Second derivatives estimate of suitable solutions to the 3D Navier-Stokes equations

TL;DR

This paper improves the regularity estimates of second derivatives of suitable weak solutions to the 3D Navier-Stokes equations by refining Lorentz space bounds using blow-up techniques and De Giorgi iteration.

Contribution

It introduces sharper Lorentz space bounds for second derivatives and higher derivatives of vorticity, advancing understanding of solution regularity without pressure assumptions.

Findings

Second derivatives are in local $L^{4/3, q}$ for any $q > 4/3$

Enhanced bounds for higher derivatives of vorticity

Methodology avoids pressure assumptions in regularity analysis

Abstract

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier-Stokes equations in dimension three. We show that it is locally $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the current result $L ^{\frac43, \infty}$. Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.