On Vortex Alignment and the Boundedness of the Lq-Norm of Vorticity in Incompressible Viscous Fluids

We show that the spatial L q -norm ( q > 5/3) of the vorticity of an incompressible viscous fluid in ℝ 3 remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in space, and that the space-time L q -norm of vorticity is finite. The Hölder index depends only on q . This serves as a variant of the classical result by Constantin-Fefferman (Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. J. Math. 42 (1993), 775–789), and the related work by Grujić-Ruzmaikina (Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. J. Math. 53 (2004), 1073–1080).