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

Abstract

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.

Explicit solutions to the 3D incompressible Euler equations in Lagrangian formulation

“…In the recent decades these solutions have been generalized using harmonic maps, see for example [1,2,3,7]. All these solutions were of the form where the time component and the spatial component could be separated, and finally in [15] we systematically found probably all possible solutions of the separation of variables type to the 2D Euler equations in Lagrangian formulation, including ones that could not be found with harmonic maps.…”

“…The simple proof can be found in [15]. The problems we study typically have several families of solutions.…”

“…This is the third paper, following [13,15], in our research of finding explicit solutions to the incompressible Euler equations in the Lagrangian framework. The Lagrangian framework is one of the two main approaches to describing fluid motion.…”

“…Anyway in [13,15] we showed that even in two dimensions the situation is best analyzed using real functions, and thus a priori it seemed natural to believe that similar techniques should be applicable also in three dimensional case. In the present paper we show that this is indeed the case: even though certain classes of solutions can equivalently be analyzed with complex formulation, one can find many more large classes of solutions operating with real functions.…”

“…One does not need any new methods to deal with the complicated cases, the difficulty comes just from the sheer size of the problem. Unlike in the 2D case we studied in [15], here we will not consider all possible families of solutions due to their vast number. Instead, we concentrate on those cases which to us seem to be the most promising from the physical point of view.…”