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.
Categories: math.AP