A Fractional Memory Kernel Approach to Regularity in the 3D Navier-Stokes Equations: Insights from the Quantized Fractal Equation (QFE) Framework

We investigate a regularization strategy for the 3D incompressible Navier-Stokes equations inspired by the non-local memory kernel of the Quantized Fractal Equation (QFE)Revised. Replacing the classical time derivative with a Caputo fractional derivative oforder α ≈ 0.85 introduces a spatiotemporal smoothing effect.A fractional Burgers’ equation is analyzed as a toy model, and a preliminary studyof the 3D Navier-Stokes equations shows that the memory kernel leads to a fractionalenergy inequality and bounds the enstrophy growth. Numerical simulations with sharpinitial gradients demonstrate that the maximum velocity gradient remains bounded, incontrast to the classical case.These results suggest that the QFE memory mechanism offers a promising perspectiveon the global regularity question of the Navier-Stokes equations.