Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation
Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) 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 non-negative; the case $m(ξ) = |ξ|$ is essentially Navier-Stokes. It is folklore (see e.g. \cite{kp}) that one has global regularity in the critical and subcritical hyperdissipation regimes $m(ξ) = |ξ|^α$ for $α\geq \frac{d+2}{4}$. 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 non-decreasing function $g: \R^+ \to \R^+$ such that $\int_1^\infty \frac{ds}{sg(s)^4} = +\infty$. In particular, the results apply for the logarithmically supercritical dissipation $m(ξ) := |ξ|^{\frac{d+2}{4}} / \log(2 + |ξ|)^{1/4}$.