Analysis & PDE

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

Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Terence Tao

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Let d3. 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:+×dd and p:+×d, where u0:dd is smooth and divergence free, and D is a Fourier multiplier whose symbol m:d+ 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)4g(|ξ|) for all sufficiently large ξ and some nondecreasing function g:++ such that 1ds(sg(s)4)=+. In particular, the results apply for the logarithmically supercritical dissipation m(ξ):=|ξ|(d+2)4log(2+|ξ|2)14.

Article information

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

Received: 16 June 2009
Revised: 22 September 2009
Accepted: 23 October 2009
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Navier–Stokes energy method


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.

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