested logarithmically weakened condition $\|(-Δ)^{s/2}u_0\|_{L^q(\mathbb{R}^3)} \leq \frac{C_0}{\prod_{j=1}^{n} (1 + L_j(\|u_0\|_{\dot{H}^s}))^{δ_j}}$ for some $s \in (1/2, 1)$, where $L_j$ represents $j$-fold nested logarithms, then the corresponding solution exists globally in time and is unique. The proof introduces a novel sequence of increasingly precise commutator estimates incorporating multiple layers of logarithmic corrections. We establish the existence of a critical threshold function $Φ(s,q,\{δ_j\}_{j=1}^n)$ that completely characterizes the boundary between global regularity and potential singularity formation, with explicit asymptotics as $s$ approaches the critical value $1/2$. This paper further provides a rigorous geometric characterization of potential singular structures through refined multi-fractal analysis, showing that any singular set must have Hausdorff dimension bounded by $1 - \sum_{j=1}^n \frac{δ_j}{1+δ_j} \cdot \frac{1}{j+1}$. Our results constitute a significant advancement toward resolving the global regularity question for the Navier-Stokes equations, as we demonstrate that with properly calibrated sequences of nested logarithmic improvements, the gap to the critical case can be systematically reduced.