Global Well-Posedness of the 3D Navier-Stokes Equations in the Limiting Case: Infinitely Nested Logarithmic Improvements
Rishabh Mishra
TL;DR
This work tackles the global regularity problem for the 3D Navier-Stokes equations by introducing infinitely nested logarithmic improvements to the regularity criteria. The authors define critical function spaces $\dot{H}^{1/2,q,\{\delta_j\}}$ governed by an infinite product of nested logarithms and prove a global well-posedness result at the critical threshold $s=1/2$ for initial data in $L^2 \cap \dot{H}^{1/2}$ satisfying $\|(-\Delta)^{1/4}u_0\|_{L^q} \le C_0 \Psi(\|u_0\|_{\dot{H}^{1/2}})$. Central to the argument are infinitely nested commutator estimates, the analysis of a limiting ODE, and energy inequalities that remain subcritical in the limit, which collectively yield a zero Hausdorff-dimension bound on any potential singular set. The results bridge subcritical and critical regularity, offering a new pathway toward resolving global regularity questions for NSE and highlighting the role of deeply nested logarithmic controls in nonlinear PDEs.
Abstract
This paper establishes a complete framework for infinitely nested logarithmic improvements to regularity criteria for the three-dimensional incompressible Navier-Stokes equations. Building upon our previous works on logarithmically improved and multi-level logarithmically improved criteria, we demonstrate that the limiting case of infinitely nested logarithms fully bridges the gap between subcritical and critical regularity. Specifically, we prove that if the initial data $u_0 \in L^2(\mathbb{R}^3)$ satisfies the condition $\|(-Δ)^{1/4}u_0\|_{L^q(\mathbb{R}^3)} \leq C_0Ψ(\|u_0\|_{\dot{H}^{1/2}})$, where $Ψ$ incorporates infinitely nested logarithmic factors with appropriate decay conditions, then there exists a unique global-in-time smooth solution to the Navier-Stokes equations. This result establishes global well-posedness at the critical regularity threshold $s = 1/2$. The proof relies on infinitely nested commutator estimates, precise characterization of the critical exponent function in the limiting case, and careful analysis of the energy cascade. We also derive the exact Hausdorff dimension bound for potential singular sets in this limiting case, proving that the dimension reduces to zero. Through systematic construction of the limiting function spaces and detailed analysis of the associated ODEs, we demonstrate that infinitely nested logarithmic improvements provide a pathway to resolving the global regularity question for the Navier-Stokes equations.