Arnold's variational principle and its application to the stability of planar vortices

TL;DR

This paper explores variational principles based on Arnold's stability criteria for 2D Euler equations, providing new insights into vortex stability and extending analysis to low-viscosity Navier-Stokes flows.

Contribution

It offers a detailed functional-analytic study of stability conditions, revisits classical vortex stability with a new proof, and examines viscous effects on Gaussian vortices.

Findings

New stability proof for Oseen's vortex with geometric insights

Conditions under which second variations inform stability analysis

Impact of viscosity on vortex stability in Navier-Stokes equations

Abstract

We consider variational principles related to V. I. Arnold's stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the the Navier-Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen's vortex, providing a new stability proof with stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex.