Construction of Maximal Functions associated with Skewed Cylinders Generated by Incompressible Flows and Applications

TL;DR

This paper introduces a maximal function linked to skewed cylinders derived from mollified fluid flow trajectories, providing new tools for analyzing fluid equations like Navier-Stokes.

Contribution

It constructs a maximal function associated with skewed cylinders and proves its boundedness properties, offering an alternative approach to derivative estimates in fluid dynamics.

Findings

Maximal function is of weak type (1,1) and strong type (p,p).

Provides an alternative proof for higher derivatives estimates in Navier-Stokes.

Introduces a new geometric approach to analyzing fluid flow trajectories.

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.