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 $$\partial_t u + (u\cdot\nabla)u + \nabla p = \nu \Delta u + Lu, \qquad {\rm div}\, u = 0,$$ where $L$ is a self-adjoint Fourier multiplier whose symbol is comparable to $-|\xi|^{2\alpha}$ 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 $s>\frac52$. We then show that the Lions exponent $\alpha=\frac54$ remains the critical energy-growth threshold in this nonlocal setting: if $\alpha\ge \frac54$, 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 $T_*$ in a continuation norm $X$, then the corresponding regularized family cannot remain uniformly bounded in $X$ on any interval approaching $T_*$. This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.