Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System

We study the three-dimensional incompressible Navier-Stokes system on ${\mathbb{R}}^3$ with an additional dissipative nonlocal term $${\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u+\mathrm{\nabla }p=\nu \mathrm{\Delta }u+Lu,\mathrm{d}\mathrm{i}\mathrm{v}u=0,$$ where $L$ is a self-adjoint Fourier multiplier whose symbol is comparable to $-\mathrm{\mid }\xi {\mathrm{\mid }}^{2\alpha }$ for some $\alpha >1$. We first identify a sharp Fourier-symbol criterion distinguishing lower-order convolution perturbations from genuinely regularizing nonlocal corrections. In the resulting hyperdissipative class we prove the exact $L^2$ energy identity, global weak solvability for every $\alpha >1$, and local strong well-posedness in $H^s({\mathbb{R}}^3)$ for $s>\frac{5}{2}$. We then show that the Lions exponent $\alpha =\frac{5}{4}$ remains the critical energy-growth threshold in this nonlocal setting: if $\alpha \ge \frac{5}{4}$, every $H^s$ solution is global, while for every $\alpha >1$ one has global strong solvability for sufficiently small $H^s$ data. Finally, for the vanishing-hyperdissipation approximation of the classical three-dimensional Navier-Stokes equations, we prove a near-singular divergence principle: if the classical flow blows up at a first singular time $T_{\ast }$ in a continuation norm $X$, then the corresponding regularized family cannot remain uniformly bounded in $X$ on any interval approaching $T_{\ast }$. This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.