Dynamics of two-dimensional time-periodic Euler fluid flows

Abstract

This paper investigates the dynamics of time-periodic Euler flows in multi-connected, planar fluid regions which are ``stirred'' by the moving boundaries. The classical Helmholtz theorem on the transport of vorticity implies that if the initial vorticity of such a flow is generic among real-valued functions in the $C^k$-topology ($k \geq 2$) or is $C^ω$ and nonconstant, then the flow has zero topological entropy. On the other hand, it is shown that for constant initial vorticity there are stirring protocols which always yield time-periodic Euler flows with positive entropy. These protocols are those that generate flow maps in pseudoAnosov isotopy classes. These classes are a basic ingredient of the Thurston-Nielsen theory and a further application of that theory shows that pseudoAnosov stirring protocols with generic initial vorticity always yield solutions to Euler's equations for which the sup norm of the gradient of the vorticity grows exponentially in time. In particular, such Euler flows are never time-periodic.