Global Well-Posedness for the 3D Navier-Stokes Equations under Logarithmically Improved Criteria: Connections to Turbulence Theory

Abstract

This paper introduces a novel class of initial data for which the three-dimensional incompressible Navier--Stokes equations yield unique global-in-time solutions. Building on a logarithmically improved regularity criterion, we impose a logarithmically subcritical condition on the initial data. Specifically, if \[ u_0 \in L^2(\mathbb{R}^3) \quad \text{and} \quad \|(-\Delta)^{s/2}u_0\|_{L^q(\mathbb{R}^3)} \le \frac{C_0}{\Bigl(1+\log\bigl(e+\|u_0\|_{\dot{H}^s}\bigr)\Bigr)^{\delta}}, \] for some $s \in (1/2,1)$ under appropriate scaling, then the corresponding solution exists globally and is unique. The proof employs refined commutator estimates for the fractional Laplacian together with new energy methods that exploit this logarithmic improvement to prevent singularity formation. Furthermore, we establish links between these improved criteria and turbulence theory. We derive precise relationships connecting the regularity conditions with turbulent intermittency, showing that the logarithmic enhancements correspond to anomalous scaling exponents in the turbulent energy spectrum. Additionally, we characterize the local structure of potential singularities and provide tight bounds on the energy flux in turbulent cascades. This approach bridges the gap between subcritical and critical regularity for the Navier--Stokes equations and offers a robust mathematical foundation for key phenomena observed in turbulence.