2023/04/16 by Inci, Hasan
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2304.07784
In this paper we consider the ``symplectic'' version of the Euler equations studied by Ebin \citeebin. We show that these equations are globally well-posed on the Sobolev space Hs(ℝ2n) for n ≥ 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 Dsω(ℝ2n) of Sobolev type Hs preserving the symplectic form ω=dx1 \wedge dx2 + … + dx2n-1 \wedge dx2n. We show that Dsω(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 Dsω(ℝ2n) as an analytic second order ODE in the manner of the Euler-Arnold formalism \citearnold. 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 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.