Vol. 2, No. 3, 2009

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 8, 1841–2148
Issue 7, 1587–1839
Issue 6, 1343–1586
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Terence Tao

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

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

tu + (u )u = D2u p, u = 0,u(0,x) = u 0(x)

for u : + × d d and p : + × d , where u0 : d d 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.

Keywords
Navier–Stokes, energy method
Mathematical Subject Classification 2000
Primary: 35Q30
Milestones
Received: 16 June 2009
Revised: 22 September 2009
Accepted: 23 October 2009
Published: 9 February 2010
Authors
Terence Tao
University of California
Department of Mathematics
Los Angeles, CA 90095-1555
United States
http://www.math.ucla.edu/~tao/