ARNOLD’S VARIATIONAL PRINCIPLE AND ITS APPLICATION TO THE STABILITY OF PLANAR VORTICES

Thierry Gallay, Vladimír Šverák

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)681-722
Number of pages42
JournalAnalysis and PDE
Volume17
Issue number2
DOIs
StatePublished - 2024

Bibliographical note

Publisher Copyright:
© 2024 MSP (Mathematical Sciences Publishers).

Keywords

  • constrained optimization
  • stability
  • two-dimensional flows
  • variational principle
  • vortices

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