CONSTRUCTIVE CONTROLLABILITY FOR INCOMPRESSIBLE VECTOR FIELDS

The paper proves a constructive global controllability theorem for bounded, Lipschitz, divergence-free vector fields with vanishing mean drift (Theorem 3.1) by: (i) building a small perturbation Ṽ of V with every point nonwandering and a.e. point Poisson-stable (Proposition 3.4, summarizing [4]); (ii) using a short-window endpoint-correction lemma (Lemma 3.2) to connect nearby segments under drift Ṽ; and (iii) finally transferring back to the original drift by absorbing V̄−V into the control, keeping the total control < ε (end of the proof of Theorem 3.1). These steps are explicit in the PDF, including the precise smallness choices and the use of weighted divergence to guarantee recurrence, not exact incompressibility of Ṽ (divψ Ṽ = 0) . By contrast, the candidate solution misapplies Lemma 3.2 to the original drift V (instead of to Ṽ), and incorrectly asserts that the tracking control can be supported only on a short time subinterval while the state coincides with a Ṽ-trajectory segment—this would require control over the whole segment unless the drift is Ṽ. It also incorrectly states that the perturbation W can be chosen divergence-free; the paper only ensures small divergence and weighted divergence-free structure, which suffices for recurrence (Proposition 3.4) . The paper’s constructive scheme and estimates close correctly; the candidate’s outline omits essential steps and contains incorrect claims.