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Arnold's variational
principle and its application to the stability of planar
vortices
Thierry Gallay and Vladimír Šverák |
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Vol. 17 (2024), No. 2, 681–722
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Abstract |
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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
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Mathematical Subject Classification
Primary: 35Q30, 35Q31
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Milestones
Received: 14 November 2021
Revised: 1 June 2022
Accepted: 11 July 2022
Published: 6 March 2024
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Authors |
| Thierry Gallay | |
| Institut Fourier Université Grenoble Alpes Gières France |
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| Vladimír Šverák | |
| School of Mathematics University of Minnesota Minneapolis, MN United States |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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