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

Abstract

We show that the spatial $L^q$ ($q>5/3$) 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 Hölder continuous in the space variable, and that the space--time $L^q$ norm of the vorticity is finite. The Hölder 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).

arXiv comment9 pages

Cited by in corpus (1)

• Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions