In recent decades, the \(\alpha \) -models and the velocity-vorticity formulation of the three-dimensional Navier–Stokes equations have been extensively studied as promising approaches to alleviate some of the analytical and computational difficulties inherent in the three-dimensional Navier–Stokes equations of incompressible fluid flows. In this paper, we propose a new regularization of the velocity-vorticity formulation of the three-dimensional Navier–Stokes equations with fractional diffusion, which we call a family of the three-dimensional velocity-vorticity- \(\alpha \) -like-models. This model includes the \(\alpha \) regularization term added to the momentum equation in velocity-vorticity form, but with no regularizing term in the vorticity equation. The paper aims to study the well-posedness and long-time dynamics results for this model under periodic boundary conditions. Specifically, we investigate well-posedness using the Faedo–Galerkin approximation method and study the long-time behavior of solutions using uniform global attractors, trajectory attractors, and their properties based on the evolutionary system theory developed by Cheskidov, Foias, and Lu in [6–9, 30]. Finally, we also study the number of determining modes in this paper.
