Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations

We study weak solutions of the 3D Navier-Stokes equations in whole space with $L^2$ initial data. It will be proved that $\nabla^\alpha u$ is locally integrable in space-time for any real $\alpha$ such that $1< \alpha <3$, which says that almost third derivative is locally integrable. Up to now, only second derivative $\nabla^2 u$ has been known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$L_{loc}^{4/(\alpha+1)}$. These estimates depend only on the $L^2$ norm of initial data and integrating domains. Moreover, they are valid even for $\alpha\geq 3$ as long as $u$ is smooth. The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study. For the local study, we use De Giorgi method with a new pressure decomposition. To handle non-locality of the fractional Laplacian, we will adopt some properties of the Hardy space and Maximal functions.