Global Well-Posedness of the 3D Navier-Stokes Equations under Multi-Level Logarithmically Improved Criteria

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 $|(-\Delta)^{s/2}u_0|_{L^q(\mathbb{R}^3)} \leq \frac{C_0}{\prod_{j=1}^{n} (1 + L_j(|u_0|{\dot{H}^s}))^{\delta_j}}$ 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 $\Phi(s,q,{\delta_j}{j=1}^n)$ 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 $1 - \sum_{j=1}^n \frac{\delta_j}{1+\delta_j} \cdot \frac{1}{j+1}$. 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.