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

Abstract

We study weak solutions of the 3D Navier Stokes equations with L-2 initial data. We prove that del(alpha)u is locally integrable in space time for any real a such that 1 < alpha < 3. Up to now, only the second derivative del(alpha)u was 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 the initial data and on the domain of integration. Moreover, they are valid even for alpha >= 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. (C) 2013 Elsevier Masson SAS. All rights reserved.