Enhanced dissipation for two-dimensional Hamiltonian flows

The paper proves the universal upper bound λ(ν) ≤ C ν^{1/3} for dissipation-enhancing autonomous 2D Hamiltonian flows b=∇⊥H under H∈C^1∩W^{2,p} (p≥2). The argument is rigorous and proceeds via a quantitative vanishing-viscosity estimate on a good invariant domain, deriving ‖ρν(t)−ρ(t)‖^2 ≤ C ν(1+t)^3, and combining this with the enhanced-dissipation hypothesis to force t ≳ ν^{-1/3} before any rapid decay can occur, yielding λ(ν) ≤ C ν^{1/3} (see Theorem 1 and its proof, including the energy estimate (3.2)–(3.4) and the construction of action–angle coordinates/regular Lagrangian flow to control gradient growth ). The candidate solution independently recovers the same ν^{1/3} scaling via a resolvent–quasimode construction localized to a regular band of closed streamlines and tests at frequency ξ=ω(h0). Its main steps are standard and consistent with the paper’s framework. However, a few technical justifications are missing as written: (i) the resolvent inequality should be applied on the operator domain, so one must ensure the quasimode belongs to D(Lν) or argue by density; (ii) the Laplacian estimate in action–angle variables requires enough regularity of the angle map θ (C^2 locally, or an approximation argument) to control Δfδ; and (iii) differentiability of ω(h)=1/T(h) at the chosen h0 must be stated precisely (a.e. on regular bands). These are repairable with mild additional hypotheses or mollification and do not change the conclusion. Net: the paper’s result and proof are correct; the model’s proof gives the same exponent with a different, essentially valid strategy but with some technical gaps to polish.