Existence of a conjugate point in the incompressible Euler flow on a three-dimensional ellipsoid

TL;DR

This paper investigates the existence of conjugate points in the incompressible Euler flow on a 3D ellipsoid, linking geometric stability criteria to fluid dynamics solutions and demonstrating conjugate points' existence.

Contribution

It introduces a class of stationary solutions with positive Misiolek curvature and proves the existence of conjugate points on a 3D ellipsoid.

Findings

Positive Misiolek curvature for a class of solutions

Existence of conjugate points on a 3D ellipsoid

Connection between curvature and Lagrangian stability

Abstract

The existence of a conjugate point on the volume-preserving diffeomorphism group of a compact Riemannian manifold M is related to the Lagrangian stability of a solution of the incompressible Euler equation on M. The Misiolek curvature is a reasonable criterion for the existence of a conjugate point on the volume-preserving diffeomorphism group corresponding to a stationary solution of the incompressible Euler equation. In this article, we introduce a class of stationary solutions on an arbitrary Riemannian manifold whose behavior is nice with respect to the Misiolek curvature and give a positivity result of the Misiolek curvature for solutions belonging to this class. Moreover, we also show the existence of a conjugate point in the three-dimensional ellipsoid case as its corollary.