This note concerns the global regularity for the hyperdissipative periodic Navier-Stokes equations \(\begin{aligned} \text {(NS)} \left\{ \begin{aligned} \partial _t u + (u \cdot \nabla ) u&= -D^2 u - \nabla p, \\ \nabla \cdot u&= 0, \\ u(0, x)&= u_0(x), \end{aligned} \right. \end{aligned}\) where \(u : {\mathbb {R}}^+ \times {\mathbb {T}}^3 \rightarrow {\mathbb {R}}^3\) is the velocity field, \(p : {\mathbb {R}}^+ \times {\mathbb {T}}^3 \rightarrow {\mathbb {R}}\) is the pressure function and \(u_0 : {\mathbb {T}}^3 \rightarrow {\mathbb {R}}^3\) is the smooth and divergence-free initial data. D is the Fourier multiplier defined by \(a_n : {\mathbb {T}}^3 \rightarrow {\mathbb {R}}^+\) . In this paper, we prove the existence of global solutions for equations under the condition that \(a_n\ge |n|^{5/4} / g(|n|)\) for all sufficiently large n, where \(g: {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+\) is a non-decreasing function, satisfying \(\int _{1}^{\infty } \frac{ds}{sg(s)^4} = +\infty\) .
