  In this paper, we show the global regularity and the optimal decay of weak
solutions to the generalized Leray problem with critical dissipation. Our
method is based on the maximal smoothing effect, $L^{p}$-type elliptic
regularity of linearization, and the action of the heat semigroup generated by
the fractional powers of Laplace operator on distributions with Fourier
transforms supported in an annulus. As a by-product, we shall construct a
self-similar solution to the three-dimensional incompressible Navier-Stokes
equations, and more importantly, prove the global regularity and the optimal
decay without additional requirement of existing literatures.
