By using a new bilinear estimate, a pointwise estimate of the generalized Oseen kernel, and an idea of a fractional bootstrap, we establish optimal local smoothing and decay estimates of solutions to the Navier-Stokes equations with fractional dissipation. We also show that solutions are analytic in space variables.
"Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations." Commun. Math. Sci. 7 (1) 67 - 80, March 2009.