The Relations Between Hölder Continuity Assumptions on the Direction of Vorticity and Energy Equality

In their seminal work, Constantin and Fefferman (Indiana Univ Math J 42(3):775-789, 1993) pioneered the study of relations between between beta-H & ouml;lder continuity in the direction of vorticity and the regularity of solutions to the Navier-Stokes equations. Building upon this foundational research, a more general result was later presented by the first author of these notes in collaboration with Berselli (Differ Integral Equ 15(3):345-356, 2002). Another pivotal area of research in the realm of incompressible Navier-Stokes equations is the energy equality for Leray-Hopf weak solutions. Numerous recent studies have shed light on this issue. In this paper, we start the study of the energy equality from a novel perspective, inspired by the aforementioned research. Our focus lies in the relations between beta-H & ouml;lder continuity assumptions on the direction of vorticity and energy equality.