Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation

Abstract

Let $d\ge 3$. 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 $u:{\mathbb{R}}^{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ and $p:{\mathbb{R}}^{+}\times {\mathbb{R}}^d\to \mathbb{R}$, where $u_0:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is smooth and divergence free, and $D$ is a Fourier multiplier whose symbol $m:{\mathbb{R}}^d\to {\mathbb{R}}^{+}$ 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 $\alpha \ge \frac{d+2}{4}$. We improve this slightly by establishing global regularity under the slightly weaker condition that $m(\xi )\ge |\xi {|}^{(d+2)/4}/g(|\xi |)$ for all sufficiently large $\xi$ and some non-decreasing function $g:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that ${\int }_1^{\infty }\frac{ds}{sg(s{)}^4}=+\infty$. In particular, the results apply for the logarithmically supercritical dissipation $m(\xi ):=|\xi {|}^{\frac{d+2}{4}}/\log (2+|\xi |{)}^{1/4}$.

7 pages, no figures, submitted, Analysis & PDE. Some slight corrections