Poisson structures for PDEs associated with diffeomorphism groups

Abstract

We study Poisson and Lie-Poisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear PDEs. In particular, we provide a precise analytic sense in which the time t map for the Euler equations of an ideal fluid in a region of R^n (or on a smooth compact n-manifold with a boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphisms. The key difficulty in finding a suitable context for that arises from the fact that the integral curves of Euler equations are not differentiable on the Lie algebra of divergence free vector fields of Sobolev class Hs. We overcome this obstacle by utilizing the smoothness that one has in Lagrangian representation and carefully performing a non-smooth Lie-Poisson reduction procedure on the appropriate functional classes. This technique is generalized to an arbitrary diffeomorphism group possessing a smooth spray. The applications include the Camassa-Holm equation on S^1, the averaged Euler and EPDiff equations on the n-manifold with a boundary. In all cases we prove that time t map is Poisson on the appropriate Lie algebra of Hs vector fields, where s > n/2 + 1 for the Euler equation and s > n/2 + 2 otherwise.