Enhanced dissipation for two-dimensional Hamiltonian flows

Let $H\in C^1\cap W^{2,p}$ be an autonomous, non-constant Hamiltonian on a compact $2$-dimensional manifold, generating an incompressible velocity field $b=\nabla^\perp H$. We give sharp upper bounds on the enhanced dissipation rate of $b$ in terms of the properties of the period $T(h)$ of the close orbits $\{H=h\}$. Specifically, if $0<\nu\ll 1$ is the diffusion coefficient, the enhanced dissipation rate can be at most $O(\nu^{1/3})$ in general, the bound improves when $H$ has isolated, non-degenerate elliptic point. Our result provides the better bound $O(\nu^{1/2})$ for the standard cellular flow given by $H_\mathsf{c}(x)=\sin x_1 \sin x_2$, for which we can also prove a new upper bound on its mixing 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 $b$.