## Analysis & PDE

• Anal. PDE
• Volume 2, Number 3 (2009), 361-366.

### Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Terence Tao

#### Abstract

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

$∂ t u + ( u ⋅ ∇ ) u = − D 2 u − ∇ p , ∇ ⋅ u = 0 , u ( 0 , x ) = u 0 ( x )$

for $u:ℝ+×ℝd→ℝd$ and $p:ℝ+×ℝd→ℝ$, where $u0:ℝd→ℝd$ is smooth and divergence free, and $D$ is a Fourier multiplier whose symbol $m:ℝd→ℝ+$ is nonnegative; the case $m(ξ)=|ξ|$ is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes $m(ξ)=|ξ|α$ for $α≥(d+2)∕4$. We improve this slightly by establishing global regularity under the slightly weaker condition that $m(ξ)≥|ξ|(d+2)∕4∕g(|ξ|)$ for all sufficiently large $ξ$ and some nondecreasing function $g:ℝ+→ℝ+$ such that $∫ 1∞ds∕(sg(s)4)=+∞$. In particular, the results apply for the logarithmically supercritical dissipation $m(ξ):=|ξ|(d+2)∕4∕log(2+|ξ|2)1∕4$.

#### Article information

Source
Anal. PDE, Volume 2, Number 3 (2009), 361-366.

Dates
Revised: 22 September 2009
Accepted: 23 October 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513798040

Digital Object Identifier
doi:10.2140/apde.2009.2.361

Mathematical Reviews number (MathSciNet)
MR2603802

Zentralblatt MATH identifier
1190.35177

Subjects

Keywords
Navier–Stokes energy method

#### Citation

Tao, Terence. Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation. Anal. PDE 2 (2009), no. 3, 361--366. doi:10.2140/apde.2009.2.361. https://projecteuclid.org/euclid.apde/1513798040

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