Well-posedness of strong solutions to the 3D generalized Navier–Stokes equations with nonlinear damping term

This paper devotes to studying the local and global well-posedness of the 3D generalized Navier–Stokes equations with fractional dissipative term [Formula: see text] and nonlinear damping term [Formula: see text]. First, we establish the local existence of the solution by using the Fourier truncation method in low regularity Sobolev spaces. Then, we prove the global existence when the power of damping term is suitable large. Moreover, when the index of the damping term is small, the existence of global solution is obtained provided that the initial data is small in [Formula: see text]-norm. Finally, we study the convergence property of the solution as the damping term disappears.