A solution to parabolic system with the fractional Laplacian

The existence of a solution to the parabolic system with the fractional Laplacian (-Delta)(alpha/2), alpha > 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an application, the existence of a solution to the generalized Navier-Stokes equations is proven, which decays at different rates along different time sequences going to infinity. The generalized Navier-Stokes equations are the equations resulting from replacing -Delta in the Navier-Stokes equations by (-Delta)(m), m > 0. At last, a similar result for 3-D incompressible anisotropic Navier-Stokes system is obtained.