A Panoramic View of Exponential Attractors

The paper’s Theorem 2.6 establishes the exact equivalence between (i) existence of a T-discrete exponential attractor and (ii) the geometric covering condition NV(S(kT)B, a q^k) ≤ b h^k, including the rate ξ < (1/T)ln(1/q) and the bound dim_f ≤ log_{1/q}h, with a complete, rigorous construction of E0 ensuring S(T)E0 ⊆ E0 and continuous-time exponential attraction to M0 = cl E0 (cf. Theorem 2.6, its Step 1 and Step 2, and (2.4), (e1), (e2) in the proof ). The candidate solution captures the high-level equivalence but makes key mistakes: (a) it asserts compactness/invariance of ω-limits (Lemma 1) without the precompactness hypothesis used in the paper’s Theorem 2.3; (b) it tries to prove S(T)M0 ⊆ M0 by an asymptotic-closedness limit passage that does not justify S(T)E0 ⊆ A; and (c) it only establishes discrete-time attraction to M0 and then shifts to a different set M = ⋃_{s∈[0,T]}S(s)M0 for continuous times, whereas Theorem 2.6 proves continuous-time exponential attraction to the same M0. Hence, the paper is correct and complete; the model’s proof is incomplete/incorrect on these points.