On Vortex Alignment and Boundedness of $L^q$ Norm of Vorticity

We show that the spatial $L^q$ ($q>5\mathrm{/}3$) norm of the vorticity of an incompressible viscous fluid in ${\mathbb{R}}^3$ or ${\mathbb{T}}^3$ remains bounded uniformly in time, provided that the direction of vorticity is H"older continuous in the space variable, and that the space–time $L^q$ norm of the vorticity is finite. The H"older index depends only on $q$. This serves as a variant of the classical result by P. Constantin and Ch. Fefferman (Direction of vorticity and the problem of global regularity for the Navier–Stokes equations, Indiana Univ. J. Math., 42 (1993), 775–789).