Regularity of the Geodesic Flow of the Incompressible Euler Equations on a Manifold

Abstract

In this paper we investigate the regularity in time of the volume-preserving geodesic flow, which is associated with the incompressible Euler equations on a compact d-dimensional Riemannian manifold with boundary. Our result, which completes the local-in-time well-posedness theory of Ebin and Marsden (Ann Math 92:102–163, 1970), states roughly that the time smoothness of geodesic curves is only limited by the smoothness of the manifold. Such regularity is measured in a broad class of ultradifferentiable functions, which includes the real analytic and Gevrey classes. A by-product of this simple and constructive proof is new ideas to design high-order semi-Lagrangian methods for integrating the incompressible Euler equations on a manifold.