Well-Posedness and L²-Decay Estimates for the Navier-Stokes Equations with Fractional Dissipation and Damping

The generalized three dimensional Navier-Stokes equations with damping are considered. Firstly, existence and uniqueness of strong solutions in the periodic domain T-3 are proved for 1/2 < alpha < 1, beta + 1 >= 6 alpha/2 alpha-1 is an element of (6, + infinity). Then, in the whole space R-3, if the critical situation beta + 1 = 6 alpha/2 alpha-1 and if u(0) is an element of H-1 (R-3) boolean AND (H) over dot(-s) (R-3) with s is an element of [0,1/2], the decay rate of solution has been established. We give proofs of these two results, based on energy estimates and a series of interpolation inequalities, the key of this paper is to give an explanation for that on the premise of increasing damping term, the well-posedness and decay can still preserve at low dissipation alpha < 1, and the relationship between dissipation and damping is given.