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Rigorous derivation of the leapfrogging motion for planar Euler equations

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arxiv 2311.15765 v2 pith:MBZNN764 submitted 2023-11-27 math.AP physics.flu-dyn

Zineb Hassainia , Taoufik Hmidi , Nader Masmoudi This is my paper · ORCID
classification math.APphysics.flu-dyn
keywords vortexpatchestimeconcentratedequationseulerfourleapfrogging
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The main goal of this paper is to explore the leapfrogging phenomenon in the inviscid planar flows. We show for 2d Euler equations that under suitable constraints, four concentrated vortex patches leapfrog for all time. When observed from a translating frame of reference, the evolution of these vortex patches can be described as a non-rigid time periodic motion. Our proof hinges upon two key components. First, we desingularize the symmetric four point vortex configuration, which leapfrogs in accordance with Love's result \cite{Love1893}, by concentrated vortex patches. Second, we borrow some tools from KAM theory to effectively tackle the small divisor problem and deal with the degeneracy in the time direction. Our approach is robust and flexible and solves a long-standing problem that has remained unresolved for many decades.

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Cited by 4 Pith papers

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  1. Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves

    math.AP 2025-09 conditional novelty 8.0 of 10

    Existence of small-amplitude, linearly stable, time quasi-periodic traveling solutions for 3D pure gravity water waves in finite depth on tori, for generic lattices and most depths.

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    math.AP 2025-05 accept novelty 8.0 of 10

    Smooth compactly supported vortex layers around any closed analytic curve evolve, in the zero-thickness limit, according to the Birkhoff-Rott equations.

  3. Stability for multiple Lamb dipoles

    math.AP 2025-07 conditional novelty 7.0 of 10

    Finite sums of Lamb dipoles in the half-plane, with ordered speeds and well-separated initial positions, are Lyapunov stable under the 2D Euler equations.

  4. Stability of oppositely-propagating pair of Hill's spherical vortices

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    An odd-symmetric pair of Hill's spherical vortices is globally stable in 3D axisymmetric Euler flow, with propagation speed close to the single-vortex speed and a sharp linear error estimate.

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