Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Abstract

AbstractLet $$H\in C^1\cap W^{2,p}$$ H ∈ C 1 ∩ W 2 , p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field $$b=\nabla ^\perp H$$ 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 $$0<\nu \ll 1$$ 0 < ν ≪ 1 is the diffusion coefficient, the enhanced dissipation rate can be at most $$O(\nu ^{1/3})$$ O ( ν 1 / 3 ) in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound $$O(\nu ^{1/2})$$ O ( ν 1 / 2 ) for the standard cellular flow given by $$H_{\textsf{c}}(x)=\sin x_1 \sin x_2$$ 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.