Electronic Journal of Differential Equations (Oct 2010)
Regularity for 3D Navier-Stokes equations in terms of two components of the vorticity
Abstract
We establish regularity conditions for the 3D Navier-Stokes equation via two components of the vorticity vector. It is known that if a Leray-Hopf weak solution $u$ satisfies $$ ilde{omega}in L^{2/(2-r)}(0,T;L^{3/r}(mathbb{R}^3))quad hbox{with }0<r<1, $$ where $ilde{omega}$ form the two components of the vorticity, $omega =operatorname{curl}u$, then $u$ becomes the classical solution on $(0,T]$ (see [5]). We prove the regularity of Leray-Hopf weak solution $u$ under each of the following two (weaker) conditions: $$displaylines{ ilde{omega}in L^{2/(2-r)}(0,T;dot {mathcal{M}}_{2, 3/r}(mathbb{R}^3))quad hbox{for }0<r<1,cr abla ilde{u}in L^{2/(2-r)}(0,T;dot {mathcal{M}}_{2, 3/r}(mathbb{R}^3))quad hbox{for }0leq r<1, }$$ where $dot {mathcal{M}}_{2,3/r}(mathbb{R}^3)$ is the Morrey-Campanato space. Since $L^{3/r}(mathbb{R}^3)$ is a proper subspace of $dot {mathcal{M}}_{2,3/r}(mathbb{R}^3)$, our regularity criterion improves the results in Chae-Choe [5].