Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Let H is an element of C1 boolean AND W2,pbe an autonomous, non-constant Hamiltonian on a compact2-dimensional manifold, generating an incompressible velocity fieldb= del H-perpendicular to.We give sharp upper bounds on the enhanced dissipation rate ofbin terms of theproperties of the periodT(h)of the closed orbit{H=h}. Specifically, if 0<nu(-1)is the diffusion coefficient, the enhanced dissipation rate can be at mostO(nu(1/3))ingeneral, the bound improves whenHhas isolated, non-degenerate elliptic points.Our result provides the better boundO(nu(1/2))for the standard cellular flow given by Hc(x)=sinx(1) sinx(2), for which we can also prove a new upper bound on its mixingrate and a lower bound on its enhanced dissipation rate. The proofs are based onthe use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by b.