Asymptotic behavior of solutions for the velocity-vorticity model of the three-dimensional generalized Navier–Stokes equations with exponential damping

We propose the velocity-vorticity model of the three-dimensional (3D) generalized Navier–Stokes equations with exponential damping terms and then study the asymptotic behavior of solutions in a periodic bounded domain. First, we study the global well-posedness using the Faedo–Galerkin approximation method. Then, we investigate the asymptotic behavior of weak solutions via attractors and their properties using the theory of the evolutionary system which was recently developed by Cheskidov, Foias and Lu. Finally, we investigate determining wavenumbers.