Nonlinear Stability for Steady Vortex Pairs |
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Authors: | Geoffrey R Burton Helena J Nussenzveig Lopes Milton C Lopes Filho |
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Institution: | 1. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK 2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária – Ilha do Fund?o, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ, Brazil
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Abstract: | In this article, we prove nonlinear orbital stability for steadily translating vortex pairs, a family of nonlinear waves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation of Kelvin’s variational principle, maximizing kinetic energy penalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulation has the advantage that the functional to be maximized and the constraint set are both invariant under the flow of the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence yields a wide class of examples to which our result applies. |
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