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 ${\nabla }^{\alpha }u$ is locally integrable in space–time for any real $\alpha$ such that $1<\alpha <3$. Up to now, only the second derivative ${\nabla }^{2}u$ was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$L_{\mathrm{loc}}^{4/(\alpha +1)}$. 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.