The group of Symplectomorphisms of $\mathbb{R}^{2n}$ and the Euler equations

Abstract

In this paper we consider the ``symplectic'' version of the Euler equations studied by Ebin \cite{ebin}. We show that these equations are globally well-posed on the Sobolev space $H^s(\mathbb{R}^{2n})$ for $n \geq 1$ and $s > 2n/2+1$. The mechanism underlying global well-posedness has similarities to the case of the 2D Euler equations. Moreover we consider the group of symplectomorphisms $\mathcal{D}^s_\omega(\mathbb{R}^{2n})$ of Sobolev type $H^s$ preserving the symplectic form $\omega=dx_1 \wedge dx_2 + \ldots + dx_{2n-1} \wedge dx_{2n}$. We show that $\mathcal{D}^s_\omega(\mathbb{R}^{2n})$ is a closed analytic submanifold of the full group $\mathcal{D}^s(\mathbb{R}^{2n})$ of diffeomorphisms of Sobolev type $H^s$ preserving the orientation. We prove that the symplectic version of the Euler equations has a Lagrangian formulation on $\mathcal{D}^s_\omega(\mathbb{R}^{2n})$ as an analytic second order ODE in the manner of the Euler-Arnold formalism \cite{arnold}. In contrast to this ``smooth'' behaviour in Lagrangian coordinates we show that it has a very ``rough'' behaviour in Eulerian coordinates. To be precise we show that the time $T > 0$ solution map $u_0 \mapsto u(T)$ 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.