Let . We consider the global Cauchy problem for the generalized Navier–Stokes system
for and , where is smooth and divergence free, and is a Fourier multiplier whose symbol is nonnegative; the case is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes for . We improve this slightly by establishing global regularity under the slightly weaker condition that for all sufficiently large and some nondecreasing function such that . In particular, the results apply for the logarithmically supercritical dissipation .
"Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation." Anal. PDE 2 (3) 361 - 366, 2009. https://doi.org/10.2140/apde.2009.2.361