Global Regularity for The Hyperdissipative Periodic Navier-Stokes Equations

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 $$\int _{1}^{\infty } \frac{ds}{sg(s)^4} = +\infty$$ .