On the Long Time Behavior of Fluid Flows

AbstractThis work is devoted to the study of the long time asymptotics of solutions of the Euler equations in a bounded 2-dimensional domain. Experiments and numerical simulations indicate the presence of an attracting set in the space of incompressible velocity fields. In this work this attractor is described, and its attracting property is established in an extended dynamics where the time is replaced by the ‘long time’ taking values in the Alexandroff line. The attracting property in the usual sense remains a conjecture

High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.Comment: 26 pages, latex2

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.Comment: 37 pages, 5 figure