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

摘要

We study weak solutions of the 3D Navier-Stokes equations with L~2 initial data. We prove that ▽~αu is locally integrable in space-time for any real α such that 1 < α < 3. Up to now, only the second derivative ▽~2u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-L_(loc)~(4/(α+1)). These estimates depend only on the L~2-norm of the initial data and on the domain of integration. Moreover, they are valid even for α ≥ 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.