A NOTE ON EQUIVALENCES BETWEEN VARIOUS MIXING SCALES

The paper’s main equivalence (Theorem 3.1) matches the candidate’s goal. However, in the paper the arguments for 1 ⇒ 3 and 1 ⇒ 4 misuse quantifiers (“for any φ ∈ L2” uniformly in φ) and do not justify the uniform-in-x control needed to conclude G(ρ(t);κ) ≤ ε for all x at large times, so those two implications are not proved as written. This gap can be repaired by a standard finite-ε-net/translation-continuity argument, but it is not present in the manuscript. The other directions, especially 3 ⇒ 1 via Lemma 3.2 (Crippa), are sound and carefully proved in the note, including the boundary-layer estimate that yields uniform-in-x control from small-radius averages . The candidate solution correctly establishes 1 ⇔ 2 (with a clear Fourier tail argument and H−1–H1 duality) and gives a strong, quantitative implication 2 ⇒ 4 using mollification and H−1 duality (not in the paper). But it contains a substantive error in the step 3 ⇒ 1: it assumes that if r = 2 G(ρ;κ) then the ball-averages at radius r are small, which is exactly the defect of the geometric mixing scale illustrated in the paper’s Example 2.1 (smallness at G need not persist at larger radii) . Thus the model’s claimed chain to full equivalence is also incomplete without invoking the Crippa lemma or working with SG. Conclusion: both the paper and the model require nontrivial repairs (paper: fix 1⇒3/4; model: fix 3⇒1); neither argument, as written, is fully correct.