Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L p 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 construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures.