Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Abstract Let H C¹ W^2, p H ∈ C 1 ∩ W 2, p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field b= ^ H 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\ H = h. Specifically, if 0 0 ν ≪ 1 is the diffusion coefficient, the enhanced dissipation rate can be at most O (^1/3) O (ν 1 / 3) in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound O (^1/2) O (ν 1 / 2) for the standard cellular flow given by H₂ (x) = x₁ x₂ H c (x) = sin x 1 sin x 2, 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.