#### Vol. 2, No. 3, 2009

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

### Terence Tao

Vol. 2 (2009), No. 3, 361–366
##### Abstract

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 $u:{ℝ}^{+}×{ℝ}^{d}\to {ℝ}^{d}$ and $p:{ℝ}^{+}×{ℝ}^{d}\to ℝ$, where ${u}_{0}:{ℝ}^{d}\to {ℝ}^{d}$ is smooth and divergence free, and $D$ is a Fourier multiplier whose symbol $m:{ℝ}^{d}\to {ℝ}^{+}$ is nonnegative; the case $m\left(\xi \right)=|\xi |$ 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 $\alpha \ge \left(d+2\right)∕4$. We improve this slightly by establishing global regularity under the slightly weaker condition that $m\left(\xi \right)\ge |\xi {|}^{\left(d+2\right)∕4}∕g\left(|\xi |\right)$ for all sufficiently large $\xi$ and some nondecreasing function $g:{ℝ}^{+}\to {ℝ}^{+}$ such that ${\int }_{1}^{\infty }ds∕\left(sg{\left(s\right)}^{4}\right)=+\infty$. In particular, the results apply for the logarithmically supercritical dissipation $m\left(\xi \right):=|\xi {|}^{\left(d+2\right)∕4}∕log{\left(2+|\xi {|}^{2}\right)}^{1∕4}$.

##### Keywords
Navier–Stokes, energy method
Primary: 35Q30