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

We study weak solutions of the 3D Navier–Stokes equations with L2L2 initial data. We prove that ∇αu∇αu is locally integrable in space–time for any real α   such that 1<α<31<α<3. Up to now, only the second derivative ∇2u∇2u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-Lloc4/(α+1). These estimates depend only on the L2L2-norm of the initial data and on the domain of integration. Moreover, they are valid even for α⩾3α⩾3 as long as u is smooth. The proof uses a standard approximation of Navier–Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced.