Velocity-vorticity formulation of the three dimensional generalized Navier-Stokes equations: Effect of algebraic calming on well-posedness, convergence, determining wavenumber and regularity criteria

We propose a modification to the nonlinear term of the three dimensional velocity-vorticity systems with fractional diffusion (3D GVV) by using a calming function. We obtain the three dimensional calmed generalized velocity-vorticity system (3D calmed-GVV) and the three dimensional calmed rotational generalized velocity-vorticity system (3D calmed-rGVV). First, we study the global well-posedness of the weak and regular solutions to our models via the Faedo Galerkin approximation method. Under suitable conditions on the calming function, we show that the solutions to 3D calmed-GVV or 3D calmed-rGVV converge to the solutions of 3D GVV. Second, we study a time-dependent determining wavenumber for 3D GVV. Finally, we establish a regularity criterion on the time interval $ (\tau, T] $ of 3D GVV.