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

Thierry Gallay and Vladimír Šverák

Vol. 17 (2024), No. 2, 681–722
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 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.

Keywords
two-dimensional flows, vortices, stability, variational principle, constrained optimization
Mathematical Subject Classification
Primary: 35Q30, 35Q31
Milestones
Received: 14 November 2021
Revised: 1 June 2022
Accepted: 11 July 2022
Published: 6 March 2024
Authors
Thierry Gallay
Institut Fourier
Université Grenoble Alpes
Gières
France
Vladimír Šverák
School of Mathematics
University of Minnesota
Minneapolis, MN
United States

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