Nonstationary vortex flows of an ideal incompressible fluid

Abrashkin, A. A.; Yakubovich, E. I.

doi:10.1007/bf00908966

Cited by 16 publications

(30 citation statements)

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“…• there exist z 1 = z 2 in 0 and an open interval I ⊂ (0, ∞) such for every t ∈ I the two complex vectors 2 are linearly independent.…”

Section: Labelling By Harmonic Mapsmentioning

confidence: 99%

“…The univalence issue is not addressed in [1] or in the more detailed exposition [2]. Choosing a convex domain 0 whose boundary does not contain line segments and analytic functions ν 1 , ν 2 whose derivatives extend continuously to 0 and which satisfy Re ν 1 > |ν 2 | throughout 0 , univalence is ensured by Proposition 4.3.…”

Section: Proposition 54 Let Be a Complex Invertible 2 × 2-matrix Wimentioning

confidence: 99%

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Harmonic Maps and Ideal Fluid Flows

Aleman

Constantin

2011

Arch Rational Mech Anal

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Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in C n with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.

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“…• there exist z 1 = z 2 in 0 and an open interval I ⊂ (0, ∞) such for every t ∈ I the two complex vectors 2 are linearly independent.…”

Section: Labelling By Harmonic Mapsmentioning

confidence: 99%

Section: Proposition 54 Let Be a Complex Invertible 2 × 2-matrix Wimentioning

confidence: 99%

Harmonic Maps and Ideal Fluid Flows

Aleman

Constantin

2011

Arch Rational Mech Anal

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“…Apparently no really new explicit solutions were found before the paper by Abrashkin and Yakubovich in 1984 [5]. These solutions were a generalization of both Kirchhoff's and Gerstner's solution.…”

Section: Introductionmentioning

confidence: 96%

On the Explicit Solutions of Separation of Variables Type for the Incompressible 2D Euler Equations

Saleva

Tuomela

2021

J. Math. Fluid Mech.

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We study explicit solutions to the 2 dimensional Euler equations in the Lagrangian framework. All known solutions have been of the separation of variables type, where time and space dependence are treated separately. The first such solutions were known already in the 19th century. We show that all the solutions known previously belong to two families of solutions and introduce three new families of solutions. It seems likely that these are all the solutions that are of the separation of variables type.

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“…As already mentioned in Introduction, the common structural property of the known explicit solutions to the governing equations (Kirchhoff's vortex [16], Gerstner's wave [9], the Ptolemaic solutions [1]) is that the labelling map (3) is harmonic at every fixed time t. Our aim is to explicitly find all solutions having this property. Most ideas in this section are inspired by [2] and are included for the sake of completeness.…”

Section: Harmonic Labellingsmentioning

confidence: 99%

“…Several publications in the applied mathematics, engineering and physics research literature (see [6,7,[10][11][12]15,18] and references therein) exploited a remarkable feature shared by some celebrated explicit solutions to the two-dimensional incompressible Euler equations, in Lagrangian variables (such as Kirchhoff's elliptical vortex [16] found in 1876, Gerstner's flow [9] found in 1809 and re-discovered in 1863 by Rankine [19], and the Ptolemaic vortices found in 1984 by Abrashkin and Yakubovich [1]), namely that in all of them the labelling map is harmonic at all times.…”

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confidence: 99%