2D incompressible Euler equations: New explicit solutions

María J. Martín; Jukka Tuomela

doi:10.3934/dcds.2019187

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2019, Volume 39, Issue 8: 4547-4563. Doi: 10.3934/dcds.2019187

This issue Previous Article Shallow water models for stratified equatorial flows Next Article On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

2D incompressible Euler equations: New explicit solutions

María J. Martín^1, , and
Jukka Tuomela^2,

1.
Departamento de Análisis Matemático, Universidad de La Laguna, 38200 San Cristóbal de La Laguna, S/C de Tenerife, Spain
2.
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland

^* 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.

Received: August 2018

Revised: January 2019

Published: May 2019

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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Abstract

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.

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

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