Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation

Abstract

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 $m(\xi) = |\xi|$ is essentially Navier-Stokes. It is folklore (see e.g. \cite{kp}) that one has global regularity in the critical and subcritical hyperdissipation regimes $m(\xi) = |\xi|^\alpha$ for $\alpha \geq \frac{d+2}{4}$. We improve this slightly by establishing global regularity under the slightly weaker condition that $m(\xi) \geq |\xi|^{(d+2)/4}/g(|\xi|)$ for all sufficiently large $\xi$ 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 $m(\xi) := |\xi|^{\frac{d+2}{4}} / \log(2 + |\xi|)^{1/4}$.