The group of symplectomorphisms of R2nand the Euler equations

Abstract

In this paper we consider the “symplectic” version of the Euler equations studied by Ebin[7]. We show that these equations are globally well-posed on the Sobolev space Hs(R2n) for n≥1 and s>2n/2+1[jls-end-space/]. The mechanism underlying global well-posedness has similarities to the case of the 2D Euler equations. Moreover we consider the group of symplectomorphisms Dωs(R2n) of Sobolev type Hs preserving the symplectic form ω=dx1∧dx2+…+dx2n−1∧dx2n[jls-end-space/]. We show that Dωs(R2n) is a closed analytic submanifold of the full group Ds(R2n) of diffeomorphisms of Sobolev type Hs preserving the orientation. We prove that the symplectic version of the Euler equations has a Lagrangian formulation on Dωs(R2n) as an analytic second order ODE in the manner of the Euler-Arnold formalism[1]. In contrast to this “smooth” behavior in Lagrangian coordinates we show that it has a very “rough” behavior in Eulerian coordinates. To be precise we show that the time T>0 solution map u0↦u(T) mapping the initial value of the solution to its time T value is nowhere locally uniformly continuous. In particular the solution map is nowhere locally Lipschitz. © 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.