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

TL;DR

This paper proves that under certain regularity conditions on vorticity direction, the $L^q$ norm of vorticity remains bounded over time for incompressible viscous fluids in 3D, extending classical results.

Contribution

It establishes boundedness of the $L^q$ vorticity norm assuming H"older continuity of vorticity direction, generalizing prior classical results.

Findings

$L^q$ vorticity norm remains bounded over time

Boundedness depends on H"older continuity of vorticity direction

Extension of classical results by Constantin and Fefferman

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 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).