GLOBAL REGULARITY AND ASYMPTOTIC BEHAVIOR OF MODIFIED NAVIER-STOKES EQUATIONS WITH FRACTIONAL DISSIPATION

This paper is concerned with the Cauchy problem of three-dimensional modified Navier-Stokes equations with fractional dissipation v(-Delta)(alpha)u. The results are three-fold. We first prove the global existence of weak solutions for 0 < alpha <= 1 and global smooth solution for 3/4 < alpha <= 1. Second, we obtain the optimal decay rates of both weak solutions and the higher-order derivative of the smooth solution. Finally, we investigate the asymptotic stability of the large solution to the system under large initial and external forcing perturbation.