A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces

Abstract

We exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to $L^{2/\mathrm{(2}-\mathrm{r)}}\left(\left(0,T\right);ℳ\left({\dot{H}}^r\left({ℝ}^3\right)\to {\dot{H}}^{-r}\left({ℝ}^3\right)\right)\right)$, where $ℳ\left({\dot{H}}^r\left({ℝ}^3\right)\to {\dot{H}}^{-r}\left({ℝ}^3\right)\right)$ is the multipliers between Sobolev spaces whose definition is given later for , then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.