Derive matching enhanced-dissipation estimates for finite-order Hamiltonian flows

Obtain matching upper and lower viscosity-dependent decay estimates for zero-streamline-average solutions of the advection-diffusion equation generated by two-dimensional Hamiltonian flows satisfying assumptions (A1)–(A2), and determine precisely how those estimates depend on the inviscid $(1+t)^{-1}$ mixing law.

Background

The paper studies inviscid transport and establishes inverse-time mixing estimates under finite-order Hamiltonian structure, separation from equilibria and infinite-period trajectories, and nonvanishing period gradients. The perspectives section proposes extending this framework to the advection-diffusion equation with diffusivity $u$ and Neumann boundary conditions.

Although enhanced dissipation for zero-streamline-average data in two-dimensional Hamiltonian flows has been studied previously, the authors leave unresolved the derivation of matching viscosity-dependent upper and lower decay rates for the broader finite-order class (A1)–(A2), together with their precise relationship to the inviscid mixing rate.