Lagrangian aspects of the axisymmetric Euler equation

TL;DR

This paper explores the geometric properties and blowup criteria of the axisymmetric 3D Euler equations with swirl, using Lagrangian formulations and Riemannian geometry to deepen understanding of singularity formation.

Contribution

It extends existing blowup criteria by incorporating Lagrangian perspectives and geometric interpretations, linking local flow geometry with the structure of the diffeomorphism group.

Findings

Extended blowup criteria related to pressure Hessian sign assumptions.

Provided geometric interpretation in terms of flow trajectories and Riemannian geometry.

Connected local flow behavior with the global structure of volume-preserving diffeomorphisms.

Abstract

In this paper we are interested in geometric aspects of blowup in the axisymmetric 3D Euler equations with swirl on a cylinder. Writing the equations in Lagrangian form for the flow derivative along either the axis or the boundary and imposing oddness on the vertical component of the flow, we extend some blowup criteria due to Chae, Constantin, and Wu related to assumptions on the sign of the pressure Hessian. In addition we give a geometric interpretation of the results, both in terms of the local geometry along trajectories and in terms of the Riemannian geometry of the volume-preserving diffeomorphism group.