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2009 Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation
Terence Tao
Anal. PDE 2(3): 361-366 (2009). DOI: 10.2140/apde.2009.2.361


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.


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Terence Tao. "Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation." Anal. PDE 2 (3) 361 - 366, 2009.


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

zbMATH: 1190.35177
MathSciNet: MR2603802
Digital Object Identifier: 10.2140/apde.2009.2.361

Primary: 35Q30

Keywords: Energy method , Navier–Stokes

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.2 • No. 3 • 2009
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