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 \begin{equation*} (-\Delta)^{\alpha}V- \frac{2\alpha-1}{2\alpha}V+V\cdot\nabla V-\frac{1}{2\alpha}x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from study self-similar to Naiver-Stokes in $\mathbb R^3$. Firstly, by making use vanishing viscosity and developing non-local effects fractional diffusion operator, we prove uniform estimates $V$ weighted Hilbert space $H^\alpha_{\omega}(\mathbb R^3)$. Via differences characterization Besov spaces bootstrap argument, improve solution R^3)$ $H_{\omega}^{1+\alpha}(\mathbb This result, together linear theory Stokes system, lead pointwise allow us obtain natural property constructed \cite{LXZ}. In particular, an optimal decay estimate classical means special structure Oseen tensor. answers question proposed Tsai \cite[Comm. Math. Phys., 328 (2014), 29-44]{T}.