2018/12/31 by Holger R. Dullin, H R Dullin, James D. Meiss +3 · 5 citations
Engineering · Mathematics · Physics and Astronomy · #Discrete Poisson equation #Euler equations #Euler method #Euler's formula #Fourier transform #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Nonlinear system #Poisson distribution #Semi-implicit Euler method #Stability and Controllability of Differential Equations #math-ph #math.DS #math.MP
paper · pdf · doi:10.1088/1751-8121/ab3363
published in Journal of Physics A Mathematical and Theoretical 52(36), 365501 (Institute of Physics)
openalex created_date 2019/01/01 · arxiv created 2019/06/24 · openalex publication_date 2019/07/18 · arxiv updated 2020/09/07 · openalex updated_date 2026/08/06
Abstract We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ordinary differential equations by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the energy-Casimir method fails.