Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Let \(H\in C^1\cap W^{2,p}\) $H\in C^1\cap W^{2,p}$ be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field \(b=\nabla ^\perp H\) $b={\mathrm{\nabla }}^{\perp }H$ . We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T(h) of the closed orbit \(\{H=h\}\) $\{H=h\}$ . Specifically, if \(0<\nu \ll 1\) $0<\nu \ll 1$ is the diffusion coefficient, the enhanced dissipation rate can be at most \(O(\nu ^{1/3})\) $O({\nu }^{1/3})$ in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound \(O(\nu ^{1/2})\) $O({\nu }^{1/2})$ for the standard cellular flow given by \(H_{\textsf{c}}(x)=\sin x_1 \sin x_2\) $H_{\mathsf{c}}(x)=sin{x}_{1}sin{x}_2$ , 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.