GLOBAL REGULARITY FOR A LOGARITHMICALLY SUPERCRITICAL HYPERDISSIPATIVE NAVIER–STOKES EQUATION

Abstract

Let d ≥ 3. We consider the global Cauchy problem for the generalized Navier–Stokes system (Formula Presented) for u (Formula Presented) where u0 (Formula Presented) is smooth and divergence free, and D is a Fourier multiplier whose symbol m (Formula Presented) is nonnegative; the case m (Formula Presented) j is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes (Formula Presented) for (Formula Presented). We improve this slightly by establishing global regularity under the slightly weaker condition that (Formula Presented) for all sufficiently large ξ and some nondecreasing function g V RC !RC such that (Formula Presented). In particular, the results apply for the logarithmically supercritical dissipation (Formula Presented)