Global regularity of weak solutions to the generalized leray equations and its applications

We investigate a regularity for weak solutions of the following generalized Leray equations (-Δ)αV - 2α - 1 2α V + V V - 1 2α x V + P = 0, which arises from the study of self-similar solutions to the generalized Navier- Stokes equations in R3. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions V in the weighted Hilbert space Hα ω (R3). Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from Hα ω (R3) to H1+α ω (R3). This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of V which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981-1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29-44.