Enhanced dissipation for two-dimensional Hamiltonian flows

TL;DR

This paper establishes sharp upper bounds on the enhanced dissipation rates for two-dimensional Hamiltonian flows, with improvements for specific flow structures like cellular flows, using action-angle coordinates and invariant domains.

Contribution

It provides the first sharp bounds on enhanced dissipation rates depending on orbit periods, especially improving bounds for cellular flows and flows with elliptic points.

Findings

Enhanced dissipation rate is at most O(ν^{1/3}) in general.

Improved bound of O(ν^{1/2}) for cellular flows.

New bounds on mixing and dissipation rates for specific flow structures.

Abstract

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 $b$.