Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Abstract

Let $d\ge 3$. We consider the global Cauchy problem for the generalized Navier–Stokes system

$${\partial }_{t}u+\left(u\cdot \nabla \right)u=-D^{2}u-\nabla p,\nabla \cdot u=0,u\left(0,x\right)=u_0\left(x\right)$$

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 nonnegative; the case $m\left(\xi \right)=\left|\xi \right|$ is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes $m\left(\xi \right)=|\xi {|}^{\alpha }$ for $\alpha \ge \left(d+2\right)∕4$. We improve this slightly by establishing global regularity under the slightly weaker condition that $m\left(\xi \right)\ge |\xi {|}^{\left(d+2\right)∕4}∕g\left(\left|\xi \right|\right)$ for all sufficiently large $\xi$ and some nondecreasing function $g:{ℝ}^{+}\to {ℝ}^{+}$ such that ${\int }_1^{\infty }ds∕\left(sg{\left(s\right)}^4\right)=+\infty$. In particular, the results apply for the logarithmically supercritical dissipation $m\left(\xi \right):=|\xi {|}^{\left(d+2\right)∕4}∕log{\left(2+|\xi {|}^2\right)}^{1∕4}$.