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      Nonlinear stabilitty for steady vortex pairs

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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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          Most cited references18

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          Ordinary differential equations, transport theory and Sobolev spaces

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            The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1. * *Mp denotes the Marcinkiewicz space or weak Lp space

            P.L. Lions (1984)
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              A family of steady, translating vortex pairs with distributed vorticity

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                Author and article information

                Journal
                22 June 2012
                Article
                10.1007/s00220-013-1806-y
                1206.5329
                de89bab7-54b1-4eb8-90a2-5cf64aa2c4a8

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                76B47, 35Q31
                Comm. Math. Phys. 324 (2013) 445-463
                25 pages
                math.AP physics.flu-dyn

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