A Recurrence Theorem on the Solutions to the 2d Euler Equation

In finite dimensions, the Poincaré recurrence theorem can be proved from the basic properties of a finite measure. In infinite dimensions, it is difficult to establish a natural finite measure, especially by extending a finite dimensional finite measure. A natural alternative is the Banach norm which can be viewed as a counterpart of the probability density. An interesting problem is to study the Poincaré recurrence problem for 2D Euler equation of fluids [3]. Nadirashvili [4] gave an example of Poincaré non-recurrence near a particular solution of the 2D Euler equation defined on an annular domain. The proof in [4] was fixed up in [3]. We believe that the Poincaré recurrence will occur more often when the 2D Euler equation is defined on a periodic domain. The theorem to be proved in this article is a result along this line. On a periodic domain, the 2D Euler equation is a more natural Hamiltonian system than e.g. on an annular domain. One final note is that any solution of the 2D Euler equation defines a nonautonomous integrable Hamiltonian two dimensional vector field [2]. The trajectories of this vector field are the fluid particle trajectories. This is the so-called Lagrangian coordinates. The integrability was proved in the usual extended coordinates of two spatial coordinates, the stream function, and an extra temporal variable. Due to the extra temporal variable, most of the invariant subsets are of infinite volume. Only in special cases e.g. the solution of the 2D Euler equation is periodic or quasi-periodic in time, one can find invariant subsets of finite volume. This indicates that in the Lagrangian coordinates, recurrence is a rare event [5].