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2D incompressible Euler equations: New explicit solutions

  • * Corresponding author: M. J. Martín

    * Corresponding author: M. J. Martín 

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin.

The first author was supported by UAM and EU funding through the InterTalentum Programme (COFUND 713366). She also thankfully acknowledges partial support from Spanish MINECO/FEDER research project MTM2015-65792-P.

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  • There are not too many known explicit solutions to the $ 2 $-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the $ 19 $th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -in the $ 1980 $s- obtained new explicit solutions with a similar feature.

    We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family.

    In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.

    Mathematics Subject Classification: 76B03, 35Q31, 13P10.


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