Construction of maximal functions associated with skewed cylinders generated by incompressible flows and applications

Abstract

We construct a maximal function associated with a family of skewed cylinders. These cylinders, which are defined as tubular neighborhoods of trajectories of a mollified flow, appear in the study of fluid equations such as the Navier–Stokes equations and the Euler equations. We define a maximal function subordinate to these cylinders and show it is of weak type $(1,1)$ and strong type $(p,p)$ by a covering lemma. As an application, we give an alternative proof for the higher-derivatives estimate of smooth solutions to the three-dimensional Navier–Stokes equations.