Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data

TL;DR

This paper proves the global regularity of 3D axi-symmetric Navier-Stokes equations with large anisotropic initial data and analyzes the dynamic growth behavior of solutions.

Contribution

It establishes global regularity results for a new class of large anisotropic initial data in the 3D axi-symmetric Navier-Stokes equations.

Findings

Global regularity for large anisotropic data

Bounded solutions in terms of initial data in $L^p$ norm

Dynamic growth behavior due to angular velocity and vorticity interaction

Abstract

In this paper, we study the 3D axi-symmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some $L^p$ norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.