Anal. PDE 2 (3), 361-366, (2009) DOI: 10.2140/apde.2009.2.361
KEYWORDS: Navier–Stokes, Energy method, 35Q30

Let $d\ge 3$. We consider the global Cauchy problem for the generalized Navier–Stokes system

$${\partial}_{t}u+\left(u\cdot \nabla \right)u=-{D}^{2}u-\nabla p,\phantom{\rule{1em}{0ex}}\nabla \cdot u=0,\phantom{\rule{1em}{0ex}}u\left(0,x\right)={u}_{0}\left(x\right)$$

for $u:{\mathbb{R}}^{+}\times {\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$ and $p:{\mathbb{R}}^{+}\times {\mathbb{R}}^{d}\to \mathbb{R}$, where ${u}_{0}:{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$ is smooth and divergence free, and $D$ is a Fourier multiplier whose symbol $m:{\mathbb{R}}^{d}\to {\mathbb{R}}^{+}$ is nonnegative; the case $m\left(\xi \right)=\left|\xi \right|$ is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes $m\left(\xi \right)=|\xi {|}^{\alpha}$ for $\alpha \ge \left(d+2\right)\u22154$. We improve this slightly by establishing global regularity under the slightly weaker condition that $m\left(\xi \right)\ge |\xi {|}^{\left(d+2\right)\u22154}\u2215g\left(\left|\xi \right|\right)$ for all sufficiently large $\xi $ and some nondecreasing function $g:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\int}_{1}^{\infty}ds\u2215\left(sg{\left(s\right)}^{4}\right)=+\infty $. In particular, the results apply for the logarithmically supercritical dissipation $m\left(\xi \right):=|\xi {|}^{\left(d+2\right)\u22154}\u2215log{\left(2+|\xi {|}^{2}\right)}^{1\u22154}$.