Global Regularity for a Logarithmically Supercritical Hyperdissipative Navier-stokes Equation

Let d ≥ 3. We consider the global Cauchy problem for the generalised Navier-Stokes system ∂tu+ (u · ∇)u = −D u−∇p ∇ · u = 0 u(0, x) = u0(x) for u : R×R → R and p : R×R → R, where u0 : R d → R is smooth and divergence free, and D is a Fourier multiplier whose symbol m : R → R is non-negative; the case m(ξ) = |ξ| is essentially Navier-Stokes. It is folklore (see e.g. [5]) that one has global regularity in the critical and subcritical hyperdissipation regimes m(ξ) = |ξ| for α ≥ d+2 4 . We improve this slightly by establishing global regularity under the slightly weaker condition that m(ξ) ≥ |ξ|/g(|ξ|) for all sufficiently large ξ and some non-decreasing function g : R → R such that ∫ ∞ 1 ds sg(s)4 = +∞. In particular, the results apply for the logarithmically supercritical dissipation m(ξ) := |ξ| d+2 4 / log(2 + |ξ|).