Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

for $u : ℝ^{+} \times ℝ^{d} \to ℝ^{d}$ and $p : ℝ^{+} \times ℝ^{d} \to ℝ$ , 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 $m (ξ) = | ξ |$ is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes $m (ξ) = | ξ |^{α}$ for $α \geq (d + 2) ∕ 4$ . We improve this slightly by establishing global regularity under the slightly weaker condition that $m (ξ) \geq | ξ |^{(d + 2) ∕ 4} ∕ g (| ξ |)$ for all sufficiently large $ξ$ and some nondecreasing function $g : ℝ^{+} \to ℝ^{+}$ such that $\int_{1}^{\infty} d s ∕ (s g {(s)}^{4}) = + \infty$ . In particular, the results apply for the logarithmically supercritical dissipation $m (ξ) : = | ξ |^{(d + 2) ∕ 4} ∕ log {(2 + | ξ |^{2})}^{1 ∕ 4}$ .

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/

Vol. 2, No. 3, 2009

Terence Tao

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors