2d incompressible Euler equations: New explicit solutions

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 19th 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 1980s-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.

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

“…Anyway it seems that all the previously known solutions reduce either to the situation described in Section 3.1 (this could be called the Kirchhoff type case) or to the family of solutions given in Theorem 5.1 (the Gerstner type case). Solutions of these types can be found using harmonic maps, and even though there have previously been hints that even more complicated solutions exist [6,18], we show in this paper how they can all be reduced to these cases. Also, as far as we know, the families of solutions in Theorems 3.3, 5.7 and 5.9 are new, the first of which is a generalization of the Kirchhoff type.…”

“…Straightforward computations show (see for example [18] for details) that we get the following conditions. Theorem 2.2.…”

“…However, while for the spatial part one could find solutions using complex functions, there was no natural role for complex functions for the time dependent part. Also in [18] it was shown that also in the spatial domain the complex functions were not as essential as was previously thought.…”

“…Since complex functions were used in the description of the solutions, it was natural to also consider harmonic functions. In [18] we showed that if the map in the plane is both area preserving and harmonic, then it is necessarily affine. So a harmonic Lagrangian solution is like Kirchhoff's solution.…”

“…We will continue our analysis of explicit solutions of the incompressible Euler equations which was started in [18]. For a general overview of various aspects of Euler equations from the mathematical point of view we refer to the survey [12].…”