Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Let H∈C1∩W2,p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field b=∇⊥H. We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T(h) of the closed orbit {H=h}. Specifically, if 0<ν≪1 is the diffusion coefficient, the enhanced dissipation rate can be at most O(ν1/3) in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound O(ν1/2) for the standard cellular flow given by Hc(x)=sinx1sinx2, for which we can also prove a new upper bound on its mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by b.