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

TL;DR

This paper establishes that weak solutions to the 3D Navier-Stokes equations have locally integrable fractional derivatives of order between 1 and 3, extending regularity understanding beyond second derivatives.

Contribution

It provides new estimates for fractional derivatives of weak solutions, showing local integrability for derivatives of order up to nearly three, using advanced approximation and harmonic analysis techniques.

Findings

Almost third derivatives are locally integrable for weak solutions.

Sharp estimates depend only on initial data's L^2 norm.

Results hold even for derivatives of order ≥ 3 if the solution is smooth.

Abstract

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.